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Correlation: An often overlooked scope measurement

11 Jun 2015  | Arthur Pini

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The correlation function is a valuable signal-analysis tool that engineers often overlook. Its formidable equation, which you have probably not thought about since your undergraduate signals and systems course, is:

where f1 and f2 are real functions of time (t) and τ represents a delay.

You can forget the pain that this equation evoked in your earlier life because modern oscilloscopes and third-party math software easily perform all the computations and make this powerful function available to everyone. Correlation can be classified into either of two functions, auto-correlation or cross-correlation, depending on the number of inputs. In this article, we'll show some common applications for both cross-correlation and auto-correlation.

Correlation functions
Correlation functions were added to the available math functions in oscilloscopes to support two optional disc drive measurements, ACSN (auto-correlation signal to noise ratio) and NLST (non-linear transition shift). While these measurements may not be of general interest, their presence makes the correlation function available for more general applications.

Auto-correlation is the correlation of a signal with itself (single waveform). It provides a measure of the similarity between observations as a function of the time lag between them. It is an analysis tool for finding repeating patterns, like the presence of a periodic signal buried in noise.

Cross-correlation measures of the similarity of two waveforms as a function of a time delay between them. Cross-correlation is used to search for a known short signal in a longer signal (detection) or to measure a time delay between two signals with a common source.

Auto-correlation example
Auto-correlation is typically used to detect periodicity within a signal. In figure 1, the top grid (channel 1) contains the input signal. It is a 10Mbit/s, NRZ (non-return-to zero) PRBS (pseudorandom bit stream) with a PRBS7 pattern that repeats every 127 clocks. It is pretty obvious that there is a repetitive pattern. The next grid down contains the auto-correlation of that PRBS7 signal.

Figure 1: The auto-correlation of a PRBS7 in the upper trace shows the waveform with good SNR. The second trace from the top is the auto-correlation function showing peaks spaced 127 clock periods apart. The third trace is the same waveform with a greater level of additive vertical noise. Note the auto-correlation function of this input, shown in the bottom trace still shows the periodicity of the waveform as 127 clock periods even in the presence of noise.

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