Performance analysis of diversity combining for frequency-hop communications under partial-band and multitone interference

PE R FO R M A N C E ANALYSIS OF D IV E R SIT Y

CO M BINING FOR FR E Q U E N C Y -H O P

COM M UNICATIONS U N D E R

PARTIAL-BAN D A N D M ULTITONE IN T E R FE R E N C E

by G A N G LI

B.Sc., University of Science and Technology of China, Hefei, 1982 M.Sc., Northwest Telecommunication Engineering In stitu te, X i’an, 1984 _ A D issertation S ubm itted in P artial Fulfillment of th e

A C C E Requirem ents for th e Degree of

CULTY OF GRADUATE STUDI ES M &

D O C T O R OF P H IL O SO P H Y

in the D epartm ent of Electrical and C om puter Engineering

DEAN

rr

---We accept this dissertation as conforming to th e required stand ard

Dr. Qiang Wang, Supervisor (D epartm ent of ECE)

Dr. V ija$K .< 4’ argava, Cc-Superv(sc^ (D epartm ent of ECE)

____________\j_________________________________________________________ Dr. Nikitas J. Dimopoulos, D epartm ental M em ber (D epartm ent of ECE)

---Dr. M icaela Serra, O utside M em ber (D epartm ent of CS)

Dr. J a m e s ^ /f Ritcey, E xternal M em l^j/(U niversity of W ashington)

© G A N G LI, 1992 U niversity of V ictoria

All lights reserved. This dissertation may not be reproduced in whole or in part, by photocopying or other means, without the permission o f the author.

PE R FO R M A N C E ANALYSIS OF D IV E R SIT Y

CO M BINING FOR FR E Q U E N C Y -H O P

CO M M UNICATIO NS U N D E R

PARTIAL-BAN D A N D M ULTITONE IN T E R FE R E N C E

by

G a n g L i

Supervisors: P r o f e s s o r Q ia n g W a n g and P r o f e s s o r V ija y K . B h a r g a v a

A B ST R A C T

This dissertation is concerned with perform ance analysis of diversity combining schemes in frequency-hop spread spectrum com m unications under th e worst case partial-band noise and m ultitone jam m ing.

Perform ance of a ratio-threshold diversity combining scheme in fast frequency hop spread spectrum systems w ith M -ary frequency shift keying m odulation (F F H /A /F S K ) under partial-band noise (PB N ) and band m ultitone jam m ing w ith­ out and with the additive white Gaussian noise (AWGN) is analyzed. T h e analysis is based on exact b it error probabilities, instead of bounds on th e bit erro r probabil­ ities. A m ethod to com pute th e b it error probability for ratio-threshold combining on jam m ing channel is developed. Relationship between the system perform ance and the system param eters, such as ratio-threshold, diversity order, and therm al noise level, is illustrated. The perform ances under ban d m ultitone jam m ing and under p artial-band noise jam m ing are com pared. For binary FSK m odulation, the perform ance under the two types of jam m ing is alm ost the sam e, b u t for 8-ary FSK m odulation, tone jam m ing is m ore effective against com m unications. The structu re of the com biner is very simple and easy to im plem ent. A nother m erit of this com biner is th a t its o u tp u t can be directly fed to a soft-decision F E C decoder.

M axim um-likelihood diversity combining for an F F H /A fF S K spread spectrum system on a PBN interference channel is investigated. T h e stru ctu re of m axim um

likelihood diversity reception on a PBN channel with AWGN is derived. It is shown th a t signai-to-noise ratio and the noise variance a t each hop have to be known to im plem ent this optim um diversity combining. Several sub-optim um di­ versity com bining schemes, which require th e inform ation on noise variance of each hop to operate, are also considered. The perform ance of the maximuin-likelihood combining can be used as a standard in judging th e perform ance of other sub­ optim um , b u t m ore practical diversity combining schemes. T he performance of the optim um com bining scheme is evaluated by sim ulations. It is shown th a t the A daptive G ain Control (AGC) diversity combining actually achieves the optim um perform ance when interference is not very weak. B ut th e perform ance difference between some of th e known diversity combining schemes, which do not require channel inform ation to operate, and the optim um scheme is not small when the diversity order is low.

An error-correction scheme is proposed for an M -ary sym m etric channel char­ acterized by a large error probability pe. Perform ance of the scheme is analyzed. T he value of pe can be close to, bu t sm aller than, 1 — 1/A/ for which the channel capacity is zero. Such a large pe may occur, for exam ple, in a jam m ing envi­ ronm ent. T h e coding scheme considered consists of an outer convolutional code and an inner repetition code of length m which is used for each convolutional code symbol. A t th e receiving end, the vn inner code symbols are used to form a soft-decision m etric, which is subsequently passed to a soft-decision decoder for th e convolutional code. Em phasis is placed on using a binary convolutional code due to the consideration th a t there exist commercial codecs for such a code. New m ethods to generate binary m etrics from M -ary ( M > 2) inner code symbols are developed. For th e binary sym m etric channel, it is shown th a t the overall code rate is larger th a n O.6R 0, where Rq is the cutoff rate of th e channel. New union bounds on th e b it error probability for systems with a binary convolutional code on 4-ary and 8-ary orthogonal channels are presented. Owing to the variable m which has no effect on the decoding procedure, this scheme has a clear opera­

tional advantage over some other schemes. For a BSC and a large m , a m ethod presented for BER approxim ation based on th e central lim it theorem .

Examiners-Dr. Qiang Wang, Supervisor (D epartm ent of ECE)

Dr. V ijly 1^ Bhargava, Co-SupeWJsor (D epartm ent of ECE)

^ -

---Dr. Nikitas J. Dimopoulos, D epartm ental M em ber (D epartm ent of ECE)

Dr. M icaela §erra, O utside M em ber (D epartm ent of CS)

C on ten ts

T itle P age i

A b stra ct ii

Table o f C on ten ts v

L ist o f Tables vii

List o f Figures viii

A cknow ledgm ents xvii

D ed ica tio n xviii

1 In trod u ction 1

1.1 Frequency-Hop C o m m u n icatio n s... 1 1.2 Intelligent J a m m in g ... 3 1.3 Diversity Reception in FII Com m unications under Jam m ing . . . . 3 1.3.1 Diversity Com bining Using R atio-Th-eshold Test Technique 5 1.3.2 O ptim um Diversity Reception in PBN with A W C N 6 1.4 C oncatenated Coding for High-Error-Rate C h a n n e ls ... 6 1.5 D issertation O utline ... 8 2 Perform ance o f R atio-T h reshold D iv ersity C om bining Schem e in

F F H /F S K S y stem s under P artial-B and N oise Jam m in g 9

Contents vi

2.2 Ratie-Threshold Diversity C o m b in in g ... 10

2.3 Average C om putation Model for Bit Error P r o b a b i l i t y ...11

2.4 Performance in A dditive W hite Gaussian N o i s e ... 16

2.5 Performance under P artial-B and Noise J a m m in g ... 29

2.6 C o n clu sio n ... 42

2.7 Proof of Equivalence between Two States Model and th e Average Single S tate M o d e l ... 42

3 Perform ance o f R atio-T h reshold D iv ersity C om bining Schem e in F F H /F S K S y stem s under M u ltito n e Jam m in g 50 3.1 In tr o d u c tio n ... 50

3.2 Perform ance under B and M ultitone Jam m ing w ithout T herm al Noise 51 3.2.1 Binary C a s e ... 51

3.2.2 M-ary C a s e ... 60

3.3 Perform ance under B and M ultitone Jam m ing w ith T herm al Noise . 69 3.4 Comparison of Performances in P artial-B and Noise Jam m ing and M ultitone J a m m in g ... 90

