# How to interpret results of FFT/DFT?

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Dan Richter on 26 Jan 2023
Commented: Star Strider on 4 Feb 2023
I generate a stepped sinusoidal signal of 5 kHz each 2 µs from 100 data samples on the 32-bit MCU (ATSAMD21). How to properly interpret the results of the FFT/DFT calculated from an Excel CSV file acquired on the oscilloscope? I am mainly interested to find out the 5 kHz peak in the frequency characteristic. Do I need to somehow normalize the data before processing?

Star Strider on 26 Jan 2023
Edited: Star Strider on 27 Jan 2023
The time vector needs to be scaled correctly if the frequency vector is to be correct.
Try this —
T1 = 1200×2 table
Var1 Var2 ____ _____ 0 0.284 1 0.292 2 0.288 3 0.276 4 0.28 5 0.276 6 0.276 7 0.268 8 0.272 9 0.264 10 0.276 11 0.26 12 0.264 13 0.256 14 0.264 15 0.256
t = T1{:,1}*0.5E-6; % Added: 't' in 0.5 µs Steps
s = T1{:,2};
figure
plot(t, s)
grid
xlabel('Time (units)')
ylabel('Amplitude (units)')
Ts = mean(diff(t));
Fs = 1/Ts;
Fn = Fs/2;
L = size(T1,1);
NFFT = 2^nextpow2(L);
FTs = fft((s-mean(s)).*hann(L),NFFT)/L;
Fv = linspace(0, 1, NFFT/2+1)*Fn;
Iv = 1:numel(Fv);
[smax,idx] = max(abs(FTs(Iv))*2);
Frq = Fv(idx)
Frq = 4.8828e+03
figure
plot(Fv, abs(FTs(Iv))*2)
grid
xlabel('Frequency')
ylabel('Magnitude')
xline(Frq, '-r', sprintf('Frequency = %.4fx10^3 Hz',Frq*1E-3))
xlim([0 1E+4]) % Zoom To See Detail
EDIT — (27 Jan 2023 at 13:50)
Scaled ‘t’ to be appropriate by multiplying it by so that the sampling interval ‘Ts’ is .
.
Star Strider on 4 Feb 2023
As always, my pleasure!

