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NAVAL POSTGRADUATE SCHOOL Monterey, California

DTIC ELECTE

M 2 11987

THESIS ^ AN ANALYSIS OF COHERENT AND DIFFERENTIALLY COHERENT DIGITAL RECEIVERS IN THE PRESENCE

OF COLORED NOISE INTERFERENCE

by

Barry L. Shoop

September 1986

Thesis Advisor: Daniel Bukofzer 1

Approved for public release; distribution is unlimited

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' i TITLE (include Security CliSSificition) AN ANALYSIS OF COHERENT AND DIFFERENTIALLY COHERENT DIGITAL RECEIVERS IN THE PRESENCE OF COLORED NOISE INTERFERENCE \i PEPSONAL AUTMOR(S) Shoop, Barry L. 3d TYPt OF REPORT

Master's Thesis 13b TIME COVERED

FROM TO 14 DATE OF REPORT (Yetr. Month. Oiy)

198 6, September 15 PAGE COUNT

115 '6 SUPPLEMENTARY NOTATION

COSATI CODES

PiELO GROUP SUB GROUP

18 SUBJECT TERMS (Continue on revene if neceutry tnd identify by block number)

Optimum Colored Noise Jamming; MPSK; DPSK; MQAM

"9 ABSTRACT (Confinu» on reverie if neceutry »nd identify by block number)

Optimum receivers for detecting digital signals in additive white Gaussian noise are analyzed when operating in the presence of both white noise and colored noise interference. Modulation schemes, such as coherent M-ary Phase Shift Keying (MPSK), Minimum Shift Keying (MSK), Differentially Coherent Phase Shift Keying (DPSK), M-ary Quadrature Amplitude Modulation (MQAM) and 16-state AM/PM are analyzed. Optimum power constrained colored noise interference spectra are developed for each modulation technique analyzed so as to maximize the receiver error probability.

Receiver performance, quantified by bit and symbol error probabilities is numerically evaluated and graphically displayed as a function of signal- to-noise ratio ard interference-to-signal ratio to demonstrate the effec- tiveness of this interference in terms of the receiver performance degradation.

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Ua NAME OF RESPONSIBLE INDIVIDUAL Prof, Daniel Bukofzer

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DO FORM 1473,84 MAR 83 APR edition may be used until exhausted

All other editions are obsolete 1

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Approved for public release; distribution is unlimited.

An Analysis of Coherent and Differentially Coherent Digital Receivers in the Presence of Colored Noise Interference

by

Barry L. Shoop Captain, United States Army

B.S., The Pennsylvania State University, 1980

Submitted in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE IN ELECTRICAL ENGINEERING

from the

NAVAL POSTGRADUATE SCHOOL September 1986

Author:

Approved by: ELJ Q-i rry L. Shyoop

r ^^jianTel Bi^ofzer,^Thesis Advisor

Paul H. Moose, Second Reader

WJOJXMA — ~^- Harri^ett B. Rigas, Chairman. Department of

)uter Engineering Electrical and Compul

John N. Dyer, Dean of Science and Engineering

.v.W- v •■•v -.-s ■.. . ..\.:,.\^.. ■'i -N , tÜäSSÜ^

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r ABSTRACT

V Optimum receivers for detecting digital signals in addi-

tive white Gj^issian noise are analyzed when operating in

the presence of both white noise and colored noise inter-

ference. Modulation schemes, such as coherent M-ary Phase

Shift Keying (MPSK), Minimum Shift Keying (MSK), Differen-

tially Coherent Phase Shift Keying (DPSK) , M-ary Quadrature

Amplitude Modulation (MQAM) and 16-state AM/PM are analyzed.

Optimum power constrained colored noise interference spectra

are developed for each modulation technique analyzed so as

to maximize the receiver error probability.

Receiver performance, quantified by bit and symbol

error probabilities is numerically evaluated and graphically

displayed as a function of signal-to-noise ratio and inter-

ference-to-signal ratio to demonstrate the effectiveness of

this interference in terms of the receiver performance

degradation. — ., ; «^ Accession For

"NTIB GRA&I DTIC TAB Unannounced Justification-

D

By Distribution/

Availability Codes

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TABLE OF CONTENTS

I. INTRODUCTION 10

II. COHERENT M-ARY PHASE SHIFT KEYING 14

A. SIGNAL DETECTION IN THE PRESENCE OF COLORED NOISE JAMMING 14

B. OPTIMIZATION OF THE COLORED NOISE JAMMER 33

III. SPECIAL CASES OF COHERENT MPSK: QUADRATURE PHASE SHIFT KEYING, OFFSET QUADRATURE PHASE SHIFT KEYING AND MINIMUM SHIFT KEYING 38

A. QPSK RECEIVER PERFORMANCE 38

B. OFFSET QPSK RECEIVER PERFORMANCE 44

C. MSK RECEIVER PERFORMANCE 47

IV. DIFFERENTIALLY COHERENT PHASE SHIFT KEYING 53

A. DPSK RECEIVER PERFORMANCE IN COLORED NOISE JAMMING 53

B. OPTIMIZATION OF THE COLORED NOISE JAMMER 63

V. M-ARY QUADRATURE AMPLITUDE MODULATION 65

A. 16 QAM RECEIVER PERFORMANCE 65

B. 64 AND 2 56 QAM RECEIVER PERFORMANCE 76

C. 32 QAM RECEIVER PERFORMANCE 81

VI. A SPECIAL CASE OF QUADRATURE AMPLITUDE MODULATION: 16-STATE AM/PM SIGNALING 87

A. RECEIVER PERFORMANCE IN COLORED NOISE JAMMING 87

VII. CONCLUSIONS 101

APPENDIX A: DETAILED INVESTIGATION OF THE PRODUCT OF $,(-£) AND $2(f) . 109

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\ LIST OF REFERENCES 113

1 INITIAL DISTRIBUTION LIST 114

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LIST OF TABLES

7.1 COMPARATIVE RECEIVER SYMBOL ERROR PROBABILITIES ~ 106

7.2 SNR PENALTY FOR MAINTAINING PR{e} = lO" IN JAMMING 107

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LIST OF FIGURES

2.1 Optimum MPSK Receiver for AWGN Channel 15

2.2 4-PSK Receiver Performance 29

2.3 8-PSK Receiver Performance 30

2.4 16-PSK Receiver Performance 31

2.5 32-PSK Receiver Performance 32

3.1 Optimum QPSK Receiver Structure 39

3.2 Signal Space Diagram for QPSK Signaling 42

3.3 QPSK Receiver Performance 45

3.4 Optimum Offset QPSK Receiver Structure 46

3.5 Optimum MSK Receiver Structure 49

4.1 Optimum DPSK Receiver Structure 55

4.2 Signal Space Representation of Received DPSK Signals 61

4.3 DPSK Receiver Performance 64

5.1 Signal Space Diagram for 16 QAM Signaling 67

5.2 Optimum 16 QAM Receiver Structure 69

5.3 16 QAM Decision Regions 72

5.4 16 QAM Receiver Performance 75

5.5 Signal Space Diagram for 64 QAM Signaling 77

5.6 64 QAM Receiver Performance 79

5.7 256 QAM Receiver Performance 80

5.8 Signal Space Diagram for 32 QAM Signaling 82

5.9 Translated Type I Decision Region for 32 QAM 84

5.10 32 QAM Receiver Performance 86

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6.1 Signal Space Diagram for 16-State AM/PM Signaling 88

6.2 Optimum Decision Regions for 16-State AM/PM Signaling 90

6.3 Suboptimum Decision Regions for 16-State AM/PM Signaling 91

6.4 Receiver Structure for Suboptimum 16-State AM/PM Signaling 92

6.5 16-State AM/PM Receiver Performance 99

6.6 Performance Comparison of 16 Level Signaling 100

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ACKNOWLEDGMENTS

I wish to express my sincere appreciation to my thesis

advisor Professor Daniel Bukofzer for his guidance, patience

and friendship during the research, composition, and

completion of this thesis. This thesis is dedicated to

my family; my wife, /for her support and continual

encouragement throughout my graduate education, and to my

parents, and particularly my father whose

wisdom, insight and counseling directed me into this field

of study.

I

I. INTRODUCTION

Of the many different digital modulation methods that

exist, each is designed and utilized so as to improve a

particular feature of the communication system. Some such

methods provide increased spectral efficiency while others

improve overall system performance. Still others simplify the

receiver structure and thereby reduce hardware costs. In

order to prudently select the best modulation technique for

a specific application, corresponding receivers must be

analyzed and their performance compared using appropriate

channel models. Although analyses of digital receiver per-

formances abound in the literature, treatment is usually

restricted to the case of additive white Gaussian noise (AWGN)

as the channel interference. The AWGN model, although proba-

bly the easiest to analyze, seldom accurately describes an

actual communication channel. For military applications,

frequently a jamming environment must be assumed in which

case the AWGN model is inadequate. While a great deal of

effort has been devoted to the analysis and design of spread

spectrum communication systems, virtually no documented re-

ceiver performance results exist for systems designed to

operate in an AWGN interference that in practice must operate

in the presence of a "smart" jammer.

10

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In this thesis the performance of receivers operating in

the presence of both AWGN and colored noise jamming is analyzed

for several digital modulation techniques. Mathematical ex-

pressions for the receiver error rate performance are derived

and optimized jamming techniques are developed for each re-

ceiver structure analyzed.

The receiver structures considered throughout are assumed

to be optimally designed, in the sense of minimum probability

of error performance in an AWGN environment. The colored noise

jammer is modeled as Gaussian, power limited, uncorrelated with

the white channel noise and with pov/er spectral density deter-

mined as part of the optimization procedure.

Chapter II presents the derivation of symbol error proba-

bility for a coherent M-ary Phase Shift Keyed (MPSK) receiver

operating in noise and jamming. Receiver performance curves

are then presented for several values of M with jamming-to-

signal ratio (JSR) as an independent variable to show the

effectiveness of the optimized jammer. The optimum colored

noise jammer applicable to this problem is developed in detail

in this chapter.

In Chapter III the analysis carried out in Chapter II is

applied to three very important special cases of MPSK signal-

ing, namely Quadrature Phase Shift Keying (QPSK), Offset

Quadrature Phase Shift Keying (OQPSK) and Minimum Shift Keying

(MSK). The analysis presented here ser\as only to highlight

11

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the differences between the MPSK techniques and the specific

special case being considered.

The analysis in Chapter IV changes focus from coherent

Phase Shift Keying to Differentially Coherent Phase Shift

Keying (DPSK). In contrast to the purely mathematical develop-

ment conducted in Chapter II, a geometric approach is used in

order to simplify the analysis. Also, only the binary signal-

ing case is considered. As in previous chapters, the mathe-

matical expression for receiver performance is derived, error

rate performance curves are presented and an optimized colored

noise jammer is developed.

In Chapter V M-ary Quadrature Amplitude Modulation (MQAM)

techniques are analyzed in the presence of colored noise jam-

ming. Symbol error probability expressions are derived for the

standard values of M = 16, 64 and 256 as well as for less

typical M = 32. Receiver performance curves and optimized

jamming waveforms are also included for each case so as to

complete the analysis.

Chapter VI is devoted to the analysis of a digital radio

transmission technique not treated in classical communication

theory literature, namely a 16-state AM/PM signaling technique

recommended by The International Telegraph and Telephone

Consultative Cormittee (CCITT). Receiver performance is

analyzed in the presence of AWGN both with and without colored

noise jamming. The analysis of this signaling technique differs

from those previously considered in that a suboptimum receiver

12

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structure is assumed. The suboptimum receiver was chosen

based on intuitive as well as practical considerations involv-

ing the implementation of the receiver logic. Also, due to the

mathematically involved expressions for the symbol error

probability, no jammer optimization is attempted. Compari-

sons between this 16 level signaling scheme and other (better

known) 16 level schemes are presented.

Finally, Chapter VII provides performance comparisons

and conclusions to be drawn from the analysis and graphical

results obtained for the modulation methods considered in this

thesis.

13

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II. COHERENT M-ARY PHASE SHIFT KEYING

Coherent M-ary Phase Shift Keying (MPSK) is a digital

signaling technique that provides bandwidth efficiency, con-

stant signal envelope, relatively good error rate performance

and simple receiver structures [Ref. 1]. MPSK is a signaling

scheme that achieves its bandwidth efficiency at the expense

of signal power.

A. SIGNAL DETECTION IN THE PRESENCE OF COLORED NOISE JAMMING

In MPSK modulation, the source transmits one of M signals,

s.(t), i = 1,2,...,M every T seconds. The transmitted signal

is of the form

si(t) = /2Es/Ts cos [2TTf0t +2Tr(M"1) +OL] (2.1)

0ltlT' i=l,2,...,M

where E is the average signal set energy, a is an arbitrary,

yet fixed phase and the information is contained in the

2iT(i-l)/M phase term.

The optimum receiver for recovering the MPSK signal in

the presence of AWGN is the maximum a posteriori (MAP) corre-

lator receiver shown in Figure 2.1. This receiver is optimum

in the sense of minimum probability of error.

14

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r(t)

4)2(t)

m

Choose Largest

NOTE: S.., i = 1,2,...,M, j = 1,2 represent 1-, multiolication factors

Figure 2.1 Optimum MPSK Receiver for AWGN Channel

15

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Because the signal is transmitted over a channel assumed

to be corrupted by both white Gaussian noise and statistically

independent colored noise jamming, the input to the receiver

front end is the signal r(t)/ where

r(t) = si(t) + nw(t) + nc(t) 0 < t < Ts; (2.2)

i = 1,2,...fM

Here, n (t) is a sample function of a white Gaussian noise w

process with zero mean and two-sided power spectral density

level Nn/2 watts/Hz, and n (t) is a sample function of a

colored Gaussian noise process having autocorrelation function

K (T). Since the colored Gaussian noise is generated by a

jammer operating independently of the AWGN in the channel,

it is reasonable to assume that n (t) and n (t) are uncorre- c w

lated random processes. Because n (t) and n (t) have Gaussian c w

amplitude statistics, the processes are statistically

independent.

The receiver takes advantage of the fact that the trans-

mitted signal s.{t) can be expressed as a linear combination

of two orthonormal basis functions (j), (t) and (K(t) where

(i>l(t) = /2/Ts cos (2TTf0t+a) and ^(t) = v/2/Ts sin(2TTf0t+a;

(2.3)

Therefore,

16

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2 si(t) = ^ Sin*n(t) i»l/2,...,M (2.4)

n=l

where

T s

Sin = ^ si(t)(j)n(t)dt n = 1,2; i=l,2,...,M (2.5)

In the MPSK case being considered, the two basis functions

were found by inspection. In more complicated modulation

schemes it may be necessary to use techniques such as the

Gramm-Schmitt orthonormalization procedure in order to deter-

mine the basis functions that allow a signal expansion of the

form given by equation (2.4).