3.5 C o n clu sio n ... 93

3.6 Derivation of F'c(0) and Fe{0) under Band M ultitone Jam m ing . . 93

3.7 Derivation of F F u n c t i o n s ... 95

3.7.1 C om putation of Fco ... 95

3.7.2 C om putation of F a ... 96

3.7.3 C om putation of Feo ...100

3.7.4 C om putation of Fe\ ... 101

4 M axim um -L ikelihood D iv ersity C om bining in P artial-B an d N o ise 109 4.1 In tr o d u c tio n ... 109

4.2 A ssu m p tio n s... 110

4.3 Maximum-Likelihood D iversity C o m b in in g ... 112

C ontents vii

4.4 Some Sub-optim um Combining S c h e m e s ... 118

4.5 Com parison of Several Diversity Combining S c h e m e s ...12;> 4.6 S u m m a r y ... 127

5 R ep ea ted C onvolutional C o d e s for H igh -E rror-R ete C hannels i3 0 5.1 I n tr o d u c tio n ... 130

5.2 T heoretical Analysis for th e B S C ...132

5.3 C om putational Resuits for the B S C ...138

5.4 M - ary Sym m etric C h a n n e l... 140

5.4.1 M - ary M e t r i c ...143 5.4.2 B inary M etric G e n e r a t i o n ... 154 5.4.3 Sim ulation R e s u l t s ... 160 5.5 Concluding R e m a r k s ... 163 5.6 F u rth er Analysis of O ne-bit-error B r a n c h ...163 6 C onclusions 166 6.1 Sum m ary of the T h e s i s ... 166

6.2 F u tu re R e s e a rc h ...168

A List o f S ym b ols 170

B L ist o f A b b reviation s 172

VI!1

List o f T ables

3.1 The cross point of h i{x , 1) with 0.5m, uo... 55 5.1 C d ( X , Y ) for the constraint-length-7 O denwalder code... 157

ix

List o f F igures

1.1 The FH noncoherent FSK spread spectrum system ... 2.1 T he Channel M o d e l... 2.2 Perform ance of F H /B F SK with diversity m = 4 ch ip s/b it and R-'i

diversity com biner with threshold 0 in additive w hite Gaussian noise. 2.3 Perform ance of F H /B F S K with diversity m = 5 ch ip s/b it anti R-T

diversity com biner with threshold 6 in additive w hite Gaussian noise. 2.4 Perform ance of F H /B F S K with diversity m = 9 ch ip s/b it and R-T

diversity com biner with threshold 0 in additive w hite Ga”.ssian noise. 2.5 Eb/No required to achieve B E R = 10-5 versus 1/0 for F H /B F SK sys­

tem s w ith diversity m in additive white Gaussian noise. Eb/No = 13.35 dB for m - 1... ... 2.6 Perform ance of FH /8F SK with diversity m = 4 ch ip s/b it and R-T

diversity com biner with threshold 0 in additive w hite G aus'ian noise. 2.7 Perform ance of F H /8F S K with diversity m — 5 ch ip s/b it and R-'I'

diversity com biner with threshold 0 in additive w hite Gaussian noise. 2.8 Perform ance of F H /8F S K with diversity m = 9 ch ip s/b it and R-T

diversity com biner with threshold 0 in additive w hite Gaussian noise. 2 9 Eb/No required to achieve B E R = 10-5 versus I/O for FH /4FSK sys­

tem s w ith diversity m in additive w hite Gaussian noise. Eb/No = 10.61 dB for m = 1... 2 12 11) 20 21 22 23 24 25 26

List o f Figures x

2.10 Eb/No required to achieve B E R = 10-5 versus 1/0 for F H /8F S K sys­ tem s w ith diversity m in additive w hite Gaussian noise. Eb/No = 9.10 dB for m —.1... 27 2.11 Eb/No required to achieve B E R = 10-5 versus 1/0 for FH /16FSK

system s w ith diversity m in ad itive w hite G aussian noise. Eb/No = 7.07 dB for m = 1... 28 2.12 Perform ance of F H /B F S K with diversity m = 4 ch ip s/b it and R-T

diversity com biner w ith threshold 0 in worst case partial-band noise jam m ing. E\./N0 = 17 dB. Upper: BER. Lower: th e worst jam m in 0 p aram eter pwc... 33 2.13 Perform ance of F H /B F SK with diversity m = 5 ch ip s/b it and R-T

diversity com biner w ith threshold 0 in worst case p artial-band noise jam m ing. Eb/N0 = 17 dB. Upper: BER. Lower: th e worst jam m ing param eter pwc... 34 2.14 Perform ance of F H /B F S K with diversity m = 9 ch ip s/b it and R-T

diversity com biner w ith threshold 0 in worst case p artial-band noise jam m ing. Eb/No = 17 dB. Upper: BER. Lower: th e worst jam m ing

param eter pwc... 35 2.15 Perform ance of F K /4F SK with diversity m = 4 ch ip s/b it and R-T

diversity com biner w ith threshold 0 in worst case partial-band noise jam m ing. Eb/No = 15 dB. Upper: BER. Lower: th e worst jam m ing

param eter pwc... 36 2.16 Perform ance of F H /4F SK with diversity m = 5 ch ip s/b it and R-T

diversity com biner with threshold 0 in worst case p artial-band noise jam m ing. Eb/N0 = 15 dB. Upper: BER. Lower: th e worst jam m ing param eter pwc... 37

List o f Figures xi

2.17 Performance of F H /4F 3 K with diversity m — 9 chips/bit and R-T diversity com biner with threshold 6 in worst case partial-band noise jam m ing. E b/N 0 — 15 dB Upper: BER. Lower: the worst jam m ing

param eter pwc... 38 2.18 Perform ance of FH /8F SK w ith diversity m = 4 ch ip s/b it and R-T

diversity com biner with threshold 0 in worst case partial-band noise jam m ing. E'0/No = 13 dB. Upper: B ER. Lower: the worst jam m ing param eter pwc... 39 2.1? Performance of F H /8 F S It w ith diversity m — 5 ch ip s/b it and R-T

diversity com biner w ith threshold 0 in worst case partial-band noise jam m ing. Eb/No — 13 dB. Upper: BER. Lower: the worst jam m ing param eter pwc... 40 2.20 Perform ance of F H /8F SK with diversity m = 9 ch ip s/b it and R-T

diversity com biner with threshold 0 in worst case partial-band noise jam m ing. E b /N 0 = 13 dB. Upper: BER. Lower: the worst jam m ing param eter pwc... 41 2.21 Perform ance of F II/16F S K with diversity rn = 4 chips/bit, and R-T

diversity com biner w ith threshold 0 in worst case partial-band no;se jam m ing. Eb/No = 12 dB. Upper: BER.. Lower: th e worst jam m ing param eter pwc... 43 2.22 Perform ance of F H /16FSK w ith diversity m = 5 ch ip s/b it and R-T

diversity com biner w hh threshold 0 in worst case p artial band noise jam m ing. Eb/No = 12 dB. Upper: BER. Lower: the worst jam m ing param eter pwc... 44 2.23 Performance of FH /16FSK with diversity m = 9 ch ip s/b it and R-T

diversity com biner with threshold 0 in worst case partial-band noise jam m ing. Eb/No = 12 dB. Upper: BER. Lower: the worst jam m ing param eter pwc... 45

List o f Figures xii

2.24 E b / N j required to achieve B ER = 10~5 versus 1/6 for th e binary

system in worst case partial-band noise jam m ing. E b / N o — 17 dB. . 46 2.25 E b / N j re^ .ed to achieve B E R = 10~5 versus 1/6 for th e 4-ary

system in worst case partial-band noise jam m ing. E b / N o = 15 dB. . 47 2.26 E b / N j required to achieve B E R = 10-5 versus 1 / 6 for th e 8-ary

system in worst case partial-band noise jam m ing. E b / N o = 14 dB. . 48 3.1 h i(x ) and h2{ x , 6) with 6 as param eter versus 1/x . m = 4; 6 =

