ENHANCED PERFORMANCE OF THE MIMO CHANNEL BY USING THE BEAMFORMING TECHNIQUE AND THE RACK RECEIVER
A Thesis
Presented to the
Faculty of
California State University, Fullerton ______
In Partial Fulfillment
of the Requirements for the Degree
Master of Science
in
Electrical Engineering ______
By
Mohanad Saleh Mohammedhasan Alaamer
Thesis Committee Approval:
Hamidian, Karim, Department of Electrical Engineering, Chair Chaudhry, Maqsood, Department of Electrical Engineering Shiva, Mostafa, Department of Electrical Engineering
Summer, 2018
ABSTRACT
In recent years, the demands for high data rate and reliable transmission have increased in wireless communications systems due to the wide use of web applications.
The key technology that achieves these requirements is the application of Multiple Input
Multiple Output (MIMO). The MIMO system exploits the multipath propagation by using multiple transmitted antennas to send the signal to multiple received
The MIMO system classification depends on whether the focus of the MIMO processing is on improving reliability, by creating spatial diversity, or on maximizing throughput, by performing spatial multiplexing. There are different ways to achieve spatial multiplexing, which are Layered Space Time coding and beamforming technique.
In this thesis, I will investigate and analyze the performance of the MIMO channel by creating the combination of using the beamforming technique and the rack receiver to achieve both spatial multiplexing, which improves the data rate, and the diversity, which increases the reliability of the MIMO channel. The proposed technique uses DSSS-CDMA with-BPSK and QPSK modulation in a Slow Rayleigh flat fading environment. To apply the beamforming, I assume the transmitter has knowledge of the channel state information, that will decouple the MIMO channel to sub-independent parallel channels. It is shown that by using the combination of the beamforming and the rack receiver at each sub-channel, the signal-to-noise ratio for each sub-channel will increase, so the reliability of the MIMO channel will improve. Also, the capacity of the
MIMO channel will increase.
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TABLE OF CONTENTS
ABSTRACT ...... ii
LIST OF TABLES ...... vi
LIST OF FIGURES ...... vii
ACKNOWLEDGMENTS ...... x
Chapter 1. INTRODUCTION ...... 1
2. WIRELESS COMMUNICATION ...... 5
Introduction to Wireless Communication ...... 5 Application of Wireless Communication ...... 5 RF Propagation ...... 6 Impulse Response of Wireless Communication Channel ...... 8 Multipath Channel Parameter ...... 9 Delay Spread (Tm) ...... 10 Coherent Bandwidth (Bc) ...... 10 Doppler Spread (Bd) ...... 11 Coherent Time (Tc) ...... 11 Type of Small-Scale Fading ...... 12 Flat Fading ...... 12 Frequency Selective Fading ...... 12 Fast Fading ...... 13 Slow Fading ...... 13 Statistic of Small- Scale Fading ...... 14 Rayleigh Fading ...... 14 Rician Fading ...... 14
3. MIMO COMMUNCATION SYSTEM ...... 16
Introduction ...... 16 SISO, MISO, and SIMO Antenna Configuration ...... 18 Single - User and Multi - User of The MIMO System ...... 21 The Concept of Diversity ...... 22 Diversity Order and Diversity Gain ...... 23
iii The Concept of Spatial Multiplexing...... 23 Capacity of The MIMO System ...... 25 Diversity – Multiplexing Tradeoff ...... 30
4. MIMO BEAMFORMING TECHNIQIE ...... 32
Introduction ...... 32 Singular Value Decomposition (SVD) ...... 34 Precoding of The MIMO Channel ...... 36 The Input-Shaping Matrix ...... 37 The Beamforming Matrix ...... 38 Spatial Multiplexing by Beamforming Technique ...... 38 Optimal Allocated of Power (Waterfilling Algorithm) ...... 42 Maximal Ratio Receive Combining (MRC) ...... 45 Maximal Ratio Transmit (MRT)...... 48 Maximum Likelihood Detection with MRRC ...... 49 Single – Mode Beamforming ...... 51
5. SPARATE SPECTRAM MOUDLATION ...... 56
Introduction ...... 56 Direct Sequence Spread Spectrum (DSSS) ...... 57 Pseudo - Noise Sequence PN ...... 61 Partial Correlation of PN Sequence ...... 62 PN Signal ...... 63 Multipath Channel Model ...... 66 The Rack Receiver ...... 68 Performance of The Rack Receiver with MIMO Beamforming Technique ...... 72
6. ANALYSIS THE RESULT OF USING BEAMFORMING TECHNIQUE WITH THE RACK RECIEVER...... 75
(N×N) DSSS-BPSK MIMO Communication System Preform Result ...... 75 (2×2) MIMO Communication System Performance Result of DSSS-BPSK 78 (3×3) MIMO Communication System Performance Result of DSSS-BPSK 94 (N×N) DSSS-QPSK MIMO Communication System Performance Result ...... 99 (2×2) MIMO Communication System Performance Result of DSSS-QPSK 99 (3×3) MIMO Communication System Performance Result of DSSS-QPSK 103 (2×Nr) DSSS-BPSK MIMO Communication System ...... 105 (2×3) MIMO Communication System Performance Result of DSSS-BPSK 106 (2×4) MIMO Communication System Performance Result of DSSS-BPSK 110 (3×4) MIMO Communication System Performance Result of DSSS-BPSK ...... 114
iv 7. CONCLUSION AND FUTURE WORK ...... 120
Conclusion ...... 120 Future Work ...... 122
APPENDICES ...... 123
A. NORMALIZ THE SYSTEM EQUATION OF THE MIMO SYSTEM ...... 123 B. MEAN AND VARIANCE OF THE CORRELATION PROCESS AT THE RACK RECEIVER ...... 126
REFERENCES ...... 129
v
LIST OF TABLES
Table Page
3.1. Relationship of The MIMO Concepts...... 18
3.2. Channel Capacity Formulas of SISO, SIMO, and MISO Communication System...... 21
5.1. Value of μ for Binary Modulation Scheme ...... 74
5.2. Value of 휇 for Selected QPSK and 4-Ary DPSK Modulation Assuming Gray Code ...... 74
vi
LIST OF FIGURES
Figure Page
2.1. Propagation Mechanism in Wireless Communication Channel...... 7
2.2. The dependence of received power over a multipath channel ...... 7
2.3. Wireless channel model...... 8
2.4. An example of the time-varying discreet impulse response for a multipath channel...... 9
2.5. The diagram shows the condition of type fading...... 15
3.1. A MIMO technique for spatial diversity...... 17
3.2. A MIMO technique for spatial multiplexing...... 18
3.3. Shows classifications of multiple antenna configuration...... 19
3.4. The diagram of MIMO system that used spatial multiplexing...... 24
3.5. Taxonomy for MIMO Application...... 30
4.1. Transmit beamforming 3 × 2 MIMO channel...... 33
4.2. Linear precoding structure as multi-mode beamformer...... 37
4.3. Precoder match both the input code structure and channel...... 38
4.4. Decupling MIMO channel to sub-channels ...... 40
4.5. Scheme of MIMO parallel channel...... 41
4.6. Scheme of waterfilling algorithm ...... 44
4.7. Receive diversity combiner...... 45
4.8. Scheme of transmitter diversity combiner...... 48
vii 4.9. Comparison of theoretical capacity ...... 55
5.1. . Transmitted and receive scheme of DS-SS system...... 58
5.2. Pulse function...... 59
5.3. Spread and despread the signal and noise interference...... 60
5.4. Coherent detection of BPSK modulation...... 61
5.5. Multipath channel scheme ...... 67
5.6. Multipath channel model ...... 67
5.7. Rack receiver scheme...... 69
5.8. General rack receiver scheme ...... 71
5.9. MIMO rack receiver scheme ...... 73
6.1 Bit error probability of N×N MIMO system (Rayleigh channel) ...... 76
6.2. The capacity of different MIMO system category...... 77
6.3. The capacity of different MIMO configuration in 5dB and 20dB...... 77
6.4. SISO DSSS modulation with BPSK...... 78
6.5. MIMO sub-channel one performance with rack receiver L=4 ...... 91
6.6. MIMO sub-channel two performances with the rack receiver L=4 ...... 92
6.7. The capacity of 2×2 MIMO system...... 93
6.8. The detection block diagram of 2×2 MIMO system...... 93
6.9. Performance of sub-channel one in 3×3 MIMO system...... 96
6.10. Performance of sub-channel two in 3×3 MIMO system...... 96
6.11. Performance of sub-channel three in 3×3 MIMO system...... 97
6.12. Detection block diagram of 3×3 MIMO system...... 98
6.13. The capacity of 3×3 MIMO system for BPSK Modulation...... 99
viii 6.14. The bit error probability of sub-channel one of 2×2 MIMO system...... 100
6.15. The bit error probability of sub-channel one of 2×2 MIMO system...... 101
6.16. The capacity of MIMO channel for different modulation...... 102
6.17. The bit error probability of sub-channel one of 3×3 MIMO system...... 103
6.18. The bit error probability of sub-channel two of 3×3 MIMO system...... 104
6.19. The bit error probability of sub-channel three of 3×3 MIMO system...... 104
6.20. The BER of the sub-channels 3×3 MIMO system in 5dB and 20 dB...... 105
6.21. The performance of 2×3 MIMO for sub-channel one...... 107
6.22. The performance of 2×3 MIMO for sub-channel two...... 108
6.23. The detection block diagram of 2×3 MIMO system...... 108
6.24. The 2×3 MIMO beamforming capacity...... 110
6.25. The performance of 2×4 MIMO system sub-channel one...... 112
6.26. The performance of 2×4 MIMO system sub-channel two...... 112
6.27. The detection block diagram of 2×4 MIMO system...... 113
6.28. The capacity of 2×4 MIMO beamforming system...... 114
6.29. The performance result of 3×4 MIMO system of sub-channel one...... 116
6.30. The performance result of 3×4 MIMO system of sub-channel two...... 117
6.31. The performance result of 3×4 MIMO system of sub-channel three...... 117
6.32. The detection block diagram of 3×4 MIMO system...... 118
6.33. The capacity of 3×4 MIMO beamforming system...... 119
ix
ACKNOWLEDGEMENTS
I would like to sincerely and profoundly thank my professor, Dr. Hamidian
Karim, who was the academic advisor on my thesis, for his guidance, support, attention, and invaluable notes that he offered on my thesis work.
Many thanks to my thesis defense committee members, professor Dr. Chaudhry
Maqsood and professor Dr. Shiva Mostafa, for accepting to be members of the defense exam and for their time.
I also would like to thank all of the professors and staff in the Electrical
Engineering department for the immense help and advice that they provided during my studies. Additionally, I would like to deeply thank the graduate office and graduate learning specialists specifically, Mr. Michael Itagaki, Miss Cristy Sotomayor, and the university reader, Eliot Cossaboom, for their help in my study plan and reviewing my thesis.
A special thanks to my family and my parents, whom without their encouragement and support I would not be as I am. And for my lovely wife, Alfaez
Zainab, who supported me in my studies and graduation. I really appreciate all my friends who helped me either directly or indirectly to complete my studies and graduation.
Finally, thanks to my Financial Supporter, the Higher Committee Education
Development in Iraq, who gave me the opportunity to obtain my master’s degree at
California State University, Fullerton.
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CHAPTER 1
INTRODUCTION
In wireless communication systems, the demands of high data rate and link quality have increased in recent years. The technological revolution has contributed to the development of communications through several technologies such as, Space Time
Coding (STC), Multi Input Multi Output (MIMO), Orthogonal Frequency Division
Multiplexing (OFDM) and other techniques, so that revolutionized human life.
Researchers take extra attention to wireless communication because it is offering many benefits, such as connecting with each other, accessing the internet and performing live surgery over long distance.
The simplest wireless communication system consists of one transmitter and one receiver antenna, which allows two- way communication with high quality and reliable information transmission. This system was good for short distance communication.
However, it is not good for long distance communication, because of the limitation of the distance between the transmitter and receiver and more power consumption. That led to the development of wireless communication by exploiting the new communication techniques.
One of these techniques is MIMO, which is referring to using multiple transmitter antennae and multiple receiver antennae. Using this technique, it is possible to enhance one of two aspects: either the capacity or reliability of the communication channel. The
2 first mention of this new technique was in 1999 by Peter Driessen and Gerry Foschini when they published a paper that analyzed the theoretical capacity of a communication system with multiple transmitter and multiple receiver antennae [19]. In 2001, Iospan
Wireless Inc. introduced commercial MIMO technology.
The MIMO technique can be used to achieve high reliability of data transfer through a communication channel by using the concept of transmitted diversity. There are many types of transmitted diversity, such as frequency diversity, time diversity, and spatial diversity. The space time coding is considered one of the new techniques that achieves spatial diversity. In 1998, Tarokh, Seshadir, and Calderbank published the first paper to develop the space time coding criteria [20]. Another paper published by
Alamouti shows another way to achieve transmitted diversity with a simple signal processing technique at the receiver [24]. All modern wireless standards employ MIMO techniques that use Alamouti code, which is the special case of space time coding. The
Alamouti technique has the following advantage over alternative strategies because it is requiring CSIR only (as opposed to requiring both CSIT and CSIR). Also, it does not involve any bandwidth expansion. However, Alamouti coding has relatively low computational complexity at the receiver [11].
On the other hand, the MIMO technique could be used to achieve spatial multiplexing (SM). The key point of spatial multiplexing is to increase the data rate without extending the channel bandwidth by transmitting multiple independent data stream over multiple channels by exploiting multipath propagation [11]. In the MIMO technique, there are two diverse ways to achieve spatial multiplexing. First, spatial multiplexing is achieved using a concept called layered space time (LST) coding, where
3 the layer simply refers to the data stream from a single transmitter antenna. The LST codes include many kinds, such as Bell laboratory layered space time (BLST) family techniques, which is classified as Vertical BLAST (V-BLAST), Horizontal BLAST (H-
BLAST), and Diagonal BLAST (D-BLAST). Second, spatial multiplexing can also be achieved by using eigenbeamforming technique. Eigenbeamforming is a practical SM technique that is used in most modern wireless communication systems. It is generally not regarded as being a LST coded SM method [11].
In this thesis, I will focus on the second method to achieve spatial multiplexing, which is the beamforming technique. Currently, the beamforming technique can achieve the highest capacity of the communication channel by applying the waterfilling algorithm. At the same time, the MIMO channel decouples to sub-individual channels, and these channels are SISO channels. These SISO channels have first order diversity, so they are suffering from high probability of error that will affect the reliability of the
MIMO channel. I aim to improve the reliability of the MIMO channel in addition to achieve the highest capacity, so I will combine the beamforming technique with the rack receiver at detection. In a multipath model of communication channels, the time diversity has been generating naturally due to the delay of the multipath components. Most often, these components accumulate destructively at the receiver. I will take advantage of this phenomenon, which naturally generates time diversity (multipath diversity) by using the rack receiver, which will extract the multipath components and add them constructively at the receiver.
This thesis is organized as follows; Chapter 2 introduces wireless communication channels and channel fading, Chapter 3 discusses the concept of MIMO capacity and
4 compares it with SISO system capacity, Chapter 4 discusses the concept of beamforming technique and MRC, Chapter 5 presents the performance of the rack receiver with beamforming technique, Chapter 6 provides analysis and simulation results of the rack receiver with beamforming technique, Chapter 7 provides the conclusion and future suggestions to improve the reliability of the MIMO channels. Appendix A explains the mathematical expression of a normalized channel matrix process. Appendix B explains the mean and variance of the correlation process at the rack receiver.
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CHAPTER 2
WIRELESS COMMUNICATION
Introduction to Wireless Communication
The first experience of wireless communication over long distance was in 1897, by Guglielmo Marconi, which demonstrated radio’s ability to provide continuous contact with ship sailing the England channel [23]. After that, a new witless communication method and serves were established by enthusiastic people throughout the world. During the past ten years, there has been an evolution of technology which drastically enhanced wireless communication. As a result, today, there are many applications of wireless communication.
Application of Wireless Communication
One of the most famous forms of wireless communication is the mobile communication system, or cellular radio system. The development for first mobile communication system, the analog cellular system, started in Chicago in 1983, and was available for use by 1984 [7]. The second generation of cellular systems, which is the current generation, were digital. In addition to voice communication, these systems provided email and voice mail. Due to the new large-scale circuit integration technology, mobile radio equipment became smaller, cheaper, and more reliable. Today, the mobile communication system provides many communication services, which expand to features beyond basic voice communication, such as internet browsing and social media.
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Because of the expansion of services provided by the mobile communication system, the demand for a higher data rate has increased in recent years. The Multiple-
Input Multiple-Output (MIMO) technique is one of the communication techniques which can provide the mobile communication system with a higher data rate and reliable data transfer by exploiting multipath propagation. Another application of wireless communication is satellite communication, which is emerging as a major component of the wireless communications infrastructure. Satellite systems can provide broadcast services over very wide areas and are also necessary to fill the coverage gap between high-density user locations [7]. In addition, there are other applications of wireless communications, such as Wireless Local Area Network or wireless LANs, Bluetooth, the
Global Positioning System or GPS, and more.
RF Propagation
In any wireless communication system, there is always a direct or indirect propagation path between transmitter and receiver antennas. Direct propagation path refers to the transmission of energy of the radio frequency (RF) signal along a direct path that does not involve any reflection, diffraction, or scattering mechanism. The direct propagation is called free space propagation or line of sigh path (LOS), and the signal undergoes free space attenuation [11]. In contrast, indirect propagation involves one or more of these mechanisms, so the RF energy propagates over the multipath between transmit and received antennas. Multipath can also occur on the light-in-sigh (LOS) path which has impotent impanation for utility of MIMO. Figure 2.1. depicts the various propagation mechanisms.
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Figure 2.1. Propagation mechanism in wireless communication channel.
