June 2, 2021
There are two definitions of coherence in literature: complex and ordinary.
- Sxx(f) is the PSD of x
- Syy(f) is the PSD of y
- Syx(f) is the CSD of x and y
ObserVIEW currently offers complex coherence. To calculate ordinary coherence, take the magnitude squared of the complex coherence.
Of the two definitions, ordinary coherence is more common in literature. However, ordinary coherence is the magnitude squared of complex coherence.
Thus, ordinary coherence is a special case of the more general complex coherence.
Ordinary coherence has two main practical uses.
- Coherence is a measure of confidence that a peak observed in a transfer function is a resonant frequency of the device under test and not a spike due to measurement noise. For example, suppose that a peak at frequency f in a transfer function is observed. Then:
|γyx2 ≥ 0.70||or|||Cyx|≥ 0.85||
High degree of confidence
|γyx2 ≤ 0.50||or|||Cyx| ≤ 0.707||
|0.50 < γyx2 < 0.70||or||0.707 < |Cyx| < 0.85||
- Coherence is a measure of linearity of the system under test at frequency f.
An ordinary coherence value near 1 indicates that the system is linear-time-invariant (LTI) at frequency f. An ordinary coherence value near 0 indicates that the system is non-linear, statistically changing with time or both.
Coherence drops within a resonance, and one must evaluate surrounding coherence values to make a sound judgment.
Coherence as a Measure of Linearity
In many cases, the signal x may represent an input to a system and the signal y may represent the output. Suppose that an input to a system results in a response h[n] and that the DTFT of the response is H(ω). If the system is LTI, then Syx(ω) = H(ω) Sxx(ω) and Syy(ω) = |H(ω)|2 Sxx(ω).
Also, the coherence will be:
|by definition of coherence Cxy|
Thus, for an LTI system, the ordinary coherence γyx2(f) will be 1 for all frequencies. In contrast, if the coherence is less than 1, then it is not LTI. Further, if γyx2(f) is 1 at certain frequencies, then the system is LTI at those frequencies.
Coherence as a Measure of Confidence in Transfer Function H1
If a system is LTI with a frequency response of Syx(ω) = H(ω) Sxx(ω), then H(ω) = Syx(ω)/Sxx(ω), which is the transfer function H1. Thus, if one wants a measure of how good an estimator H1 is likely to be, they can check the ordinary coherence function γyx2(f).
If γyx2(f) is 1 for all or nearly all frequencies, then Hyx1(f) is likely a good Hyx1 estimator. If γyx2 is close to 0, then one should be much less confident about the quality of the Hyx1 estimator.
Coherence as a Measure of Distance Between H1 and H2
You can also view the ordinary-coherence function γyx2(f) as a measure of the distance between Hyx1 and Hyx2.
As 0 ≤ γyx2(f) ≤ 1, then 0 ≤ |H1/H2| ≤ 1. Therefore, |H1(f)| ≤ |H2(f)|.
When the true H is LTI, then H1=H2—meaning the distance is 0— and γyx2=1. If H is not LTI, then the distance will be greater than 0 and γyx2<1.
In the above example, the distance between |H1(435)|=19 and |H2(435)|=607 is great. This is reflected in the following equation:
γyx2(435) = |Cyx(435)|2 = (0.007)2 =(0.0098)2 = (19/607)2 = (|H1|/|H2|)2
Note of Caution
As the name implies, estimators yield estimates. In general, these estimates contain some error compared to the true value. It would be difficult to find a more striking example of this situation than the K=1 Welch estimate of coherence.
Welch Estimate of Coherence
Compare the rectangular window K=1 Welch estimate of coherence with the true coherence of uncorrelated white noise signals. If:
- x[n] and y[n] are zero-mean white noise sequences
- x[n] and y[n] are uncorrelated with each other
- The estimate Ĉyx (ω) is the K=1 Welch estimate of x and y
Then, Ĉyx(ω)= 1.
|by definition of ordinary coherence γyx(ω)|
|by Welch algorithm|
Thus, the K=1 Welch estimate of coherence is always 1. It is an estimate, but is it a good estimate? To answer this question, we can compute the true (non-estimated) coherence.
|by definition of ordinary coherence γyx(ω)|
|by definition of CSD Syx(ω)|
|by definition of cross-correlation Ryx(m)|
|x and y are uncorrelated|
|x[n] and y[n] are zero-mean|
|because the Fourier transform F of 0 is 0|
The true value of the coherence is γyx(ω)= 0, but the Welch estimate is γyx(ω)= 1. In many cases, Welch will likely yield a good estimate. However, as the above example demonstrates, this is not always the case, and some caution is necessary.
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