Cross Correlation (Trace Data)
trace data is a two-channel measurement that is most useful for determining time delays of a common signal between two different paths. For example, to determine the speed of sound one microphone could be placed at a sound source and another microphone a known distance away. The cross correlation would show a peak at the time delay between the two microphones. It would also show peaks for each path. The width of each peak is inversely proportional to the bandwidth of the signal. For example, random noise produces a very narrow peak.
This trace data is available only when more than a single input channel is selected.
Overview
Cross correlation is similar to autocorrelation in its derivation:
Ryx(t) = IFFT Inverse Fast Fourier Transform (F(y) ´ conj [F(x)])
where: IFFT = Inverse FFT Fast Fourier Transform: A mathematical operation performed on a time-domain signal to yield the individual spectral components that constitute the signal. See Spectrum.
conj = conjugation
F(x) = FFT of channel M data
F(y) = FFT of channel N data
´ = multiplication
For theoretical information and for tips, see online help for autocorrelation—the tips and theoretical discussion there also applies to cross correlation.
Cross Correlation and Averaging
The following formulas show how the VSA calculates cross correlation for different averaging functions:
| Averaging Type | Cross Correlation trace data |
|---|---|
| No averaging | c = I { F {t2} ´ conj ( F {r ´ t1} ) } |
| RMS Average | c = I { F {t2} ´ conj ( F {r ´ t1} ) } |
| RMS Expon. Average | c = I { F {t2} ´ conj ( F {r ´ t1} ) } |
|
Peak Hold or Continuous Peak Hold Average |
c = I ( F [t2] ´ conj (F [r ´ t1]) ) |
| Time Average |
AC [n] = I { F {AT2 [n] } ´ conj ( F {r ´ AT1 [n] } ) }
|
| Time Expon. Average |
AC[n] = I { conj( F {r ´ AT[n]} ) ´ F {AT[n]} } 
and 1 £ n £ number of averages |
Key:
F = Fast Fourier Transform (FFT)
I = Fast Fourier Transform (IFFT)
AC = Averaged correlation
AT1 = Averaged time, channel N
AT2 = Averaged time, channel M
t1 = Instantaneous time, channel M
t2 = Instantaneous time, channel N
c = Instantaneous correlation
r = 1/2 width rectangular window
´ = multiplication
n = Average number
