scipy cross-correlation: interpretation

Question

I want to compute the phase shift between two 1-D signals of same frequency, but before I'm trying to compute the time shift between. The cross-correlation function seems to be ideal for that but I'm confused on how to interpret scipy cross-correlation. Let's take two sinus with a frequency f0 = 200 Hz, a sample frequency fs = 10000 Hz, playing during 0.1s and with a phase difference of pi.

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

# We build the two sinus
t = 0.1
f = 200
fs = 10000
x = np.arange(0, t, step = 1/fs)
sin1 = np.sin(2 * np.pi * f * x)
sin2 = np.sin(2 * np.pi * f * x + np.pi)

# We compute the scipy cross-correlation    
scorr = signal.correlate(sin1, sin2)
plt.plot(scorr)
plt.title('Scipy Correlation')
plt.show()

With this last graphic output I get:

Now a few questions:

1. We have a x-axis spanning on 2 * fs, as the function is hermitian, I guess that we have the hermitian symmetry? In this case we could just shift it around zero (like fftshift) and only consider the positive axis, right ?

2. If 1) is ok, does my x time vector could fit the x-axis of my cross-correlation ?

3. Finally, and not necessarily related to previous questions, how to read x and y axis ? There is not much practical documentation on cross-correlation product, the only thing I know is that we have to look where the function takes its maximum in order to get the time lag between the two signals. For me the y-axis is just the result of the product of the two signals as in the formula (cross-correlation) (but I don't get why the product of two sinus with amplitude 1 could ouput 500 ...), and the x-axis gives the indice corresponding to the time difference ( and in this case, the indice where the function takes its max corresponds to the time shift I am searching for, hence the utility of plotting the cross-correlation with a fitted x time vector). Is it right or do I misunderstand something ?

Answer 1

1. No, the output is len(x)*2-1 long, an odd number
2. I don't understand the question
3. The x axis is the delay in samples, and the y axis is the cross-correlation. The number of x samples is odd, and the middle sample represents 0 delay.

If you cross-correlate the sin with itself, you will see a peak at sample 999, which is the middle sample, which represents 0 delay. With the signal you've shown, the peak is at sample 1024, which represents a time delay of 1024-999 = +25 samples. There is also a peak at sample 974, which represents a time delay of 974-999 = −25 samples. Sine waves are repetitive and will show multiple peaks as they line up with themselves at different lags.

If you do it again with a white noise signal (randn()), you will see only a single peak and it will be clearer.