Overall, the signal over the wireless communication channel suffers from a different kind of fading, called shadowing, which causes lower received power at the receiver. Figure 2.2. illustrates the dependence of received power on the distance between transmitter and receiver antennas over a multipath channel.
Figure 2.2. The dependence of received power over a multipath channel [11].
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Three characteristic component scales were observed and are shown is Figure 2.2.
For the first component, the dash line refers to the linear trend towards the lower received power component, which indicates that the power tends to decrease when the verse of the distance is raised. The second component is large -scale fading, which is shown in the figure as a light solid curve, which occurs due to a large object along the propagation path which blocks some transmitted signal energy. The third component is the small- scale fading, which is seen in the figure as a thick curve, which is caused by constrictive and destructive interference among signals arriving at the receiver from different paths due to different propagation mechanisms. The focus will be on the last two components because they affect MIMO’s performance. The large-scale fading has an impact on the signal-to-noise value (ρ), and the small-scale fading has an impact on the statistics of a communication channel (H) [11].
Impulse Response of Wireless Communication Channel
In a multipath environment, the received signal arrives over multiple paths with different gains, {αi(t)} and delays {τi(t)}. based on this definition, the communication channel models a filter with linear time -variant (LTV) impulse response function, c (t;
τ), so the received signal is equal to the convolution of the transmitted signal with an impulse response of the channel at the given time t. The Figure (2.3) displays a low pass model of the relationship between the received signal and impulse response.
Figure 2.3. Wireless channel model.
9 r(t)=c(t;τ) ⁎ u(t) (2.1)
The impulse response function must have the following expression.
푐(푡, 휏) = ∑ 훼 (푡) 푒푥푝{−푗2휋푓 휏 (푡)}훿 휏 − 휏 (푡) (2.2)
The multipath channels are time-variance channels, so the impulse response has two-timed variables, t and τ, which means that for the received signal for each specific time t there are replicas of the arrived signal with different delay spreading times, τ, which can be seen in Figure 2.4.
Figure 2.4. An example of the time-varying discreet impulse response for a multipath channel.
Multipath Channel Parameter
There are four parameters used to characterize the properties of a multipath channel.
All these parameters depend on the channel itself, and they do not relate to the nature of the signal which is being transmit through the channel. Delay spread, and coherent bandwidth are the parameters that relate to time dispersion properties of the channel τ, which spread out the signal in the τ time dimension because of the multipath channel.
10
Doppler spread, and coherent time parameters are similar to the delay spread and coherent bandwidth, but they characterize the frequency desperation properties of the channel which spread out the signal in the frequency dimension.
Delay Spread (Tm)
Delay spread is a measure of the range of propagation delays which are experienced by the multipath component of the signal when it is transmitted over a time-dispersive multipath channel [11]. The delay spread is normally specified either by the mean excess delay (τ) or rms excess delay (στ) which are describe the time dispersive natural of the channel in local area. We can find these parameters by using the power delay profile, so they formally defined as follow:
∑ ( ) τ = (2.3) ∑ ( )
στ = τ − (τ ) (2.4)
( ) τ = (2.5) ∑ ( )
Coherent Bandwidth (Bc)
The coherent bandwidth is a measure of the range of frequencies over which a communication channel passes all spectral components of a signal with equal gain and delay, so it is relative to spread delay. It is typically defined in one of two ways, if the coherent bandwidth is defined as bandwidth over which the frequency correlation function which is about 0.9. or 0.5, then the coherent bandwidth is called the 90th or 50th percentile coherent bandwidth, respectively [10].
Bc,90 =1/ (50 στ) (2.6) or
11
Bc,50 =1/ (5στ) (2.7)
Doppler Spread (Bd)
Doppler spread refers to the broadening of a transmitted signal’s bandwidth as it propagates through a multipath channel due to relative motions between the transmitter, scatters, and receiver. For example, if the transmitted frequency is fc, the received signal spectrum in the range fc-fd to fc+fd, where fd is the Doppler shift, a function of the relative velocity (v) between the transmitter and receiver is the angle between the direction of the motion of the object and the direction of the arrived signal, then the received bandwidth
=2fd [10] This demonstrates that the spectral broadening has occurred. The Doppler spread then becomes:
Bd = 2fd (2.8) where: fd = v (fc /c) cos(θ) (2.9)
Coherent Time (Tc)
The coherent time is the time domain dual of Doppler spread and it used to characterize the time varying nature of frequency dispersiveness of the channel in the time domain. The maximum Doppler spread, and coherent time are inversely proportional to one another [9] [23]. That is,
Tc ≈ 1/Bd (2.10)
Where: Bd ≈ fm
For more explanation, we can say the coherence time is the time duration over which two received signals have a strong potential for amplitude correlation. If the reciprocal bandwidth of the baseband signal is greater than the coherence time of the
12 channel, then the channel will change during the transmission of the baseband message, thus causing distortion at the receiver [9].
If the coherent time is defined as the time over in which the time correlation function is above 0.5, then the coherent time is approximately,
Tc ≈ 9/16π fm (2.11)
Where: fm is maximum Doppler shift given by fm = fd = v (fc /c)
Type of Small-Scale Fading
Flat Fading
If the communication channel has a constant gain and linear phase response over bandwidth (Bc), which is greater than the bandwidth of the transmitted signal (w), then the received signal will undergo flat fading which is the most common type of fading described in technical literature. Therefore, the received signal will change with time due to the fluctuation of the channel caused by the multipath [23]. To summarize, a signal will undergo flat fading if,
Bs << Bc (2.12) and,
Ts >> στ (2.13)
Where Ts is the symbol period and Bs is the bandwidth of the signal.
Frequency Selective Fading
If the channel possesses a constant gain and linear phase response over a bandwidth that is smaller than the bandwidth of the transmitted signal, then the signal will undergo frequency selective fading. The frequency selective fading is due to time desperation of
13 the transmitted symbol. Thus, the channel induces inter symbol interference (ISI) [23].
To summarize, a signal will undergo frequency selective fading if,
Bs > Bc (2.14) and,
Ts < στ (2.15)
A rule thumb for digital modulation like BPSK signal is that a channel is flat fading if Ts ≥10 στ and a channel is frequency selective if Ts ≤ στ., where Ts is the signal duration.
Fast Fading
Depending on how rapidly the transmitted signal changes in comparison to the rate of change of channel, fast fading can occur. In fast fading the channel impulse response changes rapidly with the symbol duration signal. This causes frequency dispersion due to Doppler spreading, which leads to signal distortion [23]. To summarize, the signal will undergo fast fading if,
Bs < Bd (2.16) and,
Ts >Tc (2.17)
Slow Fading
In slow a fading channel, the channel impulse response changes at a rate much slower than the transmitted baseband signal. In this case, the channel may be assumed to be static over signal duration interval. In a frequency domain, this implies that the
Doppler spread of the channel is much less than the signal bandwidth, therefore the signal undergoes slow fading if,
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Bs >> Bd (2.18) and,
Ts < It should be noted that when a channel experiences fast or slow fading, it does not imply whether the channel is flat or frequency-selective fading in nature. Fast and slow fading are only related to the rate of change of the channel due to motion. Figure 2.5. shows the condition of different types of fading depending on Delay spread and Doppler spread. Statistic of Small- Scale Fading Rayleigh Fading The Rayleigh fading often occurs in multipath environment where there is no direct path component between the transmitter and receiver. In the Rayleigh fading channel, the magnitude of impulse response is a Rayleigh random variable and the signal has a Rayleigh distribution envelope as a function of time [11]. Rician Fading Rician fading occurs when in addition to multipath there is a direct path between the transmitter and receiver. In other words, the received signal consists of two components: a random portion that fluctuates in time and non- fluctuate part. From this definition, the Rician distribution should become identical to Rayleigh fading if there is no direct path, and all received signal energy is associated with scattering, which is an assumption underlying the Rayleigh fading channel [11]. 15 Figure 2.5. The diagram shows the condition of type fading. [23] 16 CHAPTER 3 MIMO COMMUNCATION SYSTEM Introduction Multiple Input Multiple Output is one of recent techniques in wireless communication. MIMO is a collection of signal processing techniques that have been developed to enhance the performance of wireless communication systems [11]. The MIMO technique refers to using multiple antennas at the transmitter, receiver, or both, and it is improving the communication performance by either combating or exploiting multipath communication channels. If the MIMO technique is used to combat multipath channels, it creates what is called spatial diversity. Or if the MIMO technique is used to exploit multipath channels, it creates what is called spatial multiplexing. The characteristic of the communication system depends on whether the focus of the MIMO technique is on creating spatial diversity, which improves reliability of communication channels, or it is the focus on increasing the data rate by performing spatial multiplexing (SM). Figures 3.1. and 3.2. show block diagrams of a generic MIMO communication system. In figure 3.1, the information bit is encoded and modulated prior to be undergoing some form of space-time coding (STC). At the receiver, space time decoding is preformed, followed by demodulation and error decoding [11]. In figure 3.2, the information bit is encoded and passed through a serial-to-parallel converter. The outputs 17 stream is modulated before being transmitted over separate antennas. At the receiver, each antenna receives mixed signals from all transmitters. Therefore, it is necessary to strip off each signal before demodulating by using an SM decoder or spatial demultiplexing. There are different types of special demultiplexing schemes such as zero forcing and linear minimum mean square error. At this point, there are two concepts to be clear about. First, special diversity techniques that are used to improve the reliability of the communication link. The second concept are special multiplexing techniques that are used to increase the capacity without increasing the required bandwidth by exploiting the multipath components. Table 3.1. shows the relationship between different terms that are used in the MIMO technique [11]. Figure 3.1. A MIMO technique for spatial diversity. 18 Figure 3.2. A MIMO technique for spatial multiplexing. Table 3.1. Relationship of The MIMO Concepts [18]. MIMO technique Purpose Approach Method Spatial diversity Improved reliability Combat fading Space-time coding Spatial multiplexing Increase capacity Exploit fading Spatial demultiplexing SISO, MISO, and SIMO Antenna Configuration There are different ways to classify the antenna configuration. These techniques enhance the performance of wireless communication system. The figure 3.3. shows the classification of multiple antenna configurations. The basic communication system is single input single output (SISO), which uses one transmitted antenna and one received antenna. The communication channel model has a single input and a single output. The data rate that transfers through a channel is related with signal- to -the noise ratio at the receiver. According to the Shannon theorem, the maximum date rate of the channel is equal to the capacity of the communication channel as given in Eq. (3.1) 19 CSISO = log2(1+ρ) ( ) (3.1) Where: ρ is the signal - to - noise ratio Figure 3.3. Shows classifications of multiple antenna configuration. The reliability of data transfer over SISO, under the assumption that the receiver has knowledge of channel (CSIR) information a is little bit low. To improve the performance of the communication system, multiple antennas are used at the receiver or at the transmitter to enhance the reliability or the capacity of channel. In a single input multiple output communication system (SIMO), the communication channels models as a single input and multiple output. Due to the multipath propagation, the signal which is transmitted from a single transmitter arrives at 20 each receive antenna through different paths. This implies that the receiver has replicas of the transmitted signal. This is called receive diversity. This technique can reduce the probability of errors because the signal was received from the path which has less fading of the signal than other paths. Therefore, increasing the number of received antennas reduce the fading of the transmitted signal, which means increasing the reliability of the wireless communication channel. Also, the capacity of the communication channel logarithmically increases with the number of the receive antennas, Nr, when the signal to noise ratio, ρ, is large [18]. The SIMO capacity can be calculated by, CSIMO = log2(1+ρ Nr) ( ) (3.2) The capacity of the SIMO wireless communication system under the assumption that the communication channel has both (CSIR) and (CSIT) is the same to the capacity when the channel has (CSIR) only see Table 3.2. The reason of having the same capacity under these two assumptions is the disadvantage of the channel knowledge at the transmitter because there is a single transmit antenna. In multiple input single output (MISO) technique, the communication channel models as a multiple input and single output. If the channel has the knowledge of the CSIR only, the receiver only receives one signal that was transmitted over Nt transmitted antenna. In this case, the performance of MISO - CSIR only is the same as the SISO communication system due to the single received antenna. The reliability and the channel capacity are the same for both MISO-CSIR only and SISO. On the other hand, if the channel information is available at both the transmitter and receiver which are the CSIR and CSIT, the transmitter can decide which one of the Nt transmit antennas will be used to transmit data to the single receiver antenna. The selected channel has a large value of 21 the signal-to-noise ratio ρ. The reliability of the MISO system will increase as the channel capacity will logarithmically increase with Nt increased. The MISO capacity can calculate by, CMISO = log2(1+ρ Nt) ( ) (3.3) It is clear that Eq. (3.2) is the same as Eq. (3.3) except that we replace Nr with Nt. Table 3.2. shows the formula of the channel capacity under the assumption CSIR only and CSIT and CSIR for different communication systems. Table 3.2. Channel Capacity Formulas of SISO, SIMO, and MISO Communication System [11]. CSI type SISO SIMO MISO CSIR only log2(1+ρ) log2(1+ρ Nr) log2(1+ρ) CSIR and CSIT log2(1+ρ) log2(1+ρ Nr) log2(1+ρ Nt) Single - User and Multi - User of the MIMO System There are two classes of MIMO communication systems that are used in wireless communication. The first class is a single-user MIMO (SU-MIMO), such as the communication system used in LTE and WiMAX, refers to using one transmitter node and one receiver node. Each node has multiple antennas as showed in Figure 3.3. The second class is a multi-user MIMO (MU-MIMO), such as a mobile cellular user. The base station processes the signal coming from each individual mobile as if they were coming from multiple transmit antennas on single node, so the multiple user can transmit data simultaneously over same bandwidth [11]. The focus will be on the SU-MIMO technique, however, the applications for a single-user is valid for a multi-user also. 22 The Concept of Diversity Diversity is described as powerful wireless communication technique that mitigates the effect of fading of the signal in most wireless communication environment. The basic principle of diversity is that receiver has more than one replica of the same signal over fading channel that each replica fades independently of each other. The probability of fading all the replicas at the same time decreases when the number of replicas gets larger. At receiver, combining the replicas reduces their adverse effects because the fades do not occur at the same time. There are many ways to generate the replicas of the signal so when the signals are transmitted with different frequencies that creates frequency diversity. Another diversity technique is called time diversity, and it means transmit the signal at the different time, in multipath environment that happens naturally because the signal arrives with delay at receiver[b1]. An interesting kind of diversity is called special diversity which refers to transmitting same information over different physical paths between transmitter and receiver. Special diversity is created by transmitting a signal from one transmitter antennas and receiving it by using multiple receiver antennas [11]. As there are different ways of generating replicas, there are different ways of combing the replicas at the receiver. The most common type of combining technique is called maximal ratio combining (MRC) which will be explained in chapter three. In (MRC), the replicas adding together at the receiver, starts with the first scaled in proportion to the signal-to-noise ratio of each replica. 