UNIVERSITY OF CONCORDIA

FINAL PROJECT

ELEC6141 - WIRELESSCOMMUNICATIONS

Simulation of A Communications System UsingMatlab

Author:Le Thanh TAN

ID: 6389953INRS-EMT

Instructor:Dr. Yousef R. SHAYAN

University of Concordia

June 16, 2011

1

CONTENTS

I Introduction 1

II System Models and Theoretical Analysis 2II-A The communications system over an AWGN channel . . . . . . . .. . . . . . . . . 2II-B BCH coding scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3II-C A slow fading channel - Rayleigh model . . . . . . . . . . . . . . . .. . . . . . . 5II-D An interleaving scheme . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 7

III Numerical Results 8

IV Conclusion 11

Appendix A: Code for a AWGN channel 11

Appendix B: Code for a AWGN channel and a BCH coding 13

Appendix C: Code for a Rayleigh channel 17

Appendix D: Code for a Rayleigh channel and a BCH coding 18

Appendix E: Code for a Rayleigh channel, a BCH coding and an interleaving scheme 20

References 22

I. I NTRODUCTION

In a communications system, the channel is affected by an additive white Gaussian noise (AWGN)and a fading due to a distance between a transmitter and a receiver. Especially, there are many kinds ofchannel fadings. Depending on the moving speeds of transmitters or receivers, a fading type can be a slowfading or a fast fading (i.e., the product of 0.1 and coherence time than smaller or larger than the symbolperiod of signal are corresponding to fast and slow fadings). Moreover, a channel can be referred as aselective fading or a flat fading corresponding to the product of 0.1 and coherence bandwidth than smalleror larger than the bandwidth of signal. These above effects can suffer received signals at a destination.Hence the performance of received signals in term of bit-error-rate (BER) is much degraded.

In order to overcome these issues, communications systems would be carefully designed. In detail,application systems operating over the AWGN channels would use coding schemes to combat an additivewhite noise. However, if environment is affected by fading,coding techniques only solve a fast fading.It implies that, coding schemes degrade received signals when they go through slow fading channels. Inthis case, an interleaving technique would be added to a communications system. In order to overcomethe fading channels, besides, using an interleaver as above, we can exploit the diversity of multi-path. Itimplies that the effects of fading can be combated by transmitting the original signals over multiple paths(experiencing independent fading) and then combining all received signals at the receiver. There are manykinds of diversities to mitigate this issue, such as diversity in time, frequency, and space. Correspondingly,a lot of state-of-art methods are given, viz. diversity receiving and transmitting, OFDM, space-time blockcodes, MIMO, Cooperation and etc.

In summary, the main scope of this report is modeling a communications system. First, I create abasic communications system, where it includes the modulation/demodulation using a QPSK modulation,a channel type is an AWGN channel. Secondly, a coder/decoder scheme is added to a transmitter/receiver toimprove received signals. Thirdly, the fading channel is considered when a receiver/transmitter is moving.It means that the slow fading is mentioned. The performance is shown to prove that the received signal

2

is degraded whether a coding scheme is used or not. Finally, an interleaver/deinterleaver is used to solvethis problem.

Besides, the performance in terms of BER is used to verify a validity of these above techniques in acommunications system.

II. SYSTEM MODELS AND THEORETICAL ANALYSIS

A. The communications system over an AWGN channel

The practical communications systems can be modeled by two following cases such as passband andbaseband models corresponding to Figs. 1 and 2. It is referred that both models are equivalent [2], hencefrom now, the baseband model is analyzed. Let’s summarize the operation of this model. First, an originaldata streamsk(t) = s0, s1, s2, s3, . . . is an input of a QPSK modulation, where bipolar pulses are presentedfor bit sequences. A mapping scheme divides a data stream to an in-phase stream,dI (t), and a quadraturestream,dQ (t), given as

sI (t) = s0, s2, s4, . . . (even bits) , (1)

sQ (t) = s1, s3, s5, . . . (odd bits) . (2)

In Fig. 1, an orthogonal realization of a QPSK waveform is presented as

sTrans (t) =1

√

(2)sI (t) cos (2πf0t)−

1√

(2)sQ (t) sin (2πf0t) , (3)

while an output symbol in Fig. 2 is given as

sSymb = sI (t) + jsQ (t) . (4)

QPSK demodulation

Mapping +

I

jQ

Input

data

stream+

AWGN

Detection

Output

data

stream

QPSK modulation

+

+

cos 2 cf t

sin 2 cf t !

LPF+

cos 2 cf t

+

sin 2 cf t !

LPF

I

Q

Fig. 1. The passband communications model.

