Shift in noise immunity of FSK signal at different frequency spacing

The code below is an FSK modulated signal noise immunity script that shows the signal immunity to noise at different frequency spacings.

One of the characteristics of the FSK signal's noise immunity is that it will improve at frequencies that are multiples of 1/T. This is explained by the fact that the scalar product of cosine waves with frequencies that are multiples of 1/T is equal to zero and, accordingly, the signals minimally influence each other during FSK modulation.

So, for a given signal length T = 0.1, noise immunity will theoretically be maximum at separation frequencies of 10, 20, 30, etc. However, if you run this code, the maximum noise immunity values ​​will be shifted by approximately 1 Hz. Also, for different values ​​of T, the bias maintains a scale of about one tenth of 1/T Hz. In this case, the period for achieving the best noise immunity is 1/T Hz, which is correct.

I can't find the reason for this shift. Please give me ideas on what the problem might be, maybe even in a simple display of results, but I couldn’t find it.

close all;
clc;
clear;

% Signal parameters
Fs = 100000;
bit_rate = 10000;
T = 0.1;
num_bits = T * bit_rate;
samples_per_bit = Fs / bit_rate;
num_samples = T * Fs;
t = (0:num_samples - 1) / Fs;
SNR = 10;

% indicating the frequency spacing of bits 0 and 1
deviation_range = linspace(5, 54, 50); 
ber_accumulated = zeros(length(deviation_range), 10); 

% Generation of random signals followed by modulation, demodulation and error counting
% 100 values ​​are given for averaging
for iter = 1:100 
    data = round(rand(1, num_bits));
    
    for k = 1:length(deviation_range)
        
        Fc0 = 4000;
        Fc1 = Fc0 + deviation_range(k);
        
        % FSK signal modulation
        fm_signal = zeros(1, num_samples);
        for i = 1:num_bits
            index_start = (i-1) * samples_per_bit + 1;
            index_end = i * samples_per_bit;
            if data(i) == 0
                fm_signal(index_start:index_end) = cos(2*pi*Fc0*t(index_start:index_end));
            else
                fm_signal(index_start:index_end) = cos(2*pi*Fc1*t(index_start:index_end));
            end
        end
    
        Eb = (sum(abs(fm_signal).^2) / num_bits) / Fs;
        noise_power = sqrt(Eb/(10^(SNR/10))/2);
        noisy_signal = fm_signal + randn(size(fm_signal)) * noise_power;
        
        
        % FSK signal demodulation
        errors = 0;
        for i = 1:num_bits
            index_start = (i-1) * samples_per_bit + 1;
            index_end = i * samples_per_bit;
            sample = noisy_signal(index_start:index_end);
            ref0 = cos(2*pi*Fc0*t(index_start:index_end));
            ref1 = cos(2*pi*Fc1*t(index_start:index_end));
            corr0 = sum(sample .* ref0);
            corr1 = sum(sample .* ref1);
            received_bit = corr1 > corr0;
            errors = errors + (data(i) ~= received_bit);
        end
        
        ber_accumulated(k, iter) = errors / num_bits;
        
    end
end

ber_average = mean(ber_accumulated, 2);
ber_theory = berawgn(SNR,'fsk',2,'coherent') * ones(size(deviation_range));

figure;
plot(deviation_range, ber_average, 'b-o');
hold on;
plot(deviation_range, ber_theory, 'r--');
xlabel('Frequency deviation (Hz)');
ylabel('BER');
title('Dependence of BER on frequency separation');
legend('experimental BER', 'theoretic BER');
grid on;

Answer 1

[Suraj Kumar]

Hi Melifaro,

I understand you are experiencing issues with the Bit Error Rate (BER) analysis related to frequency deviation in your FSK modulation.

To resolve these issues, you may refer to the following steps and the attached code snippets:

1. You can expand and refine the frequency deviation range to a broader range which will be crucial in capturing the optimal frequency spacings to reduce the BER.

% Frequency deviation range aligned with multiples of 1/T 
deviation_range = 10:2:100;  
ber_accumulated = zeros(length(deviation_range), 100); 

2. You can use ‘randi’ function in MATLAB for generating random binary sequences which is a more reliable and consistent method for generating binary data.

data = randi([0, 1], 1, num_bits); % Random binary data 

3. You can plot the results to visualize how frequency deviation impacts the BER.

figure; 
plot(deviation_range, ber_average, 'b-o'); 
hold on; 
plot(deviation_range, ber_theory, 'r--'); 
xlabel('Frequency deviation (Hz)'); 
ylabel('BER'); 
title('Dependence of BER on Frequency Separation'); 
legend('Experimental BER', 'Theoretical BER'); 
grid on; 

You may refer to the output below for better understanding:

For more information on ‘randi’ or ‘berawgn’ function in MATLAB, kindly refer to the following documentations:

• https://www.mathworks.com/help/matlab/ref/randi.html
• https://www.mathworks.com/help/comm/ref/berawgn.html

Hope this is helpful!