From s.(t) and the above definition, it is easily shown that

„ 2TT (i-1) T _ „ /-I ^ S., = cos — i = 1,2,...,M (2.6)

Si2 = - sin27T(^"^- i = 1,2,...,M (2.7)

Assuming equal prior probabilities for the transmitted

signals, the receiver of Figure 2.1 computes and bases its

decisions on

£. = S.,Y, + S.oy0 (2.8) i il 1 i2 2

where

17

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T s

Y. = / r(t)(t..(t)dt j = 1,2 (2.9) 3 0 J

Defining now

,, ä iliiiil (2#1o) i M

then

£i = Y, cose. - Y» sin9. = Vcos{ei+n) (2.11)

i = 1,2, ... ,M

where

V = V/Y^ + Y^ ; H = tan"1(Y2/Y1) (2.12)

A determination of the performance of the receiver of

Figure 2.1 i' terms of the probability of decision errors is

now made by fir. t observing that Y, and Y- are conditionally

Gaussian random va- ;ables. Thus the mean of Y, and Y- condi-

tioned on the signal t (t) being sent is

EiY,|s.(t)} - /Ec S.. (2.13) i' i s il

E{Y0 Is. (t) } = /E 3.,, (2.14)

18

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and similarly, the conditional variance of Y, and Y- is

T T

N s s a" , -i. + / / K (t-x)*,(t)^,(T)dt dx (2.15) ^1 z 0 0

T T

7 N. S S

a^ = " + / / K (t-T)(Mt)(j),(T)dt dx (2.16) x2 0 0 t z

In Appendix A we demonstrate that

T T T T s s s s

/ / K (t-T)(J), {t)(j>, (T)dtdT = / / K (t-T)(j). (t)(})„(T)dt dx 00 00 * t

= a2 (2.17) c

so that the conditional variance of Y, and Y_ are identical.

Another important parameter which is an indication of how

Y-, and Y2 are statistically related is the conditional cross

covariance of Y, and Y~, namely

E{ [Y1-E{Y1}] [Y2-E{Y2}]} = E{nw nw } + E{nc nc } (2.18)

where

T T s s

n = J n (t)(J).(t)dt ; n = / n (t)(j).(t)dt (2.19) wj 0 w ] Cj 0 :

j = 1.2

19

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It is simple to demonstrate that

E{n n } - 0 (2.20) wl w2

while the second term in equation (2.18) which takes on the

form

T T s s

E{n n } = / / K (t-T)^, (t)(MT)dt dT (2.21) cl c2 0 0 J. ^

needs more detailed analysis. First, we let

4). (t) 0 < t < T i i i - - s

:t) = ! *•(t) = [ (2.22)

otherwise I

so that (jK(t) is defined for all time t. Let *'. (f) and S (f)

be the Fourier transforms of ^ (t) and K (t), respectively.

Thus we can show that

T T S S oo

/ f K (t-T)4)1(t)())2(T)dtdT = / S (f)<t,1(-f)$2(f)df (2.23)

0 0 . i _oo

From equation (2.3) it is clear that

$j(f)| = \^2(f)\ (2.24;

20

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r ■ • '

and

2 '7 ac = I Sc(f) |<I>1(f) Tdf (2.25)

This result will prove useful in obtaining the optimum jamming

spectrum, S (f) .

We demonstrate in Appendix A that

E{n n } =0 (2.26) c1 c2

so that

E{[Y1-E{Y1}] [Y2-E{Y2}]} = 0 (2.27)

which proves that Y, and Y- are uncorrelated. Since both

Y-, and Yj are conditionally Gaussian random variables, they

are therefore statistically independent.

This fact makes the joint probability density function of

Y-, and Y2 mathematically tractable. Using the general form

for the probability density function (p.d.f.) of an N-dimen-

sional Gaussian vector x, namely

fx^) = l0 ,N/2|A|l/2 exp{4(x-mx)y (x-mx)} (2.28) — (2 TT) ' A

where

21

V.vV-v

. iL ■ i ■ ^ j ii ■ jiiiii'rj»■!'rv.»^.i \v}^ ßi'.'.' JJ J .'j .».v. ^j J;.■.^.v v ■■■ v■.'v■.' ■:» '.|' '> v v ■."'J■> v ■."'wry ■.'• ■>'."pyj■>'

m = E {x} (2.29)

and

A = E{(x-m )(x-m ) } (2.30)

the joint p.d.f. of Y, and Y- can now be specified.

The case of MPSK under consideration is a two-dimensional

problem where

m —x

s il

s i2 J

i = 1,2,...,M (2.31)

and

r 2 a 0

L 0 o J

2 N0 2 a = ^- + V (2.32)

so that

A ~x

N r 0 , 2,2 4 [-5- + ac] = a (2.33)

From equation (2.32),

-1 4; (2.34)

22

'fy£f.'J'frrf,rfTJ:ir:*:'*:Tr,r*.'VV>'V /■>'^■'^J»■;>•.'»■;■«■/»:JT^''.,y.TV».,5V^"'.V.rf,l^'.'»'l-'VV',nT'-< WT ' -v v'rivwwinv^.fnm«vi^r- v^vi

where I is a correspondingly dimensioned identity matrix.

Thus, the conditional p.d.f. of Y^ and Y2, given that s.(t)

was transmitted, becomes

^^la.Ct)^'8^ —yexp 2™

L Y.-^TS., 1 S ll

•2 s i2 J

iT

1/a'

I/o'

Y,-'/E~S.. 1 S ll

2 s i2 J) (2.35)

i = 1,2,...,M

A double random variable transformation is now introduced,

that is.

V = ^+Y| ; n = tan"1(Y2/Y1) (2.36)

This transformation leads to a new conditional p.d.f., namely

^Hls.^^'V1^ = vfY]fY2lsi(t)(VCOSn'VSinn'Si(t))

+ vf Y 1 ... (-vcosn,-vsinn|si(t)) (2.37!

v>o, o<n<Tf

where from equation (2.35) we obtain

23

!£:&>^>;^>';>>-^ .V.

fV,H|8i{t)(^'8i(t,) = ^^H

1 T

I/o'

vcosn-^Cs., s 11

vsinn-t^~S. - s 12

voosn-v^Ts., s ll

I/o' vsinn-'/E~s s Jl

f-^-exp J-i 27ro 6 "

l/o^

-vcosn-v^Ts.,

s 1Z

0 1/a

-vcosn- ̂ ul) -vsinri->^~S. - s i2

(2.38)

v>0, 0<ri<TT, i = 1,2,...,M

This p.d.f. can be simplified to the form

1 r 2 A/«1= fM^I3^5 = —^yexE.i--^-[v-2v^"cos(e.+n)+Ec]} v,tt|siit; i 27TOz 2oz s j s

1 r 2

2TT7 2? s D s (2.39)

v>_0, 0_<n^TT

It is apparent from the range of n that the two exponential

terms in equation (2.39) can be replaced by a single exponen-

tial term by allowing n to range from 0 to 27T. Therefore we

have finally,

24

.N..V."- V '-.-•■■■'. v\-.■-■' '-" V---■ < l-'-« - * -'■■ i-'> - ■ -'■ -'■ -'■» -*» ■-* -'■ -"■ -'* *-'-* --,« -**

^JWfVfviTKfwjr^Hjirji'- » "3^.ir»tf»T?-wv^v^\r<l^vitl^lfWVVv^\nivvuiivii^^wirai^>yi^v\iw\>«-\J^R.ivi>yi^r»^i-^n^-»^n»qniv»^.T«ITä^•

1 r. 2 ^T v\a i^^fM^M) = -^-expC-^ylv-2v^Fbos(9.+Ti)+Ej} (2.40) -V/H|si(t) i 27TO^ 2a^ S 3 s

v>0, 0<n<2Tr

The probability density function of n conditioned on

s.(t) can be obtained by integrating equation (2.40) over

the range of V, namely.

fH|= ^(n|s. (t)) = / -^5-exE>{-^T[v2-2v/rcos(e.+n)+E<5]}dv (2.41) H|si(t) * 0 2TT7 27 s ] s

This integration results in

fuU ^\(n|s. (t)) = ^rescpC i sin (6 .+n)} [e}cp{ ^oos (e.+n)} H|si(t) ' i 2^ ^ 2o2 3 ^ 2o2 j

+ \/2TTEg/a cos(6.+n)erfÄ{\/Es/?cos (e.+n))! (2.42)

where

erf*{a} = / — exp{-x2/2}dx = 1 - erfc*{a} (2.43)

Recalling from equation (2.11) that the decision rule was

based on evaluating

«.. = V cos(e.+n) i = 1,2,...,M (2.44)

25

jukaSajLfc^k^jalaLlaL^ii k^^^vi's^f'vi-VAi-i.ANVi .■> v.. . .^. '^v.-^.*' ■.'v'.»..-! JCv.\;vI'^'v. ^^^»•^•^••^Iv/n-^^jf.v-j

WJ.^Yr'w»,^*'-'/'^^^^

and determining which of them is largest. Therefore, if

s.(t) was sent, a correct decision will be made by the

detector if

V cos (6 i+n) > Vcos(e.+n) j = 1,2,...,M; i ^ j (2.45)

Since cos(x) is a maximum when |x| is a minimum, the decision

rule can be modified to become

i.+nj < |9.+ri| •*■ decide s.(t) was transmitted (2.46)

From equation (2.10) we have

ei = ^^"^ i = 1,2,...,M (2.47)

so that the condition of equation (2.46) is satisfied if

-e. -iSr < n < -e. +J (2.48; l M 1 M

Therefore the probability of making a correct decision given

that s.(t) was sent is 1

-e. +TT/M

Pr{c|si(t)} = / fHls (t)(4;'si(t))d,|;

-e.-ir/M ' i

TT/M

/ /MfH|s.(t)(ß-eilSi(t))dß (2-49)

-TT/M ' 1

26

■■ .•■ .•-.-■ -■• .N -•■ .-■ -^ .--^ .•-J.--V->--J.\.-'V-Vv,>-.'' .■-.,•-_.•- .■■ .M>V\ .■• V- ^. -V.' •■■.V>"V->\v v •''."'VO.I

•,* -.^..■V.""'' * •\w^Ti^:\m^'J.mK.'' ■Ur-:wV ■ V ■ 'J ▼ I "A1' J TV« t^ il."TT-t.-TKTK^TLT!ürvT«.^«.i wn.1 ^.-mn viKT. •-T.VI Vl.^.i '. ^ ■«.-i •- - -.-T-WI-VI -.. •.. l/Ti 1 ^■•VW V» W^W-U

From equation (2.42) we obtain

TT/M , E 2 EQ 2

Pr. J|S. (t)} = / ^expf lsin^ß}texp{—loos 8} 1 -TT/M ^^ 2o 2a

+ V^Eg/a oosßerfÄ{ x/Eg/a2 oosß}]dß (2.50)

and defining

Rn = 2E /N : signal power to Noise power Ratio S (SNR) (2.51)

RT = a /E : jamming power to Signal power Ratio C S (JSR) ' ' (2.52)

Since each signal was assumed to be equally likely, the

desired result is

1 "^ */M ^___ Pr{c} = M^W+V/ ^-./M 'V2^1^^005 ß

2 ,-R^sin ßv

X eXpi2(l+RnR ) f ^^V^Vj COSß} dß (2-53)

This result specifies the performance of an MPSK receiver

operating in an environment corrupted by both AWGN and colored

noise jamming.

The probability of a symbol error is defined as

Pr{e} = 1 - Pr{c} (2.54)

27

vjFyiytyfinxv.v* w*.],'*. '.^'^ "^.y wv^ aw wi.^ i* ^ ^T* i.^. i^ t^ '.•* ^' fv ■.■■' '.^i,-.1 '.^ '.^'i-v.-i '.v.-. ^ ^ \y ^^rr.Tr^.^.-»':'-''.^".''\^.-:-^v.'^,^.--'

Observe that if the colored noise is not present, R = 0,

2 a = N-/2, and equation (2.53) becomes the classical expression

for the performance of an MPSK receiver in the presence of

AWGN [Ref. 2], that is

, E TT/M E - Pr{c} = rrexpl-T^-} + / /E /TTN,, cosßexp{- r^- sin ß}

M N0 -TT/M s 0 N0

x erf*{/2Es/H0cosß}d6 (2.55)

Furthermore, if the jamming power were assumed to become

infinitely large.

lim Pr{c} = i (2.56)

J

which, as expected, is the minimum value one can expect for a

set of M equiprobable signals.

The results of equation (2.54) were numerically evaluated

and plotted for the cases M = 4, 8, 16 and 32 in Figures 2.2

through 2.5, respectively. In each graph, the special case

of JSR 2 0.0 is included as a convenient reference to the

performance of the particular received in AWGN interference

only.

Now that an expression for the performance of an MPSK

receiver in the presence of colored noise jamming has been

derived, we next optimize the jammer in such a way as to

28

4-PSK RECEIVER PERFORMANCE

-2.0

Oo »H :

1 ̂ ■■" '"""^: ; " ; : '" 1 1 -^^^v^ y w- ..:■ .Q ^1

1 ^^V^^-^ o -

•{^{{E{^^'-^f;"^{{-\'v{u-{--t^^\;^3^^ ; ; Xlir ; 1

• ^\ ■ 1

HO 1 i \^ ..1^ = ■^vV--;---- ■-:

O = I .-y • Y -i ::::::::::::;::::;:;:::::::::;::;:::;::;::::;::::::::::::::;:X:^::::::::::

P S " i • ' \\

Pr. T-( Z i H\

o - I | ::.::::::^:: V:::

>- ; ; ..:. \ -\ H | '; i \

rn 0_ : <^^

LEGEND o JSR= 0.0 o JSR= 0 DB A JSR = -10 DB + JSR = -20 DB

m^^A m . .. A .1 z*^ "A o ::..\.J o-. -

'o i \

\ ^—1 z •i-*

- • •. A

\ \

'o. 1 , ^ , 2.5 7.0 11.5 16.0

SIGNAL TO AWGN RATIO (RD=2ES/N0)

Figure 2.2 4-PSK Receiver Performance

29

.v.. - <. J^A^jf-V. n*. «'-'l'-V^ f.\'- ,•', U^>J>£££*JL

Ö-PSK RECEIVER PERFORMANCE

PQ O

OH

'o.

o JSR = 0 DB A JSR = -10 DB + JSR = -20 DB

-2.0 4.0 10.0 16.0 22.0

SIGNAL TO AWGN RATIO (RD=2ES/N0)

Figure 2.3 8-PSK Receiver Performance

30

^JIT* '■> *> ' > * '' '■V'*.'^',y,■^TT,VT?^ VT g^ V ^ '. » V* l1^V* ^ 'J*'1-1^ IT» L^ V* '/I V1^ V^ L^VX gl'■'% V^'V'TIV»UWfUmtni WA-Wl.-« L-»vn-L-mrwL^ ir»<?wir»in[ i,-^mt-i

16-PSK RECEIVER PERFORMANCE

---•---'*■ IM Q ■ • /^' j )

-2.0 5.5 13.0 20.5 28.0

SIGNAL TO AWGN RATIO (RD=2ES/N0)

Figure 2,4 16-PSK Receiver Performance

31

•V'vv.-V- ■^^*'y^y.xs:/:s:/:/j^>':^

W7^^^^^^HW^^^!^^^^Ä^.V.\W:V^!WW^Jr^WTO?'^^?*W.mTIiVWRVWWWATV V «.''».M." VT. T.'TT.^'t^v-» ; ^ ■-, -

32-PSK RECEIVER PERFORMANCE

>He

H

PQ < PQ O

PU1-

\'^\'.'CT,l.l.l,','.'.l.r^A','..'.'!.T.'..T.'.'Ä.'.'.'..'.'.'.'.'.'i5i k

LEGEND a JSR= 0.0 o JSR = 0 DB A JSR = -10 DB + JSR = -20 DB

-2.0 7.0 16.0 25.0 34.0

SIGNAL TO NOISE RATIO (2ES/N0)

Figure 2.5 32-PSK Receiver Performance

32

•y^^■;".:■.:<.;<<'->.:■>.:■.; .:••.: :-\«••,:-•.%•.: yoc\ 'V-./\-V^":.:A:^\\ v %•■-. vvv:-.-A >. :->\ s

-..r-i^1J»T-w--t^|.^LWIYf|fl^lH»y»^IHH|.^.f..»^. ■—V..-r.-.~.-rw1^....-.r-....—^.T ^ l 1

cause maximum degradation to the receiver performance while

maintaining power constrained conditions on the jammer.