1 ,1 .5 ,2 ,4 , anu 10... 57 3.2 Perform ance of F H /B F S K w ith diversity m — 2 ch ip s/b it and

R-T diversity combiner with threshold 6 in worst case n = 1 band m ultitone jam m ing w ithout noise. Uoper: BER. Lower: th e worst jam m ing param eter a wc... 61 3.3 Perform ance of F H /B F S K with diversity m = 4 ch ip s/b it and

R-T diversity com biner with threshold 6 in worst case n = 1 band m u ltitone jam m ing w ithout noise. Upper: BER. Lower: th e worst jam m ing param eter a wc... 62 3.4 Perform ance of F II/B F S K with diversity rr. — 9 ch ip s/b it and

R-T diversity combiner with threshold 6 in worst case n = 1 band m u ltitone jam m ing w ithout noise. Upper: BER. Lower: th e worst jam m ing param eter a wc... 63 3.5 Relationship between jFc, Fe, Pc, P c x , P e x , P e and a for a fixed 6. 65 3.C Perform ance of F H /4F S K w ith diversity m = 2 ch ip s/b it and

R-T diversity combiner with threshold 6 in worst case n = 1 band m u ltitone jam m ing w ithout noise. Upper: BER. Lower: th e worst jam m ing p aram eter a wc... 70

List o f Figures

3.7 Perform ance of FH /4F SK with diversity in = 4 chips/bit and R- T diversity combiner with threshold 6 in worst rase n — 1 band m ultitone jam m ing w ithout noise. Upper: BER. Lower: th e worst jam m ing p aram eter a wc... 3.8 Perform ance of FH /4F SK with diversity m = 9 chips/bit, and R- T diversity com biner with threshold 0 in worst case n = 1 band m u ltitone jam m ing w ithout noise. Upper: BER. Lower: the worst jam m ing p aram eter a wc... 3.9 Perform ance of F H /8F S K w ith diversity rn = 2 chips/b it and R-

T diversity com biner with threshold 0 in worst case n = 1 band m ultitone jam m ing w ithout noise. Upper: BER. Lower: the worst jam m ing p e rim e te r a wc... 3.10 Perform ance of F H /8F SK with diversity in = 4 chips/bit and R-

T diversity com biner with threshold 0 in worst ease n = 1 band m u ltitone jam m ing w ithout noise. Upper: BER. Lower: the worst jam m ing p aram eter a wc... 3.11 Perform ance of FH /8F SK with diversity in = 9 chips/bit, and R-

T diversity com biner with threshold 0 in worst case n — 1 band m u ltito n e ja m m in g w ith o u ti oi.se. Upper: BER. Lower: th e worst jam m ing p aram eter a wc... 3.12 T h e bit erro r rate of F H /B F S K system with diversity in — 4 ch ip s/b it and R-T diversity com biner with threshold 0 in the worst band m ul­ tito n e jam m ing w ith n = 1. E b / N 0= l l , \ 3 dB; 0 =2,4,6,10... 3.13 T h e bit erro r rate of F H /B F S K system w ith diversity rn = 4 ch ip s/b it

and R-T diversity com biner with threshold 0 in the worst band m ul­ tito n e jam m ing w ith n = 1. jFi,/Aro=15,17 dB; 0 =2,4,6,10...

List o f Figures xiv

3.14 The worst a for F H /B F SK system with diversity m = 4 ch ip s/b it and R-T diversity combiner with threshold 8 m band m ultitone ja m ­ ming w ith n = l. Eb,'N0= l l , l 3 dB; 6 =2,4,6,10... 82 3.15 The worst a for F H /B F S K system w ith diversity m = 4 ch ip s/b it

and R-T diversity combiner with threshold 6 in band m u ltitone ja m ­ ming w ith n = 1. Upper: Eb/N0=15 dB. Lower: Eb/No—17 dB. 6 =2,4,6,10... 83 3.16 The bit error rate of F H /8FSK system w ith diversity m = 4 ch ip s/b it

and R -T diversity combiner with threshold 6 in th e worst band m ul­ titone jam m ing w ith n = 1 . # 6 /./Vo= j. 0 , 1 2 , 1 4 dB; 0 = 2 ,4 ,x0... 84 3.17 The worst a for F H /8FSK system w ith diversity m = 4 ch ip s/b it

and R-T diversity combiner with threshold 0 in band m ultitone ja m ­ ming with n — 1. J?*/A^o=10,12,14 dB; 0 =2,4,10... 85 3.18 E b /N j required to achieve BER = 10-5 versus 1/0 for th e binary

system in worst case band tone jam m ing. Eb/No= 17 d B ... 87 3.19 E b /N j required to achieve BER = 10-5 versus I/O for th e 8-ary

system in worst case band tone jam m ing. Eb/No= 14 d B ... 88 3.20 The E b /N j required to sustain B E R = 10-5 for F H /F S K w ith diver­

sity 77i = 4 ch ip s/b it and R-T diversity com biner w ith threshold 0 = 2 in n = 1 band m ultitone jam m ing. Binary system: E b/N 0=17 dB. 8-ary system: Eb/No—12 dB ... 89 3.21 The bit error probability of F H /B F SK system w ith diversity m = 4

chip s/b it and R-T diversity combiner w ith threshold 0 in th e worst partial-band noise and n = 1 band m ultitone jam m ing. Eb/No=17 dB; 0 = 2,4,10... 91.

List o f Figures XV

3.22 T he b it error probability of F H /8FSK system with diversity rn = 4 ch ip s/b it and R -T diversity combiner with threshold 0 in the worst partial-band noise and n = 1 band m ultitone jam m ing. Eb/N o —l i dB; 0 =2,4,10... 92 4.1 T he F F H noncoherent M FSK spread spectrum r e c e iv e r ...I l l 4.2 B ER of F H /B F S K m aximum -hkelihood diversity receiver with dif­

ferent diversity order in worst case partial-band noise interference. E b/No = 13.35 dB ... 116 4.3 B ER of F H /8F S K maximum-likelihood diversity receiver w ith dif­

ferent diversity order in worst case partial-band noise interference. E b/ N 0 = 9.09 d B ...119 4.4 Com parison of functions \nIo{x), x and x 2/ 4 ...120 4.5 B E R of F H /B F S K maximum-likelihood diversity receiver and sub­

optim um diversity receiver in worst case partial-band noise interfer­ ence. E b/No = 13.35 dB . ^ = 2 and 4... 122 4.6 Com parison of BER of F H /B F S K with different diversity receivers

in worst case p artial-ban d noise interference, m = 2 chips/bit. E b/ N 0 = 13.35 d B ...124 4.7 B ER of F H /B F S K maximum-likelihood diversity receivers and

self-norm alizing diversity receiver in worst case partial-band noise inter­ ference. m = 4 ch ip s/b it. E b/No = 13.35 d B ... 126 4 8 B ER of F H /B F S K maximum-likelihood diversity receivers and

self-norm alizing diversity receiver in worst case partial-band noise inter­ ference. m = 2 ch ip s/b it. E b/N o = 13.35 d B ... 128 5.1 Union bounds for the repeated Odenwalder code over th e BSC using

th e first term , the first four term s, and the first nine term s of the transfer function, respectively, for m = 3, 7, and 15...134

