A Thesis
entitled
Performance Analysis of Asymmetric Constellation in
Concatenation with Trellis Coded Modulation for use in
Intelligent Systems
by
Abbas Firoz Saboowala
Submitted to the Graduate Faculty as partial fulfillment of the requirements for the
Master of Science Degree in Electrical Engineering
Dr. Kim Junghwan, Committee Chair
Dr. Mohammed Niamat, Committee Member
Dr. Kim Dong-Shik, Committee Member
Dr. Patricia R. Komuniecki, Dean College of Graduate Studies
The University of Toledo May 2011
An Abstract of
Performance Analysis of Asymmetric Constellation in Concatenation with Trellis Coded
Modulation for use in Intelligent Systems
by
Abbas F. Saboowala
Submitted to the Graduate Faculty as partial fulfillment of the requirements for the
Master of Science Degree in Electrical Engineering
The University of Toledo
May 2011
This thesis is on trellis coded modulation (TCM) schemes with asymmetrical constellation. This modulation technique can be easily adapted to the intelligent systems like Cognitive Radio and Software Defined Radio (SDR) with improved error performance. Different types of asymmetric constellation methods for QPSK, 8-PSK and
16-PSK are used, which result in better performance compared to the case of standard symmetrical constellation assignment. In a cognitive radio environment, the channel conditions change frequently. Thus it suggests the requirement for a non-linear adaptive modulation technique with variable parameters. This requirement can be met by using trellis coded modulation with asymmetrical constellation [15,16].
The approach to this thesis is to use digital modulation techniques namely M-ary
Phase Shift Keying (MPSK), in combination with convolutional codes of specific code rate, and then combining these convolutional codes to a TCM mapper, and finally
iii carrying out TCM encoding and decoding. Towards this, the following steps are performed; designing the convolutional encoder based on Ungerboeck‟s design scheme, using a signal mapper to map the modulation scheme, making use of Ungerboeck‟s set partitioning technique to obtain the maximum distance between signals [2], obtaining the trellis which maximizes the performance, and carrying out decoding using the Viterbi algorithm [18]. The method used in this thesis yields an improvement in error, minimum distance and coding gain as compared to those of conventional trellis coded modulation technique.
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Acknowledgements
I would firstly like to thank the All Mighty God, for venturing me through thick and thin. Also, I would like to express my most sincere gratitude and gratefulness to His
Holiness Dr. Syedna Mohammed Burhanuddin saheb, whose blessings and guidance have helped me at every stage of my life. I owe my deepest gratitude to my advisor, Dr.
Junghwan Kim, who has helped and supported me throughout my research and academic program. I would also like to thank Dr. Niamat Mohammed and Dr. Dong-Shik Kim, for serving as members on my thesis defense committee.
I am also very thankful to the Engineering Technology Department, Scott Park
Library and Law School for having provided a wonderful ambience to work in.
A special thanks to my wonderful friends Taher Kagalwala, Hozefa Jodiyawala,
Mohammed Taskeen, Aliasgar Presswala, Widian Abi Saab, Alafiya Nasrulla, Javed
Mapkar, Desikan Sundarajan, Shachi Mistry and Jessica Stewart for their constant support and help. An extended thanks to Taher Kagalwala and Wang Chong for helping me with my research.
My deepest gratitude goes to my parents and family who have always been my strength and source of encouragement, especially my mother, Munira Saboowala and sister, Fatema Saboowala. Also, I express my sincere gratitude to Zainab Kothari and the entire Kothari family for always being by my side throughout my Master‟s program.
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Table of Contents
Abstract …...... ………………………………………………………………………….iii Acknowledgements …………………………………………………………………….v Table of Contents...... vi
List of Figures ...... viii
1 Introduction…………………………………………………………………………..….1 1.1 Digital Communication ...... 1 1.2 Cognitive Radio (CR) ...... 2 1.2.1 Adaptive Modulation and Coding in Cognitive Radio ...... 5 1.3 Trellis Coded Modulation ...... 7 1.3.1 TCM Concepts ...... 9 1.4 Outline of Thesis ...... 12 2 Modulation and Asymmetric TCM…………………………………………………….13 2. 1 Modulation Technique...... 13 2.2 Asymmetric Constellation in TCM ...... 17 2.2.1 System design ...... 19 2.2.2 TCM Encoder ...... 21 2.2.3 Mapping ...... 25 2.2.4 Viterbi Decoding ...... 28 3 Generalized System Performance Analysis ……………………………………………31 3.1 System Performance ...... 31
3.2 Relation Between dfree and BER ...... 36 3.3 Relation Between Phase Jitter and Error Probability ...... 37 4 Performance Improvement for PSK Schemes…………………………………………40 4.1 Asymmetric 4-PSK, 2 State, Rate 1/2 ...... 40
vi
4.2 Asymmetric 4-PSK, 4 State, Rate 1/2 ...... 47 4.3 Asymmetric 4-PSK, 8 State, Rate 1/2 ...... 51 4.4 Asymmetric 8-PSK, 4 State, Rate 2/3 ...... 54 4.5 Asymmetric 8-PSK, 8 State, Rate 2/3 ...... 57 4.6 Asymmetric 8-PSK, 16 State, Rate 2/3 ...... 60 4.7 Asymmetric 16-PSK, 8 State, Rate 3/4 ...... 63 5 Results and Discussions………………………………………………………………..67 5.1 Simulation Results ...... 67 6 Conclusion and future work ……………………………………………………………83 6.1 Conclusion ...... 83 6.2 Future Work ...... 84 References………………………………………………………………………………..85
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List of Figures
Figure 1.1: Logical diagram contrasting traditional radio, software radio, and cognitive
radio ...... 3
Figure 1.2: Functional portions of a cognitive radio, representing reasoning and learning
capabilities ...... 4
Figure 1.3: Block diagram for Adaptive Modulation based Cognitive Radio ...... 6
Figure 1.4: 8 PSK constellation and squared Euclidean distances between symbols ...... 9
Figure 1.5: Euclidean and sequence Hamming distance ...... 10
Figure 2.1: Bit error rate (BER) of BPSK ...... 14
Figure 2.2: BER simulation of QPSK ...... 15
Figure 2.3: BER simulation of 8-PSK ...... 16
Figure 2.4: Symmetric 8PSK Constellation ...... 18
Figure 2.5: ASYMMETRIC 8PSK Constellation ...... 19
Figure 2.6: System block diagram for asymmetrical TCM ...... 19
Figure 2.7: A general trellis coded modulation ...... 22
Figure 2.8: Constellation doubling in TCM showing a QPSK signal transmitted using a
8PSK constellation ...... 23
Figure 2.9: Signal doubling that takes place in TCM represented with an example for
BPSK, QPSK and 8-PSK...... 24
viii
Figure 2.10: Set Partitioning of Asymmetric 8PSK ...... 27
Figure 2.11: Example of a Trellis Diagram ...... 28
Figure 4.1: Set partitioning of asymmetric 4-PSK ...... 41
Figure 4.2: Trellis diagram of Asymmetric 4-PSK, 2 State, Rate 1/2 ...... 42
Figure 4.3: Trellis structure of 4-PSK, 4 State, Rate 1/2 TCM ...... 47
Figure 4.4: TCM encoder for 4-PSK, 4 State ...... 48
Figure 4.5: Trellis diagram for 4-PSK, 8 State, Rate 1/2 ...... 52
Figure 4.6: Trellis diagram of Asymmetric 8-PSK, 4 State, Rate 2/3 ...... 54
Figure 4.7: TCM encoder for 8-PSK, 4 State ...... 55
Figure 4.8: TCM encoder for 8-PSK, 8 State, Rate 2/3 ...... 57
Figure 4.9: Trellis structure of 8-PSK, 8 State, Rate 2/3 ...... 58
Figure 4.10: TCM encoder for 8-PSK, 16 State, Rate 2/3 ...... 60
Figure 4.11: Trellis structure for 8-PSK, 16 State ...... 62
Figure 4.12: TCM encoder for 16-PSK, 8 State, Rate 3/4 ...... 64
Figure 4.13: Trellis structure for 16-PSK, 8 State ...... 66
Figure 5.1: Asymmetric Constellation for 4-PSK, 4 State, Rate 1/2 ...... 68
Figure 5.2: Comparison between symmetric and asymmetric constellation for 4-PSK, 4
State, Rate 1/2 ...... 69
Figure 5.3: Asymmetric Constellation for 8-PSK, 4 State, Rate 2/3 ...... 70
Figure 5.4: Comparison between symmetric and asymmetric constellation for 8-PSK, 4
State, Rate 2/3 ...... 71
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Figure 5.5: Asymmetric Constellation for 8-PSK, 8 State, Rate 2/3 ...... 71
Figure 5.6: Comparison between symmetric and asymmetric constellation for 8-PSK, 8
State, Rate 2/3 ...... 72
Figure 5.7: Asymmetric Constellation for 8-PSK, 16 State, Rate 2/3 ...... 73
Figure 5.8: Comparison between symmetric and asymmetric constellation for 8-PSK, 16
State, Rate 2/3 ...... 74
Figure 5.9: Asymmetric Constellation for 16-PSK, 8 State, Rate 3/4 ...... 75
Figure 5.10: Comparison between symmetric and asymmetric constellation for 16-PSK, 8
State, Rate 3/4 ...... 75
Figure 5.11: Comparison between symmetric and asymmetric constellation for 4-PSK, 4
State, Rate 1/2 TCM with respect to uncoded scheme ...... 76
Figure 5.12: Comparison between symmetric and asymmetric constellation for 8-PSK, 4
State, Rate 2/3 with respect to uncoded scheme...... 77
Figure 5.13: Comparison between symmetric and asymmetric constellation for 8-PSK, 8
State, Rate 2/3 with respect to uncoded scheme...... 79
Figure 5.14: Comparison between symmetric and asymmetric constellation for 8-PSK, 16
State, Rate 2/3 with respect to uncoded scheme...... 79
Figure 5.15: Comparison between symmetric and asymmetric constellation for Rate 3/4,
8 PSK with respect to uncoded scheme ...... 80
Figure 5.16: Comparison of all Symmetric PSK schemes...... 81
Figure 5.17: Comparison of all Asymmetric PSK schemes ...... 82
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Chapter 1
Introduction
This chapter briefly reviews the concept of an intelligent system, namely cognitive radio, and the importance of adaptive modulation scheme in intelligent systems, with an overview of the recent trends in progress. An outline of the thesis is also presented in this chapter.
