Plotting Wigner-Ville distribution with pytftb on python 3

I am testing wigner ville distribution to see if it works for the estimation of original amplitude of a signal with noise.

The pytftb provides a wigner ville function that works well with their examples. I use it like this:

tfr = WignerVilleDistribution(prepack[0])
tfr.run()
tfr.plot(show_tf=True)

And it looks like it's working:

However, I can't really understand what's happening here, so i'd like to alter the frequency axis, and probably print regarding to absolute frequency rather than normalized? I don't really understand Wigner Ville Distribution though, so if am looking at it wrong i'd appreciate some help!

Here is some code to plot the Wigner-Ville Transform (WVT) using the absolute frequency. I use the python library: https://pypi.org/project/tftb/, version 0.1.1

First I create a signal:

import tftb
import numpy as np
import matplotlib.pyplot as plt
import scipy.signal as sig


T = 2  # signal duration
dt = 1/500  # sample interval/spacing
freq_s = 1/dt  # sampling frequency
N = T / dt  # number of samples
ts = np.arange(N) * dt  # times

#  constructing a chirp multiplied by a Gaussian
t0 = T/2
freq = np.linspace(10, 30, int(N))
sigma = 0.1
signal = np.cos((ts-t0) * 2 * np.pi * freq) * np.exp(-(ts-t0)**2/(2*sigma**2))/np.sqrt(sigma)

# adding some noise
signal += np.random.randn(len(signal))*0.5


#  plotting the signal
plt.figure()
plt.plot(ts, signal)
plt.show()

Here is a plot of the signal

For reference, I will also construct a spectrogram before the WVT:

# first looking at the power of the short time fourier transform (SFTF):
nperseg = 2**6  # window size of the STFT
f_stft, t_stft, Zxx = sig.stft(signal, freq_s, nperseg=nperseg,
                           noverlap=nperseg-1, return_onesided=False)

# shifting the frequency axis for better representation
Zxx = np.fft.fftshift(Zxx, axes=0)
f_stft = np.fft.fftshift(f_stft)

# Doing the WVT
wvd = tftb.processing.WignerVilleDistribution(signal, timestamps=ts)
tfr_wvd, t_wvd, f_wvd = wvd.run()
# here t_wvd is the same as our ts, and f_wvd are the "normalized frequencies"
# so we will not use them and construct our own.

Now plotting the heatmaps:

f, axx = plt.subplots(2, 1)

df1 = f_stft[1] - f_stft[0]  # the frequency step
im = axx[0].imshow(np.real(Zxx * np.conj(Zxx)), aspect='auto',
          interpolation=None, origin='lower',
          extent=(ts[0] - dt/2, ts[-1] + dt/2,
                  f_stft[0] - df1/2, f_stft[-1] + df1/2))
axx[0].set_ylabel('frequency [Hz]')
plt.colorbar(im, ax=axx[0])
axx[0].set_title('spectrogram')


# because of how they implemented WVT, the maximum frequency is half of
# the sampling Nyquist frequency, so 125 Hz instead of 250 Hz, and the sampling
# is 2 * dt instead of dt
f_wvd = np.fft.fftshift(np.fft.fftfreq(tfr_wvd.shape[0], d=2 * dt))
df_wvd = f_wvd[1]-f_wvd[0]  # the frequency step in the WVT
im = axx[1].imshow(np.fft.fftshift(tfr_wvd, axes=0), aspect='auto', origin='lower',
       extent=(ts[0] - dt/2, ts[-1] + dt/2,
               f_wvd[0]-df_wvd/2, f_wvd[-1]+df_wvd/2))
axx[1].set_xlabel('time [s]')
axx[1].set_ylabel('frequency [Hz]')
plt.colorbar(im, ax=axx[1])
axx[1].set_title('Wigner-Ville Transform')
plt.show()

Here is the result:

Because the signal is real, there is activity in the positive and in the negative frequencies, just like for the FFT. In addition, there is the interference between the 2 terms, at frequency 0 (i.e., in the middle between the 2). This is also what you saw in your graph. What you had above the middle lines are the negative frequencies and what is at the very top is frequency 0.

For an analytical signal, the image is simpler. Here I construct the signal using:

signal = np.exp(1j * (ts-t0) * 2 * np.pi * freq) * np.exp(-(ts-t0)**2/(2*sigma**2))/np.sqrt(sigma)

And here is the resulting spectrogram and WVT:

Let me know if you need any additional clarifications.