University of Central Florida STARS
Retrospective Theses and Dissertations
1984
AWQPSK : an optimum modulation technique for spread spectrum communication
Madjid A. Belkerdid University of Central Florida
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STARS Citation Belkerdid, Madjid A., "AWQPSK : an optimum modulation technique for spread spectrum communication" (1984). Retrospective Theses and Dissertations. 4728. https://stars.library.ucf.edu/rtd/4728 AWQPSK: AN OPTIMUM MODULATION TECHNIQUE FOR SPREAD SPECTRUM COMMUNICATION
by
Madjid A. Belkerdid
A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Electrical Engineering and Communication Science at the University of Central Florida Orlando, Florida
July 1984
Advisor: Dr. Brian Petrasko ABSTRACT
Quadrature phase shift keying (QPSK) and minimum shift keying (MSK) are the two most used M-ary modula tion techniques in Direct-Sequence (DS) Spread Spectrum
Communication systems. This thesis introduces a new modulation technique that can compete well with QPSK and
MSK in many applications. This new modulation technique, made up of a superposition of one QPSK signal and two amplitude weighted QPSK signals, is called Amplitude
Weighted Quadrature Phase Shift Keying (AWQPSK). It is found to have the same probability of error as QPSK and
MSK techniques. It has a higher bandwidth efficiency in bits/sec/Hz than QPSK and MSK. It has 99.99 percent of its energy within the null bandwidth ·and its sidelobes are 63 db down from the main lobe. Intersymbol inter ference (ISI) was simulated on an HP 9845 computer and was shown to be smaller than the ISI in a QPSK or an MSK signal. Two different implementation schemes are pre sented. ACKNOWLEDGMENTS
I wish to express my deep gratitude to my advisors
Dr. Brian Petrasko and Dr. Donald Malocha. I also wish to thank Laura Wiechel for her typing. Finally many special thanks to my wife and son for their patience and collaboration.
iii TABLE OF CONTENTS
LIST OF FIGURES • • vi
INTRODUCTION 1
I. DIGITAL COMMUNICATION SYSTEMS. 6
A. System Model 6
Transmitter and Receiver Model• • • . 6 Optimum Receiver • • • • • • . ••• 7 Matched Filter Receiver. . • • . • • 9 Correlator Receiver. • • • . • • • . 9
B. MODULATION SCHEMES USED IN SPREAD SPEC- TRUM COMMUNICATIONS. • ••••••• 9
PS K • . • • · • • • • • • . 10 CPFSK • • • • • • • • • • • • • • • • 10 QPS K . . • • . • • • • . • . • . 11 OQPSK. • • • . • • • . . 12 MS K • • • • • • • • • • • • • • • 13 A comparison of MSK and QPSK waveforms • 15
C. PERFORMANCE EVALUATION OF QPSK AND MSK • 19
Overview .•.... • . 19 Spectral Efficiency •• • 19 Probability of Error •••• • • • • 21 Intersymbol Interference ..•. . • • . 21
II. DESCRIPTION OF THE PROPOSED MODULATION TECH NIQUE: AWQPSK ••••••••••.•••• 30
A. Window Functions Overview. . 30
Raised Cosine Function •••••.••• 31 Blackman Function. • • • • . 32
B. Eigen Function • • 3 2
Time Domain ••• 32 Frequency Domain • • 3 3
iv C. AWQPSK ...... • 37 Quadrature Modulation •.•••• . •••• 37 Analysis of AWQPSK ••..• 39
III. EVALUATION OF AWQPSK • • • 4 1
A. Spectral Efficiency. • • • 41 B. Probability of Error • • • • 4 1 C. Intersymbol Interference • • • • 51 D. Percent Amplitude Modulation • • • • 52
IV. IMPLEMENTATIONS ••.• • • • 5 7
A. Parallel Scheme. . . .•.•.•.. 57 B. SAW Device Implementation . . ••.• 57
v. CONCLUSION • 61
Appendixes • 62
A. PROBABILITY OF ERROR FOR BINARY MODULA- TION TECHNIQUES •••••••••••.• 63
B. POWER SPECTRAL DENSITY FUNCTION OF RANDOM BINARY WAVEFORM • • . . • • • . .. 68
c. INTERSYMBOL INTERFERENCE MODEL • • 71
REFERENCES 74
V LIST OF FIGURES
1. Direct Sequence Spread Spectrum Transmitter System ...... • . . • . . • . . . 4
2. Frequency Hopped Spread Spectrum Transmitter Sys tern • . . • . . • • . • . . . . 5
3. Digital Communication System Model 8
4. QPSK Modulator Model • . 12
5. MSK Modulator Model .. • 16
6. MSK Time Domain Waveform • • 1 7
7. QPSK Time Domain Waveforms • • 18
8. QPSK Spectrum. • • • • • 21
9. MSK Spectrum • 22
10. MSK Spectrum Decomposition . 23
11. Filtered (ISI) QPSK Time Pulse (Linear Scale) . 26 12. Filtered (ISI) QPSK Time Pulse ( Log Scale) . . 27 13. Filtered (ISI) MSK Time Pulse (Linear Scale) . 28 14. Filtered (ISI) MSK Time Pulse ( Log Scale) . 29 15. Blackman Function Spectrum ...... 34 16. RF Eigen Function Time Domain Pulse ...... 35 17. Eigen Function Spectrum Decomposition 36
18. Eigen Function Spectrum. • 37
19. Phasor Diagram • • 4 3
20. Quadrature Correlator Receiver • • • 4 4
21. Filtered (ISI) AWQPSK Time Pulse (Linear Scale) • • • • • • . • • • • • • • • . • • . • 53
vi 22. Filtered (ISI) AWQPSK Time Pulse (Log Scale) • 54
23. Inphase AWQPSK Component • • 5 5
24. Quadrature AWQPSK Component • 55
25. AWQPSK Time Domain Waveform with UI = +l and u = +l 56 q ...... 26. AWQPSK Time Domain Waveform with UI = +l and u = -1 56 q ...... 27. Parallel implementation of a AWQPSK Modula- tion • • . • • . . • . • • • • • . . • • . • • 59
28. SAW Device Implementation of a AWQPSK Modula- tion • . • . • • . • • • . . . • • 60
29. Receiver Structure • • 6 3
30. Binary Correlator Receiver • 66
31. Integrate and Dump Receiver • • • • 66
32. Binary Random Waveform • 68
33. Example of a Received Pulse Train. • • 7 3
vii INTRODUCTION
Spread spectrum is a spreading of information energy in time and frequency beyond the required information bandwidth. This spreading operation brings the signal level below the noise level which makes a spread spec trum system capable of low probability of unwanted in tercept and high rejection of intentional or uninten tional jamming. Even though the signal to noise ratio is very low, the probability of error is low in a spread spectrum communication system.
There are two fundamental elements characterizing a spread spectrum communication system: a spectrally ef ficient modulation technique, and a pseudo-random pulse generator [1].
A spectrally efficient modulation technique is char acterized by its modulated waveform having most of its energy contained in a frequency band centered around the carrier frequency, and very little energy outside this band [ 2] ..