B. OPTIMIZATION OF THE COLORED NOISE JAMMER

We now investigate the dependence of Pr{c} on the jamming

to signal ratio, R , by analyzing the unsimplified form of

Pr{c} given by equation (2.49) with equation (2.41) substituted

for the conditional p.d.f., namely

IT/M «> 15 Pr{c} = 2/ / ^ exp{- ±-[x^x/P'cosß+RlHdxdß (2.57)

0 0 2TT 2 D D

where now

E RI = —,- (2.58)

V2 + °c

With the double change of variables

u = x cosß ; v = x sin Q (2.59)

equation (2.57) becomes

oo i _ o utanir/M , Pr{c} = 2/ -i-exp{- y[u-/Rl]^}/ -±-exp {-vV2 }dvdu

0 /2TT ^ ü 0 /27T

(2.60)

We notice that in terms of R- and R ,

/RD = ^/1+RDRJ {2-61)

33

k-^^lA^^^l-aL^i^I^jjl^Al^aL^rfJh^

*»^ v ■.' uymymy9ymyymT'j!*y 'yi Twvjwmrpr 'vmrmrvrwrxrnvnrv mj:rjmy'? ■> v v ■■» ■? ■> •> ^ WJ- 'j» VJ v '>-

Evaluating now the derivative of equation (2.60) with respect

to Rj, we obtain

P 00

£- Pr{c} = - i\/ —T / erfÄ{utamT/M} dRJ ^(IH-R^)3 J0

x [u-ZRp/l+RpRjlexpl- i[u-/R^7l+R^]2}du (2.62)

and with the change of variables

x = i[u -/Rp/1+Rj^Rj]2 (2.63)

we have

/ 3 VR ^ 5_ / erfÄ{[/2^

(^VJ) V2(1+RDRJ)

-x + /Rp/I+RpRj] tan TT/Mle dx (2.64!

Since both erf^Ix} and e are non-negative functions for

all x, the integration in equation (2.64) must result in a

non-negative quantity. Therefore,

~- Pr(r} < 0 (2.65) J

which indicates that the probability of making a correct

decision is a monotonically decreasing function with respect

34

fe^fcy^^-^

LTTT? ' T^y^Sr-JSTJTF W vry* VT» V-» VVk.-Ki.TmniwVirOTVWVin-Vi nr. ■vi'V» J»n nn i*n «.1 vi wx via

to Rj. As a result, in order to maximize the detrimental

effect of the jammer on the performance of the MPSK receiver,

we will want to maximize the jamming power, within assumed

power constrained limits.

Returning to equations (2.25) and (2.52), for a fixed

signal energy, R can be maximized by maximizing J

0c = / Sc(f)|$1(f)|2df (2.66)

By the Cauchy-Schwartz inequality

I CO oo

0c -yl S2(f)df / |$;[(f)|4df (2.67)

with equality if and only if

Sc(f) = K|$1(f)|2 (2.68)

where K is an arbitrary constant. Since we require that

the jamming signal be power constrained, that is

oo

P = f S (f)df < oo (2.69) c J c

— 00

then, provided that

35

>:^>:^^

;W*M 'Ri pjina niTJT'JI • j »ji ■> 'j ■.» "j "j 'j i"^ rj *J 'r -* \*.'r rw. v r?jvriPJV.^Jv.'\^KnrjTrsrj^nrrr*irjir:v\rvvnrj-ir\nn. *■. *•.'>rirT^ ra^jJTBjn^En

/ |*^(f)|2df < » (2.70}

we must select K such that the power constraint of equation

(2.69) is satisfied.

For

())1(t) = /2/Ts cos [2TTf0t+a] 0 1 t <_ Ts (2.71)

we can define

(/2/T 0 < t < T .em y <■/ ±s u _ ^ _ ^g sinirfT p(t) = j ^P(f) = /5f^exp{-JTTfTs}—^jr-

1 0 otherwise

s

(2.72)

The Fourier Transform of cos [2TTfQt+a] is

2-[6(f-f0) +6(f+f0)]exp{jfa/f0} (2.73)

so that

$^(f) = |[P(f-f0)+P(f+f0)]exp{jfoi/f0} *-* <t>^{t)

(2.74)

and

jct^f)!2 = ^[|P(f-f0)l2+|P(f+f0) |2+P(f-f0)P*(f+f0)+P*(f-f0)P(f+f0)]

(2.75)

36

' v^ ^ '^^^ ^^: • ^^>^^^"^^^^>^i^ ^^^/j^;^^»:v?^Xv:

The last two terms in equation (2.7 5) for reasonable values

of carrier frequency, fn, are negligible. Returning to

equation (2.68), we now have

sc(f) = |[|p(f-fn)|2 + |P(f+fn)|

2] (2.76)

and this results in

/ Sc(f)df = K —en

(2.77)

Therefore

Sc(f) = pc|^(f)|2 (2.78)

so that

16 Pc (2.79)

thus demonstrating that optimized jamming of an MPSK receiver

requires in essence the spectrum of the colored noise jammer

to be matched tc the spectrum of the MPSK signal.

3 7

teW.frte^^

III. SPECIAL CASES OF COHERENT MPSK; QUADRATURE PHASE SHIFT KEYING, OFFSET QUADRATURE PHASE

SHIFT KEYING AND MINIMUM SHIFT KEYING

Quadrature Phase Shift Keyed (QPSK) modulation is a

special case of MPSK modulation where M = 4. Offset QPSK

(OQPSK) is a special case of QPSK in which the in-phase and

quadrature data components are offset in time by one-half of

a symbol interval. Minimum Shift Keying (MSK) is furthermore

a special case of OQPSK in which sinusoidal pulse weighting

prior to carrier modulation produces a more compact signal

spectrum making this modulation scheme very useful in cases

where severe bandwidth constraints are imposed.

A. QPSK RECEIVER PERFORMANCE

The optimum receiver for a QPSK modulated signal is a

special version of that shown in Figure 2.1 where now M = 4

and appropriate simplifications are made as shown in Figure

3.1.

Development of this receiver's performance from equation

(2.53) with M = 4 is mathematically difficult and does not

provide any significant insight to the results derived.

Therefore, the QPSK receiver performance will be developed

by relying on res-.ults presented in the previous chapter so

as to be able to reduce the necessary development.

For the QPSK signaling scheme, the transmitted signals

are as given by equation (2.1) of the form

38

'.A

^^•'JV^n'V'J'VTW'R'^'^AVWWW^'.'W^v^T^.^^l^^^^ •jrji-JV'«ivnÄ.'vjta-wwOT«

r(t)

>0

<0

m, = 1 | T -dt

^(t) = ^7^"ccs (2^f0t+a)

t)2(t) = /2/r sin (2TTf0t+a)

"b. (t) t = nT

>0 ITU = 1

<0 I!^ = -1 1/. ^ * r i^t t = nT

Figure 3.1 Optimum OPSK Receiver Structure

39

i&&&^^

*X^F^&VV-'\V*?**W[KV.^[KVW\**tr.K\K\v:\K\^ \-.-."i-■»I

Si(t) = /2Es/Ts cos [2TTf0t +1T(l^l) +a] 1=1,2,3,4 (3.1)

We will consider the case in which all signals s.(t), i = 1,2,

3,4, are equally likely to be transmitted. The analysis

begins by assuming that signal s,(t) was transmitted. Under

this assumption, the statistics of Y, and Y2 are

E{Y, } = E{Y0} = Ea {3.2) 1 2 s

N0 2 Var{Y, } = Var{Y0} = 4^+0 (3.3)

E{ [Y1-E{Y1}] [Y2-E{Y2}]} = 0 (3.4)

Just as in the MPSK case, both Y-, and Y- are conditionally

Gaussian random variables which, due to their uncorrelated-

ness, are statistically independent.

The joint probability density function of Y, and Y- is

1 (YrEs)2 (Y2-Es)2

VYJSV7^

15!^

= 7VP{--^TT-^^-T^-} (3-5)

1 2' 1 zira to 2.0

where

2 No . 2 i-> a\ a = -j- + ac (3.6)

In order to find the probability of making a correct

decision conditioned on the assumption that signal s-, (t) was

40

asaaaa^im&flä^^

|<r>WW^W^^^^^^WtWWWWWT^T?n^^^^^iTJ^^ ''J T[ W} f."^1 ^ fi ^') WEV 'Tir: rjy;wrjyjyj rj ^.i /M tfvwuy/fwfr^i^.' miy.

sent, we first examine the QPSK signal space diagram of

Figure 3.2 so as to be able to determine this probability.

For the QPSK case, the axes of the state space diagram serve

as the perpendicular bisectors separating the decision regions

associated with each signal.

Given that s1(t) was sent, the probability of the detec-

tor making a correct decision is

00 00

Prlcls^t)} = /o/ofYrY2|S2(t)(yry2ls1(t))dy1dY2 (3.7)

Evaluation of equation (3.7) using equation (3.5) yields

the result

Pr{c|s1(t)} = [erfÄ{/RD/2(l+RDRJ)}]' (3.8)

where again

R- = 2E /Nn: Signal to Noise Ratio (SNR) (3.9) D s 0 ^

2 RT = a /E : Jamming to Signal Ratio (JSR) (3.10) J c s

A similar development for s2(t), s3(t) and s.(t) results in

Pr{c|s1(t)} = Pr{c|s2(t)} = Pr{cls3(t)}

= Pr{c|s4(t)} (3.11)

41

> ■J- -. -.. ■. - -^ -.. -J. -.. .- -.- .- -.. v/,..f. ..-.-.. , .- . ^v^v^"-^^'J^>/^,^^^■.^^^v.^^VJ,v^^,wO,^v.v?>J■^>:>^A?•>>^•^«•>,.'>/lv^XwJ•^i

^^i^iv"*L*<.4i.*!^Vi;*rVViv"iL'»y^^^

Figure 3.2 Signal Space Diagran for QPSK Signaling

42

*.-•J.^>'J.1'.'\- v v v v •.• v.V-'.-.v.\-.v,v.VMV>.v..vv^vy^^^'v-'^V

J,^-'«.-V^'

and therefore, since all signals are assumed equally likely

Pr{c} = [erf^{/RD/2(l+RDRJ)}]' = 1 - Pr{e} (3.12)

Observe that if the colored noise is not present, RT = 0, 2

a = N./2 and equation (3.12) becomes

Pr{c} = [erfA{/Es/N0}r (3.13)

which is the probability of making a correct decision for a

QPSK modulated signal transmitted over a channel corrupted

by AWGN.

Furthermore, if the jamming power grows without bound,

lim Pr{c} = j (3.14)

V00

which is as expected the minimum value of the probability of

a correct decision for a set of 4 equiprobable signals.

The optimization of the jammer is identical to that

derived for MPSK signaling and as such, the spectrum of the

optimum colored noise jammer is given by

S (f) = P K (f) |2 (3.15) c c i

where

al = yi- P^ (3.16) c 16 c

43

^^T^^yifff^w^y^^^in■ ..i n 'i»■.■>vi'iy.n j». ^'ji'j^rjg'j y.iyj^•,^rTrTT7^i^riTi^y^:y,r^r^:,^r^rJ^^JV'iv^irrrrliK.~v,rf,ivrr.

The corresponding probability of symbol error is plotted in

Figure 3.3 as a function of SNR for various jamming-to-signal

power ratios.

B. OFFSET QPSK RECEIVER PERFORMANCE

QPSK signaling techniques as previously pointed out are

attractive from a bandwidth efficiency point of view. For

an unfiltered QPSK signal, phase transitions occur instan-

taneously resulting in a constant amplitude envelope signal.

However, phase changes for filtered QPSK signals result in a

varying envelope amplitude. Offset QPSK signaling, in which

the in-phase and quadrature data bits are offset or staggered

by one-half of a symbol interval results in a more constant

amplitude envelope even after filtering. When a bandlimited

offset QPSK signal is transmitted through an amplitude-limiting

device, there is only partial regeneration of the spectrum

amplitude back to the unfiltered level. For QPSK under the

same circumstances, however, there is almost complete regener-

ation to the unfiltered level. [Ref. 3]

Figure 3.4 shows the structure of the optimum offset QPSK

receiver.

