Frequency Analysis from the Basics (18) - "Transfer Functions and Coherence Functions"

Last time, we discussed the "transfer function," which is the most important and practically valuable component in an FFT analyzer. This time, we will discuss the coherence function, which is often used as an indicator of the accuracy of the transfer function estimation. The coherence function represents the ratio of the power of the signal component based on the signal input to the system to the power of the overall signal component output from the system, and it shows how much noise signal is mixed into the output signal.
Furthermore, I will explain other definitions of transfer functions besides the one explained last time, and discuss their relationship with the coherence function.

As in the previous example, let v(t) be the output of the transfer system h(t) with the input signal x(t) added in Figure 1, and let y(t) be the output of the superimposed disturbance noise n(t).

　　　　　　y (t) = v (t) +n (t)

The Fourier transform of equation (1) is:

　　　　　　Y ( f ) = V ( f ) + N ( f )

Here, in Figure 1, v(t) is the output of the system;

The Fourier transform of equation (3) is:

　　　　　　V ( f ) = X ( f ) H ( f )

This means that equation (4) shows that v(t) is a signal that depends only on the input x(t) and is not affected by external noise. However, in reality, only x(t) and y(t) can be measured, and v(t) cannot be measured.

Next, the signal strength, or power spectrum, of v(t) is:

As explained last time, the estimated value of the transfer function obtained from the measured values x(t) and y(t) is:

Therefore, substituting equation (6) into equation (5) gives:

It will be.

We will take the ratio of the power spectrum of v(t) to the power spectrum of the total output signal y(t). Since this ratio has the dimension of the square of the amplitude, γ ²_xy Let (f);

Therefore, it can be calculated from the measurable input signal x(t) and output signal y(t). Equation (8) is the ratio of the power of the signal that depends only on the input signal (the phase-coherent component) to the total output signal power, and is therefore called the coherence function (or correlation function).
In Figure 1, the value of the coherence function is 1 when there is no disturbance noise, and 0 when there is no input-dependent signal v(t), so clearly;

　　　　　　0  ≤ γ ² _xy ( f ) ≤  1

It will be.

In equation (8), the coherence function can be obtained from the averaged spectrum, just like the transfer function in the previous example (Figure 2).

The main reasons why the coherence function becomes less than 1 are:

1. When external noise is introduced into the output system
2. When the system is nonlinear
3. If the output response is longer than the time window length, leakage error will occur.
4. If there is an extreme time delay in the output response

These are some possibilities.

Now, the usual formula for calculating the transfer function is given by equation (6), but by multiplying both sides of the definition by the Fourier spectrum of the output, we can also consider the transfer function given by the following equation.

The estimation method in equation (6) is called _H1 estimation, and the estimation method in equation (10) is called _H2 estimation. The definitions of both equations are as follows:

​_H1 estimation is mainly suitable when the output contains a lot of disturbance noise or when linear approximation of a nonlinear system. In contrast, _H2 estimation is suitable when the input contains a lot of disturbance noise or when leakage error at the resonance point is to be reduced.

Now let's compare _H1 and _H2. The transfer function can be divided into a gain component and a phase component. First, the phases of the two transfer functions are the same as the phases of the cross spectrum, and from equations (11) and (12), it is clear that they are equal. Next, let's consider the ratio of the gain components.

Therefore, the ratio of the two transfer functions _H1 and _H2 is the coherence function. Clearly from equation (9), the relationship between the gains of the two transfer functions is:

This is how it works. As the coherence function decreases, the gain difference between _H1 and _H2 increases, and when it is 1, they coincide.

Furthermore, a direct method for estimating the gain of the transfer function is to approximate it by the ratio of the power spectra of two measured signals, the input signal x(t) and the output signal y(t). Therefore, if the phase of the transfer function is θ(f), the estimated transfer function is:

The estimation method for equation (15) is called H-_V estimation. Since the phase is equal to the phase of the cross spectrum, we can rewrite the phase component part as follows:

It can be defined as follows.

Here, we find the relationship between the gain estimates of the three transfer functions H _V, H ₁, and H ₂.

If we take the ratio of the squares of the gains of _H1 and _HV, then:

This is equal to the coherence function. By comparing equation (13) and equation (17);

From this, we can see that the gain of H _V is equal to the geometric mean (or arithmetic mean on a logarithmic scale) of the gains of H ₁ and H ₂.

From equations (14) and (18):

We can see that this is the case. Here, equality holds when the coherence function is 1.

Generally, H ₂ The estimate is a bit of an overestimation, H ₁ The estimate tends to be an underestimate, H _V While estimation is likely to be the closest to the true value, considering the theoretical clarity and the fact that many models have noise added to the output, and also by always measuring the coherence function simultaneously, the actual value is...
In most measurement examples, H ₁ Estimating the transfer function should be sufficient.

Now, let's return to the topic of coherence functions and discuss some of their applications.

Firstly, it can be used to check the accuracy of the transfer function estimation. When the coherence function is less than 1,
Possible causes include noise contamination, nonlinearity, and the effects of leakage errors.
Secondly, we calculate the contribution power. This allows us to determine how much of the output power is related to the input power component (coherent output power, COP).

Thirdly, we calculate the signal-to-noise ratio (SNR).

Figures 3 and 4 show the transfer function of the transfer system of a low-pass filter including a resonant system as measured by an FFT analyzer.
In the example of the measured coherence function, both figures show:

Top panel: Transfer function of _H1 estimation

Middle section: Transfer function of _H2 estimation

Bottom panel: Coherence function

In the example in Figure 3, we can see that the frequency resolution is insufficient, resulting in leakage errors near the resonance point and a decrease in the coherence function. In the example in Figure 4, by increasing the frequency resolution eightfold, the _H1 estimation shows a nearly correct resonance point value.
Near the semi-resonant point, the _H2 estimate appears to be overestimated.

Finally, here's a summary.

1. The ratio of the power of the total signal component output from the system to the power of the signal component based on the input signal to the system is called the coherence function, and it can be used to check the accuracy of the transfer function estimation.
2. The coherence function, like the transfer function, is the averaged power spectrum and
   It can be calculated from the measured cross-spectrum values.
3. The coherence function is a function normalized by the power of the input and output, so its value will be between 0 and 1.
4. The main causes of a decrease in the coherence function include noise, nonlinearity, leakage errors, and time delays.
5. There are several methods for estimating the transfer function, such as H1 estimation, H2 estimation, and Hv estimation. However, by simultaneously determining the coherence function and measuring it so that it approaches 1, H1 estimation is practically sufficient.
6. Other applications of the coherence function include calculating the signal-to-noise ratio (SNR), which involves determining only the power component related to the input from the total output power.

【keyword】

Transfer function, coherence function, relevance function, nonlinear, leakage error, _H1 estimation, _H2 estimation, _HV estimation, contribution, coherent output power, COP, S/N ratio

[Reference materials]

1. "Digital Fourier Analysis (2) - Advanced Level -" by Kenichi Kido, Corona Publishing Co., Ltd. (2007)
2. "Spectral Analysis of Sound and Vibration," by Hiroshi Kanai, Corona Publishing Co., Ltd. (1999)

(Excerpt from the email newsletter issued on November 20, 2014)