List o f Figures xvi

5.2 BER based on th e Gaussian approxim ation and th e union bound for th e repeated Odenwalder code over th e BSC. m = 3, 7, 15, and 31...137 5.3 BER based on sim ulation, the union bound, and th e G aussian ap­

proxim ation for th e repeated Odenwalder code over th e BSC. m = 3, 5, 7, 15, and 31... 139 5.4 Comparison of th e cutoff rate Rq of the BSC and th e overall code

rate r of th e repeated Odenwalder code over th e BSC to sustain Pb = 10~4... 141 5.5 R atio of th e overall code ra te r of the repeated O denwalder code

over th e BSC t j sustain Pb = 10-4 to th e cutoff rate Rq of th e BSC. 142 5.6 Model of M - ary Sym m etric Channel ... 143 5.7 Union bounds for th e repeated Trum pis and Odenwalder codes w ith

three kinds of m etrics over 4-ary sym m etric channels. m =3,7,15,

and 31... 147

5.8 Union bounds for th e repeated Trum pis and Odenwalder codes w ith two kinds of m etrics over 8-ary sym m etric channels. m =3,7,15, and 31... 149 5.9 Union bounds for th e repeated Trum pis and Odenwalder codes w ith

8-ary m etrics over 8-ary s; m m etric channels. m =3,7,15, and 31 for the Trum pis code; m =4,10,22, and 46 and m = 5 ,ll,2 3 , and 47 for the Odenwalder code... 150 5.10 M onte Carlo sim ulation B ER perform ance for a 4-ary sym m etric

channel using a repeated Odenwalder code w ithout interleaving and direct-generation, approxim ation, and conversion m etrics. . . . 161 5.11 M onte Carlo sim ulation B ER perform ance for an 8-ary sym m etric

channel using a repeated Odenwalder code w ithout interleaving and direct-generation and conversion m etrics...162

A ck n ow led gem en ts

I would like to th a n k my supervisors, Professor Q iang Wang and Professor Vijay K. Bhargava for th e ir guidance, inspiration and encouragem ent throughout my g raduate education a t the U niversity of V ictoria. I am grateful to Comm unications C anada for th e ir financial support through a research contract granted to Professor Vijay K. Bhargava and Professor Qiang Wang. I am also grateful to the University of V ictoria for the support in the form of an University of V ictoria Fellowship.

I would also like to th an k Professor M icaela Serra and Professor Nikitas J Dimopoulos for serving on m y supervisory com m ittee, and Professor Jam es A. Ritcey for serving as th e external exam iner in my Ph.D . oral exam ination. 1 owe m any thanks to Mr. Ron K err for proofreading p arts of the dissertation.

Finally, m y special thanks to my wife, Pam Xiaoping Li, for her support and consideration throughout this work.

To

M Y PA R EN TS

and

C h ap ter 1

In tro d u ctio n

In m any com m unication system s, in addition to background therm al noise, there are some other forms of disturbance to th e tran sm itted signals. The disturbance is called interference to th e com m unications. One type of interference is intentional, and this type of interference is called jam m ing. In order to provide adequate perfor­ m ance of th e com m unication link in an environm ent w ith interference or jam m ing, com m unication system s have to be designed with th e capability of working prop­ erly in such a severe environm ent. Spread spectrum is a technique which can be used to increase th e anti-jam capability of com munication systems. One of two m ajor classes of spread spectrum techniques is frequency hopping. This disserta­ tion is concerned with th e anti-jam capability of frequency-hop spread spectrum commun ications.

1.1

F req uency-H op C om m u n ication s

A diagram of frequency hopped frequency shift keying spread spectrum (F H /F S K SS) com m unication system is depicted in Figure 1.1. The F II/F S K SS system is basically an ordinary FSK system with the carrier frequency hopping over the spread spectrum com m unication bandw idth, denoted as W ,s , under th e control of a psuedonoise (P N ) code. T he FSK m odulated signal is called d a ta symbol, or for short, symbol. T he bandw idth of th e FSK m odulated signal is much sm aller

Chapter 1. Introduction 2 M Non-coherent Mtfched Fill > i Received Binary Data PN PN JAMMER FH DEMOD

Figure 1.1: T he FH noncoherent FSK spread spectrum system

than W aa. B ut by averaging over m an) hops, th e spectrum of F H /F S K signal has a bandw idth of W„. T he carrier frequency changing ra te is called hop rate. For a FH system , if the hop rate is greater th an symbol rate, th e FH system is called a fast frequency hopping (F F H ) system . O therw ise, it is called a slow frequency hopping (SFH) system . In a FFH system , one symbol is tran sm itted over m ore th an one hop. In this case, one hop is also called one chip. Let the duration of one symbol be Ta, and th e duration of one hop, oi chip, Tc. Suppose one symbol is tran sm itted over m (m > 1) hops, th en Ta = m T c. Since one symbol is tran sm itted over different tim e intervals, th e transm ission is a tim e diversity transm ission. T h e transm ission is also a frequency diversity transm ission since one symbol is tran sm itted o"er different frequency by carrier frequency hopping. The p aram eter m is called diversity order. T h e u n it of m is chips/sym bol. W hen the energy of a symbol is fixed, there is usually an optim um diversity order, in th e sense th a t there is a value of m which requires th e least signal-to-noise ratio to achieve a given final bit error probability, Ph.

Chapter 1. Introduction 3

1.2

In tellig en t Jam m in g

An intelligent jam m er is a jam m er which can adjust its jam m ing power distribution over th e entire spread spectrum signal bandw idth 1F„ to maximize its jam m ing effect based on th e certain am ount of inform ation about the stru ctu re of the com­ m unication system to be jam m ed. In other words, the jam m er can find and use th e best strategy to m axim ize its jam m ing effect. From the com m unicator’s point of view, this kind of jam m ing is th e worst case jam m ing. In this dissertation, we are concerned w ith the perform ance of FII com munication system s under worst case jam m ing. We assume th a t th e jam m er knows everything about th e system except th e PN code which controls the carrier frequency hopping sequence. We also assum e th a t th e total jam m ing power of the jam m er is finite and fixed.

Two types of intelligent jam m ing are considered, partial-band noise (PBN ) and m ultitone jam m ing. P artial-band noise jam m ing concentrates the to tal jam m ing power in a fraction of th e spread spectrum signal bandw idth, and injects jam m ing power into a receiver in th e form of additive Gaussian noise. M ultitone jam m ing injects th e to tal jam m ing power into a finite num ber of tones, which coincide with some of th e FSK signal tones used by the com m unicator. According to the distribution of th e jam m ing tones, m ultitone jam m ing can be divided into two classes: band m ultitone jam m ing and independent m ulti tone jam m ing. It is known th a t th e worst case m ultitone jam m ing usually occurs when there is only one tone in a jam m ed frequency hop slot (or a M -ary FSK symbol bandw idth) [1, 2].

1.3

D iv ersity R e c e p tio n in F H C om m u n ication s

under Jam m in g

Since th e jam m ing power distribution in th e spread spectrum bandw idth is non- uniform , tim e variant, and unknown to th e receiver, then after FH dem odulation, th e channel is statio nary only in every hop interval Tc, and th e channel is nonsta­

Chapter I. Introduction 4

tion<iry for th e symbol duration Ts. Several diversity receiver structures for this kind of channel have been proposed [3, 4, 5, 6] in last decade.

T h e inform ation on th e channel condition for a hop, w hether jam m ing is present or not, is called side inform ation. Some diversity combining schemes require side inform ation, or some o th er channel inform ation. Soft decision linear energy com­ bining (or soft decision square-law envelope combining) needs side inform ation to work well in strong jam m ing. If no side inform ation is available, the perform ance is very poor [7]. Clipped-linear combining needs inform ation on the signal am pli­ tude [5]. A daptive G ain Control (AGC) combining requires th e noise variance of each hop [8]. Providing side inform ation or o th er inform ation increases th e system complexity, and the im perfection of th e inform ation m ay cause some perform ance loss. Therefore, diversity combining schemes which do not require side inform a­ tion or inform ation o ther th an outp uts of M non-coherent m atched filters are more attractive.

