How to make a visible side bands in the waterfall of AM signal with Fc = 440 and Fs = 10?

Hey,

I need to do a waterfall for an AM signal for 3 modulation indexes. The problem is that the carrier frequency should be 440Hz and the frequency of the source signal is 10Hz. I've tried a lot of parameters, but every time I print the waterfall I don't see the sidebands. My math is good, so the problem in the visibilty probably comes from the low frequency of the source signal (the side-bands should be around 430Hz and 440Hz). What can I do to actually see the side bands? to see the modulation!

This is my code:

close all; % close all : close all open figures

% whose 'HandleVisability' property set to 'on'

clear; % removes all variables from the current workspace,

% releasing them for system memory

clc; % clears all the text from the Command Window,

% resulting in a clear screen

% Values for the spectrogram

M = 1024;

L = 200;

window = bartlett(M);

ndft = 4096;

Fc = 440; % Frequency of the carrier wave [Hz]

Fs = 10; % Frequency of the modulated wave signal [Hz]

As = 1;

Ac = 1;

modulationFactors = [0.5, 1, 1.5]; % Modulation factors

sampleRate = 16000; % Samples per second

signalLengthInSeconds = 2;

time = linspace(0,signalLengthInSeconds,sampleRate*signalLengthInSeconds+1); % Time Vector +1 Sample

time(end)=[]; % Remove extra sample

x_carrier = Ac*sin(2*pi*Fc*time);

x_signal = As*sin(2*pi*Fs*time);

for i = 1:length(modulationFactors)

if (abs(modulationFactors(i)) > 1)

disp(append(string(modulationFactors(i)) ," is a bad modulation factor"));

end

% Generating AM Signal

amSignal = (Ac + (modulationFactors(i))*x_signal) .* x_carrier;

% Plotting synthetic digital sinus audio signal

figure;

subplot(5,1,1);

plot(time, x_signal)

title('Source signal of 10 Hz');

xlabel('Time in seconds');

ylabel('Amplitude');

subplot(5,1,2);

plot(time, x_carrier)

title('Carrier signal of 440 Hz');

xlabel('Time in seconds');

ylabel('Amplitude');

subplot(5,1,3);

plot(time, amSignal);

title(append("Amplitude modulation with a modulation factor of ", string(modulationFactors(i))));

xlabel('Time in seconds');

ylabel('Amplitude');

sound(x_signal, sampleRate);

pause(4);

sound(amSignal,sampleRate);

pause(4);

WAVfileName = sprintf('%s_%d%s', 'EX_02_01_2Audio', i, '.wav');

% WAV file will be created in the directory of the script

if exist(WAVfileName, 'file')==2

delete(WAVfileName);

end

% Normalize AM signal to range [-1, 1] before saving

AMsignalNormalized = amSignal / max(abs(amSignal));

audiowrite(WAVfileName, AMsignalNormalized, sampleRate);

% Creating spectrogram

figure;

spectrogram(amSignal, window, L, ndft, sampleRate, 'yaxis');

[s,f,t] = spectrogram(amSignal, window, L, ndft, sampleRate, 'yaxis');

title(['AM Signal with Modulation Factor = ', num2str(modulationFactors(i))]);

figure;

% Waterfall plot of spectrogram

waterfall(f,t,abs(s)'.^2)

set(gca,XDir="reverse",View=[30 50])

xlim([350 550]);

title(['AM Signal with Modulation Factor = ', num2str(modulationFactors(i))]);

xlabel("Frequency (Hz)");

ylabel("Time (s)");

zlabel('Power');

end

WAVfileName_original = 'OriginalAudio02_01_2.wav';

% WAV file will be created in the directory of the script

if exist(WAVfileName_original, 'file')==2

delete(WAVfileName_original);

end

audiowrite(WAVfileName_original, x_signal, sampleRate);

[s,f,t] = spectrogram(x_signal, window, L, ndft, sampleRate, 'yaxis');

figure; spectrogram(x_signal, window, L, ndft, sampleRate, 'yaxis');

title(['Original Signal']);

figure; waterfall(f,t,abs(s)'.^2)

set(gca,XDir="reverse",View=[30 50])

xlabel('Frequency [Hz]');

ylabel('Time (s)');

title(['Original Signal']);

I'll appreciate any help. Thanks.

