5.4 Consider a general M-ary orthogonal system in which all the trans forms havi

Question

5.4 Consider a general M-ary orthogonal system in which all the trans forms having energy E are modeled by s,(t) -./E ,(t), 0 t T, i-1 , , M, where the set of {0,(t), 0 t T, 1-1 ,,.., M} constitutes an orthonormal system of function. Then in signal vector space, each transmitted signal waveform s,(t) can be represented by s-E [0, , 0, 1 , 0, , T, where all the components of the vector are zeros except the ith. Assume all the signals have equi-probability of trans- mission and the channel is AWGN with two- sided spectral density of No/2. The coherent receiver decides the jth hypothesis from the largest value of the outputs of the M correla- tors (or matched filters) matched to each sig nal waveform, (t). If the transmitted signal is s, then the outputs of all the correlators are zero-mean Gaussian r.v.'s of variance No/2, except the jth output is also a Gaussian r.v. of mean E and also of variance No/2. Then the probability of a correct decision when the jth signal was transmitted and conditioned on the noise by P(Clsn)is given by mitted wave- M-1 exp(-/No) dt Then obtain P(Cls) - ENIPCls,n)]. Find the average probability of a symbol error P(error) -1 - P(Clsj).

Explanation / Answer

%------------------------------------------------------------------------- % Mohammad Ibrahim Baker % The University of Jordan %------------------------------------------------------------------------- % The below code finds the probablity of error for M-FSK orthogonal signals. %------------------------------------------------------------------------- format long clear clc disp('please wait...This requires a few seconds'),... disp(' ') %------------------------------------------------------------------------- %------------------------------------------------------------------------- SNR = .6:.1:20; % signal to noise ratio per bit SNR_dB = 10*log10(SNR); % signal to noise ratio per bit in dB for M = [2 4 8 16 32 64]; % To find probablity of error for M orthogonal signals k = log2(M); % k is number of bits : M = 2^k i = 1; % i is a counter used to construct BER Matirx...see below %------------------------------------------------------------------------- disp((['The program is estimating the probability of error for M = ' num2str(M) ' ....Processing'])) %------------------------------------------------------------------------- for j = .6:.1:20; % j is a counter equal to SNR per bit in the range from 0.6 to 20 SNR_sym = j*k; % SNR per symbol equal to SNR per bit multiplied by Number of bits %Below is the probability of error equation for M orthogonal Siganl As it is derived in "Digital communications" Book, 4th edition, Proakis. eval(['v =inline(''(1./(sqrt(2*pi)).*(1-((1-0.5*erfc(t)./sqrt(2))).^(' num2str(M) '-1)).*exp(-.5*(t-sqrt(2*' num2str(SNR_sym) ')).^2))'',''t'');' ]); % To estimate the probablity of error numarically using "quadrature gaussian" approxiamtion with tolerance 10^-12 w = quadl(v,-10000,10000,1e-12); BER(k,i) = w; % Construct probability of error Matrix with 6 rows, corresponding to M-signals i = i+1; % counter increaments end %------------------------------------------------------------------------- disp(' Completed ') %------------------------------------------------------------------------- end %------------------------------------------------------------------------- % Plotting Probablity of error graphs %------------------------------------------------------------------------- semilogy(SNR_dB,BER) ylim([1e-6 1e-1]) grid xlabel('SNR per bit (dB)') ylabel('Probability of error') title('Probability of error for coherent detection of orthogonal M-FSK signals') legend('2-FSK','4-FSK','8--FSK','18-FSK','32-FSK','64-FSK') %-------------------------------------------------------------------------

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