1.1 Digital Communication
A communication system can be classified as one of the three types: bandwidth efficient, power efficient, and cost efficient. A bandwidth efficient system is one which transmits the data in a manner which minimizes the use of bandwidth and thus in turn maximizing the bandwidth efficiency. A power efficient system is one, which transmits data effectively using least power, and a cost efficient system is one which maximizes the task using least cost. In this age of limited spectral resources, a number of efforts are being made to develop bandwidth efficient systems, categorizing them as intelligent systems; a system which can vary its parameters according to the requirement and thus in turn maximizing the efficiency. A system of large interest in this context is Cognitive
1
Radio. Research is carried out widely to implement this, since its results are very compelling and can provide significant gain over other systems in use.
1.2 Cognitive Radio (CR)
Over the past several years, notions about radios have been evolving away from pure hardware-based radios to radios that involve a combination of hardware and software. In the early 1990s, Joseph Mitola introduced the idea of software defined radios
(SDRs) [7, 8]. These radios typically have a radio frequency (RF) front-end with a software-controlled tuner. Baseband signals are passed into an analog-to-digital (A/D) converter. The quantized baseband is then demodulated in a reconfigurable device such as a field-programmable gate array (FPGA), digital signal processor (DSP), or commodity personal computer (PC). The reconfigurability of the modulation scheme makes it a software-defined radio.
In his 2000 dissertation, Mitola took the SDR concept one step further, by introducing the concept of cognitive radio (CR) [9]. CRs are essentially SDRs with artificial intelligence, capable of sensing and reacting to their environment. Figure 1.1 graphically contrasts traditional radio, software radio, and cognitive radio.
2
Figure 1.1: Logical diagram contrasting traditional radio, software radio, and Cognitive radio [6]
In the past few years, many different interpretations of the word “cognitive radio” have been developed. Some of the more extreme definitions might be, for example, a military radio that can sense the urgency in the operator's voice, and adjust Quality of
Service (QoS) proportionally [10].
A radio may be able to sense the current spectral environment, and have some memory of past transmitted and received packets along with their power, bandwidth, and modulation. From all this, it can make better decisions about how to best optimize for some overall goal [6]. Possible goals could include achieving an optimal network capacity, minimizing interference to other signals, or providing robust security or jamming protection.
3
Figure 1.2: Functional portions of a cognitive radio, representing reasoning and learning capabilities [6]
Figure 1.2 shows the functional components of concrete cognitive radio architecture. The SDR is accessed via a CR application programming interface (API) that allows the CR engine to configure the radio, and sense its environment. The policy-based reasoning engine takes facts from the knowledge base, which acts as a memory bank, which has some memory of past transmitted and received packets along with their power, bandwidth and modulation [6]. The knowledge base shown in Figure1.2 helps in receiving information from the environment to form judgments about RF spectrum accessing opportunities. In addition to a simple policy-based engine, a learning engine observes the radio's behavior and resulting performance, and adjusts facts in the knowledge base used to form judgments.
However a fundamental problem with a system like this is its complexity. Can the proposed learning and reasoning be done in near real time, to keep up with an ever
4 changing RF environment? Can we come up with a simple set of metrics that can perform well without being overly computationally complex? Seeing the advances in smart radio technology, the FCC began researching ways in which CRs could use licensed bands, provided they didn't interfere with existing licensees. A motion to allow operations to operate was recently approved and adopted by the FCC [11], and allows cognitive radios to operate in certain frequency bands.
1.2.1 Adaptive Modulation and Coding in Cognitive Radio
In a scenario where channel conditions fluctuate dynamically, systems based on fixed modulation schemes do not perform well, as they cannot take into account the difference in channel conditions. In such a situation, a system that adapts to the worst case scenario would have to be built to offer an acceptable bit-error rate. To achieve a robust and a spectrally efficient communication, adaptive modulation is used, which adapts the transmission scheme to the current channel characteristics. Taking advantage of the time-varying nature of the wireless channels, adaptive modulation based systems alter transmission parameters like power, data rate, coding, and modulation schemes, or any combination of these in accordance with the state of the channel [12]. If the channel can be estimated properly, the transmitter can be easily made to adapt to the current channel conditions by altering the modulation schemes while maintaining a constant
BER. This can be typically done by estimating the channel at the receiver and transmitting this estimate back to the transmitter. Thus, with adaptive modulation, high
5 spectral efficiency can be attained at a given BER in good channel conditions, while a reduction in the throughput is experienced in degrading channel conditions. [13]
Adaptive modulation techniques can modify transmission characteristics and waveforms to provide opportunities for improved spectrum access and more intensive use of spectrum while working around other signals that are present. A CR could select the appropriate modulation type for use with a particular transmission system to permit interoperability between systems.
Figure 1.3: Block diagram for Adaptive Modulation based Cognitive Radio [14]
Figure 1.3 provides a detail view of the whole adaptive modulation system with all the necessary feedback paths. It is assumed that the transmitter has a perfect knowledge of the channel and the channel estimator at the receiver is error-free and there is no time delay. The receiver uses coherent detection methods to detect signal envelopes.
The adaptive modulation, Mary PSK, M-QAM, and M-ary AM schemes with different
6 modes are provided at the transmitter. With the assumption that the estimation of the channel is perfect, for each transmission, the mode is adjusted to maximize the data throughput under average BER constraint, based on the instantaneous channel SNR.
Based on the perfect knowledge about the channel state information (CSI), at all instants of time, the modes are adjusted to maximize the data throughput under average BER constraint [14].
1.3 Trellis Coded Modulation
Trellis modulation, which is also known as trellis coded modulation, or simply TCM, is a modulation scheme which allows highly efficient transmission of information over band-limited channels such as telephone lines. Trellis modulation was invented by Gottfried Ungerboeck in the 1970‟s, but went largely unnoticed until a prominent paper was published in 1982 [2].
TCM is basically a digital modulation technique combined with convolutional
푟 code of rates , where „r‟ can be considered the length of the input bits, and „r+1‟ is 푟+1 the length of the output. Ungerboeck's unique contribution is to apply the parity check to a symbol for modulation instead of the older technique of applying it to the bit stream and then modulating the bits. The key idea is „Mapping by Set Partitions‟ [2]. This idea was to group the symbols in a tree-like fashion and then separating them into two limbs of equal size. At each limb of the tree, the symbols are further apart. Suppose the symbols are located at [1, 2, 3, 4, ...]. Then take all odd symbols and place them in one group, and the even symbols in the second group. He next proposed a method of assigning the
7 encoded bit stream onto the symbols in a very systematic procedure. Once this procedure was fully described, next step is to program the algorithms into a computer and let the computer search for the best codes. The results are astonishing. Even the simplest code
(of 4 state) produced error rates nearly 1,000 times lower than an equivalent uncoded system.
TCM uses many diverse concepts from signal processing. In simplest terms, it is a combination of coding and modulation, hence its name Trellis Coded Modulation.
Whereas we normally talk about coding and modulation as two independent aspects of the communications link, in TCM they are combined. It uses ideas from modulation and coding as well as dynamic programming, lattice structures, and matrix operation.
Communications theory says that it is best to design codes in long sequences of random messages. The receiver can then make a decision between sequences using their statistics rather than on symbol-by-symbol basis. When decoding is done, the probability of error is an inverse function of the sequence length. In general, the probability of error between sequences is given by the expression,
−푑2 /2휍2 푝푒 ~ 푒 푚푖푛 (1)
2 Where, dmin is the Euclidean distance between sequences and σ is the noise power. We measure the performance of TCM (and many other schemes) by Asymptotic
Coding Gain (ACG). This is the gain obtained over some baseline performance at high
SNR in a Gaussian environment. In this thesis, this gain will be calculated with reference to the uncoded and symmetrically coded schemes.
8
1.3.1 TCM Concepts
Euclidean Distance
A straight line distance between any two points is called the Euclidean distance.
The Euclidean distance is an analog concept. For signals, we define this distance in the I-
Q plane. In Figure 1.4 we have a 8PSK signal constellation. The radius is equal to 1 and represents the maximum amplitude. Each point of the constellation is a certain combination of a particular amplitude and phase. The calculated Euclidian distance between the signal points is also given as shown in Figure 1.4. These distances are squared and then are called Squared Euclidean Distance (SED). The smallest of these distances is called the Minimum Squared Euclidean Distance (MSED), designated as
2 d min for a particular constellation.
Figure 1.4: 8 PSK constellation and squared Euclidean distances between symbols [19]
Distance between sequences
We can also talk about Euclidean distances between sequences by comparing distances between corresponding points of the sequences. Let‟s take for example an
9
8PSK signal that consists of a sequence of these symbols. The symbols denoted in Figure
1.5 are for a 8-PSK constellation, which means that it will have 8 signal points, labeled from S0 to S7.
S2 S1
S3
S0
S7 S4
S6
S5
Figure 1.5: Euclidean and Hamming distance [19]
The Euclidean distance for this sequence is the distance between each symbol in this sequence and a reference sequence. If we designate the all-zero-symbols as the reference sequence, then the squared Euclidean distance (SED) is the distance between each one of these symbols and the symbol S0.
S0 to S0 = 0.0,
S0 to S1 = 0.586,
S0 to S2 = 2.0,
S0 to S3 = 3.414
In summary, the major features of TCM [19], are as follows;
1. TCM is bandwidth efficient modulation combined with convolutional coding.
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2. It conserves bandwidth by doubling the number of constellation points of the signal.
This way, the bit rate increases but the symbol rate stays the same.
3. Convolutional coding constrains allowed symbol transitions, creating sequence
coding.
4. Unlike conventional convolutional coding, not all incoming bits are coded.
5. Increasing the constellation size reduces Euclidean distances between the constellation
points, but sequence coding offers a coding gain that overcomes the power
disadvantage of going to the higher constellation.
6. Performance is measured by coding gain over an uncoded signal.
7. The decoding metric is the Euclidean distance and not Hamming distance.
8. Ungerboeck [2], originally proposed TCM which used set-partitioning and small
number of states with code rates that varied with the input signal type.
9. Pragmatic TCM [20] uses a less than perfect rate ½ convolutional code with constraint
length equal to 7 or 9. This is a widely available code and it makes TCM less complex
to implement.
10. The constellation mapping in set partitioning is based on natural numbering,
whereas Gray coding is preferred in pragmatic TCM.
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1.4 Outline of Thesis
In Chapter 1, we introduce the topic, which mainly deals with the basic concept about intelligent systems, in particular Cognitive Radio, and the need for adaptive modulation in the system. It also briefly mentions the basic concepts for Trellis Coded
Modulation, which is the primary technique used to provide modulation and coding together. Chapter 2, is devoted to the basic modulation techniques which form the core for TCM and describe the use of asymmetric constellation, and mention in detail the system design including the encoder and decoding technique employed in our research.