Answer 1

[Neelanshu]

Hi Lior,

To make the sidebands visible in the waterfall display, you need a smaller frequency resolution, which is given by the equation:

where represents frequency resolution, represents the sampling frequency and represents the window size. According to the following paper, the minimum window size to distinguish two closely spaced components is given by :

https://www.scirp.org/journal/paperinformation?paperid=71142

Hence, in this case, with a sampling rate of 16KHz and closest frequency components being 10 Hz apart, you will require a window size of at least 3200 samples. Additionally, the sidebands will be around 430 Hz and 450 Hz, not 440 Hz and 430 Hz.

Here is the code and obtained output for window size of 4096 samples :

close all; % close all : close all open figures
           % whose 'HandleVisability' property set to 'on'
clear;     % removes all variables from the current workspace,
           % releasing them for system memory
clc;       % clears all the text from the Command Window,
           % resulting in a clear screen
format long
% Values for the spectrogram 
M = 1024*4; % for sideband display  
L = 200; 
window = bartlett(M);  
ndft = 4096*4; %for smooth graph
Fc = 440; % Frequency of the carrier wave [Hz]
Fs = 10; % Frequency of the modulated wave signal [Hz]
As = 1; 
Ac = 1; 
modulationFactors = [0.5, 1, 1.5]; % Modulation factors
sampleRate = 16000; % Samples per second
signalLengthInSeconds = 2;
time = linspace(0,signalLengthInSeconds,sampleRate*signalLengthInSeconds+1); % Time Vector +1 Sample
time(end)=[]; % Remove extra sample 
x_carrier = Ac*sin(2*pi*Fc*time); 
x_signal = As*sin(2*pi*Fs*time); 
for i = 1:length(modulationFactors)
    if (abs(modulationFactors(i)) > 1)
        disp(append(string(modulationFactors(i)) ," is a bad modulation factor"));
    end
    % Generating AM Signal
    amSignal = (1 + (modulationFactors(i))*x_signal/As) .* x_carrier;
    % Plotting synthetic digital sinus audio signal
    figure;
    subplot(5,1,1);
    plot(time, x_signal)
    title('Source signal of 10 Hz');
    xlabel('Time in seconds');
    ylabel('Amplitude'); 
    
    subplot(5,1,2);
    plot(time, x_carrier)
    title('Carrier signal of 440 Hz');
    xlabel('Time in seconds');
    ylabel('Amplitude'); 
    
    subplot(5,1,3);
    plot(time, amSignal);
    title(append("Amplitude modulation with a modulation factor of ", string(modulationFactors(i))));
    xlabel('Time in seconds');
    ylabel('Amplitude'); 
    
    sound(x_signal, sampleRate);
    pause(4);
    sound(amSignal,sampleRate);
    pause(4);
    WAVfileName = sprintf('%s_%d%s', 'EX_02_01_2Audio', i, '.wav');
        % WAV file will be created in the directory of the script
    if exist(WAVfileName, 'file')==2
    delete(WAVfileName);
    end
    
    % Normalize AM signal to range [-1, 1] before saving
    AMsignalNormalized = amSignal / max(abs(amSignal));
    audiowrite(WAVfileName, AMsignalNormalized, sampleRate);
    % Creating spectrogram 
    figure; 
    spectrogram(amSignal, window, L, ndft, sampleRate, 'yaxis');
    [s,f,t] = spectrogram(amSignal, window, L, ndft, sampleRate, 'yaxis');
    title(['AM Signal with Modulation Factor = ', num2str(modulationFactors(i))]); 
   
    figure; 
    % Waterfall plot of spectrogram
    waterfall(f,t,abs(s)'.^2)
    set(gca,XDir="reverse",View=[30 50])
    xlim([350 550]);
    title(['AM Signal with Modulation Factor = ', num2str(modulationFactors(i))]);
    xlabel("Frequency (Hz)");
    ylabel("Time (s)");
    zlabel('Power');
  
end 
    
WAVfileName_original = 'OriginalAudio02_01_2.wav';
        % WAV file will be created in the directory of the script
if exist(WAVfileName_original, 'file')==2
    delete(WAVfileName_original);
end
audiowrite(WAVfileName_original, x_signal, sampleRate);
[s,f,t] = spectrogram(x_signal, window, L, ndft, sampleRate, 'yaxis');
figure; spectrogram(x_signal, window, L, ndft, sampleRate, 'yaxis');
title(['Original Signal']);
figure; waterfall(f,t,abs(s)'.^2)
set(gca,XDir="reverse",View=[30 50])
xlabel('Frequency [Hz]');
ylabel('Time (s)');
title(['Original Signal']);         

Hope this helps.