In most of the previous works on analysis of diversity receivers, th e influence of the background th erm al noise is ignored. T his approxim ation is accurate when the jam m ing is so strong th a t th e worst case jam m ing is simply full band jam m ing (distributing jam m ing power uniform ly over th e full spread spectrum bandw idth kP„). B ut when the jam m ing is not very strong, this approxim ation is no longer valid, due to th e noncoherent combining loss in background noise, which is modeled as additive w hite G aussian noise (AWGN), an d lower signal-to-noise ratio at the chip level. T h e influence of these two factors increases as th e diversity order increases. A nother reason to consider background noise is th a t it is th e existence of background noise th a t makes th e extraction of accurate side inform ation difficult. Thus, perform ance of diversity combining schemes w ithout th e requirem ent of side inform ation should be analyzed in an environm ent where th e influence of the background noise is not small. Therefore, in a com plete analysis for strong, m oderate, and weak jam m ing or interference, background th erm al noise should be considered.

Chapter 1. Introduction 5

1.3.1

D iversity Combining Using Ratio-Threshold Test

Technique

V iterbi [9] first proposed to use ratio-threshold test technique to com bat jam m ing in F H /F S K spread spectrum communication .systems. By using a ratio-threshold test, an estim ation of channel condition can be obtained. T his estim ation can then be used to form a soft decision decoding metric. V iterbi analyzed th e performance of this technique in a F H /F S K system with coding bu t w ithout diversity under m u ltitone and PB N jam m ing, using the com putational cutoff rate technique [9, 10]. T he results indicated th at certain performance improven ent is possible.

Since then, th ere have been further works in this dilection. Chang analyzed the ratio-threshold (R-T) anti-jam technique from the inform ation theory point of view and used game theory approach in the analysis [11]. She showed the perform ance im provem ent by R-T technique over a hard decision receiver. Clifford and Schonhoff analyzed th e perform ance of R-T anti-jam technique in m ultitone jam m ing channel w ith background noise [12]. T heir analysis is also based on cutoff rate. T h e above analyses concentrated on system s w ithout diversity. There are several ways to use th e R-T technique in diversity combining [13, 14]. The scheme proposed in [14] is fu rth er studied in this dissertation.

Since th e R -T diversity combining scheme has a very simple structure, it is qu ite easy to im plem ent. T his type of diversity combining does not require side inform ation b u t with an adjustable param eter. T h e purpose of our research is to investigate th e perform ance of ratio-threshold test technique when used in diversity com bining and th e influence of the param eter of threshold to th e system perfor­ m ance. This dissertation analyzed th e perform ance of th e R-T diversity combining scheme under p artial-band noise jam m ing and m ultitone jam m ing with the influ­ ence of background therm al noise taken into account. T h e analysis is based on th e exact bit error probability instead of bounds on bit error probability or cutoff rate. T hus m ore accurate and direct results on system perform ance are obtained

Chapter 1. Introduction 6

in this dissertation. The perform ance comparison of th e various diversity combin­ ing schemes is m ore accurate when it is based on th e exact bit error probabilities rather th a n bounds.

1.3.2 Optim um D iversity R eception in P B N w ith AWGN

According to signal detection theory, the optim um diversity receiver stru ctu re can be derived given th e properties of th e transm ission channel. A lthough m any pro­ found works have been done in design and analysis of diversity receivers in jam m ed channels, th e optim um structures of diversity receiver on partial-band noise and m ultitone jam m ing channel with background therm al noise are unknown. Keller considered optim um diversity reception on PBN channel w ith perfect side infor­ m ation and no background therm al noise [13]. In this case, th e optim um diversity reception in PBN can be treated as th e problem of th e optim um diversity reception in additive w hite Gaussian noise. T h e optim um diversity reception on PB N chan­ nel with therm al noise is investigated in this dissertation. A lthough th e optim um stru ctu re may require even more channel inform ation, and th us is im practical, the perform ance of optim um diversity reception can provide a upper bound on the perform ance of a non-optim um diversity receiver on PBN channel. A nd the upper bound can be used to judge non-optim um but practical diversity receivers. The im practical stru ctu re may provide guidance when designing practical diversity receiver.

1.4

C o n ca ten a ted C od in g for H igh -E rror-R ate

C hannels

F.’om a channel coding theory point of view, since one symbol is independently tran sm itted several times in a fast frequency hop system , there is repetition cod­ ing inherently in th e system . If th e tran sm itted d a ta are encoded w ith an error correction code, th en th e entire system is equivalent to a concatenated coding

Chapter 1. Introduction 7

cheme w ith a repetition code as the inner code.

T he inner m ost transm ission channel, i.e., the channel in which th e d ata with repetition coding are transm itted , can be qu ite different depending on, such as, th e m odulation, th e presence or absence of interference, and the types of interference. For different inner channel models, differ nt analysis m ethods have to be used to analyze th e system performance.

If th e channel is modeled as a soft decision non-coherent A/FSK channel, then decoding of repetition code is a form of diversity combining. However, if the inner m ost transm ission channel is m odeled as hard decision non-coherent A/FSK channel, or Af-ary orthogonal channel, th en th e channel belongs to the A/-ary sym m etrical channel (A/SC). The case where th e error rate of the com m unication channel is very high (as high as in the order of 0.1) is considered. This high error rate m ay be due to strong interference a n d /o r a relatively low signal-to-jam m ing ratio at th e chip level (i.e., chip energy-to-jam m ing power spectral density ratio). Since th e jam m ing is assum ed to be very strong, the worst case jam m ing is full- band jam m ing. T hus, th e channel is assum ed to be stationary. Low rate coding has to be em ployed in a channel w ith such high error rate. C oncatenated coding with a repetition code as the inner code provides a way to im plem ent low rate coding. As we m entioned before, this coding stru ctu re is inherent in coded FF?I/A /FSK system s. On M -ary sym m etric channel, th e maximum-likelihood decoding (MLI)) of repetition code is by a simple m ajority vote [15, page 60]. But in order to im plem ent soft decision decoding of th e concatenated code, a m ethod m ust be found for th e decoder of inner repetition code to generate soft decision m etrics for th e soft decision decoder of outer code. Furtherm ore, the outer code and the inner code are r o t necessarily in the sam e dom ain. For exam ple, th e outer code m ay be a binary code, and the inner code may be an A/-ary repetition code. Thus, it is interesting to investigate the perform ance of a code w ith decoding m etrics generated from a different dom ain. A convolutional code concatenated w ith repetition code as the inner code is called repeated convolutional code. In

Chapter 1. Introduction 8

this dissertation, a new coding scheme with a binary convolutional code as th e outer code and an M -ary repetition code as th e inner code is proposed. M ethods of generating th e decoding m etrics for this coding scheme are developed, and new generating functions of a binary convolutional code are found. The perform ance of the new scheme is analyzed in various conditions.

1.5

D isse r ta tio n O utline

In C h apter 2 and 3, the performance of a F F H /M F S K system w ith R -T diversity com bining scheme is analyzed under partial-band noise jam m ing and band m ulti- tone jam m ing, respectively. By using an average com putation model developed in C hapter 2, the exac! bit error probability of th e F F H /M F S K system under worst case P B N and m ultitone jam m ing including therm al noise influence is com puted. Effects on the system performance of different system param eters, such ac diversity order, ratio-threshold, and signal-to-therm al-noise ratio, and jam m ing param eters are also analyzed.

C hapter 4 discusses maximum-likelihood diversity combining on a PB N chan­ nel. T h e stru ctu re of maximum-likelihood diversity combining in PB N , and some sub-optim um combining schemes are investigated. The optim um perform ance is obtained by M onte Carlo simulation. The results show th a t the AGC diversity com bining scheme is actually optim um w ith regard to th e system perform ance under w orst case jam m ing when the PB N jam m ing is noc too weak.

