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-1. THE Sff SKBLIIT TED IS PARTIAL FULFILLMEST

OF THE REQUIREVESTS FOR THE DEGREE O F

~ ~ S T E R OF -APPLIED SCIESCE

in ;be Schcol of

Engineering Science

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autorisation.

APPROVAL

Name: Tony Chan

Degree: Master of Applied Science

Title of thesis: The Error Performance Evaluation of Coded Sys-

tems in Radio Fading Channels

Examining Committee: Dr. John Jones, Chairman

Dr. Paul Wo Senior Super7;isor

Dr. Jacques tiaisey Supervisor

Dr. James Cavers Examiner

Date Approved: March 31, 1993

I hereby p t to Simon Fmsa University the right to lend my thesis, project or extended essay

(fie title of which is show2 k b v j to users of the Simon Fraser University Libraq, and to make

pafiid or ,&ngle copies only for such users or in response to a request from the library of any

other university, or otfia educational institutbn, on its own behdf or for one of its users. I

fu&er a p e that permission for multiple copying of this work for scholarly purposes mEy be

granted be me or the Dean of Graduate Studies. It is understood that copying or publication of

this work for financia! g& shd1 not be allowed without my mitten permission.

"The Error Performance Evafuahon of Coded System in Mobile Radio Fading Channel"

Author:

The mobile radio channel is a frequency selective fading channel which increases the

bit error rate of data +,ransmission when compared to the frequency non-selective

(or flat) fading channel. The emerging Korth American digital cellular system uses

convolutional code in conjunction with 2-QPSK to improve the data transmission

quality.

In this thesis, we attempt to improve the error performance of the existing coding

system used in cellular applications. ?Te will evaluate the error performance of two

coded systems: trellis coded modulation (TCM), and convolutional codes. Specifi-

cally, we will investigate the feasibiEty of using TCM to replace convolutiona1 codes

in celfular applications.

We first investigated the error performance of uncoded QPSK transmitted over

a frequency selective fading channel and this channel is modelled as a 2-ray channel

with independent Rayleigh flat fd ing in each ray. It was found +hat the relative delay

between the two arrival rays introduces a diversity effect which helps to improve the -

error performance I ~ I ~ I I compared to the Bat fading channei. hen, we designed

and stadied the emor perfom-mce of coherent trellis-coded I-nPSK modulations in 4 3

the additi.r-e white Gaussian noise (AWGS) and the Rqleigh flat fading channels.

The basic idea is to use multiple trellis coded modulations. Several good codes are

designed with thraughpur eqaal ro 1.5 birs/synbof and 1 bit/s.=mboi. One of the codes

bas a 44. dB coding gain compared to ancoded QPSK in ar! .?itVGS cfrarmcl while

another one p r o ~ r d e a 5% d e r dirersizy eifect in a flat fading cilaniiei. [Ye have

also compwed the enor performance of the trellis codes with conventional systems

that rise c o n v o f ~ t l o d code ir?, crtnjuodon with 2-QPSK rnoddation. We found that

trelPis codes pedora at %east as tsw4 as convolutional codes. In some cases, the coding

gain can be as !xge zs 2ddE. Einzlls we combined the results from the two previous

studies and invetigzied the enor frerformacce of coded systems in frcyuenty selective

fading channels. For an sncoded ~vstem. it, was found that frequency selective fading

introduces a diwfsiry e-ifect which improves the error perfornrance. However, this is

not true for coded sptems. For both trellis codes and co~xofutional codes, the best

performance belongs to the frequency nc 3 selective fading channeI. Nevertheless,

trellis codes once again perform at least as well as convolutionaf codes in frequency

selective fading cha~rsels.

ACKNOWLEDGMENTS & * d*,,

I would like to express my sincere appreciation to Dr. Paul Ho for providing the

subject of this thesis aod the pidance throughout the course of this research. I

::-oufd also like to thank Dr. P.J.McLane for the many insightful discussions during

the course of the thesis.

Financial support from the Canadian hstitute of Te1ecommunicaf;ion Research (CITR),

titrough Dr. P.J.McLax of Qaen's University, is gratefully acknowledged.

Finally, special thaoks tto my falllily and friends for their encouragement during the

preparation of the thesis.

CONTE

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ACKKOWLEDGM ESTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1 Backgrouod and Literature Review . . . . . . . . . . . . . . . . . . .

1.1.1 Frequency Selective Fading Channel . . . . . . . . . . . . . . .

1.1.2 Treilis Coded Modulation . . . . . . . . . . . . . . . . . . . .

1.1.3 Coded Systems Operating in a Frequency Selective Fading Chan- nel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2 Contributions of the Thesis . . . . . . . . . . . . . . . . . . . . . . .

1.3 Thesis Outfioe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 The Error Performance of Uacoded QPSK . . . . . . . . . . . . . . . . . .

2.i System Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2 Uncoded QPSK with Perfect Channel State Information . . . . . . .

2.2.1 Systerrr %ode1 . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2.2 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . 2.2.3 Example: Two-Ray Model

v

xi

xii

1

3

3

6

. . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 NumericalResults

. . . . . . . . . . . . . . . . . . 2.3 Imperfect Channel State Information

. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Error Anak~sis

. . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 3umerizal Results

. . . . . . . . . . . . . . . . . . . . . . . 2.4 The effect of Pulse Shaping

. . . . . . . . . . . . . . . . . . . . . 2.4.1 The Raised Cosine Pulse

. . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Simulation Model

. . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Numerical Results

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Summary

. . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Trellis-Coded Modulation

. . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Trellis-Coded 2-QPSK

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Design Procedure

. . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Rate 3/2 Codes

. . . . . . . . . . . . . . . . . . . . 3.2.2 Rate 3/3 and 212 Codes

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Simulation Results

. . . . . . . . . . . . . . . . . 3.4 Comparisons with Convolutional Codes

3.5 64 State. Trellis Codes and Comparison with the IS44 Convolutional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Code

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Summary

4 Performacce of Coded Systems in Selective Fading Channels . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . 4.1 Derivahn of the Decoding Metric

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Simulation Results

4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusions

5.1 Conc!usions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LO1

5.2 Suggestions for Further Research . . . . . . . . . . . . . . . . . . . . 103

A Effect of the Pox-er Spflt Rztio . . . . . . . . . . . . . . . . . . . . . . . . 1%

B Trellis structure of the 8 statet rate 3 f 2 code . . . . . . . . . . . . . . . . . lllS

C Trellis structures crf 64 state codes . . . . . . . . . . . . . . . . . . . . . . . 111

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

... f i i f

T OF FIGU

2.1 The Block Diagram of a Communication System . . .. . . . . . . . . .

X n A "t 2.3 The rnatcUcu m:er rrespo~se . . . . . . . . . . . . . . . . . . . . - . .

2.4 TreiGs Biagrm ;or the t'iterbi Decoding of QPSK in a 2 - i q ~ Channel

2.5 The Approximated Bit Ezor Probability of QPSK \~itr! El = E2 . 2.6 The Approximated 131:. Error Probability of QPSK it-ith El = 16E2 .

2.1 The Xpprfrximsted Biz Error Probability of QPSK at E = 20 d B and with power sp& ratio as a parzmeter . . . . . . . . . . . . . . . . . .

2.8 The Xpprftxiaatd Bit Error Probability of QPSK at E = -10 dB and with power spEi:; ratio as a paale ter . . . . . . . . . . . . . . . . . .

2.9 The Simdzkd Bit Ermr Probability of QPSK with El = E2 . . . . . 2-10 The Sina,z.fated Bi: Error Probability of QPSK at E = 20 dB and ~ 5 t h

power s p k ratio as a parameter . . . . . . . . . . . . . . . . . . . . . 2.31 The Xppraximaied Bir Error P~rsfjability of QPSK is-i t t El = E2 . .

2.22 The Approximated Bit Error Probability of QPSK with El = 16E2 . 2. f 3 The .4pproximated Bit Error Probability of QPSK at E = 20 dB . .

'3 -. 1-2 The Approximated Bit Error Pmbability of QPSK at E = 40 dB and it-.;& p';-"r s&r fat,io as 2 p=~=eter . . . . . . . . . . . . . . . . . .

2-16 Tire Sirnuked Bit Error Probability of QPSK at E = 20 dB and with . . . . . . . . . . . . . . . . . . . . . power s p 5 ~ r a i o zs a parameter 46

2.18 The Treliis Dizgram for the 1-iterbi Equalizer with Prilsc Shaping. (0: f : 2, 3; are ::he _t'uz&ep .%sso&tpd v;itfi Q f ) S f < Sig;jais S h m n

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . in Figue2.4. 53

2.19 The Simulzted Bit Error PtobabiIit:; of QPSK with El = E2 a d Pulse - - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shaping 22

- 2.20 The Sirnuiated Bit Lrror Probability of QPSK with Pulse Shaping at

E = 25 dB and with the poil:er sp?it ratio as a parararler . . . . . . . 56

2.21 The Sinaiazd Bit Ermr P ~ o b ~ b i l i t ~ of QPSK with Pulse Shapizlg at E = 25 dB a d xi& Equal Po-xrr Split . . . . . . . . . . . . . . . . . 37

3.1 The %-QPSft= Siwd Gnnstefiztrion. . . . . . . . . . . . . . . . . . . . 6%

3.2 X 4 state, rate 3 j2 code ~ 5 t h 1 gara!lel transitiorxs betiwen pairs of states 65

3.3 An 8 sraee. rzte 3T2 code with 2parallel transitions bctwcn pairs of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . states 65

3.4 Set partitioning ai the $-QPSK signal set over a 3 symboi inrervai . GG

3.6 The Error Perfomace of Differeat Codes in the AtYGS Char~iiel . . 7.5

3.1 The Error Perforname of DiEerent Codes in the Flat Fading Channel 76

3.8 A Coqmrison of the Bit Error 23E.ribrrnance of Coded 9yster:ts :.i.itfi a . . . . . . . . . . . . . . . . . . . . . . throu&prtt of 1.5 bi~sJsymboH 80

3.10 The Error Perfurmznce ci 54 state Codes in the Rityicigh I-*;:ding Chanrtel 83

The Error Pedwmmce of the 16 State, Rate 2/2 Code in a Frequency Selective Fading Channel Usiog a Rectangular Pulse and with Delay

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . as a Parameter

The Error Performance ctf 16 State, Rate 2f2 Code in a Frequency Sefectir-e Fading Channel Csing Rectangular Pulse and with the Power

. . . . . . . . . . . Split h r i c as a Pararzeier. The Bit SSR is 8 dB

The Error Penir,mmce of 16 state; rate 212 Code in a F'requency Se- lective Fading Channel Using Raised Cosine Pulse and with a Rolloff Factor c+f 0.35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Error Perfo-masce of the 16 State, Rate 212 Code in a Frequency Selective Fading GSLmrrel TIsing a Raised Cosine Pulse with a Rolloff Factor of 0.35 and i ~ i t h itbe Power Split Ratio as a Parameter. The Bit

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SNR is &3 dB

The Error Perictmance Comparison between the 16 State, Rate 212 Trellis Code and the 8 State: Rate 1 f2 Convolutional Code. Both use the Rectangdaz Pulse . . . . . . . . . . . . . . . . . . . . . . . . . .

The Error Performance Comparison between the 16 State, Rate 212 Trellis Code and the 16 State, Rate 1/2 Convolutio~al Code using a Raised Cosioe Pnise with a Roiloif Factor of 0.35 . . . . . . . . . . .

The Error Pericfmance of 64 state, rate 2/2 Code in the 2-ray Fre- quency Selective Fading Channel Gsing a Raised Cosine Puke with a Rolloff Factor of 0.35 . . . . . . . . . . . . . . . . . . . . . . . . . . .

The E m f Performance Cornparison between the 64 state, rate 212 Code and the 32 state, rate 1/2 convolutional code in the h a y Fre- quency Seiectiw Fading Chamef Esing a Raised Cosine Pulse with a R d o E Factor of 0.35 . . . . . . . . . . . . . . . . . . . . . . . . . .

F TABLE

2.1 The irrtersydml icierference Magnitude at different ticlays for the raised cosine pulse of a rctffof factor of 0.35 . . . . . . . . . . . . . . . 52

3.1 A P e d o r ~ a x e Summary for Digereat T d i s Codes . . . . . . . . . . 72

mm CHAP IBR 1

Introduction

The demand for celldar coamunications has increased dramatically since its introduc-

tion in the early 1980's. Iiesearch has shown that the use of cellular communications

can achieve a direct cost saving in a company, achieve a more rapid and adequate

response to emergencies, and especially improve the competitive advantage in the

marketplace [I]. The nely emerging Sorth American Cellular system will be digital

and it has several ad:-mtages over its analog counterpart. First, regenerative repeaters

dong the transmission path can detect a signal and retransmit a new noise-free signal.

These repeaters prevent accumulation of noise along the path. This is not possible

in analog communications, Second, digital signals can be coded to yield extremely

iort- error rates and high fidelity Finally, digitd communication is inherently more

efficient than anahg in realizing the exchange of signal-to-noise ratio (SNR) for band-

wi&h 12). For exampk, the digital cellular system will be three times more bandwidth

efficient than the existing analog counterpart [33.

Digital transmission over cellular communication channels has become an impor-

tant research area and a major consideration is the bit error performance of the

communication system under different channel conditions. In mobile radio systems,

a received signal is normally modified by its environment such as the Doppler fre-

quency caused by the motion of the vehicle and the presence of multiple paths. Such

an environment leads to frequency selective fading which can cause degradation of

the bit error rate (compared to its non-selective fading counterpart). Frequerlcy se-

lective fading means that different frequency components of the transmitted signal

nill be subjected to different fading gains, and these fading gains can be correlated.

One method of improving the error performance in such an environment is to employ

combined equalization and channel coding techniques.

This brings us t o the central theme of the thesis: to study the error performance

of various combined equalization and channel coding techniques operating in the fre-

quency selective fa&g environment. Specifically, the frequency selective fading chan-

nel will be modelled as a %ray channel with independent Rayleigh fading in each ray.

In addition to using conventional channel coding techniques, we will design trellis

coded modulation (TCX'I) schemes that are matched to the digital cellular system's

modulation format, i.e. f-shifted QPSK. Our ultimate goal is to see if these coded

modulation schemes are superior to convolutional codes in the cellular application in

the sense of providing better bandwidth efficiency. In all our studies, perfect channel

impulse respame estimation is assumed. A soft decision metric that takes into ac-

courtt interleaving and the time varying nature of the channel impulse response is also

derive& This decoding metric is applied to both the convaiutional codes specified in

the digital celldar standard and the TCM schemes we designed.