23 Diversity Order and Diversity Gain The two parameters related with special diversity may appear different. Diversity order (Nd) refers to the number of signal replicas that are being combined at the receiver. Diversity gain (Gd) refers to the slope of bit error probability versus Eb/No. In most cases, in Rayleigh fading environment, the diversity gain and diversity order are numerically equivalent especially at specific modulation types when MRC is used [11]. The MIMO system is said to have achieved full diversity when the diversity order or diversity gain are equal to the product of the number of transmitter and receiver antennas. Full diversity = Nd = Gd =Nt Nr (3.4) The Concept of Spatial Multiplexing Spatial multiplexing (SM) is described as transmitting multiple independent data streams over a multipath channel by exploiting multipath [16]. Spatial multiplexing is like the other type of multiplexing such as frequency division multiplexing (FDM) and time division multiplexing (TDM) where the multiple signal is assigned to the frequency slot in (FDM) or to the time slot in (TDM). In spatial multiplexing, the data signals are transmitting at the same time and same frequency but at different spatial channels [11]. Therefore, spatial multiplexing technique is better than FDM and TDM because it does not suffer from bandwidth expansion like these techniques. Figure 3.4. shows the block diagram of SM-MIMO system. There are three components related with spatial multiplexing. The first component is a precoder. It maps the multiple data stream to the set of transmit antennas. In general, the number of data streams is less than or equal to the transmit antennas [11]. One type of precoding 24 component is eigenbeamforming, which will be explained in Chapter 3. The second component is postcoder which processes the signal at the receiver and generates the estimate of the original signal that went to the input of the precoder. The postcoder must be able to extract the data stream of each transmit antenna from the sum of the data streams at the received antennas. The Zero Forcing (ZF) and linear minimum mean square error estimate (LMMSE) are the most common techniques of the postcoder. The third component of spatial multiplexing is the communication channel itself. If the communication channel suffers from scattering, that will generate a multipath which lets spatial multiplexing work, so the channel must have significant amounts of multipath to exploit by spatial multiplexing. Figure3.4. The diagram of MIMO system that used spatial multiplexing. The maximum number of data streams, Nstream, can be supported by MIMO system using spatial multiplexing increases linearly with number of antennas and it is given in Eq. (3.5) Nstream = min (Nt,Nr) (3.5) Capacity of the MIMO System To define the MIMO capacity, we need to know what the capacity of a conventional single-input single is-output (SISO) communication system. Basically, we 25 will start with defining the average mutual information I(x; y). It is the average amount of information about x that is conveyed by knowing the value of y [11]. I(x; y) can be considered the average amount of transmitted information over a communication channel when the input and output of a communication channel are x and y. The communication channel does not have control of the information examination at the receiver, but the transmitters have a control distribution of the information and it generates the Px(x) [11]. Based on this consideration, we can define the capacity of SISO system as follows: CSISO = max I(x; y) (3.6) {Px(x)} Where the maximization is performed under the following consideration: 푃 (푥) =1 And, 0 ≤ Px(x) ≤ 1 To define the capacity of MIMO system, we need to describe the relationship between the transmitted and received signal. Assume the signal is a narrowband communication signal, which suffers from flat fading because its bandwidth is narrow compared with coherent bandwidth of the channel. We start with following parameters. hij= channel response between jth transmit antenna and ith receive antenna ri = received signal at ith receive antenna sj = symbol transmitted from the jth transmit antenna zi = noise signal at ith receive antenna So that the received signal at receiver over flat fading channel is follow that: 26 푟 = ℎ 푠 + 푧 푖 =1…………. 푁푟 (3.7) In the matrix form Eq. (3.7) becomes: r = Hs + z (3.8) Where each of this parameter as matrix T s =[s1,…….sNt] T z=[z1,…….zNr] T r =[r1,…….rNr] ℎ1,1 ⋯ ℎ1, 푁푡 H= ⋮ ⋱ ⋮ ℎ푁푟,1 ⋯ ℎ푁푟, 푁푡 The dimension of the channel matrix (H) is Nr × Nt. when we reversed the diminution to channel matrix, the s, z, and r vector change to the row vector instead of column vector and Eq. (3.6) becomes r = Hs + z. We need to also know the relationship between the average mutual information I (x; y) and the average of information source, which is called entropy H(x). Thus: I(x; y) = H(x) – H(x\y). (3.9) where: H(x); average amount of information in x prior to the observation of y. H(x\y); average amount of information in x after y is observed. Also, it indicates the average amount of information that has been lost on communication channels. We can get the capacity of MIMO system by generalizing the Eq. (3.3) by replacing X with s and Y with r [8]. The capacity of MIMO system becomes: 퐶 = 푚푎푥 {퐻(푟) − 퐻(푟\푠)} (3.10) ( ⋯ 푁푡 ) To simplify the capacity of MIMO system substitute eq. (3.8) in to Eq. (3.10). H (Hs + z\s) because of H and Hs are fixed and the only random variation in H(r\s) is the noise z. Also, under the condition of the noise that has Gaussian distribution with 27 mean (μ) and variance (σ 2) [11], we can replace H(r\s) by H(z). and the capacity of MIMO system becomes: 퐶 = 푚푎푥 {퐻(푟) − 퐻(푧)} (3.11) ( ⋯ 푁푡 ) where: H(r): average amount of information in r. H(z): average amount of noise in z. The probability density function of the Gaussian vector (z) can be denoted as: 푃푧(푧) = 푒푥휌(− (푧 − 휇) 푅 (푧 − 휇) (3.12) √ | | Where: 푅 is the covariance matrix of the noise. Assume 휇 = 0, then one can verify that the value of 푅 = 휎 퐼 ,in which 퐼 indicates an identical matrix with dimension (푁 ×푁 ). Thus H(z) can be expressed using a logarithmic version as H(z)= -E{log2(Pz(z)} (3.13) After applying the equation (3.13) in the expectation of equation (3.12), H(z) is 퐻(푧) = 푙표푔 (2휋푒) + 푙표푔 휎 퐼 (3.14) In addition, the value of H(r) that maximizes the MIMO capacity as shown in equation (3.11) can be obtained using the entropy – maximizing theorem (EMT), which is proven in [4]. The H(r) is maximized when r is a Gaussian vector with independently and identically distribution elements that are zero mean and with covariance matrix 푅 . Therefore, using equation (3.8), the covariance matrix of (푅 ) is obtained as follows: 푅 = 퐻푅 퐻 + 휎 퐼 (3.15) So that the max (H(r)) can be written as: max {H(r)} = 푙표푔 (2휋푒) + 푙표푔 퐻푅 퐻 + 휎 퐼 (3.16) 28 Substituting Eq. (3.14) and Eq. (3.16) in Eq. (3.11), we will get the capacity of MIMO system for real valued signals as: CMIMO = 푙표푔 퐼 + 퐻푅 퐻 ( ) (3.17) In general, Eq. (3.16) represents the capacity for real signal, so the capacity of complex signal is twice the capacity of real signal because the complex signal consists of two parts, real and imaginary, that are able transmit independent data stream. Also, instead of using the transpose operation of channel matrix, we used Hermitian operation for complex signal. Telatar’s paper [5] shows that for complex signals the Hermitian operation, which can be seen below, should be used instead of the transpose operation. CMIMO = 푙표푔 퐼 + 퐻푅 퐻 ( ) (3.18) Under the assumption that the receiver has the knowledge of channel state information only (CSIR), the capacity of MIMO system can be calculated by Eq. (3.23) after normalizing the MIMO system equation as shown in Appendix A. To normalize the Eq. (3.18), let us express the physical channel matrix as 퐻 = 휌퐻 , where 퐻 is the normalized channel matrix, and E { ℎ }=1 for random 퐻 (3.19) ‖퐻 ‖ = 푁 푁 for fixed 퐻 (3.20) E {|푧 | }= 휎 = 1 , (3.21) E{|푠 | } = 휎 = (3.22) 푅 = 휎 퐼 The term of ‖ . ‖ refers to the Forbenius norm, which ‖퐻 ‖ = 훴 훴 퐻 . CMIMO = 푙표푔 퐼 + 퐻 퐻 ( ) (3.23) 29 Using 퐻 = 퐻 , where 퐻 is normalized or it explicitly contains the signal to noise ratio ( 휌). Thus, the MIMO capacity is CMIMO= 푙표푔 퐼 + 퐻퐻 ( ) (3.24) We can obtain 퐻퐻 by using an eigenvalue decomposition. Thus, 퐻퐻 = 푈퐷푈 , where 푈 is the unitary matrix, and 퐷 is diagonal matrix. Thus, 퐷 = 푑푖푎푔{휆1 , 휆2 , 휆3…… 휆r,0,0,0}, where λi is the Eigenvalue of the channel matrix and r is the rank of channel matrix , r = rank(퐻퐻 )= rank (퐻). Therefore, the MIMO capacity becomes CMIMO = 푙표푔 (1 + 휆 ) ( ) (3.25) From Eq. (3.25), it is clear that the capacity of MIMO channel with CSIR only can be expressed as the sum of r SISO channels, each channel has the power gain, 휆 , 푖 =1,…. 푟, where the effective transmitted power of a SISO channel is times of actual transmit power [11]. On the other hand, the capacity of the MIMO system under the assumption of both CSIR and CSIT, which leads to the concept of eigenbeamforming, will be as follows: 퐶 = 푙표푔 (1+ 퐸{|푠 | }휌휆 ) (3.26) where: |푠 | is variance of transmit symbols. 휌 is signal-to -noise ratio (SNR) 휆푖: is a eigen value at ith receiver 30 Diversity – Multiplexing Tradeoff MIMO technique is used to enhance performance of communication system. There is always a tradeoff between the technique used to get spatial diversity and spatial multiplexing, so if we used space time coding, we will get the maximum diversity (Gd), but we will get also minimum capacity (rs), spatial rate represents the capacity of the MIMO system. On the other hand, if we use spatial multiplexing techniques, such as VBLAST, HBLAST, we will get maximum capacity, but we will also get minimum spatial diversity. Figure 3.5. illustrates the tradeoff relationship between the spatial diversity and spatial multiplexing. Figure 3.5. Taxonomy for MIMO Application. We can see from Figure 3.6. that a taxonomy of different types of techniques is used with the MIMO system. At the left-hand side of figure, there are two category codes: space time block code (STBCs) and space time trellis codes (STTCs). These codes achieve maximum diversity gain, but they have a small spatial rate which means a minimum capacity. Also, the orthogonal space time block code OSTBC has advantage in 31 comparison to the STTC. The advantage is simple decoding, which is different than STTC’s complex decoding. However, the STTC can achieve coding gain in addition to the diversity gain, which means that it has a smaller bit error rate than OSTBCs. In contrast, in Figure 3.6, the right most side of taxonomy lists techniques to achieve maximum capacity (rs=Nt), however, they all have low diversity gain (Gd = Nr). Spatial multiplexing includes H-BLAST, V-BLAST, and D-BLAST, as well as multi-group space time coding MGSTC, which can achieve higher spatial diversity in comparison to other spatial multiplexing techniques and achieve the maximum capacity. In addition to the code and techniques that we explained, the hybrid codes attempt to compromise between the spatial diversity and capacity. These codes have spatial rate which falls between the minimum, 1 and the maximum, which is Nt. Also, the diversity gain of these codes is given by Gd = Nr min {Nt, P} [11] [26]. 32 CHAPTER 4 MIMO BEAMFORMING TECHNIQIE Introduction We know the MIMO technique can be used to achieve spatial diversity gain by using space time coding (STC). Also, it can be used to achieve spatial multiplexing gain according to the concept of layer space time (LST) coding which has various types, such as (BLAST) family technique, which I previously mentioned in Chapter 3. With both techniques, we assume the channel state information is available at receiver only (CSIR only) to calculate the capacity of communication channel. Using MIMO beamforming technique, we consider a system that assumes both transmitter and receiver have the knowledge of channel state information, that is CSIT and CSIR to achieve diversity gain or spatial multiplexing gain [11]. The main limiting factor is the cumbersome overhead which acquires CSI at the transmitter. The solutions are categorized into FDD (Frequency Division Duplex) and TDD (Time Division Duplex) approaches [15]. Because of the limited feedback in FDD, the number of antennas cannot be large, which is one of the major drawbacks of transmit beamforming, but it may not be a drawback for the TDD model due to the use of channel reciprocity [14] [3]. Also, because beamforming required feedback information, which is channel state information (CSI) at the transmitter, the MIMO channel is modeled under the assumption of slow fading and flat fading channel when the time coherent (Tc) is 33 larger than the signal time and signal time much bigger than the delay spread of the multipath channel. In this chapter, we will discuss point – point MIMO channel (single – beamforming). Beamforming can be applied to the MIMO system by exploiting the multiple antennas at the transmitter and receiver [15]. It is well known that the phase array antenna can form one directional radiation pattern (i.e., beam) to enhance transmit (or receive) signal energy in a desired direction. The directivity is obtained by constructive interference among multiple antenna signals in the desired direction. This is called transmit (or receive) beamforming as shown in Figure 4.1. [15]. In modern cellular system, we use the term of smart antennas instead of MIMO technology. The smart antennas described ways to dynamically generate beams at the cellular base station that point in the desired direction of mobile users. It is used in a much broader sense to include not only dynamic beamforming and antennae nulling, but also spatial multiplexing and spatial diversity [11]. However, we can say that the MIMO technology is a subset of smart antenna technology. Figure 4.1. Transmit beamforming 3 × 2 MIMO channel. 34 MIMO beamforming technique is a practical spatial multiplexing technique that is used in most modern wireless communication systems, and it has been proposed in open literature and standard bodies, such as 3GPP, IEEE802.11n, IEEE802 16d/e [21]. Moreover, single – mode beamforming refers to the practical form of beamforming in which information is only transmitted over single eigenmode, which has the largest eigenvalue to achieve full diversity gain [11]. The principle of MIMO beamforming and the ideal beamforming is singular value decomposed (SVD) beamforming. Singular Value Decomposition (SVD) For any complex valued Nt × Nr channel matrix (H) of rank r, we can write this matrix in terms of the product of three matrices UNt×Nt , VNr×Nr , and diagonal matrix Dr×r . The number of singular value is equal to the number of rank. Unlike the eigenvalue decomposition, which exist only for square matrix, the singular value decomposition exists for any demotion of the channel matrix. Thus, we can find the SVD of the MIMO channel matrix (H) as follows H= U D VH (4.1) 푉 휎 ⋯ 0 ⎡ ⎤ 푉 H= [ U1 U2 …….Ur ] ⋮ 휎 ⋮ ⎢ ⎥ (4.2) ⎢ : ⎥ 0 ⋯ 휎 ⎣푉 ⎦ Where U and V are unitary matrices and D is the diagonal matrix with i- th singular value σi i= 1……r. Also, the U matrix has Nt columns U1….UNt and the V matrix has Nt rows, where each of these columns and rows are orthogonal, so H H 푈 =1, and U U =I if the rank r =Nt UU =I (4.3) H H ‖푣 ‖ =1, and VV =I, V V =I (4.4) 35 To find the optimal SVD, the singular value, σi, should have the following properties. σi has a positive value, that is, σi ≥ 0 , i = 1…….r (4.5) Also, the singular values should satisfy the following ordered form σ1 ≥ σ2 ≥ σ3 ≥ σr ≥ 0 (4.6) Example: ℎ ℎ ℎ Find the SVD for the following channel matrix H = ℎ ℎ ℎ We know that H= UDVH when the channel matrix is complex matrix, but now the channel matrix is real where the Hermitian operation becomes Transpose operation. Any real m × n matrix H can be decomposed uniquely as H= UDVH U is m×n and column orthogonal (its columns are eigenvectors of HHH) (HHH = UDHH VDUH = UD2UH ) V is n×n and orthogonal (its columns are eigenvectors of HHH) (HHH = VDUH UDVH = VD2VH) D is n×n diagonal (non-negative real values called singular values) D = diag ( σ1, σ2, . . . , σn) ordered so that σ1 ≥ σ2 ≥ . . . ≥ σn In our case H is (2×3), so U = (U1 U2 U3) and V = (푉 푉 푉 ) Where U1 and 푉 are column vector and row vector 푢 푈 = (4.7) 푢 푣 푉 = 푣 (4.8) 푣 36 휎 ⋯ 0 H=[ U1 U2 U3 ] ⋮ 휎 ⋮ [푉 푉 푉 ] (4.9) 0 ⋯ 휎 H=∑ 휎푖 Ui 푉 (4.9) In our example n=3 The sum in Eq. (4.9) goes from 1 to r where r is the rank of H Precoding of The MIMO Channel There are many types of precoding techniques used with the MIMO channel, such as Linear precoding and Tomlinson-Harashima precoding. In this thesis, we will discuss the linear precoding designed and its related with beamforming technique. The precoder connects between the encoder and the channel, depending on the code used, the encoder produces codewords with a certain covariance Q. We assume the (CSI) is available at the transmitter. Consider the singular value decomposition of the precoder matrix. F = UF D VF (4.10) Where the UF and VF are unitary matrices and D is diagonal matrix. The orthogonal beam directions are the left singular vectors, UF, where each column represents a beam direction (pattern). Note that UF is also the eigenvectors of the product FFH, thus, the structure is often referred to as eigen-beamforming. The beam 2 power loadings are the squared singular values D . The right singular vectors VF mix the precoder input symbols to feed into each beam, and hence, is referred to as the input shaping matrix [25]. Figure 4.2. shows the linear precoding structure. 