Since offset QPSK uses the same principles, waveforms and

receiver structure as those used in QPSK with the exception

that now, one channel is offset in time with respect to the

other one by one-half of a symbol interval, it should not be

surprising to find that the performance of the receiver of

Figure 3.4, with or without jamming is identical to that of

44

::^>^>::^

FWW^^WÄF^^TP^^^^W^WVj^T-ÄV*-iÄfJlW,Ä"*MWTJWflÄWVF"'J7?r.T^ WTnnnra Tmr xv -n

m o K

QPSK RECEIVER PERFORMANCE

LEGEND □ JSR= 0.0 o JSR= 0 DB A JSR = -10 DB + JSR = -20 DB

-2.0 2.5 7.0 11.5 16.0

SIGNAL TO AWGN RATIO (RD=2ES/NO)

Figure 3.3 QPSK Receiver Performance

45

^ sv '«:<-: tä&Stääü^^

^^^^Wf*f^^^), - u' ■.-^ v^ >.. ^ -M-t-v-F it1 ^ t »'i'^ VPil*r'.Vi'*ii'*ji*li<'.'j*> "T'j; ',w .^"ji^j.,1 fj '»JL1 r^jiry^jf v r> rji r^^-vüpr^jjijgL-jCTwrjt-ji-jM

-(EHE r(t)

> 0 m. = 1

< 0 Ji • -i

t = nT

^(t) = ^27T^cos(2TTf0tH-a)

i^ > 0 nu

< 0 n^

1

-1 ■is- 1 T?2 it)

nT t =

SZTT sin (2TTf0t+a)

Figure 3.4 Optimum Offset QPSK Receiver Structure

46

<.'•". •/■.■' <". -' % -••."' v. - - v ■ ^ '- -f- ■ - - - v ^ ■ - - - -^ ■ -^ - - - • IMA i.^i^l ^^^■^

^^^■■•;.-^:o:.,-: ■^

^^f^^^^W^^^^^^^^^^T^-*.-* -v T^ r*r*-J "J mJ "J "Jl fM 9JI *jn.Vy ^ HWV W S^ '.^\^r.T-'l."vv%'\Trl,v.,*wv,«i«n -v ■ - -j--v -v ^^

the QPSK receiver of Figure 3.1. The only difference is that

the offset QPSK receiver operates on delayed data so that one

correlator must offset its integration interval accordingly.

Therefore, the probability of error is as given in equa-

tion (3.12), namely

Pr{e} = 1 - [erfÄ{/RD/2(l+RDRJ)}]' (3.17)

and the corresponding performance curves are obviously identi-

cal to those shown in Figure 3.3.

C. MSK RECEIVER PERFORMANCE

The logical progression from QPSK to offset QPSK suggests

that further suppression of out-of-band interference in band-

limiting applications can be obtained if the offset QPSK signal

format is modified to avoid phase transitions altogether.

Minimum Shift Keying (MSK) is a constant envelope modulation

with continuous phase at the bit transition times which pro-

vides the desired sideband suppression. The MSK signal can

be considered to be an offset QPSK signal with sinusoidal

pulse weighting. [Ref. 4]

The transmitted signals are of the form

s. (t) = /2E Ac aT (t) cos (2TTf, t) cos [27Tf .t-tct ] 1 S ^ 1 -L U

+ /2E^Tsa (Usin^f-jUsin^TTfgt+a] 1=1,2,3,4 (3.18)

47

M"/J'ÄWWJPJIPJKfW*!ViWiJTJrj»:'.'»J";»-J• ^ry^y^'•.'''.''■'^r^C^T^T*.■'/-"'."^^.■vv*XTTTyjr?\*T'"\v'^ -v ■' '.v.^.-.v

where aT(t) and a0(t) are the in-phase and quadrature binary

data. The orthonormal basis functions necessary to represent

the MSK signals as an equivalent orthogonal series now take

on the form

^(t) = /4/T cos {27rf1t)cos(2TTf t+a] ;

(3.19)

<t>2(t) = /4/Tg sin (2TTf1t)sin[27Tf0t+a]

The optimum receiver for the recovery of the MSK signals

is shown in Figure 3.5.

Assuming that all signals s.(t), i = 1,2,3,4 are equally

likely to be transmitted and given a priori knowledge that

signal s,(t) was transmitted, the received signal is

r(t) = s, (t) + n (t) + n,(t) (3.20)

where again n (t) is a sample function of a white Gaussian

noise process with zero mean and two-sided power spectral den-

sity level N0/2 watts/Hz, and n (t) is a sample function of

a colored Gaussian noise process having autocorrelation

function K (T ) . c

The statistics of Y, and Y- are given by

E{Y1|s1(t)} = E{Y2|s1(t)} = Es (3.21)

Var{Y1Is,(t)} = E{n2 } + E{n2 } (3.22) i' i w, c.

48

^<<^ ^ ■: : ■•//^^:<-.; \-^^

^^^^^^^^W^^^^WWWWWTWWWWTW*!*!!»V^jrjl 'jf'i^ . rg»? ^v.i'j n p^mrjK'vjrv w\ TB^ J«T; ^. ^; ',' T. TP^V T¥ m Wt'

r(t)

> 0 at = 1

< 0 »j = -1

^(t) = /4/Tscos (2Mf1t)cos[27Tf0t+a]

>0 a=l

< 0 a. Q = -1

(})2(t) = /4/Tssin {2-nf1 it) sin [2iTf-t+ct]

•TS (t)

t = nT

♦ a, (t)

t^-f

Figure 3.5 Optimum MSK Receiver Structure

49

^^^^MfäN&a^

^^^^p^^^^^^F^^^^"V-»»-w-vV«'.• "^"^ -j -.1 -.i ■.«■>■ »»j■>->'■".¥■->■--*»¥«v"»\i""vrvTir"."."■■k'vif VTfc"« wu".v»Twir"ji"^"ji'jr»^^. •."••".■

Var{Y2|s1(t) } = ECn2, } + E{n2 } w2 c2

(3.23)

where

T 3T /2 s s

nw = / n (t)(j) (t)dt ; nw = / n^ (t) (()_ (t)dt (3.24) wl 0 W2 T /2 W ^ s

and

3T /2 s

nc » / nc{t)())1{t)dt ; nc = / nc(t) (j)2 (t)dt (3.25) '2 Ts/2

Also

£{[¥,-£{¥,}][¥.-£{¥-}]} = E{n n } + E{n n } (3.26) i. L 2. 2. W-, W- C, C-

Evaluating the first term in equation (3.26) yields

, Ts

3TS/

2

^SS1 = TO nw(tHl(t)dt ^^ nw(T)VT)dT;

s N, = / -j (t)1(t)(()2(t)dt =

T /2 s'

(3.27)

From this result we observe that

N ^v E{n" } W2

50

(3.28)

The second term in equation (3.26) leads to

T 3T /2 s s

E{n n } = / / K (t-xjct), (t)4).(T)dt dx (3.29) Cl C2 0 T /2 c i "

s

Using techniques similar to those used in the MPSK case, it

can be shown that

E{n n } = 0 (3.30) cl c2

Therefore

E{[Y1-E{y1}][Y2-E{Y2}]} = 0 (3.31)

so that the remaining analysis for the performance of the MSK

receiver is identical to that of the QPSK receiver. There-

fore, the probability of a symbol error is

Pr{e} = 1 - [erfj,{/RD/2(l+RDRJ)}]" (3.32)

and the spectrum of the optimum colored noise jammer has the

same mathematical form as that of MPSK, QPSK and OQPSK,

namely

S (f) = P U (f)|2 (3.33) C C -L.

51

'**** v M v\' i ■»«».■ i.« w,■;.«.,• w.- j.- v»v H.-1.. H,-..-i;-»^^^^^••.^'/^^^^v^^^^.v^^^v.^ ,%-^^^i^n,^j;-v..^>n.<>^>-.>^>^

except that now, due to the modification of ^^(t) as indi-

cated by equation (3.19), the spectrum of the colored noise

jammer is given by

-, 4P E sc(f) = pclvf>r = -V

COS 7T (f-fn)T„ 0 s

[l-(2(f-f0)Ts)2]2

COS TT (f + f,»)? 0 S

[l-(2(f+f0)Ts)2]2

(3.34)

as opposed to the spectrum of the colored noise jammer for

QPSK given by

sc(f) = P.I^Cf) E P s c

\lsinnf-f0)TsY

[\ ^f-f0)Ts I

/sin^f+f0)Ts\2

\ ^f+f0)Ts/ (3.35)

52

M&Xtä&Ym^MS^^^^ •v.-'

yw.-j- — .'-.- ^--P '^ V "7 W W '^ V« '^ V ^ ^> V Vtf'lW M f^i TV t^v^vv^r^r^v^v^vinncy^nryp ^ ^ T^ VT v^ V»vw y» vwiu-iri.Tr v^» tn« T« k~p wi

IV. DIFFERENTIALLY COHERENT PHASE SHIFT KEYING

Differentially Coherent Phase Shift Keying (DPSK) is a

signaling technique which eliminates the need for phase

synchronization of the local carrier to the received signal

by using a delayed version of the received signal as the local

reference during demodulation. At the transmitter, the digi-

tal information is encoded into phase differences between two

successive signaling intervals and then modulated onto the

carrier using conventional PSK techniques. DPSK allows the

use of simpler and therefore less costly receiver structures

at the expense of only a slight performance degradation as

compared to coherent PSK signaling. [Ref. 5]

A. DPSK RECEIVER PERFORMANCE IN COLORED NOISE JAMMING

Contrasted to the coherent signaling techniques previously

analyzed, DPSK analysis poses a mathematically formidable

task, even in the case of binary signaling over an AWGN cor-

rupted channel. Consequently, the evaluation of the receiver

performance will use the geometric approach first developed by

Arthurs and Dym [Ref. 6], extended here t' the case where

the channel interference consists of additional colored noise

jamming.

The signal set for DPSK signaling is identical to that for

MPSK. signaling as given by equation (2.1). Define T. to be the

ith signaling interval so that t e T. + (i-l)T < t < iT ,

53

te&S^^a^^&^^^^^^ , ±^£££&ü&££8i

v:v,',T^',.VWW.N'im.^y* VW'* -iv'.> rn'ji 'r r.-» rffrrTirj '■;'»"; A'^-T-r/'/f-n si s-k-.T.^fm.^xTurv^.'rj »-u»-.w-.ir-vr-ji-j «viruw-jw-.riruv-ik-.'inrw

i = 1,2,... . The transmitted DPSK signals are therefore

given by

S(t) = /2E /T cos[2TTfnt +e(:L) +aj t e T. (4.1)

S S U I 1

where

e(i) = 9(i-l) + e^ (modulo 271) (4.2)

The phase information is denoted by 9 and a is an arbitrary,

yet fixed phase. The possible values that 9 can take on are

(i) _ 2TT(j-l) ■_■,■> w IA r,\ = jjj ] = 1,2,...,M (4.3)

The optimum receiver structure for such a signaling scheme

is shown in Figure 4.1. In Figure 4.1, ß represents the re-

ceiver phase ambiguity. In much the same way as in noncoherent

demodulation, no attempt is made to phase lock the receiver

in such a way that ß = a.

The received signal is

r(t) = s(t) + n (t) + n (t) , t € T. (4.4) w c i

where the noise components are the same as those used in

previous analyses.

In order to evaluate the DPSK receiver performance, the

probability density function of n must be found. To this

end, the statistics of Y, and Y^ are required.

54

V Wll Wli ^1IVV«.1 '.'S'.V fJ'.VW LN fV l^J U^ BK V\ CVVTVmmmn vwrnvrnv vvw \r*irmir-\.-* i

r(t)

ll*

^(t) = v^7T^COS[2TTf0t+ß] -1 h tan N-f.)

Yl

(i)

(})2(t) = /2/Tssin[27Tf0t+ß]

Figure 4.1 Ontinun DPSK Receiver Structure

.*.>■. • . P . ■ ■* . " ■ ■ v V" ■. N

Y, = /E cos[a-ß+e(l)] + n + n (4.5) i s w, c.

Y0 = -/IT sin[a-ß+e(l) ] + n + n (4.6) 2 s w~ c_

where

T T s s n = / nM(t)(t). (t)dt ; n^ = / n^, (t) (}). (t)dt (4.7) w. o w ^ cj 0 c ^

j = 1,2

The expected value of Y, and Y2 is

(i) E {Y,} = /E~ cosU + 9^'] (4.8)

E{Y9} = -/E~ sin[({)o + e(l)] (4.9)

where (j) is the phase error defined by

4>e = a - ß (4.10)

As shown for the MPSK case, a similar result holds here

in that

E{[YT-EIY,}][Y0-E{Y0}]} = E{n n } + E{n, n } = 0 (4.11] I 1 z z w-j w_ c, c~

so that

56

•

N ■ o.

0 2 2 c = a (4.12)

and

E{ [Y1-E{Y1}] [Y2-E{Y2}] } = 0 (4.13)

Again we find that Y, and Y- are independent Gaussian

random variables and therefore, the joint probability density

function of Y, and Y2 is

s-v^2'= ^^ H r(Y1-E{YI})

2 (Y2-E{Y2})2

(4.14)

In order to obtain the p.d.f. of n we will use a double

transformation of random variables, namely

V (i) N/YTT? 2 ' (l) = tan~1(Y2/Y1) (4.15)

resulting in a joint p.d.f. for V and n given by

f ii\ ti\iy,r]) = vfv v (vcosri,vsinn) + vf^ v (-vcosri,-vsinn) (4.16) V(1),HU) Yr2 Yr2

v>0; 0<n<TT

from the p.d.f. of equation (4.14) we obtain

57

vvv.^>>:w-^y-:^.y^>>.>^^

?*T*WiSfiSWi9Wf'WmfTm^3V*f*V*iFi!*-iimii*vmy*}'*y,ymy,r'± •> ■> •> v •»v»'.»v "jir^ ">■ '^-^ v».-^nFT?^yTWJvJ-Ä,-v -^ä

VU,,H^' 2TTO 2a s e s

+ -^expi-^-W -2v^E~cos[T]+<p +QU'+n]+Ej] (4.17) 2TTO 2a S e S

v >^ 0; 0 <_ n £ IT

Since the second term in equation (4.17) is equivalent to the

first with the exception of the TT radian offset, it is possible

to eliminate the second term by allowing n to range from 0

to 2TT . Thus ■

f /.x /.n (v,n) = —^^expf--iT[v2-2wfrcos[n+<t) +e(l)]+EJ (4.18) V(1),H(1) 2™/ 2? s e S

v >^ 0; 0 <_r\ <2i\

The p.d.f. of n is now obtained by integrating equation (4.18)

over the range of V, namely

f m (n) = / f M w-n (v,n)dv (4.19)

This integration leads to the result

, -E sin ii -E cos ip f M^(n) = ^-exp{—5-5 }[exp{—^ }

HUJ ^ 2a 2a

+ v^EVa cosij; erfÄ{ VE/a2 cos t^}] (4.20)

58

«■".Jf njT" W^sr^VWVWVWirn VI vw WT» 4-WM-W virir« v»wwirwi.-»uwu-»u-«-w«

where

ijj = n + <t)Ä + e(l) (4.21) e

and

2 N0 2 a = ^ + a (4.22) 2 "c

(i-D A similar expression is valid for the p.d.f. of n

In order to obtain the probability of error performance

of the DPSK receiver, it is necessary to evaluate

PrU} = pr{|n(i)-n(i"1)-(e(i)-e(i"1)) | > TT/M) (4.23)

First, observe that as long as $ remains constant over two

consecutive bit intervals, the mathematical expression of

equation (4.2 3) remains unchanged so that it is possible to

set 4) = 0 without loss of generality. Furthermore, for the

binary case with equally likely signals 6 -9 =0 and

e(i)_Q(i-l) = ^ each with pj-obabiüty o.5.