1.1 Background and Literature Review

This thesis is divided into three parts. First, we investigate, via analytical means, the

error performance of uncoded systems transmitted over a frequency selective fading

channel. This study is relatively simple but it enables us to understand the behaviour

of digital modufations operating in the frequency selective fading environment. Then

in the second part, we design new TCM schemes which are compatible with the -

QPSK format. The code design procedure assumes a flat (i.e. non-frequency selec-

tive) fading environment. This means that the design criterion is the free Hamming

distance [12] (at the symbol level) of the coded modulation scheme. The objective

here is to come up with new TCM schemes that are more bandwidth efficient and/or

power efficient than the convolutional code (in conjunction with the $-QPSK scheme)

specified in the digital cefldar standard. Finally, we will investigate the error per-

formance of our TCM schemes, as welf as that of convolutional codes with the same

throughput and compiexity, in the frequency selective fading environment. Below is

a literature review for each of these parts of the thesis.

1.1.1 Frequency Selective Fading Channel

During the last few years, a number of studies have been carried out to investigate the

error performaace of digital communication systems operating in frequency selective

fading channels. These studies can be classified into 3 categories: the characteristics of

the channel, the error performance of coherent detection and the error performance of

differential detection. An experiment ming a multi-tone technique has been designed

to determine the degree of frequency selectivity of a mobile radio channel [4]. The

multi-tone method basically transmits a group of closely spaced tones to a mobile

receiving system. By the measuremerat of each tone and comparison of the relative

effects of the mobile channel on each of these tones, it is possible to determine the

response of the frequency selective fading channel. A measure of the selectivity is

the correlation between two tones; for example, a perfect correlation corresponds to

the absence of frequency selectivity. From the results of the experiment, the most

disturbing factors in a selective fading channel are hills and the number of reflectors

present. For a small channel bandwidth, a large number of reflectors does not seem

to destroy the signal from the point of view of frequency selectivity.

Digital transmission over a mobile frequency selective fading channel can be de-

tected either coherently or differentially. In the case of coherent, detection, a theoret-

ical comparison of the average bit error rates of four digital modulation techniques

subjected to Rayleigh distributed two-ray multipath fading has been performed in

151-161. The four digital modulation techniques are BPSK (binary phase-shift keying),

QPSK (quadxture phase-shift keying), OQPSK (offset phase-shift keying) and MSK

(minimum phase shift keying). Results show that BPSK is the most tolerant to multi-

path distortion among the four modulation techniques. This arises from the fact that

BPSK suffers from no cross channel interference between the in-phase and quadra-

ture components, while the other three modulations suffer from severe cross channel

interference, in addition to the intersymbol interference from adjacent data symbols.

However, BPSK has a poor spectral efficiency. In 171, Mazo considered the case of

coherent detection of BPSK and 4QAM (4 level quadrature amplitude modulation)

in a 2-ray itayieigb fading channel. Specifically, he derived an exact matched filter

bound for the error probability.

In the case of differential coherent PSK (DPSK), the effect of baseband pulse shap-

ing on the error performance in a Rayleigh fading channel is studied in [8],[9]. The

authors showed that the performance of DPSK over a frequency selective Ftayleigh

fading channel can be closely approximated by parameters which are obtained from

the measurements of the channel. An example of such a parameter is the normal-

ized root mean squared multipath spread. An approximation technique, based on

these parameters, was developed for obtaining performance bound for DPSK over

frequency selective fading channels. This approximation can dso be applied to the

more general frequeacy selective Rician fading channels. In [lo] and ill], the bit error

rate performance of % - DQPSK modems in the cellular mobile channel was derived

and analyzed. The chamel is modelled as a frequency selective fast Rayleigh fading

channel corrupted by additive white Gaussian noise (AWGN) and co-channel inter-

ference (CCI). The probability density function of the phase difference between two

consecutive symbols of a M-ary DPSK signal is first derived. Based on the probability

density function, the bit error rate of ;-DQPSK is derived in closed form and calcu-

Iated directly. The numerical results show that for 2 - DQPSK, the bit error rate is

?ominattd by the CCI. This analysis in j lD] can be extended to investigate the error

performiznce of a frequency selective fading channel with multiple CCI.

All the studies mentioned above only considered uncoded modulations. As men-

tioned earlier, coding (together 5i th interleaving and equalization), is a necessity for

good error performance in the frequency selective fading envi~onment. Below is a

literature review on coded modulation schemes.

Io the past decade: one of the most poplar coding techniques is TCM, which vas first

ictroduced by t-~gerboeck jf2j. TC3f is known for its abiliti. to achieve a significant

coding gain i ~ i i h ~ ~ i sacriEcing data rate or requiring extra bandxidth. Tile n~ain

idea beffind 'TCx n:+ -=o- Gcm&& expansion to FrGvi& r ~ u l ; ~ a l ; C 5 ' for tocfiiig,

and to design codiag mci signd-mapping functions jointly so as to directly masirriizr

the minimum EncIidean distance (or free hamming distance) bet~s?;eert coded signal

sqtiences. The x d t i ~ g Eiiclirlean distance of the coded sequence significantly ex-

ceeds the mini=== 2 i t s n c ~ b e ~ r ~ e e ~ ancoded modufation signals at the same infor-

mation ratet fra~~dwi&%, a d sfgziai paiyer. For example, consider the result in (121

where Ungerbcteck compand his &sratx, 8-PSK Trellis code with the uncoded 4PSK

scheme. Both systems t rwmit tm idormittion bits per modulation interval. How-

ever, the free hamming &:lsta~ce for w m d e d 4-PSK is only 1.414 whereas the free

Emming distance for Trellis-coded &state, 8PSK is 2,141. Such an improvement in

the ffee H m b g &srmce restt4ts in a coding gain of 3.6 dB over uncoded 4-PSfi in

ssn additive white Ga*;ssijl;lti: coke tbiznnei.

fn 1988, Wilson 1151 introduced a new TCM technique that was based on Unger-

boeck's 8 state, &PSK treflis code. Wilson assumed that 8-PSK modulation is used

twice per treflis interval (forming a Wary set), but that this se5s coded with a 5/6

trellis code. Thus, the spectral efficiency (5 bits/2 symbol) is better than the &PSK

(2 bits/symbol) trellis code. Wilson showed that his 8 state code provides a 6.2 dB

gain on the Gaussian noise channel over uncoded &PSK, while sacrificing only 16%

in spectral efficiency. This modulation scheme of Wilson was then generalized by

Divsalar and Simon to create a technique called multiple trellis coded modulation

(MTCM). Later on, Divsalar and Simon considered the performance of multiple trel-

lis codes in a Ricim fading environment f16-181 and showed that the design of TCM

is guided by factors [in particular, the length of the shortest error event path and

the product of branch distances along that path) other than the squared Euclidean

distance when used in a Rician fading channel with interleaving/deinterleaving [17].

The common drawback in the above studies was that all the results were obtained via

computer simulation. Although simulation is capable of reflecting the actual system

performance, it is a time-consuming process. In addition, simulation studies cannot

provide much insight into the understanding of the behaviour of the system.

The first analytical result on trellis coded MPSK modulation transmitted over

a fading channel was reported by Divsalar and Simon [19], where they applied the

Chernoff bound technique to obtain an upper bound on the pairwise error probability.

By making use of the pairwke error prolxbility bound and the transfer function of

the pair-state transition diagram, an upper bound of the average bit error probabil-

ity was obtained. Later on, Divsalaf and Simon [20] used a similar technique and

exteaded their analysis to indude trellis-coded multilevel diiferential phase shift key-

ing (MDPSK), Howxer, the upper bounds obtained were too loose over the normal

CHAPTER 1. INTRODUCTION 8

range of signal to noise ratios of interest. Besides, the pair-state transition diagranl

approach may be a tedious task when the number of states in the trellis diagram

becomes large. By using the characteristic function and the numerical Gauss-Konrod

integration rule, McKay et a1 1211 were able to evaluate an exact pairwise error event

probability for TChf in Rayleigh and Rician fading channels. Although his results

were satisfactory, the numerical evaluation of this upper bound is quite complicated.

By employing the characteristic function and the residue theorem, Cavers and Ho

[22] obtained an exact and easily computed expression for the pairwise error event

probability of TCM operating in Rayleigh fading channels. This expression is quite

general and includes not only trellis coded MPSK, but also trellis-coded quadrature

amplitude modulation (QAM) with perfect channel state information (CSI), differen-

tial detection, or pilot tones. Acclirate average bit error probabilities were obtained

by considering only a small set of short error events.

All the studies cited in this subsection on TCM consider only the additive white

Gaussian noise (AWGN) channel or the Rayleigh flat fading channel. There are rela-

tively few studies of coded systems and their performance in frequency selective fading

channels. A survey of studies in this latter category is given in the next subsection.

1,1.3 Coded Systems Operating . , a Frequency Selective

Fading Channel

One of the inter2st.s in frequency selective fading channels is the decoding metric used

for coded systems. it is well known that maximum-iikebfiood sequence estimation

(MLSE) [23] is the optimal decoder for coded systems operating in a frequency se-

lective fading channel. However, the computational compfexity of MLSE increases

exponentially with the channel memory. Moreover, it is complicated by the presence

of interleaving and de-interleaving that. is usually required for good performance in

coded systems. Consequentiy, a number of studies have been carried out to find other

decoding metrics which can reduce the decoding complexity and still retain the same

bit error rate a% a given signal-to-noise ratio. In [24], a sub-optimum variant of the

optimum soft decisim equalizer is presented, namely the soft decision Viterbi equal-

izer (SDVE), which is similar in structure and in implementation effort to the classical

zL-state Viterbi equalizer with L bit path history. Moreover, the complexity of

SD\IE is less tha;: that of the soft-decision output Viterbi a!goritfm (SOYA) proposed

by Hagenauer f26f.

In f27], Haeher, starting from the optimum structure, derived a reduced com-

plexity receiver. Under the constraint of similar complexity, Boeher showed that the

modified symbol-by-symbol estimator outperforms the sequence estimator when the

signal space is nonbinary. He considered both trellis-coded 8 PSK [12] and uncoded

4PSK (as the reference system) and shox-ed that the simplified version of the symbol-

by-symbol algorithm is more powerful than the sequence estimator.

Other studies on reduced comp!eiit__s. sequence estimation that can be applied to

the frequency sdedive fading channel icclude the one reported in E28f. There the

reduced state sequence estirnaror (=SE) uses a conventional Viterbi algorithm with

decision feedback to search for a reduced-state "subset trellis" which is constructed

wing set; partitiof;ng p~heipks [f2]. T o e set partitioning pfincipies systematicdy

reduce the cornpiexiif: of the seqrtence esiimator, due to the length of the channel

memory and the size of the signal set. An error probability analysis sllows that a

good performancejcomp1exity tradeoff c m be obtained. Other reduced compirsity

equafization techxiiques include the 11-algori~hrn of Anderson and IIohan f29f and the

T-afgoritftrn of Sirnmclns 1301.

*- En this thesis7 we w!f d e r k a decodizig rnetric similar to that ir: [24] to i::~ts- --

tigate the error performance of our new TChf schemes in the frequency selective

fading channels. X suboptimal version of this decoding metric isill also be presented.

This suboptima! mer;ri-.,c dl intmfve the truncation of the channel impulse responsc

estimate, as well as the removal of the exponential operator in the metric given ir t

E24j -

1.2 Contributions of the Thesis

The major contributions of this thesis cur be sunmarized a f'tlows:

1. An investigation of the error ped~rmance of uncoded sptems transmitted ovcr

a frequency selective RayTeigh fading channel. We found that the channel provides

an implicit diversity effect that hdps to improve the error performance over that of a

Bat ;'ding channel.

2. The d e s i s af new TCf f schemes that are compatible $vit h the S-QPSK mod-

ulation format chasen far digital cellular application. It was found that these rlcw

methods perform at leas as well as c~nvo~utionaf codes in the fiat fading environ-

ment, In same cases, our ex TCM schemes are 1 to 2 dB more energy efficient than

con~wTrrtiond codes with similar iirrctugbpnt and c;ornplexitt;.

3. The evaltrztion ui rhe error performance of coded systems operating in frequency

seiective fading channels. The decoding metric for coded systems is derived. As in

the Bat fading cha~ne!, it was found that trellis codes still perform at least; as well as

con~wlutional codes, and in some cases, significantly better. It was also found that

the frequency selective fading channel does not provide s diversity effect in coded

systems.

1.3 Thesis Outfine

Chapter 2 desc-rIbe the system model used in this study itnd azlalyzec the error per-

formance of uncocfed QPSK in the frequency selective fading channel; both analytical

and simulation restrits dl 'be gke=. In addition, we have also investigated the effects

of pulse shaping on the error performance. Chapter 3 presems a design procedure for

trellis-coded 2-QPSK schemes a d compares the error performance of the designed

codes with conr.entio~aI coding systems using convolutionai codes. The results given

in chapter 3 apply u d y to the Bat fading channel. In Chapter 4, we evaluate the

error perfommice of coded sys~ems in the frequency selectiw fading cirannel. We will

also derive in xhis chapter the decoding metric used in the frequency selective fading

channel. Simrrfaiion resdts will be presented. Finally, conclusions of this study will

be given in Chapter 5.

The Error Performance of

Uncoded QPSK

fn mobile radio comiffunaicztions: a signal seceix-ed by a vehicle is significantly ~rlodified

by its environment. First, the rnotior! of the vehicle causes a Doppler effect tl lat shifts

its frequency. Seconrlli., mzq- rays due to reflection and diifraction of the signal on

various objects s- togerher to form ac interference pattern. As the vehicle moves

throrrgh this pattern, the received sipah will suffer from frequency selective fading.

Frequency selectivity egects are important: especially in digital cornmunicatior~s where

it leads to ictersq;mbai interkrecce tirat can cause degradation iii bit error rate [31].

state informatio~ QCSf). Afi exact expression for the pairwise error probability of the

deader wilf be pre~ite2. As the reader will discover, the analysis given in this chap-

ter is a more genera1 version of that; given in [TI. Although conventional tlncoded

QPSK is being consideredt the analq-tical and numerical results presented will dso

hold for f -QPSK.

