Page 63 - Prosig Catalogue 2005

SOFTWARE PRODUCTS
UNDERSTANDING THE CROSS CORRELATION FUNCTION

d + d = 39
1 2
d – d = -11
Training & Support
2 1
Solving gives d = 14 feet and d = 25 feet which are the correct results.
2 1
Note that if we correlate x1 with x2 as the reference then the delay is 11
msecs as shown below.

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

In this case we have

d + d = 39
1 2

d – d = 11
1 2
Solving again gives d = 25 and d = 14.
1 2
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 Software
function H(f) is defined by

where G is the auto spectrum of the “excitation” and G is the cross
xy
xx
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 For reference purposes we have also shown the Coherence function. Hardware
response function.

System Packages

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.

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