In C h apter 5, a repeated convolutional code for an M -ary sym m etric channel (M SC ) characterized by a large error probability pe is considered. Em phasis is placed on using a binary convolutional code. T he effect of finite quantization and m ethods to generate binary metrics for M > 2 are investigated. New union bounds on the b it error probability for systems with a binary convolutional code on i-ary and 8-ary orthogonal channels are presented.

C h ap ter 2

P erform an ce o f R atio-T h resh old

D iv ersity C om bining Schem e in

F F H /F S K S y stem s under

P a rtia l-B a n d N o ise Jam m ing

2.1

In tro d u ctio n

The R atio-threshold technique, proposed by V iterbi [9], can be used in fast fre­ quency hop M -ary frequency shift keying (F F H /M F S K ) spread spectrum com­ m unication system s to com bat jam m ing. V iterbi analyzed th e perform ance of this scheme in coded F F H /M F S K systems under m ultitone and partial-band noise ja m ­ ming using th e com putational cutoff rate technique [9] [10]. His results hidicated th a t certain perform ance im provem ent is possible.

W hen diversity is used in a frequency hop system , with or w ithout an outer error correction code, the ratio-threshold technique can be used in d>versit,y com­ bining. This, in fact, is equivalent to im plem enting soft decision decoding of a repetition code. Laufer and Reichm an [14] discussed this scheme, and analyzed the perform ance of th e scheme in worst case partial-band noise (PB N ) jam m ing. However, their em phasis is on the effects of non-ideal :nterleaving.

In this chapter, we first give a brief description of the ratio-threshold (R-T) di­

Chapter 2. R - T Com bining in P B N Jam m ing 10

versity combining scheme. Then a m ethod for com putation of th e b it error prob­ ability of a F F H /M F S K system w ith ratio-threshold diversity combining under jam m ing is introduced. Next the b it error probabilities of th e F F H /M F S K system with ratio-threshold diversity combining under worst case p artial band noise are com puted. Some results are com pared w ith th e perform ance of th e F F H /M F S K systems with diversity utilizing the soft decision m etric w ith side inform ation. Ef­ fects on the system perform ance of different system param eters, such as diversity order, ratio-threshold, and signal-to-therm al-noise ratio, and jam m ing param eters are also analyzed.

2.2

R a tio -T h resh o ld D iv ersity C om bining

The stru ctu re of ratio-threshold diversity combining scheme is quite sim ple and is described as follows.

T h e ratio-threshold test is m ade on every m atched filter o u tp u t at each hop, and a hard decision is m ade with a quality bit. T he hard decision of each M -ary symbol is m apped back to A = log2 M b it binary symbols. T he o u tp u ts of all hops (in the form of binary symbols) aie accum ulated w ith good quality bits and with poor quality bits, respectively. If th ere is at least one hop decision w ith a good quality bit and there is a m ajority decision w ithout a tie, then th e o u tp u t of the com biner is th e m ajority decision with a good o u tp u t quality b it attached. If there is a tie between 0 and 1 (or “space” an d “m ark” ) w ith good quality bits, or there is no hop decision w ith a good quality bit, a decision is m ade based on hop decisions with poor quality bits (if there is a tie, flip a coin), and this o u tp u t of the com biner is attached w ith a poor o u tp u t quality bit.

If th ere is no outer error correction code, th e ou tp u ts of diversity com biner are also th e o u tp u t of the receiver, and th e o u tp u t quality b it has no use and can be discarded. However, if there is an outer error correction code, th e soft decision decoding can be carried out based on the o utp u ts of th e diversity combiner.

Chapter 2. R -T Com bining in P B N Jam m ing 11

2.3

A verage C o m p u ta tio n M o d el for B it Error

P ro b a b ility

Consider a F F H /M F S K spread spectrum system (Figure 1.1) with a R-T diversity combiner. Assume th a t the diversity order, the num ber of chips or hops per d ata symbol, is m. Let th e o u tp u t of th e non-coherent m atched filter for the ith symbol a t th e fcth hop be Xik, where i = 0,1, • • •, M , and k = 1,2, ■ • •, m. Let the ratio threshold be 9, which is a p aram eter chosen by th e com m unicator. At first we assume th a t th ere is only system therm al noise, b ut no jam m ing.

F irst we consider a binary F F H /F S K system . T he hop decision is made based on which non-coherent m atched filter o u tp u t is larger, and th e quality bit q is set according to

= ( 0 (good) if f e p > *, ( 1 (poor) otherwise, where

X max(k) — max{Xofc, Xi*;},

A^m,'n (^') — min{ Xok, .Xi/j}.

and k = 1,2, • • • ,m . This results in a channel w ith binary inputs and quaternary ou tp u ts, as shown in Figure 2.1. T he transition probabilities of the channel are known to be [10]: Pc = Fc (0), P c x = F c ( l ) ~ F c ( 0 ) , Pe x = Fe(1) — Fe(0), Pe = Fe(0), where Fc {9) = P r { X Qk > 9 X n \ “0” sent} , Fe(0) = P r { X i k > 9Xok\ “0” sent} .

Chapter 2. R -T Com bining in P B N Jam m ing 12 Input 0 c x E X E X C X 1 Output Quality 0 0 Good

01

1 1

10

Poor Poor Good

Figure 2.1: The Channel Model

Fc{0) and Fe(&) can be derived from th e distribution of th e two m atched filter outputs, Xok and Xik- T he bit error probability a t th e o u tp u t of R -T com biner can be derived from P c , Pc x, Pe x and Pe accordingly.

Let I be the o u tp u t of th e diversity com biner before flipping th e two-sided coin. There are five possible values for /:

I =

00 decision is 0, good quality,

01 decision is 0, poor quality,

x tie

11 decision is 1, poor quality,

10 decision is 1, good quality.

y Q h ,fc2,fei,fc4 p k i p k 2 p k ? x P g x SK,ki>h

The conditional distribution of 1 given th a t “0” is sent is

P r(/ = 00|0 sent) = P r(/ = 0110 sent) = P r(l = 1110 sent) = (2.1) y Q k t l k 2 , k 3 ,k 4 p k j p k 2 p ^ P g X (2 2) Sh-,k2=ki ,k3>ki £ C t k2'k3MPcl P kE2Pc3x P t x (2-3) SjC,fc2=fcl ,k3<k\

Chapter 2. R - T Com bining in P B N Jam m ing 13

P r ( / = 10|0 sent) = £ C%'k M tP #P j? P & P & x (2-4)

SkM <k2

P r(/ = x|0 sent) = £ C%’k M * Pfr P k> P & P kE'x (2.5) Sh-,k2=ki ,k3=kt

where

Sk = {^i, k2, k3, fc4|0 < h , k2, k3, k4 < m, ki + k2 + k3 + k4 = m ) and C ^ ’k2'k3,k* is m ultinom ial num ber, which is given by

ml (Jki,k2,k3,ki _

h \ k 2\k3\k4l

Suppose th a t th e two symbols are equiprobable, then the b it error probability at th e diversity com biner o u tp u t is

Pb = P r(/ = 1110 sent) + P r(/ = 10|0 sent) + ^ P r ( / = x|0 sent). (2.6)

For M -ary F F H /F S K system s, th e hop decision is m ade by choosing th e largest o u tp u t of M m atched filters. Suppose th a t, for th e fcth hop,

JCjf* — m ax Nik J 0<t<M—1

th en th e quality b it can be assigned as:

, , . _ f 0 good if ^ > 0 for all i j , \ 1 poor otherwise.

Assume th a t th e K = log2 M binary inform ation bits associated with each trans­ m itte d M -ary sym bol are ideally interleaved. Then th e channel model is th e same as th e binary case shown in Figure 2.1. B ut th e transition probabilities are changed into [10]:

Chapter 2. R -T Com bining in P B N Jam m ing 14

where

Fc (0) = P r{X jfc > 6X ik for all i ^ j \ j sent},

Fe(&) = Pr{X„A; > OXik for a specific n ^ j and all i ^ n \ j sent }.

