I am familiar with the continuous Fourier transform yet the more I try to understand FFT, the more I'm confused. If we've got a discrete N sample signal, FFT is a faster way to calculate DFT. If calculating the DFT is independent of frequency bins why do they prop up? For example in the code below that I took from webpage, we're clearly filtering frequencies, in this case the daily temperature, but again I do not understand how can you partition an already discrete sequence in further frequencies called bins. Is there a practical purpose for bins and their interpretability?
from numpy.fft import rfft, irfft, rfftfreq
def low_pass(s, threshold=2e4):
fourier = rfft(s)
frequencies = rfftfreq(s.size, d=2e-3 / s.size)
display(1/frequencies)
fourier[frequencies > threshold] = 0
return irfft(fourier)
how can you partition an already discrete sequence in further frequencies called bins? If the original time-domain signal is $x[n]$ then its DFT is $X[k]$. The value $X[k]$ is the frequency-domain value at the $k^{\rm th}$ bin. The termbinis just used to indicate the frequency domain index $k$. $\endgroup$