NON-COHERENT DETECTION OF UNMODULATED SINUSOID IN AWGN CHANNEL

PROBLEM SETUP:

Non-coherent detection is a problem that comes up all the time in Radar signal processing. Alternate hypothesis is a noise corrupted random signal where the randomness in the signal, in this problem, is due to random phase of an unmodulated sinusoid. In a practical setting, this would correspond to the case where you are trying to detect stationary targets. The problem can be made more complex by introducing randomness in frequency and amplitude corresponding to moving targets. The null hypothesis corresponds to noise alone, whose statistics are known apriori.

MATLAB CODE:

clc;

clear all;

close all;

scale=0.05;

var1=scale*50;

var2=scale*100;                             %DIFFERENT NOISE VARIANCE ==> DIFFERENT SNRs

var3=scale*200;

sigma1=sqrt(var1);

sigma2=sqrt(var2);

sigma3=sqrt(var3);

i=0:9;

incmp=transpose(cos(i*0.2*pi));   %OPTIMUM RX ARCHITECTURE REQUIREMENT (DERIVED FROM BAYE'S MIN RISK ANALYSIS)

quadcmp=transpose(sin(i*0.2*pi)); %OPTIMUM RX ARCHITECTURE REQUIREMENT (DERIVED FROM BAYE'S MIN RISK ANALYSIS)

i1max=10000;                                %NUMBER OF EXPERIMENTS

threshold=-500:10:500;                      %THRESHOLD RANGE (THEORETICAL RANGE FROM -INFINITY TO +INFINITY)

for j=1:length(threshold)

    numdetect1(j)=0;       

    numfa1(j)=0;

    numdetect2(j)=0;                        %NUMBER OF DETECTIONS FOR DIFFERENT SNRs

    numfa2(j)=0;                            %NUMBER OF FALSE ALARMS FOR DIFFERENT SNRs

    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 i1=1:i1max

       

        teta=2*pi*rand(1);                  %GENERATE RANDOM PHASE FOR EACH EXPERIMENT FROM A UNIFORM DISTRIBUTION

        s=transpose(sin(0.2*pi*i+teta));    %GENERATE RANDOM SIGNAL EACH EXPT. BASED ON GENERATED RANDOM PHASE

       

        n1=sigma1*randn(10,1);

        n2=sigma2*randn(10,1);

        n3=sigma3*randn(10,1);

       

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

       

        if sgnlpresent==1

            notst(j)=notst(j)+1;

            r1=s+n1;

            r2=s+n2;                        %PROBABILISTIC TRANSITION MECHANISM FOR ALTERNATE HYPOTHESIS

            r3=s+n3;

        else

            notsnt(j)=notsnt(j)+1;

            r1=n1;

            r2=n2;                          %PROBABILISTIC TRANSITION MECHANISM FOR NULL HYPOTHESIS

            r3=n3;

        end

       

        inproc1=transpose(r1)*incmp;

        quadproc1=transpose(r1)*quadcmp;

        inproc2=transpose(r2)*incmp;        %PART OF SUFFICIENT STATISTIC

        quadproc2=transpose(r2)*quadcmp;    %PART OF SUFFICIENT STATISTIC

        inproc3=transpose(r3)*incmp;

        quadproc3=transpose(r3)*quadcmp;

       

        inprocsqr1=power(inproc1,2);

        quadprocsqr1=power(quadproc1,2);

        inprocsqr2=power(inproc2,2);

        quadprocsqr2=power(quadproc2,2);

        inprocsqr3=power(inproc3,2);

        quadprocsqr3=power(quadproc3,2);

       

        suffstat1=inprocsqr1+quadprocsqr1;

        suffstat2=inprocsqr2+quadprocsqr2;  %COMPUTING SUFFICIENT STATISTIC FOR DIFFERENT VALUES OF SNR

        suffstat3=inprocsqr3+quadprocsqr3;

       

        %THRESHOLDING FOR DECESION DEVICE AND COMPUTING Pr(False alarm) AND Pr(Detection) FOR DIFFERENT SNRs

       

        if suffstat1>threshold(j)

            detect1=1;

        else

            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

        if suffstat2>threshold(j)

            detect2=1;

        else

            detect2=0;

        end

       

        if detect2==1 && sgnlpresent==1

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

        end

        if detect2==1 && sgnlpresent==0

            numfa2(j)=numfa2(j)+1;

        end

        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

    end

    pd1(j)=numdetect1(j)/notst(j);

    pf1(j)=numfa1(j)/notsnt(j);

    pd2(j)=numdetect2(j)/notst(j);

    pf2(j)=numfa2(j)/notsnt(j);

    pd3(j)=numdetect3(j)/notst(j);

    pf3(j)=numfa3(j)/notsnt(j);

end

figure(1);

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

xlim([0 1]);

ylim([0 1]);

xlabel('PROBABILITY OF FALSE ALARM');

ylabel('PROBABILITY OF DETECTION');

legend('d^2=2','d^2=1','d^2=0.5');

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

grid;

ROC PLOT:

SOME PHILOSOPHY:

Bayesian Risk analysis cannot be applied directly to partition the observation space in this case since the probabilistic transition mechanism is not constant for all vectors in the alternate hypothesis. However, since the phase distribution is known, the unknown parameter can be averaged out and subsequently, we can come up with some sort of an average for the probabilistic transition mechanism. Consequently, we can derive an optimal detection rule based on that. However, the ROC curve for this would only lie below that of coherent detection (this is also intuitive since in the coherent case, you have more knowledge about the signal that you want to detect). 

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.