CODEWITHSUNDEEP
pythonnumpysignal-processingfft
lll
lll

Reputation: 1109

python numpy.fft.rfft: why is when NFFT included or not, outputs are very different

I am trying to understand to the meaning of NFFT in numpy.fft.rfft. But I get confused why when NFFT included or not, the outputs get very different. Please see the example below.

numpy.fft.rfft([0, 1, 0, 0, 4.3, 3, 599], 8)
array([ 607.3         +0.j        ,   -5.71421356+600.41421356j,
   -594.7         -4.j        ,   -2.88578644-597.58578644j,
    599.3         +0.j        ])

numpy.fft.rfft([0, 1, 0, 0, 4.3, 3, 599])
array([ 607.3         +0.j        ,  369.55215218+472.32571033j,
   -133.53446083+578.34336489j, -539.66769135+261.30917157j])

Upvotes: 0

Views: 2025

Answers (1)

SleuthEye
SleuthEye

Reputation: 14577

The FFT is an efficient implementation of the Discrete Fourier Transform (DFT), which is a discrete function of frequency. It is also related to the Discrete-Time Fourier Transform (DTFT), itself a continuous function of frequency. More specifically, the DFT corresponds exactly to the DTFT evaluated at the discrete frequencies of the DFT.

In other words, when computing a Discrete Fourier Transform with numpy.fft.rfft, you are essentially sampling the DTFT function at discrete frequency points. You can see this by plotting transforms of different lengths on the same graph with the following:

import numpy as np
import matplotlib.pyplot as plt

x = [0, 1, 0, 0, 4.3, 3, 599]

# Compute the DTFT at a sufficiently large number of points using the explicit formula
N = 2048
f = np.linspace(0, 0.5, N)
dtft = np.zeros(len(f), dtype=np.complex128)
for n in range(0,len(x)):
  dtft += x[n] * np.exp(-1j*2*np.pi*f*n)

# Compute the FFT without NFFT argument (NFFT defaults to the length of the input)
y1 = np.fft.rfft(x)
f1 = np.fft.rfftfreq(len(x))

# Compute the FFT with NFFT argument
N2 = 8
y2 = np.fft.rfft(x,N2)
f2 = np.fft.rfftfreq(N2)

# Plot results
plt.figure(1)
plt.subplot(2,1,1)
plt.plot(f, np.abs(dtft), label='DTFT')
plt.plot(f1, np.abs(y1), 'C1x', label='FFT N=7')
plt.plot(f2, np.abs(y2), 'C2s', label='FFT N=8')
plt.title('Magnitude')
plt.legend(loc='upper right')

plt.subplot(2,1,2)
plt.plot(f, np.angle(dtft), label='DTFT')
plt.plot(f1, np.angle(y1), 'C1x', label='FFT N=7')
plt.plot(f2, np.angle(y2), 'C2s', label='FFT N=8')
plt.title('Phase')
plt.legend(loc='upper right')

plt.show()

enter image description here

Upvotes: 2

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