A pseudo-random pulse generator is used as a carrier in spread spectrum communication system. This generator introduces an element of unpredictability or randomness
(Pseudo-Randomness) in each of the transmitted waveforms 2
which is known to the intended receiver only [3]. ,.,
When this pseudo-random pulse generator is used in conjunction with a phase modulator, the resulting modu lated signal is called a direct sequence (DS) or a pseu do-noise (PN) spread spectrum signal [4]. When the modulator is a binary or an M-ary FSK, the resulting transmitted signal is called a frequency hopped (FH) spread spectrum signal [3]. ADS and an FH spread spec trum transmitter systems are depicted in Figure 1 and
Figure 2 respectively.
A new modulation technique is introduced. Its per formance parameters will be evaluated and compared to the same parameters of QPSK and MSK modulation tech niques. The parameters used for evaluation are spectral efficiency, probability of error, and intersymbol inter ference. Spectral efficiency is very important in a crowded spectrum. The new modulation technique has
99.99 percent of energy within the null bandwidth, the sidelobes are 63 db down which makes spectrum spillage to other cochannels very minimum.
The probability of error using a coherent correlator receiver is derived for this modulation technique and was found to be the same as for QPSK and MSK. Intersym bol interference arises from the spectrum truncation associated with finite bandwidths which generate time domain sidelobes (tails). ISI is generated by the over- 3
lapping tails of other pulses adding to the particular pulse which is examined at any one sampling time [1].
Intersymbol interference is simulated on an HP 9845 computer and was found to be less than the ISI produced by QPSK and MSK. -
Finally a parallel implementation and a surface acoustic wave (SAW) implementation of a modulator are presented. 4
DS Spread Spectrum Signal
BINARY _____,_,. MODULATOR DATA
Pseudo-Rand. Pulse Generator
Figure 1. Direct Sequence Spread Spectrum Trans mitter System. 5
FH Signal
BINARY DATA MODULATOR
FREQUENCY SYNTHESIZER
PSEUDO-RANDOM PULSE GENERATOR
Figure 2. Frequency Hopped Spread Spectrum Trans mitter System. I. DIGITAL COMMUNICATION SYSTEMS
A Spread Spectrum Communication system consists of a transmitter and a receiver. The transmitter consists of a modulator followed by a spectrum spreading circuit.
The receiver consists of a circuit which unspreads the spectrum followed by a demodulator.
The modulation technique used is evaluated by some key parameters, such as the bandwidth efficiency of the transmitted signal, intersymbol interference, and prob ability of error.
A. System Model
Transmitter and Receiver Model
The basic elements of a Digital communication system can be described by the block diagram depicted in Figure
3.
The modulator is the most important part of the transmitter. Examples of modulation techniques used in spread spectrum communication systems are given in Sec tion II.B.
6 7
·The transmitted signal is given by
s (t-(k-l)T) if = 0 1 bk z ( t) = (t-(k-l)T) if = 1 s 2 bk for (k-l)T < t < kT (t) and (t) are orthogonal energy signals having a s 1 s 2 time duration T=l/R where R is the symbol rate (BAUD rate) [5].
Optimum Receiver
The function of a receiver in a digital communica tion system is to differentiate between orthogonal sig nals in the presence of noise. The performance of the receiver is associated with the probability of error of the system. The receiver is referred to as optimum if the probability of error is minimized. A structure of an optimum receiver is presented in Appendix A. The matched filter receiver, and the correlator receiver are implementations of the optimum receiver. 8 \
BINARY DATA TRANSMITTER MODULATOR > FILTER
n(t)
CHANNEL
"1 tt RECEIVER DEMODULATOR z (t)+n(t) FILTER "O"
Figure 3. Digital Communication System Model. 9
Matched Filter Receiver
If the channel noise is white and Gaussian the above optimum receiver is a matched filter receiver [1]. The probability of error is a monotonically decreasing func tion of the signal to noise ratio. A matched filter by definition is a filter that maximizes the signal to noise ratio. A matched filter minimizes the probability of error and is hence an optil!!..~ receiver.
Correlator Receiver
A correlator receiver is also another form of an optimum receiver. As shown in Appendix A, it consists of a product-integrator or correlator for the binary case, or a bank of product-integrators for the M-ary case supplied with a set of coherent local reference signals. A correlator receiver, also referred to as an
Integrate and Dump (ID) receiver, is a very close ap proximation of a matched filter [1].
B. Modulation Schemes Used in Spread Spectrum Commun1cat1on
There are many modulation techniques that have been used in Spread Spectrum Communication Systems, some of which are, PSK, CPFSK, QPSK, OQPSK and MSK. PSK and
CPFSK are referred to as binary modulation techniques, 10
whereas QPSK, OQPSK and MSK are refered to as M-ary
(Multi-symbol) modulation techniques.
The most popular techinques used in spread spectrum are QPSK and MSK and hence the performance of the new modulation technique will be compared to the performance of QPSK and MSK. QPSK and MSK are presented below. The presentation is followed by a development of their per formance characteristics relative to the key parameters.
PSK
Phase-shift keying (PSK), also referred to as dis crete phase modulation, is a technique of transmitting binary data via an analog carrier over a bandpass chan nel. A PSK modulated signal can be represented by X(t)
= D(t) Cos 2xf t where D(t) is the binary data, and f C C is the carrier frequency. A binary one is transmitted as
A Cos 2xf t and a binary zero is transmitted as -A Cos2n C ft. C
CPFSK
Continuous-phase frequency-shift keying (CPFSK) or
FSK is another technique of transmitting binary data over a bandpass channel. Unlike PSK, in FSK the infor mation carried by the transmitted carrier is contained in its frequency rather than its phase. A binary one is transmitted as A Cos 2x(fc + Af)t and a binary zero is transmitted as A Cos 2x(fc - Af)t, where fc is the car- 11
rier frequency, and Af is the peak frequency deviation.
Both PSK and FSK require a transmission bandwidth of at least 2rb HZ, where rb is the binary data rate. In bandwidth limited situation, M-ary modulation techniques are used in order to increase transmission rates over fixed bandwidth at the expense of higher probability of error and more complex designs. QPSK, OQPSK and MSK are examples of M-ary modulation techniques.
In QPSK (Quadrature Phase Shift Keying) the infor mation carried by the transmitted wave is contained in the phase. The phase of the carrier has four possible values [51, such as ~/4, 3~/4, 5~/4, and 7~/4 as shown by
A Cos [ ( 2 ,tfct + ( 2 i -1 ) i)] 0 < t < T
0 otherwise
for i = 1, 2, 3, 4.
Using a simple trigonometric identity the above e 11uation can be written as
ACos [ ( 2i-1) ~1 Cos( 2 ~ f t) - C 4 - A Sin [ ( 2 i-1 ) ~1 Sin ( 2 ~f t) 0 < t < T - C 4 s. (t)= 1 0 otherwise 12
or it can be written as
Si(t} = A f u Cos ( 2 Jt f t} 1rect [ttz] C
- u rect Sin ( 2 Jt f t} J for O < t < T q [tt2] C
and 0 otherwise
where & can take on UI uq + 1 1 t-T/2 0 < t < T and rect [ = T 0 otherwise
From the above expression of QPSK, it is easily seen that the transmitted pulse is a rectangular pulse, whose fourier transform is a sine function. This pulse will be compared to the pulse used in MSK and the pulse used in the proposed modulation technique. A QPSK modulator is depicted in Figure 4.