In Chapter 3, we analyze the generalized performance of an asymmetric constellation used in our research. Chapter 4 deals with specific TCM techniques to which the asymmetric constellation is applied and the theoretical performance comparisons with the symmetric and uncoded schemes are are also provided. The results and simulations corresponding to the techniques mentioned in Chapter 4 are shown in Chapter 5. Finally, the conclusion and future work is presented in Chapter 6.
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Chapter 2
Modulation and Asymmetric TCM
2. 1 Modulation Technique
In this work, we employ Phase Shift Keying (BPSK, QPSK, 8-PSK). A few important factors that need to be considered before selecting a digital modulation technique can be highlighted as follows;
- High spectral efficiency, high power efficiency, robust to multipath effects, low cost
and ease of implementation, low cost and ease of implementation, low carrier-to-co
channel interference ratio, low out-of-band radiation and constant or near constant
envelope.
The basic error performance of BPSK, QPSK and 8-PSK schemes are shown in Figures
2.1, 2.2 and 2.3 respectively.
Some important parameters for performance evaluation are defined below;
. Eb = Energy-per-bit
. Es = Energy-per-symbol = nEb with n bits per symbol
. Tb = Bit duration
13
. Ts = Symbol duration
. N0 / 2 = Noise power spectral density (W/Hz)
. Pb = Probability of bit-error
. Ps = Probability of symbol-error
Bit Error Rate, M=2 0 10 Theoretical Simulated -1 10
-2 10
-3
10 BER
-4 10
-5 10
-6 10 0 5 10 15 Eb/N (dB) o
Figure 2.1: Bit error rate (BER) of BPSK
Figure 2.1 shows the calculated bit error rate (BER) for Binary Phase shift Keying
(BPSK). The bit error probability for BPSK can be given as,
1 퐸푏 푃푏 = 푒푟푓푐 2 푁0
14
For BPSK, the bit error is the same as the symbol error, thus the symbol error probability for BPSK is the same.
Bit Error Rate, M=4 0 10 Theoretical Simulated -1 10
-2 10
-3
10 BER
-4 10
-5 10
-6 10 0 5 10 15 Eb/N (dB) o
Figure 2.2: BER of QPSK
Figure 2.2 shows the calculated BER for Quadrature Phase Shift Keying (QPSK). The bit probability of error for QPSK can be given as,
1 퐸푏 푃푏 = 푒푟푓푐 2 푁0
15
And the symbol error probability can be given as,
퐸푠 푃푠 = 푒푟푓푐 2푁0
Bit Error Rate, M=8 0 10 Theoretical Simulated -1 10
-2 10
-3
10 BER
-4 10
-5 10
-6 10 0 5 10 15 Eb/N (dB) o
Figure 2.3: BER of 8-PSK
Figure 2.3 shows the BER for 8-PSK under AWGN channel conditions.
The probability of bit error for 8-PSK can be represented as,
1 푘퐸푏 휋 푃푏 = 푒푟푓푐 푠푖푛 푘 푁표 8
16
Where k, is defined as (where M=8 for 8-PSK),
푘 = log2 8
Also, the probability of symbol error can be represented as,
퐸푠 휋 푃푠 = 푒푟푓푐 푠푖푛 푁표 8
2.2 Asymmetric Constellation in TCM
Trellis coded modulation offers higher power and bandwidth efficiency in
Additive White Gaussian Noise (AWGN) channels. With changing industry trends, TCM has come to be used in channels such as mobile radio and wireless networks. In recent times, the usage of this channel is also being made for data transmission, in addition with speech. The acceptable speech error rate is 10-3 while data error rate is 10-5. In this research, asymmetric constellations of Phase Shift Keying (PSK) are combined with
TCM schemes in order to obtain higher performance than that achievable with symmetric
PSK [16]. By designing asymmetric MPSK signal constellations, and combining them with optimized trellis codes, the performance can be significantly improved without increasing the average or peak power or without changing the bandwidth limit and parameters. Also, important to note is that the other specifications like the constant envelope, number of dimensions, etc., are not changed upon incorporating asymmetry.
For MPSK, the optimum asymmetric 2n+1- point constellation is the combination of a symmetric 2n- point constellation with a phase rotated version of itself. Figure 2.4 and
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Figure 2.5 show the difference in constellation structure for an asymmetric 8-PSK. As observed in the Figure 2.4 the points are equally spaced and thus making this constellation the conventional choice. Figure 2.5 shows the modified constellation, making it asymmetric, and this thesis will deal with a constellation as represented in
Figure 2.5. Also, the term asymmetric is misleading to a certain extent, as the important point to be noted is that, asymmetry with respect to signal constellation actually implies non-uniformity between the spacing of the signal points, in such a way, that the phase is not equally distributed, as can be observed from comparing Figure 2.4 to Figure 2.5.
Figure 2.4: Symmetric 8PSK Constellation [15]
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Figure 2.5: Asymmetric 8PSK Constellation [15]
This condition of signal spacing has come to be known as asymmetry with regard to constellations, and indeed the signal constellations will show symmetrical properties across defined co-ordinate axes, but with unequal spacing.
2.2.1 System design
An asymmetric constellation with a TCM scheme is employed in this thesis.
n k Input r k message (u k) TRELLIS CODE ASYMMETRICAL VITERBI v k
SIGNAL SET DECODER
Input message (asymmetrical
constellation) (x k) Figure 2.6: System block diagram for asymmetrical TCM
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Figure 2.6 represents the basic block diagram of a asymmetrical TCM. The input message is represented by uk which, after being modified to an asymmetrical constellation, can be represented by xk. The code rate is represented by rk which is given as (n / n+1). Finally the output is represented by vk.
Modern communication systems commonly use symmetric modulation schemes to transmit information across wireless channels. Constant envelope, phase-shift keying
(PSK) signals are used to keep signal recovery inexpensive and minimally complicated.
Standard PSK constellations are constructed of symbols with equal energy that are evenly distributed about the origin of the normalized unit circle in the complex plane. The constellation structures of both of these signals lend themselves to ambiguity in phase and are limited in their ability to adapt in dynamic environments. Coherent recovery of these signals requires accurate frequency and phase measurements, requiring some form of synchronization. Quick synchronization can be attained through data-aided methods or pilot tones, both of which have the disadvantage of reducing throughput efficiency since additional bandwidth is occupied for synchronization. Non-data aided methods of synchronization, like phase-locked loops or discrete-time signal processing algorithms, are throughput efficient, but they require more time for synchronization [1]. Receiver capabilities may vary by several dBs (in received signal power). There are a number of ways to achieve different QoS for different receivers. The role of asymmetric modulation comes in use here. This method is the most efficient method to achieve unequal error protection for uncoded systems, where depending on the difference between the users, the constellation points can be placed asymmetrically to achieve the required performance
[4].
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In this thesis, the method employed for developing an asymmetric M = 2n+1 signal set is, adding together an M/2-signal point set with a rotated version of itself. Also another effective way of obtaining an asymmetrical constellation would be to partition the M-point signal set into two M/2-point signal sets and then performing rotation as required.
2.2.2 TCM Encoder
According to the principle from channel coding, additional redundant bits for error control are transmitted in the code, and as a consequence, wider bandwidth is needed to keep the same data rate with lower BER can be obtained for the same SNR or the SNR to obtain the same BER can be decreased. This process used in convolutional coding, shows a trade-off between bandwidth efficiency and power efficiency. In order to avoid this, and to mediate between these two important factors which conflict with each other, we make use of trellis coded modulation. For example, a code rate 2/3 TCM encoder combining a 2-bit input (k) and outputting a 3-bit output (n) convolutional coding scheme, with a 8 (23) PSK modulation scheme is depicted in Figure 2.9. In the encoder shown in Figure 2.9, the 8-PSK signal mapper generates one of the eight PSK signals depending on the 3-bit symbol.
The functions of a TCM consist of a Trellis code and a constellation mapper as shown in Figure 2.7. The block diagram depicted in Figure 2.7 combines the functions of a convolutional coder of rate R =k/k+1 and a M-ary signal mapper that maps M = 2k input points into a larger constellation of M = 2k +1 constellation points.
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Figure 2.7: A general trellis coded modulation [19]
For k = 2, we have a code of rate 2/3 that takes a QPSK signal (M = 4) and puts out a 8-
PSK signal (M = 8). So instead of expanding the bandwidth as the signal goes from
QPSK to 8PSK, it instead doubles the constellation points. It is kind of an upgrading system, where we take a chosen signal and upgrade it to another with larger number of constellation points as shown in Figure 2.8. As shown in Figure 2.8, we begin with 4 signal points, which, after being passed through a TCM encoder, are appropriately mapped as 8-signal points.
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Figure 2.8: Constellation doubling in TCM showing a QPSK signal transmitted using a 8PSK constellation [19]
TCM is a general concept and by varying k, we can create a QPSK, 8PSK or higher level signals. Figure 2.9 represents a diagrammatic view of signal doubling that takes place in TCM with the help of an example for BPSK, QPSK and 8-PSK.
A few things that need to be mentioned about TCM are:
1) When we compare the number of signal points with the uncoded case, it is
observed that the signal points in the constellation are larger. For example, the
number of signal points in the constellation of the TCM or the modulation order
of the TCM for the previous figure is 2k+1 = 22+1 = 8, for k = 2. This number of
signal points is twice as larger compared to the signal points in QPSK
constellation that would be actually used to send the message of two bits. This
increase in the modulation order, without any change in the SNR decreases the
minimum distance among the code words, thus the Bit Error Rate may suffer to
some extent.
23
Figure 2.9: Signal doubling that takes place in TCM represented with an example for BPSK, QPSK and 8-PSK [19]
24
2) However TCM has some measures that can be taken accordingly which will
combat the possible BER performance degradation.
- The convolutional coding generally allows only certain sequences (paths) of
signal points so that the free distance among different signal paths in the trellis
can be increased.
- The decoding used by the TCM decoder tries to find a path with minimum
Euclidian distance through the trellis path. Thus the trellis code design is
employed with trying to maximize the Euclidian distance among the code words.
- As will be shown later, a technique called “Mapping by Set Partitioning” is used
which lets the more significant message bits to have a larger Euclidian distance
from its compliment for there are less likely errors in the uncoded message bits
than those in the coded bits.
- We employ a trellis scheme for encoding purpose and the decoding can be carried
out using Viterbi Algorithm.