### More Answers (2)

Dan Richter on 26 Jan 2023
But how to scale the time vector to get frequency vector correctly? It looks like that time step for 1 200 data points is 500 µs.
Picture is my DFT calculation in C++ VS2022:
C++ code:
// https://www.geeksforgeeks.org/discrete-fourier-transform-and-its-inverse-using-c/
#include <iostream>
#include <math.h>
#include <stdio.h>
const int len = 1200;
// Function to calculate the DFT
void calculateDFT()
{
int xn[len] = { 840, 920, 880, 760, 800, 760, 760, 680, 720, 640, 760, 600, 640, 560, 640, 560, 520, 640, 600, 480, 520, 480, 560, 440, 480, 440, 440, 480, 400, 480, 440, 360, 440, 360, 360, 440, 400, 360, 360, 440, 360, 440, 440, 360, 400, 400, 440, 360, 360, 480, 440, 360, 480, 440, 480, 400, 440, 480, 440, 560, 440, 560, 560, 480, 520, 600, 600, 520, 560, 640, 680, 600, 640, 760, 760, 680, 680, 840, 800, 880, 760, 880, 880, 960, 880, 000, 040, 960, 040, 960, 80, 160, 160, 80, 160, 240, 280, 200, 240, 400, 320, 400, 360, 520, 520, 440, 440, 640, 640, 560, 600, 760, 720, 800, 760, 920, 880, 920, 880, 80, 80, 040, 040, 240, 200, 240, 200, 320, 320, 400, 320, 520, 560, 480, 480, 680, 640, 760, 640, 800, 880, 800, 800, 920, 960, 040, 960, 80, 80, 200, 200, 120, 240, 360, 320, 400, 320, 520, 520, 440, 480, 720, 600, 680, 640, 840, 840, 760, 800, 000, 920, 000, 880, 120, 80, 160, 80, 280, 280, 240, 160, 400, 400, 320, 360, 560, 560, 480, 480, 680, 680, 600, 600, 760, 800, 720, 680, 880, 840, 920, 800, 000, 920, 040, 040, 920, 040, 120, 80, 040, 120, 160, 200, 120, 160, 320, 200, 320, 200, 400, 360, 280, 320, 440, 400, 320, 360, 480, 400, 480, 400, 520, 520, 440, 440, 560, 480, 560, 480, 560, 560, 480, 480, 600, 600, 480, 600, 520, 600, 520, 520, 600, 520, 600, 520, 600, 560, 480, 480, 560, 440, 520, 480, 560, 400, 520, 520, 400, 400, 480, 440, 360, 400, 320, 320, 400, 360, 240, 240, 320, 320, 160, 160, 240, 200, 80, 80, 160, 160, 000, 000, 80, 80, 880, 920, 960, 960, 800, 800, 880, 840, 680, 760, 680, 760, 560, 560, 640, 640, 440, 440, 520, 480, 320, 360, 280, 360, 200, 240, 120, 200, 120, 80, 000, 80, 000, 000, 840, 920, 840, 880, 680, 720, 760, 720, 520, 640, 520, 600, 400, 360, 480, 400, 240, 240, 320, 320, 80, 80, 160, 160, 920, 880, 960, 000, 720, 840, 720, 800, 600, 640, 600, 640, 480, 520, 400, 480, 280, 320, 240, 240, 320, 160, 120, 200, 80, 040, 960, 000, 920, 920, 800, 800, 920, 800, 680, 760, 640, 720, 520, 560, 520, 560, 400, 480, 400, 440, 280, 280, 360, 280, 160, 120, 240, 240, 040, 120, 040, 120, 960, 920, 000, 960, 840, 880, 800, 880, 760, 720, 840, 800, 680, 640, 720, 640, 720, 680, 600, 640, 560, 520, 560, 600, 480, 520, 440, 520, 480, 520, 400, 440, 480, 480, 400, 400, 440, 440, 360, 440, 360, 360, 440, 360, 400, 400, 360, 320, 440, 440, 360, 480, 360, 360, 440, 440, 400, 360, 480, 480, 400, 400, 480, 440, 520, 520, 440, 480, 600, 520, 560, 560, 600, 560, 640, 680, 600, 600, 720, 680, 760, 680, 760, 880, 760, 800, 840, 840, 960, 880, 960, 920, 040, 000, 040, 000, 160, 80, 160, 120, 280, 200, 280, 200, 400, 320, 400, 320, 520, 480, 440, 480, 640, 600, 680, 600, 760, 760, 720, 760, 920, 880, 960, 880, 040, 120, 000, 040, 200, 240, 160, 160, 280, 320, 400, 320, 480, 480, 560, 560, 480, 600, 720, 720, 640, 760, 880, 800, 920, 880, 040, 040, 960, 000, 200, 120, 200, 200, 320, 360, 280, 320, 520, 440, 520, 480, 680, 600, 720, 640, 840, 840, 760, 760, 960, 880, 000, 920, 160, 160, 040, 80, 240, 200, 280, 200, 440, 360, 400, 360, 560, 560, 480, 480, 640, 600, 680, 600, 760, 800, 720, 800, 760, 800, 960, 840, 920, 920, 000, 960, 000, 960, 120, 040, 120, 040, 200, 120, 240, 160, 320, 240, 320, 240, 400, 280, 360, 280, 