The mapping figure is shown in Fig. 3, whered = 1, Eb = 1, henceEs =1

4(4 ∗ 2d2) = 2Eb. We also

note that a modulation and demodulation using the common function presented in Table IIn this subsection, an AWGN channel is considered as Fig. 4. The received signal is

r = s+ n, (5)

wheres andn are the transmitted signal and Gaussian noise, respectively. It is shown in Appendix A thatn is zero mean Gaussian variable with variance ofN0

2for each I and Q components. Theoretical BER of

QPSK modulation is given as [2]

PAWGNb = Q

(

√

2Eb

N0

)

, (6)

whereEb/N0 is signal-to-noise ratio, SNR. Due to AWGN, received signals are lost. Hence a codingscheme is added in order to correct the error at the receiver.

3

QPSK demodulation

Mapping +

I

jQ

Input

data

stream+

AWGN

Detection

Output

data

stream

QPSK modulation

Fig. 2. The baseband communications model.

TABLE ITHE COMMON FUNCTION USED FOR MODULATION/DEMODULATION

x y I Q

0 0 d d1 0 -d d1 1 -d -d0 1 d -d

Fig. 3. The mapping of a QPSK modulation.

B. BCH coding scheme

(15,11,1) BCH coding scheme (i.e.,n = 15, k = 11) is mentioned in this report. In Fig. 5, BCH coderis added before modulator and BCH decoder is after demodulator. The relation between the bit rate andthe symbol rate when using BCH coding scheme is given as

Rc =n

kRb =

15

11Rb, (7)

while

Rs =Rc

logM2

=Rc

2. (8)

4

QPSK demodulation

Mapping +

I

jQ

Input

data

stream+

AWGN

Detection

Output

data

stream

QPSK modulation

Comparison to find BER

Fig. 4. The communications model over a AWGN channel.

Hence,

Rs =15

22Rb. (9)

The new signal-to-noise ratio isEc

N0

= kEb

nN0

. Hence, theoretical BER is calculated as

PBCHb =

1

n

n∑

j=t+1

j

(

n

j

)

Pjc (1− Pc)

n−j, (10)

wherePc = Q(√

2Ec

N0

)

, t = 1.

QPSK demodulation

Mapping +

I

jQ

Input

data

stream+

AWGN

Detection

Output

data

stream

QPSK modulation

Comparison to find BER

Coder Decoder

Fig. 5. The communications model using a coding scheme over a AWGN channel.

In the followings, an implementation of coding scheme is presented.

5

First, the coder is shown, where the generator matrix of (15,11,1) BCH coding scheme is generated as

G =

111110000000000

011101000000000

101100100000000

110100010000000

111000001000000

001100000100000

010100000010000

011000000001000

101000000000100

100100000000010

110000000000001

, (11)

whereG =

[

P...Ik

]

, P is the parity matrix,Ik is the eye matrix. It is also noted that the creation ofG is

illustrated in Appendix B. I omit how to create G due to the limited pages.Secondly, decoder is briefly shown. The syndrome can be received as

S = rHT , (12)

whereH =

[

In−k

...PT

]

which is calculated fromGHT = 0 (note that the creation ofH is shown in

Appendix B), r is the received signal,H is the parity check matrix. Then syndrome table is created,where it includes zero, one and two error patterns (i.e.,t = 1). Besides, the syndrome table has 16rows and2n−k = 16 syndromes. Based on the syndrome table, each received codeword is referred to thecorresponding error pattern on the table. Hence, the received signal can be corrected.

C. A slow fading channel - Rayleigh model

In Global System for Mobile Communications (GSM), the most concerned issue is a fading effect.There are many kinds of fading such as selective fading vs flatfading and slow fading vs fast fadingwhich are clearly presented in [2], [3]. In this report, Rayleigh fading in Fig. 6 is considered, wherethe magnitude of a signal that is transmitted on this channelrandomly varies following the Rayleighdistribution. In other words, the received signal is the combination of multipath reflected signals exceptthe direct signal.

Fig. 7 illustrates the communications system over slow Rayleigh fading channel which is the main pointfor fading effects in this report. The following is a procedure to recover the original transmitted signaland calculate theoretical BER for the case that communications channel is a Rayleigh fading type. Theenvelope of channel response follows the Rayleigh distribution and given as

Pr (r) =2r

σ2exp

(

−r2

σ2

)

, r ≥ 0. (13)

The received signal is

r = h ∗ s+ n, (14)

where s is the transmitted signal,n is AWGN, andh is the multiplicative noise caused by Rayleighfading of the channel. It is shown in Appendix C thath is zero-mean complex Gaussian variable withunit variance (each a half is for each dimension). Furthermore, the number of coefficient channels would

6

Fig. 6. Rayleigh fading channel.