In addition to the study n-ith perfect CSI , we will also consider the effect of im-

perfect CSI. hforwer? the effect of pufse shaping on the error performance will also

be studied.

This chapter is organized zs foffoiiis. Section 2.1 presents the general model of our

communication system. Section 2.2 analyzes the bit error probability of the uncoded

QPSK modulation using a rectangular pulse transmitted over a %ray frequency selec-

tive Rayleigh fading charnel with perfect channel state information- Both analytical

and simulation r a d t s are presented. In Section 2.3, the case of imperfect channel

state information will be investigated and once again the baseband pulse has a rect-

angular shape. Both analytical and simulation results will be shown. Section 2.4

discusses the effects of pdse shaping. Due to the complexity of the system model,

theoretical analysis is very hard to perform and hence only simulation results will be

gisen. Finafly, a summary of the chapter is given in Section 2.5.

2.1 System Overview

The general communication system block diagram is shown in Figure 2.1. Throughout

this thesis, baseband signal representation is used.

Figure 2.1: The Block Diagram of a Communicaiion System

The input to the encoder is a sequence of binary digits denoted by bk and the

output is ir sequence of complex coded PSX symbols whose amplitudes are normalized

to unity. In order xo disperse possible deep fades in the channel, the PSK symbols are

passed to an interlever. The interleaved PSK symbols are then fed to a pulse shaping

filter. f't is assmEii aha: the cuiiibimd trmsmitter and receiver puke shape satisfies

the Nyquist criterion far zero intersymbol interference (IS1 j. The output signal from

CHAPTER 2, THE E M O R PERFORtVANCE OF UNCODED QPSK 15

the pulse shaping filter is denoted by t ( t ) = C x k p ( t - kT), where xk is the transmitted

'PSK sgmbot in the kt%intemal, 1/T is the symbol rate, A is a constant, and p ( t ) is

the transmitter puise shape.

The output signal from the puke shaping filter is then sent to the channel. The

channel here can be either a frequency selective or a flat (i.e. non-frequency selective)

fading channel. The channel will introduce fading and noise to the transmitted signal.

For frequency selective fading, the channel will also introduce intersymbol interference

(ISI) to the transmitted signal. The received signal is then passed through a matched

filter. The output from the matched filter will be sampled a t baud rate.

The filtered signal samples are then passed to a channel estimator as well as to

Decoder 1. In the case of coded modulation, Decoder 1 generates soft. decision met-

r i c ~ and these soft decisions are deinterleaved and passed on to Decoder 2 (for the

coded modulation scheme]. In this chapter, we are only interested in uncoded modu-

lation. Consequently, Decoder 1 is simply a Viterbi equalizer and the interleaver/de-

interleaver and Dewder 2 are not considered.

2.2 Uncoded QPSK with Perfect Channel State

In this section, we will investigate the error performance of uncoded QPSK modu-

lation in frequency selec'tix-e fading channels. Here, we assume that the receiver has

perf& channel state idomation (CSI). -h exact expression for the pairwise error

probability of the demder wil be defived. Both theoretical and simulation results

CHAPTER 2. THE ERROR PERFORAfANCE OF UNCODED QPSK 16

will be presented.

2.2.1 System Model

A general mathematical model for a frequency selective Rayfeigh fading channel is

the tapped delay fine model shown in Figure 2.2.

Figure 2.2: Tapped Delay Line Channel Model

Here, xr; represents the transmitted PSK symbol in the kih interval. For conve-

nience, we assume that the zk's take on one of the M possible values from the set

(&?zn/tZ.i. , n = 0, I,. . . , A4 - 1). According to Figure 2.2, the corresponding received

symbol yk is

where the hkjfst j = 0,. . . ,N, are correlated, zero-mean complex Gaussian random

CHAPTER 2. THE ERROR PERFORMANCE OF UNCODED QPSK 17

variables with a certain correlation matrix, and nk is a zero mean complex Gaussian

random variable with a normalized variance of unity. The hkjYs represent the effect

of frequency selective fading while the nk's represent the effect of the channel's white

Gaussian noise.

The kfh received sample in (2.1) can be written in matrix form as

where

is the k f h channel state vector and

is the kth data vector. In this study, we assume that the receiver has perfect chan-

nel state information, i.e., each Hk is known to the receiver. In practice, reasonably

accurate CSI can be obtained through channel sounding techniques such as inserted

pilot symbols f311-[32]. ?Ye will show later on in this thesis that imperfect channel

state estimation will lead to degradation in the bit error rate. With perfect CSI, the

optimal decoder is a Viterbi decoder that selects the sequence 2 = (21, iz,. . . , iN)

that minimizes the conditional probability density function (pdf) of the received se-

quence p(xly), where y = (yl, 92,. . . , yN) is the received sequence. Given the channel -cY

state estimation (H,,, Hz,. . . , nN), this is equivalent to selecting the sequence 2 that

CHAPTER 2. THE EiRROR PERFORMANCE OF UNCODED QPSK 18

We assume here that the precursor symbols

are known to the receiver. The decoder in (2.5) can be implemented via the Viterbi

algorithm. Now, gken that the transmitted sequence is x = (xl, 52,. . . , xlV), if the

metric J (1) is less than J(x) , then a decoding error will occur. In the next subsection,

we will derive an exact expression the pairwise error probability of the Viterbi decoder

in (2.5).

2.2.2 Error Analysis

Given that the transmitted sequence is x = (xl, 2 2 , . . . , x,), the metric J ( x ) is

On the other hand, for any sequence 1 = (it1, it2, . . . , itN), the metric J(A) is

where

Now, let the random variable D be defined as

where

is a zero mean complex Gaussiar, random variable. The ak's, in general, are correlated.

Let the random vector A be defined as

where

and

This implies that the covaziitnce matrix of the ak7s is

a& t is the He-mGtian traaspoose of the matrix. As shown ia [34], the pair~vise error

probabiiith P{x -+ P j Qi-e. the probability that the random variable D i ~ i (2.9) is less

than zero), c m be expressed in terms of the eigenvalues of the covariance matrix in

(2.1 5) . Speci6caBy7

where arz(s) is the ehzrzcteristic famction of D and the sum is taken over all the

in 1341, the chmaicteris'tic &'unction is

where X k is tbe kt' eigemahe of the covariance matrix of the uk's in f2.15), and the

product is taken over the set of q of k for which dk in (2.8) is not equal to zero.

Substituting (2.18) arrd (2.19) into (2.117) allows us to calculate the pairwise error

event probabilityB which in turn enables us to calculate the bit error probability. This

be given in the next subsection.

2,2.3 Example: Two-Ray Model

KTe consider in t&s sef:&~ the error performance of uncoded QPSK transmitted over

a %ray fiequency sekcti~e Rq-lei& fading channel. For shplicity, vie assume a

CHAPTER 2. THE ERROR PERFURLMANGE OF UNCODED QPSK 2 1

rectangular pulse shape with a duration equal to one symbol interval. The case of

pulse shaping will be investigated later in Section 2.3, through simulations.

The transmitted QPSK signal can be written as

where A is a constant, p(t) is a unit energy rectangular pulse with a duration of T,

and the xkfs are complex QPSK symbols taken from the set (I, j, -1, -j]. For a 2-ray

channel, the corresponding received signal is

where gl(t) and g2(t) are two independent fading processes with variances a: and a:

respectively, .r is the relative delay, and n,(t) is the complex envelope of the channel's

white Gaussian noise. The double-sided power spectral density (PSD) of n,(t) is No.

In this study, we assume that the relative delay T is less than or equal to one symbol

duration.

If the fading processes are slow enough, they can be assumed constant over a

symbol duration. In this case, the received signal z(t) in (2.21) can be written as

where glk and g2k represent the fading gains that affect the kfh symbol and both glk

and g2k are complex Gaussian variables. The above received signal will pass through

a matched mter &ose impulse response is equal to

The concatenation of Ap(tj and q(t) is the pulse shape r(t) as shown in Figure 2.3

Figure 2.3: The matched filter response

After matched filtering, the resultant signal can be written as

where n(t) is the filtered noise with a variance equal to

The filtered signal y(t) is sampled at instance kT, k being integers. The k th sample is

where

and Ag2'k-x

hkl = ,/'N,

P'

CHAPTER 2. THE ERROR PERFORlMAiVCE O F UhiCODED QPSK 23

Note that the variance of the random variable hko is

where

In this study, the tot& received signal energy is denoted by

Similarly, the variance of hkl is

Finally, for any pair of k and j

where Jo(*f is the Bessel function of zero order and fD is the maximum Doppler

frequency. ft should be pointed out that the autocorrelation function of the fading

processes gl ft) and g2ft) are respectively a; Jo(2afDr) and o,2 J0(2r fD7) [31], where 7

is t.he delay variable.

compit*iag (2.26) arid j2.3), the channel state vector becomes

This implies tkax the cmzdatioi; matrix gHH in (2.15) can be written as

where

In most commuxGcation systems, we are interested in the bit error probability

rather than the individual pairwise error probability. For the Viterbi decoding of

uncoded QPSK transmitted over a 2-tap channel, the bit error probability cart be

approximated by :

where

1. P,[x -., 1) is f he pairwise error probability of the kih error event,

2. ak[x,2) is the P_amt;ring distance between the two information sequences that

generate the sequences x and jl;

3. m = 2 is the number of infarmation bits per symbol, and

4. the sum is over the set of error events that merge in no more than 3 szeps in

the trellis diagram obtained from the intersymbol interference pattern. This trellis

diagram is show^ in Figwe '5.4.

There are 4 states in tire ~reEs; one for each of the 4 s_vrnbofs. According to the

tretlis diagram, there are 3 md 9 error events of length 2 and 3 steps respectively. As

;a result, there ate 12 exor events being used in bit error approximation in (2.12). It

should be pointed out that it-, our alcarfation: the transmitted sequence is assl~med to

be the all-zero sequence in the trellis diagram. We have incorporated error events of

merge length 4 into the bit- error probability approximation. HaxseverF they cause only

minor changes to the end results. Consequently, we believe that the approximation

reffects the actual bit error rate ~easnabfy r d l .

! State I Symbol 1 Information Bits

Figure 2.4: Trellis Diagpm far the ti'iterbi Decoding of QPSK in a %tap Charmel

CHAPTER 2. THE ERROR PERFORXfANCE O F UNCODED QPSK 27

22.4 Numerical Results

We show in Figures. 2.5-2.10 the numerical results for the 2-ray channel in Section

2.2.3, Figures 2 3 and 2.6 show the bit error probability a s a function of the total

received enere- E (see equations (2.31)-(2.33)) and with the relative delay T of the -

2 rays as a parameter. l'wo different power split ratios are considered: El = Ez

and El = 16E2. In both cases, the maximum Doppler frequency fDT is equal to

0.01. Also shos-o in these figures are the results for the frequency non-selective or

flat fading (one-ray) cbannel (r = 0). f t is observed that the performance of uncoded

QPSK in the %ray chairnei are better than that in the one-ray channel. The results

in Figures 2.7 and 2.8 are subsets of those shown in Figures 2.5 and 2.6. In these

two figures, we plot the approximated bit error rate as a function of the relative delay

T and with the power split ratios as a parameter. Two cases are presented: a total

energy of 20 and 10 dB. It is found that the best performance is obtained when the

arrival rays have equal energy; the reader can refer to Appendix A for explanation.

These results appeaf suspicious at first glance. However, recall that we have a 2-ray

channel with independent fading and that we assume the receiver has perfect channel

state information. -2s a rmult, some form of diversity effect is introduced because

of the frequency sefectivits of the fading. For example, when the delay is equal to

one-symbol intend, a &.it-ersity order of 2 is achiesed; refer to Appendix A as well

as f3, eqf23)E. In other :t.ords, the error curve decays two order of magnitude as the

signal to noise increases every fO dB. The simdation results for Figures 2.5 and 2.7

are shown in Figares 2.3 axid 2.10. It is observed that the simulation results agree

with the theory.

Signai-to-Noise Ratio, TfdE3)

Figure 2.5: The Approximated Bir Error Probability of QPSIi with Er = f;;2

(1) - Delay, p = 0-0

(2) - Delay? p = 0.2

(3) - Delay? p = 0.6

(4) - Delay7 p = 1.0

CHAPTER 2. TNE ERROR PERFORMAIVCE OF UNCODED QPSK

............. ............ ............. ................ ................................. l. , r.. : , .......... : ............................................. ................................................................................................................................ -

......... ................ ................................. ................................. ... .............. -.... .......,.. _....___..I____I__._____________I_______ _....___..I____I__._____________I_______ _....___..I____I__._____________I_______ _....___..I____I__._____________I_______ _....___..I____I__._____________I_______.._....___..I____I__._____________I_______ ,...,, .............................................................. - ................................................................. ....................

............................ . . . ................ .............................. ..,....... ... ... ......... .....

................ . . . . . . . . . . ........... : ............... ............... : i

................ ..............

............. ........... ............... ................ .............. ................ ................ ................ .............. ................ ................ - t 1 ............... ;

..... .............. ................ - : : ................ ............ .............. . . . . . . . .

....................................................................................................................................... - ................ ............. ................ . . . . . . . . ................ - .............. < ................ & < g. j ................ ; z i

............... ...................................................................... - .................................................. ................................. . ................. .............. - < i.... ................................... < 8

........ ................ ........... ................ ................................. .............. ................ - < +.. ............... 5

- ............................................................................................................................... -

lo4 0 0:1 012 0:3 014 0:s 0:6 0:7 018 d 9 1

The Normalized Relative Delay, p

Figure 2.7: The Approximated Bit Error Probability of QPSK at E = 20 dB and with power split ratio as a parameter

(1) - Power Split Ratio = 16: El = 16&

(2) - Power Split Ratio = 4: El = 4E2

(3) - Power Split Ratio = I: El = E2

CHAPTER 2. THE ERROR PERFORMANCE OF UNCODED QPSK

The Normalized Relative Delay, p

Figure 2.8: The Approximated Bit Error Probability of QPSK at E = 40 dB and with power split ratio as a parameter

(1) - Power Split Ratio = 16: El = 16Ez

(2) - Power Split Ratio = 4: El = 4Ez

(3) - Power Split Ratio = 1: El = Ez

CHAPTEB 2. THE ERROR PERFORl%fAs\TCE OF UNCOIfED QPSK

Figure 2.10: The Simulated Bit Error Probability of QPSK at E = 20 dB and with power split ratio as a parameter

Solid Line - Analytical Results

(1) - Power Split Ratio =

(2) - Power Spfit h t i o =

(3) - Power Split Ratio =

2.3 Imperfect Channel State Information

fn the last section, we =sume that the receiver has perfect channel state information.