37 Figure 4.2. Linear precoding structure as multi-mode beamformer. A linear precoder is composed of an input shaping matrix, a beamforming matrix, and the power allocation over these beams. The optimal input shaping matrix is determined by the input code alone, the beamforming matrix by the CSIT alone, and the power allocation by both [25]. The Input-Shaping Matrix The encoder shapes the covariance of the codeword input to the precoder; in response, the precoder chooses its input-shaping matrix to match this covariance. Suppose the input codeword covariance matrix Q has the eigenvalue decomposition Q = UQΛQUQ, the optimal input-shaping matrix is then given by Eq. (4.11). This optimal input-shaping matrix results directly from the predetermined input code covariance Q, which is not an optimization variable nor involved in the power constraint. The covariance Q characterizes the code chosen for the system. By matching the input codeword covariance, the precoder spatially de-correlates the input signal and optimally collects the input energy [25]. 38 VF = UQ (4.11) The Beamforming Matrix Unlike the input-shaping matrix, which is independent of the CSIT, the beamforming matrix is a function of the CSIT. Because the transmitter has the knowledge of channel matrix, we can find the Singular Value Decomposition (SVD) of MIMO channel and decouple the MIMO channel to sub-individual channel. The optimal precoder matrix for perfect CSIT, under all criteria and at all SNRs, has the left and right singular vectors determined separately by the eigenvectors of the channel gain HHH and the input codeword covariance, Q, respectively. Therefore, the precoder spatially matches both sides [25]. Figure 4.3. displays the precoding beamforming MIMO channel. Figure 4.3. Precoder match both the input code structure and channel. Spatial Multiplexing by Beamforming Technique By using a beamforming technique, the MIMO channel can be modeled as sub- parallel channels under the assumption that the transmitter has the knowledge of channel state information. A beamforming technique includes the exertion of singular value decomposition of channel matrix (H). The MIMO communication system can be modeled as. 푟 = 퐻푠 + 푧 (4.12) where: 39 r: receive vector 푠: data vector H: channel matrix 푧: noise vector Substituting the channel matrix with singular value decomposition SVD of channel matrix we have r = [U D VH] s+ z (4.13) Since the transmitter is assumed to have knowledge of H, it can compute V and perform the following precoding operation that have been done before the transmission process. 푠 = 푉푠̃ (4.14) Where: s is the transmitted vector. U and V are unitary matrices, so their rows and columns are orthogonal. The assumption is that number of receiver antennas is larger than the number of transmitted antennas, Nr ≥ Nt . Also, the channel rank (r) is equal the number of transmit antenna Nt. We can write the received vector as. 휎 ⋯ 0 H 푟 = U ⋮ 휎 ⋮ V V 푠̃ + 푧 (4.15) 0 ⋯ 휎 Where: VHV =I and UHU =I so, at the receiver, we will multiply the incoming vector by UH to calculate 푟̃. 푟̃ = UH r (4.16) 휎 ⋯ 0 H H 푟̃ = U U ⋮ 휎 ⋮ 푠̃ + U (4.17) 0 ⋯ 휎 40 휎 ⋯ 0 푟̃ = ⋮ 휎 ⋮ 푠̃ + 푧̃ (4.18) 0 ⋯ 휎 From Eq. (4.18) we can write the received signal at ith receiver. 푟 ̃ = 휎 푠 ̃ + 푧 ̃ (4.19) 푟 ̃ = 휎 푠̃ + 푧 ̃ (4.20) 푟 ̃ = 휎 푠 ̃ + 푧̃ (4.21) According to equation (4.16), we convert the MIMO channel into independent and parallel additive-white-noise channels, and the number of these channel equals the minimum number between the numbers of transmit and receive antennas [5]. The parallel channels can be processed independently, each with independent modulation and coding, allowing per-mode rate control and simplifying receiver processing [25] [16]. This process is considered spatial multiplexing as shown in Figure 4.4. Figure 4.4. Decupling MIMO channel to sub-channels. [16] To prove that each sub-channel has independent additive-white-noise, at the receiver, the received signal is multiplied by UH. Also, the noise component multiplied by UH that 푧̃ = 푈 푧, so the variance of the noise component in Eq. (4.17) will be as follows: 41 E{푧̃푧̃ } = E{푈 푧푧 푈} (4.22) = UH E{푧푧 }U (4.23) H 2 = U σz I U (4.24) 2 H = σz U U (4.25) 2 = σz INt (4.26) 2 2 E{푧̃푧̃ } = σž = σz (4.27) We can treatment each of these sub-channels as a SISO channel. Figure 4.5. illustrates the scheme of each SISO parallel channel. However, the signal-to -noise ratio of the ith sub-channel (SNR) will be as follows: (SNR)i = (4.28) Figure 4.5. Scheme of MIMO parallel channel. The maximum transmission rate of each sub-channel will be the same as the capacity of the SISO channel, which is the Shannon capacity as we show in Eq. (4.29) for ith parallel channel. 42 Ci = log2 (1+ SNRi) (4.29) The maximum MIMO channel capacity will be the sum of all parallel sub-channels capacity. CMIMO = ∑ (1+ 푆푁푅 ) (4.30) 2 푃푖휎푖 CMIMO = 1+ 2 (4.31) 휎푧 Where: Pi is the power of the signal at the ith sub-channel. 휎 is the gain at each sub-channel. 휎 is the variance of AWGN for each sub-channel. Eq. (4.31) shows that the capacity of the MIMO channel is r times the capacity of the SISO channel, with equal power distribution over transmitter antenna and with same bandwidth. In fact, we need to allocate the power over these sub-channels to achieve optimal total capacity of MIMO channel. The best way to allocate transmitted power is the waterfilling algorithm. Optimal Allocated of Power (Waterfilling Algorithm) The sum of power of each stream (sub-channel) should be equal to the total power transmitted, we cannot allocate more power than the total power transmitted. ∑ 푃 = P =1 (4.32) where P is the total transmitted power. We maximize the capacity of MIMO channel with constrains of the power allocated algorithm, which is called the waterfilling algorithm. The principle of waterfilling is that the higher power is allocated to the beams corresponding to known 43 strong channel directions. Also, reduced or no power is allocated to the weaker beams. The function we seek to optimize is beamforming channel capacity in Eq. (4.29), which is function of {Pi}, and the constraint given by Eq. (4.32) [11]. Thus, let us call this function F. 2 푃푖휎푖 F = 1+ 2 + λ (P -∑ 푃 ) (4.33) 휎푧 Where λ is a constant called Lagrange multiplier. We can maximize the function (F) by differentiating Eq. (4.33) respect to the power of each sub-channel 푃 and set it to zero. = 0, (4.34) + 휆(−1) =0 (4.35) = 휆 (4.36) =1+ (4.37) Multiple the Eq. (4.37) by ,thus = + 푃 (4.38) From Eq. (4.38), we can find the optimal power sing for each sub-channel as follows: 2 휎푧 푃 = + 2 (4.39) 휎1 2 휎푧 푃 = + 2 (4.40) 휎2 44 2 휎푧 푃 = + 2 (4.41) 휎3 Since the power is always positive, the expression of (.)+ means 푥 푖푓 푥 >0 푋 = (4.42) 0 푖푓 푥 ≤0 Thus, if the power of sub-channel Pi is less than zero, the power will be set to zero. All sub- channels’ power depends on quantity , and we still must satisfy the constraint condition given in Eq. (4.32), substituting Eq. (4.39), (4.40), (4.41) in to Eq. (4.32). 2 휎푧 + 2 = 푃 (4.43) 휎푖 We find the parameter by the power transmitter constraint, and this is called the waterfilling algorithm, as shown in Figure 4.6 Figure 4.6. Scheme of waterfilling algorithm. Assume that, P i = , i = 1,2 ….r (4.44) Figure 4.6. shows the waterfilling algorithm, where r =5, shows there are 5 parallel channels. By finding the level , the power allocated for channel one {P1} is 45 positive because it is less the level , that is also true for P2, P3, but the power allocated for channel four and five are equal to or greater than the level . Using the expression of (.)+ and Eq. (4.43), we will discard these modes. The waterfilling algorithm is a more selective power allocation scheme. At low SNRs, weak modes tend to have a high error rate; therefore, dropping these modes and allocating power to stronger modes leads to better overall system error performance [25]. Maximal Ratio Receive Combining (MRRC) One of the methods that was used to achieve received diversity is maximal ratio combiner. In general, a receiver that employs maximum ratio combiner (MRC) has the architecture shown in Figure 4.7, where the output of combiner, 푠̃, consists of linear weighted combinations of the input [11]. Figure 4.7. Receive diversity combiner. Thus, ∗ 푠̃ = ∑ 푤 푟 or 푠̃ = ∑ ℎ 푟 (4.45) Where { 푤 } denotes complex weights and {푟 } the received signal on each diversity channel (diversity path). With MRC, each combing weight is equal to the complex 46 conjugate of the respective channel gain (ℎ∗ ) [11]. According to Eq. (4.42), combining the signal received at (Nr) receiver antenna will be as follows: ∗ ∗ ∗ 푊 푟 + 푊 푟 +⋯+ 푊 푟 (4.46) 푟 ⎡ ⎤ 푟 ∗ ∗ ∗ ⎢ : ⎥ 푠̃ = [푤 푤 ⋅⋯ 푤 ] ⎢ ⎥ (4.47) ⎢ : ⎥ ⎣푟 ⎦ We can write the output of combiner in vector representation form as follows: 푠̃ = 푤 푟̅ (4.48) The process of linearly combining the signal with a complex weight at the output of received antenna is known as the beamforming technique, and the vector 푤 is known as beamformer in a multiple antenna received system. 푤 ⎡ 푤 ⎤ ⎢ ⎥ Where the 푤 = ⎢ ⋮ ⎥ (4.49) ⎢ ⋮ ⎥ ⎣푤 ⎦ We know that the received signal models as 푟̅ =ℎ 푠 + 푧̅ (4.50) Where: 푟̅ is received vector, ℎ is fading coefficient vector, 푧̅ is AWGN. The output of combiner substitute Eq. (4.50) in Eq. (4.48) 푠̃ = 푤 ℎ 푠 + 푤 푧 ̅ (4.51) Where the signal component is 푤 ℎ 푠, and the noise component is 푤 푧̅, Signal to noise ratio of combining received signal SNR = (4.52) From Eq. (4.50), the signal power = 푤 ℎ 푃 (4.53) 47 The noise power of combining received signal is obtained as follows: Noise = 푤 푧,̅ because the noise is Gaussian random variable = E {(푤 푧̅ )( 푤 푧)̅ *} (4.54) = E{푤 푧̅푧̅ 푤 } (4.53) = 푤 E{ 푧̅푧̅ }푤 (4.54) = 휎 푤 푤 (4.55) = 휎 ‖푤 ‖ (4.56) Thus, SNR= 2 (4.57) 휎푧 Note, the SNR depends on the 푤 , so we choose a 푤 to maximize the signal to noise ratio. Let us choose 푤 푤 = 1 , ‖푤 ‖ = 1 (4.58) SNR= (4.59) The dot product between 푤 푎푛푑 ℎ is maximum when the cosθ =1, happens when angle (θ) between these two vectors is equal to zero, so 푤 = C ℎ , Where; C is a constant from Eq. (4.58) ‖푤 ‖ = 퐶 ℎ 2=1 (4.60) ℎ So, C = , 푤 = (4.61) ‖ ‖ ‖ ‖ The optimal beamforming vector 푤 that maximize SNR will be, ℎ 푤 = (4.62) ‖ℎ‖ 48 This technique is known as MRC. In fact, the optimal vector is the scaled version of fading coefficient over multiple antenna receivers. From the idea of matching filter, the optimal vector 푤 matches the fading coefficient over space, which is called spatial match. Substitute Eq. (4.61) in (4.59) (SNR)Max = ℎ 2 (4.63) 휎푧 Maximal Ratio Transmit (MRT) One of most important methods that achieves transmitter diversity gain is maximal ratio transmit (MRT). Figure 4.8. shows the architecture of a communication system with a transmitter diversity combiner. There are different techniques used to achieve transmitted diversity, such as space time coding and single- mode beamforming, based on that MRT method [11] [15]. Figure 4.8. Scheme of transmitter diversity combiner. Figure 4.8. shows that there are replicas of transmitted signal arrived at received antenna with different fading coefficients. To achieve better performance, we combine these signals by applying the maximal ratio combiner technique, thus the received signal at the receiver antenna will be expressed as. 49 푟 = ∑ ℎ 푠 + 푧 (4.64) Based on the beamforming technique, which requires the channel state information at transmitter (CSIT), assume that we transmit the same signal over multiple transmitted antennas. Thus, 푟 = ∑ ℎ 푠 + 푧 (4.65) Using the same equation as the MRC, the signal at the output of the combiner. 푠̃ = 푤 (ℎ 푠 + 푧 ) (4.66) Where { 푤 } denotes complex weights, {ℎ } channel gain, and { 푧 } i.i.d noise. The maximum signal to noise ratio at the output of combiner that we mentioned previously will be. (SNR)Max = ℎ 2 (4.67) 휎푧 Maximum Likelihood Detection with MRRC As we have seen, the Eq. (4.51) indicates the output of MRRC. In general, the output of MRRC has the following form: 푠̃ = 퐾푠 + 푧 (4.68) Where 푠̃ indicates the output of the combiner, K is a constant that depends on the channel gain, s is the signal that we want to detect, and z is the complex Gaussian noise with zero mean and variance equal to 휎 . The maximum a-posteriori probability (MAP) estimate of s, which we indicated by 푠̂, is expressed as follows: 푠̂ = argmax 푃 (푠 ∕ 푠̃) (4.69) {푠} 50 Where P(s/푠̃) denotes the probability that symbol s was sent, given the output of the combiner 푠̃. The symbol that maximizes this probability is the MAP estimate of the transmitted signal. Using Bays theorem, if the a-priori probabilities (APP) of the signals being transmitted are all the same, so that P(s/푠̃) is equal to P(푠̃/푠) then Eq. (4.69) can be rewritten as follows: 푠̂ = argmax 푃 (푠̃/푠) (4.70) {푠} Here P (푠̃/푠) is called the likelihood probability and Eq. (4.70) defines the Maximum likelihood criterion. Since the noise is complex Gaussian noise with zero mean and variance equal to 휎 . Using Eq. (4.71) one can see that 푠̃/푠 is Gaussian, thus P(푠̃/푠) = 푒 | ̃ | ∕ (4.71) Therefore, the value of the transmitted signal that maximizes P(푠̃/푠) is the symbol that minimizes the exponent in this equation, that is; 푠̂ = arg min|푠̃ − 퐾푠| (4.72) { } The preceding ML detection rule can be rewritten as follows: 푠̂ = argmin[|푠̃ − 퐾푠| ] (4.73) {푠} = argmin [(s − Ks)(s − Ks)∗] (4.74) {푠} = argmin [|푠̃| − 푠̃퐾푠∗ − 퐾푠푠̃∗ + 퐾 |푠| ] (4.75) {푠} = argmin [퐾 |푠| − 푠̃퐾푠∗ − 퐾푠푠̃ ∗] (4.76) {푠} = argmin [퐾|푠| − 푠̃푠∗ − 푠푠̃ ∗] (4.77) {푠} 51 We used the fact that |푠̃ − 푠| = |푠̃| − 푠̃푠∗ − 푠푠̃∗ + |푠| (4.78) which implies that − 푠̃푠∗ − 푠푠̃∗ = |푠̃ − 푠| − |푠̃| − |푠| (4.79) Using this result, we can write the Eq. (4.77) as follows: 푠̂ = argmin [퐾|푠| + |푠̃ − 푠| − |푠̃| − |푠| ] (4.80) {푠} 푠̂ = argmin [(퐾−1)|푠| + |푠̃ − 푠| ] (4.81) {푠} It is straightforward to apply the ML detection rule in Eq. (4.81) to MRRC. The Eq. (4.51) represents the output of the combiner of the rack receiver with the L finger, where K = 푤 ℎ . Therefore, the ML detection rule for a MRRC receiver with combiner output 푠̃ is as follows: 푠̂ = argmin ( 푤 ℎ −1)|푠| + |푠̃ − 푠| − |푠̃| (4.82) {푠} Single – Mode Beamforming Single-mode beamforming is a case of the beamforming technique in which information is only transmitted over a single eigenmode, the one with the largest eigenvalue. At low SNR, the capacity of beamforming technique based on allocate transmitted power, which is the waterfilling algorithm, will reduce to the capacity of single – mode beamforming technique [11]. This means instead of transmitting in multiple spatial dimensions, we transmit over one spatial dimension by using the same eigenbeamforming technique that we used before when we wanted to achieve spatial gain, such as precoding and pre-multiplying at 52 the receiver. With the single-mode beamforming technique, because we transmit one transmitted symbol over an array of transmitted antennas, the transmitted vector will be: 푠̃ = [푠 ̃ 0,0…….,0] (4.83) The first eigenmode is used to transmit the symbol, 푠̃ , thus, we assume the transmitter has the knowledge of channel state information, CSIT. After the precoding technique, the transmitted data vector is given by: 푠 = 푉푠̃ (4.84) 푣 ⋯ 푣 푠 ̃ 푠 = ⋮ ⋱ ⋮ ⋮ (4.85) 푣 ⋯ 푣 0 푣 푠 ̃ 푠 = ⋮ (4.86) 푣 푠 ̃ T Where the vector 푣 =[푣 ⋯ 푣 ] is the abstract of the direction in N dimensional space along which {s} is being transmitted. Single- mode eigenbeamforming is similar to the eigenbeamforming technique, which applies complex weights to each antenna element in an array in eigenbeamforming, the weights are chosen to point the beam in a desired physical direction; however, with single-mode beamforming, the weights are chosen to match the channel matrix [11]. The receive signal can be express as follows: 휎 ⋯ 0 H 푟 = U ⋮ 휎 ⋮ V V 푠̃ + 푧 (4.87) 0 ⋯ 휎 휎 ⋯ 0 푠 ̃ 푟 = U ⋮ 휎 ⋮ ⋮ + 푧 (4.88) 0 ⋯ 휎 0 Where U is a unitary matrix, and it has Nt or r (rank) orthonormal column vector, thus, we can write it as follows: U = [푈 푈 ⋯ 푈 ] (4.89) 53 휎 ⋯ 0 푠̃ 푟 = [푈 푈 ⋯ 푈 ] ⋮ 휎 ⋮ ⋮ + 푧 (4.90) 0 ⋯ 휎 0 푟 = 푈 휎 푠 ̃ + 푧 (4.91) Where, 푈 can be employed for maximal ratio combiner (MRC). At the receiver, we pre-multiply the received signal by 푈 , where 푈 represents the dominant direction of the received signal. Thus, 푟 =푈 푈 휎 푠 ̃ + 푈 푧 (4.92) 푟 = 휎 푠 ̃ + 푧 ̃ (4.93) Where: 휎 : maximum singular value 푠 ̃ : transmitted symbol 푧 ̃ : noise In fact, the received signal shown in Eq. (4.93) is similar to the received signal in the SISO communication system, but there is a gain that is a maximum singular value of the channel {휎 }. One might ask, what benefit is there in only transmitting data using one eigenmode? The answer is that it enables a higher throughput than can be achieved with the SISO system at the same signal to noise ratio [11]. From Eq. (4.93), the signal to noise ratio is given by: 푆푁푅 = (4.94) Where: {σ } is the largest singular value, and it is also a gain associated with the dominant mode. By using the beamforming technique, the channel effectively becomes the SISO channel with the maximal gain {σ }.This scheme of MIMO beamforming is 54 termed as the maximal ratio transmission (MRT). Let us express signal to noise ratio of the SISO system by ρ. 