Therefore it is necessary to only compute

Pr{e} = Pr{|n(i)-n{i':L) | >TT/2} (4.24)

however it must be remembered that by considering only

e^'-e^ ^ = o, two cases are in fact being analyzed, namely

59

.".\VV_S,V.,VV.-.V.VV.-.V 's- >'^'^ ^ VV/.-.vv: v.v. v^^'V^<-^^^

jyiVV.i* "V"" >" i" '' i'J *f\*rji*y*$*y*S\v*\is.*y*\^\PT'Vr\vw\rr^^^?^j^v,^ ■■■ v M

0(i)_e(i-l) = 0 or e(i)_e(i-l) m ^^ Regardless of which of

these cases occurs, the behavior of the angle n -n

remains unchanged. Since these two cases occur with the same

probability, we assume without loss of generality that

e^'-e'1"1' = o.

Figure 4.2 shows a typical signal space representation of

received DPSK signals.

Assuming that n is known. Figure 4.2 shows a line

perpendicular to the vector corresponding to the assumed phase

n , in order to highlight the region where the next

received vector could lie and result in no receiver error.

In each case we have the vector /E 4, transmitted and a s—1

noise vector n = n + n added to it to form the corresponding

received vector. The statistics of n are

E{n} = 0 (4.25;

N0 2 Var{n} = ^ + a (4.26)

A receiver error will be made if the component of n along

<£, exceeds /E cosn . Since n is a Gaussian vector,

regardless of the coordinate system chosen, the components of

the noise along dimensions 0, and $~ will be zero mean, inde- 2

pendent with variance o where

N 2 n 2 (j = -^ + 0^ (4.27

60

MÄ&£&aS^^

^m^^- ■ " - '-•^^^w?*^T^r._' *_' '.-X' .,- >.• n." V." I.« f." 1 ■ I." V^ (." •." I." ».' '." i." w.i'r • " - lr.~ ^.- %^ '.:' 1,- - ">-. -s ■ -. m^mmmmm^m^mmm^mmm—■■!

Figure 4.2 Signal Space Representation of Received DPSK Signals

61

f^WT.^'f '^L'^'.^'.^ '.V.^'.AT^ '.'• 'A ."• '.^ '.^■.■«"■.'■ '.^'.">'.^ '.'•,W^TT'.". ;'.T:"VTrT-.-'T-.•.-.-.-' -.-\r.--..-- .- -,-, .%'.-»T.TT.-»vinuTn-wt-w^t-> .-«\rm

Consequently

Pr{Error|n(i'1)} = Prfr, > SE-cos n(i"1,} i s

~2

= / —^-exp{ yldn. (4.28)

9

where n, is the component of the noise vector n along $.

Furthermore

Pr{e} = / Pr{Error|n(i_1)}f (i-D (n (l"1) )dn (i'1) (4.29) -oo H

Using the p.d.f. of n , with 0=0 and 6 ^ = 0,

carrying out the integration indicated in equation (4.'29)

yields

Pr<E) - l^p'-rrmr' ,4■30, D J

where

RD = 2Es/N0 ; Rj = 0^' (4.31)

Observe that if no colored noise is present, the performance

of a binary DPSK receiver in AWGN results, namely

i E

Pr{e} = y exp{- ^} (4.32!

62

-w-r^^"?^!,1.,' 'l^^W,WVTJTJ>'.^r.-^-^.-^rÄrjl^> V"Ji VFj|^PHIU^w^CTV^Y¥VwwirAnr7^vwi?vv^.'w*^ >,. iw^m^mmmmmm^mm^mmmmmmjm

Furthermore, if R becomes infinitely large

lim Pr{e} = j (4.33)

J

which is to be expected for a binary signaling scheme with

equiprobable signals.

B. OPTIMIZATION OF THE COLORED NOISE JAMMER

Analyzing again the problem of designing an optimum

jammer which will cause the greatest performance degradation

to the DPSK receiver subject to a power constraint, we begin

by taking derivatives of equation (4.30) with respect to R

Prte} = J> Texp -UH .rT» r) > 0 for all R, (4.34) 4(1+RDRJ)-

cUT"-' - ^^Z^eXp-{2a+Ripj)} - 0 £orallRj

Therefore, Pr{e} is a monotonically increasing function with

respect to the jamming-to-signal ratio, R . Therefore, just

as for MPSK signaling

and

Sc(f) = Pc|<I>1(f)|2 (4.351

al = TT p^ (4-36) C 16 c

The resulting DPSK receiver performance is plotted in

Figure 4.3 for different values of R .

63

wi-.T'vw-i'-.T-v-.- -.--.-' %- -.'.". ,A,".-l.vni?."/l'.T'^'Tsr» l-f T» V« v» ■.-'VVVT.T-nrrnr-vT.nvm*-* -jrw rw -4P T^..^, w,/lr.^^-w..rw.Jlr3 ,- ,, ^ ..„.,„,._

DPSK RECEIVER PERFORMANCE

■o.

O t-.

K

O

HJ^E:

<

PU LEGEND a JSR= 0.0 o JSR = 0 DB A JSR = -10 DB + JSR = -20 DB

-2.0 2.5 7.0 11.5 a16Ci

SIGNAL TO AWGN RATIO (RD=2ES/N0)

Figure 4.3 DPSK Receiver Performance

64

^ *. ^ -"•.-._% .%. _> J: .'- jN- _*. . • .'• . - j- - • -_JI

-M-" «"* V>~" v^ -^ J» ■"■ y-* ^ L'^ r» V< VT» VV1^ ^^ in VlV^ U^ vwynrwini vw^y^ y IT» -y irwirwi.-» irw w^w if-■-»-■ irw

V. M-ARY QUADRATURE AMPLITUDE MODULATION

The types of spectrally efficient signaling techniques

discussed in this thesis provide bandwidth efficiency propor-

tional to k = log-M, where k is the number of information

bits per symbol and M is the number of signaling waveforms.

It is well known that for MPSK signaling over an AWGN channel,

for every doubling of signal phases beyond eight phases,

approximately a 6 dB increase in average transmitte'l signal

power is required in order to maintain the same error '-ate

performance. Quadrature Amplitude Modulation (QAM) is a

signaling technique that can reduce this penalty by using a

combination of signal amplitudes and phases in order to trans-

mit the M symbols consisting'of k bits each.

A. 16 QAM RECEIVER PERFORMANCE

The waveforms of the 16 QAM signaling scheme can be

represented by

x (t) = A1m1(t)cos[2nf0t+a] + A2m2 (t) sin [2TTf 0t+a] (5.1)

where A, = A- = a and a is an amplitwde parameter, a is the

transmitter phase uncertainty which ..s modeled as fixed but

arbitrary and m,(t) and m2(t) are the digital data signals of

duration T seconds having amplitudes

65

•. v •._•..-. . •. . -^ ••_. •_. "J. -.. ,. v '.• v/.-.v.v. .-.v. /. r. y.<\ ^A •v•V•.J'v'^/■/..•^/•.'V.-■^•^.••.. :o:-.": .*•.% _•• *• •»

W7V •"Vr'V'Ä'Wr"^ V 'Urw'.^V^.»"w3r—IT" ir^-."F>~w k"" V"Wli"» VV^ V"" '."< 1-» 1.-* '.">'."« V- - '.-W-.-I.-T-.' -. i ".T-•'-•>>■» ■* i--■ T..-« w v-ynn-»\-. rp*-. ^-.-w.-^ -.I-WHSBSBS

m1(t) = ±1, ±3

m2(t) = ±1, ±3

0 < t < T (5.2) — — s

If we now define

(^(t) a /2/Ts cos [2T:f0t +a] (5.3)

(})2(t) = /27f^ sin [2Trf0t +a] (5.4)

and

A = a /T/2 (5.5) s

then the signals of the 16 QAM signal set can be expressed as

si(t) = Ain1(t)(()1(t) + Ain2(t)())2(t) i = 1,2,...,16 (5.6)

where the four values that m, (t) and nu (t) can individually

take, generate 16 unique signals that can be represented as

vectors on a two-dimensional plane as shown in Figure (5.1).

The average energy of the signal set is

E = 10 A2 (5.7) s

so that the parameter A in terms of E becomes c s

66

> .■• -•• ."■ > "^. • _» '.»'■-' "-•'.-■.•".'".■• •.■■.■■.■'■.•'".• '.ö.. ■.• ■"J-'V.> ".»"v "> ^ »"^y

•■V^VTH* ir"jr» v» »r»v^-»v»".r» -^ --»v^ r» -.-■ vn .-• -. r,t,^^T^,^l,"» LTf' ~VWBrrvTr-~r^^wcTri '**rmi

Figure 5.1 Signal Space Diagrair1. for 16 QAM Signaling

67

»^>^VJW/Z,^^^<^XNfVC^^

JJnOT^i^W^'WWfWWT^A^IIIW^l^V.Ji'W

= >/E /lO - (5.8) s

The optimum receiver in minimum error probability sense

for a 16 QAM scheme operating in AWGN is shown in Figure 5.2

When jamming is present, the received signal for this scheme

is

r(t) = s.(t) + n (t) + n (t) (5.9) i w c

The statistics of random variables Y, and Y- under these condi-

tions can be shown to be

E{Y1} = Sil (5.10)

E{Y2} = Si2 (5.11)

2 2 N0 , 2 A 2 ,c i,x a = ov = ^- + o = a (5.12) xl 12 * c

E{ [Y1-E{Y;L}] [Y2-E{Y2}]} = 0 (5.13)

where

S.. = Am.(t) j = 1,2 ; i = 1,2,...,16 (5.14)

Again, we find that Y, and Y? are statistically independent,

conditionally Gaussian random variables. Therefore the joint

conditional p.d.f. of Y, and Y2 is

68

r(t)

< -2a X!L=-3

-2a<y1<0 ir^-l

0<y1<2a nyO

> 2a m,=l

|JTS ^(t)

^(t) = /2^cos(2TTf0t+ct)

JT -dt

< -2a "b11"^

-2a y2 0 nu*5-!

0 y2 2a ^S

> 2a m2=l

/N

m^t)

(J)9(t) = /2/T sin(2TTf t+a)

Figure 5.2 Optimum 16 QAM Receiver Structure

69

fr^::tf^^<>^>>^^^

^^^^»^"^T"rT"?r'^TTT"rTT'r?rT"»TTvrrTr-^7r^^ ■.-» t-. -.-«

1 i {Y1-E{Y1})2-(Y2-E{Y2})

2

V2l«i(t),yri'2|si(t,) * i^* i? i (5-15)-

and the probability of making a correct decision, given that

signal s.(t) was transmitted is

PrUls.a)} = Pr{Lu <YllLlu/L2£ <Y2 <L2u} (5.16)

where the upper and lower limits L. , L.n, j = 1,2 must be rr ju jj, J

determined for each of the signals in the signal set. In

general

Llu , . (VS.,)2

Pr{c|s.{t)} = / -^^_exp{ 1 i }dY1 Luy^7 20

L2u , (VSi2)2

x/ —L-expi ±| }dY2 (5.17)

which can be simplified to the form

g2 ^ Pr{c|s.(t)} = / — exp{-Z2/2}dZ / — expl-W2^^ (5.18)

g, /in h, /hi

where the limits of integration are defined as

L]iL ' Sil g1 = £ g Xl (5.19)

70

.v ^vvs .-. .-> '.■■.-■■ ,•. .">>■..-•.-. i", , V--W-J,

V V*\«"> '.mV k^lTU^'VLTVSI L^^WIV^V.V.W^W VT WULIV^ l"' I" I-rvirgirt^n^-jr.-^i »-j ■ \i •! tnmrvrr»-ni.m-wwvWM,

L, - S. ■, ^2 " ~ a (5-20)

L2Ji - Si2 hj^ = ^^ a 1^ (5.21)

h, = 2u - l2 (5.22) 2 a

In order to evaluate Pr{c|s.(t)}, we must first define

the decision regions for each of the signals in the signal

space. Fortunately, all of the signals in the 16 QAM signal

space have decision regions which can be described by one of

the subsets of the two-dimensional plane as described in

Figure 5.3.

It can be shown that the conditional probabilities of

correct decision associated with these regions are

Pr{cil} = [erf*(Y)]2 (5.23)

Pr{c|ll} = [1-2 erfcÄ(Y)]2 (5.24)

Pr{c|lll} = erf*(Y)[1 - 2 erfc^(Y)] (5.25)

where

A A Y a /VT°rrTVV (5-26)

and

71

J'l"J',1'WI'V VV"V*i".y W". ■ JFli" V •>PJV«i'■> '7-'jr\^p^^^*f^^r.j:vrrm-r.rjwTrciriwvri7ntrr*mmvjiu*m.-rj-*jiwrvi.^ni.r,-Lr,irwumi

(()2(t)

-A

^(t)

(a) Translated Type I Decision Region

(j)2(t)4

A *1(t)

(b) Translated Type II Decision Region (t)2(t)

-A - (t)1(t)

(c) Translated Type III Decision Region

Figure 5.3 16 QAM Decision Regions

72

"^ "^ ■- -- .»-■. "S. ^. ". ". L% .t . _% .-- .% .-* .% fc% .-. .-. /. .%. „^ , - Ä-w k; .-w , -_. ,_, ^j, •_, v -_. ^ v .^ .j. \v VJV.V/^V/^V_V.V.V.V.V*A

HÜTV^-VT'JW^J^WVFy^V V'WTJirETO/TV*V^.r», ^ IT« %■■ tnurrr» i TU -H VT ^ v*w CWT.-. L-» .-. L-. -. -.. » - ■- • -«

RD = 2E /N0 (Sigr al to Noise Ratio) (5.27)

2 _ RT = a /E (Jamming to Signal Ratio) J c s

(5.28)

Assuming all signals are equally likely to be transmitted, we

have

Pr{e} = 1 - {^erfJ(Y) +[^-erfci,(Y)]2 + erfÄ(Y)[i-erfc^(Y)]} (5.29)

Observe that if no jamming is present, equation (5.29)

becomes

Pr{e} = 1 - {ierf*(^75N^) + [|-erfcVk(^75N^)]2

+ erf+(/E /5NJ [^ -erfcA(/E /5Nn)]} V 0' l2 s' ""O' (5.30)

which is the probability of error for a 16 QAM signaling

scheme in AWGN. Furthermore, if the jamming power becomes

infinitely large,

„ i \ 15 lim Prlel = *-?r (5.31)

which is expected for a signaling set with 16 equiprobable

signals.