In reality, perfect channel state information is almost impossible to achieve. 1x1 this

section, we mill investigate the effects of the estimation error on the error performance

of the uncodeci QPSK system.

2.3.1 Error Analysis

Let

be the estimated charnel state ~ e c t o r in the kth interval. In the ideal case, H~ = Hk.

Since the focus of this study is not on channel estimation, we modelled the estimation

error through the introduction of random Gaussian noise to Hk. In other words,

where

CHAPTER 2. THE ERROR PERFORMANCE OF UNCODED QPSK 35

is a random Gaussian vector with a covariance matrix of aEE = a;I, where I is a

( L + 1) x (L + 1) identity matrix and (L + l)a: is the power of the estimate noise.

tiVe assume €?k is independent of Hk.

The Viterbi decoder will select the sequence P = 2 2 , . . . , ? N ) that mini~i~izes

the metric -

Given the transmitted sequence x = ( ~ 1 ~ x 2 , . . . , x,), the metric J(x) is

For any other sequence 2, the metric J(k) is

If vie define the random variable D to be

CHAPTER 2. THE ERROR PERFORMANCE OF UNCODED QPSK 36

where

then a decoding error will occur if the random variable D is less zero. Thus? random

variable D can be written as

where

CHAPTER 2. THE ERROR PERFORMANCE OF UNCODED QPSK 37

and I is the identity matrix.

From (2.28) and (2.591, the matrices U and V are related by

V = U + A ( H + E )

where

E =

and A and H are defined earlier in (2.12)-(2.14).

Now, the covariance matrix of G can be written as

where aUu, Quv, avU and Bvv are the covariances between U and V. Specifically,

the covariance matrix of U can be written as

where

and

Recall that, 5: and ( L + ljt?,2 are the variances of the noise and the estimated error

respectively.

Similarly, we can easily determine BUv, Qrvu and Qrvv. If we define the matrix W as

then, it. can be shown that

and @HIf is defined in (2.40).

Equations (2.60j-Q2.56] can he substituted into (2.59) to determine aGG. Once

@GG is known , 1%-e can determine the characteristic function of D [39] in (2.52) as

follows

and Xk is the iEfh eigemdue of @G@. Substituting @o(s) into /2.17), we can calcufate

the pairwise error event probability- It should be emphasized that the error analysis

method presented in this section c m be ex-tended to include the case of correlated

estimation errors.

2.3.2 Numerical Results

Figares 2.11 to 2.15 show :he r,urr,erIca! results for the 2-ray channel with imperfect

chutnel state infarmation(GS1). fa each figure, solid curves represent the result for

perfect CSI whereas isthe dashed curves represent the results for imperfect CSI. The

Doppler frequency for ail the figures is 0.01 and the variance of the estimated error, a:,

is 0.5. This corresponds to an estimate noise of unity (since L = 1). The actual channel

noise power also a s s t m a tbe szrne value, In other words, we assume that the channel

noise is irreducible iio the channel estimation process. Figures 2.11 and 2.12 show the f .. tz;E-error probabiky zs a fx=tctioa of the to id received energy E and with the relative

delay 7 of the 2 rays as a parameter. It is observed that the performance of uncoded

QPSK in the %ray charinel is still better than that in the one-ray channel. However,

%he performance of pedect CSi is about 3 dB better than that of the imperfect CSI.

This is predictable since ;be estimator noise is equal to the channel noise,

The results shown in Fi,wes 2.13 and 2.14 are the bit error rate as a function of

the delay T and with rhe power split ratia as a parameter. Two cases are presented: a

total energy of 20 dB and 40 dB. It is found that the best performance is still obtained

when the afrid rays have equd energ;. These two figures obviously show that the

system xi& perfect chzmef staie inforrssztion performs better than the one without.

All of these resufts c m be interpreted as z loss in energy. Theoretical and simulation

resrrfts of the uncoded QPSK madnlatiox system with perfect and imperfect CSI are

shown Figures 2-15 a d '2.16. From these two figures, it is found t!mt the simulation

results match it-h &e therjry ;.when the signal-to-noise is above 20 dB. However, the

sinrdation results show rEzr rHe p e r k t CSf system is oniy about 2 dB better than

the imperfect CCI system (the tbeoreticd results show a 3 dB difference).

CHAPTER 2. THE ERROR PERFORMAXCE OF GNCODED QPSK

Signal-to-Noise Ratio, T(dB)

Figure 2.11: The -4pproGmifted Bit Error Probability of QPSK with El = Ez

Figure 2.12: TEe AppraxErnrtt,ed Bit Error Probability of QPSK with El = 16E2

The Nom&ed Relative Delay, p

Figure 2.13: The Approximated Bit Error Probability of QPSK at E = 20 dB

CHAPTER 2. THE ERROR PERFORA.fA?JCE OF UNCODED QPSII'

The Normalized Relative Delay, p

Fiewe 2.14: The Approximated Bit Error Probability of QPSK at E = 40 dB and with power split ratio as a parameter

SoEd Curse - Perfect Chamel State Information

(1) - El = E2: Estimated Error Variance, (T: = 0.0

(2) - El = 4E2, Estimated Error Variance, a: = 0.0

Dashed Curve - Imperfect Channel State Information

f 3) - El = E2? Estimated Error Variance, a: = 0.5

(4) - El = 4E2, Estimated Error Variance, a: = 0.5

CHAPTER 2. THE EBROR PERFORMANCE OF UNCODED &PSI<

..........

.....................................................................................................................................................................

10-5 f I I

0 5 10 15 20 25

Signal-to-Noise Ratio, E(dB)

Figure 2.15: The Simulated Bit Error Probability of QPSK with El = Ez

Solid Curves - Perfect Chvlnel State Idormation

flf - Delayfp) = 0.6, Estimated Error Variance, a: = 0.0

(2) - Delay(p) = 0.0, Estimated Error Variance, a: = 0.0

Dashed Curves - Imperfect Channel State Information

(3) - Delay(p) = 0.6, Estimated Error Variance, a: = 0.5

(4) - Delay(p) = 0.0, Estimated Error fiariance, a: = 0.5

CHAPTER 2. THE ERROR PERFORiK4NCE OF UNCODED QPSK

The Normalized Relative Delay

Figure 2.16: The Simulated Bit Error Probability of QPSK at E = 20 dB and with power split ratio a s a parameter

Solid Curve - Perfect Channel State Information

(I) - El = E2, Estimated Error Variance, a: = 0.0

(2) - El = 4E2, Estimated Error Variance, a: = 0.0

Dashed C w e - Imperfect Chamel State Information

(3) - El = E2, Estimated Error tTariance, 0; = 0.5

(4) - El = 4Ez, Estimated Ermr Variance, a," = 0.5

CHAPTER 2. THE ERROR PERFORA4.4il;rCE OF UNCODED QPSK 47

2.4 The effect of Pulse Shaping

In the previous two sections, we assumed that the baseband sign& has a rectangular

pulse shape with a duration equd to one symbol interval. There is one basic prob-

lem associated with this pulse shape: it is not bandwidth eificient. Specifically, the

transmitted signal's power spectrum (at baseband) is of the form

&fz) where sinc(2) = : and f is the frequency variable. This signal spectrum is clearly

not bandlimited. For better bandwidth efiiciency, bandlimited pulses such as the

squared root raised cosine pdse should be used.

In this section: the effects of pulse shaping on uncoded QPSK is invest-igated. The

pulse used in here is the r&ed cosine pulse with a 35% excess bandwidth and 35%

excess bandwidth is the cment digital cellular requirement. Due to the complexity

of the system model, only simulation results are provided.

2.4.1 The Raised Cosine Puke

The raised cosine pulsez \r%ch satisfies the Xyquist criterion of zero intersymbol inter-

ference, is widely used in digital trztnsmission on band-limited channels. The matched

filter response (i-e. the combined transait and receive fi!ters response) of a raised

cosine pulse, rft), can Ire expressed as

sin af jT cos p ~ t / T r(i) =

at/T f - 4B2t2/T2

CHAPTER 2, TBE ERROR PERFORJIANCE OF UNCODED QPSK 48

where p is called the rolloff parameter. Figure 2.17 shows the matched filter response

r ( t ) for several -v;ilues of 3. Kote that r ( t ) = 0 for all t = kT, where k is any integer,

A larger value of ,& corresponds to a narrower pulse.

4.4 '- -42' -3•‹F -2T -T 0 T 2T 3T 4T

Figure 2.11: The Matched Filter Response of a Raised Cosine Pulse

CHAPTER 2. THE ERROR PERFORMANCE OF UNCODED QPSK 49

2.4.2 Simulation Model

Fiom Figure 2.17, it can be deduced that the presence of a delay arrival ray will

introduce intersymbol interference which in turn causes degradation in the bit error

rate. In this case, the corresponding received sample, yk, can be written in the form

The infinite sum in the above expression causes two problems in our study. First, an

infinite number of terms is not realizable and cannot be generated by a computer.

Consequently, a finite number of interference terms must be determined and we trun-

cate the channel impulse response to 16 symbols. A detail explanation of this choice

of truncation length will be given later in this section. The second problem is the

decoding complexity of the Viterbi decoder. The decoding complexity of the Viterbi

decoder is proportional to 4L, where L is the length of the channel impulse response

and 4 is the size of the QPSK constellation. As we can see from (2.71), if the length of

the channel sequence is infinite, the decoding complexity of the Viterbi decoder will

be infinite as well. If the length of the impulse response is 16, the decoding complexity

will be equal to 4'". Clearly, such a Viterbi equalizer is still not realizable. Thus, the

receiver has no choice but to assume that the channel memory is short. The Viterbi

equalizer used in this study assumes a memory of 2, or 16 states in the decoding trellis.

ff we choose a memory of 3, then there are 64 states io the decoding trellis and the

decoding complexity is increased by a factor 4. Such a large number of states is not

practical in simulation. On t.he other hand, a reduced complexity decoder will yield

gmd performance at moderate to fnw channel SNR but will generate an irreducible

CHAPTER 2. THE ERROR PERFORAMNCE OF UNCODED QPSK 50

error floor due to the truncation of the channel response. This is because the residual

energy in the truncation signal acts as irreducible noise and therefore, an irreducible

error floor will occur. In summary, we use in our signal generation the mudel

and use for the Viterbi decoder the metric

Table 2.1 lists the intersymbol interference magnitude at different delay T for a

raised cosine pulse with a rolloff factor of 0.35. The first column in Table 2.1 is the

time interval of the transmitted symbol and the second to the fifth columns are the

matched filter response at diRerent delays of the second arrival ray. In other words,

where r(t) is the raised cosine pulse in (2.70). From Table 2.1, the reader will observe

that the magnitude of the intersymbol interference diminishes as the time difference

between the signal and the interference term increases. After 8 time intervals, the

magnitude is almost zero. Therefore, we truncate the pulse to 16 symbols (8 on

each side) in the simulation. Note that the residual energy at the truncation point

is 0.0037%, 0.0097% and 0.0089% for relative delays equal to 0.25, 0.5 and 0.75,

respectively.

i n the simulationT we assume that the fading processes are slow enough that they

can be assumed constant over 16 symbol duration. In this case, the received signal yk

CHAPTER 2. THE ERROR PERFORMANCE OF UNCODED QPSK

can be written as

where glk and gzk represent the fading gains that affect the bth symbol , T is the

relative delay and r(k) was defined in (2.74). In addition, glk and g2k are independent

of each other.

Now comparing (2.72) and (2.75), it is observed that

It should be pointed that hko and hkj are correlated.

The Viterbi decoder will perform detection based on the trellis diagram shown

in Figure 2.18. There are 16 states, one for each combination of the 2 previous

transmitted symbols. In other words, each state represents (xk-Z, x ~ - ~ ) . There are 16

states because the Viterbi equalizer assumes the length of the channel memory to be

2. If the Viterbi equalizer assumes the length of the memory to be 3, then the trellis

diagram will have 64 states.

CHAPTER 2. THE ERROR PERFORAIANCE OF UArCODED QPSK

Table 2.1: The intersymbol interference Magnitude at different delays for the raised cosine pulse of a rdlofF factar of 0.35

CHAPTER 2. THE ERROR PERFOR-tIi\A7CE OF UNCODED QPSK 53

0 . - Figure 2-18: The Trellis Uiagmm for the Viterbi Equdizer with Pulse Shaping. (0. I , 2: 3) are the Sunher Assaciated with the QPSK Signals f hown in Fi,gure 2.4

2.4.3 Numerical Results

Figures 2.19-2.21 &O~S the simulation results for the 2-ray chamel 1%-ith pulse shaping.

The pulse used is the raised cosine pulse with the rolloff factor, .3, equal to 0.35 and the

fade rate of t h e e simulation resrtlts is 0.01. Figure 2.19 shows the bit error probability

as a function of the xotd received energy E and with tile relative delay uf the 2 rays

as a parameter. The simulation result is obtained under the condition that the 2 rays

have equal p o w (power split ratio = I). It is found that the performance of the 2

ray channel is stif1 better than that of the flat fading channel (one ray channel). For

delays fess &zm m e s y ~ b i j : duratim, we found that the larger the relative delay, tile

better the periormace.

Figure 2.20 illwrrates the effect of the relative delay as iri-ell as the power split ratio.

The simulation red-rf-ts are o5tained whe;i the total energy of the 2 ray is ey ual to 25

dB. From this figare: it is observed that 'the best performance is obtained when tltc

arrival rays haye e q d eeIierrgy= Io addition, Figure 2.20 also supports the cor~ciusiori

of Figure 2.19: the higber the relatk-e delay. the better the performa~ce.

fin all^., Figure 2.21 demctnstrates the effect of the ro fM factor, 3, on the bit error

rate. The simakiricro was ca-xied atit at a total received energy equal to 25 dB and

with equal power split. F m Figwe 2.31, we can see that tile higher rdfoff factor,

he better the error pedo-mmce. The aos t significant improvement is seen when the

roifaff factor increases &om Q to 0.4.