2.2.3 Mapping
This work makes use of the mapping by set-partitioning technique to assign signals to the trellis code [2]. The idea behind this technique is the successive partitioning of the 2n+1-ary signal sets in smaller subsets, such that the Euclidian distance between the signals is maximized. This helps in limiting the transitions occurring, only along the path
25 with largest Squared Euclidian Distance (SED). The set partitioning is carried out in the
Figure 2.10.
In Figure 2.10, the signal set consists of 8 signals. In the first step of mapping, the original signal set is divided into 2 subsets, each consisting of 4 signals. The first signal set consists of signals (0,2,4,6) and the second set consists of (1,3,5,7). The advantage of this method is that the Euclidian distance between the signals in the subsets is higher than the Euclidian distance in the original 8-signal set. We continue partitioning the subsets until the minimum Euclidian distance desired is obtained. In general it is not necessary to carry out the partitions, until only signal remains in the subset. If desired, the partitions only need to be performed until the minimum Euclidian distance between the signals at the final level of partitioning is equal to or larger than the desired minimum Euclidian distance of the TCM scheme to be designed [3].
26
Figure 2.10: Set Partitioning of Asymmetric 8PSK [15]
27
2.2.4 Viterbi Decoding
In order for decoding the trellis to obtain the correct information at the decoder, we make use of a technique called Viterbi decoding. This technique is the most reliable and effective in generating output at the decoder with minimum number of errors.
To understand the concept of Viterbi decoding, let us consider an example of a trellis diagram, as shown below in Figure 2.11. The analysis of how the decoding is carried out with respect to Figure 2.11 is described below:
Figure 2.11: Example of a Trellis Diagram [17]
A Viterbi algorithm consists of the following three major parts:
1. Branch metric calculation – calculation of a distance between the input pair of
bits and the possible “ideal” pairs (e.g. “00”, “01”, “10”, “11”).
28
2. Path metric calculation – for every encoder state, calculate a metric for the
survivor path ending in this state (a survivor path is a path with the minimum
metric).
3. Traceback – this step is necessary for hardware implementations that don't store
full information about the survivor paths, but store only one bit decision every
time when one survivor path is selected from the two.
For a soft decision decoder, a branch metric is measured using the Euclidean distance.
Let x be the first received bit in the pair, y – the second, x0 and y0 – the “ideal” values.
Then branch metric is,
2 2 Mb=(x-x0) + (y-y0)
Path metrics are calculated using a procedure called ACS (Add-Compare-Select). This procedure is repeated for every encoder state.
1. Add – for a given state, we know two states on the previous step which can move
to this state, and the output bit pairs that correspond to these transitions. To
calculate new path metrics, we add the previous path metrics with the
corresponding branch metrics.
2. Compare, select – we now have two paths, ending in a given state. One of them
(with the greater metric) is dropped.
As there are 2K-1 encoder states, we have 2K-1 survivor paths at any given time.
If we decode a continuous stream of data, we want our decoder to have finite latency. It is obvious that when some part of path at the beginning of the graph belongs to every
29 survivor path, the decoded bits corresponding to this part can be sent to the output. Given the above statement, we can perform the decoding as follows:
1. Find the survivor paths for N+D input pairs of bits.
2. Trace back from the end of any survivor paths to the beginning.
3. Send N bits to the output.
4. Find the survivor paths for another N pairs of input bits.
5. Go to step 2.
Where, D is an important parameter called decoding depth. A decoding depth should be considerably large for quality decoding, no less than 5K.
Where, N specifies how many bits are being sent to the output after each traceback [17].
30
Chapter 3
Generalized System Performance Analysis
3.1 System Performance
For a code rate of n/(n + 1), with n information bits to begin with, the TCM encoder produces n + 1 output coded symbols. As described earlier, set partitioning is employed for encoding purposes, and as a result, the symbols are assigned to a unique
n+1 member of the asymmetric 2 signal set. If we assign a variable sk to the state of the encoder, then each transmitted signal represented by the variable xk, at a specific time unit k, is a non-linear function f, of the state of the encoder and the n information bits which are input in the system denoted by the variable uk.
This can be represented as [15],
푥푘 = 푓 푠푘 , 푢푘 (3)
The next state of the encoder can be defined by a combination of the present state of the encoder and the input to the encoder at that specific time. Thus the next state sk+1 can be observed to be a non-linear function g of the present state of the encoder and the input uk.
31
This can be represented as [15],
푠푘+1 = 푔(푠푘, 푢푘 ) (4)
The received signal at a specific time k can be represented as [15]
푟푘 = 푥푘 + 푛푘 (5)
2 2 Where, nk is a sample of a zero mean Gaussian noise process with 휍 , and 휍 is variance.
As mentioned previously, the decoding technique we use is Viterbi decoding. Thus to obtain the average bit error probability of the Viterbi decoder, the first step needs to be finding the pair-wise error probability 푝 (푥 → 푥 ) existing between the actual coded sequence 푥푘 and the sequence that is estimated to be correct, i.e. 푥 푘 . The coded sequence and the estimated sequence are assigned as 푥 and 푥 respectively here.
2 Let us assume that 푥푘 = 1, then applying the Bhattacharyya bound accordingly to this case, we have [5],
푝 푥 → 푥 ≤ 퐷∆ (6)
2 ∆ = 푘 훿 (푆푘, 푈푘 ) (7)
2 Also, 훿 (푆푘, 푈푘 ) can be expressed as [15],
2 2 훿 (푆푘, 푈푘 ) ≜ 푓 푠푘, 푢푘 − 푓(푠 푘, 푢 푘 (8)
32
In the above equation, 푠 푘 and 푢 푘 represent the estimates of the state of the decoder and the input symbols respectively.
In Equation (6), D is the Bhattacharyya bound and it can be expressed as,
1 퐷 = exp ( − ) (9) 8휍2
Where 휍2 is the variance of the system, which depends on the bit energy-to-noise ratio of the system.
Thus, the Bhattacharyya bound can be calculated accordingly, by recognizing that,
2퐸 휍2 = ( 푠 )−1 (10) 푁0
Where, Es is the symbol energy of the M-ary system.
We have „n‟ input bits, in the case of a n / (n+1) rate encoder. Each of these „n‟ input bits have a bit energy of Eb. These „n‟ bits provide an output with n + 1 bits accordingly, which in turn represent an M-ary symbol of energy Es.
Thus,
퐸푠 = 푛퐸푏 (11)
Using, Equation (11) in Equation (9), we get,
푛퐸 퐷 = exp( − 푏 ) (12) 4푁0
33
From Equation (8),
Sk is the pair state, and Uk is the pair-information symbol.
Sk and Uk can be defined by [5] and [21],
푆푘 ≜ 푠푘 , 푠 푘 (13)
푈푘 ≜ 푢푘 , 푢 푘 (14)
The concept of Viterbi decoding checks if, at a specific time, we are in a correct state or an incorrect state. This is done by observing Equation (13),
where, 푠푘 represents the transmitted symbol, and,
푠 푘 represents the outcome of that transmitted symbol.
Thus the system is in a correct state if,
푠 푘 = 푠푘 (15)
And the system is in an incorrect state if,
푠 푘 ≠ 푠푘 (16)
Keeping in mind the above conditions, it can be shown that,
1 푑 푃 ≤ 푇(퐷, 푧) 푧 = 1 (17) 푏 푛 푑푧
34
A tighter approximation on the bound can be shown as [15],
2 1 푛퐸 푑 2 푑 푏 푓푟푒푒 −푑푓푟푒푒 푃푏 ≤ 푒푟푓푐 퐷 푇(퐷, 푧) 푧=1 (18) 2푛 푁0 4 푑푧
The transfer function can be represented as,
1 푇 퐷, 푧 = 푉푡 퐼 − 퐴 −1푊 (19) 푚
The derivative of Equation (19) can be shown as [15],
푑 1 푡 −1 −1 푇 퐷, 푧 = 푉 퐼 − 퐴 1 퐴 (1) 퐼 − 퐴(1) 푊 (20) 푑푧 푧=1 푚
Where, m is the number of correct states identified.
V and W are defined as vectors whose dimension is m2 + m and the elements of that vector take values of either 0 or 1.
A(i) can be defined as a matrix with a dimension of (m2 + m) * (m2 + m) with elements
(Sk, Sk+1).
Thus, a(Sk, Sk+1) can be represented as,
1 2 푤(푈푘 ) 훿 (푆푘 ,푈푘 ) 푢푘 ∈푈푘 푛 푧 퐷 ; 푖푓 푈푘 푖푠 푛표푛 − 푒푚푝푡푦 푠푒푡 푎 푆푘 , 푆푘+1 = 2 (21) 0; 표푡푕푒푟푤푖푠푒
Where,
푈푘 = 푢푘 , 푢 푘 푠 푘 , 푢 푘 ≠ (푠푘 , 푢푘 ), 푆푘 ⊄ 푆푑 , 푆푘+1 퐺 푆푘 , 푈푘 ⊄ 푆푡 (22)
35
Where, 푆푡 and 푆푑 are a representation of all the true and dummy sets of the correct pair states respectively.
And,
퐺 푆푘 , 푈푘 ≜ (푔 푠푘 , 푢푘 , 푔 푠 푘 , 푢 푘 ) (23)
Eventually, the free Euclidian distance of the code can be represented as [6],
푇(2퐷,1) 푑2 = lim log (24) 푓푟푒푒 퐷→0 2 푇(퐷,1)
3.2 Relation Between dfree and BER
In general, the relationship existing between dfree and BER is that: for large signal- to-noise ratios, the dfree is synonymous with minimizing the average bit error probability.
This relation is true, as long as the distance between the separate points in a signal set are not reduced considerably and aren‟t too small. When using asymmetric constellations, the optimization of the asymmetry condition might lead to producing signals sets in which the signal points on the constellation might draw extremely close to each other, thus resulting in the signals to merge together as the SNR approaches a higher value, thus resulting in the distance between the signal points to worsen as the SNR approaches infinity. In case of such an event, the outcome is a trellis code which is completely undesirable. The merging of the signals provides a trellis code with multiple longer paths than the length of the paths actually required. The longer paths have a squared distance equal to or larger than the squared free distance. This implies that the error probability
36 can no longer be approximated correctly by the first approximation. Also, since the error probability and the free distance are inter-related, it implies that the free distance can no longer serve as the only parameter in determining the coding gain.
Another important factor that needs to be kept in mind when the signal points keep drawing closer and eventually tend to merge together with the increasing SNR, besides the undesirable trellis codes is, that there are other adverse practical affects too.