We now wish to maximize the jammer's effect on the 16

QAM signaling technique. Taking derivatives of equation

(5.29) with respect to y yields

73

\£&&^^

WTOWW^MMMMM'.MH'inHn'^'i.U1;1 "^«'^''^

d n-I-^ _ 3 ..._ r .2

2/1? 37 Pr{e} ■ - —^expi-Y /2}[3erfik(Y) -1] (5.32)

Since y is always positive and erf^iy) takes on values rang-

ing from 0.5 to 1.0 for y between 0 and o", respectively.

|- Pr{e} < 0 for all y (5.33)

Therefore, in order to maximize Pr{e}, y must be made as

small as possible, which further implies making R or equi- 2 —

valently a as large as possible for fixed E . Recalling that c s

oo

o2c = I Sc(f) |$j(f) |2df (5.34)

we again have for a power-constrained jammer

Sc(f) = Pc|$^(f)|2 (5.35)

so that

"c = nrpc (5-36)

Figure 5.4 shows the performance for the 16 QAM receiver as

a function of SNR for fixed values of JSR.

74

. -A. • . «V - « - . - \. - \ * k -\ - _ *» - _ «^ r. «"_ w. ifM m\ if. Ai ••_ ^. •■. «W. .\ fm .\ ^ -,•« /. r. • . -\ • - - . < . •» * . i

fT^s"* V \»-r^Wnxnjrr^Tgr-^-^n^nit-sr^ÄTWTlFrwTwra- r^v.rwi r

16-QAM RECEIVER PERFORMANCE

5H

i—i

CQ <

O

Pu

<0

'o.

L

LEGEND JSR= 0.0

o JSR = 0 DB A JSR = -10 DB + JSR = -20 DB

0.0 8.5 17.0 25.5 34.0

SIGNAL TO AWGN RATIO (RD=2ES/N0)

Figure 5.4 16 QAM Receiver Performance

75

■^ttttttte*^^

•B*J^5^^!^5^^^^^W.il,J." L • l!,,!fJ ■! ■ V";i ■ f»L"?'»'. «j. P5 • i1 ■3.l'j.,«\'y»jr'>""j--,J--,.l-,^">-,>,'> •>'} ->:> ■^j-Tjrr^j-T.Jti

B. 64 AND 256 QAM RECEIVER PERFORMANCE

The concepts just developed for 16 QAM signaling can

most easily be extended to 64 and 256 QAM systems.

The signal space diagram for 64 QAM signaling is shown in

Figure 5.5. The signal space diagram for 64 QAM signaling

contains the same Type I, Type II and Type III decision regions

as in 16 QAM except that a different number of each of these

exist.

Also, the average energy of the signal set is now

E = 42 A2 (5.37) s

and as a result, y of equation (5.26) is given by

Y = ^ = /RD/42(1+RDRJ) (5.38)

The total probability of a correct decision now becomes

Pr{c} = gj[4Pr{c|l} + 36Pr{c|lI} + 24Pr{c|lII}] (5.39)

where Pr{c|l}, Pr{c|ll} and Pr{c|lII} are defined by equa-

tions (5.23), (5.24) and (5.25) respectively with y defined

by equation (5.38).

The symbol error rate performance for 64 QAM is now

Prle} = 1 - ^erf2(Y)+9[j-erfcjt(Y)]2+3erfVk(Y)li-erfc^Y)]} (5.40)

76

j- .-'.*'* J.- .- . .-. -- ••.••..". w.

7\ VI ^i O V^UTX^mx^ muuncrs.WTV ^wwnru xrnBBBBMBBHB^BBBa

<j.2(t)

. 7A. .

• 5R' "

• 3A- • •

• A-

I I -'SA ' -JA ' 4 *-?£-*■ JA ' 7A . n -7A

•-A"

•-3A- •

-5ft- •

•-7A- •

4>1(t

Figure 5.5 Signal Space Diagram for 64 QAM Signaling

77

i .• >. «•«.-■» i •■ *».■!•.• i.^ ..» ^« i.« «.* «_s im «_" »j <riKA v.«y" •.1. v1 "•" ^.■■■,•.

Receiver performance curves for 64 QAM signaling described

by equation (5.40) are shown in Figure 5.6.

The same technique can be used to find the error per-

formance of 256 QAM in colored noise jamming.

The average signal set energy is now

— 1359 2 Es = ^^ k (5.41)

and

Y = - = /8RD/13 59(1+1^^) (5.42)

so that the symbol error probability for 256 QAM. signaling

becomes

Pr{e} = 1 - Ij|^erf^(Y)+7[i-erfc*(Y)]2+erf*(Y)[|-erfcÄ(Y)] ! (5.43)

The corresponding receiver performance for 256 QAM signaling

is shown in Figure 5.7.

It can easily be shown that the symbol error probabilities

for both 64 and 256 QAM systems yield expected results under

limiting conditions. That is if no jamming is present,

classical AWGN performance results are obtained and if the

jamming power grows without bound, the error probability tends

toward the maximum value for a set of M equiprobable signals,

namely (M-l)/M.

78

"1

1 b.

64-QAM RECEIVER PERFORMANCE

;:-".:-v.^^ygtBr^^^ r i'

).0 i

| r .TS^iv^^- ;

Nv^V**,i-x»^, ! \J "-Ki^,—!_ ^

j i\ | i »H Z

l X - 1 o - 1 K - •; wo

b:::::::::::::::::\::::::::::^ : • J.....V

• ■ \ •

: ^ V ■

1 1 t I o :

l! CQ

1 w,o_

••••U;üUü-UUUllü:U:-;Uü-l:lUlüUU:5l-l;-l-^

i O :

1 ^ : i—•

CO O

;;::::::;:;:;J:::;:i:;:;::::;;;;:;:;;::i::;:;:;^ • -. \—i

■ ' \ '

i ; \ ;

1 i \\ <^ = PQ : o - « -

i '©_ ! 1—1 z

:;::l;:::;l^u;::;;UU:l;;:--h;:;";;"--;;::;":;a";;-:"::"";..;:.:

LEGEND a JSR= 0.0 o JSR= 0 DB A JSR = -10 DB + JSR = -20 DB

Ffv5= . ... .:

li I

Y 1 i TH 1 1 1 1 1 | 0.0 10.0 20.0 30.0 4(

| SIGNAL TO AWGN RATIO (RD=2ES/N0)

Figure 5.6 64 QAM Receiver Performance

79

^<A«fl^itf«^;!y^

^1

256-QAM RECEIVER PERFORMANCE

} * * m"Hb~4 T * ' f \. i

1 ! \ i

Kb.

i:;if3:^^i}ff=ifi^^i;i?:i^ffii?i;3if:3iifi::i^3:ii;iiifiiifi;ii:; J :::::::::::::::::::x::::::::::::^^ ;■••, V

A

1 w ^ j J -

g« - 1 s2-

::::::::::::;::::::^::::;::::;;::::::;^

1 CO :

SH O-

::::;::::::::;:::::p::;:;::::::::::::^:::::::::::::::::::^;::::i::::;::::::

• ■ ' i

1—1

•—•

m O-

■::::::::::::::::::-i:::::::::::::::^

Pu -

b. ^—i =

1 b

.-.•.•.•

LEGEND o JSR= 0.0 o JSR= 0 DB A JSR = -10 DB + JSR = -20 DB

mmmmm ......;;.......... r. 1

■» 1

: : '• ih •^-,~ 1 I I -

O.C 10.0 20.0 30.0 4(

SIGNAL TO AWGN RATIO (RD=2ES/N0) ).0 i

Figure 5.7 2 56 QAM Receiver Performance

80

For the 64 and 256 QAM systems, jammer optimization re-

quires finding the derivative of the symbol error probability

expressions, equations (5.40) and (5.43) with respect to RT

or equivalently y. For the 64 QAM system the derivative yields

^-Pr{e} = -^-exp{-Y2/2}[7erf,.(Y)-3] < 0 for all y (5.44) dY 8/5?

and similarly for the 256 QAM system the result is

^-Pr{e} = ^^-exp{-Y2/2}[15erf,(Y)-7] < 0 for all y (5.45) aT 32/27

so that the mathematical expression for the optimized colored

noise jamming spectrum is the same for both 64 and 256 QAM

signaling as that developed for the 16 QAM system, namely

Sc(f) = Pc|*;[(f)|2 (5.46)

C. 32 QAM RECEIVER PERFORMANCE

Unlike the 16, 64 and 256 QAM cases, 32 QAM signaling

does not have all the same decision region types previously

considered. This can be observed by examining the signal

space diagram for 32 QAM signaling shown in Figure 5.8.

As can be seen, the perpendicular bisectors which define

the optimum decision regions have changed slightly due to

x.he lack of "corner" signals. However, there still exist

only three distinct types of decision regions, two of which

81

t^&&tt**:>:&a*::^^

r :'^v:^"f.'^^^•■^^".'•"l^■^.'>^■yT«:liTs-"i■Ty,

(t)2(t) \

I V 1 / \

i \ • . 5A . I • |

• / \

i \ 1

1 / i

1 1 • • • 3A , L • 1 • | r j

1 L . — — — | — mm mm ~ , - -» ^ — r

i

— — ^ 1 '■— "" *~

• i • • A • • i

i

• *

-SA | -3A 4 A i

3/ 5A ^^t)!

* ' • • -A ■ • i • | •

— — —| . — — T

— — — . i

_ _ . "1

j • 1 • • -3A • • • • * i

• — _ __ X w M_ «. J - ^ -— ^^ -L .» ^m —N 1 ! / i ^ 1

\ j \ j

1 / • •-5A j • I • \

j / 1 \ 1

Figure 5.8 Signal Space Diagram for 32 QAM Signaling

82

Rft*^a^^^^

are identical to those considered previously. The Type II

and Type III decision regions shown in Figure 5.3 are consis-

tent for 16, 32, 64 and 2 56 QAM cases. A new Type I decision

region must be considered for the 32 QAM signaling case.

Figure 5.9 shows this new Type I decision region.

From the geometry of this Type I decision region, the

probability of making a correct decision given that the trans-

mitted signal lies in a Type I decision region in the absence

of noise is

Pr{c|l} = Pr{-A ^N^-A <N2 < (Nj+ZA)} (5.47)

which results in

00

Pr{c 11} = erf^ (-) - / — exp r-x /2}erfcÄ (x +—)dx (5.48) 0 -A/a ^rT a

The average energy of the signal set is

jo that now

E = 20 A2 (5.49) s

A Y = ^ = /RD/2 0(1+RDRJ) (5.50)

The total probability of a correct decision is now

83

. VH'_.'.. ^>ji*^'J"j-J.>>>_.X.^/v/-y^^V^JN>>_.v.%_.%..N>v'\.h/v.>-:/ >/..%•. .\.. •.•'."'",>■>

WVJ'V^.'WWVnT^WV iW.V.V'kW.yvv'F'riVVT'r-r'y '.«'^ ^ r:»v'^Tr^^r^T^i^^rr^^/L'^^s^-^^v^T^T^Teyv^^yriiriL-ra

Figure 5.9 Translated Type I Decision Region for 32 QAM

84

i^v:^>rt^^:^^^

Pr{c} = ^-[8Pr{c|l} + 16Pr{c|ll} + 8Pr{c|lIl}] (5.51)

where Pr{c|ll} and Pr{c|lll} are defined by equations (5.24)

and (5.25) respectively with y given by equation (5.50).

The probability of a symbol error is

00

Pr{e} = 1 - y[2+llerfJ(Y)-9erfw(Y) - / ~exp{-x2/2}erfc*(x+2Y)dx]

(5.52)

Performance curves for a 32 QAM receiver structure

operating in colored noise jamming are shown in Figure 5.10.

The optimized jammer is found by first taking the deriva-

tive of equation (5.52) with respect to y This derivative

shows

|-Pr{e} = — a23erfÄ(Y)-10Jexp{-Y2/2}-KiJxp{-Y2}l < 0 (5.53)

for all Y

and therefore we again have for a power constrained jammer

Sc(f) = Pc|<D]_{f)|2 (5.54)

so that

^ = TIT ^ (5-55) C ib C

85

r. -*. -V-'o . "V-\-". ■'. •'. ■/. •'. •'- ■". -". -*• -V. • V-""- •\-'- ■"• •'• -"O"^-v.''. •"• ■■• •

^^^^n"1" r.TJ ^5 ''."f'i"^." r:*.' rl ?:*■; «-.' i-yJiTrfrprfyf; ^.-rrv^ji r.'rr.'^JV.^ rj1 WTf^I '." T -jr-s-s.-JMrr? rjt rjv-j\.-v. nr rj-ju.. r^ rv w rj-.

32-QAM RECEIVER PERFORMANCE

< m o Pi. DH1-

LEGEND a JSR= 0.0 o JSR = 0 DB A JSR = -10 DB + JSR = -20 DB

-2.0 5.0 12.0 19.0 26.0

SIGNAL TO AWGN RATIO (RD=2ES/N0)

Figure 5.10 32 QAM Receiver Performance

86

lJy^^^'^*v^*^.' ^J *Jt ""/TIP j rvrv nfiwv* Bj'tva V^XT ^ MftA.»^ vgvw zmnvM-nmrr ¥^, r ■■■■■IBHBiBiVnB'^H'Fi'B'WS ««I w fM wtv «H ■

VI. A SPECIAL CASE OF QUADRATURE AMPLITUDE MODULATION: 16-STATE AM/PM SIGNALING

The 16-state AM/PM signaling scheme is a CCITT recommended

technique used in 9600 bit/sec. voice band modems [Ref. 7].

This technique can be considered to be a modification of the

16 QAM signaling previously considered. Because of its

potential application in digital radio transmission, its

performance in the presence of AWGN and colored noise jamming

is analyzed.

A. RECEIVER PERFORMANCE IN COLORED NOISE JAMMING

The signal space diagram for 16-state AM/PM signaling is

shown in Figure 6.1.

The waveforms of this signaling technique can be expressed

in terms of the quadrature components, namely

xc(t) = A1m1(t)cos[27Tf0t+a] + A2m2(t)sin[2TTf0t+ot] (6.1)

where A, = A- = a, a is the transmit phase uncertainty and

m, (t) and m_(t) are the digital data signals of duration T

seconds with amplitudes

m1(t) = ±1, ±3, ±5

m2(t) = ±1, ±3, ±5

0 < t < T (6.2] - — s

87

■ I——T I , ,. 1

Figure 6.1 Signal Space Diagram for 16-State AM/PM Signaling

88

*» ^^ ■'- - ^ ■ ^ - *> i

• V \. •. N • "

•-■-■-»-■-'-■'- -. - . ■•/-•. -N^'<

^*.*-,.,r-\mmymVV','*V*V9V1V*V*V%VViniV9VIVVV*\im\\1UMmmmmmmam*mmmmmmmmmmmmmmmmmmmmmmmmmmmmm~mm~

The 16 signals can be expressed as

si(t) = Am1(t)(j»1(t) + Am2(t)(l)2(t) i = 1,2,...,16 (6.3)

where ^^.(t), 4)2(t) and A are defined by equations (5.3),

(5.4) and (5.5) respectively.