Signal-to-Noise Ratio, dB

Figwe 2.19: The Simulated Bit Error P~rabability of QPSK with El = E;, and Pulse Shaping

The ?;ormalized Delay

Figwe 2.20: Tae Simulated Bir Error Pmbability af QPSK with Pulse Shapirrg at - * E = 25 dB and xiah the pofrer spirt rzrk as a parameter

(1) - Posver S p k Ratio = 86: E3 = I&& (2) - Puwer Split Ratio = 4: El = 4E2 (3'1 - Power Split Ratio = f : Ei = Ez

CHAPTER 2. THE ERROR PERFORMANCE OF UNCODED QPSK

?lie Rolloff Factor

Figure 2.21: The Simutated Bit Error Probability of QPSK with Pdse Shaping at E = 25 dB and with Equd Pm-er Split

CHAPTER 2. THE ERROR PERFORMANCE O F UArCODED QPSK ti8

2.5 Summary

We studied in this chapter the error performance of a Viterbi receiver used for decoding

PSK signals transmitted over a frequency selective Rayleigh fading channel. We found

that in the case where the frequency selective fading is introduced by the presence

of 2 propagation paths, the channel provides an implicit diversity effect that helps

t.o improve the error performance over that of a flat fading channel. 1% studied

the uncoded system with both perfect and imperfect channel state information. In

the imperfect CSi case: we assume an independent estimated error with the sarne

magnitude (varia~cej i ~ f the channel xoise. The analytical and simulation results

show that the channel still provides an implicit diversity effect. However, the error

performance with imperfect CSI is about 3 dB worse than the perfect one, as expected.

Furthermore, the eEects of pulse shaping are also investigated. The pulse used is a

raised cosine puke with a rolloff factor of 0.35. It is found that the raised cosine pulse

pelforms better than the rectangular pulse. In addition, with pulse shaping, the Zray

channel stiff performs better than the one-ray (flat fading) channel.

APTER 3

Trellis- Coded Modulation

The conventional way to improve the error performance of a communication system

is to apply forward enor correction codes (FEC). However FEC Ieads to bandwidth

expansion which is not desiraMe in bandlimited applications. Alte-mativel51, we can use

TCM which can provide coding gain without bandwidth expansion. This is achieved

through consteflation expansion, rather than signal dimensionality expansion found

iri conventional FEC systems. TCM Eras gained popularity since its introduction in

f 982 by Ungerbmk [I 2;.

The perfomaoce of any coded system mainly depends on its free Euclidean dis-

tance, which is the minimum Edidean distance between paths that diverge from any

state and remerge at the sane state after some delay. As mentioned earlier, TCM ac-

complishes the coding gain without Sandedth expansion by coding onto an expanded

G g z d co~~teEatioii sa that the fie E.sc2bea &stance is m a e e d . For an additive

white Gaussriu noise cHasoelF the cuatrudion of close to optimal trellis codes can

CHAPTER 3. TRELLIS-CODED MODULATION

be performed on the basis of the following heuristic rules:

(1) Parallel transitions (when they occur) in the encoder trellis are assigned to

signal points separated by the maximum Euclidean distance.

(2) The transitian originating from and merging into any state is assigned the

same set of signals.

(3) The signal points from each subset should occur an equal number of times in

each state.

Rules (1) w d (2) guarantee that the Euclidean distance associated with sing!e

and multiple paths that diverge from any state and remerge at the same state has a

maximum free EucEdean distance. Rule (3) guarantees that the trellis code will have

a regular structure. X ~ r e w e r , rules (1) and (2) will involve the concept of "mapping

OF set partitioning, which d l be discussed later in this chapter.

Since the introduction of the 2-Dimensional TCM by Ungerboeck[l2], extensions

were made to obtain multidimensional TCM [35]-[38]. Trellis codes based on 4, 8, and

16-dimension& s i s a l consreffations have been constructed and some of these codes

ha= been implemented in commercid_v available modems 1353. One of the advan-

tages of multidimensionaf TCM is that 1r.e can use smaller constituent 2-dimensional

signal consteBations that d o w a tradeoff between coding gain and irnplement,ation

complexity Recently? a sew design technique for TGM based r,q lattice and co-sets

of a sublattice has been de-t-eloped

The m&hin parposes of this chapter are to design new TCM schemes fox digital

ceEdar corrrmmicatlms, to coffipare the new trellis codes with conventionaI coding

CHAPTER 3. TRELLIS-CODED MOD ULATI0.N 6 1

systems that use convoiutionaf code in conjunction with :-QPSK modulation and,

specifically, to investigate the feasibility of using TCM to replace convolutional codes

in digital cellular applications.

The chapter is orgmized as follows: Section 3.1 presents the general trellis coded

5-QPSK T system. Section 3.2discusses the design procedure of our new trellis coded

modulation schemes. Three new treflis codes are designed: the 8 state, rate 312 code;

the 8 state, rate 31.3 code; and the 16 state, rate 212 code. Tbe code rate in this thesis

is defined as the rsumber of input bits per channel symbol. The simulation results for

these three trellis codes are given in Section 3.3. Their comparisons with convolutional

codes are given in Section 3.4. In addition, we present in Section 3.5 two 64 state

codes designed with the same approach as those in Section 3.3, and compare their

performance wifb the 32 state corn-olutional code proposed for the digital celluIar

application. Finally2 a summary of tbis chzpter is given in Section 3.6. It should

be pointed out that xhe resuits presented in this chapter apply only to flat fading

channels. The e m r perfamance of the %ray frequency selective fading channel will

be given in the next chapter.

3.1 Trellis-Coded %-QPSK

:i fdormat5ioil Bits i Phase Change ir

00 I X - i 4

i ; 3" 4

18 l i

F@e 3.1: The $-QPSK Signal Consteilation.

Althou& it is u eight p i n t ccr.zstftllation, signal points i ~ i i h even and odd Iabels

are wed in dtemizte sigaaBiing Ineerrds. This signalling strategy avoids the signal

transitions thmrrgb the m-igin and id cfinsequently I d to a more coristant enretope

msduhticr~l scheme,

CHAPTER 3. TRELLIS- CODED MOD LTLATION

For convenience, we use

to represent the two subsets of signal points used in the alternate intervals. This

implies that over a period of 2 symbol intervals, the signal set S is simply the cartesian

product of X and 3'. Specificall_v,

where @ denotes the cartesian product of signal sets. In order to achieve the desired

spectrum shaping as -rd as channel coding, we can design trellis codes based on the

sign& set S shown above. The codes thus obtained are examples of multiple trellis

coded modulation fMTC31) !18], since there are multiple symbols in each transition.

3.2 Design Procedure

The first; step in desIgC-g a ~ r & k code is to specify the desired code rate and the

r,reliis structure. For eonsecience in o w later discussion, lye will introduce a s m d

CHAPTER 3. TRELLIS-CODED i2IODEILATION 64

distinction between the code rate and the throughput. The code rate will always be

given as a rational number. 4 rate k/n code is one that encodes E bits of information

into n modulation symbols in each encoding interva:. The throughput, on the other

hand, is the code rate expressed as a real number. By adopting this convention,

we can conveniently distinguish the structures of codes with the same throughput.

With ;-QPSK being the modulation format, the throughput must be lower than 2

bits/symbol in order to achieve a significant coding gain. The code rates considered

in this chapter are 3/2? 3/3, and 2/2, which translate into throughputs of 1.5 and 1

bits/symbol.

3.2.1 Rate 3 / 2 Codes

The two trellis structrrres shown in Figures 3.2 and 3.3 can be used to encode 3 bits

of information into 2 modulation symbols using 2-QPSK. The detail explanation of

the 8 state, rate 3/2 code will be given in Appendix B. The encoder in Figure 3.2

has four states with 4 parallel transitions between pairs of states. On the other hand;

the encoder in Fieme 3.3 has 8 states but only 2 parallel transitions between pairs of

.-a ~bates. + Both encoders exhibit regularity and symmetry I121 and all the transitions in

the two encoders receive symbol-pairs from the set S in (3.2).

CHAPTER 3. TRELLIS-CODED MODULATION

Figure 3.2: A 4 state, rate 3 f 2 code with 4 parallel transitions between pairs of states

Al l A12 A21

B1, 12 B21 B22

A12 All A22 A~~

B12 BIl B22 B21

A21 A, A l l A12

3 2 , B, E l , 312

A22 A,,

Figure 3.3: An 8 state, rate 312 code with 2 parallel transitions between pairs of states

CHAPTER 3. TRELLIS-CODED MODULATION 66

The second step in designing a trellis code is to partition the signal set S into

subsets in accordance with the trellis structure. Based on the ideas presented in 1121

and [16], we arrive at the set partitioning in Figure 3.4.

Figure 3.4: Set partitioning of the 2-QPSK signal set over a 2 symtol interval

As shown in Figure 3.4, the original signal set has an intra-set squared Euclidean

distance of A; = 2: while subsets in the next three layers have iatraset distances of

A: = 4, 0; = 4, axid 4,; = 8. Atthough there is no gain in distance from the first

level to the second, this is of no concern in this study because of the trellis structures

used; see Figures 3.2 and 3.3. We also note that the symbds assigned to the same

subsets in the second and third Seveis of the partitioning tree are all different. This

guarantees that trellis structures with 2 or 4 parallel transitions between any pair of

states will have a free Hamming distance (at the symbol level) of 2. As pointed out

in [I '71, the order of diversity provided by a trellis code is aspptoticafly equd to its

free Hamming disia~ce. Thk is the performance criterion emphasized in this study.

Finally, to complete the design of a trellis code, we h a ~ e to appropriately assign

the subsets in Figure 3.4 to the transitions of the encoder. Once again, we adopt

Ungerboeck's heuristic code design rdes 1121: transitions originating and terminating

at the same state must receive signals from the same subset, and pardel transitions

should receive signals from subsets in the lowest level of the set partitioning tree.

The signal mapping for the 4 and 8 state codes are shoxt-n in Figures 3.2 and 3.3

respectively. We can deilrrce em$!)- from these figures that the minimnm squared

Euclidean bistacce, 6: is 4E or 6E6 for the 4 state code2 itad 822 or 12Eb for the 8

state code, where E is the symbol energy and Eb is the bit energy When compared

to nncoded QPSK whose mioimum squared distance is SEj, the 4 state rode provides

a coding gain of 1.2 dB in the AIYGS chamel while the 8 state code pro\~ides a 4.8

dB gain. 111 addition, these rwo codes provide a second order diversity effect in a

fuify interleaved fading c'nmnef, since both have a free Hamming distznce, d ~ , of 2.

Finatiy, it should be pinred out that the 8 state code compared favorabfy against a

sirnifar code in the hzerath~e for the XWGX channel application. The 8 state, rate

312, SPSIC code ia [16j has a squared Euclidean distance of 5.17E, which is 2 dB

less energy eEcIenr &an our code in the * W G N charme!. Ifowever, tile codc i l l j iG]

can provide a third order dii-ersity effect for fading applicatiorns. o m more tliari that

provided by otlr 8 srare code. This loss in diversity is the price we have to pay in

order to achieve spectrum shaping via :-QFSK.

It is clear &om the set partitioning tree in Figure 3.4 that in order to have a

rate 3/2 code with 2 squared distance greater than 8E , there must not exist any

parallel transitions in the trellis bizgram. This together with the requirerntrit that

transitions originating zad terminzting at the same state must receive sigtiafs from

the same subset implies that at !east 64 states are required :o improve the squared

distance to beyond 8E. The encoder of a 64 state, rate 3 f2 code with no paraltc!

transitions will consist of 3 inpnr branches tvith two 1-bit delay eIcntcnts in each

branch. Conseqi;entl> there exists error events of length equal to 3 steps i11 thcb

trellis. The sqaared Euciidean distances of these error events h a ~ c a loiwr i~ourtd

equal to 4E + 2E -+ 3-23 = 10E. \shere 4E is the intraset distance for sigrial subsets

A and B, and 2E is the Zntesset & s t a c e between A and B- This a 6-I stale code

can provide a coding gain of 5.7 dl3 over QPSK in the AJVG?; channei. It is possible

to increase the s i p r e d Eilstmce further to 12E. To achieve this, a 51 2 state code is

required. This grecttly iinc~ezses the decoding complexity.

3.2.2 Rate 3 f 3 and 212 Codes

To encode i1iifim2tion 2: a throughput ui I bitfs_vmbol, 5:-e can use codr~~ with rates

q3zl 3/3 md 29, 3 Faie 3/3 c d e car, he &tzincd fWm a rate 3 / 2 by

repeatjizg one symbol in each tramition of the rate 312 code. Specificajly, if we &ride

the ',rcffis of a r&Ee 3/? code into periods of 2 intervals and labelled the 4 symbols

sent in each period by

[ ~ I ~ C ~ ~ C ; , C ; ) (3 .3 )

then the 6 symbols smt- by the corresponding rate 313 code in the same period is

A rate 3 / 3 code obtrldmed &is way clearly has a periodic sipal mapping (the period is

equai to 2 encoding interds) . This is unavoidable if we are to conform to the 2-QPSIC

modulation format i.;itfi an odd number of symbols in each encoder transition.

1% have applied the technique mentioned above to obtain a rate 3 f 3 code from

tbe 8 state, rate 3p2 cade in Figwe 3.3. The tatter will be referred to as the mother

code. The squared Eucfidezn distance oi the rate 3 / 3 code is 10E, with E = Eb.

Consequentllv, a 4 dB co&g gain is achieved by the mother code. Moreover, this

cock can provide a bird arder time dirersity in fading applications: one more than

that provided by rkie morhez code.

Zr is also possible t-o achiei-e a throtrghput of 1 bit/syrnholb:; using rate 212 cotics.

Tlfe rate 212 code shoiw in Figure 3.5 has 16 states with no parailct traasitio~ls

between pairs of states. irs bas the same decoding complesir_r 3s the rate 333 code

rneationed in tk last paragraph. It should be pointeJ out that, for a rate k/ t l trellis

code with i't: m n b e s of states: the decoding complexity is proportio~lai to the total

mimber of channel symbols in all the encoder transitions [lSj. i.e.

n C = - operations/information bit.

A-

where n is the n - ~ ~ ~ b e r of infomatimi bits, k is the number of transmitted symbols

and XS is the numSer of states.