The QPSK transmitted signal is either A Cos 21tfct,
A Sin 2nf t, -A Cos 2nf tor -A Sin 2nf t. C C C
OQPSK
Offset QPSK, also referred to as staggered QPSK, is a slight variation of QPSK. The difference is that the inphase dibit, u , is delayed by half a symbol duration 1 from the quadrature dibit, Uq· OQPSK is sometimes more desirable than QPSK in a non-linear enviionment [6].
All other parameters are exactly the same as for QPSK. 13
QPSK Signal u,
BINARY BIT TO DIBIT DATA CONVERTER
Uq
A5in (Wet)
Figure 4. OPSK Modulator Model.
MSK
MSK (Minimum Shift Keying) is a special case of
CPFSK (Continuous Phase Frequency Shift Keying) or FSK
[7,6]. The information carried by the transmitted wave is contained in both the frequency of the carrier and its continuous phase from symbol to symbol.
MSK is an M-ary FSK. Binary ones and zeros in an FSK system are transmitted as (t) or (t) with s 1 s 2 14
- ) A Cos 2~(f + Af)t for O < t < (t) C Tb s 1 - lo otherwise and A cos 2 Jt ( f - 4 f) t for O < t < = l C Tb (t) s 2 0 otherwise s 1 (t) and s 2 (t) are orthogonal signals over the interval 0 to Tb, where Tb= 1/rb = bit duration.
The orthogonality property is characterized by
s 2*(t) dt = O, where* denotes complex
~ I conjugate.
Therefore
Tb 2 f A Cos 2~(f + Af)t Cos 2~(f - Af)t dt = 0 Q C C
with f = k 1/T; i.e., there is an integer number C of carrier cycles per bit period). The above equation yields Sin 4~AfTb = 0 which leads to 2AfTb = n, where n is a positive non-zero integer. The spectrum of the FSK signal depends on Af. The larger Af the wider the spec trum. For maximum spectral efficienty, n equals one.
Therefore
n = 1 .r or 2Af Tb = 1/2. ) .,._.... ~,t/ 0 Since the above relation for the minimum value of n yields the minimum spectral width, the CPFSK wave- 15
form under this condition is referred to as Minimum
Shift Keying (MSK). With - f = l/4Tb the MSK signal can be written as
Si(t) = U Cos [2,r (fc ~ l/4Tb)t]
- { +l where U - to provide the carrier phase -1 continuity.
With a simple trigonometric identity Si(t) can be writ ten as
Si(t) = U Cos _2! t Cos w t + Sin 2-!t Sin wt. C C 4Tb 4Tb
Let T = 2Tb, where T is the symbol duration. The ex- pression becomes
C {1t t) ( Jr t) Si(t) = os-- Cos w t + u Sin Sin w t UI C q C T T where and Uq are the inphase and quadrature symbols u1 that can equal~ 1. In MSK the transmitted pulse is a sinusoidal pulse [8,9].
An MSK modulator is shown in Figure 5.
Comparison of MSK and QPSK waveforms.
MSK has a continuous phase and a constant envelope.
As seen in Figure 6 there is no amplitude modulation in the MSK signal.
A QPSK signal is also a constant envelope signal (no
AM) but phase is discontinuous, as seen in Figure 7. 16
vco
SPF Acos( ..t) fc + .~.. ...---~--&:aJ
A:os(,ft)
vco u, r, = ..!. T
llT TO 11nary Data DIBIT CONVIltTER
Uq
Figure s. MSK Modulator Model. 17
0 1 1 0 1 1 Binary I ➔ t Data I I I u, l I ➔ t Uq . I I l ~t
iAttV ~
Quadrat. Signal
Figure 6. MSK Time Domain Waveform. 18
0 0 0 0 0
Binary Data __l0
u, ------i------+----
Uq
In phase Signal Quadrat. Signal
QPSK /\ /\ ~ /\ /\ /\ /'. /\ /\ , Signal I\J\)V\.I\JV VV\
Figure 7. OPSK Time Domain Waveforms. 19
C. Performance Evaluation of QPSK and MSK
Overview
The key evaluation parameters for various modulation techniques are spectral efficiency, probability of error and intersymbol interference. Spectral efficiency is a measure of the compactness of the power spectrum of the transmitted signal [10]. A measure of the compactness of a modulation waveform's spectrum is the transmission bandwidth which contains most of the total power. The transmission bandwidth used in this dissertation is the bandwidth that contains 99.99% of the total power. The primary objective of spectral efficiency is to maximize the ratio of the data rate rb in bits/sec to the trans mission bandwidth BT in Hz. This ratio is defined as the bandwidth efficiency in bits/sec/Hz [10]. The pro bability of error is a measure of the performance in a noisy environment. The intersyrnbol interference is a measure of performance in high speed transmission and jitter in the receiver circuitry.
Spectral Efficiency
The baseband power spectral density functions of QPSK and MSK are respectively given by [11, 10, -12] 20
8MSK {f)
However, as shown in Appendix B, the Fourier transforms of the rectangular and sinusoidial pulses of QPSK and
MSK give sufficient information about the spectral ef ficiency of these modulation techniques. These pulse spectra obtained from Appendix B, are given by
FQPSK{f) = AT Sinc{fT)
FMSK {f) = 0.5AT Sinc{fT 0.5) +0.5AT Sinc{fT + 0.5)
An HP9845 computer is used to generate the spectrum of various modulating functions. A carrier frequency of
100 MHZ, and a symbol duration of .4 µsecs were used for illustration purposes. Figure 10 depicts the spectrum decomposition of the MSK waveform. The QPSK and MSK pulse spectra are depicted in Figures 8 and 9 respec tively. The first sidelobes of QPSK are only 13 db be low the main lobe, whereas in MSK the first sidelobes are 23 db down from the main lobe.
The parameter that describes the spectral effi ciency, is the bandwidth efficiency in bits/sec/Hz. An
HP 9845 computer was used to compute the above quantity.
It was found that the 99.99% bandwidth for QPSK is 10/Tb
Hz, and the bandwidth efficiency is rb/ 10/Tb = 1/10 21
bits/sec/Hz. The 99.99% bandwidth for MSK was found to be 2.75/Tb Hz, and the bandwidth efficiency is rb/2.75
/Tb= 1/2.75 bits/sec/Hz. The bandwidth efficiency is higher for MSK, and therefore MSK is a more compact and spectrally more efficient than QPSK.
There are many bandwidth criteria used in communi cation. The 99.99% bandwidth is the criterion used in this disseration.
Probability of Error
Using a correlator receiver model, the probability of error for a Quadrature Phase Shift Keying (QPSK), an
Offset QPSK (OQPSK), a Minimum Shift Keying (MSK) modu lation techniques can be shown to be [13, 2, 14]
The Q function is defined in Appendix A.
This result was expected since QPSK, OQPSK and MSK signals have identical phasor diagrams [13].
