Plotting power spectrum in python

Question:

I have an array with 301 values, which were gathered from a movie clip with 301 frames. This means 1 value from 1 frame. The movie clip is running at 30 fps, so is in fact 10 sec long

Now I would like to get the power spectrum of this “signal” ( with the right Axis). I tried:

 X = fft(S_[:,2]);
 pl.plot(abs(X))
 pl.show()

I also tried:

 X = fft(S_[:,2]);
 pl.plot(abs(X)**2)
 pl.show()

Though I don’t think this is the real spectrum.

the signal:
enter image description here

The spectrum: enter image description here

The power spectrum :

enter image description here

Can anyone provide some help with this ? I would like to have a plot in Hz.

Asked By: Olivier_s_j

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Source

Answers:

From the numpy fft page http://docs.scipy.org/doc/numpy/reference/routines.fft.html:

When the input a is a time-domain signal and A = fft(a), np.abs(A) is
its amplitude spectrum and np.abs(A)**2 is its power spectrum. The
phase spectrum is obtained by np.angle(A).

Answered By: glormph

if rate is the sampling rate(Hz), then np.linspace(0, rate/2, n) is the frequency array of every point in fft. You can use rfft to calculate the fft in your data is real values:

import numpy as np
import pylab as pl
rate = 30.0
t = np.arange(0, 10, 1/rate)
x = np.sin(2*np.pi*4*t) + np.sin(2*np.pi*7*t) + np.random.randn(len(t))*0.2
p = 20*np.log10(np.abs(np.fft.rfft(x)))
f = np.linspace(0, rate/2, len(p))
plot(f, p)

enter image description here

signal x contains 4Hz & 7Hz sin wave, so there are two peaks at 4Hz & 7Hz.

Answered By: HYRY

Numpy has a convenience function, np.fft.fftfreq to compute the frequencies associated with FFT components:

from __future__ import division
import numpy as np
import matplotlib.pyplot as plt

data = np.random.rand(301) - 0.5
ps = np.abs(np.fft.fft(data))**2

time_step = 1 / 30
freqs = np.fft.fftfreq(data.size, time_step)
idx = np.argsort(freqs)

plt.plot(freqs[idx], ps[idx])

enter image description here

Note that the largest frequency you see in your case is not 30 Hz, but

In [7]: max(freqs)
Out[7]: 14.950166112956811

You never see the sampling frequency in a power spectrum. If you had had an even number of samples, then you would have reached the Nyquist frequency, 15 Hz in your case (although numpy would have calculated it as -15).

Answered By: Jaime

Since FFT is symmetric over it’s centre, half the values are just enough.

import numpy as np
import matplotlib.pyplot as plt

fs = 30.0
t = np.arange(0,10,1/fs)
x = np.cos(2*np.pi*10*t)

xF = np.fft.fft(x)
N = len(xF)
xF = xF[0:N/2]
fr = np.linspace(0,fs/2,N/2)

plt.ion()
plt.plot(fr,abs(xF)**2)
Answered By: Abhi

You can also use scipy.signal.welch to estimate the power spectral density using Welch’s method.
Here is an comparison between np.fft.fft and scipy.signal.welch:

from scipy import signal
import numpy as np
import matplotlib.pyplot as plt

fs = 10e3
N = 1e5
amp = 2*np.sqrt(2)
freq = 1234.0
noise_power = 0.001 * fs / 2
time = np.arange(N) / fs
x = amp*np.sin(2*np.pi*freq*time)
x += np.random.normal(scale=np.sqrt(noise_power), size=time.shape)

# np.fft.fft
freqs = np.fft.fftfreq(time.size, 1/fs)
idx = np.argsort(freqs)
ps = np.abs(np.fft.fft(x))**2
plt.figure()
plt.plot(freqs[idx], ps[idx])
plt.title('Power spectrum (np.fft.fft)')

# signal.welch
f, Pxx_spec = signal.welch(x, fs, 'flattop', 1024, scaling='spectrum')
plt.figure()
plt.semilogy(f, np.sqrt(Pxx_spec))
plt.xlabel('frequency [Hz]')
plt.ylabel('Linear spectrum [V RMS]')
plt.title('Power spectrum (scipy.signal.welch)')
plt.show()

[fft[2]
welch

Answered By: Noam Peled