QPSK demodulation

Mapping +

I

jQ

Input

data

stream+

AWGN

Detection

Output

data

stream

QPSK modulation

Comparison to find BER

Rayleigh

fading

Devided by

channel

coefficients

Fig. 7. The communications model over a slow fading channel.

be carefully mentioned. Recall that bit rate isRb = 1Mbps, carrier frequency isfc = 5.4545GHz andmaximum moving speed of mobile user isv = 81km/hour. Hence the coherence time and symbolduration are

Tc =0.423

fm=

0.423c

vfc= 1034µsec, (15)

Ts = 2Tb =2

Rb

= 2µsec, (16)

wherefm = vλ= vfc

cis Doppler spread. So, the ratio ofTc

Ts= 517. As the results, there are 517 symbols

in the one block of slow fading. It is assumed that the perfectknowledge of channel state information isapplied. It implies that the channel estimation has no errorand hence the channel coefficients is perfectivelyknown at the receiver side. So the signal input of QPSK demodulator is obtained by a received signaldivided to coefficient channels.

Now theoretical BER of a communications system over Rayleigh fading channel is approximated as[1]

PRayleighb =

1

2

1−

√

√

√

√

Eb

N0

Eb

N0

+ 1

, (17)

7

or

PRayleighb =

1

4

Eb

N0

. (18)

Especially, a coding scheme in Fig. 8 is not exploited to overcome a fading problem. This issue will befurther explained in the simulation results in Section III.Also in Appendix D, there is a minor changein the number of symbols in one box of a slow fading channel, which is given as

Tc

T codeds

= 705, (19)

whereTc = 1034µsec (same as in II-C),T codeds = 11

15Ts = 1.4667µsec. However, fading effects can be

QPSK demodulation

Mapping +

I

jQ

Input

data

stream+

AWGN

Detection

Output

data

stream

QPSK modulation

Comparison to find BER

Rayleigh

fading

Devided by

channel

coefficients

DecoderCoder

Fig. 8. The communications model using a coding scheme over a slow fading channel.

reduced by using diversities which are further discussed innext subsection.

D. An interleaving scheme

As being known that fading effects can be reduced by exploiting diversities. There are many kinds ofdiversities including time, frequency and space diversities. However, due to limit number of pages, thissubsection only consider one simple case in terms of an interleaving scheme which is illustrated in Fig. 9.As known in BCH coding scheme, all the bits of one codeword see the same fading coefficient. If it is

QPSK demodulation

Mapping +

I

jQ

Input

data

stream

+

AWGN

Detection

Output

data

stream

QPSK modulation

Rayleigh

fading

Devided by

channel

coefficients

DecoderCoder 1b 2b 3b

16b 17b 18b

31b 32b 33b

15 14db 15 13db 15 12db

. . .

. . .

. . .

15b

30b

45b

15db

. . .

. . .

. . .

. . .

. . .

. . .

d r

ow

s

CW1

CW2

CW3

CWd

Codeword read

into interleaver by

rows

Codeword read

out of interleaver

by columns

1b 2b 3b

16b 17b 18b

31b 32b 33b

15 14db 15 13db 15 12db

. . .

. . .

. . .

15b

30b

45b

15db

. . .

. . .

. . .

. . .

. . .

. . .

CW1

CW2

CW3

CWd

Codeword read into

interleaver by columns

Codeword read out of

interleaver by rows

Fig. 9. The communications model using a coding scheme and an interleaving technique over a slow fading channel.

8

affected by a specific deep fading box, all of them are corrupted. Because (15,11,1) BCH code can onlycorrect 1 error bit in each codeword. Furthermore, one fading box include 705 symbols. Hence if onespecific box is in deep fading, all symbols inside this duration are affected. So this code would add evenmore errors and the performance significantly degraded.

Motivated from this above feature, the smart scheme is proposed. It is good if different fading coefficientsare applied into different bits within a codeword. Hence, some of bits are affected by deep fading and theothers are still remained. As the results, error coding in terms of BCH coding can recover the receivedsignal. As seen in Fig. 9, each bit will be separated from the others in the same codeword byd− 1 bits.It is concluded that if this separation is larger than the channel coherence time, the fading affecting tocodewords is independent. It is handled by the following condition

d >Tc

T codedb

, (20)

whered is the deep of interleaving. So in a communications system, an interleaver is added before amodulator and a deinterleaver is added after a demodulator.

III. N UMERICAL RESULTS

0 1 2 3 4 5 6 7 810

−4

10−3

10−2

10−1

Eb/N

0, dB

Bit

Err

or R

ate

Bit error probability curve for QPSK

theory−QPSKsimulation−QPSK

Fig. 10. BER performance of a communications system using a QPSK modulation over an AWGN channel.

