具有意外结果的频谱图（Python，scipy.signal）

Question

I'm looking for a clarification of what I get wrong in my understanding or implementation of spectogram plotting.

To be sure that I get everything correct I started with a toy example, so I generated a very simple signal (sum of three periodic signals of different frequency and phase shift, no noise) as fallow:

import numpy as np
sampling_rate = 1.0 / 1000
duration = 2
t = np.arange(0, duration, sampling_rate)

# Signal 1 
A_1 = 0.8 # amplitude of the cosine wave
f_1 = 100 # frequency of the cosine wave
phase_1 = 30  #desired phase shift of the cosine in degrees
phi_1 = phase_1*np.pi/180
s1 = A_1*np.cos(2*np.pi*f_1*t+phi_1)

# Signal 2 
A_2 = 0.3 # amplitude of the cosine wave
f_2 = 8 # frequency of the cosine wave
phase_2 = 45  #desired phase shift of the cosine in degrees
phi_2 = phase_2*np.pi/180
s2 = A_2*np.cos(2*np.pi*f_2*t+phi_2)

# Signal 3 
A_3 = 0.1 # amplitude of the cosine wave
f_3 = 60 # frequency of the cosine wave
phase_3 = -10  #desired phase shift of the cosine in degrees
phi_3 = phase_3*np.pi/180
s3 = A_3*np.cos(2*np.pi*f_3*t+phi_3)

# Result
x = s1 + s2 + s3

I expected that the spectrogram from the "x" signal would be three horizontal lines corresponding to the three frequencies of the generated signal. The signal doesn't change over time so I expected results similar to FFT, but in a different representation.

But after plotting it with the code below:

from scipy import signal
import matplotlib.pyplot as plt

freqs, times, spectrogram = signal.spectrogram(x)

plt.figure(figsize=(8, 6))
plt.imshow(spectrogram, aspect='auto', cmap='hot_r', origin='lower')
plt.title('Spectrogram')
plt.ylabel('Frequency band')
plt.xlabel('Time window')
plt.tight_layout()

I get something quite unexpected:

The found frequencies do not make sense.

So my question is: where I made a mistake? My expectations were wrong? My implementation is spoiled somehow? Maybe someone can recommend me a good source of knowledge on this topic?

Thank you very much in advance for your time and help.

Answer 1

I find an answer before I ask this question, so instead of deleting the whole post, I will share my result with you. I hope somebody can benefit from it.

The plot is good but the axis are badly signed. If you strain your eyes, you will notice that there are in fact three bars, (the weakest is around 15). The visibility of the line is proportional to the amplitude value of the component signal, which makes sense because of the color codes for the amplitude. So now is quite easy to notice that the y axis is for some reason divided by 4. A similar problem is with the x-axis, which is even more understandable when you realize that we are not giving information about the sampling frequency anywhere.

So now we can correct this problems, and changes color palette as fallow:

freqs, times, spectrogram = signal.spectrogram(x,fs=1./sampling_rate)

plt.figure(figsize=(8, 6))

plt.pcolormesh(times, freqs, spectrogram, shading='gouraud')
plt.ylabel('Frequency [Hz]')
plt.ylim([0,110])
plt.xlabel('Time [sec]')

plt.tight_layout()
plt.show()

Which brings us to the expected result(Amplitude for s2 multiplied by two):

I hope that's helpful.