Understanding The Cross Correlation Function

To illustrate the use of the cross correlation function, a source location example is shown below. For this, it is assumed that there is a noise source at some unknown position between 2 microphones. A cross correlation technique and a transfer function like approach were used to determine the location.

To simulate the noise a broad band Gaussian signal was bandpass filtered from 500 to 1500Hz. This random signal, s(t), was generated at 10000 samples/second. Two delayed signals, p₁(t) and p₂(t), were then formed. Assuming the speed of sound in air is 1000ft/second then p₁(t) was formed from s(t) with a 25 msec delay by ignoring the first 250 values. Similarly, p₂(t) was formed with 14msec delay by ignoring the first 140 values. In order to represent dispersion and other specific path effects, two unrelated Gaussian broad band signals, n₁(t) and n₂(t), were also generated, each with approximately 20% of the overall energy of the original signal. The microphone simulation signals, x₁(t) and x₂(t), were then formed from

x₁(t) = p₁(t) + n₁(t)

x₂(t) = p₂(t) + n₂(t)

A section of x₁(t) and x₂(t) is shown below

Correlating x₂ with x₁ as reference gave a peak in the cross correlation at -11 msecs as shown below

The distance between the microphones was 39 feet. A delay of -11 msecs represents a distance of -11 feet.

Thus if we let the distance of the source from microphone one be d₁ and from microphone two be d₂ then we have

d₁ + d₂ = 39

d₂ – d₁ = -11

Solving gives d₂ = 14 feet and d₁ = 25 feet which are the correct results.

Note that if we correlate x₁ with x₂ as the reference then the delay is 11 msecs as shown below.

In this case, we have

d₁ + d₂ = 39

d₁ – d₂ = 11

Solving again gives d₁ = 25 and d₂ = 14.

An alternative approach is to carry out a form of transfer function analysis between the 2 microphones responses. This is not a strict transfer function which is normally an excitation and response to that excitation. However, the delay information is still contained in the signals. Now the transfer function H(f) is defined by

where G_xx is the auto spectrum of the “excitation” and G_xy is the cross spectrum of the response with respect to the excitation. Once H(f) is known then the impulse response function h(t) may be found by inverse Fourier transforming H(f). The way in which h(t) is formed means that positive time delays are from 0 time to the mid time point and negative delays are from the last time point back to the midpoint. An option is available to carry out the time shift automatically.

The graphs below show the two auto spectra, the cross spectrum, the pseudo transfer function, the coherence function and the impulse response function.

The two auto spectra above are very similar as would be expected but differences are discernable. Also, the broadband nature of the uncorrelated noise is evident at around the 62dB level.

The cross spectrum and the transfer function are also very similar except for scale. In both cases, the phase has been unwrapped which shows the effective linear delay.

For reference purposes, we have also shown the Coherence function.

As would be expected in this case the coherence outside the effective bandwidth of the signal is essentially zero. The coherence indicates that everything between about 500 to 1500 Hz is safe to believe but outside that region, there is little or no relationship.

Finally, the “Impulse Response” function was formed by calculating an inverse FFT of the “transfer function” and selecting the timeshift option to get positive and negative time shown.

The peak is just before zero. Showing an expanded view reveals the time delay as -11 msec as expected.

If we had just inverse transformed the cross spectrum we would have just got the correlation function. Dividing by the input auto spectrum is useful however because it effectively normalises the data. Another helpful scheme is to multiply the cross spectrum by the coherence as this effectively eliminates the unrelated parts. This is not necessary here as we have a sufficient signal to noise ratio.

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Dr Colin Mercer

Chief Signal Processing Analyst (Retired) at Prosig

Dr Colin Mercer was formerly at the Institute of Sound and Vibration Research (ISVR), University of Southampton where he founded the Data Analysis Centre. He then went on to found Prosig in 1977. Colin retired as Chief Signal Processing Analyst at Prosig in December 2016. He is a Chartered Engineer and a Fellow of the British Computer Society.

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