COMPARISON OF DIFFERENT CHANNEL ESTIMATION TECHNIQUES FOR CP-OFDM SYSTEMS OPERATING IN A MULTIPATH RAYLEIGH FADING CHANNEL WITH EXPONENTIAL POWER DELAY PROFILE

SYSTEM SPECIFICATIONS:

   1)  128 Subcarriers used

   2)  Guard tones in first 8 and last 7 subcarriers to keep check on adjacent channel interference

   3)  DC guard at subcarrier 64 to keep check on PAPR

   4)  8 tap Rayleigh fading channel with exponential PDP

   5)  Cyclic prefix length of 16

   6)  BPSK modulation

   7)  Pilot spacing of 4 subcarriers

MATLAB CODE:

clc;

clear all;

close all;

for j=1:8

    wf(1,j)=(1)./exp(j-1);          %FOR EXPONENTIAL PDP

end

a=1;                                %BPSK MODULATION OF FIXED POWER

N=128;                              %NUMBER OF SUB-CARRIERS

SNRstart=-10;

SNRend=30;                          %DEFINITION OF SNR RANGE OVER WHICH WE SIMULATE

snr=SNRstart:SNRend;

for SNR=SNRstart:SNRend

    for chnlno=1:power(10,2)        %TIME VARYING CHANNEL FOR EVERY SNR

       

        databits=rand(1,128);       %DATA BITS

        

        for i=1:length(databits)

            if databits(i)<0.5

                sym(i)=a;

            else                    %MAPPING BITS TO BPSK CONSTELLATION

                sym(i)=-a;

            end

        end

       

        for i=1:8

            sym(i)=0;               %GUARD TONES

        end

       

        for i=121:128

            sym(i)=0;               %GUARD TONES

        end

       

        for i=9:4:117

            sym(i)=a;               %PILOTS WITH SPACING OF 4 SUBCARRIERS

        end

       

        sym(64)=0;                  %DC GUARD

       

        pilotvector=a*ones(1,28);

        X=diag(pilotvector);        %REPRESENTING PILOT IN MATRIX FORM FOR EQUALIZATION ALGORITHMS

       

        %SYMBOL SEQUENCE GENERATED WITH GUARD TONES AND PILOTS

       

        xi=ifft(sym,128);

        for i=113:128

            cp(i-112)=xi(i);

        end

        x=horzcat(cp,xi);

       

        %INSERTED CP. READY FOR TRANSMISSION

       

        hi=((1/sqrt(2))*randn(8,2));  %1/sqrt(2) to make variance of each dimension 1/2

        for j=1:length(hi)

            hi2(1,j)=hi(j,1)+sqrt(-1)*hi(j,2);

        end

        h=(wf).*(hi2);

       

        %FREQ. SELECTIVE RAYLEIGH FADING CHANNEL GENERATED WITH EXPONENTIAL PDP

       

        chf=fft(h,128);

        chfp=chf(9:4:117);              %CHANNEL FREQUENCY RESPONSE AT PILOT LOCATIONS

       

        ynn=conv(x,h);

        ynnc=cconv(xi,h,128);

       

        sigma=(1)/(sqrt(2*(power(10,(SNR/10)))));

        noisei=(sigma)*randn(length(ynn),2);

        for j=1:length(ynn)

            noise(1,j)=noisei(j,1)+sqrt(-1)*noisei(j,2);

        end

        y=ynn+noise;

                     

        %SIGNAL TRANSMITTED THRU RAYLEIGH AWGN CHANNEL

       

        for i=17:144

            yhi(i-16)=y(i);

        end

       

        %CP REMOVED

       

        yh=fft(yhi);

       

        %DATA DEMODULATED

       

        flag=0;

        for i=9:4:117

            flag=flag+1;

            Yp(flag,1)=yh(i);

        end

       

        %DATA EXTRACTED FROM PILOT TONES

       

        D=diag(a*ones(flag,1));

        cefpi=(D')*(Yp);        %  ' STANDS FOR HERMITION. THIS CASE, D and D' ARE JUST IDENTITY MATRICES. (a=1)

        cet=ifft(cefpi);

               

        for i=17:length(cet)

            cet(i)=0;               % REMOVING NOISE SINCE WE KNOW THAT CP LENGTH IS ONLY 16

        end

        cefproc=fft(cet,28);        % DFT BASED LS ESTIMATE

       

        %TIME DOMAIN PROCESSING DONE TO SUPPRESS NOISE

              

        Fmi=dftmtx(128);

        flag=0;

        for i=9:4:117

            flag=flag+1;

            Fmi2(flag,:)=Fmi(i,:);

        end

        nullingmatrx=zeros(128,8);

        for i=1:8

            nullingmatrx(i,i)=1;

        end

        Fm=Fmi2*nullingmatrx;                        %Modified DFT Matrix for MMSE equalizer

       

        Hcapfdls(:,chnlno)=X\Yp;       %Freq. domain LS. Same as Yp since X is identity for our choice of pilot

        Hcapmmse(:,chnlno)=pinv(Fm)*Yp;              %MMSE estimate.

        Hcapdftmmse(:,chnlno)=pinv(Fm)*cefproc;      %DFT Based MMSE.