The average energy of the signal set is

Ec = -^ A2 ■+ A = v/2E 72 7 (6.4)

The optimum decision regions defined by the perpendicular

bisectors of the signals in the signal space diagram produce

four unique segments of the two-dimensional space as Figure

6.2 shows.

As can be seen from Figure 6.2, the optimum decision

regions are unusually shaped. The receiver logic necessary

to determine whether a received signal is within its corres-

ponding decision region would be extremely complex. Since

such an optimum receiver would be either impractical or

uneconomical to implement, we consider instead suboptimum

decision regions associated with this scheme as shown in

Figure 6.3. Such decision regions could be implemented in

logic very easily by first determining the received signal

component along ^(t) and ^-(t).

This is demonstrated by the receiver structure shown in

Figure 6.4, where amplitude and phase information of the

received signal is extracted.

89

•^;raa^y:fea^^ ;., UiM^M^iiMä^^M^iiMM^

•J"^ W.WtVWy.VUVmtrj W^'V 'J ' /."'.T"^ 7VV'J"r7'rjv:nn-^'THrrj"l«-u« roirvÄ -■» .'■wiir.»'Vkrmn»^Mn»w»n»^ii>i»n.«nif«iv»j(i>«m» •»»»-»w-««».-»»».'«.«

<l>2(t) i /

5A i

/

3A i • /

• . II X A • [ /

<*1it)

-A • \ II1 IV

-3A( > • \ x

Figure 6.2 Ootimum Decision Regions for 16-State AM/PM Signaling

90

s-s-:^:v:v:-:^vS-:-:^:-:<.^:^.^

V^V'^V.W'li.^Wl1" H'H.'.K"^^k,"V»U*l \T*"U TKBin'iwxT«7i.«n 10sjutw».-«

(t)2{t)l I

/

1 5A ' »

/

3A ( ► / •

II ^

i

/ I \^ ̂

| A ' " / *

^-^ III i IV

h(t) i

i "A ' • ^^Z ̂

-3A * • • ^

Figure 6.3 Suboptinun Decision Regions for 16-State AJ1/PM Signaling

91

i ■ ■ ^ ■ ^ ■ ^M n«^ ■ ' }>ff^^^?T^^f^^ '.^l '.'l 'A'A'.^'A'.^'A l.^'.^^.'. '.v^',». ».■■ \: '^ »^ '.v.». ■.». .v.-•^,■^^ .v.TrTT".y?r.',.'.,V"iTl.'n.i

r(t)

. dt

^(t) = ^7T^cos(2TTf0t4a)

. dt

V = ^f

n = -1 Y2 tan l{^)

Arplitude Infonration

Phase Infonration

$2(t) = /27T^sin(2TTf0t+a)

Figure 6.4 Receiver Structure for Subontimuin 16-State AH/PM Signaling

"">*'^>V'l."-ü.>V-Vv \ ( (■>r/^^^^W/?.'?«*^^^^«''-^v<i<^^^'^>C^,fO'«C'*^f« •"-/s\';'<','C,f/"f^^vV'v«.J'-l'r%,r^- ""^■CK"'j^

P.'V.'i^Wi T.'m1 r'T'rrTmrjTn-frrr*. -w-ui rouw^.i^-ifK» F^ IWH« « « » »T. «T, ,, uTi,Ti, ^.~» *rm vw »-«-w^-i -» m

The statistical description of the random variables Y,

and Y- is similar to that developed for QAM signaling. That

is,

where

E{Y1} = Sil (6-5)

E{Y2} = Si2 (6.6)

2 2 N0 . 2 A 2 ,, _,

E{[Y1-E{Y1}][Y2-E{Y2}]} = 0 (6.8)

S.. = Am.(t) j =1,2 i = 1,2,...,16 (6.9

Again, Y, and Y- are statistically independent conditionally

Gaussian random variables.

Just as was carried out for MPSK signaling, a double

random variable transformation is used so as to generate

random variables V and n having conditional probability den-

sity function given by

fvulo /4.x (v,ri|s. (t)) = VAGxp{ 5r[v-v^7cos(n-6.)] 1 ^'H|si(t) 1 2m 2aZ 1 1

exp{- i[E.sin2(n-3J]} (6.10) 2a 1

v>_0, 0<_ri£2iT

93

s^;^^i:^^-c^^^^^;^^

where

v = ^l + Y2 ; n = tan"1(y2/Y1) (6.11)

and

Ei = ^il + s2i2 {6-12)

ßi = tan"1(Si2/Sil) (6.13)

We now compute the probability of a correct decision

associated with each of the four types of decision regions

described in Figure 6.3.

For the Type I decision region

Pr{c|l} = Pr{0 £V i2A/2,g- <n <^} (6.14)

which with the use of the conditional p.d.f. of equation

(6.10) results in

Pr{c|l} = „expl-d"} exp{-5d } / exp{4d cosiJ;}d^

P TT/8 2 2 + — d / cosijjexp{-d sin ijj} [erf^(^dcosijj)-erfc^(^dr2-cosip}) ]d>j;

/n 0

(6.15)

94

-.■'■• •'• ''•L.'"«'". ■r-.,r- '• '•-."'• ■'-."'•.'r ".■*"• ''• "'• '"• ■-' -,' Si" s' -." •.' N" -.' ■.""-' •/ V •-' -.*^l' V ^NP"«.' «»VC «." •." •" s ^*\" v"> '•.'•.*• -r- ■*..■<•'■/••• '

TSXimWU WnWH^J n mi-jrj-wjwuw-jw-rw w: v •«- rw, ,»■.>*. r-w, *.,», »,

where

d - I - /2RD/27(1+RDRJ) (6.16)

For the Type II decision region

Pr{c|ll} = Pr{2A/2 <V <",! <n <i!} (6.17)

which yields

TT /8

Pr{c|ll} = -exp{-13d2} / exp{12d2 cosi(j}d^ 71 0

+ — d / cosiJjexp{-9d sin 4j}erfc#{/2d(2-3cos^) }di|; (6.18) y^f 0

where again equation (6.10) has been used to obtain equation

(6.18). Equation (6.10) is also used to evaluate

Pr{c|lll} = PriO <V <4A,-^ <n ijl (6.19)

fOT the Type III decision region, thus yielding

1 0 2 1 2C 2 ^f 2 Pr{c|lll} = -g-expl- 4d } exp{—jd }/ exp{12d cosnldn

fi ^^ Q 2 2 + d/ cosnexp{ -=d sin nl[erf^(3dcosn)-erfcA(d{4-3cosn})]dn

•^7 0 * (6.20)

95

^.>^>^>^:>>>:.:,>^>^ovr>>^^^^>:v^:^ , ^ /^^^-i^^ii^i^

rttttryr.^^-WWüü^

and finally, for the Type IV decision region

PricllV} = Pr{4A <V <»,-£■ <n <5-} (6.21) v_V 2. '"7 _'' 1.Q

for which

1 41 2 7T/8 2 Pr{c|lV} = -exp{—=-d } / exp{20d cosnldn TT ^ 0

10 f"^' 25 2 2 + dj cosnexp{--^d sin ri}erfcÄ{d{4-5 cosn})dn (6.22)

/2? 0 z

Assuming equal prior probability of transmission for all

signals, the total probability of a correct decision is ob-

tained by evaluating

Pric) = -£[4Pr{c|I} +4Pr{c|II} +4Pr{c|III} +4Pr{c|IV}]

(6.23)

which after some simplification becomes

96

1 '

Pr{c} = -^-expC-d2}+^-exp{-|d2}-^exp{-d2]/ exp{-8d2sin2 |}d^

- -fifxpi- jd2}j exp{-24d2sin2 J}dn+^exp{-d2}/ exp{-24d2sin2 |}d^

i i 5 ^Z8 o i + ^-e/vp{- ^6r)\ exp{-40d sin^ ^}dn

d TT/8 2 2 + -2- / cos ^exp{-d sin i^} [erfÄ(/2 cx)s^)-erfcA(i/2d[2-cos^] )]di|;

2/if 0

6d ?/* 9 2 2 / CXDS nexp{-yd sin n) [erf^Od cosn)-erfcÄ(d[4-3 cosn]) ]dn 4/2? 0

10d ^^ 25 2 2 + / cosn exp{- ^-d sin n}erfcilr(d[4-5 cosn])dn 4^? 0

+ -22./ cos i);exp{-9d sin ^}erfc^(/2d[2-3cosiJj])dip (6.24) 4/if 0

This rather lengthy expression can be analyzed for the case

in which the jamming power increases without bound. From

equation (6.16) under these conditions d tends toward zero

and

lim Pr(c} = T^ (6.25) d-0 16

97

"--'- /-•■■.•■.••..•.• .- V.V -■ V "JVV V •.-", • "-" '* '•' '-" '-- .• ".- V 's •.- V ".' ".- "J- •_- ■.- '.- ■.- -.- '.., •> ».i -.- •.- • -1

which is the expected result for a signal set with 16 equi-

probable signals.

Performance curves for the suboptimum 16-state AM/PM

receiver are shown in Figure 6.5.

Since the expression for the probability of a correct

decision given by equation (6.24) is so mathematically involved,

no attempt has been made to optimize the colored noise jammer

against the 16-state AM/PM receiver. However, based on the

results of Figure 6.5, it seems reasonable to assume that the

optimum jamming spectrum should match the transmitted signal

spectrum, just as encountered in the previous signaling

schemes analyzed.

For a performance comparison of the 16-state AM/PM scheme.

Figure 6.6 shows receiver performance curves for 16-QAM, 16-

state AM/PM and 16-PSK signaling in an AWGN environment.

98

> >.•->

*

^1

16-AM/PM RECEIVER PERFORMANCE

I

).0

Sb.

ii j 0 n O---V-Q oooo' ''Tf,+«l^~^t_^r__ 1 '

m ' ^^^^^/^'"■^-A. [

;ÜäiKt ; ^^ *""—t* Ai

• ^V\ ' ' 1 \\l

SYM

BO

L E

RR

(

1

I

I

i

I I

I I

i

•....•.■.•.\V^-.-.-.-.-;.VAV^-.^. .•.■•.■.•.•.!.•.•.■.•.•.•.•. .VA-.?X 1.............. I \.. • ... .T^v, r r \r '>«<—•••• m T*S

{V

■ ■ \

: i : \

i I \ O - >H - H - i—i

J - 1—1

<

O'o

■ : i : Y 1 1 1 \ r r r V

*

• • \ i ! i V

LEGEND a JSR= 0.0 o JSR= 0 DB A JSR = -10 DB + JSR = -20 DB

: :::::::::::::::::::\ .,;....... :

: [

I ! I

•«—• i i i i

2.0 5.0 12.0 19.0 2e

SIGNAL TO AWGN RATIO (RD=2ES/N0)

-

Figure 6.5 16-State AM/PM Receiver

99

Performance

•nc-.r. n w * r u- n.™rTjrwv:KTvm- w': v yv wt^ HV ' > fj rß v.' '.■ j ?rr*.\Tr>\Tr jri';' 7ryrA'^r^.v.'Jv\'.^.nJc,.>:7.wy:,>.ra(v^]

COMPARISON OF 16 LEVEL SIGNALING

-2.0 5.0 12.0 19.0 26.0

SIGNAL TO A¥GN RATIO (RD=2ES/N0)

Figure 6.6 Performance Comparison of 16 Level Signaling

100

••."- _■ • .■»

w«."-! ^yr« T mr% m/T», i ■:.- « Fi i ' wwi *f*.mnmin*u\msn.-m.* n ■% ■ -vir ^rf TU « MT «:-i « A ■ •% ».-v «tn «_n. «ä mL^«.-i.*,-i «t/vJ»JI v.'bK.i ^^-^ Kn iwv^.'T ^.n.^.^ m»".\ JVI

VII. CONCLUSIONS

In this thesis the performance of receivers assumed to

be operating in the presence of both AWGN and colored noise

jamming has been analyzed for several digital modulation tech-

niques. In all except the 16-state AM/PM case, the receivers

considered are optimum for discriminating amongst M signals

received in an AWGN environment in the sense of minimum proba-

bility of error. Receiver symbol error probability was used

throughout as the measure of receiver performance.

In addition to receiver performance analysis, optimized

jamming techniques were also developed. The colored noise

jammer was modeled as power limited, uncorrelated with the

white channel noise and Gaussianly distributed with power

spectral density determined as part of the optimization proce-

dure. The intent of the optimization was to maximize the

receiver symbol error probability while making efficient use

of the jammer's available power.

In Chapter II, the MPSK receiver structure was analyzed

in the presence of colored noise jamming. Figures 2.2 through

2.5 corresponding to the M = 4, 8, 16 and 32 cases respec-

tively, all conclusively show significant receiver performance

degradation in the presence of relatively low levels of colored

noise jamming. Typically only a -10 dB jamming-to-signal ratio

was required to increase the symbol error probability a

101

jT-nr—H- -v -rf> rv w -j "j -J -j-jr^jr-jnar^in'jn'.vy y^Jf ^-M ■ ina ^M *. vv ^irnifiTW WUrriPI ipi'lf» y WT» V« VTT^ liV* fv ■.•a'VTJ '.-y '.^ ^r^'.'- 'A ^ '.^ vn

minimum of one order of magnitude for SNR values in the range

of 15 dB to 30 dB. The M = 4 and M = 8 cases showed superior

performance in the presence of jamming when compared to the

M = 16 and M = 32 cases. This is not surprising since the

optimum decision regions defined by the perpendicular bisec-

tors of adjacent signals in the signal space, mathematically

described by the angle 9, where - rr < 9 < —, become smaller MM

as M increases. The smaller the decision regions become, the

more likely it is that the additive interference will produce

an observation vector at the receiver which lies outside of

the correct decision region thereby causing a decision error.

For the MPSK modulation method, the optimum power constrained

colored noise jammer was found to have a power spectral den-

sity which mimicked the power spectrum of the MPSK signal.

The general results developed in Chapter II were then

applied to three special cases of MPSK signaling in Chapter

III. QPSK, OQPSK and MSK are all special cases of 4-ary PSK

modulation with OQPSK and MSK providing improved performance

in bandlimited applications over QPSK modulation. Although

the purpose and implementation differed for each, all three

were found to perform identically in the jamming environment.

This was not too surprising since these three signaling

schemes also perform identically in AWGN-only interference.