In this thesis we deal with Phase Shift Keying, and considering a case of PSK, it can be noticed that, when the spacing between the signals is close and the decoding takes place in such a scenario, the system is highly prone to be affected by phase jitter. The primary reason, phase jitter can be introduced in such a scenario is due to imperfect carrier synchronization. Also, the drawing of signal points extremely close could cause a delay in the detection of the correct path pattern for the trellis, thus resulting in a significant increase in the time to build up the distance along the error path. This condition can eventually result in the requirement of a large trellis memory in order to accommodate the increase in the time for the detection of the path, thus requiring larger buffer size, which in turn increases the complexity and cost of the system [15].
3.3 Relation Between Phase Jitter and Error Probability
As mentioned earlier, the phase jitter can have an adverse effect on the decoding process thus in turn resulting in detrimental error probability results. One way of analyzing the extent to which the phase jitter is affecting the error probability is by analyzing two cases, one where the bit rate is the lowest and the second where the bit rate
37 is the highest. Once we find these two figures, we can obtain a set of variable and possible data rates by interpolating these two extremes.
From observation, it is concluded that the lowest data rate occurs when the result of the product of phase loop bandwidth BL and time period of each symbol, also known as symbol time TS, is extremely larger than one, which means the product results are very large, i.e.,
퐵퐿푇푠 ≫ 1 (25)
퐸푏 Let us consider that, 푃푏( ) is the bit error probability under normal 푁0 circumstances, which means that phase jitter has no significant affect on its outcome, then, using the Tikhonov function, the average bit error probability can be shown to be as
[15],
2 퐸푏 퐼1 (휌) 푃푏 = 푃푏 2 (26) 푁0 퐼0 (휌)
Where, 퐼푛 (∙) for n = 0, 1, are the zeroth and first order modified Bessel functions,
휌 : loop SNR
As mentioned earlier, the low data rates are a result of the product of bandwidth and symbol time. Similarly, the high date rate occurs due to the product of the loop bandwidth and the time taken by the decoder to decide, which can be represented as TM, is extremely small, which implies that the product has to be very less than one, thus,
퐵퐿푇푀 ≪ 1 (27)
38
Wherein, the decoder decision time, TM, can be represented as the product of the symbol time and the buffer size of the decoder, it terms of symbol size,
Therefore,
TM = TS x buffer size of decoder (in terms of symbol size)
Thus after considering the parameters mentioned above, the bit error probability can be represented as [15],
휋 퐸푏 2 exp (휌 푐표푠휃 ) 푃 푏 = 푃푏 푐표푠 휃 푑휃 (28) −휋 푁0 2휋퐼0(휌)
Where, 휃 is the phase error
Thus as mentioned previously, the possible bit error probability for different and variable data rates can be found out by interpolating the equations (27) and (28).
39
Chapter 4
Performance Improvement for PSK Schemes
4.1 Asymmetric 4-PSK, 2 State, Rate 1/2
The signal set partitioning for an asymmetric 4-PSK can be shown in Figure 4.1.
As can be seen in Figure 4.1, the original signal set consists of four symbols, two of them separated by an angle ∅. The signals are labeled from 0 to 3. Upon set partitioning we obtain the signal subsets with just one signal in each of them, thus indicating that the distance between the signals has been maximized.
We consider that each input generates two outputs, thus the trellis emerging from each state has two choices to reach a certain signal point. There could be transitions between similar states where the trellis reaches the same state after emerging, or there could be transitions between different states where the trellis reaches a different state after emerging. If the trellis emerges and reaches the same state the shortest event path, meaning the path which a trellis chooses to start from a specific state and reach that state
2 again, will be limited to one. In such a case the maximum value of 푑푓푟푒푒 is the Euclidian distance between the pair of signal points.
40
2 4 1
∅ ∅ 4 푠푖푛2 2 3 0 ∅ 4 푐표푠2 2
1 1
0 3
1 2
0 3 11 00 10 01
Figure 4.1: Set partitioning of asymmetric 4-PSK
As can be seen in Figure 4.2, there are two states emerging from every point. If we consider the event of a parallel path, wherein the state trellis emerging from state 0 goes back to state 0, in such a case the squared distance between the signal points of that pair
(which can be either between 0,2 or 1,3) is 4.0 (This can be observed from the Figure 4.1 where the signals are set partitioned).
41
The state transition matrix for the trellis of Figure 4.2 can be given as,
S0 S1
S0 0 2 T =
S1 1 3
The trellis structure is represented as shown below,
0 0 S 0 0 0 0
2
1 1
3 S1 1 1
Figure 4.2: Trellis diagram of Asymmetric 4-PSK, 2 State, Rate 1/2
If the trellis emerging from state 0 goes to state 1, where there is no prallel path, the shortest distance for error path is two, since it requires transition from state 0 to state 1 and back to state 0. If this transition is followed, the error event path chosen (signal 2 will follow by signal 1) is shown in Figure 4.2. On observing, it is clear that the value of
2 2 푑푓푟푒푒 is larger in this case, since the very first path that is choosing signal 2 has a 푑푓푟푒푒
2 equal to 4. In addition to this, the 푑푓푟푒푒 for choosing signal 1 will be added too. Thus, this
42 choice of transition is better compared to the parallel path transition, where the distance
2 푑푓푟푒푒 is maximized.
On observing Figure 4.1, it can be noted that the data rates on the two channels are equal and the symbol transition times are aligned, but the powers are unbalanced. The ratio of powers between the I and Q cannel can be related to the angle ∅ that defines the asymmetry [15].
If we denote the ratio of the powers on channel I and Q by 훼, then,
∅ 훼 = 푡푎푛2 (29) 2
Where, ∅ is the angle of asymmetry.
The transfer function bound for the trellis code can be represented as [15],
푧퐷4(1+2훼)/(1+훼) 푇 퐷, 푧 = (30) 1−푧퐷4/(1+훼)
Where D is the bhattacharya bound and can be defined by equation (12).
Substituting the transfer function from Equation (30) into Equation (18), we get the bound on the bit error probability as [15],
0.5 푒푟푓푐 − 4(1+2훼)/(1+훼) ln 퐷 푃푏 = 2 (31) 1−퐷4/(1+훼)
The radius of the circle in Figure 4.1 is one, which implies that,
PI + PQ = 1 (32)
43
Where, PI and PQ denote I channel and Q channel power respectively.
The value which optimizes 훼 can be given as the value which minimizes the bound on Pb of Equation (31). Thus this value can be given as [15],
훼 ≈ −4(ln 퐷/ ln 3) − 1 (33)
Substituing the optimumvalue of 훼 obtained from Equation (33) in Equation (31), we get the optimum upper bound for the 4-PSK constellation.
9 2퐸푏 푃푏 ≤ 푒푟푓푐 − ln 3 (34) 8 푁0
If we consider the case of symmetrical constellation and apply those parameters
(∅ = 휋 2 , 훼 = 1) to the bit error probability bound of Equation (31), we obtain the bit error probability for a symmetric constellation, which is given as [15],
0.5푒푟푓푐 ( 3퐸 2푁 푏 0 (35) 푃푏 ≤ 퐸 2 1−exp (− 푏 ) 2푁0
Also, the bit error probability for an uncoded PSK can be given as,
1 푃 = 푒푟푓푐 퐸 /푁 (36) 푏 2 푏 0
In addition, the denominator of Equation (36) can be approximated to unity at very high
Eb/N0 [15]. Thus the bit error probability of symmetric constellation can be given as,
푃푏 ≤ 0.5푒푟푓푐( 3퐸푏 2푁0 (37)
44
The gain of the coded (symmetrical) 4-PSK over uncoded 4-PSK can thus be represented as,
3 10 log ( ) = 1.76 푑퐵 (38) 10 2
Now let us determine the gain of asymmetric PSK over uncoded PSK. The important parameter for this case, will be the free distance.
2 Let us denote 훿푗 as the squared distance from a reference signal point which, in our case is point 0, to the other signal points, which in the case of 4-PSK are 1, 2 and 3.
Then, the squared distances between the reference signal point and the other signal points can be given as [15],
∅ 훿2 = 4 푠푖푛2 (39) 1 2
2 where 훿1 is the distance between signal point 0 and 1.
2 훿2 = 4 (40)
2 where, 훿2 is the distance between signal point 0 and 2.
∅ 훿2 = 4 푐표푠2 (41) 3 2
2 Where, 훿3 is the distance between signal point 0 and 3.
2 From the trellis of Figure 4.2, we observe that for length 2, we have, the 푑푓푟푒푒 as the addition of distance to signal point 2 and signal point 1.
45
2 Thus, 푑푓푟푒푒 for asymmetric constellation can be represented as,
2 2 2 푑푓푟푒푒 = 훿2 + 훿1 (42)
Thus, substituing the values from Equation (39) and Equation (40), we get,
2 ∅ 2 ∅ 푑2 = 4 푠푖푛 + 4 = 4(1 + 푠푖푛 ) (43) 푓푟푒푒 2 2
By substituting Equation (29) into Equation (43) we get a generalized equation for any asymmetric case [15],
훼 푑2 = 4(1 + ) (44) 푓푟푒푒 1+훼
휋 Also, if we assume an angle of in Equation (43) for the symmetrical case, we have, 2
2 푑푓푟푒푒 = 6 (45)
2 Thus the improvement can be shown as ratio of 푑푓푟푒 푒 for asymmetrical PSK to symmetrical PSK. This can be represented as,
2(1+2훼) 10 log (46) 10 3(1+훼)
8 푔푎푖푛 = 10 log = 1.25 푑퐵 (47) 푎푠푦푚 /푐표푑푒푑 10 6
8 푔푎푖푛 = 10 log = 1.76 푑퐵 (48) 푎푠푦푚 /푢푛푐표푑푒푑 10 4
46
4.2 Asymmetric 4-PSK, 4 State, Rate ½
Figure 4.3 represents the trellis diagram for a 4-PSK, 4 State, Rate 1/2 TCM scheme. The trellis structure represents 4 states, and their corresponding transitions at each length interval, thus providing the desired Euclidian distance from the paths (Path I and Path II) in Figure 4.3.
