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Random Data Analysis Part 7: Cross-Coherence

February 5, 2016 By: Jonathan Dudley

In my last post, I discussed the use of Short-Time Fourier to obtain amplitude, time, and frequency content.  In this post, I would like to introduce the concept of Cross-Coherence and discuss some benefits of performing this analysis

So far, I have dealt with extracting information from a single time series. In this blog, I will discuss the method for relating 2-time dependent signals as a function of frequency. This type of analysis is referred to as the level of “Coherence” between the signals. The Cross-Coherence of two such signals is given by Equation 1 where represents the cross-spectra of the two signals and is normalized by the square root of each signal’s respective auto-spectra.

If the signals are identical, the correlation coefficient is unity, and if they are unrelated the correlation coefficient is zero. If the signals are identical, but the phase is shifted by exactly 180 degrees, then the correlation coefficient is -1. 

Figure 1. Cross-Coherence of a baseline configuration and a geometrically modified scenario sampled at the same points in space and frequency. The FFTs (at the same point) are embedded in the upper right hand corner.

The coherence plot for the cases shows strong coherence with coefficients exceeding 0.80 for most peak tones in the baseline case. Since the pressure signal is being correlated, this tells us that the pressure at the aft-sensing point is strongly correlated with what is happening as you progress away from that point (as x/L decreases). Conversely, for the altered case, the magnitudes of the coherence fall significantly, coinciding with a broadening and lowering of the peaks presented on the embedded FFT spectra. This suggests that the feedback mechanism thought to be responsible for generating these peaks has been disturbed.

As shown, the use of Cross-Coherence plots is yet another method for extracting useful information from your time series data, but their use requires at least two points with the same sampling rate and length. When two independent signals are compared in the time domain, the procedure is known as cross-correlation. Autocorrelation is a special case of cross-correlation where the same signal is compared to itself.

In the next post in this series, I will discuss correlation analysis in the time domain and introduce the reader to an efficient numerical algorithm for post-processing using convolution. If you have further experiences with this method, or have questions please feel free to add a comment.