numpy/fft.py

Output:

sin(t)
fft
real 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
imag 0 -10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10
rfft
real 0 0 0 0 0 0 0 0 0 0 0
imag 0 -10 0 0 0 0 0 0 0 0 0

sin(t) + sin(4t)
fft
real 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
imag 0 -10 0 0 -10 0 0 0 0 0 0 0 0 0 0 0 10 0 0 10
rfft
real 0 0 0 0 0 0 0 0 0 0 0
imag 0 -10 0 0 -10 0 0 0 0 0 0

With our understanding of the discrete Fourier transform we see clearly that:

the signal is being decomposed into sinusoidal components
because we are doing the Discrete Fourier transform of a real signal, for the fft, $X_{k} = \overset{ˉ}{X_{N - k}}$ so there is redundancy in the. We also understand that rfft simply cuts off and only keeps half of the coefficients

numpy/fft.py

#!/usr/bin/env python
import math
import numpy as np

def printi(X):
    # Round to int otherwise there is a lot of small numerical noise.
    print('real ' + ' '.join(str(int(np.real(x))) for x in X))
    print('imag ' + ' '.join(str(int(np.imag(x))) for x in X))

def analyze(x):
    # FFT
    X = np.fft.fft(x)
    print('fft')
    printi(X)

    # Real FFT
    Xr = np.fft.rfft(x)
    print('rfft')
    printi(Xr)

N = 20

print('sin(t)')
x = np.array([math.sin(i * 2 * math.pi / N) for i in range(N)])
analyze(x)
print()

print('sin(t) + sin(4t)')
x = np.array([math.sin(i * 2 * math.pi / N) + math.sin(i * 2 * math.pi * 4 / N) for i in range(N)])
analyze(x)
print()

 Ancestors

 Incoming links

Discrete Fourier transform