Sim ilar to the binary case, Fc{0) and Fe(0) can be derived from th e distributions of M m atched filter ou tp u ts X o k , Xi k , • • • , X(m-i)k- T hen b it error probability at

N ote th a t th e above results are for channels w ithout jam m ing. T he statistics of these channels are stationary. A channel w ith jam m ing present, however, is nonstationary. For one hop, the signal is either jam m ed or not jam m ed. Therefore the channel has two states. One is th a t a hop is being jam m ed; and th e o ther one is th a t a hop is free of jam m ing.

Let Q c , Qei QcXi and Qe c be th e transition probabilities of th e channel sta te

w ithout jam m ing. They can be com puted by

the o u tp u t of th e com biner can be com puted by using (2 .6).

Q c

Q c x = [ F c o i V - F c o m + ^ Y - ^ l F E o M - F E o m ,

Qe x = y [ ^ o ( l ) - F £oW ],

Qe = f F EO(0), where

Chapter 2. R - T Com bining in P B N Jam m ing 15

Feq(9) = P r{ X njt > OXik for a specific n ^ j and all i n |j sent, w ithout jam m ing}.

Similarly, let R c , Re, Rc x, and Re x be th e transition probabilities of the channel sta te w ith jam m ing. And they can be com puted in the sim ilar way as Q c , etc. by using

Fci(6) — Pr{Ajfc > OXik for all i ^ j \ j sent, with jam m ing}, Fe i(9) — Pr{A"njt > OXik f°r a specific j and all i ^ n \ j sent,

w ith jam m ing}.

Assume th e probability of a hop being jam m ed is p, then th e conditional prob­ ability distribution of I given th a t “0” is sent is

Pr(Z = 00|0 sent) = f 2( k i , k 2, k 3, k 4, Q c , Q E , Q c x , Q E X , (2 . 11 ) Sk , k , = k 2 , k z > k i R c , Re, R e x , Re x- p) 1 P r(/ = 1110 sent) = ^ f 2{ k i , k 2, k 3, k 4, Q c , Q E , Q c x , Q E X , (2.12) SKM=k2,kz<ki Rc, Re, Rc x, Re x, p) P r(/ = 10|0 sent) = ^ f 2( k i , k 2, h , k4, Q c , Q e , Q c x , Q e x , (2.13) <A.'2 Rc, Re, Rc x, Re x,p) I P r(/ = ®|0 sent) = f 2{ h , k 2, h , k4, Q G, Q e , Q c x , Q e x , (2.14) Rc, Re, Rc x, Re x, p) (2.15) where

Chapter 2. R -T Com bining in P b N Jam m ing 16

k i k j k 3 k 4

_ ^ y ' £ i l i , h , h , U < k i —l i , k 2 —l 2 , k 3 —l 3 , k 4 — U h = 0 l 2 = 0 1 3 = 0 U = 0

x (1 - p)il +'2+'3+'4pm~h $ ql*ql*ql*r f1 ~h ~ hr*3 - '3r ki~u .

In order to simplify th e com putation, it can be shown th a t equations ( 2.11) - (2.15) can be com puted w ith (2. 1) - (2.5) by sub stituting P c , P e , P c x, and P e x with th e average transition probabilities (see Section 2.7 for detail) :

Pc = (1 — p )Q c + p R c (2.16)

Pe = (1 •• p) Qe + pRe (2-17)

P c x = (1 — p ) Q c x + p R c x (2.18)

Pe x = (1 — p) Qe x + pRe x (2.19)

Further, th e average transition probabilities Pc, etc. can be com puted w ith (2.7) - (2.10) by su bstitu ting F c ( 0 ) and Fe( 0 ) w ith averaged version of F c { 0 ) and Fe{0)

which are given by:

Fc (0) = (1 - p)Fco(0) + pFci(0) (2.20) Fe(0) = ( 1 -p) Fe o(0) + pFe i(0) (2 .2 1 )

Thus, even th e jam m ing channel considered here is a tw o state channel, the bit error probability can be com puted by using an equivalent single s ta te channel which is characterized by «ne average of statistical properties of th e two channel states. To simplify th e notations, F c and Fe will be ju st w ritten as F c and Fe respectively in th e following p art of this chapter and the next chapter.

2.4

P erform an ce in A d d itiv e W h ite G au ssian

N o ise

We first analyze th e perform ance of th e R-T diversity com biner in additive w hite Gaussian noise (AWGN) only. This is th e sim plest case, and it also represents th e

Chapter 2. R -T C om bining in P B N Jam m ing 17

asym ptotic situ ation when jam m ing is very weak.

Let Eb be th e energy per inform ation bit, No th e power spectral density of AWGN. Then th e energy per hop to noise spectral density ratio is I\Eb/rnNo.

F c ( 0 ) and Fe{ 0) are:

A-/-1 POO t x f V

Fc{0) = / Ps + n{ x) / p n ( y ) d y

Jo Jo

roo f x/& t xiv

Fe{0) = p n { x ) J^ p 1+n( y ) d y J^ p n{ z ) d z d x , rx/0 M- 2 d x, where *2 + 2 £ & \ p ,+„(x) = x e x p I j / . ( * \ [ ^ ) . Pn( x ) = I e x p

and I o ( x) is the modified zeroth order Bessel function. Carrying out th e integrals, we can obtain M -i / F o W = i + E ( - ! ) M *=1 \ m - 1 \ e 2 k + o2exp , 2 2 2 > \ k + 02 m N 0) ' and Fe{6) = M- 1 '*+' ( M k ‘ ) e2 k M - 1 ’ V k j k + e2 k + o2 - l k + e2 - i K E b\ x e x p { — T T ¥ - ^ N o ) - (2.23)

Eqs. (2.22) and (2.23) can also be obtained from equations (3.9) and (3.10) in [10] by su bstitutin g p = 1 and changing E , into Eb/ m, and N j into N 0, respectively.

F igure 2.2, F igure 2.3 and Figure 2.4 show th e bit error rates of th e binary system s w ith m = 4,5, and 9, respectively. Bit error rates of th e binary systems

Chapter 2. R -T C om bining in P B N Jam m ing 18

w ithout diversity are also plotted in these figures for reference ( it is known th a t tim e diversity can not generate any gains in AWGN, and in contrast, there are some diversity com bining losses). It can be seen th a t 0 = 2 is roughly optim um in all th ree cases. T h e perform ance of a system w ith very large 0 (0 > 10) is alm ost the sam e as th e perform ance of a system with h ard decision only (0 = 1). To see th e perform ance dependence on 0, th e Eb/No required to achieve B ER = 10-5 versus reciprocal of 0 is plo tted in Figure 2.5. For a fixed m , it can be seen th a t the left end (0 —> oo) and right end (0 = 1) of th e curve have th e sam e height. This m eans th a t th e perform ance of a system w ith sufficiently large 0 is th e sam e as th a t of a system with 0 = 1. This is because large 0 causes the decision o u tp u t for a hop to be alm ost always w ith bad quality bit, form ing a two quantization level, instead of four. T hus it is equivalent to h ard decision system . It also can be seen th a t th e optim um 0 which requires the least Eb/No to achieve B E R = 10-5 is less th an 2 and the exact optim um 0 depends on diversity order m. For m = 2 with optim um 0, th e m inim um Eb/No required to have a B E R of 10-5 is about 1.5 dB m ore than th a t for th e system w ithout diversity.