휌 = (4.95) The capacity of single mode eigenbeamforming will be: C Single -Mode = log2(1+ 휌 σ ) (4.96) CSISO = log2(1+ 휌 ) (4.97) In comparing Eq. (4.96) with Eq. (4.97), we see that the capacity of the single-mode eigenbeamforming system is greater than that of the SISO system with the same signal to noise ratio. Furthermore, when the MRT is used, the capacity is optimal, even at a low signal- to-noise ratio. At a low signal-to-noise ratio, we transmitted the signal along one dominant mode instead of few dominant modes, which is single-mode eigenbeamforming. In addition, the MIMO beamforming is simplistic transmission and reception scheme in comparison to other MIMO techniques, such as VBLAS, ZF, and MMES, which require more computations for inverse channel matrix (H) at receiver. The single- mode eigenbeamforming technique based on the maximum ratio transmitter can achieve complete diversity order which is (Nt×Nr). Figure 4.9. shows the capacity of communication channels among different communication techniques. We find that the performance of beamforming techniques depends on the signal-to-noise ratio and the value of Nt and Nr. 55 Figure 4.9. Comparison of theoretical capacity [11] The results seen on Figure 4.9. are achieved based on the assumption that the number of receiver antennas is greater than or equal to the number of transmit antennas, when the channel is full rank, r =Nt. In this case, the channel can support up to Nt data Strems. Figure 4.12 shows computer generated curves of the MIMO capacity as a function of the signal-to-noise ratio for different MIMO techniques when the Nt = 3 and Nr = 4. Also, these results are generated by a randomly chosen channel matrix H1 (3×4) [11]. 0.46 + 푗0.31 −0.07+ 푗0.18 0.43− 푗0.68 −0.04 + 푗0.66 0.95− 푗0.5 −0.77− 푗0.51 H = (4.98) 1 0.45 + 푗0.46 0.90+ 푗0.56 0.46− 푗1.87 −0.13 − 푗0.19 0.90− 푗073 1.04+ 푗0.91 56 CHAPTER 5 SPREAD SPECTRUM MOUDLATION Introduction Spread spectrum (SS) is a system initially developed with military guidance and a communication system. In fact, SS’s purpose is to provide safe communication as it spreads the signal to a wide frequency band [6]. Also, when this technique is used with a communication system it will become highly jammer resistant. Later, SS become the basic component of the coding division multiple access (CDMA) system [2] A communication system is considered as a SS system if the transmitted signal satisfies the following conditions [10]: The bandwidth of the transmitted signal must be much greater than the message bandwidth. The transmitted signal bandwidth must be determined by some function that is independent of the message and known to the receiver. The amount of performance improvement that is achieved using SS system is called process gain of SS. That is, process gain is the difference between the communication system performance using SS modulation technique and system performance not using SS technique [10]. The SS system has many advantages. First, it is considered as an anti-jammer system (narrowband jamming). Second, it is also an interference rejection system and has low probability of intercept. Third, because of using PN sequence to spread the transmitted signal, the SS system could be a multiple access system, such as the CDMA 57 system. Finally, we could use the SS technique to reject multipath component or exploit the multipath by using the Rack receiver, as we will explain later in this chapter, to improve the performance of communication channel in MIMO technique. There are different types of SS modulation techniques. We can classify the SS system by the following modulation types: Direct Sequence (DS) Frequency Hopping (FH) Time Hopping (TH) Chirp Hybrid mothed (combining two or more of the above modulation) In this chapter, we will focus on one type of SS modulation, Direct Sequence Spread Spectrum (DS-SS) technique [10]. Direct Sequence Spread Spectrum (DSSS) One method of spreading the spectrum of a data modulated carrier signal {xd(t)} is to modulate the signal a second time using wideband spreading signal {c(t)}. The second modulation usually is some form of digital phase modulation [10]. The spreading signal c(t) is chosen to have properties which facilitates the demodulation of the transmitted signal by intended receiver, which makes demodulation by the unintended receiver very difficult. These same properties will also make it possible for the intended receiver to discriminate between communication signal and jamming signal. The bandwidth spreading by the direct modulation of date modulated carrier by wideband signal is called DS-SS technique. The simple form of DS-SS employed BPSK as the spreading modulation. Specifically, for DS-SS-BPSK, we can spread the message signal, m(t), before the BPSK modulate, or we can do the opposite. Figure 5.1. shows the general concept DS-SS system. 58 Figure 5.1. Transmitted and receive scheme of DS-SS system [10]. Assume the binary data signal is a sequence of rectangular pulse signal with amplitude {-1, +1} and duration Tb, so, it is a narrowband signal. We called this data signal d(t). Also, it is assumed that the PN code signal (pseudo noise bits), or spreader signal, c(t), is a sequence of rectangular pulse signal with amplitude {-1, +1} and has duration of Tc, which is called chip time, where Tc<< Tb, so it is a wideband signal. The mathematical expression of DS -SS -BPSK signal is: 푑(푡) = ∑ 푑 푃 (푡 − 푘푇) (5.1) Since 푑 ∈ {±1} and 푃 (푡 − 푘푇) is pulse signal with Tb (see Figure 5.2) 59 Figure 5.2. Pulse function. Similar to Eq. (5.1) the PN code signal c(t) is 푐(푡) = ∑ 푐 푃 (푡 − 푛푇 ) (5.2) 푐 ∈ {±1} and 푃 (푡 − 푛푇 ) is pulse signal with 푇 seconds duration Thus, BPSK signal and the transmitted DS-SS-BPSK are as follows 푠 (푡) = 2푝 푑(푡) 푐표푠(푤 푡) (5.3) 푠(푡) = 2푝 푑(푡)푐(푡) 푐표푠(푤 푡) (5.4) Where, d(t) is data signal (information), c(t) is PN code single. 푝 is the power of transmitted signal 푤 is the angular frequency of the carrier of the transmitted signal. At the receiver, we receive the transmitted signal with some propagation delay and noise. If we assume there is no delay of propagation, the received signal will be as follows: 푟(푡) = 푠(푡) + 푧(푡) (5.5) Where, z(t) is noise signal. The noise signal could be AWGN and or international Jammer. The de-spreading of the receiving signal requires the perfect synchronization of PN code signal at receiver with that of the transmitter. At the same time of de-spreading the desired signal, we will 60 spread the noise signal by multiplying it with the PN code signal c(t). After this point, we use the bandpass filter to extract the data modulated signal 푠 (푡) with a very small portion of the noise signal. The data modulating signal is BPSK signal. After demodulating the BPSK signal, we used the Maximum likelihood detection to estimate the data signal. As shown in the following equation, 푟(푡) ={푠(푡) + 푧(푡) } × c(t) (5.6) 푟(푡)= 2푝 푑(푡)푐 (푡) 푐표푠(푤 푡) + 푧(푡)c(t) (5.7) Since, 푐 (푡) = 1 After Bandpass filter (BPF) with limited bandwidth, the noise interference will be neglected and the received signal will only be the date modulate signal. The Figure 5.3. shows the spreading and de-spreading operation of the signal in the presence of the noise interference [17]. 푟(푡)= 2푝 푑(푡) 푐표푠(푤 푡) (5.8) Figure 5.3. Spread and despread the signal and noise interference. 61 There are different ways to detect the BPSK signal, if we assume the coherent detection is used to detect the BPSK signal, the received signal will be as shown in Figure 5.4. Figure 5.4. Coherent detection of BPSK modulation. 푟(푡)= 2푝 푑(푡) 푐표푠(푤 푡)× 2푝 푐표푠(푤 푡) (5.9) 푟(푡)= 2푝 푑(푡)푐표푠 (푤 푡) (5.10) 푟(푡)= 푝 푑(푡) [1 + 푐표푠(2푤 푡)] after LPF (5.11) 푟(푡)= 푝 푑(푡) (5.12) Where, 푝 is transmitted power, and 푑(푡) is data signal. Pseudo - Noise Sequence PN PN sequence consists of a sequence of {+1, -1}, that processes certain specified autocorrelation properties [10]. A PN sequence could be: Aperiodic which does not repeat itself An Aperiodic Sequence A periodic sequence, such as the Barker sequences, exists only for few values of N, where N is the duration of the PN sequence. That is, when N = 1,2,3,4,5,7,11, and 13, no longer sequences have been found. Thus, such sequences are normally too short for use as a spreading sequence in spread spectrum [10]. The periodic sequence consists of an infinity sequence of {-1, +1}, divided in to block of length N in which the sequence in each block is the same [10]. In fact, PN 62 sequence acts like a noise, but it is deterministic, and it used for bandwidth spreading of signal energy. The selection of good code is important because the type and length of code set bound on system capability. Periodic sequence is called pseudorandom sequence, but it is a not real random sequence because it is periodic, and a random signal cannot be periodic. The autocorrelation of PN sequence has properties similar to those of white noise [17]. The PN sequence has three main randomness properties: Balance: In every period, the number of +1 and -1 differs by exactly one. 푁 + 푁 =N, and 푁 − 푁 =1 Run properties: a run is a sequence of a single type of binary digits. All ones or all zeros in each period are desirable so that one- half of the runs are of length 1, about one- fourth of runs are of length 2, one- eighth are of length 3, and so on [17]. Correlation: the autocorrelation of a periodic PN sequence is defined as the number of agreements less the number of disagreements in a term by term comparison of one full period of the sequence with cyclic shift of the sequence itself [17]. We can represent the discreet autocorrelation of full period N in two values as 푁, 푘 =0 푐(푘) = ∑ 푎 푎 = (5.13) −1, otherwise Partial Correlation of PN Sequence The subject of partial correlation is extremely important in the SS system in which the duration of the period of the PN code sequence is much larger than the duration of a message bit time, that is NTc >> Tb. Let us assume, Tb =M Tc, and let N be a sequence period with N >>M, that is NTc >> MTc=Tb. Because the correlation in the receiver must be taken over a bit time, the correlation over the full period N in Eq. (5.13) of PN sequence cannot occur [10]. Therefore, the discreet partial correlation is defined as 63 푟(푘) = ∑ 푎 푎 M < N (5.14) We assume {푎 }, which is a random variable, is a starting point of a random stationary process. If we model the PN sequence as a random process and assume the {푎 } values in each sequence are statistical independent. Assume 푎 휖 [+1,−1], the probability of random variable 푎 is P (푎 ) = From the discrete random properties, thus, E {푎 } = 0 , 퐸{푎 }=1 (5.15) Because the partial correlation (PC) is a random function, we will calculate the mean and variance of the correlation sequence. Thus, 푀, 푘 =0 퐸{푟(푘)}= 퐸{푎 푎 } = (5.16) 0, otherwise The mean of PC is zero when the sequences are not lined up and has a maximum when they are lined up. Therefore, the variance of the PN sequence is the opposite (see Appendix B). 0, 푘 =0 Var { 푟(푘)}= (5.17) 푀, otherwise PN Signal So far, we considered the sequence of {+1, -1} a discrete sequence. However, what is used in the SS system is a time function like c(t) which is called PN signal. Also, we express the PN signal in discrete form c(k), where k indicates the time delay or shift interval. Therefore, using the discrete form of full correlation for the PN signal over a full period N, thus, 푓푟 (푘) = ∑ 푐 (푖)푐 (푖 + 푘) , 1≤ i ≤ N (5.18) 64 Where: 푓푟 (푘) is full correlation,푐 (푖) is chip sequence of {+1, -1} with probability P(푐 (푖))= , and each 푐(푖) is independent of 푐(푗). Because the full correlation of the PN sequence cannot occur, we define the discrete partial correlation function of the PN signal as, 푟 (푘) = ∑ 푐 (푖)푐 (푖 + 푘) M < N (5.19) Where the {푐 (푖) } is a random function, thus, E{푐(푖)} = 0 , 퐸{푐 (푖) }=1, and the correlation function is a random function [17].Thus, the expected value of the discrete autocorrelation function or cyclic shift of the same chip sequence given by 퐸{푟 (푘)} = 퐸{푐 (푖)} 퐸{푐 (푖 + 푘)} (5.20) Since 푐 (푖) 푎푛푑 푐 (푖 + 푘) are independent random process, 퐸{푟 (푘)} = ∑ 0 = 0 The expected value of any shift correlation k is zero. With a large sequence, the pseudo random acts like a truly random sequence, therefore, the variance of partial correlation is 푉푎푟{푟 (푘)} = 퐸{푟 (푘)}− 퐸 {푟 (푘)} (5.21) 푉푎푟{푟 (푘)} = 퐸{푟 (푘)} (5.22) 푉푎푟{푟 (푘)} = 퐸{ 푐 (푖) 푐 (푖 + 푘)} 퐸{ 푐 (푗) 푐 (푗 + 푘)} (5.23) = 퐸{푐 (푖)} 퐸{푐 (푖 + 푘)}퐸{푐 (푗)} 퐸{푐 (푗 + 푘) (5.24) When 푖 ≠ 푗, the expected value of 푐 (푖) = 0 , and the chip sequences are independent. Thus, the 푉푎푟{푟 (푘)} =0, but when 푖 = 푗 푉푎푟{푟 (푘)}= , 퐸{푐 (푖)} 퐸{푐 (푖 + 푘)} (5.25) 65 Since 퐸{푐 (푖) }=1, 푉푎푟{푟 (푘)}=퐸{푟 (푘)} = This variance represents the self-noise of the PN signal which is called also code noise, and that plays a key factor in the SS system because reducing the code noise depends on maximizing the length of M [10]. On the other hand, the self-correlation of the PN signal occurs when the k=0. 푟 (0) = 푐 (푖) (5.26) When 푐 (푖) is a sequence of {+1, -1}, 푐 (푖) =1 푟 (0) = ∑ 1 = =1 (5.27) Thus, we can define the PN correlation function as follows: 1 푖푓 푘 =0 푟 (푘) = (5.28) 퐸{푟 (푘)} = 0, 푉푎푟{푟 (푘)} = 푖푓 푘 ≠ 0 The cross-correlation properties of the PN signal are defined as the following, 푐 (푖) 1≤ i ≤ M 푐 (푖) 1≤ i ≤ M 푐 (푖) 푎푛푑 푐 (푖) are independent chip sequences. The definition of discrete cross partial correlation of the PN signal can be expressed as 푟 (푘) = ∑ 푐 (푖)푐 (푖 + 푘) (5.29) The cross-correlation is a random function, so we need to calculate the expected value and variance of cross correlation of the PN code signal. 퐸{푟 (푘)} = 퐸{푐 (푖)} 퐸{푐 (푖 + 푘)} (5.30) 66 At any value of k, even when k=0, the two-chip sequences are independent so that the expected value of cross correlation will be zero. 퐸{푟 (푘)} = 0 (5.31) The variance of cross-correlation of two PN signals (chip sequences) are as follows 푉푎푟{푟 (푘)} = 퐸{푟 (푘)} (5.32) 푉푎푟{푟 (푘)} = 퐸{ 푐 (푖) 푐 (푖 + 푘)} 퐸{ 푐 (푗) 푐 (푗 + 푘)} (5.33) = 퐸{푐 (푖)} 퐸{푐 (푖 + 푘)}퐸{푐 (푗)} 퐸{푐 (푗 + 푘) (5.34) When 푖 ≠ 푗, E{ 푐 (푖)}=0 , and the chip sequences are independent at any value of shifting k Thus, the 푉푎푟{푟 (푘)} =0, but when 푖 ≠ 푗 푉푎푟{푟 (푘)}= , 퐸{푐 (푖)} 퐸{푐 (푖 + 푘)} (5.35) Since 퐸{푐 (푖) }=1, 퐸{푐 (푖 + 푘)}=1 푉푎푟{푟 (푘)}=퐸{푟 (푘)} = (5.36) This variance represents the noise power of the chip sequence and it becomes zero when the length of the sequence M becomes larger. Multipath Channel Model As stated in Chapter 2, the signal propagates through a communication channel from different paths due to the reflection, diffraction, or scattering mechanism which affect the energy of the signal at the receiver. In this thesis, the communication channel is modeled as a multipath channel to exploit the multipath component of the signal and add them constructively to improve the reliability of the communication channel. Figure 5.5. illustrates the scheme of the multipath channel. 67 Figure 5.5. Multipath channel scheme There are L independent paths between the transmitter and receiver, so the channel will act like a filter. The output of this channel depends on the current symbol and past delayed symbol due to different paths of propagation. Therefore, the signal arrives at the receiver with an independent delay {L} and independent time-variant gain {h}. Figure 5.6. shows the delay of the multipath components which arrived at the receiver [12]. Figure 5.6. Multipath channel model. According to multipath model of communication channel, the received signal in discrete form will be as follows: 68 푟(푖) =ℎ(0)푠(푖) +ℎ(1)푠(푖 −1)………. ℎ(퐿 −1)푠(푖 − 퐿 −1) + 푧(푖) (5.37) 푟(푖) = ∑ ℎ(푑)푆(푖 − 푑) + 푧(푖) (5.38) Where; ℎ(푑) is time- variant gain related with a different path, 푆(푖) is the transmitted signal,푧(푖) is the Additive White Gaussian Noise signal. Because SS modulation is used, the transmitted signal is a wideband. If one assumes that the receiver received only the first component the channel would be considered a flat fading channel but, if the different path components are received, the channel would be considered a frequency selective fading channel. In this case, the convolution modulation techniques require an equalizer which combated the inter symbol interference (ISI) between adjust symbols [10]. There are different kinds of equalizer techniques used to combat the ISI at the receiver, such as Zero Forcing equalizer. The Rack Receiver The rack receiver is used specifically in CDMA cellular systems. In this thesis, the rack receiver has been used with the DS-SS communication system. The rack receiver can combine the multipath components, which are time-delayed versions of the original signal transmission. These components can be used to improve the signal to noise ratio (SNR). The rack receiver attempts to collect the time shift version of the original signal by providing a separate correlation receiver for each of the multipath signals. This can be done because the multipath components are practically uncorrelated with each another when their relative propagation delay exceeds a chip period [12]. Figure 5.7. shows the rack receiver scheme. 