In the previous article, we discussed the specific method for calculating transfer functions. This time, we will explain the coherence function, which is important for measuring the accuracy of transfer function estimation.
If the output signal y(t) is a combination of the output signal v(t) obtained by inputting signal x(t) into the transfer system and an unrelated signal (i.e., noise) n(t), then
y (t) = v (t) + n (t) -------------------------(1)
This is the result. Here, only x(t) and y(t) can actually be measured, and v(t) cannot be measured. Also, if the transfer function of the system is H(f),
V (f) = H (f) X (f) -----------------------------(2)
Therefore, the power spectrum of the output signal v(t) due only to the x(t) component is
GVV (f) = |H (f)|2 GXX (f) ---------------------(3)
It can be expressed as follows.
Here, if we calculate the ratio γ2(f) of the power spectrum of equation (3) and the power spectrum of the total output y(t),
γ 2 (f) = |H (f)|2GXX (f) /GYY (f)
(Substitute H(f) = GXY(f) / GXX(f))
= |GXY (f) | 2 /(GXX (f) GYY (f)) ----------(4)
This is the result. This γ2(f) is called the coherence function (or relevance function), and it represents the ratio of the power of the component based on the input signal (i.e., the component linear to the input signal) within the total output power.
Thus, the coherence function is a ratio that expresses the strength of the relationship between the input and the output, so its value is between 0 and 1. If γ² (f) = 0, it means that the input and output are completely unrelated, and if γ² (f) = 1, it means that the output consists entirely of components contributed by the input.
Generally, a coherence function being less than 1 can be caused by one or a combination of the following reasons:
- If the system is not linear
- When external noise unrelated to the measurement system is introduced.
- When leakage error (resolution bias error) occurs
(When the impulse response of the transfer system is longer than the time window length)
- When signals other than the input signal x(t) of interest are input to the system
And so on.
In reality, the coherence function also degrades when the output power spectrum is very small (i.e., when the output does not respond).
The coherence function is used to evaluate the reliability of the transfer function estimation. In practice, frequency components for which a coherence function of 0.9 or higher has been calculated can be considered to have a reliable estimated transfer function.
(Excerpt from the email newsletter issued on October 17, 2003)
