Coherence Mathematics
March 22, 2019
Back to: Mathematics for Understanding Waveform Relationships
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 crossspectral 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:
Ordinary Coherence 

Complex Coherence 
Description 
• γxy2(f) ≥ 0.70 
or 
Cxy ≥ 0.85 
A high degree of confidence that the peak is a resonant frequency. 
• γxy2(f) ≤ 0.50 
or 
Cxy ≤ 0.707 
Not likely that the peak is a resonant frequency. 
•0.50 < γxy2(f) < 0.70 
or 
0.707 < Cxy < 0.85 
Somewhat likely that the peak is a resonant frequency. 
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 timeinvariant at frequency f. An ordinary coherence value close to zero indicates that the system is either nonlinear, 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 WalkThrough
 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.
 To determine the spectral relationship between the x and y accelerations, use the transfer function estimators Ĥ1 and Ĥ2.
 Around 435Hz, both Ĥ1 and Ĥ2 exhibit remarkable gains. Initially, it may be assumed that 435Hz is a resonant frequency for Hxy.
 However, complex coherence equals 0.007, indicating that 435Hz is probably not a resonant peak.
 Conversely, the peak at 5330Hz does appear to indicate a resonant frequency, as there is a high coherence value in that vicinity.
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 discretetime Fourier transform (DTFT) of this response is H(ω). If the system is linear timeinvariant (LTI), then:
(3)
The coherence will be:
(4)
(5)
(6)
(7)
(8)
For a system that is linear timeinvariant, 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 timeinvariant 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 ordinarycoherence 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 ordinarycoherence function, the distance between H1(435)=19 and H2(435)=607 is very great. This is reflected in the following equation:
(11)