69 Figure 5.7. Rack receiver scheme. The communication channel consists of many copies of the originally transmitted signal, which have different amplitudes, phases, and delays. The rack receiver uses the principle of multipath diversity. When the signal component arrives at a duration of more than one chip apart from each other, the rack receiver can resolve and combine them [14]. Eq. (5.38) expresses the received signal at the rack receiver under the assumption that the channel is a slow frequency selective fading channel, but we use the equalizer to combat inter symbol interference (ISI) with the adjacent symbol. A component of the transmitted signal, which is the SS signal, will correlate with the cycle shift correlator in the rack receiver, according to its paths, so the output of each correlator, which is called rack finger, will combine by using the maximum ratio combiner to maximize the SNR. A correlator at the rack receiver could use the match filter. Also, by increasing the rack finger the probability of error will be reduced. This will increase the reliability of the communication channel. The output of each finger is 푟푟(0) = 푟(푖) 푐 (푖) (5.39) 푟푟(0) = {∑ ℎ(푑)푆(푖 − 푑) + 푧(푖) } 푐 (푖) (5.40) Since; the signal 푆(푖 − 푑 ) is DS-SS BPSK signal. Thus, 70 푆(푖 − 푑) = 푠푐(푖 − 푑) (5.41) Where: 푠 is the data signal, and 푐(푖 − 푑)is the PN code signal 푟푟(0) = ∑ ℎ(푑)푠 푐 (푖 − 푑) 푐 (푖) + 푧̃(0) (5.42) Where 푧̃(0) is the noise relate with path (0), and it is equal to 푧(푖) 푐 (푖). Using the PN sequence properties, the auto correlation of PN sequence will be one and the cross-correlation will be zero, the output of each finger which represent the different path is given as follows: 푟푟(0) =ℎ(0)푠 + 푧̃(0) (5.43) 푟푟(1) =ℎ(1)푠 + 푧̃(1) (5.44) 푟푟(퐿 −1) =ℎ(퐿 −1)푠 + 푧̃(퐿 −1) (5.45) We can write these equations in vector form 푟푟 = 푠ℎ + 푧̃ ̅ (5.46) By using maximum ratio combiner, which is described in Chapter 4, we are adding the weight of each finger. Thus, the signal at the output of the combiner is given as follows: ∗ ∗ ∗ 푟 = 푤 푟푟(0) + 푤 푟푟(1) +⋯+ 푤 푟푟(퐿 −1) (5.47) Or 푟 = 푤 푟푟 (5.48) We will choose this weight gain to maximize the SNR, 푤 = (5.49) ‖ ‖ By substitute Eq. (5.49) in Eq. (5.48) the received signal after MRC will be as follows: 푟 = {푠ℎ + 푧̃}̅ (5.50) ‖ ‖ 푟 = ℎ 푠 + 푧̃ ̅ (5.51) ‖ ‖ ‖ ‖ 71 푟 = ‖ℎ‖푠 + 푧 (5.52) Note z is additive Gaussian noise with zero mean and variance = , so the correlation with noise signal will be as follow: 푧̃(0) = 푧(0) 푐 (푖) (5.53) Because 푐 (푖) 푧(0) are a random signal, we will find the mean and variance of 푧̃(0) E{푧̃(0)}= 퐸{푧(0)} 퐸{ 푐 (푖)} = 0 (5.54) Var{푧̃(0)}= 퐸{푧 (0)} 퐸{퐶 (푖)} (5.55) Var{푧̃(0)}= (5.56) The SNR of the rack receiver is given as. ‖ ‖ 푆푁푅 = (5.57) Where: P is the power of the modulation signal, and ℎ is the fading coefficient of the multipath channel. From Eq. (5.57), the increase of the SNR depends on the length of the PN sequence (M). Figure 5.8. shows the general scheme of the rack receiver is corrupted by ISI. Figure 5.8. General rack receiver scheme [13]. 72 The rack demodulator that was previously described is an optimum demodulator that is based on the condition that the bit interval Tb>> Tc. Also, there is an assumption that there is a negligible ISI. When this condition is not satisfied, the rack demodulator, which is a match filter to the channel response, is corrupted by ISI, thus, an equalizer is required to suppress the ISI as shown in Figure 5.8. [13]. Performance of The Rack Receiver with MIMO Beamforming Technique In this thesis, the assumption is made that the transmitter has the knowledge of the channel matrix. using beamforming technique, which is requires the channel state information at the transmitter (CSIT), the MIMO channel can be decoupled into independent and parallel additive -white noise channels. The number of parallel channels is equal to the minimum number between the transmitter and receiver antenna. In fact, these parallel channels are established by applying the singular value decomposing (SVD) of the channel matrix. The parallel channel can be processed independently, each with independent modulation and coding [25]. The different data streams can be transmitted through out these parallel channels, and this is nothing but a special multiplexing technique. Our aim in this thesis is to evaluate ways to increase the throughput and the reliability of the MIMO channel. We know from Chapter 3 that the capacity of the MIMO channel by using beamforming technique is a sum of the SISO channel capacity. Using spread spectrum modulation over these parallel channels with different codes sequence and using the rack receiver can improve the reliability of the parallel channels. In fact, we will exploit the multipath component to achieve received diversity that will reduce the probability of 73 errors in each parallel channel, and that is called multipath diversity. Figure 5.9. shows the multiple input multiple output scheme with the -rack receiver. Figure 5.9. MIMO rack receiver scheme [13] In Figure 5.9, the block diagram of the chip match filter and the block diagram of the bank of desperation represent the rack receiver structure, and {r1j} is the number of the output finger of each rack receiver. However, we can use the correlate receiver with a shift cyclic correlator, which is equivalent to the match filter of the rack receiver. The rack demodulator with perfect (noiseless) estimate of the channel tap weight of each parallel channel is equivalent to the maximal ratio combiner of the channel with Lth- order diversity. Thus, when all the tap weights have the same mean-square value, the performance of the rack receiver is given by Eq. 5.58 where the modulations are BPSK, 74 coherent orthogonal BFSK, non-coherent DPSK and non-coherent orthogonal BFSK. However, we used Eq. 5.59 if the modulations were QPSK, or 4-ary DPSK. In practice, a relatively good estimate of channel tap weight can be obtained if the channel fading is sufficiently slow Tc/T >>100 [13]. 2−1+푘 Pb = (1−휇) [ (1+휇) ] (5.58) 푘 2푘 Pb = 1− (5.59) 푘 Table 5.1. Value of μ for Binary Modulation Scheme [11] Modulation MRRC Alamouti 휌̅ 휌̅ BPSK 1 + 휌̅ 2 + 휌̅ 휌̅ 휌̅ Coherent orthogonal BFSK 2 + 휌̅ 1 + 휌̅ 휌̅ 휌̅ Non-coherent DPSK 1 + 휌̅ 2 + 휌̅ 휌̅ 휌̅ Non-coherent BFSK 2 + 휌̅ 4 + 휌̅ Table 5.2. Value of 휇 for Selected QPSK and 4-Ary DPSK Modulation Assuming Gray Code [11] Modulation MRRC Alamouti 휌̅ 휌̅ QPSK 1 + 휌̅ 2 + 휌̅ 휌̅ 휌̅ 4-DPSK 1 + 휌̅ 2 + 휌̅ Note 휌̅ denotes the time average SNR. 75 CHAPTER 6 ANALYSIS THE RESULT OF USING BEAMFORMING TECHNIQUE WITH THE RACK RECIEVER (N×N) DSSS-BPSK MIMO Communication System Preform Result This section presents the results of using different MIMO system categories with the beamforming technique and the rack receiver combination. First, I illustrate the three examples of Nt=Nr=N. Then, these results will compare with the result of using beamforming technique only, which uses MRRC and ML detection at the receiver. The bit error probability is plotted under the assumption that the channel is a Slow Rayleigh fading channel, and the modulation is DSSS-BPSK for three examples: a (2×2), a (3×3), and a (4×4) MIMO communication system. These results are based on the computer simulation which is N= 9,000,000 information bits. This implies there are 4,500,000 iterations in the 2×2 MIMO communication system example, 3,000,000 iterations in the 3×3 MIMO communication system example, and 2,250,000 iterations in the 4×4 MIMO communication system example. The channel matrixes have been chosen randomly in these examples. Also, the received vector involved random noise. To see the difference of using the beamforming technique and LST coding technique, see Figure 6.1. which shows the BER performance of three MIMO systems. These techniques have increased the throughput of the MIMO system comparing with SISO communication system, but there is no effect on the transmission reliability of the MIMO channel. 76 Figure 6.1. Bit error probability of N×N MIMO system (Rayleigh channel). The result shows the bit error probability of these systems is somewhat similar when the MIMO system used the Layer Space Time coding concept, which include the VBLAST, HBLAST, and DBLAST techniques, so that the diversity order approaches to Nd=2. This implies that if the difference between the transmitted and received antennas is constant, the transmission reliability will be the same even though Nt=Nr =N increased. However, increasing the dimension of the square channel matrix will increase the capacity of the communication system as shown in Figure 6.2. Also, Figure 6.3. shows the capacity of a different configuration of the MIMO system at low signal-to-noise ratio around 5dB and at high signal-to-noise ratio around 20dB. 77 Figure 6.2. The capacity of different MIMO system category. Figure 6.3. The capacity of different MIMO configuration in 5dB and 20dB. 78 (2×2) MIMO Communication System Performance Result of DSSS-BPSK In this thesis, initial we will analyze the 2×2 MIMO communication system which it has two transmit antennas and two receive antennas. However, what we applied for this MIMO system is valid for (Nt ×Nr) MIMO communication system. To use the rack receiver, we must apply the direct sequence spread spectrum modulation (DSSS). By using the beamforming technique, which is mentioned in Chapter 4, the MIMO channel will decouple to (r) sub-channel where (r) is the rank of channel matrix. These sub- channels are individual and independent. We modulate each sub-channel with the same carrier frequency but different code sequences. These codes are orthogonal. However, each sub-channel has a different data modulated signal that is transmitted at the same time from different transmitted antennas. This is a special multiplexing technique that will achieve a higher capacity than the SISO communication system. On the BPSK system, we can modulate the data signal first then spread the signal by using the PN sequence or vice versa. Figure 6.4. shows the DSSS modulation with BPSK for SISO channel. Figure 6.4. SISO DSSS modulation with BPSK. 79 The data transmitted from sub-channel one d1(t) is the sequence of {+1, -1}. After the phase modulation, which is BPSK, the data modulate signal Xd1(t) will be as follows: 푋푑 (푡) = √2푃 푑 (푡) 푐표푠 2휋푓 푡 (6.1) We spread the data modulate signal by the second modulation with the code sequence one c1(t), so the transmitted signal from sub-channel one, which is a SISO channel is: 푋푡 (푡) = √2푃 푑 (푡) 푐표푠 2휋푓 푡 푐 (푡) (6.2) We can write the transmitted signal as follows: 푋푡 (푡) = 푠 (푡)푐 (푡) (6.3) The same process is done with sub-channel two, so the transmitted signal is 푋푡 (푡) = 푠 (푡)푐 (푡) (6.4) We transmitted two data streams through these two parallel sub-channels individually, and that will increase the throughput of the MIMO channel. We will represent the transmitted signal from each sub-channel with a discrete form as follows: 푋푡 = 푠 푐 (6.5) At the receiver, we will despread the receiving signal and apply phase demodulation techniques to extract the data signal of each sub-channel. In our case, we used the rack receiver to extract the multipath components of the transmitted signal and combined them constructively to reduce the probability of error at detection for each sub- channel. This will improve the performance of the MIMO channel overall. Our goal is to enhance the performance of the MIMO channel by using the rack receiver with beamforming technique. We will compare that result with using the beamforming technique only without rack receiver. To apply the beamforming technique, we assume 80 the transmitter has knowledge of the channel matrix (H). That will happen when the channel is Slow and Rayleigh fading as we have mentioned in Chapter2. In this system, we assume the channel matrix is h h 퐻 = (6.6) h h Using the singular value decomposition method to decouple the MIMO channel from the sub-individual channels, as we have shown in Chapter 4 the equation is: H= U D VH (6.7) Where U and V are the unitary matrix and D is the diagonal matrix. By doing the pre-coding, which is multiplying the signal with matrix V at the transmitter, and post-coding which is multiplying the received signal with U , at each receiver antenna, we will receive the signal coming from the specific transmitted antennae. The equation of the received signal under assumption that the channel is Slow Rayleigh fading and the condition of Nr ≥ Nt will be as follows (see Chapter 4) 푟 =H푠 + 푧 (6.8) r = [U D VH] s+ z (6.9) 휎 0 푟 = U VH V 푠̃ + 푧 (6.10) 0 휎 휎 0 푟̃ = UHU 푠̃ + UH z (6.11) 0 휎 휎 0 푟̃ = 푠̃ + 푧̃ (6.12) 0 휎 Similar to what we have explained in Chapter 4 in Eq. (4.19), (4.20), and (4.21), we can see the MIMO channel consists of two parallel sub-channels. These channels have independent AWGN from each other: 81 푟 ̃ = 휎 푠 ̃ + 푧 ̃ (6.13) 푟 ̃ = 휎 푠̃ + 푧 ̃ (6.14) Each sub-channel acts like SISO channel but it has beamforming gain which is singular value (휎 ), so the capacity of the sub-channel is more than the SISO channel. Overall, the capacity of the MIMO channel is a summation of the capacity of the sub- channels. Eq. (5.72) shows that the beamforming technique suppresses the interference between these sub-channels. In fact, there is an interference between these sub-channels due to the small physical distance between these transmitted antennae. To analyze the interference between sub-channels, we model each sub-channel as the multipath channel. We modulate each channel with DSSS-BPSK at the same frequency and different code sequence. For a single transmitter and receiver system, the multipath model of the communication channel is: 푟 = ∑ ℎ(푑)푠 + 푧 (6.15) Where L is the number of the path between the transmitter and receiver. 푠: transmitted signal. ℎ(푑): complex weight relates with each path. 푧: AWGN relate with each path In our case, we have two parallel channels. We model each one with a multipath module so the received signal at each receiver without interference from other channel will be as follows: 푟 ̃ = 휎 ∑ ℎ (푑) 푠 ̃ + 푧 ̃ (6.16) 푟 ̃ = 휎 ∑ ℎ (푑) 푠̃ + 푧 ̃ (6.17) 82 Now assume there is interference between these channels. In general, the discrete form of the received signal at each receive antenna will be as follows: 푟 (푚) = ∑ 휎 ℎ (푑) 푠 푐 (푚 − 푑) + 푧 (푚) (6.18) Where K is the number of sub-channels. 푠 푐 (푚 − 푑) is the transmitted spread spectrum signal. 푧 (푚) : is the AWGN for each sub-channel. At the rack receiver, we use a match filter or correlator receiver, which correlates the incoming signal with a cyclic shift correlator of PN sequence, to extract the data signal. From the definition of the partial correlation of the PN sequence, the general equation will be as follows: 푟푟 (푑) = ∑ 푟 (푚)푐 (푚) (6.19) Where 푟푟 (푑) is the correlation process with path (d) of sub-channel (k) M is the length of PN sequence 푟 (푚) is the received signal 푐 (푚) is PN code signal In this MIMO system, we have two parallel sub-channels. We used the rack receiver of for each sub-channel consisting of L fingers, so we will explain the correlation of each path of these two channels as follows. The correlation with path (0) of sub-channel number (0) is: 푟푟 (0) = ∑ 푟 (푚)푐 (푚) (6.20) Substitute Eq. (6.18) in Eq. (6.20) 83 rr (0) = 휎 ℎ (푑)푠 푐 (푚 − 푑)푐 (푚) + ∑ 푧 (푚) 푐 (푚) (6.21) The correlation with path (1) of sub-channel number (0) is as follows: rr (1) = 휎 ℎ (푑)푠 푐 (푚 − 푑)푐 (푚 −1) + ∑ 푧 (푚) 푐 (푚 −1) (6.22) The correlation with path (L-1) of sub-channel number (0) is as follows: rr (퐿 −1) = 휎 ℎ (푑)푠 푐 (푚 − 푑)푐 (푚 − 퐿 −1) + ∑ 푧 (푚) 푐 (푚 − 퐿 −1) (6.23) Each correlation process (finger) of the rack receiver consists of the four components desired signal L multipath components of the desired signal KL multipath component of the sub-channel interference Noise We can rewrite each equation of the correlation process at the rack finger to show these four components is as follows: rr (0) = ∑ 휎 ℎ (0) 푠 푐 (푚)푐 (푚) + ∑ 휎 ℎ (푑)푠 푐 (푚 − 푑) 푐 (푚) + ∑ 휎 ℎ (푑)푠 푐 (푚 − 푑)푐 (푚)+ ∑ 푧 (푚) 푐 (푚) (6.24) We will analyze each of these components separately. 