Intersymbol Interference As stated in Appendix c, ISI is due to the time domain sidelobes generated by filtering the communication channel. ISI is measured by sidelobe levels, and the zero-crossing spacing of these sidelobes 22
RECT rUNCTION CQPSK PULSE> SPECTRUM ..... r------~------1~ -11.ee -
-21.ee •
-31.ee
.; -.e.ee . -~ I -!e.ae - M i -11.ee • -11.ee •
-ee.ee •
-te.ae •
-aee.ae L---.a..•...____._.--J---,1•...__._L-•-....•----•J.-___.,.___.__.. ____ •__ _ &I.I II.I 76.8 14.I sz.e 1a,.e ••·• 11&.1 124.I 132.1 14e.e rR£0l£NCY Oltz>
Figure 8. OPSK Spectrum. 23
MSK SPECTRUM
-10.ee
-28.80
-Je.ee
,.. ◄ B.10 -m.,, ~ -se.ee i -1e.ee
-ee.ae
-se.ae
-1e0.ee se.e se.e 1s.e &4.e 12.e 100.e 1ee.e 11s.e 124.e 132.0 1 ◄ 0.e f"REQUENCY (Pl-lz)
Figure 9. MSK Spectrum. 24
MSK SPECTRUM DECOMPOSITION . +.28
+.18
+.15
+.13
+.18
I.... +.BB i +.116
+.13
+.81
-.12
-.M 18.I 18.1 7&.I 84.8 12.1 1aa.1 1•.1 Ill.I 124.I 132.1 l4e.e rREQlENCY (,tu)
Figure 10. MSK Spectrum Decomposition. 25
of the time domain pulse. The sidelobe generation was simulated on a VAX 750, for QPSK, MSK and AWQPSK. The spectra of these signals were truncated by an ideal
(Brick wall) filter at their respective null bandwidths.
The null bandwidth was chosen to demonstrate the relative effect of filtering. An Inverse Fast Fourier
Transform (IFFT) algorithm was used to create the time domain pulses with their respective.generated sidelobes.
Figure 11 depicts the simulated QPSK pulse in a linear scale whereas Figure 12 depicts it in a logarithmic scale. Figures 13 and 14 depicts the MSK pulse, both in linear and logarithmic scales. Note that the QPSK pulse sidelobes are about only 27 db down, whereas the MSK pulse sidelobes are about 31.7 db down. The sidelobe zero-crossings of the MSK pulse are about twice as fast as the QPSK pulse. These two parameters show that MSK performs better than QPSK in a limited channel size environment where Intersymbol Interference is critical
[ 10] • ISi CALCtn.ATIONS for QPSk I • 898 1 a a u ■ a a a a ■ ■ : ■ a a a a a ■ a a ■ a : u : a a : a &Q
.833
~ E L A .667 T · uI E A .509 H p L Tl .333 u c- E .1,1
• 880 1 C 2 I I -,.eee -J.333 -1.667 .eee 1.667 3.333 5.111 Tl"E I" SECONDS Figure 11. Filtered (ISi) QPSK Time Pulse (Linear
Scale) • I\) I 0) ISi CALCULATIONS for QP9K • 89 , , , , , , , , , , , , , , , m , , , , , , , , , , , , , , 1
-8.33
-16. 6,7 C A I H c2s.ee ft d 8-JJ.33
-41.67
•59 • 88 I I t I ti II I t I I I I t Ill 4 I I t t I t t I t 9 Ill e 11 I I e I II It e I e 1 -5.809 -J.333 -I • 667 • 881 I • 667 J.333 ,.... Tl"E 111 SECONDS
Figure 12. Filtered (ISi) OPSK Time Pulse (Log
Scale). I\.) '-.) 1.888 ISi CALCULATIONS for "SK Fi i I i I I I i i ■ • I I I a i .. ••I I Ii I I I I I iGi
.833 R E L A .667 T I IJ E A .500 f1 p L TI .JJ3 u D E .167
• 888 , , , , , , , o , a : a -- -s.eee -3.333 -t.667 .eee 1.,,1 3.333 s.111 Tl"E In SECONDS Figure 13. Filtered (ISi) MSK Time Pulse (Linear l'v Scale)., CX) ISi CALCULATIONS for "SK .ee, ' ' I I I I ' I I I I I ' 'MI I I I I i I I I I ' I I I I
-8.33
-1,.61 C A I H .-25.99 I n d 8-33. JJ
-41.67
..59. 88 ■ , , , , ■ , , , e , II , , II , , ,, , , , ,, , , II, , II, e , , , 1 , , , , 1 -5.889 -3.333 -l.667 .eee 1.,,1 3.333 s.na Tl"E In SECONDS Figure 14. Filtered (ISi) MSK Time Pulse (Log
Scale). I'\) ~ II. DESCRIPTION OF THE PROPOSED MODULATION TECHNIQUE: AWQPSK
In most modulation techniques, the baseband pulse shape influences the bandwidth efficiency and inter symbol interference. The difference between QPSK and
MSK is the baseband pulse shape. The new modulation technique optimizes the bandwidth efficiency, by using an optimum pulse shape. The pulse shaping function is a window function.
A. Window Function Overview
Window functions are very popular in digital signal
processing [15]. They are also used in pulse shaping in
baseband communication systems. The purpose of window
functions in pulse shaping is to produce no intersymbol
interference at the receiver.
All windows used in signal processing are even func
tions of time when centered around the origin. They were
introduced to minimize a phenomenon called "leakage".
Leakage is produced by the need to truncate a long signal
for practical considerations and finite memory limita
tions of the processing system. This truncation process
is equivalent to a multiplication of the finite time
window function with the indefinitely long input signal
30 31
in the time domain. Multiplication in the time domain is
equivalent to the convolution of the respective spectra
in the frequency domain.
The spectrum of the finite window function has in
herent sidelobes. The convolution of these sidelobes
with the spectrum of the original signal causes spreading
or leakage of the signal spectrum into adjacent frequency
bands. Examples of windows used are the Hamming window,
the Hanning window, the Blackman window, the Dolph
Chebyshev window to name a few. The Hanning and the
Blackman windows are very close to the function used to
generate the new modulation technique. The analytical
description of these two functions are given below.
Raised Cosine Function
The Hamming window, also referred to as raised cosine function, is defined by [15].
\ 0.54 + 0.46 Cos 1.i!,_t -T < t < T 2 2 f(t) = O T l otherwise
The spectrum of the Hamming function is- given by 0 46 F(f) = 0.54T Sinc(fT) + • T [Sinc(fT-1) 2
+ Sinc(fT +l)] 32
Blackman Function
The Blackman function is given by [16).
0.42 + 0.5 Cos -r2 ,r t f(t)= + 0.08 Cos .!2L_t -T/2 < t < T/2 T
0 otherwise The spectrum of the Blackman function is depicted in
Figure 15 and is given by
F(f) = 0.42T Sinc(fT) +~ T [Sinc(fT -1) 2
+ Sinc(fT + 1)) + ~ T [Sinc(fT-2)) 2
+ Sine ( fT + 2) ] •
B. Eigen Function
Time Domain
The proposed modulation technique will have a baseband pulse shape P(t). It is called the Eigen function [17, 18]. The difference between the Blackman and Eigenfunctions is only in the coefficients. Even though the difference is small, the spectrum and the 33
sidelobes are quite different . The Eigenfunction was
introduced for bandpass filter synthesis of · surface
acoustic waves ( SAW) devices. The Eigenfunction P(t) is given by
0.44 + 0.5 Cos 2 1t t -T/2 < t < T/2 T + 0.07 Cos 4 1t t T P(t) = 0 otherwise
An RF Eigen pulse is shown in Figure 16. This RF
pulse is modulated by a 100 MHZ carrier for illustration
purposes.
