Representation of Sampling Frequency in the Fast Fourier transform

Question

I have 21600 data in the time domain. What Sampling frequency should I use? can someone explain the effect of showing the result by selecting different sapling frequencies? Cheers

Answer 1

For a N sample FFT, the samples are distributed from DC (f=0) up to one sample less than the sampling rate.

I have the plots below demonstrating this (the FFT is an algorithm that gives us the DFT or Discrete Fourier Transform), where the last sample is shown as the sampling rate, but as mentioned above, in this example it would be sample number $101$ not shown that would be exactly $f_s$. What is also shown is the frequency periodicity property of the DFT, where the frequencies above $N/2$ are also equivalently the negative frequencies (where in Matlab / Octave and Python scipy.signal we can use the fftshift command if we want to view the spectrum centered about 0.

The common ways to represent the frequency axis are listed in the plot below, showing how they all relate to the sampling rate. The top row with frequency given in cycles/sec or Hz would be our typical representation in the continuous time domain, as well as radian frequency given as radians/sec which is not shown but would extend from $0$ to $2\pi F_s$. The second and third row are "normalized frequency" where we divide the Hz or radians/sec representations by the sampling rate, which is most commonly used for representing discrete time signals. The bottom row is the frequency index of the DFT, where $k$ represents the bin number for the $N$ samples in the DFT, ranging from $k=0$ to $k=N-1$.