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10.3: Fourier Transforms

  • Page ID
    84391
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    By Carey A. Smith

    The code in this section is in file: FFT_demo_1.m

    FFT_demo_1.m

    The Fourier Transform is defined as the integral of a signal multiplied by sine and cosine functions of frequencies which are multiples of a fundamental frequency.

    A common use of FFT's is to find the frequency components of a signal.

    % Computers compute the Fourier Transform of a signal vector using an algorithm called the "Fast Fourier Transform" or FFT.

    MATLAB's function is fft(). The input is a vector y of signal values sampled at times specified by a vector t. For a signal vector, y, and a time vector, t, the syntax to properly scale the frequencies is:

    y_FFT = (sqrt(t(end)))*fft(y);

    Example \(\PageIndex{1}\) Spectral Analysis 1

    %% This example shows the use of the Fourier Transform for spectral analysis.

    clear all; close all; format compact; format short; clc;

    % Create a time vector

    nt = 256; % Vector length (A power of 2 is most efficient)

    dt = 0.01; % (sec) Sampling interval

    t = (0:1:(nt-1))*dt; % (sec) Time vector

    %% Create a Signal vector

    bias = 0; % Average value

    f1 = 4.0 % (Hz) 1st frequency

    a1 = 4.0 % Magnitude of the 1st frequency

    f2 = 10.0 % (Hz) 2nd frequency

    a2 = 2.0 % Magnitude of the 2nd frequency

    y = bias + a1*sin(2*pi*f1*t) + a2*sin(2*pi*f2*t);

    y_power = mean(y.^2) % power in the signal

    %% Plot y vs. t

    figure;

    plot(t,y)

    title('Original Signal vs. Time')

    xlabel('Time (s)')

    grid on;

    %% Compute the Fourier Transform

    y_FFT = (sqrt(t(end)))*fft(y); % Scale the FFT

    f_max = (1/dt) % (Hz) Maximum frequency = Sample frequency

    df = (1/dt)/nt % (Hz) Freq. interval

    freqs = (0:1:(nt-1))*df; % Frequencies of the FFT

    %% Plot the FFT of y

    % The FFT result vector has complex values

    figure;

    plot(freqs,real(y_FFT),'b');

    hold on;

    plot(freqs,imag(y_FFT),'r');

    legend('Real(y\_FFT)','Imag(y\_FFT)');

    title('FFT(y))');

    xlabel('Frequency (Hz)');

    grid on;

    %% Convert the complex Fourier Transform to magnitude squared.

    y_FFT2 = (df/nt^2)*(y_FFT.*conj(y_FFT)); % (df/nt^2)is a scaled factor

    y_FFT2_power = sum(y_FFT2)

    % 10.02

    figure;

    plot(freqs,y_FFT2);

    title('FFT2 Power)');

    xlabel('Frequency (Hz)');

    grid on;

    %% The Nyquist–Shannon sampling theorem says

    % you can only measure frequencies up to 1/2 the sampling frequency.

    f_Nyquist = f_max/2

    % You can see that the power plot is symmetric about the center frequency.

    % The frequencies > f_Nyquist are equivalent to negative frequncies.

    % We double the lower frequncies (except for 0 Hz)

    % and delete the high frequencies.

    y_FFT3 = [y_FFT2(1), 2*y_FFT2(2:(nt/2)), y_FFT2(nt/2+1)];

    y_FFT3_power = sum(y_FFT3)

    % 10.02

    freqs3 = freqs(1:(nt/2+1));

    figure;

    plot(freqs3,y_FFT3);

    title('y\_FFT3 Power Spectrum');

    xlabel('Frequency (Hz)');

    grid on;

    clipboard_e572e7cff6056b04795110075dbb8fcaf.png

    Figure \(\PageIndex{1}\): Signal vs. time

    clipboard_e4a4af1d906f08229f5fb573dbfc4b50b.png
    Figure \(\PageIndex{1}\): The FFT's real and imaginary components
    clipboard_e3403e846d51abf55cd10402a8902318a.png
    Figure \(\PageIndex{1}\): Power Spectrum, 0 to 100 Hz
    clipboard_ea962bc0379c0d760a4fef5029582b48c.png
    Figure \(\PageIndex{1}\): Power Spectrum, folder onto 0 to 50 Hz

    .

    Exercise \(\PageIndex{1}\) Spectral Analysis 2

    Modify the code in the above example as follows.

    Change the 2 frequencies to 6 Hz and 16 Hz.

    Call this signal y0. (This is without noise)

    Use randn() to add a noise vector to y0 with a mean of 0 and a standard deviation of 4. 

    y  = y0 + 4*randn(size(y));

    This adds noise of all frequencies. Such noise is called “white noise”.

    The Power Spectrum should be strong at the 2 main frequencies. A small amount of random power should be at all the other frequencies.

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{2}\) Spectral Analysis with large noise

    Using your code from the previous exercise (Spectral Analysis 2), increase the noise to a point where the noise makes it somewhat hard to see the periodic part of signal in the plot of y vs. t.

    Describe what the Fourier Transform plot shows.

    Are the frequencies of the 2 sine waves identifiable?

    Answer

    Add texts here. Do not delete this text first.


    This page titled 10.3: Fourier Transforms is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Carey Smith.

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