Frequency Domain
The spectrum of the Eigenfunction is easily obtained
by taking the Fourier Transform of P(t) and is given by
P(f) = 0.44T Sinc(fT) + 0.5 T [Sinc(fT -1) 2
+ Sinc(fT + 1)] + 0.07 T [Sinc(fT - 2) -2- + Sinc(fT +2)]
P(f) has three main components. These components
are shown in Figure 17. It is easily noted that the
sidelobes have the same zero crossings and ~re out of
phase, such that the sum of these three components
yields sidelobes that are 63 db down or lower from the main lobes. 34
BLACKMAN FUNCTION SPECTRUM +e.B0
-10.00
-20.00
-30.90
m -1e.00 -"D - ...~ -se.ee i -10.ee
-70.00
-ee.ee
-90.80
-iae.e:'0•0 sa.e 76 .e e ◄ .e 12.e 100.e 1ae.e 11s.e 12 ◄ .e 132.0 1 ◄ 0.0 rREQutNCY (tl-tz >
Figure 15. Blackman Function Spectrum. 35
+l .50
+l.2e
•.90 -
+.60 ~ ,,, !!llll1 I111,h ,.. , .,•h1!! ..,, I ~ l •.JC I' ·,: , I.• .. , r,., l'ti~ 1 l'}l!'fI • •·~ • ,, '"' ia/, I 1 , ~ ,1 1 i' t" =~• -••:, ··:- '~" ' • I •0.00 :, 1,l 1'i,:lli !;11M1!:!ll!l!!•!1!ilji'{f I ~/,': 11 11,, •, ,, 1,,. ,, 11•· i ' f ' ' I'.,,, ,\ .,.' 11,i t ~; I I -.JC ~
-.60 ~ j 'ij 1111:
-.,e - 11 Ill· -1.2e -
-J.SC ' J . -.5 -.◄ -.l -.2 -.s I.B .J .2 .J •◄ .5 TlH[ (alcl"OHC)
Figure 16. RF Eigen Function Time Domain Pulse. 36
EIGEN rUNCTION SPECTRUM DECOMPOSITION +. 18
+.19
+.17 •••
+.15 In ,
I MI +.84 '
I I I +.12
+.11 __ .,
-.12
-.83 se.e 11.1 1s.1 1 ◄ .1 12.e 1ee.e 1ee.e 11s.1 12 ◄ .1 1J2.e 1 ◄ e.e F'REOLCNCY Oltz>
Figure 17. Eigen Function Spectrum Decomposition. 37
Another advantage that the Eigenfunction has is that the first two sidelobes a r e about 80 db down as se en in Figure 18. This makes the Eigenfunction desir able in a crowded spectrum, where ,co-channel interfer ence is critical.
C. AWQPSK
AWQPSK modulation is achieved by using the Eigen function as its baseband pulse.
Quadrature Modulation
A quadrature modulated signal can be modeled by g(t) = _+ P(t) Cos wt+ P'(t) Sin wt, C - C where P(t) is the Inphase baseband pulse shape, and
P'(t) is the quadrature baseband pulse shape.
It was seen that for QPSK,
P(t) = P'(t) = rect [t-T/2] T for MSK
P(t) = Cos(~) rect [t-T/2] T T and
P'(t) = Sin(~ rect [ t-T/2] • T T
Similarly the new modulation technique can be modeled the same way with 38
.... EIGENFUNCTION SPECTRUM
-11.ee
-21.ee
-31.10 m -48.IB -~ ...I ~e.ae i -1, .•
-78.IB
-ee.ee
-98.IB
-110.ee II.I 18.1 76.8 14.I 12.1 lie.I IE.I Jl&.I 124.I 132.8 l ◄ e.e F'R£0ll:NCY Ctttz)
Figure 18. Eigen Function Spectrum. 39
P ( t ) = [ 0 • 4 4 + 0 • 5 Co s 2 ,r t T
+ 0 • 0 7 Cos 4 ,r t rect [t-T/2] • T T and
P'(t) = [0.44 - 0.5 Cos~ T
+ 0. 07 Cos !.2!...t, ] rect [ t-T/2] T T
Analysis of AWQPSK
The quadrature modulation signal obtained using the
Eigenfunction can be written as g(t) = Ur P(t) Cos wet+ uq P'(t) Sin wet where Ur and uq can take on+ 1. Substitution in the above expression for P(t) and P'(t), leads to
g(t) = f0.44 Cos w t + 0.5 Cos ( 2 ,r t) Cos w t Ur C C T
+ 0.07 Cos (~) .Cos w t l C T
+U f0.44 Sin w t - 0.5 Cos ( 2 ,rt ) Sin w t q C C T
+ 0.07 Cos ( 4,rt) Sin w tJ. C T
The function g(t) can be decomposed into three signals; 40
where g ( t) = 0.44 { u Cos wet+ 1 1 uq Sin wet}
g2(t) = 0.5 { u Cos (~) Cos w t 1 C T
- uq Cos ( 2 Jtt) Sin wet} T
= 0.5 Cos 2Jtt u Cos w t -u { 1 C q Sin wet} -T
g3(t) = 0. 07 { Cos ( 4 Jtt) u1 Cos wet T
+ Cos {4Jtt) uq Sin wet} T
g 1 (t) is a quadrature phase shift keying signal (QPSK). g (t) and g (t) are two cosine weighted quad 2 3 rature phase shift keying signals.
The proposed modulation technique is a superposition of the above modulation techniques and henc·e the assign ed name: Amplitude - Weighted - Quadrature - Phase -
Shift Keying or AWQPSK. III. PERFORMANCE EVALUATION OF AWQPSK
A. Spectral Efficiency
The envelope of the AWQPSK Spectrum is given by the spectrum of the Eigenfunction shown in Figure 18. The sidelobes are at least 63 db down. The HP 9845 computer was used to calculate the transmission bandwidth BT that contains 99.99% of the total power. The transmission bandwidth BT was found to be 1.15/Tb, which yields a bandwidth efficiency of rb/1.15/Tb = 1/1.15 bits/sec/Hz.
AWQPSK has a higher bandwidth efficiency than QPSK and
MSK.
B. Probability of Error
The output of a QPSK modulator is one out of the four following signals:
= A rect (t-T/2) Cos wt C T
(t-T/2) Sin wt C T
s (t) = -A rect (t-T/2) Cos wt 3 T c s (t) = -A rect (t-T/2) Sin wt 4 T C with w = k2~ where k is a positive integer, i.e. there c T is an integer number of carrier cycles per symbol.
41 42
Similarly the output of a AWQPSK modulator is one out of the four following signals, during the interval
(0,T).
