FFT / DFT

Symbols

Table 1 fft_abbrev

name	alt	symbol	unit
sample rate	sample frequency	fs	Hz or 1/s
discretization quantity	bin size	Δt = 1/fs	s
samples		N	(no dimension)
window length	pts (points with no dimension)	wl = T = N * Δt	s

N corresponds to time measured/recorded or used; for a sampling rate of 1024 Hz and different window length you get a corresponding time:

2048 pts @1024Hz = 2s = N/fs
1024 pts @1024Hz = 1s
1024 pts @0.0625 (16s) = 16384s

Definitions

time Domain	symbol	frequency domain	symbol
sampling rate	fs = 1/Δt	(sampling) bandwidth	bw = fs /2
samples	N	spectral lines	N/2 + 1 (+1 == DC part)
window length	wl = T = N * Δt	frequency resolution [Hz] (bandwith)	\(Δf = \frac{bw}{N/2} = \frac{ f_s }{N}\)

Warning

Almost everywhere the DC part (or N+1) is not included.
The FFTW returns the DC part as 1^st element.
You need the DC part only for the inverse transform.
For MT and noise analysis you detrend the data (DC part is removed). And here only you have the symmetry of (for example) 1024 real input and 512 (511 + real) complex output
The FFTW returns the Nyquist frequency (which is a real number) as last element. You can not use the Nyquist frequency in your analysis.

Note

In the real world by selecting N samples = wl you already have a window length of T

The Nyquist frequency is \(f_s /2\) where a sine wave would be described with two points.
Additionally some digitizers or loggers use an anti alias filter at 80% of the Nyquist frequency. So at 1024 Hz sample rate the cut off could be 0.8 * 512 = 410 Hz.
As a good choice for data interpretation a limit of \(f_s / 4\) can be used

weiter

Power Spectral Density (PSD)

When you change the bandwidth the amplitude also changes.

For a sample rate of 1024 we use 1024 samples and 4096 samples, according to 1 s and 4 s data.
The bin size in the frequency domain has changed now \(Δf = \frac{bw}{N/2}\) == 1 Hz and 0.25 Hz frequency resolution, respectively bin sizes. So each bin or bucket of the 4096 window contain 4 times less data.

On https://community.sw.siemens.com/s/article/what-is-a-power-spectral-density-psd you find a genius picture for that:

Warning

Almost everywhere people are playing with sine waves.
In MT we have broad band noise and must scale the FFT. You want to process data with different window length but get the same amplitudes independently from the window length.