Integration of fft signal

I have the following code

m=importdata("Raw data (1).csv")
t=m(:,1)
v=m(:,2)
L = length(t);
Ts = t(2)-t(1); % Sampling Interval (sec)
Fs = 1/Ts; % Sampling Frequency
Fn = Fs/2;

Good = v- mean(v);                                   % Remove D-C (Constant) Offset
Y = fft(v)/L;
Fv = linspace(0, 1, fix(L/2)+1)*Fn;                         % Frequency Vector
Iv = 1:length(Fv);                                          % Index Vector For Plot
figure(1)
plot(Fv, 2*abs(Y(Iv)))
grid
xlabel('Frequency')
ylabel('Magnitude')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

I would like to integrate the signal. How is this possible ?

Answer 1

[Soumya]

Hi Moza,

There are many ways in MATLAB to integrate discrete data, like the FFT signal in your case. One common approach is to use the trapezoidal rule, which can be implemented in MATLAB using the ‘cumtrapz’ function. The ‘cumtrapz’ function computes the cumulative integral of a vector using the trapezoidal rule, returning a vector of integrated values at each point:

Q = cumtrapz(X,Y);

You can also use the ‘trapz’ function to compute the total integral, which provides a single scalar value that represents the integral over the entire data range:

Q = trapz(X,Y)

Please refer to the following documentation to get more information on:

• Integration of numeric values: https://www.mathworks.com/help/matlab/math/integration-of-numeric-data.html
• ‘cumtrapz’ function: https://www.mathworks.com/help/matlab/ref/cumtrapz.html
• ‘trapz’ function: https://www.mathworks.com/help/matlab/ref/trapz.html

I hope this helps!

Answer 2

[Walter Roberson]

fft() is inherently a discrete process. Integration is inherently a continuous process.

"numerc integration" assumes that the underlying data is a discrete representation of a continous function. However, with fft() being inherently discrete, it is not true that fft() is a discrete representation to a continuous function.

So... logically you cannot integrate the result of fft().

[Paul]

The elements of the Discrete Fourier Transform (DFT), which are computed by fft, are samples of the Discrete Time Fourier Transform (DTFT), which is a continuous function (of frequency) for any signal for which the DFT would apply. So, one can use cumtrapz (or whatever) to approximate the integral of the DTFT, which can be computed by freqz for any signal for which the DFT would apply, and which itself, with appropriate scaling, is an approximation of the Continuous Time Fourier Transform. Whether or not the DFT samples are suitable for approximating integral of the DTFT depends on the characteristics of the underlying time domain signal and the amount of zero padding used to compute the FFT. But the DFT elements are samples of a continuous function.

rng(100);
N = 20;
x = randn(1,N);
XDFT = fft(x);
wdft = (0:N-1)/N*2*pi;
[XDTFT,wdtft] = freqz(x,1,1000,'whole');
figure
plot(subplot(211),wdtft,real(XDTFT),wdft,real(XDFT),'o'),grid
plot(subplot(212),wdtft,imag(XDTFT),wdft,imag(XDFT),'o'),grid

[Walter Roberson]

Discrete Time Fourier Transform is not a continuous function of frequency: it is inherently discrete frequency.

[Paul]

No, that's not correct.

The Discrete Fourier Transform is a function of discrete frequency (though the definition on the Wikipedia page uses the integers k as the independent variable). To be more precise, the independent variable is defined only at discrete points on a finite domain; some treatments would define the independent variable at discrete points on an infinite domain (e.g. the integers).

The Discrete Time Fourier Transform is a function of continuous frequency (omega) that covers the entire real line. If there is a source that defines the DTFT with an independent variable that is discrete, I'd really like to see it.

This blog post might be of interest.