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I am using an implementation of the averaged cyclic periodogram (section 3.2.4 in Antoni, Jérôme. "Cyclic spectral analysis in practice." Mechanical Systems and Signal Processing 21.2 (2007): 597-630.). The cyclic spectrum is estimated based on two $L$ ln $$\hat{P}_{YX}^{(W)}\left(f,\alpha;L\right)=\frac{1}{K\Delta}\sum_{k=0}^{K-1}X_{N_w}^{k}\left(f+0.5\alpha \right)X_{N_w}^{k}\left(f-0.5\alpha \right)^*$$ where $$X_{N_w}^{k}\left(f\right)=\Delta\sum_{n=kR}^{n=kR+N_w-1}w_k[n]y[n]e^{-j2\pi fn\Delta}$$

This implementation is based on the attached scipy pull request.

Now, if I apply this implementation with the Hann window, on normal noise, I receive an artifact elevation at the modulation frequency representing the half of the window length ($\frac{2F_s}{N_w}$). Seems like a Boxcar window eliminates this problem. Is it the only and the best solution?

import numpy as np
from scipy.signal import csd, windows
import matplotlib.pyplot as plt

fs = 20000
s = np.random.normal(size=fs)
alpha = np.arange(5, 200)

# window = windows.boxcar(256)
window = windows.hann(256)

BiSpectrum = []
for alpha1 in alpha:
    x = s * np.exp(-1j * np.pi * (alpha1 / fs) * np.arange(s.shape[-1]))
    y = s * np.exp(1j * np.pi * (alpha1 / fs) * np.arange(s.shape[-1]))
    f, Pxy = csd(x, y, fs=fs, window=window)
    Pxy = Pxy[f > 0] 
    f = f[f > 0] 
    BiSpectrum.append(Pxy)

BiSpectrum = np.abs(BiSpectrum).T
plt.figure(figsize=[19, 9])
plt.pcolormesh(alpha, f, BiSpectrum, vmax=np.percentile(BiSpectrum, 99.99))
plt.xlabel('Modulation frequency $\\alpha$ [Hz]')
plt.ylabel('Carrier Frequency $f$ [Hz]')
plt.colorbar()
plt.show()

enter image description here

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1 Answer 1

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Answer found in a Boustany, Roger, and Jérôme Antoni. "Cyclic spectral analysis from the averaged cyclic periodogram." IFAC Proceedings Volumes 38.1 (2005): 166-171.

The article states that the artifact is due to leakage and it is solved by increasing the overlap between the CSD windows. Sensitivity test is presented via the envelope spectrum, gained by the sum of the bispectrum over the carrier frequency. Simulation resuts are with an agreement with the article.

enter image description here

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