0 0 0 0 S0 00 00 00 00 00
2 2
S1 01 01 01 01 01
1 2
10 10 10 10 S2 10
3 3
S3 11 11 11 11 11
Figure 4.3: Trellis structure of 4-PSK, 4 State, Rate 1/2 TCM
47
The state transition matrix, for the trellis structure represented in Figure 4.3, is given as,
S0 S1 S2 S3
S0 0 2
S1 1 3 T =
S 2 0 1 2
S3 3 1
The encoder diagram for the rate 1/2, 4 PSK scheme is as shown in Figure 4.4,
4
X1 S2 S1 U 1 P
X2 S
K
Figure 4.4: TCM encoder for 4-PSK, 4 State
48
As seen in Figure 4.4, there is one input entering the system, and eventually resulting in two outputs, thus this is a rate ½ encoder. Also, the number of state delays are
2, which means the number of states = 22 = 4. Thus the encoder in Figure 4.4 represents the rate 1/2, 4-state TCM.
As seen in Figure 4.3, there are two possibilities for a trellis to merge from a specific state. One possibility being that of, the parallel path transition, which has been discussed earlier. The second possibility is that of, emerging and going to a different state. As observed in Figure 4.3, besides the condition of parallel path, there exist two paths which can provide us with the shortest paths.
Path I: (2, 3, 3, 2)
Path II: (2, 1, 2)
The squared Euclidian distances for the paths can be given as,
∅ 푑2 2,3,3,2 = 4 + 8푐표푠2 + 4 (49) 2
And,
∅ 푑2 2,1,2 = 4 + 4푠푖푛2 + 4 (50) 2
In order to obtain the optimum value of ∅, we equate Equation (49) and Equation (50),
Therefore, we get,
∅ 푡푎푛2 = 2 , ∅ = 1.91 푟푎푑 (51) 2
49
Thus substituting the value of ∅ in Equation (49), we get,
1 푑2 = 4 + 8 + 4 = 10.67 (52) 푓푟푒푒 1+2
Also, when we assume conditions for symmetry (∅ = 휋/2), Equation (50) gives,
1 푑2 = 4 + 4 + 4 = 10.00 (53) 푓푟푒푒 2
From Equation (52) and Equation (53), the gain can be observed as,
10.67 푔푎푖푛 = 10 log = 0.28 푑퐵 (54) 푎푠푦푚 /푐표푑푒푑 10 10
Also, the gain with respect to uncoded PSK can be given as,
10.67 푔푎푖푛 = 10 log = 4.26 푑퐵 (55) 푎푠푦푚 /푢푛푐표푑푒푑 10 4
50
4.3 Asymmetric 4-PSK, 8 State, Rate 1/2
Given below is the state transition matrix for the trellis diagram of a 4-PSK, 8
State, Rate 1/2, TCM scheme. The matrix gives us the corresponding transition of the states at different lengths. The state transition matrix for a rate 1/2, 4 PSK can be represented as,
S0 S1 S2 S3 S4 S5 S6 S7
S0 0 2
S1 3 1
S2 2 0
S 1 3 T = 3
S4 2 0
S5 1 3
S6 0 2
S7 3 1
Figure 4.5 represents the trellis diagram for an 8-state, 4-PSK. There are two possibilities for a transition to emerge from every state. Besides the parallel path, there are three possible paths which can provide the shortest distance emerging from a specific state and returning back to it. The 3 possible paths for shortest distance from Figure 4.5 can be shown as,
Path I: (2, 3, 2, 2)
Path II: (2, 1, 1, 0, 2)
Path III: (2, 1, 3, 3, 0, 2)
51
The trellis structure for rate 1/2, 4 PSK, 8 State can be shown as,
0 0 0 0 0 0
S0
2 2
S 1
1 3 2
S2
2 2
S3
S4
1 0 S5
0
S6
3 3
S7
Figure 4.5: Trellis diagram for 4-PSK, 8 State, Rate 1/2
52
The squared Euclidian distance for these paths can be given as,
∅ 푑2 2,3,2,2 = 12 + 4푐표푠2 (56) 2
∅ ∅ 푑2 2,1,1,0,2 = 8 + 8푠푖푛2 − 4푐표푠2 (57) 2 2
∅ 푑2 2,1,3,3,0,2 = 8 + 8푐표푠2 (58) 2
Thus, equating Equation (57) and Equation (58) together, we can obtain the optimum value of ∅,
Therefore, ∅ = 1.23 푟푎푑.
2 Thus, substituting the optimum value of ∅ in Equation (58), we get the value of 푑푓푟푒푒 as,
2 푑푓푟푒푒 /푎푠푦푚 = 13.34 (59)
Also,
2 푑푓푟푒푒 /푠푦푚 = 12.00 (60)
Therefore,
13.34 푔푎푖푛 = 10 log = 0.46 푑퐵 (61) 푎푠푦푚 /푐표푑푒푑 10 12
13.34 푔푎푖푛 = 10 log = 4.77 푑퐵 (62) 푎푠푦푚 /푢푛푐표푑푒푑 10 4
53
4.4 Asymmetric 8-PSK, 4 State, Rate 2/3
The trellis diagram for asymmetric scheme is represented in Figure 4.6. As seen in figure, each state has 4 transitions emerging from it. All the possible transitions from the corresponding states can be as represented in the state transition matrix described below.
0 0 0
S0
4 6 2 1
1 5 S1
3 7 0 1
0 4
S2 6
2
5 7
S3 3
Figure 4.6: Trellis diagram of Asymmetric 8-PSK, 4 State, Rate 2/3
54
The state transition matrix for the trellis diagram for Figure 4.6 can be given as,
S0 S1 S2 S3
S0 0 4 2 6
T = S1 1 5 3 7
S2 4 0 6 2
S3 5 1 7 3
The corresponding encoder diagram for the rate 2/3, 8 PSK scheme is shown in Figure
4.7,
U1 X1
8
S1 S2 X2
P U2 S
K
X3
Figure 4.7: TCM encoder for 8-PSK, 4 State
55
As seen in Figure 4.7, there are two inputs entering the system, and eventually resulting in three outputs, thus this is a rate 2/3 encoder. Also, the number of state delays are 2, which means the number of states = 22 = 4. Thus the encoder in Figure 4.7 represents the rate 2/3, 4-state TCM.
The set partitioning of an 8-PSK can be shown as in Figure 2.10.
On observing Figure 4.6, we can see that the error event path leading back to State 0 (S0) after a length of interval three is,
Path I: (2,0,1)
The squared Euclidian distance for this path can be given as,
푑2 2,0,1 = 4 − 2푐표푠∅ (63)
On testing for different values of ∅, between 0 and 휋/2, we obtain the maximum value
2 for 푑푓푟푒푒 as,
2 푑푓푟푒푒 = 4 − 2 (64)
The case of 8-PSK does not provide substantial gain using the constellation as shown in [15], but on slightly rotating the constellation and maintaining the same parameters as described above, with a rotation angle of ∅ = 0.39, a sufficient amount of improvement can be obtained. The improvement in performance can be observed in
Figure 5.2.
4 푔푎푖푛 = 10 log = 3.01 푑퐵 (65) 푎푠푦푚 /푢푛푐표푑푒푑 10 2
56
4.5 Asymmetric 8-PSK, 8 State, Rate 2/3
The encoder for the rate 2/3, 8 PSK, 8 State can be given as shown in Figure 4.8.
As can be seen from Figure 4.8, there are two inputs to the encoder, eventually resulting
in 3 outputs, thus exemplifying a rate 2/3 encoder.
S1 X1
8 U1
S S X 2 3 2 P U2 S X3 K
Figure 4.8: TCM encoder for 8-PSK, 8 State, Rate 2/3
S0 S1 S2 S3 S4 S5 S6 S7
0 4 2 6 S0
1 5 3 7 S1
4 0 6 2 S2 T = 5 1 7 3 S3
2 6 0 4 S4
3 7 1 5 S5
6 2 4 0 S6
7 3 5 1 S7
57
The state transition matrix can be given as seen above. The states S0 – S7
represent the eight states of the 8-PSK, 8 State TCM scheme represented in this section.
The transition matrix shown above represents the state transition corresponding to an
input to every given state. Also the number of state delay blocks, as can be seen in Figure
4.8, are three, thus it represents a rate 2/3, 8 State encoder representation .
The trellis structure can be shown as,
0 0 0 0 S0
2 S1 0
=
6 0 S2 2 1
S3
S 4 7 6
S 5
S6
S7
Figure 4.9: Trellis structure of 8-PSK, 8 State, Rate 2/3
58
As seen in Figure 4.9, there are two survivor paths originating from the trellis diagram.
Path I: (6, 7, 6)
Path II: (2, 0, 1, 2)
The squared distance for the respective paths can be given as,
휋 ∅ 푑2 6,7,6 = 4 + 4푠푖푛2 − = 6 − 2푠푖푛∅ (66) 4 2
And,
∅ 푑2 2,0,1,2 = 4 + 4푠푖푛2 = 6 − 2푐표푠∅ (67) 2
휋 Thus, by using the above two equations, we can obtain the value of ∅ = . Thus the 4
2 푑푓푟푒푒 can be given as,
2 푑푓푟푒푒 = 6 − 2 = 4.586 (68)
2 Since the 푑푓푟푒푒 is larger than 4 (which is the case for symmetrical constellation), it can be observed that there is a gain provided by the asymmetrical constellation. Also, the gain with respect to the uncoded scheme can be given as,
6− 2 푔푎푖푛 = 10 log = 3.60 푑퐵 (69) 푎푠푦푚 /푢푛푐표푑푒푑 10 2
59
4.6 Asymmetric 8-PSK, 16 State, Rate 2/3
This section highlights a Rate 2/3, 8-PSK, 16 State TCM scheme. The asymmetric
constellation can be constructed by adding a rotated 4-PSK constellation to an original 4-
PSK constellation. The angle that maximizes the performance for the 16-State TCM
scheme, is represented in Equation (72).
S1 S2 U 1
S3 S4 U2
8
X1
P
X2 S
K X3
Figure 4.10: TCM encoder for 8-PSK, 16 State, Rate 2/3
The encoder for rate 2/3, 8 PSK can be given as seen in Figure 4.10. There are two inputs
to the encoder, corresponding to three outputs, which eventually are mapped using an 8-
PSK signal mapper. The number of state delays are four, thus Figure 4.10 represents the
TCM encoder for a rate 2/3, 16-PSK, 8 State scheme. Also, the trellis diagram for this
60
scheme can be seen as shown in Figure 4.11. From each state, there are four transitions,
eventually resulting in two survivor paths.
