频带通过python吗？

Question

I would like to filter out unwanted frequencies and keep an only 60Hz signal.

Here is what I have done so far:

import numpy as np
from scipy.fftpack import rfft, irfft, fftfreq
#
time   = np.linspace(0,1,1000)

in_sig = np.cos(54*np.pi*time) + np.cos(60*np.pi*time)  + np.sin(66*np.pi*time);
high_freq = 62;
low_freq = 58;

freqs = fftfreq(len(in_sig), d=time[1]-time[0])
filt_sig = rfft(in_sig)

cut_filt_sig = filt_sig.copy()
cut_filt_sig[(freqs<low_freq)] = 0
cut_filt_sig[(freqs>high_freq)] = 0

cut_in_sig = irfft(cut_filt_sig)

from pylab import *
figure(figsize=(10, 6))
subplot(221);plot(time,in_sig); title('Input signal');
subplot(222);plot(freqs,filt_sig);xlim(0,100);title('FFT of the input signal');

subplot(223);plot(time,cut_in_sig); title('Filtered signal');
xlabel('Time (s)')
subplot(224);plot(freqs,cut_filt_sig);xlim(0,100); title('FFT of the filtered signal');
xlabel('Freq. (Hz)')

show()

Plotted results

As I can see the filtered signal has lower amplitudes at the edges, I assume it could be due to applied rectangular window. What windows would you recommend to use to improve to output?

Answer 1

The issue likely comes from numpy's linspace() . The default mode is to include the endpoint stop . So time is 0, 1/999, 2/999, ..., 1 . On the contrary, fft , handles of signal of length N as a periodic signal sampled at 0, T/N, ... , T(N-1)/N , thus avoiding the redundancy of the endpoint. The computed DFT therefore use a frame of length T=1000/999. Hence the frequencies of the DFT are k*999/1000, not k. Since the length of the frame is not a multiple of the period of the signal (1/6s), a problem named spectral leakage occurs.

To avoid the spectral leakage, the length of the frame can be shortened to a multiple of the period, by removing the endpoint:

time   = np.linspace(0,1,1000,endpoint=False)

It returns time as 0, 1/1000, ....999/1000, handled by the DFT as a frame of length 1, that is a multiple of the period of the input signal (1/6s).

If the length of the frame is not a multiple of the period of the signal, the input signal can be windowed so as to partly mitigate the effect related to the discontinuity at the edge of the frame, but spurous frequencies still exist. Finally, the actual frequencies can be properly computed by estimating the frequency of a peak as its mean frequency wih respect to power density. See my answer to Why are frequency values rounded in signal using FFT?