Relation of zero-padding and frequency resolution

Question

Consider simple rectangular pulse and FFT of it in Python:

import numpy as np
import matplotlib.pyplot as plt

def rectangular_pulse(t, amplitude, start, stop):
    wave = np.zeros(len(t))+amplitude
    wave[0:start] = 0
    wave[stop:] = 0
    return wave

t = np.linspace(0,1,1001) # time
fs = 1/t[1] # sampling frequency
N = len(t) # number of points
func = rectangular_pulse(t, 1, 1, 100)

plt.plot(t, func)

FFT = np.fft.fft(func)
freqs_fft = np.fft.fftfreq(len(func), t[1])

if I have a look at the frequency resolution $\left(\frac{f_s}{N}\right)$, I'll get:

freqs_fft[1]

which returns $0.9990009990009989$

If I add 1000 zero points (with the same sampling):

I'll get for freqs_fft[1]: $0.49975012493753124$ - obviously, the value has been changed.

In this case, I have a question: if zero-padding doesn't improve frequency resolution, why do I get "better" frequency resolution in case of adding zero-padding to this signal.

Answer 1

why do I get "better" frequency resolution in case of adding zero-padding to this signal.

You do and you don't. Zero padding increases resolution by interpolating between existing data point. The interpolated data doesn't add any new information, it's linearly dependent on the existing data (from the non-zero padded FFT). So you create more data but not more information. That still can be useful, even just for visual representation, i.e. it often "looks nicer".

There are different ways to interpolate, zero-padding is just an efficient way of doing it.

Answer 2

Zero padding before an FFT increases the number of interpolated points to plot from the longer result, by doing a high quality Sinc interpolation.

With a higher density of plot points, the probability that one of them is closer to some random frequency isolated spectral peak increases.

If an isolated spectral peak is right at a low-res plot point, its estimation won't improve by adding some points to the left and right. If the peak is right between low-res plot points, then increasing the plot resolution by doubling the number of plot points might land on it more directly, thus appearing to be a "higher resolution" result. But the peak hasn't moved. You've just done a better job of interpolation to hunt it down.