In this section, the simulation results are presented and further discussions are provided. Recall that thecommunications system uses QPSK modulation, and (15,11,1)BCH coding scheme. AWGN channel isfirst studied. Then Rayleigh fading channel is further examined due to the fact that the users are movingunder the condition of maximum moving speed,v = 81km/hour. The bit rate of communications systemis Rb = 1Mbps and the carrier frequency isfc = 5.4545GHz.

First of all, Fig. 10 illustrates the simulation results of the communications system over AWGN channel.It is clearly seen that the simulation and theoretical BERs arefitted.

Then, the coding scheme is added to communications system inorder to improve the performance interms of BER. In Fig. 11, BER of coding scheme is decreased significantly whenEb

N0

larger than7dB.For example, atEb

N0

= 8dB, BER for coding case is 10 times less than BER for without coding. However,at low Eb

N0

, coding does not work well. For instance, atEb

N0

= 0dB, BERs for coding and without codingcases are the same values.

9

0 1 2 3 4 5 6 7 8

10−6

10−4

10−2

100

Eb/N

0, dB

Bit

Err

or R

ate

Bit error probability curve for QPSK

theory−QPSKsimulation−QPSKtheory−QPSK−codingsimulation−QPSK−coding

Fig. 11. BER performance of a communications system using a QPSK modulation and a (15,11,1) BCH coding scheme over an AWGNchannel.

0 2 4 6 8 10 12 14 16 18 20

10−6

10−4

10−2

100

Eb/N

0, dB

Bit

Err

or R

ate

Bit error probability curve for QPSK

theory−QPSK−AWGNsimulation−QPSK−AWGNtheory−QPSK−fadingsimulation−QPSK−fading

Fig. 12. BER performance of a communications system using a QPSK modulation over a slow fading channel.

Now fading is considered. Fig. 12 shows the performance of communications system. It is easilyobserved that simulation BER is fitted with theoretical BER forthe case of fading channel. Also therelation between BER andEb

N0

(SNR) for the fading channel is a linear decrease while the relation forAWGN channel is an exponential decrease. It implies that although SNR is increased, BER performancefor fading channel is not much more improved. For example, even thoughEb

N0

= 20dB, BER of fadingchannel is still larger than10−3. So the need of technique to combat the fading effects is essential.

Let’s try to use coding scheme to overcome the fading effects. In Fig. 13, the results are not expected andsurprised. BER performance using coding is even dramatically degraded and higher than BER performancewithout using coding. It is due to the fact that when using coding, some boxes of fading channel aredeeply faded. Hence, all the whole codewords within these boxes get errors. Furthermore, BCH coding

10

0 2 4 6 8 10 12 14 16 18 20

10−6

10−4

10−2

100

Eb/N

0, dB

Bit

Err

or R

ate

Bit error probability curve for QPSK

simulation−QPSK−AWGNtheory−QPSK−AWGNtheory−QPSK−fadingsimulation−QPSK−fadingsimulation−QPSK−fading−coding

Fig. 13. BER performance of a communications system using a QPSK modulation and a (15,11,1) BCH coding scheme over a slow fadingchannel.

0 2 4 6 8 10 12 14 16 18 20

10−6

10−4

10−2

100

Eb/N

0, dB

Bit

Err

or R

ate

Bit error probability curve for QPSK

theory−QPSK−AWGNsimulation−QPSK−AWGNtheory−QPSK−fadingsimulation−QPSK−fadingsimulation−QPSK−fading−codingsimulation−QPSK−interleaver

Fig. 14. BER performance of a communications system using a QPSK modulation, a (15,11,1) BCH coding scheme and an interleavingtechnique over a slow fading channel.

can only correct one bit for each codeword and cannot recoverfor the case of burst errors above. As theresults, the errors are significantly increased in this case.

Motivated from finding solution for this above issue, an interleaving scheme which is one of the mostfamous diversity schemes is applied to a communications system. As seen in Fig. 14, an improvementis remarkably received. For example, atEb

N0

= 20dB, BER performance using an interleaver is 100 timesless than BER performance without using an interleaver. Alsoat this SNR, BER performance reach10−5

that is acceptable for a practical communications system.

11

IV. CONCLUSION

In this report, the communications system is simulated by using Matlab, where AWGN and Rayleighfading channels are used. In order to combat AWGN, (15,11,1) BCHcoding scheme is exploited. However,this coding scheme does not work well for Rayleigh fading channel. Hence an interleaving scheme isproposed to handle a slow fading effect. Finally, the simulation results show the improvements of BCHcoding over AWGN channel and an interleaving scheme over Rayleigh fading channel.

