Auto Correlation (Trace Data)

(not available with all measurement types)

Auto Correlation shows the autocorrelation for the selected input channel. Autocorrelation is a form of correlation, a measure of the similarity between two signals.

Auto Correlation is not available when a digital demodulation measurement type is selected.

Here are some tips when using Auto Correlation:

Use ac coupling only. Correlation measurements are disturbed by dc offsets in the signal.
Some types of averaging may be useful -- rms averaging does not affect correlation measurements, but time averaging can be used to reduce noise, if a consistent trigger is provided. However, averaging is usually unnecessary to make good correlation measurements.
Use appropriate triggering and trigger delays. This is especially true for time averaging.
Use a random noise source for delay measurements. Correlation measurements provide the ability to resolve time differences between waveforms that appear to be random.
Waveforms on the correlation trace may not appear as they do in the time trace. This is particularly noticeable when using correlation to extract synchronous signals from noise. The different shape of some waveforms is a direct result of the mathematical definition of correlation. For example, a correlated square wave appears as a triangle wave. It's important to remember that the period of the waveform is preserved even if the correlation waveform looks different.
To avoid wrap-around effects, correlation produces a time record one-half the length of the measurement time-record.

Theory of Operation

Autocorrelation is a form of correlation, a measure of the similarity between two signals. Correlation is performed by multiplying two signals together at each instant in time and summing all the products. If the signals are identical, every product is positive and the resulting sum is large.

If, however, the two signals are dissimilar, then some of the products are positive and some are negative. In this case, the final sum is smaller because the products tend to cancel.

Autocorrelation performs a correlation of a signal with itself. This is done by multiplying the signal with time-shifted versions of itself and then integrating the result of the multiplication at each time shift. The following is the formula for autocorrelation:

$image\RxxTau_wmf.jpg$

where:

Rxx = autocorrelation function

t = amount of time shift

¥ = infinity

x(t) = signal to be correlated

intgrl = integration

conj = conjugation

t = time

´ = multiplication

That is, the autocorrelation function at time t is found by taking a signal, multiplying it by the same signal displaced (t) units in time, and averaging the product over all time.

Duality With the Power Spectrum

For simplicity and speed, the 89600 VSA performs the autocorrelation operation by taking advantage of its duality with the power spectrum:

Rxx(t) « Gxx(f)

Thus,

Rxx(t) = IFFT Inverse Fast Fourier Transform [Gxx(f)] = IFFT [conj(F[r ´ t]) ´ F(t)]

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

´ = multiplication

r = half size rectangular window (thus the result is 1/2 the original time length)

When to use Auto Correlation

Auto correlation is useful for detecting echoes in a signal. For random noise, an echo appears as an impulse -- if there is more than one echo, multiple peaks on the auto correlation trace will be seen. Keep in mind that an echo appears as an impulse only if the delayed signal has not been filtered. The impulse broadens as the original random noise signal is filtered -- in fact, the width of each peak is inversely proportional to the bandwidth of the signal.

To determine the time delay (in seconds) of an echo, move a marker to the peak of the echo. Note that there is always a correlated peak at zero lag -- this peak marks the original excitation signal. Any other peaks point out that the excitation signal also appeared at another time relative to the original signal. The amplitude value at the zero lag point is the total power in the time record.

This function is also useful for isolating low-level periodic signals from noise. A sine wave signal shows up as a sine wave in auto correlation. A square wave signal shows up as a triangular wave of the same frequency.

Auto correlation is a single-channel measurement. If the original signal is on one channel and the delayed version on another, use Cross Correlation.

Auto Correlation and Averaging

The following formulas show how the VSA calculates auto correlation for different averaging functions:

Averaging type	Autocorrelation trace data
no averaging	c = I { conj ( F {r ´ t} ) ´ F {t} }
RMSAverage	c = I { conj ( F {r ´ t} ) ´ F {t} }
RMS Expon. Average	c = I { conj ( F {r ´ t} ) ´ F {t} }
Peak Hold or Continuous Peak Hold Average	c = I { conj ( F {r ´ t} ) ´ F {t} }
Time Average	AC [n] = I { conj ( F {r ´ AT [n] } ) ´ F {AT [n]} } $image\atN_wmf.jpg$
Time Expon. Average	AC[n] = I { conj ( F {r ´ AT [n] } ) ´ F {AT [n]} } $image\atN1_wmf.jpg$ and 1 £ n £ number of averages

Key:

F = Fast Fourier Transform (FFT)

I = Fast Fourier Transform (IFFT)

AC = Averaged correlation

AT = Averaged time

t = Instantaneous time

c = Instantaneous correlation

r = 1/2 width rectangular window

´ = multiplication

n = Average number