S l ( t ) = A ' ( 0 • 4 4 + 0 • 5 Cos ~ T
+ 0 • 0 7 Cos ~ ) Cos w t T c
= A ' ( 0 • 4 4 - 0 • 5 Co s 2 ,rt T
+ 0.07 Cos~) Sin wet T
= -A ' ( 0 . 4 4 + 0 . 5 Cos 2 ,rt y-
+ 0.07 Cos 4,rt) Cos wt y- C
(0.44 - 0.5 Cos~ T
+ 0.07 Cos~) Sin w t T c
The above expressions can be written as
2,r (t) A ( 1 1.136 Cos-t s 1 = + T C 4,rt) + 0.16 OS- Cos w t T C
(t) A ( 1 1.136 Cos 2Jtt s 2 = - ~ C 4,rt) + 0.16 OS- Sin w t C T
(t) -A (1 1.136 Cos 2Jtt s 3 = + -T S. 4 Jtt) Cos w t + 0.16 in-- C T 43
2xt = -A ( 1 - 1 . 13 6 Cos -- T
+ 0.16 Cos~) Sin w t T c
where A= 0.44 A'.
These signals are well described by the following phasor diagram depicted in Figure 19.
S,
Figure 19. Phasor Diagram. 44
The correlator receiver for this Quadrature signal, requires two local reference signals A Cos(w t + X/4) C and A Cos (wt - X/4). C A AWQPSK correlator receiver is shown on Figure 20.
The outputs of correlator 1, s (T), and correlator 2, 01 S02(T), are presented below for the following cases: CASE 1. s (t) is transmitted 1 CASE 2. s (t) is transmitted 2 CASE 3. s (t) is transmitted 3 CASE 4. s (t) is transmitted 4
Acos (wt+•/4)
n(t) LTdt ~ I VOl
llC£IvtR I FILTER i I V02 LTdt ~
Acos(vt-•14)
Figure 20. Quadrature Correlator Receiver. 45
CASE 1:
If s (t) was transmitted, the output signal, s (T), 1 01 with no noise
SOl(T) = A2f + 1.136 Cos~ T
,t + .16 Cos~t) Cos w t Cos (wt +- ) dt T C C 4
4 ,t SOl(T) = A2[(Tl + 1. 136 Cos ~t + 0.16 Cos- t) Cos w t T T C
J[ J[ X [Cos w t Cos- - Sin w t Sin-] dt C 4 C 4
SOl(T) = A 2 t Cos J[ !. + Cos.!. 1 JTcos 2w t dt C 4 2 4 2 0
+ 1.136 Cos~ Cos ~t dt 2 4 T
+ 1.136 Cos!:_ fc:s -3.!:_ t Cos 2wc t dt + 0 .16 2 4 } " T -Y 0
x Cos-Jt lTCos-t 4Jt dt + 0.16 Cos~ 4 o T . ~ 4
x fc:s....!!:... t Cos 2wct dt - (iT + 1.136 Cos~ t 1a' T lolJ T 46
for wc = k 2~/T the last six intergrals are zero. Hence
where A is a constant. 0 The output, s (T), is 02 2 SO ( T ) = A 1 + 1. 13 6 Cos 2 t + 0 • 16 Cos t) 2 f[ ,r ~ 0 T T
x Cos wt Cos (wt - E) dt. C C 4
Integrating the above expression yields 2 S02(T) A T Ao = V2 -4- =
CASE 2:
If (t) was transmitted, the output, (T) is s 2 s01 2 (T) A /~1 - 1.136 Cos 0.16 Cos s01 = !,rt+ tt> 0
x Sin w ~ Cos (wt+_!_) dt. C C 4
=-A. SOl (T) = 0
The output, s (T), is 02 2 1 SO ( T ) = A ( ( 1 - 1. 13 6 Cos~ t + 0 . 16 Cos ~ t ) 2 ~ T T
x Sin w t Cos ( w t - -)~ dt. C C 4
CASE 3: If s (t) was transmitted, the output, s (T), is 3 01 47
2 4 SO l ( T ) = -A IT( 1 + 1. 1 3 6 .Cos .f.!...! + 0 • 16 Cos 7f t ) O T T
x Cos wt Cos(w t +2.) dt C C 4 or 2 s ( T) = - A T 01 V2 =-A. 4 0
The output, s (T), is 02 SO ( T ) = -A 2 JT( 1 + 1. 13 6 Cos -2 rr t + 0 • 16 Cos ---2:.4 t ) 2 o T T
x Cos w t Cos ( w t - L) d t • C C 4 2 s02(T) = - V2 A T = -Ao. 4
CASE 4:
If s (t) was transmitted, the output, s (T), is 4 01 s (T) = 2 1.136 Cos ~t + 0.16 Cos~ t) 01 -A J~1 - o T T
x Sin w t Cos ( w t + ..!...) d t • C C 4
= A • SOl (T) = 0
The output, s (T), is 02 s (T) = -A2 1.136 Cos~t + 0.16 Cos~t 02 f~1 o T T
x Sin w t Cos ( w t - ..L) d t. C C 4 2 =-'./fA T 4 48
The above analysis is summarized by the following relations.
SOl(T), S02(T)= A A for s (t) o' 0 1 SOl(T), S02(T)= -A A for s (t) o' 0 2 S01 (T), S02(T)= -A -A for s (t) o' 0 3 SOl(T), S02(T)= A -A for s (t) o' 0 4
The output of correlator 1 and correlator 2,
considering noise are given by v (T) and v (T) 01 02 respectively
VOl(T) = SOl(T) + nOl(T)
V02(T) = s02(T) + n02(T). The input noise n(t) is white and Gaussian with zero mean and a power spectral density n/2. The output noise
expressions are given by
and
It has been shown that n (t) and n (t) are 01 02 independent random variables normally distributed with
equal variance N given by [1] 0 2 N = ~n • 0 4 49
Let Peel denote the probability of error of correlator 1, and P ec2 the probability of error of correltor 2. By symmetry it is seen that p p eel = ec2 p if (t) or (t) is transmitted Peel = { vo1 < o} s1 s 4 p if s (t) or (t) is transmitted = { VOl > 0} 2 s 3 but vo1 = 8 01 + no1· Therefore
= p 8 < 0 s (t) or · s Peel { 01 + nOl I 1 4 (t>} + p 8 > 0 I s (t) or s (t)} { 01 + nOl 2 3 but 8 = A if s (t) or s (t) is transmitted 01 0 1 4 and 8 =-A if s (t) or s (t) is transmitted. 01 0 2 3 Therefore
p 0 s (t) or (t)} Peel = { Ao + nOl < I 1 s4 p 0 s (t) or s (t)} = { -Ao + nOl > I 2 3
n is a Gaussian random variable with a P.d.f. given by 01
2 - ( no 1) exp 2N 0 50
therefore
2 · -(nOl) exp
1 2N 0 using the Q function defined by
Q(x) = l/(2rr) 11 2 1:xp{- and a change of variables, the probability of error of correlator 1 is 2 p = Q (A i·'N1 ) where A =fl A T eel o V "0 0 4
P - Q I r;:5:; eel - V~
Let Pc denote the probability that the transmitted signal is received correctly, given by
2 P c = 1 - 2 P eel + P eel 51
For small probability of error P 2 can be neglected eel and
Pc= 1 - 2 Peel· Therefore the probability of error for the AWQPSK signal is given by
p = 1 - p = 1 - 1 + 2 p e C eel
=> p = 2 p 2Q e eel = (~) 2 rJ
This is the same probability of error obtained for QPSK and MSK [14] •
C. Intersymbol Interference
The ISI model used in this section is the same as the one used for the ISI discussion of QPSK and MSK in Section
II. Figure 21 depicts the simulated AWQPSK pulse. The linear scale shows no sidelobes generated. The Logarith mic scale shown on Figure 22 shows that sidelobes are 67 db below the mainlobe. This simulated pulse also shows that the zero crossings of the AWQPSK signal are twice as close together as compared to the zero-crossing spacing of the MSK signal.