84 The first component is the desired signal component of sub-channel (0) with beamforming gain. The desired signal = ∑ 휎 ℎ (푑) 푠 푐 (푚)푐 (푚) (6.25) According to the definition of the partial correlation, this represents the self-correlation which is equal to (1), so the desired signal will be as follows: desired signal = ∑ 휎 ℎ (0) 푠 (6.26) desired signal = 휎 ℎ (0)푠 (6.27) The second component is the multipath component of sub-channel (0) for the desired signal which is called Desired Channel Interference (DCI) and that will be as follows: DCI= ∑ 휎 ℎ (푑)푠 푐 (푚 − 푑)푐 (푚) (6.28) Note: that Eq. (6.28) contain the autocorrelation function of PN signal (coding signal) which is a random function, so that we will find the expected value and variance of each multipath component. See Eq. (5.19) in Chapter 5. Thus: E{DCI}= 퐸 ∑ 휎 ℎ (푑)푠 푟 (푑) (6.29) Where: 푟 (푑) is the autocorrelation function of PN signal. From Eq. (5.28) the E {푟 (푑)}= 0, thus E{DCI}=0 (6.30) The variance of DCI is Var{DCI}= E {DCI2} (6.31) =E[(∑ 휎 ℎ (푑)푠 푟 (푑)) ] (6.32) = 휎 E{푠 } |ℎ (푑)| E{푟 (푑)} (6.33) 85 Where E{푠 } is the power of desire signal, also from Eq. (5.28) the variance of the autocorrelation function of PN signal is: E{푟 (푑)}= (6.34) Thus Eq. (6.33) becomes as follows: Var{DCI}= |ℎ (푑)| (6.35) We can write Eq. (6.35) like this: Var{DCI}= |ℎ (푑)| - |ℎ (0)| (6.36) Since,‖h ‖ = |ℎ (푑)| (6.37) Var{DCI}= [ ‖h ‖ - |ℎ (0)| ] (6.38) The third component is the interference component of another adjacent sub-channel and its multipath with sub-channel (0) is called sub-channel interference (SCI) SCI= ∑ 휎 ℎ (푑)푠 푐 (푚 − 푑)푐 (푚) (6.39) From Eq. (5.29) in Chapter 5, which is the cross-correlation of PN signal and it is a random function: 푟 (푑) = ∑ 푐 (푚 − 푑)푐 (푚) (6.40) We will find the expected value and variance of the SCI because its random component, that will be as follows: E{SCI}=E{ ∑ 휎 ℎ (푑)푠 푟 (푑)} (6.41) E{SCI}= ∑ 휎 ℎ (푑)푠 E{푟 (푑)} (6.42) From Eq. (5.31) in Chapter 5, the expected value of the cross-correlation is equal to (0) E{푟 (푑)}=0, thus the E{SCI}=0 (6.43) 86 The variance of SCI is: Var{SCI}= E{SCI2} (6.44) = E ∑ 휎 ℎ (푑)푠 푟 (푑) (6.45) = 휎 |ℎ (푑)| E{푠 } E{푟 (푑)} (6.46) From Eq. (5.36) the variance of the autocross-correlation is: E{푟 (푑)}= (6.47) Thus, the variance of SCI is: Var{SCI}=∑ 푃 ‖ℎ ‖ (6.48) The fourth component is the noise component related with each path of the sub-channel (0) which is AWGN. Let us call the noise of path (0) of sub-channel (0) is n , so the noise is: 푛 (푚) = ∑ 푧 (푚) 푐 (푚) (6.49) Because the AWGN is a random process and the PN code signal is the pseudorandom signal, so we will calculate the mean and variance of the noise signal. E{푛 (푚)}= E{ ∑ 푧 (푚) 푐 (푚)} (6.50) = ∑ 퐸{푧 (푚)}퐸 {푐 (푚)} (6.51) Where 푐 (푚) is independent random process, thus 퐸 {푐 (푚)}=0 and 퐸 {푐 (푚)}=1 (6.52) E{푛 (푚)}=0 (6.53) Var{n (푚)}= E{n (푚)} (6.54) = 퐸{푧 (푚)}퐸 {푐 (푚)} (6.55) 87 Var{n (푚)}= 퐸 {푐 (푚)} (6.56) Where 퐸 {푐 (푚)}=1 (6.57) Var{n (푚)}= (6.58) The total power of the noise and the interference of path (0) for the sub-channel (0) 푧 (0)= DCI+SCI+n (푚) (6.59) 푧 (0) = ‖ℎ ‖ − |ℎ (0)| + ‖ℎ ‖ + (6.60) The total power of the noise and the interference for any path of the sub-channel (0) 푧 (푑)= ‖ℎ ‖ − |ℎ (푑)| + (6.61) In our case, we have two sub-channels at each sub-channel we used the rack receiver with L finger. We will extract the multipath component. The output of each finger of the rack receiver will be as follows: r(0) = 휎 h (0)s +z (0) (6.62) r(1) = 휎 h (1)s +z (1) (6.63) r(L−1) = 휎 h (L−1)s +z (퐿 − 1) (6.64) We will combine the output of each finger in the rack receiver by using MRC to maximize the signal to noise interface ratio (SNIR) of each sub-channel (see Chapter 4). The output of each combiner is as follows: 푟 = 푤 푟̅ (6.65) Where, w = , we chose the complex weight (w) of the maximum ratio combiner to ‖ ‖ maximize the SNIR. The output of sub-channel (0) is: 88 ∗ ∗ ∗ ∗ 푟 = 푤 푟(0) + 푤 푟(1) + 푤 푟(2) +⋯+ 푤 푟(퐿 −1) (6.65) ∗ ∗ ∗ ∗ ( ) ( ) ( ) ( ) 푟 = 푟(0) + 푟(1) + 푟(2) +⋯+ 푟(퐿 −1) (6.66) ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ To know the performance of each sub-channel, we calculate the SNIR of each sub- channel. The signal component is: ∗ ∗ ∗ ∗ ( ) ( ) ( ) ( ) Signal=휎 ℎ (0)푠 + 휎 ℎ (1)푠 + 휎 ℎ (2)푠 +⋯+ 휎 ℎ (퐿 − ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ 1)푠 (6.67) | ( )| | ( )| | ( )| | ( )| Signal = 휎 푠 + 휎 푠 + 휎 푠 +⋯+ 휎 푠 ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ = 휎 푠 (6.68) ‖ ‖ =‖ℎ ‖휎 s (6.69) The power of the received signal = E[휎 ‖h ‖ s ] = 휎 ‖h ‖ E[s ] = 휎 ‖h ‖ P (6.70) The total noise of the sub-channel (0) is the summation of all noise from all paths is: ∗ ( ) Noise = 푧 (푑) (6.71) ‖ ‖ We will find the variance of this noise, which is the power of the noise related with sub- channel (0). ∗ ( ) 휎 = 퐸{ } E{|푧 (푑)|} (6.72) ‖ ‖ 89 Where, E {|푧 (푑)|} is the variance of d path, and 휎 is the power of the noise related with sub-channel (0) from all paths. ∗ ( ) 휎 = ( ‖ℎ ‖ − |ℎ (푑)| + ) (6.73) ‖ ‖ | ( )| ∗ ( ) 휎 = ( ‖ℎ ‖ − + ) (6.74) ‖ ‖ ‖ ‖ Let, ∗ ( ) 휎 = 휎 (6.75) ‖ ‖ The total noise of sub-channel (0) is: | ( )| 휎 = ‖ℎ ‖ − + (6.76) ‖ ‖ The signal to noise interface ratio is: ‖ ‖ SNIR= (6.77) | ( )| ‖ ‖ ‖ ‖ Instead of using the power of the signal in Eq. (6.77), we will use energy of the signal over the time interval. Thus, P = E/T (6.78) Substitute Eq. (6.78) in Eq. (6.77) ‖ ‖ SNIR= 휌 = (6.79) | ( )| ‖ ‖ ‖ ‖ Note, increasing the transmitted power of the signal will increase the noise also as we mention in Eq. (6.79). To improve the system performance by improving the signal-to- interference noise, we need to increase M, which is the length of the PN sequence. 90 In BPSK modulation the time interval is the bit duration, symbol time is the bit time, so we will find the average of SNIR (휌̅), At the rack receiver, we used the maximum ratio combiner, so we need to use Table 5.1. to find the value of μ. We used the BPSK modulation that uses Eq. (5.58) to find the probability of error of each sub- channel. If we used QPSK modulation, then we will use Eq. (5.59) to find the probability of error of each sub-channel, where L in these two previous equations, represents the number of rack fingers at each sub-channel. As we have seen in Chapter 4, the Eq. (4.68) represents the general form of the MRC output. We will use the Maximum likelihood detection to extract the incoming signal of each sub-channel. The MRC output for sub-channel (0) is: ∗ ( ) 푟 =‖ℎ ‖휎 푠 + 푧 (푑) (6.80) ‖ ‖ We know that the general form of the output of MRC is 푟̃ = 퐾푠 + 푧̅ (6.81) By comparing the Eq. (6.80) with Eq. (6.81), the constant K is equal to the complex weight gain ‖ℎ ‖휎 . Similar to Eq. (4.82) in Chapter 4, the estimated signal of sub- channel (0) by using ML will be as follows: 푠̂ = argmin (‖ℎ ‖휎 −1)|푠| + |푠̃ − 푠| (6.82) {푠} Where: 푠 is the transmitted signal (symbol). At the detection, each received antenna connect to the matching filter and despreaders bank, which both together represent the rack receiver process. We will correlate the incoming signal with cyclic shift correlator and kept the desperate signal in the bank of despreaders, which is equal to the number of the finger in the rack 91 receiver. After the despread process, we will combine the output of each finger by using the MRC to maximize the signal-to-noise ratio for each sub-channel, Then We will use the ML algorithm for each sub-channel to detect the signal {푠̂ , 푠 ̂ , 푠̂ …… 푠̂ }. Figure 5.9. in Chapter 5 shows the detection scheme of using the rack receiver with MIMO technique. Overall, we improve the reliability of each sub-channel and that enhances the performance of the MIMO channel compared with using the beamforming technique only. Otherwise, we maintain the capacity of the MIMO channel that we achieved when we used the beamforming technique only without using the rack receiver. Figure 6.5. and 6.6. show the theoretical and simulation results of sub-channel one and two of the 2×2 MIMO system. Figure 6.5. MIMO sub-channel one performance with rack receiver L=4. 92 We can see the difference between using the beamforming technique with the rack receiver and using the beamforming technique only. The probability of bit error has reduced with using the rack receiver because the diversity order is equal to the rack finger L=4 for each sub-channel. In contrast, using the beamforming technique only has first order diversity because these sub-channels act like the SISO channels, so it has a higher bit error probability. Similarly, Figure 6.6. shows the performance of sub-channel two. The only difference between sub-channel one and two is the beamforming gain which is dependent on singular value decomposition (휎 ) of the channel matrix. In other words, the capacity of the MIMO channel is a little bit larger than the capacity achieved with beamforming technique only. Also, because we used the MRC at the rack receiver which is maximized the SNR of each sub-channel. Figure 6.7. shows the capacity of the 2×2 MIMO system. Figure 6.6. MIMO sub-channel two performances with the rack receiver L=4. 93 Figure 6.7. The capacity of 2×2 MIMO system. Figure 6.8. The detection block diagram of 2×2 MIMO system. 94 From Figure 6.7, we see that the capacity of the MIMO system with using the rack receiver is a little bit larger than the capacity we have achieved with using the beamforming technique only that because using the MRC at the rack receiver to maximize the SNR for each sub-channel. We have used the waterfilling algorithm based on the MRTC technique to achieve the optimal capacity, which is equal to the r SISO channel. Also, the capacity of MIMO channel using the beamforming technique and the rack receiver is better than the capacity of SISO channel. Figure 6.8. shows detection block diagram of 2×2 MIMO system. MIMO Communication System Performance Result of DSSS-BPSK Assume the MIMO communication system has 3×3 dimension, which has three transmitted antennae and three received antennae. Assume we apply the beamforming technique under the assumption the transmitter has the knowledge of the channel matrix. The MIMO channel will consist of three parallel sub-channels, where each one has a different gain. Similar to what we did on the 2×2 MIMO system, the following equations explain the beamforming technique of the 3×3 MIMO system. ℎ ℎ ℎ 퐻 = ℎ ℎ ℎ (6.83) ℎ ℎ ℎ Using singular value decomposition method to decouple the MIMO channel to sub- individual channels, H= U D VH (6.84) Where U and V are unitary matrix and D is diagonal matrix. 95 We will receive the signal coming from a specific transmitted antenna. The equation of the received signal under the assumption that the channel is a Slow Rayleigh fading channel will be as follows: 푟 = 퐻푠 + 푧 (6.85) r = [U D VH] s+ z (6.86) 휎 0 0 H 푟 = U 0 휎 0 V V 푠̃ + 푧 (6.87) 0 0 휎 휎 0 0 H H 푟̃ = U U 0 휎 0 푠̃ + U z (6.88) 0 0 휎 휎 0 0 푟̃ = 0 휎 0 푠̃ + 푧̃ (6.89) 0 0 휎 From Eq. (6.89), we can see that the MIMO channel consists of three parallel sub- channels. These channels have independent AWGN from each other. 푟 ̃ = 휎 푠 ̃ + 푧 ̃ (6.90) 푟 ̃ = 휎 푠̃ + 푧 ̃ (6.91) 푟 ̃ = 휎 푠̃ + 푧 ̃ (6.92) By using the same bandwidth, we modulate each channel with different data streams using DSSS-BPSK, so each channel has a different code signal. These three channels are independent, and they have independent Gaussian noise. We model the sub-channels as multipath channels (see Eq. 6.15), so there are interference components, which come from the multipath components of the desired signal and the other sub-channels. Eq. (6.18) represents the received signal with all these components. Similar to the 2×2 MIMO system, we will use the rack receiver to extract 96 the multipath components of the desired signal and add them constructively at detection to reduce the probability of bit error for each sub-channel. For BPSK modulation, we find the probability of bit error for each sub-channel by using the equation (5.58) and Table 5.1. Figure 6.9. ,6.10, and 6.11 show the performance of these three sub-channels. Figure 6.9. Performance of sub-channel one in 3×3 MIMO system. 97 Figure 6.10. Performance of sub-channel two in 3×3 MIMO system. Figure 6.11. Performance of sub-channel three in 3×3 MIMO system. 98 We can see from the result, sub-channel one of this system has lower probability of bit error because the beamforming gain (휎 ) is higher than (휎 ) 푎푛푑 (휎 ) in sub- channels two and three respectively. Overall, the probability of bit error of the MIMO system using the rack receiver will be less than using the beamforming technique only. That is because we enhance the performance of each sub-channel of this MIMO system by using the rack receiver. Figure 6.12 shows the detection block diagram of 3×3 MIMO system. Figure 6.12. Detection block diagram of 3×3 MIMO system. In addition, the capacity of 3×3 MIMO system using the rack receiver is larger than the capacity of this MIMO beamforming system only, which is the summation of the capacity of the three parallel sub-channels. That is because using MRC at rack receiver will maximize the SNR of each sub-channel. Figure 6.13 shows the capacity of MIMO system for BPSK Modulation. 99 Figure 6.13. The capacity of 3×3 MIMO system for BPSK Modulation. (N×N) DSSS-QPSK MIMO Communication System Performance Result In this section, we will analyze the performance of the beamforming technique with the rack receiver when the modulation is DSSS-QPSK and compare the result with the modulation of DSSS-BPSK for the ×MIMO communication system and for the 3×3 MIMO communication system. (2×2) MIMO Communication System Performance Result of DSSS-QPSK In general, using QPSK modulation impacts the bit error probability. Using the beamforming technique of a 2×2 MIMO system will generate two sub-individual channels. However, using DSSS-QPSK modulation has the advantage that it is more difficult to detect the signal in low signal to noise ratio and the system is less sensitive to some type of jammer. In addition, QPSK can conserve the bandwidth because it uses one 100 half of the bandwidth of BPSK at the same transmitted power and same bit error probability. However, in a spread spectrum system, we spread the signal, so it has a large bandwidth. We are not concerned about the signal bandwidth as much as reducing the effect of the noise and interference by spread the noise and interference at detection. Also, it is easy to implement the rack receiver, which is a correlation receiver, with a QPSK modulation because we can correlate all sets of the signals by using only two bases. functions. Figure 6.14. The bit error probability of sub-channel one of 2×2 MIMO system. 101 Figure 6.15. The bit error probability of sub-channel two of 2×2 MIMO system. Figure 6.12 and 6.13 show the result of bit error probability of sub-channel one and two of the 2×2 MIMO system. We can see that the performance of using the beamforming and the rack receiver with the QPSK modulation is better than when the BPSK modulation is used. However, the performance of sub-channel one using the beamforming and the rack receiver is better than using the beamforming technique only when both methods used the ML detection technique as shown in figure 6.14 and 6.15. However, if the bit energy of the QPSK modulation is the same, the performance of the MIMO system using QPSK is the same as using BPSK. But we have used the waterfilling algorithm technique based on the maximum ratio transmitter combiner (MRTC) to achieve the higher capacity of the beamforming technique. So, the transmitted power 102 from every transmitter antenna is not similar, so the energy of bit per signal is different from using QPSK and BPSK modulations. On the other hand, the speed of the signal processing at decoding depends on the number of bits per modulation symbol and the number of transmitted symbols during the symbol period. However, in QPSK, there are two bits per symbol while in the BPSK there is one bit per symbol. That makes the processing speed of the QPSK signal at the rack receiver twice of the BPSK modulation. This means that the capacity of the MIMO channel is affected positively when increasing the number of bits per symbol as shown in Figure 6.16. Figure 6.16 The capacity of MIMO channel for different modulation. 103 (3×3) MIMO Communication System Performance Result of DSSS-QPSK As shown in previous analysis, the beamforming technique of the 3×3 MIMO system has three independent sub-channels. We modulate each channel with DSSS- QPSK so that each sub-channel has a different code signal. We also know that the capacity of a system using QPSK is larger than using BPSK modulation. Now, we will evaluate the performance of this system with QPSK modulation for the three sub- channels and how much improvement we got from combining the beamforming technique with the rack receiver. Figure 6.17. The bit error probability of sub-channel one of 3×3 MIMO system. 104 Figure 6.18. The bit error probability of sub-channel two of 3×3 MIMO system. Figure 6.19. The bit error probability of sub-channel three of 3×3 MIMO system. 105 Figure 6.20. The BER of the sub-channels 3×3 MIMO system in 5dB and 20 dB. The result in the Figure 6.20 shows that the performance of sub-channel one is somewhat better than sub-channel two and sub-channel three respectively at the same SNR. That is because sub-channel one has a beamforming gain larger the other sub- channels. The singular value decomposition of the channel matrix of the 3×3 MIMO system follows the order form that 휎 > 휎 > 휎 that will increase the SNR for each sub- channel. Figures 6.17, 6.18, and 6.19, show the performance of the three sub-channels on the 3×3 MIMO system and the theoretical results based on using the Eq. (5.59) and Table 5.2. because we have used the QPSK modulation. Also, we can see that at the high SNR, the performance of these three sub-channels became quite similar (2×Nr) DSSS-BPSK MIMO Communication System In this section, we will analyze the performance of the MIMO channel for different MIMO configurations. Under the assumption that the number of received antenna is bigger or equal to the number of transmitted antenna, we will analyze the 106 MIMO system when the number of transmitted antenna is two. Also, I will compare this result with the result of (2×2) MIMO system by using the DSSS- BPSK modulation and using the combination of the beamforming technique and rack receiver at detection. (2×3) MIMO Communication System Performance Result of DSSS-BPSK When we apply the beamforming technique on the 2×3 MIMO system, the number of parallel sub-channels will be two, which is equal to the minimum number of the transmitter or receiver antennae. We will see the performance result of the combination of the rack receiver and beamforming technique on this system and compare with the result of 2×2 MIMO system, where both systems have two sub-channels. As we have seen in Chapter 4, we assumed the transmitter has knowledge of the channel state information to apply the beamforming technique, so we will find the singular value decomposition to decuple the MIMO channel to sub-individual channels. The following equations explain the beamforming technique of this system. The channel matrix is as follows: ℎ ℎ 퐻 = ℎ ℎ (6.93) ℎ ℎ r = [U D VH] s + z (6.94) After the precoding technique, which happens at the transmitter, substitute s with V푠̃ Where: 푠̃ is transmitted data 휎 0 v 푟 = [u u u ] 0 휎 [v v ] 푠̃ + 푧 (6.95) v 0 0 Where: u is the column vector, v is the row vector. At the receiver, we multiply the received signal by UH. 107 휎 0 0 H H 푟̃ = U U 0 휎 0 푠̃ + U z (6.96) 0 0 0 휎 0 0 푟̃ = 0 휎 0 푠̃ + 푧̃ (6.97) 0 0 0 푟 ̃ = 휎 푠 ̃ + 푧 ̃ (6.98) 푟 ̃ = 휎 푠̃ + 푧 ̃ (6.99) At the transmitter, we have used the DSSS-BPSK modulation, and we modulate each data stream with a different code signal. At detection, we used the rack receiver with each sub-channel to increase the diversity order of each sub-channel, so that we have achieved the higher reliability of MIMO channel. Figure 6.21 and 6.22 show the performance of the 2×3 MIMO system for sub-channel one and two respectively. Figure 6.21. The performance of 2×3 MIMO for sub-channel one. 108 Figure 6.22. The performance of 2×3 MIMO for sub-channel two. Figure 6.23. The detection block diagram of 2×3 MIMO system. 