plot aprove fourier transform property in matlab

To plot and verify the Fourier Transform properties in MATLAB, you can follow these steps:

1. Generate a test signal. Let's use a simple sine wave as an example:

main.m
fs = 1000; % sampling frequency
t = 0:(1/fs):1; % time vector
f = 10; % frequency of the sine wave
x = sin(2*pi*f*t); % sine wave signal
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2. Compute the Fourier Transform of the signal using the fft function:

main.m
X = fft(x);
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3. Plot the magnitude spectrum of the Fourier Transform:

main.m
frequencies = 0:(fs/length(x)):(fs-(1/fs));
magnitude_spectrum = abs(X);
plot(frequencies, magnitude_spectrum);
xlabel('Frequency (Hz)');
ylabel('Magnitude');
title('Magnitude Spectrum');
grid on;
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4. Verify the Fourier Transform properties. Here are a few examples:

• Linearity:

  main.m
  a = 2; % scaling factor
  y1 = a * x; % scaled signal
  Y1 = fft(y1); % Fourier Transform of the scaled signal
  
  y2 = 3 * x; % another scaled signal
  Y2 = fft(y2); % Fourier Transform of the other scaled signal
  
  % Verify linearity property
  is_linearity_preserved = isequal(a * X, Y1) && isequal(3 * X, Y2);
  disp(is_linearity_preserved);
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• Time Shift:

  main.m
  t_shift = 0.3; % time shift value
  x_shifted = sin(2 * pi * f * (t - t_shift)); % time-shifted signal
  X_shifted = fft(x_shifted); % Fourier Transform of the time-shifted signal
  
  % Verify time shift property
  is_time_shift_preserved = isequal(X_shifted, exp(-1i * 2 * pi * f * t_shift) * X);
  disp(is_time_shift_preserved);
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• Frequency Shift:

  main.m
  f_shift = 5; % frequency shift value
  x_shifted_freq = sin(2 * pi * (f + f_shift) * t); % frequency-shifted signal
  X_shifted_freq = fft(x_shifted_freq); % Fourier Transform of the frequency-shifted signal
  
  % Verify frequency shift property
  is_frequency_shift_preserved = isequal(X_shifted_freq, X.* exp(1i * 2 * pi * f_shift * t));
  disp(is_frequency_shift_preserved);
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• Convolution:

  main.m
  impulse_response = [1, 0.5, 0.2, 0.1]; % example impulse response
  y_conv = conv(x, impulse_response); % convolution of the input signal and impulse response
  Y_conv = fft(y_conv); % Fourier Transform of the convolved signal
  
  impulse_response_FT = fft(impulse_response); % Fourier Transform of the impulse response
  
  % Verify convolution property
  is_convolution_preserved = isequal(Y_conv, X .* impulse_response_FT);
  disp(is_convolution_preserved);
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These are just a few examples of how you can plot and verify Fourier Transform properties in MATLAB.