A s for the enor perfomJmce; the 16 state? rate 212 code has a Eucfideart distance

equal to 10E [witis E = E+j and a free Hamming distance of 5. Based on these

fipres and those sEsv,-o in the last paragraph: we can predict that the rate 21% code

aad the rate 3/3 code will haw ro11gh4~- the same error performace in the AMGGN

cbzmneL Hm-ever, the r ~ t e 2,62 code is preferred in fading applications due to its

higher diversity order. Xthoagb the rate 2,d2 code outperforms the rate 3/3 code i ~ i

fading applications, the fatter has rhe &&antage that it can be used in corijunctio~i

mi& its mother code (the 8 state? rste 3f2 code in Figure 3.3) to provide unequal

enar protection [liif. Ibis is of pa&xfr?s importance for applications such as digital

speech &ere &&rent speech elements E a ~ e different sensitivities to trai~smission

errors. Since the Fate 3/3 code is d e d d horn the rate 3 / 2 code {or vice vcrsaj, it

ge~era l sllgoritkm cslni be used far decoding in both cases.

CHAPTER 3. TRELLIS-CODED MOD tZdTIO.8 72

We summarize in Table 3.1 b e h - the performance indicators for the different

trelEs-code 2-QPSK schemes. Also shrtwr.n are the results fm rm~entional systems

that use con~oiutitlrnai codes along with 2-QPSK modulation. A discussion oia the

cora~plexity issue i:-iB be g k n in Section 3.1.

k Code Il

4 state, rate 3f2

11 8 state, rate 3/3 1 16 state, rare 2/2

/j 8 state, rate q 4 convolu tional code

convolutiond code

-.I I nrougbpui j Complexity Squared Distance 1 Diversity 1

TabIe 3.1: X Performance Summary for Different Treilis Codes

3.3 Simulation Results

?Ye have determined by sirndation the bit error performance of the 8 state, rate 3 f 2

code, the 8 state, rate 3/3 code and the 16 state, rate 212 code. Both the AWGN and

the Rayleigh fiat Ehciing citannels have been considered, and in both cases, perfect

coherent detection as6 soft decision maximum likelihood decoding were assumed.

Furthermore, far the Rqleigh fading channel, we assumed that ideal interleaving and

perfect information about the fading gains are available. It shoidd be pointed out

that in general, a mobile comr~iunica~ion channel will introdzce frequency selective

fading and it is our I~tention to ei-entuzilg test our codes in such an environment; see

Chapter 4.

The simulation res-&s oi our TCM schemes are shown in Figures 3.6 and 3.7,

along with uncmded QPSK and Ungerboeck% 8 state PSK code 1121 as om reference

systems. The Unge~boeck's 8 state PSK code has an asymptotic coding gain of 3.6

dB compared to mcoded QPSE in the AWGS channel, and it provides a second

order diversity efiecr in the fading envi:iroment. It is observed from Figure 3.6 that

for the AWGN chamel and m error rate of lo-', the 8 state: rate 312 code is about

1.5 dB mare energy efficienr thm Ungerboeck's code and 4.2 dB mofe efftcient than

unccrded QPSK. This figwe a g e s quite well with the asymptotic mlrres predicted

in Section 32.1. Howe~er~ both the 8 state, 3/3 code and the 16 state, rate 212

code are s14gbtlj= worse thar: the 8 statet rate 312 code in the -4TVGT channel. As

mentioned earlier, the rate 3/3 code and the rate 312 code can be used in conjunction

to p x i d e u ~ : e ~ ' ~ : d error pzarec&m For ti& appflcation, the bit SXR is no longer the

meanindul mesrrre af eaergy eSccienc3- because the rate 3/3 code is nohoptima1 in

the sense of bit SSR. Rather, the symbd SSR should be used instead. Ln this case,

the rate 313 code is a b m t 1 dB more efficient than the rate 312 code in the A!i!GN

chamel.

Figare 3.7 illustrates the performance of our codes in the flat fading channef. In

the fading chamel, the 8 state, rate 3J2 code is consistentl_v about 2 dB better than

Ungerboeck's code. ft should be pointed out t h a t the throughput of this codc is

only 75% that 0.f Cagerboeck's. As mentioned earlier, the rate 313 and the rate 2/2

codes perform worse than the 8 stare, rate 3 / 2 code in the AKGX channel. Hotvevcr,

they perform sigdicantiy better than the rate 3 / 2 code in the fading channe l due to

their higher diversity orders. This is especially true in the case of the rate 2f2 code

where a bit-error probabiiity of lo-' can be achieved in the fading charrncl wit11 a

signal-to-noise ratio (SKI%) as small as If dB.

CHAPTER 3. TRELLIS- CODED MUD LEATIOX

Bi; Signaf- noise Noise, Ebmo (d3)

Figure 3.6: Tire Enor Pedormasce of Different Codes io the Ati5GS Cfiannel

j l ) - 8 State, R a e 3/2 Coiie

(2) - 8 Stale, Rate 3/3 Code

(3) - I6 State, Rate "22 Code

(4) - t;ngerbwck's- 8 State Cade

(3) - Encoded QPSK

2 4 6 12 14 16 18 20

Bit Sipat-10-Noise Ratio, EbfNo (dB)

Figare 3.1: The Error Per"cif51iance of Different Codes in the Flat Fading Ciiannel

ft) - I6 State, Rate 2/2 Code

(2 ) - 8 State, Rate 3/3 CoGe

(3) - 8 State, Rate 312 Code

(4) - Vngerbaeck's 8 State Code

( 5 ) - Uncoded QPSK

3.4 Comparisons with Convolutional Codes

We want to address in rhis section the question of how well the trellis codes com-

pare with conventional systems that use convo1utionaI code in concatenation with

z-QPSK 4 rnodulatioa. frr order to obtain a fair comparison, only codes with the same

throughput and with the same decoding complexity will be considered.

Let N: be the number of states in the convolutional code, k' be the number of input

bits per encoding intervaf, and nf(assume an even number) be the number of output

bits per interval, Then the decoding complexity for the conventional system, which

is proportional to the number of symLols contained in all the encoder transitions, is:

Sotice that there I s a constant of 4 appearing in the above equation. This is due

to the fact that optimal fiecoding for a con~entional system ~ u s t incorporate into

the decoder trellis not s d y $fie states 0: the convolutionai code, but also the phase

states of the differenrial enc~der. There are altogether eight phase states, but only 4

states are reachable at. an? ioten-;ti. Consequently, the decoding complexity of

a coos-eentiond s~s tem is -2 rimes that of the underlying consdutional code,

To predict the error pedommce of a con-~dutional coded QPSK sfstern with

perfect CSI, we czm make use of the foBowing approximations. Let the underlying

c~tnvolutionai code hzs a free Hamming distance (at the bit level) of df7 then the

sqrrzed Euctidearr distance of the combined spstem is

6; z 2djE

CHAPTER 3. TRELLIS-CODED MODULATION 78

The free Hamming distance (at the symbol level), or equivalently the diversity order,

d j even d, x { 2 d j odd

These approximations can be explained as follows. If a convolution code with

a free Hamming distance of d j is used in conjunction with conventional QPSK, the

squared Euclidean distance of the combined system is simply 4 = 2PEdj. However,

if 2-QPSK is used insteadi due to the differential encoding process, the squared Eu-

clidean distance may no longer be equal to 2 E d j . Actually, our previous experience

in similar systems {for example, a partial response signalling system) leads us to be-

lieve that differential eucoding will lor. ,r the squared distance of a combined system.

For the purpose of estimating the error performance of a conventional system in the

AWGN channel, we m&e the (optimistic) assumption that the differential encoding

p r o m s in z-QPSK does not change the squared Euclidean distance. Consequently,

me have the approximation in (3.7). As for the approximation in (3.81, we made use

of the fact that in any coodutlonaffy encoded QPSK system, two encoded digits

are =rapped into a channel symbol. Consequently, the free Hamming distance (at

the syz~bol lewd) of the combined system is, in the worst case, simply half the free

distance of the convoTutiona! code. However, sometimes the diversity order of a con-

ventictnaf system is better &= fiie appmximation gix-en in (3.8). A good example is

the 8 state, rate 1/2 code in Table 3.1.

The 8 state, rate 3/% code in Figure 3.3 has a throughput of 1.3 bitsfsyrnbol

and a co;rrpTexity of 43; = Tahk 3.1. This thro*l=hwt -0-r a d comp!e:ii?y car.: &r3

be acbiesed by wing rn 8 state, rate 314 convolutional code in conjitnctiorr with

=-QPSK 4 noddifatioxz. The ~~ptimat raw 3/4 convolutional code with 8 states has a

CHAPTER 3. TRELLIS-CODED MODULATION 79

free Hamming distance of df = 4. Based on the approximations made in (3.7) and

(3.81, a conventional system that uses this code in conjunction with 2-QPSK will

have a squared distance of 8E and will exhibit a second order diversity effect in fading

channels. In other words, this conventional system has roughly the same performance

as our 8 state, rate 3/2 code in both the AWGN and the Rayleigh fading channels.

Actually, the simulation results in Figure 3.8 show that our code is slightly better in

both applications.

Finally, we want to compare the performance of trellis-coded systems and con-

ventional systems at a throughput equal to 1 bit/symbol. Fi,we 3.9 shows the error

performance of codes with a throughpnt equal to 1 bit/symbol. Both of our rate 3/3

and rate 2/2 codes have a throughput of 1 bitlsymbol and a decoding complexity

of 64. The best c ~ n ~ e n t i o n d system at this throughput and \pith this complexity is

the one based on the optimal 8 state? rate I f 2 convolutionaf code. This code has a

free Hamming distance of 6 and consequently, the conventiorial system has a square

distance of 12E6 a d a fourth order time tiiversity. By comparing these figures with

those shown in Table 3.1 for khe rate 313 and the rate 2/2 codes, we e x ~ e c t that the

rate 212 and the conwlutiu~al code have roughly the same performance in both the

AWGN and the fading cfimne'is. The 8 staxe, rate 3 f 3 code has the same performance

as the other two codes in the AWGS channef but not in the fading channel. This is

corffirmed by the sirnufation resdts shorn in Figures 3.9.

2 4 6 10 12 16 118 20

Bit Simal-to-Noise Ratio, EbfNo (dB)

Figure 3.8: X Cornpaisan d the Bit Error Performance of Cotfed S:SE~RB with a throughput of 1.5 bits/symbal

(1) - 8 State: Rate 3/2 Code in the Xfi'C-t; Channel

(2) - 8 Statet Rare 3j4 Cam-olnriond Code in the A\'tPGS f hanricl

(3) - 8 State, Rate SJZ G d e in the Fiat Fading Channel

(3) - 8 State: R a ~ e 3/4 Corn-olmtionid Code in the Flat Fading t'hirnnd

Bit f ignal-to-Xoise Ratio, EbP=rP(dB)

Figwe 3.9: -4 Comparison of the Bit Error Performance of Coded Systems with a throughput of f bii/symbol

f I) - I6 stale. Rate 212 Code in the Rayleigh Fading Chamel

(2) - 3 State, Itate :p'2 C~m-~h: iona Cede in the Rqleigh Fadi~g, Charnel

(3) - 8 State. Rare 3i3 Code in the Rqleigh Fading Channel

f 4 ) - 16 state, Rate 2 / 2 Code in the A\\-GS Channel

( 5 ) - 8 State, Raie i f 2 Convohiiond Code in the AWGS Channel

(6) - 5 State, Raw 3/3 Ca@e in :he -4ii'GS Channel

3.5 64 State, Trellis Codes and Comparison with

the IS44 Csnvolutional Code

Current digital celular communications employ the (65 , 57) 32 stale, rate 1/2 convo-

lutional code (hereafter referred to as the IS-54 code). In this section. iw compare the

peffclrmance or' tire IS-54 code with TCM schemes designed 1:-ith our approach that

have similar througbp~t and complexity. Two new TChl schemes are considered: 64

state, rate 2 / 2 64 state. rate 8 j2 codes. Both codes have single transition between

pairs of states. The 6-1 state. rate 212 code basically is an extension of the 16 state,

rate 212 code, and the 61 state: rate 3 j 2 code is an extension of the 8 state, rate 3 / 2

code. The trellis stmctrrres of these two codes are given in Appendix C.

Figure 3.10 shoxs rhe e m s ,rperfo~mance of the 64 state, rate 3/2 and the rate

2/2 trellis codes and the 32 srztte, rate 212 convolutional code. From this figure, it

is found that at an error rare of 10-% the rate 2/2 code is about 1 dB more energy

eificient than. the rate LP2cmvoiutionzi code, which is in turn about 3 dB rnore

energy efficient than the rzte 312 code. The decoding conlpiesity of the rate 3 j 2 aid

the m e 2/2 trellis codes and the rate 1/2 convoiutional code are 341.33, 512 arrtl

-712 respectively. Thus It c a be corrrludrd that the 61 stare, rate 212 treifis code is

preferred over the 32 state, raw fi-2 convc-lurionaf code. Although the rate 3/2 code

is less eoergy e%cient thzn the rate 1 f 2 convolutional code. it increases the cfmrnel

capzcity by a factor of 1.5. ia other words. the coding gain Is traded for tire chanriel . .

capac:t~ In the case of t2 sstrire, rate 3j2 trellis code.

3;: &is chapter. se pre3erzed a ~~erh.ctdcrlogy for designing xreilis-coded mod-dation

schemes that conf~mi to the ;-QPSK modulation format. The basic idea is to use

. > multiple trellis ccoies ;.;l:n z signal set which is the artesian product of the even and

odd subsets in the :-f,lPSI; signal cmste!faiion. Several good codes xith multiplicity

of 2 are designed. Airmowg them are the 8 state, rate 312 code 5%-ith a throughput

0: 1.5 bits/symbd a i d a IG state. ra te 2f2 code with a throughput of unity. The

* s

former can prwzcs z second order tiiwrsity effect in fading applications while the

!alter can provide i~ sit3 order diversity. In the AjirG;\; chamel, the rate 312 code

is about 0.8 dB more eraerg- escieat than the rate 2/2 code. Both codes perform at

!east as r~elf as jaboue "dB better! the conventional system that uses $-QPSK in

% . * - coi;jur?ction vSrh ~ o ~ ~ ~ i ~ i i o i s d codrnc, - The 8 state, rate 3/2 code is about 2 dB more

eztergj- effcienr iis.2e.1 cornpeed 3,i;iti-r the rani-olutional code ii;ith the same thoughput -

and cornpiexity. Two f2 sxate i C>i schemes are also designed: the 64 state, rate 3/52

a d the 61 state, rate 21;2 codes. These codes have throughputs of 1.5 bits/symbol

and 1 bit f s y ~ b o l respectlrek;. $%%en r he 64 state, rate 2 / 2 code is compared with

the 3.3 &te. rzte 162 ct3~a-oit~tiona: mde used in digitaf ce!iufar: it is f o n d that the

:~-e!Eis code is z b o ~ I dB nmre e n e r g e%cient than the condutional code. Rcwever,

i l. , G-! sfitfiji rate :3/2 i d k C O ~ E perfomis ii-orse than the coxoiut io~al code. This

Isss in energv can be explained 'a>- :he i~crease in channel capacity. The rate 3/2 code

ran a h b - f ..7 tirncs more users than the rate 212 TCM and the rate 112 convolutional

code.