APPENDIX ACODE FOR AAWGN CHANNEL

%% Author and his affiliation% This program is implemented by Le Thanh Tan% Ph.D Student% Institut National de la Recherche Scientifique (INRS)% University of Quebec, Montreal, QC, H5A 1K6, Canada% Email: lethanh@emt.inrs.ca%%clc;clear all;close all;pause(0.05)N all = [517*10̂ 3*ones(1,6) 517*10̂3*ones(1,5)];% we can change NEb N0 dB = [0:2:20]; % multiple Eb/N0 valuesfor ii = 1:length(Eb N0 dB)

iiN = N all(ii);s data = (1/sqrt(2))*rand(1,N)>0.5;%% Modulationip = qpsk(s data);s = ip; % normalization of energy to 1Noise1 = 1/sqrt(2)*[randn(1,N/2) + j*randn(1,N/2)];% white guassian noise, 0dB variance%% Add noisey = s + 10̂ (-Eb N0 dB(ii)/20)*Noise1;% additive white gaussian noise%% Demodulationshat = detectqpsk(y);nErr(ii) = size(find([s data- shat]),2);% couting the number of errorsa=find([s data- shat]);

end

simBer QPSK = nErr./Nall;theoryBer QPSK = 0.5*erfc(sqrt((10.ˆ (Eb N0 dB/10)))) ;

close allfiguresemilogy(EbN0 dB,theoryBerQPSK,’b.-’ );hold onsemilogy(EbN0 dB,simBer QPSK,’mx-’ );axis([0 20 10̂ -7 1])grid onlegend(’theory-QPSK’, ’simulation-QPSK’);

12

xlabel(’Es/No, dB’)ylabel(’Bit Error Rate’)title(’Bit error probability curve for QPSK’)save Quesadata simBerQPSK theoryBerQPSK Eb N0 dB% save Quesadata1 simBerQPSK theoryBerQPSK Eb N0 dB

where ’qpsk.m’ is%% Author and his affiliation

% This program is implemented by Le Thanh Tan% Ph.D Student% Institut National de la Recherche Scientifique (INRS)% University of Quebec, Montreal, QC, H5A 1K6, Canada% Email: lethanh@emt.inrs.ca%%function d = qpsk(b)% ˆ% —% 10 x — x 00 (odd bit, even bit)% —% ——-+——-> real part (I channel)% —% 11 x — x 01% —%% Input:% b = bits{0, 1} to be mapped into QPSK symbols%% Output:% d = complex-valued QPSK symbolsfor i =1:length(b)/2

if b(2*i-1)==1d(i) =1 ;

elsed(i) = -1;endif b(2*i)==1

d(i)=d(i)+j;elsed(i)=d(i)-j;end

end

and ’detectqpsk.m’ is%% Author and his affiliation% This program is implemented by Le Thanh Tan% Ph.D Student% Institut National de la Recherche Scientifique (INRS)% University of Quebec, Montreal, QC, H5A 1K6, Canada% Email: lethanh@emt.inrs.ca%%

13

function bhat = detect(r)

% Assumed mapping:%% 10 x — x 00% —% ——-+——-% —% 11 x — x 01%% Input:% r = sequence of complex-valued QPSK symbols%% Output:% bhat = bits{0,1} corresponding to the QPSK symbolsL = length(r);bhat =[];for i=1:L

if real(r(i))>=0if imag(r(i))>=0

bhat = [bhat,[1,1]];elsebhat = [bhat,[1,0]];end

elseif imag(r(i))>=0

bhat = [bhat,[0,1]];elsebhat = [bhat,[0,0]];end

endend

APPENDIX BCODE FOR AAWGN CHANNEL AND A BCH CODING

%% Author and his affiliation% This program is implemented by Le Thanh Tan% Ph.D Student% Institut National de la Recherche Scientifique (INRS)% University of Quebec, Montreal, QC, H5A 1K6, Canada% Email: lethanh@emt.inrs.ca%%clc;clear all;close all;pause(0.05)N all = [517*10̂ 3*ones(1,6) 517*10̂3*ones(1,5)];% we can change NEb N0 dB = [0:2:20]; % multiple Eb/N0 values%% Prepare for coder and decoder

k=11;

