Coherence Mathematics

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Coherence indicates how closely a pair of signals (x and y) are statistically related. It is an indication of how closely x coheres or “sticks to” y.

In other words, coherence is how much influence events at x and events at y have on one another. The magnitude of coherence will always be between zero (no influence) and 1 (direct influence).

Mathematical literature defines two coherence functions.

Complex coherence:

(1)  

Ordinary coherence:

(2)  

where:

• Sxx(f) is the power spectral density (PSD) of x
• Syy(f) is the PSD of y, and
• Sxy(f) is the cross-spectral density (CSD) of x and y

Of the two definitions, ordinary coherence is more common in literature. However, ordinary coherence is the magnitude squared of complex coherence (γxy2(f)= |Cxy(f)|2). Thus, ordinary coherence is a special case of the more general complex coherence.

Applications of Coherence

In practical use, coherence has two main uses.

First, it establishes 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:

Coherence also serves as a measure of a system’s linearity at frequency f. An ordinary coherence value close to 1 indicates that the system is linear time-invariant at frequency f. An ordinary coherence value close to zero indicates that the system is either non-linear, statistically changing with time, or both.

Note, if coherence drops within a resonance, the surrounding coherence values should be evaluated before making a judgment.

Example Walk-Through

1. Start with a vibration test setup where the accelerometers can measure acceleration in the x and y directions across a range of frequencies. Generate the graphs shown as images in Figures 2.1, 2.2, and 2.3.
2. To determine the spectral relationship between the x and y accelerations, use the transfer function estimators Ĥ1 and Ĥ2.
3. Around 435Hz, both Ĥ1 and Ĥ2 exhibit remarkable gains. Initially, it may be assumed that 435Hz is a resonant frequency for Hxy.
4. However, complex coherence equals 0.007, indicating that 435Hz is probably not a resonant peak.
5. Conversely, the peak at 5330Hz does appear to indicate a resonant frequency, as there is a high coherence value in that vicinity.

Figure 2.1. Ĥ1 transfer function graph.

Figure 2.2. Ĥ2 transfer function graph.

Figure 2.3. Cxy complex coherence graph.

Mathematical Details

Coherence as a Measure of Linearity

In many cases, the signal x may represent an input to a system (e.g., a device under test such as a mechanical device or the earth). The signal y may represent the output.

Suppose that an impulse to this system results in a response h[n] (the “impulse response”) and the discrete-time Fourier transform (DTFT) of this response is H(ω). If the system is linear time-invariant (LTI), then:

(3)  

The coherence will be:

(4)  

(5)  

(6)  

(7)  

(8)  

For a system that is linear time-invariant, the ordinary coherence γxy2(f) will be 1 for all frequencies. Conversely, if the coherence is less than 1, then it is not LTI. And, if γxy2(f) is 1 only 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 linear time-invariant with frequency response (Sxy(ω) = H(ω) Sxx(ω) ), then H(ω) = Sxy(ω)/Sxx(ω). This is the transfer function H1. To measure how “good” an estimator H1 is likely to be, check the ordinary coherence function γxy2(f). If γxy2(f) is 1 for all frequencies or close to it, then Hxy1(f) is likely a good Hxy1 estimator. If γxy2 is close to zero, then one should be much less confident about the quality of the  Hxy1 estimator.

Coherence as a measure of distance between H1 and H2

The ordinary-coherence function γxy2(f) can also be viewed as a measure of the “distance” between Hxy1 and Hxy2. This is because:

(9)  

(10)  

By using the ordinary-coherence function, the distance between |H1(435)|=19 and |H2(435)|=607 is very great. This is reflected in the following equation:

(11)