If we find the cross-correlation of a sine wave with a white noise process, why does the resulting signal also show harmonic properties with the same frequency as the input sine wave?
I would have thought that as the processes are separate and unique we would see no correlation?
The python code I'm using to generate the image is:
import numpy as np
import matplotlib.pyplot as plt
fs = 500 # Sample rate
l = int(1*fs) # Length of correlation to calculate
sine_wave = np.sin(3*2*np.pi*np.arange(0,30,1/fs))
white_noise = np.random.normal(0,1,len(sine_wave))
corr = np.correlate(white_noise,sine_wave[:-l+1],mode='valid')
plt.close('all')
fg,ax = plt.subplots()
ax.plot(np.arange(0,l/fs,1/fs),corr,'k')
ax.set_xlabel('Time [s]')
ax.set_ylabel('Amplitude')
1 second cross-correlation signal between 30 seconds of a 3Hz sine wave and a white noise process:
The background to the problem is that I'm looking at a signal processing problem for extracting harmonic behaviour from structures excited by white noise. Several methods for extracting this behaviour use the auto-correlation of the structures acceleration response. The basis for this is that there is no correlation between the white noise forcing and the past structural response.
However when I look at the residual errors (difference in measured autocorrelation and the expected autocorrelation) in the methods they exhibit this same behaviour where there is this additional correlation between the white noise and the structures response. I've mimicked these results by taking the cross-correlation of the sine wave and a stochastic response, something which I assumed would be close to zero and stochastic. However instead its showing these oscillations.
Best Answer
$$R (t_1, t_2) := \mathbb E \left[ \sin(\omega_0 t_1) \, w (t_2) \right] = \underbrace{\mathbb E \left[ \sin(\omega_0 t_1) \right]}_{= \sin(\omega_0 t_1)} \, \underbrace{\mathbb E \left[ w (t_2) \right]}_{=: \mu_w} = \color{blue}{\mu_w \sin(\omega_0 t_1)}$$