14

n=15;g1=[1 0 0 1 1];G=Gformulation(k,g1);g=[1 0 0 1 1 0 0 0 0 0 0 0 0 0 0];r=n-k;H=Hformulation(k,G,r);HT=H.’;for ii = 1:length(Eb N0 dB)

iiN = N all(ii);s data = (1/sqrt(2))*rand(1,N)>0.5;% normalization of energy to 1s data1 = reshape(sdata,11,floor(N/11))’;for i=1:size(s data1,1),

tan temp = s data1(i,:)*G;tan temp = mod(tantemp,2);bch1(i,:) = tantemp;

ends data2 = reshape(bch1’,1,size(bch1,1)*size(bch1,2));%% QPSK modulationip = qpsk(s data2);s = ip;N1 = length(s);Noise1 = 1/sqrt(2)*[randn(1,N1) + j*randn(1,N1)];% white guassian noise, 0dB variance

y = s + sqrt(n/k)*10̂ (-Eb N0 dB(ii)/20)*Noise1;% additive white gaussian noise%% QPSK demodulationshat = detectqpsk(y);%% decodershat1 = reshape(shat,15,floor(length(shat)/15))’;for i=1:size(shat1,1),

bch2(i,:)=syndtan(shat1(i,:),g,HT);bch3(i,1:11)=bch2(i,1:11);

endshat2 = reshape(bch3’,1,size(bch3,1)*size(bch3,2));%% Error number calculationnErr(ii) = size(find([s data- shat2]),2);% couting the number of errorsa=find([s data- shat2]);

endsimBer QPSK = nErr./Nall;P nocode = 0.5*erfc(sqrt((k/n*10.ˆ (Eb N0 dB/10))));for i =2:n,

Ptem(i,:)=i*nchoosek(n,i)*(Pnocode.̂ i).*((1-P nocode).̂ (n-i));endtheoryBer QPSK = 1/n* sum(Ptem);figuresemilogy(EbN0 dB,theoryBerQPSK,’b.-’ );hold onsemilogy(EbN0 dB,simBer QPSK,’mx-’ );axis([0 20 10̂ -7 1])

15

grid onlegend(’theory-QPSK-coding’, ’simulation-QPSK-coding’);xlabel(’Es/No, dB’)ylabel(’Bit Error Rate’)title(’Bit error probability curve for QPSK’)save Quesbdata simBerQPSK theoryBerQPSK Eb N0 dB

where ’Gformulation.m’ is%% Author and his affiliation

% This program is implemented by Le Thanh Tan% Ph.D Student% Institut National de la Recherche Scientifique (INRS)% University of Quebec, Montreal, QC, H5A 1K6, Canada% Email: lethanh@emt.inrs.cafunction G=Gformulation(k,g)row=k;n=length(g);col=row-1+n;A=zeros(row,col);A(end,row:end)=g;

for i=1:row-1A(end-i,row-i:end-i)=A(end,row:end);

endfor j=row:-1:2

for i=j-1:-1:1if(A(i,j) >0)C1=A(i,:);C2=A(j,:);C=C1+C2;B=mod(C,2);A(i,:)=B;

endclear C1 C2 C B

endendG=A;

’Hformulation.m’ is%% Author and his affiliation

% This program is implemented by Le Thanh Tan% Ph.D Student% Institut National de la Recherche Scientifique (INRS)% University of Quebec, Montreal, QC, H5A 1K6, Canada% Email: lethanh@emt.inrs.cafunction H=Hformulation(k,G,r)G1=G(:,end-r+1:end);G1=G1’;

16

Ir=eye(r);H=[G1,Ir];

’synd tan.m’ is%% Author and his affiliation

% This program is implemented by Le Thanh Tan% Ph.D Student% Institut National de la Recherche Scientifique (INRS)% University of Quebec, Montreal, QC, H5A 1K6, Canada% Email: lethanh@emt.inrs.cafunction[bch]=syndtan(data,G,HT)k=0;for i=1:15

if data(i)==1k=i;break

endendg=circshift(G,[0 k-1]);g1=g;c=data;for n=1:15

c=xor(c,g1);h=0;for i=1:15

if c(i)==0h=h+1;

endendif h==15

breakendg1=G;for i=1:15

if c(i)==1kk=i;break

endendif kk>11

breakendg1=circshift(g1,[0 kk-1]);

endk1=0;for i=1:15

if c(i)==1k1=k1+1;

end

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endif k1==0

bch=data;else

for i=1:4s(i)=c(11+i);

endfor j=1:15

ht(j,:)=xor(s,HT(j,:));k2=0;for i=1:4;

if ht(j,i)==0k2=k2+1;

endif k2==4

k3=j;break

endend

endfor i=1:15

e(i)=0;ende(k3)=1;bch=xor(data,e);

end

APPENDIX CCODE FOR ARAYLEIGH CHANNEL

%% Author and his affiliation% This program is implemented by Le Thanh Tan% Ph.D Student% Institut National de la Recherche Scientifique (INRS)% University of Quebec, Montreal, QC, H5A 1K6, Canada% Email: lethanh@emt.inrs.ca%%clc;clear all;close all;pause(0.05)N all = [517*10̂ 3*ones(1,6) 517*10̂3*ones(1,5)];% we can change NEb N0 dB = [0:2:20]; % multiple Eb/N0 valuesfor ii = 1:length(Eb N0 dB)

iiN = N all(ii);s data = (1/sqrt(2))*rand(1,N)>0.5;%% Modulationip = qpsk(s data);

