RECEIVER OPERATING CHARACTERISTICS (ROC) FOR BINARY HYPOTHESIS TESTING

PROBLEM SETUP:

  1)  Binary hypothesis testing problem

  2)  Null hypothesis: S = zeros(10,1)

  3)  Alternate hypothesis: S = ones(10,1)

  4)  Probabilistic transition mechanism: Multidimensional Gaussian with mean vector S and covariance matrix equal to scaled identity

  5)  SNR d² defined as S^TQ^-1S where Q is the covariance matrix of the multidimensional Gaussian distribution

MATLAB CODE:

clc;

clear all;

close all;

s=ones(10,1);

nos=zeros(10,1);

sigma=[sqrt(20) sqrt(10) sqrt(5)];

imax=10000;                %NUMBER OF EXPERIMENTS

threshold=-50:2:50;        %THRESHOLD RANGE OF LOG-LIKELIHOOD RATIO(THEORETICAL RANGE FROM -INFINITY TO +INFINITY)

for j=1:length(threshold)  %LOOP TO VARY THRESHOLD

   

    numdetect1(j)=0;       %NUMBER OF DETECTIONS

    numfa1(j)=0;           %NUMBER OF FALSE ALARMS

    numdetect2(j)=0;

    numfa2(j)=0;

    numdetect3(j)=0;

    numfa3(j)=0;

    notst(j)=0;             %NUMBER OF TIMES ALTERNATE HYPOTHESIS IS TRUE

    notsnt(j)=0;            %NUMBER OF TIMES NULL HYPOTHESIS IS TRUE

   

    for i=1:imax            %EXPERIMENT NUMBER i

       

        n=randn(10,1);

        n1=sigma(1)*n;      %GENERATE NOISE FOR i-th EXPT WITH DIFFERNT POWERS CORRESPONDING TO DIFFERENT SNRs

        n2=sigma(2)*n;

        n3=sigma(3)*n;

       

        sgnlpresent=randi(2)-1;    %RANDOM GENERATION OF HYPOTHESES

       

        if sgnlpresent==1

            notst(j)=notst(j)+1;

            r1=s+n1;               %PROBABILISTIC TRANSITION MECHANISM FOR ALTERNATE HYPOTHESIS FOR DIFFERENT SNRs

            r2=s+n2;

            r3=s+n3;

        else

            notsnt(j)=notsnt(j)+1;

            r1=nos+n1;

            r2=nos+n2;             %PROBABILISTIC TRANSITITON MECHANISM FOR NULL HYPOTHESIS FOR DIFFERENT SNRs

            r3=nos+n3;

        end                        

       

        suffstat1=transpose(r1)*s;       %COMPUTE SUFFICIENT STATISTIC FROM RECEIVED OBSERVATIONS

        suffstat2=transpose(r2)*s;

        suffstat3=transpose(r3)*s;       %STATISTIC DERIVED USING BAYE'S MINIMUM RISK ANALYSIS

       

        if suffstat1>threshold(j)

            detect1=1;

        else                                 %THRESHOLDING FOR DECESION DEVICE

            detect1=0;

        end          

        if detect1==1 && sgnlpresent==1

            numdetect1(j)=numdetect1(j)+1; 

        end                                 

        if detect1==1 && sgnlpresent==0

            numfa1(j)=numfa1(j)+1;

        end          

        pd1(j)=numdetect1(j)/notst(j);

        pf1(j)=numfa1(j)/notsnt(j);

       

        if suffstat2>threshold(j)

            detect2=1;

        else

            detect2=0;

        end          

        if detect2==1 && sgnlpresent==1

            numdetect2(j)=numdetect2(j)+1;

        end                                 %COMPUTING Pr(False Alarm) and Pr(Detection) FOR DIFFERENT SNRs

        if detect2==1 && sgnlpresent==0

            numfa2(j)=numfa2(j)+1;

        end          

        pd2(j)=numdetect2(j)/notst(j);

        pf2(j)=numfa2(j)/notsnt(j);

       

        if suffstat3>threshold(j)

            detect3=1;

        else

            detect3=0;

        end          

        if detect3==1 && sgnlpresent==1

            numdetect3(j)=numdetect3(j)+1;

        end

        if detect3==1 && sgnlpresent==0

            numfa3(j)=numfa3(j)+1;

        end          

        pd3(j)=numdetect3(j)/notst(j);

        pf3(j)=numfa3(j)/notsnt(j);

    end

end

figure(1);

plot(pf1,pd1,pf2,pd2,pf3,pd3,'linewidth',1.5);

title('RECEIVER OPERATING CHARACTERISTICS FOR DIFFERENT VALUES OF SNR');

xlabel('PROBABILITY OF FALSE ALARM');

ylabel('PROBABILITY OF DETECTION');

legend('d2=0.5','d2=1','d2=2');

grid;

ROC PLOT:

SOME PHILOSOPHY:

   1)  The statistic is derived in a generic framework using Bayes’ criterion. The result can be applied with minor modifications to problems whose null hypothesis, alternate hypothesis and probabilistic transition mechanism follow the model described in the beginning. I have tried this code to detect image signals in the presence of noise and it works properly.  

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.