The mathematical expression for the optimum colored noise

jamming waveform was also the same for these three modulation

techniques. However, since the basis functions used in the

102

.^Fi^wv^^rjT^^j^v^vvKTVT^r^r^LHi^^^^

series representation of the MSK signals are different from

those uied in QPSK and OQPSK signaling, the actual optimum

jamming spectrums differed accordingly. The optimized jamming

spectrum still mimicked the signaling spectrum except that

now, the signal spectrum of MSK modulation is different from

that of QPSK and OQPSK modulation.

Chapter IV presented the analysis and performance evalua-

tion of a receiver for DPSK modulation operating in the presence

of AWGN and colored noise jamming. As was the case for MPSK,

Figure 4.3 shows the same severe performance degradation ex-

perienced by the DPSK receiver in the presence of relatively

low levels of jamming. Figure 4.3 shows the corresponding

performance curves and demonstrates the degrading effect of

jamming. The optimum jamming spectrum for use against OPSK

modulation, as in the preceding analyses, was shown to be

matched to the transmit signal spectrum.-

The M-ary QAM techniques discussed in Chapter V, although

showing the same general performance degradation tendency

observed in the other signaling schemes, provided the best

overall performance in the presence of jamming for a given

value of M and JSR level of all the signaling techniques

analyzed. Relating again the size of the optimum receiver

decision regions to the receiver performance, QAM signals

have the largest decision regions for fixed signal energy and

value of M, with a smaller decrease in size of the decision

region for increasing values of M, when compared to other

103

i^vy^jfc£tyww{kAV.4^*Awf *.>. .-•> ^\«.- ..v«.'w. mjf^n^ *.<*JM~\ ..<VX..". «y^-r«^V- ^IV\ ^•. ^ .J .. <.y^A^iV!f. ^ <. w. .•. ^_ v. t'. .•- AV-V. ^. AL

.■".■VW".-".-'v ^.-v^ vi ^.-i ^.1^^-^;,'■: ■ •,-. «.i ■.T»." rii»/! ^n^■T«7rvr^'^ VT wT■.%"'.^~L^',.^<■^^■,'.■•^.Vl■^.^T'•'1.^'1.','•"l.*•■,.^"."' v.v;^v'V';*."^V^ VJ.T'VTy'W'.y1^

digital modulation schemes. Therefore, for a given SNR and

symbol error probability, more signaling levels and therefore

more information bits per symbol (k = log-M) can be trans-

mitted using M-ary QAM than using MPSK modulation, whether

or not jamming is present. This improved receiver performance

combined with a receiver structure implementation that is as

simple as that for MPSK signaling, explains the popularity of

QAM techniques in modern digital communication applications.

Figures 5.4, 5.6, 5.7 and 5.10 graphically display the QAM

receiver performance for M = 16, 64, 256 and 32. As before,

the jamming spectrum that optimizes the colored noise jammer

in QAM transmission cases is identical to that of the transmit

signal spectrum.

The 16-state AM/PM signaling scheme analyzed in Chapter

VI yielded several interesting results. First, the receiver

structure necessary to make optimum decisions proved too com-

plicated to practically implement. A suboptimum receiver

structure was therefore selected by modifying the MPSK re-

ceiver structure in such a way that both amplitude and phase

information about the observation signal vector are computed.

The performance analysis was then carried out assuming that

both AWGN and colored noise jamming were present ir the chan-

nel. The mathematical expression for symbol error probability

proved so involved that jammer optimization was not attempted.

However, receiver performance graphs for the 16-state AM/PM

technique using the suboptimum system described above were

104

rj ■ - ^-jw-v w^. «r-.* ^ .- ■- . -- -- ' ,- a -■ ^ '_- ^ * > ■ •-■-*-.*-»-.» k F w ^ i - ■ ^ ii" w^" '• ■ ir^ uT«nrn f* u-» L~" 'iTi w"» CTI i.""\.ni v"»VTr UTI'-"* UTHTBU"« UTIW^ WJk~«k

generated as shown in Figure 6.5. A comparison of Figure 6.5

with Figures 2.4 and 5.4 which show 16-PSK and 16 QAM receiver

performance, respectively, demonstrates that the suboptimum

16-state AM/PM receiver structure yields a performance which

lies between that of 16-PSK and 16 QAM. A performance com-

parison of these three 16-level signaling techniques in AWGN

is shown in Figure 6.6. The 16-state AM/PM suboptimum scheme

performs almost as well as the QAM system at SNR levels below

12 dB, approaches the performance of PSK for SNR values be-

tween 16 dB and 22 dB, and then performs worse than both QAM

and PSK for SNR values beyond 22 dB. This is partly due to

the use of suboptimum rather than optimum decision regions in

the 16-state AM/PM receiver. As the SNR increases, the size

of the optimum decision regions increases proportionally.

The size of the suboptimum decision regions, however, do not

increase at a similar rate causing the receiver structure to

become more suboptimum at higher SNR levels. Although no

jammer optimization was performed, based on previous results,

the optimum colored noise jammer spectrum would be expected

to mimic the 16-state AM/PM signal spectrum. Consequently,

receiver performance was determined and evaluated on the

assumption that the jammer spectrum was matched to the signal

spectrum.

Table 7.1 presents selected receiver performance results

for each of the receiver structures analyzed. The SNR was

-4 selected corresponding to a symbol error probability of 10

in AWGN. Table 7.2 identifies the SNR penalty associated with

105

a^^-aaa^^v;, v v ,^f£M£k£&^^

p^^^y^^y^^y^rery^ryTNri'i'^n'y.rarwy^n^AV'^.^Titrre^

TABLE 7.1

COMPARATIVE RECEIVER SYMBOL ERROR PROBABILITIES

Nunber of Signaling Levels M

Signaling Method

2 DPSK

4 QPSK

8 8-PSK

16 16-PSK

16-QAM

SNR JSR = 0.0 JSR = -20 dB JSR = -10 dB

12 dB 1.8 xio"4 5.35 xio"4

14 dB 3.93 xlO-4 1.52 xio"3

20 dB 1.28 xlO-4 6.77 xio'3

25 dB 5:17 xlO-4 8.88 xio"2

22 dB 1.03 xlO-4 3.25 xio"4

16-state AM/PM 26 dB 4.54 xlO-4 5.21 xio-2

2.33 xlO -2

5.37 xlO -2

2.48 xlO -1

5.44 xlO -1

1.98 xlO -2

4.27 xlO -1

32 32-PSK 31 dB 4.84 xlO-4 3.44 xio"1

32-QAM 25 dB 1.14 xio"4 8.15 xio"2

7.57 xlO -1

6.29 xlO -1

64 64-QAM 28 dB 1.86 xlO-4 2.48 xio"1 7.97 xlO -1

256 256-QAM 34 dB 2.26 xio"4 6.68 xio"1 9.42 xio"1

106

^*."V".■ ■.■ *•"v ■.■ v',«",• "^ v '."■.• v ".■ '.•".- v■.-■,■'.• *< ■.■• v /■,•■.• v■.• ■.•■.-' ■,•".• v■.■ v' •" "-■ *•• v '-*"•"'•■ •■ "•■ ■.'"•' v -""."■. <^^.Ov■:\^^^^^X:,'^^l^^^^^v^•-^^lvL:vlv^.^j,^^v^.^^^^/'.^:v^.:^^:v^v^//^^v-^

rJurT.jmtr^jr^jrF;jrF.ir^ir«r>i>r^jrRTmr" vuwmrmmmunrKrwimu \VTtuTrn.irwxTC3r**T*ii*.Tiivii*i

TABLE 7.2

-2 SNR PENALTY FOR MAINTAINING PR{e} = 10 IN JAMMING

Number Signal Levels

Of ing , M

Signaling Method

SNR Penalty for

JSR = -20 dB

2 DESK 0.5 dB

4 QPSK 0.5 dB

8 8-PSK 2 dB

16 16-PSK >6 dB

16-QAM 2 dB

16-state AM/PM >4 dB

32 32-PSK *

32-QAM >10 dB

64

256

64-QAM

2 5 6-QAM

* Asymptotically approaches a Pri0} < 10 -2

107

qy w**"jFi""Jfw^F. MF^"5'.'■^■yyv»■■"■!"■>^J.^^'J'>'>'j.■ J^^'.J■mwm■ j*JIv■ >v'>v■ JTJ?.-'.v*A-J-■ J'J■ ^'.)■'>'

-2 maintaining a symbol error probability of 10 ' in a -20 dB

JSR environment. As shown in all M-ary signaling techniques

analyzed, symbol error probability increases with increasing

M. The modulation scheme least affected by the colored noise

jamming was the QAM system.

Throughout this thesis, the optimum colored noise jammer

was found to require exact knowledge of the transmit signal

spectrum. Without this a priori information, a suboptimum

colored noise jammer would be expected to provide similar

yet less effective results since the Optimum jammer caused

such large performance degradation with low power levels.

In summary, although the AWGN receiver is considered a

general purpose receiver that performs well in many different

channel environments, the results presented here show that

significant performance degradation can be expected when the

same system is used in a colored noise jamming environment.

By the same reasoning, the colored noise jamming model used

throughout this thesis has proven to be an extremely effec-

tive, power efficient jammer for use against AWGN receivers.

108

•*.■■:■■: \ ^: &&&äü^^

>»t '.W.TnirrVWJI''J"-^r r.w~\ ■-- /-' ■ ',TjrW-j.rj»r ■ *■** ^mur» \nnrm\rmv~m*rmvm yrmvmm \-mt*mi-w± -. ^ .-.,

APPENDIX A

DETAILED INVESTIGATION OF THE PRODUCT OF ^1(-f) AND OTT

Let ({>, (t) and ty-y^ ^e given ^Y equation (2.3), namely

(J)1(t) = /2/Ts cos (2TTf0t+a) and ^2{t) = /2/Ts sin (2TTf0t+a)

(A.l)

and define

i (p.it) , 0 < t ^ T • A \ -'

^jCt) = / j = 1,2 (A.2)

( 0 , otherwise

■ i

so that the (J) . (t) are defined for all time t. Thus, $ . (t) ,

j = 1,2, can now be expressed as

i

0, (t) = p(t) cos (2TTfnt) (A.3;

(()2(t) = p(t) sin (2TTf0t) (A.4)

where

/2/Ts, 0 < t < Ts

p(t) = (A.5!

' 0 , otherwise

and a has been set to zero.

109

■•■ •».'-- \. •.^»■. ,'*^,- .• i''. ..^•.l^-»-,- .i,

The Fourier Transform of p(t), namely P(f) is given by

sinUfT ) -JTrfT -JTrfT P(f) = /2/Ts (7TfT J e s = /27T^ sinc(fTs)e

(A.6)

so that

$|(f) = 2-[P(f-f0) +P(f+f0)]

-J7T(f-f0)Ts

= j /27Tg[e w asinc(f-f0)Ts

-JTT(f+f0)Ts + e sine(f+f0)T ] (A.7)

and

*'2(f) = p-[P(f-f0) -P(f+f0)]

-JTT{f-f0)Ts

Us

JJ /27Tg[e w a sine (f-f0)Ts

-JTr(f+f )T - e u s sine(f+fn)T ] (A.8) u s

ing equations (A.7) and (A.8) the product of $.(-f) and

$_(f) becomes

T j2TTf T -j27Tf T $1(-f)$2(f) = j|[e U SS+S_+s:-S^-S+S_e U S] (A.9)

110

^<y:-y^^^

tta " -" k^'ir» WJ ■ uv ■■ u ^ ■■■■■MHff ■!■■" ■ v~m i-Ti \:->* i 'iii.-%>'T~vi.niv,/maHi^HBBaw^a -.' ■v ■%. i --'■..» -. y ^.-*r%rt-'wm^immM

■ ■

where

S+ = sinc(f+f0)Ts and S_ - sinc(f-f0)T (A.10)

Through factoring we can show

$;(-f)^(f) = ^[S_e 0 S-S+eJ 0s]lS+eJ 0s+S.e' 0s]

(A.11)

Expanding the terms in the brackets yields

T G(f) = $!(-£)$!,(£) = -|-(j[sinc2{f+f0)Ts-sinc

2(f-f0)Ts]

+ sinc[(f+f0)Ts]sinc[(f-f0)Ts]sin(2TTf0Ts)} (A.12)

and similarly

T G(-f) = $1(f)*2(-f) = -|-{-j[sinc2(f+f0)Ts-sinc2(f-f0)Ts]

+ sinc[(f-f0)Ts]sinc[(f+f0)Ts]sin(2TTf0Ts) } (A.13)

Observe that

Im{G(f)} is an odd function

Re{G(f)} is an even function

111

Therefore

/ S (f)G(f)df = / S (f) [Re{G(f)}+j Im{G(f) }]df c ; m c

— 00

= / S (f)Re{G(f)}df - 0 (A.14) C

— 00

since

sinc[(f-f0)Ts]sinc[(f+f0)Ts] - 0 (A.15)

112

s-^i^::^^;*^

■

LIST OF REFERENCES

1. Ziemer, Rodger E. and Peterson, Roger L., Digital Communications and Spread Spectrum Systems, pp. 198- 213, MacMillan, 1985.

2. Lindsey, W.C. and Simon, M.K., Telecommunication System Engineering, pp. 228-231, Prentice Hall, 1973.

3. Feher Kamilo, Digital Communications, Satellite/Earth Station Engineering, pp. 162-168, Prentice Hall, 1981.

4. Pasupathy, Subbarayan, "Minimum Shift Keying: A Spectrally Efficient Modulation," IEEE Communications Magazine, July 1979.

5. Proakis, John G,, Digital Communications, pp. 171-178, McGraw-Hill, 1983.

6. Arthurs, E. and Dym, H., "On the Optimum Detection of Digital Signals in the Presence of White Gaussian Noise—A Geometric Interpretation and a Study of Three Basic Data Transmission Systems," IRE Transactions on Communications Magazine, pp. 346-353, December 1962.

7. Smith, David R., Digital Transmission Systems, pp. 296- 301, Van Nostrand, 1985.

113

1

».•n.»-»«-»«^!. » • ■■».■ V-^WT- ^ !■-» H-t-^y. .i-i.m^.w\y irm^r ■, F^TiT i"» U"»--i"--.-»jjCKKi-p V^ BT.-ST-.I"»;«^!"!1 i-" k~" «^ k1«!.'» »"■ «Tt.>'j »'.'_WTljr,JI"*> ?Ji'> \> TV*.»".*»!

INITIAL DISTRIBUTION LIST

No. Copies

1. Defense Technical Information Center 2 Cameron Station Alexandria, Virginia 22304-6145

2. Library, Code 0142 2 Naval Postgraduate School Monterey, California 93943-5002

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Computer Engineering Naval Postgraduate School Monterey, California 93943-5000

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Computer Engineering Naval Postgraduate School Monterey, California 93943-5000

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Naval Postgraduate School Monterey, California 93943-5000

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114