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s = ip; % normalization of energy to 1Noise1 = 1/sqrt(2)*[randn(1,N/2) + j*randn(1,N/2)];% white guassian noise, 0dB varianceh = 1/sqrt(2)*[randn(1,N/2/517) + j*randn(1,N/2/517)];% Rayleigh channelh = h’*ones(1,517);h = reshape(h’,1,size(h,1)*size(h,2));%% Add noisey = h.*s + 10̂ (-Eb N0 dB(ii)/20)*Noise1;% additive white gaussian noise%% Equalizationyeq = y./h;%% Demodulationshat = detectqpsk(yeq);nErr(ii) = size(find([s data- shat]),2);% couting the number of errorsa=find([s data- shat]);

endsimBer QPSK = nErr./Nall;EbN0Lin = 10.̂ (Eb N0 dB/10);theoryBer QPSK = 0.5.*(1-sqrt(EbN0Lin./(EbN0Lin+1)));theoryBerAWGN = 0.5*erfc(sqrt(10.ˆ (Eb N0 dB/10)));close allfiguresemilogy(EbN0 dB,theoryBerQPSK,’b.-’ );hold onsemilogy(EbN0 dB,simBer QPSK,’mx-’ );hold onsemilogy(EbN0 dB,theoryBerAWGN,’r*-’ );axis([0 20 10̂ -7 1])grid onlegend(’theory-QPSK-fading’, ’simulation-QPSK-fading’,’theory-QPSK-AWGN’);xlabel(’Es/No, dB’)ylabel(’Bit Error Rate’)title(’Bit error probability curve for QPSK’)save Quescdata simBerQPSK theoryBerQPSK theoryBerAWGN EbN0 dB

APPENDIX DCODE FOR ARAYLEIGH CHANNEL AND A BCH CODING

%% Author and his affiliation% This program is implemented by Le Thanh Tan% Ph.D Student% Institut National de la Recherche Scientifique (INRS)% University of Quebec, Montreal, QC, H5A 1K6, Canada% Email: lethanh@emt.inrs.ca%%clc;clear all;close all;pause(0.05)N all = [517*10̂ 3*ones(1,6) 517*10̂3*ones(1,5)];% we can change NEb N0 dB = [0:2:20]; % multiple Eb/N0 values

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%% Prepare for coder and decoderk=11;n=15;g1=[1 0 0 1 1];G=Gformulation(k,g1);g=[1 0 0 1 1 0 0 0 0 0 0 0 0 0 0];r=n-k;H=Hformulation(k,G,r);HT=H.’;for ii = 1:length(Eb N0 dB)

iiN = N all(ii);s data = (1/sqrt(2))*rand(1,N)>0.5;% normalization of energy to 1s data1 = reshape(sdata,11,floor(N/11))’;for i=1:size(s data1,1),

tan temp = s data1(i,:)*G;tan temp = mod(tantemp,2);bch1(i,:) = tantemp;

ends data2 = reshape(bch1’,1,size(bch1,1)*size(bch1,2));%% QPSK Modulationip = qpsk(s data2);s = ip; % normalization of energy to 1N1 = length(s);Noise1 = 1/sqrt(2)*[randn(1,N1) + j*randn(1,N1)];% white guassian noise, 0dB varianceh = 1/sqrt(2)*[randn(1,floor(N1/705)) + j*randn(1,floor(N1/705))]; % Rayleigh channelta temp = h(end);h = h’*ones(1,705);h = reshape(h’,1,size(h,1)*size(h,2));h = [h ta temp*ones(1,N1-length(h))];% use when the lengths of h and data are not equal%% Add noisey = h.*s + sqrt(n/k)*10̂ (-Eb N0 dB(ii)/20)*Noise1;% additive white gaussian noise%% Equalizationyeq = y./h;%% QPSK Demodulationshat = detectqpsk(yeq);%% decodershat1 = reshape(shat,15,floor(length(shat)/15))’;for i=1:size(shat1,1),

bch2(i,:)=syndtan(shat1(i,:),g,HT);bch3(i,1:11)=bch2(i,1:11);

endshat2 = reshape(bch3’,1,size(bch3,1)*size(bch3,2));%% Error number calculationnErr(ii) = size(find([s data- shat2]),2);% couting the number of errorsa=find([s data- shat2]);

endsimBer QPSK = nErr./Nall;EbN0Lin = 10.̂ (Eb N0 dB/10);

