Coherence Mathematics

March 22, 2019

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).

In mathematical literature, two coherence functions are defined.

Complex coherence:

(1)   \begin{equation*} C_{xy}(f)=\frac{\hat{S}_{xy}(f)}{\sqrt{\hat{S}_{xx}(f)\hat{S}_{yy}(f)}} \end{equation*}

Ordinary coherence:

(2)   \begin{equation*} \gamma_{xy}^2(f)=\frac{|\hat{S}_{xy}(f)|^2}{\hat{S}_{xx}(f)\hat{S}_{yy}(f)} \end{equation*}

where:

  • Sxx(f) is the PSD of x
  • Syy(f) is the PSD of y, and
  • Sxy(f) is the 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 linearity of the system under test at frequency f. An ordinary coherence value close to 1 indicates that the system is linear time-invariant at the frequency f. An ordinary coherence value close to zero indicates that the system is either non-linear, is 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 DTFT of this response is H(ω). If the system is linear time-invariant (LTI), then:

(3)   \begin{equation*} S_{xy}(\omega)=H(\omega)S_{xx}(\omega), S_{yy}(\omega)=|H(\omega)|^2S_{xx}(\omega) \end{equation*}

The coherence will be:

(4)   \begin{equation*} |C_{xy}(\omega)|\triangleq\left|\frac{S_{xy}(\omega)}{\sqrt{S_{xx}(\omega)S_{yy}(\omega)}}\right|\text{ by definition of coherence }C_{xy}(\omega) \end{equation*}

(5)   \begin{equation*} =\left|\frac{H(\omega)S_{xx}(\omega)}{\sqrt{S_{xx}(\omega)S_{yy}(\omega)}}\right| \end{equation*}

(6)   \begin{equation*} =\frac{|H(\omega)|S_{xx}(\omega)}{\sqrt{S_{xx}(\omega)S_{yy}(\omega)}} \end{equation*}

(7)   \begin{equation*} =\frac{|H(\omega)|S_{xx}(\omega)}{\sqrt{S_{xx}(\omega)|H(\omega)|^2S_{xx}(\omega)}} \end{equation*}

(8)   \begin{equation*} =1 \end{equation*}

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)   \begin{equation*} \gamma_{xy}^2=\frac{|S_{xy}|^2}{[S_{xx}S_{yy}]}=\frac{S_{xy}S_{xy}}{S_{xx}S_{yy}}=\frac{[\frac{S_{xy}(f)}{S_{xx}}]}{[\frac{S_{yy}}{S_{xy}}]}=\frac{H_{1}}{H_{2}} \end{equation*}

(10)   \begin{equation*} \text{As 0} \leq \gamma_{xy}^2(f) \leq 1, 0 \leq |\frac{H_{1}}{H_{2}}| \leq 1, \text{then } |H_{1}(f)| \leq |{H_{2}(f)}| \end{equation*}

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)   \begin{equation*} \gamma_{xy}^2(435)=|C_{xy}(435)|^2=(0.007)^2=(0.0098)^2=\left(\frac{19}{607}\right)^2=\left(\frac{|H_{1}|}{|H_{2}|}\right)^2 \end{equation*}