109 The results show that at the low signal-to-noise ratio, the probability of bit error of a 2×3 MIMO system is less than that of a 2×2 MIMO system, despite both systems having two sub-channels. This is different because the beamforming gain for each sub- channel is larger in a 2×3 MIMO system due to the change of the dimensions of the channel matrix, that increases the singular value decomposition, which itself increases the signal-to-noise ratio. Also, we use the second antenna sharing between sub-channel one and two and combine the incoming signal using MRC and that will increase SNR, which will enhance the performance of each sub-channel. Figure 6.23 shows the detection block diagram of 2×3 MIMO system. The theoretical results in figure 6.21 and 6.22 based on using Eq. (5.58) for BPSK modulation. At the same time, the capacity of the 2×3 MIMO system is better than the capacity of the 2×2 MIMO system for the same reason. Figure 6.24 show the capacity of 2×3 MIMO system compared with capacity of the 2×2 MIMO system by using DSSS-BPSK modulation. 110 Figurer 6.24. The 2×3 MIMO beamforming capacity. (2×4) MIMO Communication System Performance Result of DSSS-BPSK In this system the dimensions of the channel matrix are 2×4, and that will affect the beamforming gain. The MIMO system has also two sub-individual channels. By applying the beamforming technique on the following channel matrix. ℎ ℎ ℎ ℎ 퐻 = (6.100) ℎ ℎ ℎ ℎ The beamforming technique process will be as follows: 휎 0 0 휎 v 푟 = [u u u u ] [v v ] 푠̃ + 푧 (6.101) 0 0 v 0 0 111 Where: u is the column vector, v is the row vector. At the receiver, we multiply the received signal by UH. 휎 0 0 휎 푟̃ = UHU + UH z (6.102) 0 0 0 0 휎 0 0 휎 푟̃ = 푠̃ + 푧̃ (6.103) 0 0 0 0 푟 ̃ = 휎 푠 ̃ + 푧 ̃ (6.104) 푟 ̃ = 휎 푠̃ + 푧 ̃ (6.105) Although each sub-channel has a different data stream, we will use the DSSS- BPSK modulation so that each sub-channel has a different PN code. At detection, we will use the rack receiver with a number of fingers L=4 to extract the multipath component of each sub-channel and combine them by using the MRRC. The reliability of each sub- channel will increase because the diversity order will increase from the first order in SISO sub-channel to the fourth order, when using the rack receiver. In addition, the beamforming gain will increase the signal-to-noise ratio also that will reduce the probability of bit error and enhance the capacity of MIMO channel. Figure 6.25 and 6.26 show the performance of 2×4 MIMO system for sub-channel one and two respectively. 112 Figure 6.25. The performance of 2×4 MIMO system sub-channel one. Figure 6.26. The performance of 2×4 MIMO system sub-channel two. 113 . Figure 6.27. The detection block diagram of 2×4 MIMO system. The results show that the bit error of the 2×4 MIMO system is less than the 2×3 and 2×2 MIMO systems at the same signal-to-noise ratio. That is because changing the dimensions of the channel matrix affected the beamforming gain, which increased the signal-to-noise ratio that improved the probability of bit error and increased the capacity of the MIMO channel. Also, the 2×4 MIMO system use two received antennas to detected one signal by using MRC which maximize the SNR of each sub-channel, and that will improve the performance of each sub-channel. Figure 6.27 shows the detection block diagram of 2×4 MIMO system. 114 Figure 6.28. The capacity of 2×4 MIMO beamforming system. Figure 6.28 show that a 2×4 MIMO system has higher capacity compared to 2×3, and 2×2 that because the beamforming gain is different between these systems. Also, the detection process different among these systems. However, the capacity of a 2×4 MIMO system is the same as the capacity of a 4×2 MIMO system. (3×4) MIMO Communication System Performance Result of DSSS-BPSK In this section, we will analyze the performance of the MIMO system which has three transmitted antennae, and four received antennae. We will compare the result with a MIMO system that has the same number of sub-channels, which is a 3×3 MIMO system. In order to apply the beamforming technique, we assume the transmitter has the knowledge of the channel state information. Similar to what we did on the 2×2 MIMO system, we will use the rack receiver with each sub-channel to extract the multipath 115 component and combine them constructively by using MRRC at detection. We are using the rack receiver with L=4 fingers. We assume the channel matrix is as follows: ℎ ℎ ℎ ℎ ℎ ℎ 퐻 = (6.106) ℎ ℎ ℎ ℎ ℎ ℎ The beamforming technique process will be as follows: 휎 0 0 v 0 휎 0 푟 = [u u u u ] v [v v v ] 푠̃ + 푧 (6.107) 0 0 휎 0 0 0 v Where: u is the column vector, v is the row vector. At the receiver, we multiply the received signal by UH. 휎 0 0 0 휎 0 푟̃ = UHU + UH z (6.108) 0 0 휎 0 0 0 휎 0 0 0 휎 0 푟̃ = 푠̃ + 푧̃ (6.109) 0 0 휎 0 0 0 푟 ̃ = 휎 푠 ̃ + 푧 ̃ (6.110) 푟 ̃ = 휎 푠̃ + 푧 ̃ (6.111) 푟 ̃ = 휎 푠̃ + 푧 ̃ (6.112) Each sub-channel in equations. (6.110), (6.111), and (6.112) have different data streams. We modulate each channel with DSSS-BPSK, so each channel has a different code signal. At the rack receiver, we used the cycle shift correlator to extract the multipath component of each signal and combined them using MRRC. The reliability of each sub-channel will increase because the diversity order increased from the first order 116 in the sub-channel using the beamforming technique only, to the fourth order when using the rack receiver. In addition, the beamforming gain will increase the signal-to-noise ratio of each sub-channel, and that will enhance the performance of the MIMO channel. The beamforming gain follows the order that 휎 > 휎 > 휎 ,which is different from one sub- channel to another. Figures 6.29, 6.30, and 6.31 show the performance of three sub- channels of a 3×4 MIMO communication system. Figure 6.29. The performance result of 3×4 MIMO system of sub-channel one. 117 Figure 6.30. The performance result of 3×4 MIMO system of sub-channel two. Figure 6.31. The performance result of 3×4 MIMO system of sub-channel three. 118 The results show that the probability of bit error of each sub-channel is different due to the different beamforming gain of each sub-channel. Also, the probability of bit error of each sub-channel in a 3×4 MIMO system is less than a 3×3 MIMO system, where both have three sub-channels. That is because at the detection, we shared some of the received antennas among these three sub-channels and combined their output signals by MRC so that will promote the performance of each sub-channel. Figure 6.32 shows detection block diagram of 3×4 MIMO system. Also, changing the dimensions of the channel matrix will affect the beamforming gain of each sub-channel. Therefore, a 3×4 MIMO system has a beamforming gain higher than a 3×3 MIMO system. Figure 6.32. The detection block diagram of 3×4 MIMO system. 119 Figure 6.33. The capacity of 3×4 MIMO beamforming system. In addition, the capacity of 3×4 MIMO system is larger than the capacity of a 3×3 MIMO system due to the different value of the beamforming gain applied of each sub- channel which are three sub-channels for both MIMO system. According to the Eq. (4.31) the capacity of MIMO system is a summation of the sub-channel capacity. The capacity of MIMO system will enhance positively because the beamforming gain will increase the capacity of each sub-channel. As we see in figure 6.32, we shared two received antennas among these three sub-channels and we combined their output at the detection by using MRC that will increase the SNR of each sub-channels and reduce the bit error probability. Figure 6.28 shows the capacity of a 3×4 MIMO system. Also, the capacity of this system is the same as the capacity of a 4×3 MIMO system because they have same number of the MIMO sub-channel. 120 CHAPTER 7 CONCLUSION AND FUTURE WORK Conclusion In this thesis, we focused on the study that improved the performance of the different classes of the MIMO technique. The MIMO communication technique can be used to increase the data rate (the capacity of the communication channel) or can be used to improve the reliability of the communication channel. There is a tradeoff between achieving both the capacity and reliability of the communication channel. However, much research has been done to improve the reliability and the capacity of the MIMO channel by using different coding techniques. My thesis tried to combine the MIMO beamforming technique, which increased the capacity, and the rack receiver, which increased the reliability. The beamforming technique, that we illustrated in Chapter 4, decoupled the MIMO channel to individual parallel sub-channels. Each of these channels has first order diversity like the SISO communication system channel, so the capacity of the MIMO channels will be a summation of the capacity of these sub -channels (SISO channels). In order to improve the reliability of each sub-channel, we used the rack receiver technique, which is a part of detection of the incoming signal. By exploiting the multipath of each sub-channel, we can improve the diversity order of each sub-channel. In general, we used the concept of the multipath diversity in a multipath environment to generate time 121 diversity of the transmitted signal. At the receiver, we extracted the delayed multipath components of the transmitted signal combined these components by using the MRC technique. This process is similar to the communication system, which consists of one transmitter and multiple receiver antennas. In order to apply the beamforming technique, we assume the transmitter has knowledge of the channel state information. In this thesis, I assumed also that the same channel is Slow and Rayleigh flat fading. The greatest advantage of this combination is to achieve both a better transmission reliability and a higher data rate. We have applied this technique on different MIMO systems. First, we have analyzed the 2×2 MIMO communication system. The MIMO channel decoupled into two sub-channels, and we used the DSSS- BPSK modulation. We compared the result of this system with a 2×2 MIMO system using a beamforming technique only. We got the low probability of error even at a low signal to noise ratio. Second, we have analyzed the same MIMO system, but we use DSSS-QPSK modulation. We got higher capacity and lower probability of bit error than using BPSK modulation. Third, we have analyzed the 3×3 MIMO system, which have three MIMO sub-channel, and we used also both modulation DSSS-QPSK and DSSS- BPSK modulation. The result showed the probability of bit error of each sub-channel was less than using the same system with beamforming technique only. Finally, we have analyzed the non-symmetric MIMO system which have a 2×3, a 2×4, and a 3×4 MIMO system and we compared the result of performance of each system to a corresponding symmetric system having the same number of beamforming sub-channel. We have seen also; the performance of the sub-channels in each MIMO system will be better if we increased the number of the fingers of the rack receiver, which will increase the diversity 122 order, or if we increased the length of the PN code sequence (M) which will affect to the signal -to- noise ratio of each sub-channel. Future Work Although this thesis achieved a good result of applying the MIMO beamforming technique, the improvement of wireless communication cannot stop especially during the last 10 years. There are many opportunities for future work that can be suggested to develop the idea of this thesis Reanalyze the MIMO system when the channel is frequency selective fading, by using MIMO - OFDM technique Testing this combination of beamforming and the rack receiver with different types of modulation such as M-PSK, QAM Reanalyze the MIMO communication system with this combination of beamforming and the rack receiver, using the spherical detection(SD) method instead of using MLD. 123 APPENDIX A NORMALIZ THE SYSTEM EQUATION OF THE MIMO SYSTEM There are various forms of the MIMO system equations due to different assumptions various authors make about how to normalize the parameters in system equations. When those normalizations are considered, one finds that all the forms are equivalent. The purpose of this appendix is to explain why the normalization (section 3.6. about the capacity of MIMO system) in Eq. 3.19-3.22 are implied by the form of system equation in Eq. (A.1) 푟 = 휌 퐻푠 + 푧 (A.1) 푟 = 퐻푠 + 푧 (A.2) In this thesis, we have used the MIMO system equation in Eq. A.2 as a basis of analysis for the received system equation in most chapters, where H in Eq. (A.2) represents the “physical channel matrix”, and that is equal to the product of ρ H in Eq. (A.1). In contrast, the term H in Eq. (A.1) represents the “normalized channel matrix”. We begin by expressing the MIMO system equation in the following general form: 푟 = 훼 퐻푠 + 푧 (A.3) Where r is a vector of the received signal, 훼 is a general scalar coefficient, s is a vector of the transmitted signal, H is a channel matrix, z is a noise vector at each of the received antenna. However, the covariance matrix of the received signal is: 푅 ≜ 퐸[푟푟 ] (A.4) 124 Substituting Eq. (A.3) in Eq. (A.4) as follows: 푅 = 퐸{(훼 퐻푠 + 푧)(훼 퐻푠 + 푧) } 푅 = 퐸{훼 퐻푠푠 퐻 + 훼퐻푠푧 + 훼푧푠 퐻 + 푧푧 } (A.5) Under the assumption that H and s are independent, we can rewrite Eq. (A.5) as follows 푅 = 훼 퐸{퐻푠푠 퐻 }+ 퐸{푧푧 } 푅 = 훼 퐸{퐻푅 퐻 }+ 퐸{푧푧 } 푅 = 훼 휎 퐸{퐻퐻 }+ 휎 퐼 (A.6) Where 휎 is a variance of the transmitted signal when the transmitted signals are uncorrelated and have equal power transmitted from each transmitter antenna, 휎 is a thermal noise received at each received antenna. Therefore, we can calculate the total received power, which is equal to the sum of the diagonal term of the received covariance matrix as follows 푃 , = 푇 {푅 } (A.7) Where the term 푇 {. } refers to the trace matrix from Eq. (A.6) and (A.7), the total received power is expressed as follows: 푃 , = 훼 휎 퐸{푇 {퐻퐻 }}+ 휎 푇 퐼 푃 , = 훼 휎 퐸{‖퐻‖ }+ 휎 푁푟 (A.8) Where the term ‖퐻‖ refer to the Forbenius norm of the channel matrix. Because we assume the received power at each received antenna are equal, so the power at each receiver will be as follows: , {‖ ‖ } 푃 = = + 휎 (A.9) The first term in Eq. (A.9) represents the power of the desired signal, and the second term represents the noise power. We can write the signal-to-noise ratio (ρ) as follows: 125 {‖ ‖ } ρ = (A.10) Eq. (A.10) gives the general form of the relationship between the variance of signal and noise, Forbenius norm of the channel matrix, the signal-to noise ratio, the coefficient 훼 , and the number of receivers. The conventional assumption is to make 훼 = 휌 , so the eq. (A.10) becomes as follows: {‖ ‖ } 1 = ( A.11) There are a set of assumptions we can make about the variances of the signal, the noise, and the statistics of the element of the normalized channel matrix that satisfy Eq. (A.11). One assumption that is called the MIMO system equation, defined by Eqs.3.60-3.64, is normalized when considering the fact of random channel matrix. This implies 퐸{|ℎ | }=1. Therefore, 퐸{‖퐻‖ }=푁 푁 (A.12) In this case, the signal and noise variance satisfy Eq. (A.11) as follows: 휎 = , and 휎 = 1 (A.13) 126 APPENDIX B MEAN AND VARIANCE OF THE CORRELATION PROCESS AT THE RACK RECEIVER We know that at the rack receiver, we correlate the incoming signal with a cyclic shift correlation to extract the multipath components of the signal. The autocorrelation and cross correlation function are random processes because of the randomness of the PN sequence. However, the purpose of this appendix is to prove the mean and variance of the correlation process at the rack receiver. We begin with the assumption that the PN sequence consists of the 푎 value. These values in each sequence are statistically independent. Thus, 푎 ∈ {−1,+1} (B.1) the probability of the random variable 푎 is P (푎 ) = (B.2) The mean and variance of 푎 is E {푎 } = 0 , 퐸{푎 }=1 (B.3) The mean value of the partial correlation of the PN sequence is defined as follows: 퐸{푟(푘)}= 퐸{푎 푎 } (B.4) Thus, 퐸{푎 } = 푀 k = 0, 푁,2푁,… 퐸{푟(푘)} = (B.5) 퐸{푎 } 퐸{푎 } = 0 , k ≠ 0, N, 2N,…. 127 The mean value of the partial correlation is zero when the sequences are not lined up and it has a maximum value (M) when they are lined up [10]. The variance of the partial correlation will be as follows: Var{푟(푘)}= 퐸{푟(푘) }−(퐸{푟(푘)}) (B.6) Var{푟(푘)}= 퐸{푎 푎 } 퐸{푎 푎 } − [퐸{푟(푘)}] (B.7) If the sequence is lined up, k = 0, N,2N…., the sequence will be 푎 = 푎 , 푎푛푑 (B.8) 푎 = 푎 (B.9) We can rewrite the Eq.B.7 as follows: Var{푟(푘)}= 퐸{푎 } 퐸{푎 } − 퐸{푎 (B.10) Var{푟(푘)} = 푀 − 푀 = 0 (B.11) If the sequence is not lined up, k ≠ 0, N,2N…. 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