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theoryBer QPSK = 0.5.*(1-sqrt(EbN0Lin./(EbN0Lin+1)));theoryBerAWGN = 0.5*erfc(sqrt(10.ˆ (Eb N0 dB/10)));close allfiguresemilogy(EbN0 dB,theoryBerQPSK,’b.-’ );hold onsemilogy(EbN0 dB,simBer QPSK,’mx-’ );hold onsemilogy(EbN0 dB,theoryBerAWGN,’r*-’ );axis([0 20 10̂ -7 1])grid onlegend(’theory-QPSK-fading’, ’simulation-QPSK-fading-coding’,’theory-QPSK-AWGN’);xlabel(’Es/No, dB’)ylabel(’Bit Error Rate’)title(’Bit error probability curve for QPSK’)save Quesddata simBerQPSK theoryBerQPSK theoryBerAWGN EbN0 dB

APPENDIX ECODE FOR ARAYLEIGH CHANNEL , A BCH CODING AND AN INTERLEAVING SCHEME

%% Author and his affiliation% This program is implemented by Le Thanh Tan% Ph.D Student% Institut National de la Recherche Scientifique (INRS)% University of Quebec, Montreal, QC, H5A 1K6, Canada% Email: lethanh@emt.inrs.ca%%clc;clear all;close all;pause(0.05)N all = [517*10̂ 3*ones(1,6) 517*10̂3*ones(1,5)];% we can change NEb N0 dB = [0:2:20]; % multiple Eb/N0 values%% Prepare for coder and decoder

k=11;n=15;g1=[1 0 0 1 1];G=Gformulation(k,g1);g=[1 0 0 1 1 0 0 0 0 0 0 0 0 0 0];r=n-k;H=Hformulation(k,G,r);HT=H.’;for ii = 1:length(Eb N0 dB)

iiN = N all(ii);s data = (1/sqrt(2))*rand(1,N)>0.5;% normalization of energy to 1s data1 = reshape(sdata,11,floor(N/11))’;for i=1:size(s data1,1),

tan temp = s data1(i,:)*G;

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tan temp = mod(tantemp,2);bch1(i,:) = tantemp;

end%% Interleavers data2 = reshape(bch1,1,size(bch1,1)*size(bch1,2));%read out by column%% QPSK Modulationip = qpsk(s data2);s = ip; % normalization of energy to 1N1 = length(s);Noise1 = 1/sqrt(2)*[randn(1,N1) + j*randn(1,N1)];% white guassian noise, 0dB varianceh = 1/sqrt(2)*[randn(1,floor(N1/705)) + j*randn(1,floor(N1/705))]; % Rayleigh channelh = h’*ones(1,705);ta temp = h(end);h = reshape(h’,1,size(h,1)*size(h,2));h = [h ta temp*ones(1,N1-length(h))];% use when the lengths of h and data are not equal%% Add noisey = h.*s + 10̂ (-Eb N0 dB(ii)/20)*Noise1;% additive white gaussian noise%% Equalizationyeq = y./h;%% QPSK Demodulationshat = detectqpsk(yeq);%% decodershat1 = reshape(shat,floor(length(shat)/15),15);for i=1:size(shat1,1),

bch2(i,:)=syndtan(shat1(i,:),g,HT);bch3(i,1:11)=bch2(i,1:11);

endshat2 = reshape(bch3’,1,size(bch3,1)*size(bch3,2));%% Error number calculationnErr(ii) = size(find([s data- shat2]),2);% couting the number of errorsa=find([s data- shat2]);

endsimBer QPSK = nErr./Nall;EbN0Lin = 10.̂ (Eb N0 dB/10);theoryBer QPSK = 0.5.*(1-sqrt(EbN0Lin./(EbN0Lin+1)));theoryBerAWGN = 0.5*erfc(sqrt(10.ˆ (Eb N0 dB/10)));close allfiguresemilogy(EbN0 dB,theoryBerQPSK,’b.-’ );hold onsemilogy(EbN0 dB,simBer QPSK,’mx-’ );hold onsemilogy(EbN0 dB,theoryBerAWGN,’r*-’ );axis([0 20 10̂ -7 1])grid onlegend(’theory-QPSK-fading’, ’simulation-QPSK-interleaver’,’theory-QPSK-AWGN’);xlabel(’Es/No, dB’)ylabel(’Bit Error Rate’)title(’Bit error probability curve for QPSK’)

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save Quesedata simBerQPSK theoryBerQPSK theoryBerAWGN EbN0 dB

REFERENCES

[1] John G. Proakis, Digital Communications, 4th Edition, McGrawHill, 2000.[2] B. Sklar, Digital communications : fundamentals and applications, 2nded., Upper Saddle River, N.J.: Prentice-Hall PTR, 2001.[3] T. S. Rappaport, Wireless communications : principles and practice,2nd ed., Upper Saddle River, N.J.: Prentice Hall PTR, 2002.