400, 360, 440, 360, 480, 480, 400, 360, 520, 520, 400, 520, 440, 480, 560, 480, 600, 600, 480, 600, 480, 600, 520, 600, 520, 520, 560, 600, 520, 600, 520, 600, 520, 560, 480, 560, 480, 440, 560, 480, 520, 520, 400, 400, 520, 480, 360, 480, 360, 440, 320, 280, 360, 400, 240, 360, 240, 280, 160, 240, 160, 280, 120, 80, 200, 160, 000, 80, 000, 80, 920, 880, 960, 960, 760, 800, 880, 840, 720, 760, 640, 760, 560, 560, 600, 640, 520, 520, 400, 400, 520, 400, 280, 360, 280, 280, 160, 160, 240, 160, 000, 000, 120, 040, 840, 920, 840, 960, 680, 720, 800, 760, 560, 640, 560, 600, 400, 360, 480, 440, 240, 320, 200, 280, 80, 160, 040, 160, 920, 920, 960, 000, 760, 840, 760, 760, 640, 600, 680, 680, 520, 520, 400, 520, 360, 360, 280, 240, 360, 240, 80, 200, 80, 120, 960, 040, 960, 040, 840, 760, 880, 840, 640, 720, 680, 760, 560, 520, 600, 560, 400, 360, 480, 440, 280, 360, 240, 320, 160, 240, 120, 240, 040, 120, 040, 80, 960, 960, 920, 000, 840, 920, 800, 880, 800, 800, 720, 800, 680, 640, 760, 720, 680, 680, 600, 560, 640, 600, 520, 600, 480, 600, 480, 560, 480, 520, 400, 440, 480, 480, 400, 480, 400, 400, 480, 360, 440, 400, 400, 360, 440, 360, 440, 400, 360, 360, 440, 360, 440, 360, 480, 440, 360, 400, 480, 440, 520, 400, 480, 440, 520, 440, 520, 480, 560, 480, 600, 520, 640, 560, 640, 640, 600, 600, 720, 640, 720, 760, 680, 800, 760, 760, 840, 800, 920, 840, 960, 880, 040, 040, 960, 000, 160, 160, 80, 120, 280, 240, 160, 200, 360, 360, 280, 320, 520, 520, 440, 480, 600, 640, 560, 600, 760, 720, 800, 720, 880, 920, 840, 840, 000, 000, 120, 040, 120, 160, 240, 160, 240, 280, 360, 320, 400, 400, 560, 480, 560, 520, 720, 640, 760, 680, 880, 880, 760, 800, 040, 960, 040, 960, 200, 200, 120, 160, 360, 360, 280, 320, 560, 520, 440, 440, 680, 600, 680, 600, 840, 760, 840, 760, 000, 000, 920, 920, 120, 80, 160, 040, 240, 280, 200, 200, 360, 320, 440, 320, 440, 440, 560, 480, 560, 600, 680, 680, 600, 640, 800, 800, 680, 760, 880, 880, 840, 880, 040, 960, 040, 920, 120, 120, 040, 040, 200, 200, 120, 120, 240, 200, 320, 200, 360, 360, 280, 280, 400, 320, 440, 320, 480, 400, 480, 400, 520, 440, 520, 440, 560, 560, 440, 560, 480, 600, 480, 520, 560, 480, 600, 520, 600, 520, 600, 600, 520, 520, 600, 600, 480, 560, 480, 480, 560, 560, 440, 480, 560, 560, 440, 400, 520, 520, 360, 400, 480, 440, 320, 400, 280, 360, 240, 240, 360, 360, 160, 240, 160, 280, 120, 80, 160, 160, 000, 80, 000, 040, 920, 880, 000, 880, 960, 880, 800, 880, 760, 720, 640, 680, 760, 560, 640, 640, 560, 560, 400, 400, 520, 480, 280, 360, 280, 320, 120, 240, 160, 160, 000, 80, 000, 040, 880, 840, 960, 880, 680, 760, 680, 760, 560, 640, 520, 640, 400, 480, 400, 480, 280, 320, 240, 320, 120, 160, 040, 160, 000, 000, 880, 960, 840, 840, 720, 840, 720, 680, 560, 600, 680, 520, 440, 400, 480, 480, 240, 280, 360 };;
float Xr[len];
float Xi[len];
int i, k, n, N = 0;
printf("Enter the number of "
"points in the DFT: ");
scanf_s("%d", &N);
for (k = 0; k < len; k++) {
Xr[k] = 0;
Xi[k] = 0;
for (n = 0; n < len; n++) {
Xr[k]
= (Xr[k]
+ xn[n] * cos(2 * 3.141592 * k * n / N));
Xi[k]
= (Xi[k]
- xn[n] * sin(2 * 3.141592 * k * n / N));
}
//printf("(%f) + j(%f)\n", Xr[k], Xi[k]);
printf("%f\n", sqrt(Xr[k] * Xr[k] + Xi[k] * Xi[k]));
}
}
// Driver Code
int main()
{
calculateDFT();
return 0;
}

Paul on 27 Jan 2023
By my calculations, if that sine wave is supposed to be 4.82 kHz, then the sampling frequency must be
Fs = 1928000 Hz
and the sampling period
Ts = 1/1928000 % sec
Ts = 5.1867e-07
which isn't close to 2 micro-sec.
Dan Richter on 27 Jan 2023
Thank you. This is I was looking for.

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