The two parameters clearly show that AWQPSK is better suited to reduce ISI, than QPSK or MSK. This is an antic ipated result since the Eigen pulse has a smooth transi tion to zero in the time domain and low sidelobe levels in the frequency domain. 52
D. Percent Amplitude Modulation
A modulated signal with AM (Amplitude Modulation) will be distorted if the channel is non-Linear. Constant enve lope modulated signals are more desirable for very high frequency channels which may be non-linear. Amplitude modulation is acceptable for linear channels. AWQPSK has about 53% amplitude modulation. The percent modulation of a modulated signal f(t) is defined as [19].
max If ( t) I - min I f ( t) I % modulation=------x 100 max lf(t)I + min lf(t)I
The AM was introduced in AWQPSK by adding the inphase and quadrature signals shown in Figure 23 and Figure 24 respectively. There are four possible signs the inphase and quadrature signals may take;++,--,+- and-+.
Since there is only 180 degrees phase shift between the
++ and -- condition, and the -+ · and+- condition, only 2 possibilities of the sign of the inphase and the quadrature components need to be considered. Figure 25 shows the percent AM when the inphase and the quadrature component of the AWQPSK signal are both positive.
Figure 26 depicts the case where the two components have opposite signs. The percent AM present in the AWQPSK waveform is independent of the sign of the -two components. I • 888 - ■ • a ■ ■ a a ■ ■ ■ a : a a ■ a a u a a a a a a ■ a u ; ; aq
.833 R E L ~ .667 I "E A .500 p "L T1 .333 u 0 E .167
• 888 I O a a I I a I a O I O C a 1 -s.eee -J.333 -t.667 .eee ,.,67 3.333 ,.... Tl"E in SECONDS
Figure 21. Filtered (ISI)AWQPSK Time Pulse ( Lin~.ar Scale) • u, w •·I Ii I I I I I I I I I I I IWU I I I I I I Ii I I I Ii I
-16.7
-33.3 C A I N i -58.8 " d 0 -66. 7
-83.3
--188 • I) I I I f I I I I 1 ft Ill II ■ ■ ■ I ft I I I I I I I I I I I I I ■ Y Ill ■ b t I I t ft t t I -,.eee -3.333 -1.667 .eea 1.667 3.333 S.181 Tl"E ln SECONDS
Figure 22. Filtered (ISi) AWQPSKTime Pulse (Log Ul Scale) • .i::.. 55
IN PHASE SIGNAL ♦ J,11 r------
•1.n
-1.n
-:.,: -.s~------.....J -.4 •,J -.l •, I I.I ,J .I ., ,4 .S Tl!£ CetcP ...c >
Figure 23. Inphase AWQPSK Component.
QUADRATURE SIGNAL
•1.20
+.90
- . 6~
•.la
-.30
-.60
-.90
-1.20 -l.50 '----'-______....___,
-.5 •,4 -.3 -.z -.I B.0 , I ,ii! ,3 .4 ,5 TU£ C ■ tc:roaecl
Figure 24. Quadrature AWQPSK Component. 56
-+!.SO
-+I .20
-+.90
+.60
-+. 30 ~ ~ -+0.00 .J ~ -.30
-.60
-. 90
- : .20
- 1. 50 -.5 - . 4 -.3 -.2 -. I 0.0 .I .2 .3 . ◄ .5 Til1E (atcroucl
Figure 25. AWQPSK Time Domain Waveform With U =+1 and Uq =+1 • I
+: .50
+1.20
-+.90
-+.60
-+ , 30 ~ i +0.00 -.30
-.60
-.90
-!.20
- : .50 .1 .2 .3 . ◄ .5 -. 5 -.◄ -.3 -.2 - . 1 0.0 ·Til1E (aicroucl
Figure 26. AWQPSK Time Domain Waveform With u1 =+1 and Uq =-1. IV. IMPLEMENTATIONS
QPFSK can be implemented either as a superposition
of the three modulation techniques, or with a SAW device.
A. Parallel Scheme
The AWQPSK signl is give by g(t) = 0.44 { u Cos wt+ U Sin wet 1 C q + 0.5 { UICos (1...!...!) Cos wt - U Cos (2n)t T c q T
x Cos wctJ + 0.07 [U 1Cos (1!!!_) Coswct T
+ U Cos (2n)t Cosw t] q T . C
A parallel implementation is shown in Figure 27.
This design is based on the superposition of one
QPSK signal, and two amplitude weighted QPSK signals with appropriate scaling factors.
B. SAW Device Implementation
A SAW device implementation is shown in Figure 28.
The first transducer is apodized with the Eigenfunction being its impulse response, the second transducer is a very wideband unappodized transducer. The input bipolar binary data is fed to an impulse generator. Theim-
57 58
pulses drive the SAW device [20,21]. The quadrature component is obtained by setting the input binary bit duration to be exactly T/2 which is half the impulse time length of the apodized transducer.
Recent progress in the SAW technology makes this im plementation cost effective [22]. The modulation with the SAW devices is much simpler, smaller, and less ex pensive than the parallel implementation. 59
Ac01 (w, t) linary Data UT TO ----...JnIBIT CONVE
VCO Aco1,Yt,
Figure 27. Parallel Implementation of AWQPSK Modu lation. VtDE BAHD APODIZED TRANSDUCER TRANSDUCER
BIPOLAR DATA IHPULSE 1GENERATOR ~ ~ I t t ~t t=i. ~T11 P~•a; 0 t· ?'• 3T111
~T=I'~
Figure 28. SAW Device Implementation of AWQPSK Modulation.
°'0 V. CONCLUSION
This thesis introduced a new modulation technique which can easily be implemented with a surface acoustic wave (SAW) device. This new modulation technique is a good competitor to QPSK and MSK for· communication ap plications where spectral containment is critical. This new modulation technique is the superposition of a QPSK signal and two cosine weighted QPSK signals, and is thus named Amplitude Weighted Quadrature Phase Shift
Keying (AWQPSK). It was shown that AWQPSK has a higher bandwidth efficiency in bits/sec/Hz than QPSK or MSK.
Its sidelobes are 63 db down from the main lobe. The probability of error using a correlator receiver is the same for AWQPSK, QFSK and MSK. Due to low sidelobe levels in the frequency domain, and smooth transition to zero in the time domain, the baseband pulse used in
AWQPSK yields a better ISI than QPSK and MSK. This new modulation technique can be produced by the superposi tion of conventional modulation techniques and appro priate scaling factors. AWQPSK has about 53% AM.
