Complex cross spectral density

Question

mlab.csd from matplotlib: http://matplotlib.org/api/mlab_api.html#matplotlib.mlab.csd can be used to get real valued cross spectral density. If I want to get the phase information from the spectral density, I need a csd calculation which returns complex values. Is there one ?

Answer 1

This is discussed e.g. in this answer: https://stackoverflow.com/a/29306730/3920342

If you use csd of the mlab library you will get complex values so you can calculate phase angles (and the real valued coherence). In the following code s1 and and s2 contain the two signals (in time domain) to be correlated.

from matplotlib import mlab

# First create power sectral densities for normalization
(ps1, f) = mlab.psd(s1, Fs=1./dt, scale_by_freq=False)
(ps2, f) = mlab.psd(s2, Fs=1./dt, scale_by_freq=False)
plt.plot(f, ps1)
plt.plot(f, ps2)

# Then calculate cross spectral density
(csd, f) = mlab.csd(s1, s2, NFFT=256, Fs=1./dt,sides='default', scale_by_freq=False)
fig = plt.figure()
ax1 = fig.add_subplot(1, 2, 1)
# Normalize cross spectral absolute values by auto power spectral density
ax1.plot(f, np.absolute(csd)**2 / (ps1 * ps2))
ax2 = fig.add_subplot(1, 2, 2)
angle = np.angle(csd, deg=True)
angle[angle<-90] += 360
ax2.plot(f, angle)

# zoom in on frequency with maximum coherence
ax1.set_xlim(9, 11)
ax1.set_ylim(0, 1e-0)
ax1.set_title("Cross spectral density: Coherence")
ax2.set_xlim(9, 11)
ax2.set_ylim(0, 90)
ax2.set_title("Cross spectral density: Phase angle")

Here the real and imaginary(!) part of the cross spectral density:

This code is taken from the question How to use the cross-spectral density to calculate the phase shift of two related signals to create two signals s1 and s2:

"""
Compute the coherence of two signals
"""
import numpy as np
import matplotlib.pyplot as plt

# make a little extra space between the subplots
plt.subplots_adjust(wspace=0.5)

nfft = 256
dt = 0.01
t = np.arange(0, 30, dt)
nse1 = np.random.randn(len(t))                 # white noise 1
nse2 = np.random.randn(len(t))                 # white noise 2
r = np.exp(-t/0.05)

cnse1 = np.convolve(nse1, r, mode='same')*dt   # colored noise 1
cnse2 = np.convolve(nse2, r, mode='same')*dt   # colored noise 2

# two signals with a coherent part and a random part
s1 = 0.01*np.sin(2*np.pi*10*t) + cnse1
s2 = 0.01*np.sin(2*np.pi*10*t) + cnse2