How can I find the solution of the integral

How Can I find the SEP values for the range of SNRdB=-5:45 dBm for M-FSK in Rayleigh channel with the integrals below.

Thanks for help,

I have tried with these codes but I think something is wrong ?

% MFSK in Rayleigh channel - Theoretical SER calculation for M=4
r1=2;%degrees oof freedom for chi-square random variable 
var1=2.*r1; %variance of chi-squared 
SNRsymbolLin_4fsk = SNRbitLin * log2(modLevel_1);
q11 = @(y) integral(exp((-(y-sqrt(2.*x)).^2)/2).*(1./(SNRsymbolLin_4fsk.*2.*var1)).*exp(-(x./(SNRsymbolLin_4fsk.*2.*var1)),0,inf,'ArrayValued',true));
theorySer_rayleigh_4fsk= (1/sqrt(2.*pi).*integral(@(y) q11(y)*(1-(1-qfunc(y)).^3),-inf,inf,'ArrayValued',true));

and the related formulas I have used is ;

Answer 1

[Himanshu]

Hey Hakan,

The issue in the previous code might be related to the incorrect calculation of SNRLin from SNRdB. The formula for converting from decibels to linear scale is correct, but ensure that the calculation is implemented properly in the code.

You may try the following to get your MATLAB code to correctly implement the given integral.

% Parameters
M = 4; % Modulation order for M-FSK
SNRdB = -5:45; % SNR range in dB
SNRLin = 10.^(SNRdB/10); % Convert SNR to linear scale

% Variance for Rayleigh channel
sigma = 1; % Assuming unit variance for simplicity

% Preallocate array for theoretical SER
theorySer_rayleigh_4fsk = zeros(size(SNRdB));

% Loop over each SNR value
for i = 1:length(SNRdB)
    SNRsymbolLin = SNRLin(i) * log2(M);
    
    % Define the inner integral function
    q11 = @(y, x) exp(-((y - sqrt(2*x)).^2) / 2) .* ...
                  (1 / (SNRsymbolLin * 2 * sigma^2)) .* ...
                  exp(-x / (2 * SNRsymbolLin * sigma^2));
    
    % Define the outer integral function
    outerIntegral = @(y) (1 - (1 - qfunc(y)).^(M-1)) .* ...
                         integral(@(x) q11(y, x), 0, inf, 'ArrayValued', true);
    
    % Compute the theoretical SER using double integration
    theorySer_rayleigh_4fsk(i) = (1 / sqrt(2 * pi)) * ...
                                 integral(@(y) outerIntegral(y), -inf, inf, 'ArrayValued', true);
end

% Plot the results
figure;
semilogy(SNRdB, theorySer_rayleigh_4fsk, 'b-o');
xlabel('SNR (dB)');
ylabel('Symbol Error Probability');
title('Theoretical SER for 4-FSK in Rayleigh Channel');
grid on;

The above code represents a mathematical expression for calculating the probability of symbol error ( P_s ) in a communication system. It involves a double integral where the outer integral is over ( y ) and the inner integral is over ( x ). The expression includes the complementary error function ( Q(y) ), the signal-to-noise ratio in linear scale ( \text{SNRLin} ), and a Gaussian function. The SNR is converted from decibels to linear scale using ( \text{SNRLin} = 10^{\left(\frac{\text{SNRdB}}{10}\right)} ), where (\text{SNRdB} = -5.45).

Hope this helps!