Finally, two unique types of implementation of AWQPSK were presented in this dissertation.
61 APPENDIXES APPENDIX A
PROBABILITY OF ERROR FOR BINARY MODULATION TECHNIQUES
In generating the probability of error for the digital communication system, the following assumptions will be made:
1) The symbols are equiprobable and independent.
2) The noise, n(t) is white and Gaussian with a
power spectral density function Gn(f) = ~/2.
3) There is no ISI.
The receiver structure shown in Figure 4 can be modeled by the structure shown in Figure 29, the re ceiver filter output can be written as vo(t) = So(t) + no(t), where so(t) is the filter response to the signal alone, and no(t) the response to n(t) alone.
Z(t) Receiver- Threshold Output
Filter Detector
To=Threshold Figure 29. Receiver Structure.
63 64
So(t) = z(t) * h(t), where* denotes convolution,
and h(t) is the receiver filter impulse response. Ap
plying the convolution integral, So(t) can be written as
So( KT) = f Zk; t ) h (t - t ) d' (k-1) T
' - (k-l)T if bk= 0 but z ( ' ) = { :: t - (k-l)T if bk= 1
therefore So(kT) can be written as
s (kT) 01 if bk= 0 S (kT) = 0 { s ( kT) 02 if bk= 1 The input noise is white and Gaussian with Gn(f) being
its power spectral density funct1on. The filter noise output is also white Gaussian noise [ 11 with a variance
2 2 N = E [ n ( t) 1 = ( f) H(f) df 0 0 J;n I I -oo where H(f) is the filter transfer function. The total
filter output can be written as
S (kT) .+ n 0 ( kT) if bk= 0 vo(kT) = 01 { s ( kT) ( kT) 02 if bk= 1
Sol and S02 are equiprobable, and hence the optimum 65
threshold for the threshold detector is given by
To =
The probability of making an error is given by
Since ones and zeros are equiprobable, P(O) = P(l) = 1/2, and the probability of error can be simplified to
802 - 801 = Q (-----) 2 N 0 where
So2 and Sol depend on the receiver filter and type of digital modulation technique. The above Pe model is the generic probability of error model that will be used
in this thesis.
The correlator receiver is depicted in Figure 30.
Figure 30 can be represented by Figure 31.
The probability of error for a PSK system, using the above model is given by [3]
= Q where Eb is the energy per bit and is the single sided power spectral density of the white caussian noise. 66
z (t)
Receiver Threshol Filter Detector
n (t)
Output
Figure 30. Binary Correlator Receiver.
To z (t)
Receiver T Threshol d t _,.,. -- Detector Filter l0 n Ct)
output
Figure Jl. Integrate and Dump Receiver. 67
Similarly it can be shown that the probability of error for an FSK modulation technique is given by
p = Q e
The correlator receiver model for a quadrature modulation technique is a variation· of the one used in this appendix. The difference would be 2 identical parallel branches at the output of the receiver filter, an inphase branch and a quadrature branch. APPENDIX B
POWER SPECTRAL DENSITY FUNCTION OF A RANDOM BINARY WAVEFORM
This appendix shows that, given a modulation format such as QPSK, MSK or AWQPSK, the envelope of the transmitted random signal spectrum is given by the
Fourier transform of a single symbol (chip). This single symbol transform is used to evaluate the spectral efficiency parameter. To evaluate the power content in any frequency band, the power spectral density function is needed.
In the random binary waveform shown below, each pulse can take on~ A, with equal probability. Each pulse duration is equal to T = 2Tb seconds.
A
. - - -, t ~T -2T -T 0 T 2T 3T 4T -A 4
Figure 32. Binary Random Waveform.
68 69
The autocorrelation function of such random signal is given by
A2 (1 - J.rj ) -T < 'f"' < T R (1")= T . X l0 otherwise By virtue of the Wiener-Khinchine theorem
s (f) = F { R (T) l X X where S (f) is the power spectral density function of X the random signal x(t).
s (f) = 1 i2 ( 1_ J2J_ ) e -J2,rf, di X -T T
2 s ( f) Sin(11fT) t X (1r fT)
2 Note that since fis: n ( rr f T J 1 d f = .!. JI:. (11" fT) T -00
A2 = Average power in the signal.
The Fourier transform of a single pulse (rectangular pulse) is given by
AT Sin(nfT) = AT Sine (fT) F( f) = nfT 2 2 2 2 Let G( f) = 1F(f,1 = A T Sinc (fT) Sinnx where SinC ( X) = nx
Sinc2 (fT). but Sx ( f) = A2T which implies that 2 s ( f) = G(f) X --T- 70
It can be easily shown that for QPSK with T = 2Tb, 2 2 Sx(f) = ~ IF (f)I = 4 Eb Sinc (2fTb)
2 where Eb= A Tb= energy per bit. Similarly it can be 2
where F' ( f) = F ff(t)l = 0.5 AT Sine (fT - 0.5)
+ 0.5 AT Sine (fT + 0.5) where f(t) is the sinusoidal pulse given by
A Cos__!!_ t -T < t < T f(t) = 2T 2- -2 . 0 otherwise
Similarly the power spectral density functions of the
AWQPSK signal can be generated by
sx ( f) = { IF ( f) I 2 where F ( f ) = F { A( 0 • 4 4 + 0 • 5 Cos 2; t
+ 0.07 Cos i" t) rect <;l}•
F(f) = AT { 0.44 Sine ( fT) + 0.5 Sine [ ( T ( f-1)) -2-
+ 0.5 Sine [ T (f + 1) ] + 0.07 Sine [ T (f-2)] -2- -2-
+ 0.07 Sine [ T ( f + 2) ] } • -2- APPENDIX C
INTERSYMBOL INTERFERENCE MODEL
At the receiver the demodulated symbol will have
time domain sidelobes generated by filtering in the communication system. The output of the demodulator is modeled by
Y ( t ) = ~ + A Pr ( t - td - k Tb ) + n ( t ) k - o where td is an arbitrary time delay, n (t) is the 0 output noise, and P (t) is the received pulse shape. r The time delay td can be eliminated by sampling at t = td + mTb' where mis an integer. The received output for the mth bit is given by Y(tm) =+A + E A Pr [(m - k) T] + n (tm) • 0 kp=m The first term is the desired output. The last two terms however, are error source terms. The last term is the channel noise term. The second term represents the residual effect of all the other transmitted pulses on the mth bit. This residual term is called intersymbol interference (ISI). !SI is generated by the overlapping tails of the other pulses.
As it can be seen on Figure 33, if sampling is done at t = kT (at the zero crossings) there is no ISI.
71 72
However, if there is a jitter in the sampling clock, ISI is generated.
There are two parameters that can be optimized in order to reduce ISI.
The first parameter is pulse shape. The pulse shape should have time domain sidelobes which are as small as possible. The second parameter is the zero crossing interval. The zero crossings of the sidelobes should be as close together as possible. 73
-3T
Figure 33. Example of a Received Pulse Train. REFERENCES
[1] K. S. Shanmugam, Digital and Analog Communication Systems. New York: John Wiley and Sons, 1979.
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