Performance of Coded Systems in

Selective Fading Channels

-4s we rneniio~ed easfier. the motile radio commu~iication rilanttt~i is, i i ~ grneritl, it

frequency seleci-i~e fading channel. in the last two chapters, we studird the wror

performance of XQ irncoded sh-stem operating in the frequenc.i- se l tdve fatling rilarlnc~l

arrd the performance of coded systems operating in the flat fadirig chait~lcl. 'I his

chapter will combine the resuits frorn the tT.i.o previous chaprers and wiii t:vaiuate ilic

error performance of coded systems in the frequency selective fading envirunrnent.

The block diagram of such a sFstem was shown earlier in Figure 2.1. it requires a

soft decisioc generz'tor before Z'iterbi decoding of the coded modulation sc1ir:rnc. 'I'hc

structure of :his soft decision device ~ i l ! be presented in Section 4 . 1 : arid the bit error

rate ctf tbe corresponding coded s_vstems are studied in Section 4.2. X stlrri~riary of

this chapter is given in Section 4.3.

4.1 Derivation of the Decoding Metric

in this section, tbe decoding metric far coded modulations operating in a frcquencj.

selectiw fading chalrrrrel is der izd first for the rectangular pulse and then later for

t h e raised cosine p:!e 1.-i:b 2 35% rolbff, The decoding metric 13 then applied to the

i.irnt&tion model =,r;here perfec~ inrerleas-ing and channel estimation are assumed.

Recall that the func~ion oi the i-iierbi decoder is to select the codex-ord

that minimizes the coodirional pru5ability density function (pdi)

here y = (yl. y2-. - - : p:cj is the sequence of receive samples and

is t he correspondins sequence of channel impulse response. Due to interleaving, the

zjk's are the scrambled. recei~ed version of the transmitted s_r.mbols

Because of the chaonei memory, the information about each data symbol x:, is

contained in a block of comxurive receixred samples Rk. The different RkYs are

nut disjoint but i i \ye assume ideal interleaving, they can be treated as disjoint and

srafistiraliy i.sdepenbezt. Clcnsequent'r~, rhe decoding metric in (4.1) becomes

where

and Wr; contains the chaznet impulse response for those received sa~r~ples in Rk.

Taking the natural log of (4.2) implies that the equivalent metric is

1%-hose additive mt-stre matches the requirement in the Viterbi decode: of the mod-

ulation scheme. Dropping the subscript k in (4.5) results in a branch metric of the

form

where c is any symbol fro= ;fie signal constellation. We will esarrii~ie 11t:xt the strrtc-

ture of M(c).

Assume that c \.;as fransmitred in the i 'q ime slot. Then for a rectartgular pulse,

the syrfibol c sr-ill appew in the block

Sote that the above equation is on!y id id for the rectangular pulse of duration ex-

actly equal to 1 symbol. Sow because of the interleavingldc-interleaving process, the

symbols xl-l and s~+l are merely random variables as far as the data symbol c is

concerned. This means (ignoring the constant term in the pdij

I , r ,.r- a study the errcx gerhrmance of the coded systems in the frequency selective fading

environment.

4 2 Simulation Resuits

\Z present in this sectiorn the numerical results for different codes in Chapter 3 in the

2-ra:~ heyuenc~. selective fading chancel. Figures 4.1-4.6 show the simulation results of

the 16 state, rare 2 / 2 TCIi and the 8 slate: rate 1/2 convolutional code in frequency

dect ive fading charnels. Figure 4.1 reports the bit error performance of the rate 212

code as a functiose of the tctal received energy and with the relative delay of the 2

paths as a parameter. I: is found that the best resuit is obiained whm the delay

is equal to O (fizt fadin% channel) aod the worst result is obtaitlned when the delay

is equal to 0.6. Figare 4.2 shows the egects of the normalized delay and with the

paver split ratio as a parameter. It is observed that the best performace is obtained

&er; the power split ratio Is equal to 20. IR general, the larger the power split ratio,

the better the error perftmance of a coded system. It is a!so observed that when

the power split ratio is eqcd to unit:;. the bit error probability of the trellis code

has a peak d u e zr d e h ~ = 0.5. Ho~:e~-er: when the power split ratio is larger, the

peak value is no ionget- existed. It c a be concluded that wheo the power split ratio

approaches infioits tho norn;,ef;zed &fa;- of the second arrli-a1 path does not aEect

* - the error perbmanre 0: rhe code. This is due to the nature of the decoding metric.

Far esarnpk. if the poiwr sptis ratio is l-ei5- large. the delay term (or the second fading

s --% = In ezc"iswei'r-e zzl?-ipk caa be eh ina i ed . T h s the delay of the second path

dues not a k t the ermr &mmitnce.

Figure 4.7 s h o ; ~ tiit. error performance of the 64 state, rate 212 trellis code usiilg

2 raised cosine pulse with a rolloff factor equal to 0.35. It is found that this codcs has

the same cfiararterkric its other t r e k codes. Figure 4.8 show the error perf'ori~lanc-c

comparison bet~xem the 64 sstaie: rate 2/2 code and the 32 state, rate 1/2 convolu-

tional code. Two cases ~ r j considered: delay = 0.2 and delay = 0.6. I t was f o t ~ d

that the rate 2 j2 code is better than the rare l f 2 convolutional cock by about, 1 d B a t

t i t error rate lo-". Therefore. we can conclude that TChl can repiace cor~volutiorlal

code in the celluiar appiiretion.

CEIA P TER 4. CODED SYSTEMS iAT SELECTIVE FADLYG CHAitrVEIS 92

Figure 4.1: The Error Perfcrmance of the 16 State, Rate 212 Code in a Frequency Selective Fading Channei Gsing a Rectangular Pulse and with Delay as a Parameter

(1) - Delay, p = 0.0

(2) - Delay, p = 0.3

(3) - Deiay, p = 0.5

(4 f - Delay, p = f .O

Kormdized Delay

Figure 4.2: The Error Performance of 36 State, Rate 2/2 Crsrfr: i r r a k-rrqiwrrt.y St*-

1ecti~:e Fading Channel Csing Eectmgitlar Pulse and with the Puwcr Split f i a t i c t rts zi

Parameter. The Bit SXR is 8 dB

( I ) - Poser Split Ratio = I: Fl = f2

Bi; Signal-:o-Soise Ratio, dB

.................... ............... . . . . . . . . . . . . . . . . 10" . . . . . . . . . . . . . . . . . . ..................................... . . i ..A i..

............................................................ - . . . . ..............................

. . - .

5 - 4 8 12

Figure 4.6: The Error Perhmanre Comparison between the 16 State; Iiate 21% 'I'rellis Code and the 16 Srate? Rate l j 2 Con;-ofutionaI Code usirtg a Raised Cosine Pulse ~ i r h 2 Rolloff Factor of 0.35

SoEd Lines - The 16 Srzre, Rzte 2J-2 Trellis Code

, . . . . . . ~

2 5 6 7 8 9 10

Bi: f igaaf-lo-Noise Ratio, dB

Figure 4.7: The Ermr Performance of 64 state, rate 2/2 Code in the %ray Frequency Selective Fading Chacnei Using a Raised Cosine Pulse with z Rolloff Factor of 0.35

. . . . . . . . , . . . . . .

Bit Signal-to-Noise Ratio, dB

Figure 4.8: The Error Performance Comparison between the 64 state, rate 212 Codc and the 32 state, rate 1/2 convolutional code in the 2-ray Frequency Selective Fading Channel Using a Raised Cosine Pulse with a Rolloff Factor of 0.35

Solid Line - 64 state: rate 2 /2 code

(1) - Delay, p = 0.2

(2) - Delay, p = 0.6

Dashed Line - 3-2 state: rate 112 convolutional code

(3) - Delay, p = Tr.2

(4) - Delay2 p = 0.6

4.3 Summary

In this chapter, we evaluated the error performance of coded systems operating in

the frequency selective fading channel. As opposed to what was found in an uncoded

system, it is discovered that the channel does not introduce a diversity effect in a

coded system. ft is observed that> in general, the error performance of coded systems

is the worst when the delay is equal to half a symbol and is the best when the channel

is closest to being a flat fading channel. Although coded systems perform differently

(compared to t h i s ltncoded counterpart) in the frequency selective fading channel,

TCM still performs better than convolutional codes in the frequency selective fading

channel.

Conclusions

5.1 Conclusions

In this them, we studied the performance of uncoded and coded PSK rnotlulations

operating in the frequency selective fading environment. The major objective is to

see if trellis coded rnoduiation (TCM) can be used to replace convolutional code in

the digital cellular application. The thesis consists of three major parts

(1) A performance evaluation of uncoded rxodulation: with pulse shaping, in tile 2-ray

frequency selective fading chamel.

(2) The design and performance evaluation of TCM schemes that conforrri to the - $-QPSK modulation format.

(3) A perfmmance esdilatiiiion of our TCM schemes, with puke shaping, in the 2-ray

hquency seiecth-e fading chanoei.

Details of the findings are given below. We found that in the case vhere the

frequency selective fading is introduced by the presence of 2 propagation paths, the

channel provides an impiicit diversity eiiect that helps to improve the error perfor-

mance over that of a fiat fading channel. We studied the uncoded system with both

perfect and imperfect channel state information. In the imperfect CSI case, we in-

troduce an independent estimator error with equal power as the channel noise. The

results show that the channel still provides an implicit diversity effect. However, the

error performance vAth imperfect Cff is about 3 dB worse than the perfect CSI case.

Furthermore, the effect of pulse shaping is also investigated. The pulse used is a

raised cosine pulse with a rolloff factor of 0.35. It is found that the raised cosine pulse

performs better than the rectangular pulse. In addition, with pulse shaping, the 2-ray

channel still performs better than the one-ray (flat fading) channel.

In Chapter 3, we pesented a methodology for designing trellis-coded modulation

schemes that conform to the :-QPSK moduiztion format. The basic idea is to use

multiple trellis codes with a signal set which is the cartesian product of the even and

odd subsets in the :-QPSIi signal constellation. Several good codes are designed, such

as the 8 state, rate 3/2 code with a throughput of 1.5 bits/symbol and a 16 state, rate

212 code with a throughput of unity. The former can provide a second order diversity

effect in fzding applications xrhite the latter can provide a fifth order diversity. In the

AIYGX channel, the rake 312 code is about 0.8 dB more energy efficient than the rate

2 / 2 code. Both codes perform at least as good as (about 1 dB better) the ccrnventional

system that uses $-QPSK in conjunction r ~ i t h convolutional coding. The 8 state; rate

3f-2 code is about 2 dB more energy effcient when compared with the convolutional

code with the same tkcp"dghput itnd complexity. Two 64 state TCM schemes are also

designed: the 64 state: raw 3f2 and the 64 state, rate 212 codes. These codes have a

5 2 Suggest ions for Further Research

2. A decoding afgvri~hms with significant reduction in computational complexity

are required for tire coded systems operating in the frequency selective fading channel.

r- ror the decoding me~ric derived in Chapter 4, we found a suboptimal version under

ti:.- large SSZR zsszfnptioil. Huwe~er, for small SNR, the decoding metric with the

c:xpor:entiai function should be used and its decoding complexity increases with the

cfaarmei memory. An efficient decoding algorithm for small SNR should be derived.

3. In Chapter 1. &re stated that Wilson 1151 has designed a rate 5/6 trellis code

v;itlr a throughput of -2.5 bitsjsymboi and that this code can achieve a significaqt

coding g a b in an AJi7G3 channel r&ik only sacrificing only 16% in spectral efficiency.

In our study, we preseored several I C l f schemes with a maximum throughput of

1.5 bitsfsymbol. Thus. higher throughput TCM scheme should be designed while

retaining the s a a e pois<er eEciency.

Effect of the Power Split

We consider in this section the error probability of the sequential detection of uricoded

QPSK transmitted eyer a 2-ray Rayjeigh fading channel. The relative delay between

the two arrival paths is exactly equal to one symbol interval. We use in the followir~g

analysis the most dominant error event associated with the treilis diagrarn ir i Figure

2.4,

Assuming the transmitted sequence is the all-zero sequence in Figure 2.4. rI'i~is

implies that the corresponding sequence of complex symbols is

From the trellis-diagram in Figure 2.4; the most probable errorteous sequence is

A. EFFECT OF T H E POfVER SPLfT RATIO 106

which represents a 2-step error event. The corresponding A matrix is (see (2.12));

When the relative delay r is equal to one symbol interval, the parameter p in (2.27)

becomes unity. Consequenrly, each submatrix aij in (2.40) is a diagonal matrix. For

the two step error event listed above, the matrix GHH is the following 4x1 matrix:

Substituting the Iast two equation into (2.15) implies

which is a diagonal matrix. The cori-esponding eigenvalues are

Finally. by substieuri~g (-%A) - (-4.8) into (2.17) - (2.19), we get the followire approx-

ha t ion of the pairwise error probability of the above 2 step error event:

It is observed from (-4.91 that becausc of t l ~ e product of t hc i w t z e~ le i .~v ~t ' r~i ls

El and Ez, the error probabilitp decreases two order of m~gr~i t~ i t i c for vvcry 10 t i l f

increase in the torai received signal ro imise raiio (ticfind as i< = El $ k'2. seti iL'.;iJl).

Thrts, the frequeflC5- p!err_jr:e frdjEu art1121 introduces a sccortr! ttr<ic>r c!it-cr.-;i?;. e!reF:lc*? 0 ---

to the system performance. Moreover. the best perforrna~lre is achit*vtri w11tw t tw

total received energy is split equally betvieen two rays, i.e. tsftrn f,'t = J.'t.

Trellis structure of the 8 state,

rate 3/2 code

This appendix shov::s the trellis strrncitire of the 8 state, rate $32 code studied in . .