T he bit error ra te of 8-ary system w ith m = 4,5, and 9 are plotted in Figure 2.6 to Figure 2.S. T h e required Eb/No to achieve B E R = 1 0-5 versus th e reciprocal of 0 for 4-ary, 8 -ary and 16-ary systems are plotted in Figure 2.9, Figure 2.10, and Figure 2.11, respectively. Similarly, the perform ance of a M -ary system (M > 4) with large enough 0 is alm ost the same as the perform ance of the system w ith hard decision in each hop (0 = 1). W hen m is small, 0 = 1 and very large 0 gives worse performance. B ut when m is large, the perform ance is q u ite sensitive to 0. T h e optim um 0 is also less th a n 2, and the specific optim um p oin t depends on m.

Chapter 2, R - T Com bining in P B N Jam m ing ₁₉ -£> ,0 10 l 10

m

= 1 -2 10

0

=

2

■3 10 _{r ~} 0 = 3 ■4 10 5

10

-6 10 0 = 6, 8, 10, 1.01 -7

10

0 5

10

15 20 Eb / N 0 (dB)

F igure 2.2: Perform ance of F H /B F S K with diversity m = 4 c h ip s/b it and R-T diversity com biner w ith threshold 0 in additive w hite G aussian noise.

Chapter 2. R -T Com bining in P B N Jam m ing 20 1 10 m = 1 -2 10 0 = 2 -3 10

₀₌₁₀

•4 10

0

=

1.01

■5 10 ■6 10 ■7 10 0 2 4 6 8 10 14 16 Eb / N 0 (dB)

Figure 2.3: Perform ance of F H /B F S K w ith diversity m = 5 c h ip s/b it and R-T diversity com biner w ith threshold 0 in ad ditiv e w hite G aussian noise.

Chapter 2. R -T Combining in P B N Jam m ing ₂₁ •e O, .0 10 l 10 0 = 6

io'2 r r

0 = 2 0 = 1 0 E 0= 1.01 -3 10 0 = 3 .-4 1 0 0 = 4 •5 10 ,-6 10 0 2 4 6 8 10 12 14 16 18 Eb / N 0 (dB)

Figure 2.4: Performance of F H /B F S K with diversity m = 9 ch ip s/b it and R-T diversity combiner w ith threshold 9 in additive w hite G aussian noise.

E

b

!N

0

(dB)

C hapter 2. R - T C om bining in P B N Jam m ing 22

20.0 19.5 19.0 18.5 18.0 _{m = 10} 17.5 17.0 m = 4 16.5 16.0 15.5

m = 5

m = 2 15.0 14.5 14.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1 / 0

F igure 2.5: E b/No required to achieve B E R =10 5 versus 1/0 for F H /B F S K systems w ith diversity m in additive w hite Gaussian noise. Eb/No = 13.35 dB for m = 1.

Chapter 2. R - T Com bining in P B N Jam m ing 23

0 = 1.01,4

6 8 10

Eb / N 0 (dB)

Figure 2.6: Perform ance of F H /8 F S K w ith diversity m = 4 ch ip s/b it and R-T diversity com biner w ith threshold 0 in additive w hite G aussian noise.

Chapter 2. R -T Combining in P B N Jam m ing 24 ,0 10 10 ■2 10 ■3 10 9 = 2 ■4 10 0 = 3 ■5 10 6 10 •7 10 2 4 6 8 10 12 14 Eb / N 0 (dB)

Figure 2.7: Perform ance of F H /8F S K with diversity m = 5 c h ip s/b it and R-T diversity com biner with threshold 0 in additive w hite G aussian noise.

Chapter 2. R - T Com bining in P B N Jam m ing 9 5 •ft

n

0 = 1.01, 10 6 8 10 12 Eb ! N 0 (dB)

F igure 2.8: Perform ance of F H /8F S K with diversity m = 9 ch ip s/b it and R-T diversity com biner with threshold 0 in additive w hite G aussian noise.

(g p ) ° N /q 3

C hapter 2. R - T C om bining in P B N Jam m ing 26

16 r ~ r

m =

10

t

4

1 / 9

Figure 2.9: Eb/No required to achieve B E R = 10 5 versus 1/0 for F H /4 F S K systems w ith diversity m in additive w hite G aussian noise. Eb/No = 10.61 dB for m = 1.

Chapter 2. R - T Combining in P B N Jam m ing 27 PQ

3

m =

10

m

=

9

m

=

4

1 / 0

Figure 2.10: Eb/No required to achieve B E R =10 5 versus 1/0 for F H /8F S K sys­ tem s w ith diversity m in additive w hite Gaussian noise. Eb/No = 9.10 dB for m = 1.

C hapter 2. R - T C om bining in P B N Jam m ing 28 14 m = 10 13

m = 9

m = 4

12

m = 5

11 m = 2 10 9

0.2

0.4 0.6 1 / 0 0.8 1.0

F igure 2.11: Eb/No required to achieve B E R = 10 5 versus 1/0 for F H /16F S K sys­ tem s w ith diversity m in additive white G aussian noise. Eb/No = 7.07 dB for m

C hapter 2. R - T Com bining in P B N Jam m ing 29

2.5

P erform an ce under P artial-B an d N lis e Jam ­

m ing

W hen a partial-band noise jamm er jam s a fraction p of the transmission band with

noise power density N j / p , F c ( 0 ) and Fe{0) cab be com puted according to (2.20) and (2.21): Fc (0) = ( l - p ) F co(0) + pFcx(0) F E (0) = ( l - p ) F Eo(O) + p F E l (O) and /•OO t x /Q F c o ( 6 ) = / P s + n ( z ) / P n ( y ) d y J o J o = /„' M - l d x X a: exp rx/Q [ y e x p ( - y 2/ 2 ) d y Jo 2 I \ V m N c J M - 1 d x M - l / k- 0 \ M - 1 \ 02 0* + kexp k K E b\ 02 + k m N 0) ’ yoo jFci(^) = / p ,+j+ „(x) / p j + n ( y ) d y J O J o M - l d x x exp

-

j

C

x / 0 I X x 2 + 2 K E b/ [ m { N 0 + N j / p ) ] \ 2 K E b m ( N 0 + N j / p ) rx/0 / e x p { - y 2/ 2 ) d y Jo M - l d x K E b too rx/0 fx/8 F e o ( 0 ) = / P n ( x ) / p s+n( y ) d y / p n{ z ) d. Jo Jo Jo 02 + k V 02 + k m ( N 0 + N j / p ) ) ’ M —2 d x

Chapter 2. R -T Combining in P B N Jam m ing 30 ro o r x f O I x e x p (—x2/ 2 ) I y e xp Jo Jo

!c±2ife * (« /!!)*

1" r x / C x \ J z e x p ( —z 2/2 ) dz M —2 dx M —2 / *=0 \ m - 2 \ e2 1 fe ; <?2 + fe^2 + fe + i r:r/0 /*x/^ exp

(

-J

roo t x/V t x/ “ ' P j + n ( x ) / p a+ j + n ( y ) d y / P j + n ( z) d. o Jo Jo 02 + k I<Eb\ 02 + fe + 1 tuNq J ’ M —2 d x f°° f xl& ( = J x e x p ( —x 2/ 2) j /*/» y e x p (f y2 + 2 K E b l [ m { N 0 + N j / p ) \ \ Jo

yCXPl

--- 2---

J

U * , ’ " eXp(_2'V 2 )^ fc=0 \ M —2 <ix M —2 k: x exp I —

fe

I

02 + fe 02 + fe + 1

fl2 + fe K E b \ 92 + k + 1 m ( N o + iV j/p) J ‘ So * c ( 0 )

1 W ( V W

(

-, fe K E b V + P exP ^ Q2 fem (]V0 + N j / p ) ) 1 fe / 02 + fe 02 + fe + 1 , ( 02 + k I<Eb\ d - p ) e x p / g2 + fe — Q J pexp ^ #2 + } m(/V0 -f N j / p ) ) K E b