       

                    

        MSEi(chnlno)=(1/length(chfp))*(sum(power((abs(cefproc-transpose(chfp))),2)));

        MSEfdlsi(chnlno)=(1/length(chfp))*(sum(power((abs(Hcapfdls(:,chnlno)-transpose(chfp))),2)));

        MSEmmsei(chnlno)=(1/length(h))*(sum(power((abs(Hcapmmse(:,chnlno)-transpose(h))),2)));

        MSEdftmmsei(chnlno)=(1/length(h))*(sum(power((abs(Hcapdftmmse(:,chnlno)-transpose(h))),2)));

    end

   

    MSE(SNR-SNRstart+1)=(1/chnlno)*sum(MSEi);   

    MSEfdls(SNR-SNRstart+1)=(1/chnlno)*sum(MSEfdlsi);   

    MSEmmse(SNR-SNRstart+1)=(1/chnlno)*sum(MSEmmsei);   

    MSEdftmmse(SNR-SNRstart+1)=(1/chnlno)*sum(MSEdftmmsei);   

end

figure(1);

semilogy(snr,MSEfdls,snr,MSE,snr,MSEmmse,snr,MSEdftmmse,'linewidth',1.5);

legend('LEAST SQUARES','DFT BASED LEAST SQUARES','MMSE','DFT BASED MMSE');

xlim([-5 30]);

xlabel('SNR IN dB ------>');

ylabel('MEAN SQUARE ERROR');

title('COMPARISON OF DIFFERENT CHANNEL ESTIMATION TECHNIQUES FOR CP-OFDM SYSTEMS');

grid;

MSE PLOTS:

SOME PHILOSOPHY:

   1)  Transmitting RF amplifier limits PAPR of the transmitted signal. The RF industry has caught up with this problem. Nevertheless, from power efficiency point of view, it makes sense to restrict the PAPR. The power amplifiers, even today, are only about 50%-60% power efficient. An easy way of limiting PAPR is to use a DC guard.

        2)       DFT based MMSE and DFT based LS perform better than MMSE alone and LS alone, respectively. This is expected since we suppress noise in DFT based estimates by exploiting the knowledge of channel impulse response length.

QUESTIONS:

   1)  I am not sure if DFT based LS will perform better than MMSE. My results say otherwise. Does anybody know for sure that MMSE performs better than DFT based LS? Any published reference that compares DFT based LS and MMSE? 

BIT ERROR RATE OF COHERENT CP-OFDM SYSTEM OPERATING OVER A RAYLEIGH FADING CHANNEL

SYSTEM SPECIFICATIONS:

1) 128 subcarriers

2   BPSK modulation

3)  10 tap Rayleigh fading channel with exponential PDP

4)  Cyclic prefix length of 32 to avoid Inter-block interference

5)  Coherent receiver

MATLAB CODE:

clc;

clear all;

close all;

k=0;

snr= -5:2:11;

snrn=power(10,snr/10);

a=1;

for j=1:20

    wf(1,j)=1./exp(j-1);                   %EXPONENTIAL POWER DELAY PROFILE

end

for SNR=-5:2:11

    k=k+1;

    error=0;

    for chnlno=1:power(10,3)

        msgbits=randint(1,128);                           % 128 SUBCARRIERS

        for i=1:length(msgbits)

            if msgbits(i)==0

                s(i)=a;

            else

                s(i)=-a;

            end

        end                                              %BPSK MAPPING DONE

       

        xi=ifft(s);

        for i=97:128

            cp(i-96)=xi(i);                                  % 32 LENGTH CP

        end

        x=horzcat(cp,xi);                              %CYCLIC PREFIX ADDED

       

        hi=((1/sqrt(2))*randn(20,2));

        for j=1:20

            hi2(1,j)=hi(j,1)+sqrt(-1)*hi(j,2);             % 10 TAP CHANNEL

        end

        h=(wf).*(hi2);

       

        ynn=conv(x,h);     

       

        sigmasqr=(1)/(2*(power(10,(SNR/10))));

        noisei=(sigmasqr)*randn(length(ynn),2);

       

        for j=1:length(ynn)

            noise(1,j)=noisei(j,1)+sqrt(-1)*noisei(j,2);

        end

       

        y=ynn+noise;

       

        for i=33:160

            yh(i-32)=y(i);                           %CYCLIC PREFIX REMOVED

        end

       

        Yeq=(fft(yh))./(fft(h,128));

        yeq=ifft(Yeq);

        yi=fft(yeq);

       

        for i=1:128

            if real(yi(i)>0)

                mh(i)=0;

            else

                mh(i)=1;

            end

        end

       

        for i=1:128

            if mh(i)==msgbits(i)

                error=error+0;

            else

                error=error+1;

            end

        end

    end

    BER(k)=error/(128*power(10,4));

end

figure(1);

semilogy(snr,BER,'linewidth',1.5);

xlabel('SNR IN dB ------>');

ylabel('BER');

title('BER PERFORMANCE OF A CP-OFDM SYSTEM');

grid;

SNR Vs BER CURVE:

SOME PHILOSOPHY:

1)  BER curve almost in sync with that of a single carrier system operating over Rayleigh flat fading channel (as expected).

2)  Do not expect waterfall curve for BER in Rayleigh fading channels. BER plot will always be a straight line in high SNR regime. Slope can be improved using diversity (time, frequency, space).

3)  To make OFDM achieve frequency diversity, you have to code across subcarriers instead of coding across time. Coding across time alone results in high latency since OFDM symbol duration is pretty high. Typical systems code across both subcarriers and time in order to achieve same diversity with lesser latency (as compared to coding just across time).

4)  While coding across subcarriers, keep in mind that not all parallel channels are uncorrelated. You need to determine the minimum frequency spacing for uncorrelatedness before you code. Otherwise, you will not get full diversity.