The transition matrix for 16 state can be given as,
S0 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15
S0 0 4 2 6
S1 1 5 3 7
S2 4 0 6 2
S3 5 1 7 3
S4 2 6 0 4
S5 3 7 1 5
S6 6 2 4 0
S7 7 3 5 1 T = S8 4 0 6 2
S9 5 1 7 3
S10 0 4 2 6
S11 1 5 3 7
S12 6 2 4 0
S13 7 3 5 1
S14 2 6 0 4
S15 3 7 1 5 Path I represented in the trellis diagram of Figure 4.11 was discovered by Ungerboeck,
but Path II shows gain over the symmetric constellation, but it is not identified until we
examine the trellis until a longer length is selected [15].
Path I: (6, 1, 7, 2)
61
Path II: (2, 0, 1, 1, 0, 1, 6)
0 0 0 0 0 0 0 S0
S1 6
S2 2 2 S3 1 S4
S5 7 1
6 S6
S 7 0 S8
S9 0
S10 1
1 S11
S12
S13
S14
S15
Figure 4.11: Trellis structure for 8-PSK, 16 State, Rate 2/3
62
The squared Euclidian distances for the paths can be given as,
2 푑푓푟푒푒 6,1,7,2 = 8 − 2(푠푖푛∅ + 푐표푠∅) (70)
And,
2 푑푓푟푒푒 2,0,1,1,0,1,6 = 10 − 6푐표푠∅ (71)
Also, on equating equations (70) and (71), together, we get,
∅ = 0.644 푟푎푑 (72)
2 푑푓푟푒푒 = 5.2 (73)
Also,
5.2 푔푎푖푛 = 10 log = 0.03 푑퐵 (74) 푎푠푦푚 /푐표푑푒푑 10 5.1
5.2 푔푎푖푛 = 10 log = 4.15 푑퐵 (75) 푎푠푦푚 /푢푛푐표푑푒푑 10 2
4.7 Asymmetric 16-PSK, 8 State, Rate 3/4
The TCM encoder for a 16-PSK, rate 3/4 scheme is shown in Figure 4.12. There are three inputs to the encoder, eventually resulting in four outputs, which can be mapped using a 16-PSK signal mapper. Also, the trellis diagram corresponding to the state transition matrix is represented as in Figure 4.13. There are 3 survivor paths, but as pointed out previously in Section 4.6, the third survivor path does not show until the trellis is examined for a larger number of lengths.
63
U1
X 1
16 U2 S1 X2
P X3 S
S3 K U3 S2
X4
Figure 4.12: TCM encoder for 16-PSK, 8 State, Rate 3/4
The state transition matrix can be given as,
S0 S1 S2 S3 S4 S5 S6 S7
S0 A0 A2
S1 A3 A1
S2 A2 A0
S3 A1 A3 T = S4 A2 A0
S5 A1 A3
S6 A0 A2
S7 A3 A1
64
Where, A0 = 0, 4, 8, 12
A1 = 3, 7, 11, 15
A2 = 2, 6, 10, 14
A3 = 1, 5, 9, 13
Figure 4.13 represents the trellis diagram for a 16-PSK, State 8, Rate 3/4 TCM scheme.
There are three survivor paths that can be identified, when we examine the trellis for a length of seven. The three survivor paths can be given as,
Path I: (2, 1, 2, 2)
Path II: (2, 15, 15, 0, 2)
Path III: (2, 1, 0, 1, 15, 0, 2)
The free Euclidian distance corresponding to the paths can be shown as,
2 푑푓푟푒푒 2,1,2,2 = 8 − 3 2 − 2푐표푠∅ (76)
휋 푑2 2,15,15,0,2 = 8 − 2 2 − 4cos( − ∅) (77) 푓푟푒푒 4
휋 푑2 2,1,0,1,15,0,2 = 10 − 2 2 − 4푐표푠∅ − 2푐표푠( − ∅) (78) 푓푟푒푒 4
On equating Equation (77) and Equation (78) we get,
∅ = 0.324 푟푎푑 (79)
2 푑푓푟푒푒 = 1.59 (80)
Thus,
1.59 푔푎푖푛 = 10 log = 0.32 푑퐵 (81) 푎푠푦푚 /푐표푑푒푑 10 1.47
1.59 푔푎푖푛 = 10 log = 4.33 푑퐵 (82) 푎푠푦푚 /푢푛푐표푑푒푑 10 0.59
65
0 0 0 0 0 0 0
S0
2 2 2 S 1 1 1 2 2 2
S2 15
2
S3
0 1
S4 15 15
S5 0 0
S 6
S7
Figure 4.13: Trellis structure for 16-PSK, 8 State, Rate 3/4
66
Chapter 5
Results and Discussions
5.1 Simulation Results
For performance evaluation, we have simulated the performance of symmetric and asymmetric TCM using Matlab. The simulation model consists of various sections relating to TCM encoder, TCM decoder, constellation assignment, and the model corresponding to the present and next states for a convolutional encoder.
The following considerations are made for simulation purposes;
1) TCM provides highly efficient transmission quality over the band-limited channels,
thus the simulation is performed under Additive White Gaussian Noise (AWGN)
conditions, to test the performance of asymmetric constellation over symmetric
constellations.
2) The code rate of a scheme is evaluated from the Octal Code Generator matrix, which is
generated from the specific encoder corresponding to an individual scheme.
3) The asymmetric constellation is formed by adding a rotated „M/2‟ constellation to the
original „M/2‟ constellation, where the rotation angle is the angle which maximizes the
performance of the asymmetric constellation.
67
4) The decoding is carried out using the Viterbi decoding algorithm, which is described
in detail in Section 2.2.4.
5) The performance is evaluated with respect to BER improvement over Eb/No.
Asymmetric Constellation: 4PSK, 4 State, Rate 1/2 1
0.8
0.6
0.4
0.2
0
Quadrature -0.2
-0.4
-0.6
-0.8
-1 -1 -0.5 0 0.5 1 In-Phase
Figure 5.1: Asymmetric Constellation for 4-PSK, 4 State, Rate 1/2
Figure 5.1 shows the asymmetric constellation employed for a 4-PSK, 4 State, Rate 1/2
TCM scheme. The angle of rotation is ∅ = 1.91 푟푎푑.
68
BER of 4-PSK, 4 State, Rate 1/2 -2 10 Symm 4-PSK, 4 State, Rate 1/2 Asymm 4-PSK, 4 State, Rate 1/2
-3
10 BER
-4 10
-5 10 3 3.5 4 4.5 5 5.5 6 6.5 7 Eb/N0[dB]
Figure 5.2: Comparison between symmetric and asymmetric constellation for 4- PSK, 4 State, Rate 1/2
Figure 5.2 shows the BER comparison of TCM scheme with reference to symmetric and asymmetric constellations for 4-PSK, 4 State, Rate-1/2. An octal generator matrix with values of [5 7] is used. There is a gain of about 0.3 dB with respect to symmetric constellation under AWGN environment.
The asymmetric constellation of 8-PSK, 4 State, with an angle of rotation
∅ = 0.39 푟푎푑 is shown in Figure 5.3.
The performance of 8-PSK, 4 State, Rate 2/3 TCM is illustrated in Figure 5.4. The difference of performance is clearly shown between the symmetric and asymmetric constellations. The constellation used is as shown in Figure 5.3, along with an octal generator matrix with values [1 0 0; 0 5 2] obtained from an encoder as shown in Figure
4.7. On performing simulation the asymmetric constellation represented by Figure 5.3
69 shows an improvement in performance compared to the symmetrical constellation under
AWGN channel conditions.
Asymmetric Constellation: 8-PSK, 4 State, Rate 2/3 1
0.8
0.6
0.4
0.2
0
Quadrature -0.2
-0.4
-0.6
-0.8
-1 -1 -0.5 0 0.5 1 In-Phase
Figure 5.3: Asymmetric Constellation for 8-PSK, 4 State, Rate 2/3
As observed in Figure 5.4, the performance of an asymmetric scheme is comparatively better than the symmetric scheme at most of Eb/No ranging from 3 dB to 7 dB. As Eb/No increases, the asymmetric scheme maintains a slightly better performance.
70
BER of 8-PSK, 4 State, Rate 2/3 -1 10 Symm 8-PSK, 4 State, Rate 2/3 Asymm 8-PSK, 4 State, Rate 2/3
-2 10
-3
10 BER
-4 10
-5 10 2 3 4 5 6 7 8 9 10 11 Eb/N0[dB]
Figure 5.4: Comparison between symmetric and asymmetric constellation for 8- PSK, 4 State, Rate 2/3
Asymmetric Constellation: 8-PSK, 8 State, Rate 2/3 1
0.8
0.6
0.4
0.2
0
Quadrature -0.2
-0.4
-0.6
-0.8
-1 -1 -0.5 0 0.5 1 In-Phase
Figure 5.5: Asymmetric Constellation for 8-PSK, 8 State, Rate 2/3
71
The asymmetric constellation for an 8-PSK, 8 State with an angle of rotation of
∅ = 1.07 푟푎푑 is as shown in Figure 5.5. As can be observed from the Figure 5.5, the signal spacing is asymmetric.
BER of 8-PSK, 8 State, Rate 2/3 -2 10 Symm 8-PSK, 8 State, Rate 2/3 Asymm 8-PSK, 8 State, Rate 2/3
-3 10
-4
10 BER
-5 10
-6 10 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 Eb/N0[dB]
Figure 5.6: Comparison between symmetric and asymmetric constellation for 8 PSK, 8 State, Rate 2/3
Figure 5.6 shows the performance of a symmetric 8-PSK, 8 State, Rate 2/3 TCM, compared to an asymmetric scheme. The constellation used for the rate 2/3, 8 state, is as shown in Figure 5.5. As observed, there is sufficient gain when using the asymmetric constellation as compared to the symmetric constellation. The octal generator matrix used for this case is [1 2 0; 4 1 2] which can be obtained from Figure 4.8.
72
The asymmetric constellation used for an 8-PSK, 16 State, Rate 2/3 TCM technique can be as shown in Figure 5.7. The angle of rotation used for the this constellation to optimize the performance of asymmetry is ∅ = 0.644 푟푎푑.
Asymmetric Constellation: 8-PSK, 16 State, Rate 2/3 1
0.8
0.6
0.4
0.2
0
Quadrature -0.2
-0.4
-0.6
-0.8
-1 -1 -0.5 0 0.5 1 In-Phase
Figure 5.7: Asymmetric Constellation for 8-PSK, 16 State, Rate 2/3
The performance of 8-PSK, 16 State, Rate 3/4 TCM under AWGN conditions for an octal matrix of [2 7 0; 5 1 2] is represented in Figure 5.8. The octal matrix is obtained from the encoder of Figure 4.10. The asymmetric scheme provides a gain of 0.3 dB over the symmetric scheme as observed in Figure 5.8. As the Eb/No increases, the performance improvement of the asymmetric scheme over the symmetric constellation is maintained.