Section 3.2. The trellis structure rs gven in term ctf the current stair?; rlie transmitted

idornlation bits, the traesmfrted - QPSK symbols, the signal subsei label, and

the next state.

Signal Subset S e s t State

1 2 3 4 1 2 Q .J

4

.- .--L

.-

r- "I

"--. - ,-

;- ,n

r*

c'-? "?

-8

hi?

I-

Signal Subset 1 Xext State

a 2 1 1 5

B 2 2 6 &I 7 B12 8 B,, , 5 a 2 2 i

4 1

3

B12

B22 5 &I 6 8 1 2 7 BII 8

Trellis structures of 64 state codes

This appzndix shfiiw the trellis s t ructure of the 61 state, rate 3 j2 and t h e ti 1 state, rate

212 codes studied in Sectfotl 3.4. t'iP first define the signal set tftscriflr i i l Figure 3.4,

and then sho:~ ihe t r e k strilctures oi t he two trellis codes iri tcrrns of progrml~rting

codes. The p r o g r ~ m ~ i n g codes are wrirren in FORTRAN and sornrnents will iw givm

along the program.

Signal Set

c Daf ine the signal set described in Figure 3.4

c Tne veriable n u e is called signal-set which is a 2 dimensional

c arrzy. The first iadex iodicstes the number of signal set and

c the second index indicates the sequence of the transmitted

c s p b a l . B pit4 QPSK sipai is assigned to each variable. For

c exanple, signal-set(1, 1) = 0 neans that the first transmiited

c QPSK signal. in, signal. set 1 is 0.

signal-set(l, 1 ) = O

s ignal - s&( l , 22 = 1

signzl-set(2, 1) = 2

signal-set(2, 2) = 3

s ignal-sei(3 , l j = 2

signal-set(3, 2 ) = 7

signal-sstf4, Ij = O

signal-s&(4, 2) = 5

signal-setCS, 15 = 4

signal-set ( 5 , 2) = 5

signal-seti5, 1) = 6

signal-set(6, 21 = I

signal-set(7, I) = 6

signal-set(7, 2) = 3

signal-setl8, I5 = 4

si.pal-set(8, 2) = 1

signal-seti9, 1) = 0

signal-set(8, 2) = 7

signal-se%[lO, 1) = 2

signal-set(l0, 2) = 1

signal-set(11, Ij = 2

sip&set(ll, 2) = 5

sig~al-set(lz, P> =

siea-setCL2, 2j = 3

C. TRELLIS STRL'CTI-RES OF 6-1 SLATE CODES

signal-setfl3, I) = 4

signal-set(l3, 2) = 3

signal-set(l4, I) = 6

signal-set(l4, 2) = 5

signal-set(25, 1) = E

signal-set(i5, 2) = 1

signal-set(l6, 1) = 4

signal- set (16, 2) = 7

Beginning of 64 state, rate 3/2 code trellis structure. The

variabie n m e is called send-out-signal-set and it is a two

dimensioaaf as vell. The first index is the state number and

the second index is the transmitted information bits. For

example, send-out-signal-set(I, 0) = 1 means that if the

curreat state is I and the transmitted information bits are 000,

then the trellis encoder will send out the QPSK signals in

signal-set I. In other words, the two transmitted pi/4 -9PSK

symbols are 0 and 1, respectively.

C. TRELLIS STRCrCTt'P,ES OF 64 STATE CODES

send-out-signal-set (1, 6) = 7

send-out-signal-set (1, 7 ) = 8

send-out-signal-set (2, 0) = 9

send-out-signal-set (2, 1) = 10

send-out-signal-set (2, 2) = I1

send-out-signal-set (2, 3) = 12

send-out,sigr1al-set(2, 4) = 13

send-out-signal-setC2, 5) = 14

send-out-sisal-set (2, 6) = 15

send-out-signal-set (2, 7 ) = 16

send,out_signd_set(3, 0) = 2

send-out-signal-set (3, 1) = 1

send,out,signal-set (3, 2) = 4

send-out -signal-set (3, 3) = 3

send-out-signal-set(3, 4) = 6

send-out-signal-set (3, 5) = 5

send-out-signal-set(3, 6) = 8

send-out-signal-set (3, 7) = 7

send-out-signhi-set (4, 0) = 10

send-out-signal-set& 1) = 9

send-out-sisal-set (4, 2) = 12

send-ont-signal-set (4, 3j = I1

send-out-signal-set (4, 4) = 14

send-out-signal-set& 5) = 13

send-out-signaLset(4, 6) = 16

send-out-signal-set(4, 7) = 15

send-out-signal-set (5, 0) = 3

send-out-signal-set (5, 1) = 4

send,out,signal-set (5, 2) = I

send-ont-signal-set(5, 3) = 2

send-out-signal-set (5, 4) = 7

send-out-signal-set (5, 5) = 8

send-out-signal-set(5, 6) = 5

send-out-signal-set (5, 7) = 6

send-out-signal-set (6, 0 ) = 11

send-out-signal-set (6, 1) = 12

send-out-signal-set (6, 2) = 9

send,out_signal_set(6, 3) = 10

send-out-signal-set (6, 4) = 15

send-out-signd-set(6, 5) = 16

send-out-signal-set (6, 6) = 13

send-out-signal-set(6, 7) = 14

send-out_si@al_set(8, 0) = 12

send-out-sigaal-set& 11 = 11

send-out-signd-set(8, 2) = 10

send-out-signaLset(8, 3) = 9

send-out-signal-set (8 4) = 16

send-out-sigad-set (8, 5 ) = 15

send,out-sipd.-set18, 6) = 14

send-out-signal-set% 7) = 13

c use loops t o generate the r e s t of the t r e l l i s

do 100 state-number = 9, 16

base - s tah = s+,ate,n~amber - 8

send-out-sigzal-set (state-number , 0)

= send-out-signal-set (baseestate , 1)

send-mt-sip&-set (state-number , 1)

= send-o~t-s i~d-setCcase~state , Oj

send-out-sipal-set (state-mmber 2 j

send-03%-sipai-set (state-number, 5)

= send,out,signal-set(base,state, 7)

~end,out-s ig@.~set (state-number, 6)

= send-out-signal-set (base-state , 4)

send-cut-signal-set (state-number, 7)

= send-out-signal-set (base-state, 5)

110 continue

do 120 s t a t e ,~wibe r = 25, 32

base-state = state ,nwber - 24

send-out-signai-set(state-amber, 0)

= send-out-signal-set (base-state, 3)

send-out-signal-set (stateenumber, 1)

= send,ot;t,signal-set (base-state, 2)

send-out-signal-set (state-number, 2)

= send-ollt-signal-set(base,state, 1)

send-out-signd-set(state-number, 3)

= send-out-signal-setbse-state, 0)

send-out-signal-set (state-number , 4)

= send-out-signal-set (base-state, 7)

send-out-signal-set (state-number, 5)

= send-out-signal-set (base-state, 6)

send-out--sipal-set (state-number , 6 )

= send,o~~%-signal-set Cbase-state, 5)

send-out-signal-set (state-number , 7)

= send-out-signal-set(base-state, 4)

120 continue

do 130 state,nu&er = 33, 40

base-stzte = state-number - 32

send-ant-signzl-set (stat e-number , 0)

= send-out-signal-set(base-state, 4)

send-out-signal-set (state-number , 1)

= send-out-signal-set (base-state, 5)

send-out-signal-set (state-number, 2)

= send-out-signal ,set (barnstate, 6)

send-cut-signal-set ( state-number, 3)

= send-out-signal-set (base-state, 7)

send-out-signdl-set(state-number, 4)

= send-out-signal-set (bascstate , 0)

send-out-signal-set (state-number , 5)

= send-ozt-signal ,set (basestate, 1)

send-out-signal-set (state-number , 6)

= send-ou%-signal-set (base-stat e , 2)

send-out-signal-set (state-number , 7)

= send-out-signal-set (base-state, 3)

130 continue

do 140 state-naber = 41, 48

base-state = stzte-number - 40

send-out-signsl-set ( s tz te - rmber , 0)

= sesd-out-signal-set(base-state, 5)

send-out-signal-set ( s t a t e-number , 1)

- - send-cnt-signal-set (base-state, 4)

send-out -signal,set ( s t a t e-number , 2)

= send-out-signal-set (base-state , 7)

send-out-signal-set (state-number, 3)

= send-out-signal-set (base-state, 6)

sen&-out-sigrtd-set (state-number, 4)

= send-out-signal-set (base-state, 1)

send-out-signal- s e t (state-number, 5)

= send-om-signal-set(5ase-state, 0)

serid-out-signal-set (state-number , 6)

= send-out-signal-set C~ase-s ta te , 3)

send-out-signal-set h t e - o m b e r , 7)

= send-out-sipaf-set(base-state, 2)

140 continue

do 150 state-rider = 49, 56

base-state = state-number - 48

send-out - s i p a l , s e t (stateenumber, 0)

= send-out-signal-set (base-state , 6)

send-out-s isal-set (state-oumber, 1)

= send-out-signal-set (base-state, 7)

send-out-s ipal-set (state-number , 2)

C. TRELLIS STR ITTI-RE5 OF 64 ST-4 TE CODES

= send-out-signal-set(base-state, 4)

send-out-signal-set (stateenamber, 3)

= send-ont -s ipa l - se t b a s e - s t a t e , 5)

send-out - s ignaLse t ( s t a t e-number , 4)

= send-ont-signal-set (base-state, 2)

send-out-signal-set (state-nnmber, 5)

= se~d-cut -s igna l -se t (base-state, 3)

send-out-signal-set ( ~ t a t e ~ ~ ~ m b e r 6)

= send-out-sipal,set(jase-state, 0 )

send-out -s i sa l - se t (state-number , 7)

= send-out-signal-set (base-state, I)

i50 continue

63 150 state-nmber = 57, 64

base-state = s ta te -nmber - 56

send-out-signal-set (s~;ate-n&"uber, 0)

= send-out-signal-set (base-state, 7)

send-oat-sip-a-set (state-nwber , 1)

= send-out-s ipal-set {biise-state, 6 )

send-out-signal-set (state-amberJ 2 )

= se?id,o~t-s igaal~set (base-state, 5)

send-ol_~t-sign&,set(state-nmber, 3)

= senti?-ont-signaI-set cease-st a t e 4)

secd-out,siga2-se-t is'.,ate-n-mber , 4)

= send-am-s ipa i -se t f 'xse-s tate , 3)

c Now, def ine the destination state of each trmsition

c End of the trellis strrrcture

c Beginning of 54 state, rate 212 code tref fis stntlcture. The

c variabfe a w e is czlfed send-out-sigaal-set a d it is a t m

c 85aensiccd zs sell. first index is the state n1mber and

c the second k B e x is the trzllsmitted infomat ion bits

c For e x a p i e , se~d-o~t,slpzL-set(l, 0) = 1 nems that

c use loops to generate the re s t of the t r e l l i s structure

do 100 s t a t e - ~ . ~ b e r = 5 , 8

base-state = state-number - 4

send-out-signal-set (state-number , 0)

= send-ozt-signal ,set (base-state, 1)

send,out-~ii;l iaI~set (state-number, 1)

= send-out-signal-set (base-state, 0)

send-oat-signal-set (state-number , 2)

= send-out-signal-set (base-state, 3)

send-cut-signal-set (state-number, 3)

= send-out,signal,set (base-state, 2)

100 cont ime

do 110 state-nmber = 9, 12

base-state = state-number - 8

secd,c\it,sipd-se.+,(state-number, 0)

= send-out-sipal,set(base,state, 2)

se~d-oat_sii;=~aP-set (stateenumber, 1)

= secd-ozt-sipd,set(5ase_state, 3)

send -o .~ t - s i p~ - se t~s t a t e~number , 2)

= send-out-signal-set (base-state, 0)

serrd,a'r;t,sigml_set (state-number, 3)

= send-oot-signal-set (base-state, 1)

110 rontinne

do 120 state-ember = 13, 16

base-stzte = s t a t e - n u b e r - 12

send-out-signal-set (state-number, 0)

= send-o.;t,signal-set (base-state, 3)

send-out-signal-set ( s t a t e -nube r , I)

= send-out-s ipal-set (base-stat e , 2)

send-out-signal-set (state-number , 2)

= seob-orit-signal-set (base-stat e , 1)

send-out-signal-set ( s t a t e-number , 3)

= send-out-signal-set (base-state, 0)

120 continue

do 130 state-number = 17, 32

base-state = state-number - 16

send-out-sigaaf-set (state-number, 0)

= send-out-signal-set (base-state, 1)

send-out-signal-set (state-number , 1)

= send-out-signal-set (base-state , 0)

send-out-signal-set (state-number , 2)

= send-out-signal-set (base-state, 3 )

send-out-signal-set [state-nwber , 3)

= send-out-s ipaf - se t {base-state, 2)

13C co;ltinlre

do 140 sttZo-n-mber = 33, 48

base-state = state-number - 32

send-oxt-sip&-set(sta2e-number, 0 )

= send-o~t-sipal-set(base-state, 2)

send,out,s4pdl,set (state-number, I)

= sezd-03%-sipal,set(base-state, 3)

send-ou t - s ipd - se t (stztexumber , 2)

= srad,oz%,sipal-set (base-stat e , 0)

send-oat-sipal-set (state-number, 3)

= seod-out-signal-set (base-state, 1)

140 continua

do 150 s ta te -nmber = 49, 84

bzse-stzte = s+,a+,e,o*x&er - 48

send-out-signal-set ( s t a t s-number , 0)

= send ,o :s+ , , s ip~-se t (base-state, 3)

send,oat;,sigsal.-set (state-number, 1)

= send-on";-sign&-setCbase-state, 2)

send-oat-s ipdl-set (state-number , 2)

= send-a~t-sip&-set(base-state, 1)

send-o~t-s ignal-set (state-number, 3)

= sem5,out-sipdl-set (base-st a t e , 0)

I50 continue

-- c ,Yo%, w e define the des t ina t ion of each t r a n s t i o n

6. TRELLIS STRPCTLTRES OF 64 S r U E CODES

do 160 state-nwber = 1, 64

do I70 trmsnit-number = 0, 3

remaioder = nod((state,number - I), 16)

next-atate(state-nmber , transmit-number)

t = reaainder*4 + transmit-number + I

170 continue

160 continue

c end af the structure trellis

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