Generalized Cross Correlation (GCC) Estimates

This project provides estimators for the generalized cross correlation according to Knapp and Carter 1976 [KC76].

Implemented Estimators (compare [KC76])

The generalized Estimator can be described by

$\hat{R}_{xy}^{(\text{g})} = \int_{-\infty}^{\infty}{\psi_\text{g}(f) G_{xy}(f)~e^{\text{j} 2\pi f \tau} df}$

where $G_{xy}(f)$ denotes the cross power spectrum of $x(t)$ and $y(t)$ . In this project, all estimates are computed in the spectral domain using the Wiener-Kinchin relations (e.g. $G_{xx}=X(f)X^{*}(f)$ ).

Following estimators are implemented:

Cross Correlation $\psi_{\text{CC}}=1$
Roth; same as the $`H_1`$ estimator describing the Wiener-Hopf filter $\psi_{\text{Roth}} = \frac{1}{G_{xx}(f)}$
Smoothed Coherence Transform (SCOT): $\psi_{\text{SCOT}} = \frac{1}{\sqrt{G_{xx}(f)G_{yy}(f)}}$
PHAse Transform (PHAT): $\psi_{\text{PHAT}} = \frac{1}{|G_{xy}(f)|}$
Eckart $\psi_{\text{Eckart}} = \frac{G_{uu}(f)}{G_{nn}(f)G_{mm}(f)}$
Hanan Thomson (HT), Maximum Likelihood estimator $\psi_{\text{HT}} = \psi_{\text{ML}} = \frac{\left|\gamma_{xy}(f)\right|^2}{\left|G_{xy}\right| \left(1-\gamma_{xy}(f)\right)^2}$ with $\gamma_{xy}(f) = \frac{G_{xy}(f)}{\sqrt{G_{xx}(f)G_{yy}(f)}}$

Insalling

This repo uses a pyproject.toml file generated with the dependency and package managing tool poetry.

This package can be installed with an up to date pip pip install .

or using poetry poetry install

otherwise use your own favorite way to install/use the code in your environment.

Example

import numpy as np
import matplotlib.pylab as plt
from gccestimating import GCC, corrlags

 # generate some noise signals
nsamp = 1024

noise1 =  0.5*np.random.randn(nsamp)
sig1 = np.zeros(nsamp) + noise1

noise2 =  0.5*np.random.randn(nsamp)
sig2 = np.zeros_like(sig1) + noise2

noise_both = np.random.randn(256)

sig1[:256] = noise_both
sig2[500:756] = noise_both

# create a lags array
lags = corrlags(2*nsamp-1, samplerate=1)

# Create the a GCC instance    
gcc = GCC(sig1, sig2)

def mkplot(est, p):
    plt.subplot(p)
    plt.plot(lags, est.sig, label=est.name)
    plt.legend()

# calculate the standard cc estimate
cc_est = gcc.cc()

# plot it using the mkplot function
mkplot(cc_est, 611)

# plot the other estimates
mkplot(gcc.scot(), 612)
mkplot(gcc.phat(), 613)
mkplot(gcc.roth(), 614)
mkplot(gcc.ht(), 615)
mkplot(gcc.eckart(noise_both, noise1, noise2), 616)

# compare cc to the timedomain based 
# implementation from Numpy
# you will see: very close (errors < 1e-13)
plt.figure()
plt.plot(np.correlate(sig1, sig2, 'full'))
plt.plot(gcc.cc())
plt.show()

References

[KC76]: Knapp and Carter, "The Generalized Correlation Method for Estimation of Time Delay", IEEE Trans. Acoust., Speech, Signal Processing, August, 1976

SiggiGue/gccestimating

Generalized Cross Correlation (GCC) Estimates

Implemented Estimators (compare [KC76])

Insalling

Example

References