73
BER of 8-PSK, 16 State, Rate 2/3 -1 10 Symm 8-PSK, 16 State, Rate 2/3 Asymm 8-PSK, 16 State, Rate 2/3
-2 10
-3
10 BER
-4 10
-5 10 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 Eb/N0[dB]
Figure 5.8: Comparison between symmetric and asymmetric constellation for 8- PSK, 16 State, Rate 2/3 Figure 5.9 shows the asymmetric constellation used in a 16-PSK, 8 State, Rate 3/4
TCM. The angle of rotation to optimize the performance for asymmetry, is given by
∅ = 0.324 푟푎푑.
74
Asymmetric Constellation: 16-PSK, 8 State, Rate 3/4 1
0.8
0.6
0.4
0.2
0
Quadrature -0.2
-0.4
-0.6
-0.8
-1 -1 -0.5 0 0.5 1 In-Phase
Figure 5.9: Asymmetric Constellation for 16-PSK, 8 State, Rate 3/4
BER of 16-PSK, 8 State, Rate 3/4 -1 10 Symm 16-PSK, 8 State, Rate 3/4 Asymm 16-PSK, 8 State, Rate 3/4
-2 10
-3
10 BER
-4 10
-5 10 2 3 4 5 6 7 8 9 Eb/N0[dB]
Figure 5.10: Comparison between symmetric and asymmetric constellation for 16- PSK, 8 State, Rate 3/4
75
Figure 5.10 represents the performance comparison between the symmetric and asymmetric performance of a 16-PSK, 8 State, Rate 3/4 TCM scheme, for an octal matrix represented by [1 0 0 0; 0 1 2 0; 0 4 1 2] which is obtained from the encoder of Figure
4.12. On observing Figure 5.10, it can be noticed that the asymmetric constellation provides a 0.32 dB gain compared to the symmetrical assignment.
Figure 5.11 shows the comparison between the symmetric and asymmetric constellation in the AWGN environments with respect to the uncoded scheme for 4-PSK,
4 State, Rate 1/2 TCM. The performance plot of the uncoded scheme is under AWGN channel conditions. As observed, the asymmetric TCM outperforms the symmetric TCM in the AWGN channel compared to the uncoded scheme. The performance with respect to uncoded scheme almost shows an improvement of 3 dB.
BER of 4-PSK, 4 State, Rate 1/2 compared to UNCODED 0 10 Symm 4-PSK, 4 State, Rate 1/2 Asymm 4-PSK, 4 State, Rate 1/2
-1 Theoretical 10 UNCODED
-2
10 BER -3 10
-4 10
2 3 4 5 6 7 8 9 10 Eb/N0[dB]
Figure 5.11: Comparison between symmetric and asymmetric constellation for 4- PSK, 4 State, Rate 1/2 TCM with respect to uncoded scheme
76
The difference between the symmetric and asymmetric constellation for an 8-
PSK, 4 State, Rate 2/3 TCM can be as shown in Figure 5.12. The performance evaluation under the Gaussian environment is compared with reference to the uncoded scheme. On observing Figure 5.12 it can be seen that the asymmetric scheme also outperforms the symmetric constellation under AWGN, but due to the presence of additive white noise of a higher variance, the performance suffers at lower Eb/No, but as the Eb/No increases the performance of the asymmetric scheme shows significant improvement compared to the uncoded scheme.
BER of 8-PSK, 4 State, Rate 2/3 compared to UNCODED 0 10
-1 10
-2
10 BER -3 10
-4 10 Symm 8-PSK, 4 State, Rate 2/3 Asymm 8-PSK, 4 State, Rate 2/3 Theoretical UNCODED
2 3 4 5 6 7 8 9 10 11 Eb/N0[dB]
Figure 5.12: Comparison between symmetric and asymmetric constellation for 8- PSK, 4 State, Rate 2/3 with respect to uncoded scheme
Figure 5.13 shows the comparison between the symmetric and asymmetric schemes under Gaussian environment for 8-PSK, 8 State, Rate 2/3 TCM. As can be
77 observed, the asymmetric scheme performs better compared to the symmetric constellation, and the performance improves at higher Eb/No.
The performance comparison for the 8-PSK, 16 State, Rate 2/3 TCM constellations, are shown in Figure 5.14. The performance of asymmetric and symmetric constellation is compared to the uncoded modulation technique. As can be seen in Figure
5.14, the asymmetric constellation provides better error performance compared to both the symmetric and uncoded technique. The performance improvement of an asymmetric constellation compared to an uncoded modulation technique is around 4.13dB.
Also, comparative performance for 16-PSK, 8 State, Rate 3/4 TCM, is shown in figure
5.15. The asymmetric constellation shows sufficient improvement in performance over the symmetric constellation in AWGN channel, and also shows significant improvement when compared to the uncoded scheme. The performance improvement when compared to an uncoded scheme is equivalent to approximately 4dB.
78
BER of 8-PSK, 8 State, Rate 2/3 compared to UNCODED 0 10 Symm 8-PSK, 8 State, Rate 2/3 Asymm 8-PSK, 8 State, Rate 2/3
-1 Theoretical 10 UNCODED
-2
10 BER -3 10
-4 10
2 3 4 5 6 7 8 9 10 Eb/N0[dB]
Figure 5.13: Comparison between symmetric and asymmetric constellation for 8- PSK, 8 State, Rate 2/3 with respect to uncoded scheme
BER of 8-PSK, 16 State, Rate 2/3 compared to UNCODED 0 10 Symm 8-PSK, 16 State, Rate 2/3 Asymm 8-PSK, 16 State, Rate 2/3
-1 Theoretical 10 UNCODED
-2
10 BER -3 10
-4 10
2 3 4 5 6 7 8 9 10 Eb/N0[dB]
Figure 5.14: Comparison between symmetric and asymmetric constellation for 8- PSK, 16 State, Rate 2/3 with respect to uncoded scheme
79
BER of 16-PSK, 8 State, Rate 3/4 compared to UNCODED 0 10
-1 10
-2
10 BER -3 10
-4 10 Symm 16-PSK, 8 State, Rate 3/4 Asymm 16-PSK, 8 State, Rate 3/4 Theoretical UNCODED
1 2 3 4 5 6 7 8 9 10 11 Eb/N0[dB]
Figure 5.15: Comparison between symmetric and asymmetric constellation for Rate 3/4, 8 PSK with respect to uncoded scheme Figure 5.16 illustrates the performance analysis of all the symmetric schemes for 4-PSK,
8-PSK, and 16-PSK TCM. The performance improvement in a scheme with higher number of states, for e.g 8-PSK, 16 State, shows improvement in error as the Eb/No increases, thus it can be noticed that the higher states yield a better error performance at a higher range of Eb/No.
80
BER comparison of 4,8 and 16-PSK Symmetric schemes -1 10 Symm 4-PSK, 4 State, Rate 1/2 Symm 8-PSK, 4 State, Rate 2/3 Symm 8-PSK, 8 State, Rate 2/3
-2 Symm 8-PSK, 16 State, Rate 2/3 10 Symm 16-PSK, 8 State, Rate 3/4
-3
10 BER
-4 10
-5 10 2 3 4 5 6 7 8 9 10 11 Eb/N0[dB]
Figure 5.16: Comparison of all Symmetric PSK schemes
Figure 5.17 shows the error performance of all the asymmetric schemes for 4-
PSK, 8-PSK and 16-PSK TCM. As can be seen from Figure 5.2, 5.4, 5.6, 5.8 and 5.10, the asymmetric scheme outperforms the symmetric constellation. In Figure 5.17 we show the comparison of all the asymmetrical schemes that have been simulated in this research.
As pointed out in Figure 5.16, the schemes with higher number of states, perform better at higher Eb/No, which can also be observed for the asymmetric constellations represented in Figure 5.17.
81
BER comparison of 4,8 and 16-PSK ASymmetric schemes -1 10 ASymm 4-PSK, 4 State, Rate 1/2 ASymm 8-PSK, 4 State, Rate 2/3
-2 ASymm 8-PSK, 8 State, Rate 2/3 10 ASymm 8-PSK, 16 State, Rate 2/3 ASymm 16-PSK, 8 State, Rate 3/4
-3
10 BER -4 10
-5 10
-6 10 2 3 4 5 6 7 8 9 10 11 Eb/N0[dB]
Figure 5.17: Comparison of all Asymmetric PSK schemes
82
Chapter 6
Conclusion and future work
6.1 Conclusion
In this thesis, the performance of asymmetric constellation in concatenation with
TCM is evaluated and compared to the conventional symmetric constellation method for
PSK schemes. The schemes used for theoretical analysis are, rate 1/2 - 2, 4, 8 PSK, rate
2/3 - 4, 8, 16 PSK and rate 3/4 - 8 PSK. Also, simulation is performed for rate 1/2 - 4
PSK, rate 2/3 - 4, 8, 16 PSK, and rate 3/4 - 8 PSK in AWGN environment and the corresponding results are observed. The parameter considered to evaluate the performance improvement is Bit Error Rate (BER). As observed in Chapter 5, the simulation results for the asymmetric schemes show an obvious improvement in performance over the conventional symmetric technique. The gain varies from 0.3 dB to
2 dB depending on the specific scheme, but the important point is that most schemes provide sufficient gain improvement over symmetric constellation, in addition to the flexibility in transmission of data. Also, the performance of the asymmetric schemes are compared to the corresponding uncoded techniques, and the results are observed. The uncoded schemes are simulated considering ideal channel conditions. In this case, the performance improvement varies from 1.8dB to 4.4dB. As, can be seen in Figures 5.10,
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5.11, 5.12 and 5.13, the asymmetric schemes in the AWGN environment outperform the uncoded schemes with a significant gain. The performance at lower Eb/No is affected by the high noise variance present in the AWGN environment coupled with the phase jitter experienced due to the asymmetric arrangement of the symbols. But with the increase in
Eb/No, the gain shows consistent improvement, thus providing an effective method of modulation at higher bit rates.
6.2 Future Work
This research can be extended in the future to test for more number of modulation schemes consisting of a large number of states. As it can be observed in the case of 8-
PSK, the gain showed a slight improvement with the increase in the number of states, thus the same criterion could be applied to other modulation techniques and the gain be calculated for higher number of states, for systems requiring complex and highly effective modulation techniques. Also, different types of constellation assignments can be tested to observe the difference in performance. For future research, asymmetric constellation concept can also be applied to higher Quadrature Amplitude Modulation techniques and the performance can be observed accordingly.
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