Last time, we discussed the cross-spectrum, which is the frequency function between two channels. This time, as a continuation of that, we will discuss the "transfer function," which is the most important and practically valuable function in a two-channel system in an FFT analyzer. We will then introduce a method for calculating the transfer function using the "cross-spectrum method," which is an important application of the cross-spectrum.
【Note】
Generally, a transfer function is a function that represents the relationship between the input and output of a system, and is defined as the ratio of the Laplace transforms of the input and output signals. However, here we will explain the frequency transfer function (frequency response function), which is a function of frequency.
As explained in the measurement column "Fourier Transform and Convolution," Figure 1 of the linear system is reproduced here.
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Figure 1. Time function in a linear system and its Fourier transform.
Here, if we add the input time signal x(t) to a linear system whose impulse response h(t) and let its output time signal be y(t), then:
................................. （1）
In other words, the output time signal can be expressed as the convolution integral of the input time signal and the system's impulse response.
Furthermore, if we denote the Fourier transforms of x(t), h(t), and y(t) as X(f), H(f), and Y(f), respectively, then by the convolution theorem:
Y ( f ) = X ( f ) H ( f )................................. （2）
It will be.
From equation (2):
................................. （3）
In equation (3), H(f) is called the transfer function. In an FFT analyzer, it is also called the frequency response function (FRF). The transfer function represents the transfer characteristics of the system on the frequency axis and is generally a function of a complex number.
In practical calculations, similar to power spectrum and cross-spectrum estimation calculations,
Averaging is necessary, but by adding averaging to equation (3);
................................. （4）
or;
................................. （5）
Here, the bar at the top represents the aggregate average (additive average).
Is the transfer function estimation calculation performed using equation (4) or equation (5)?
In actual FFT analyzers, the transfer function is not calculated using either equation (4) or equation (5) above for the following reasons.
- The average of an asynchronous Fourier spectrum cannot be calculated correctly (because the phase is random and converges to 0).
- This does not improve the signal-to-noise ratio of the system.
The actual calculation involves multiplying the numerator and denominator of the right-hand side of equation (3) by the complex conjugate X(f)* of the Fourier spectrum of the input signal x(t);
................................. （6）
It will be.
In other words, it is calculated by dividing the cross-spectrum Cxy (f) of the system's input and output by the input power spectrum Pxx (f). Furthermore, the estimated transfer function is calculated as follows:
................................. （7）
As mentioned previously, cross-spectral averaging has a noise reduction effect.
Therefore, the transfer function is usually estimated using equation (7). This is called the cross-spectral method.
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Figure 2 shows an example of a transfer system with noise components added to the output.
As shown in Figure 2, consider a transfer system h(t) in which a noise component n(t) unrelated to the input x(t) is mixed into the output, and let its output be y(t);
Y ( f ) = X ( f ) H ( f ) + N ( f ) ................................. （8）
It will be.
................................. （9）
Here, by performing sufficient averaging, Cxn(f) converges to 0;
............................... （10）
Therefore, it can be seen that the estimation method in equation (7) is effective in a transfer system like the one shown in Figure 2. Furthermore, considering a method to minimize the error, if we let E(f) be the expected value (mean value) of the error component N(f) in equation (8):
............................... （10）
Let H^(f) be the estimated value of H(f) that minimizes this;
...................... （12）
from now;
...................... （13）
Thus, the least squares approximation method also shows that equation (7) is a valid estimate.
To summarize these points, the reason for adopting equation (7) as the method for estimating the transfer function is:
- The cross-spectral method can improve the signal-to-noise ratio.
- When there is external noise in the output, averaging can minimize random errors.
- The least squares approximation method allows for linear approximation of nonlinear systems by using random signals.
- This can be easily calculated as a secondary process in spectral calculations using an FFT analyzer.
And so on.
Finally, Figure 3 summarizes the process of transfer function estimation using an actual FFT analyzer.
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Figure 3 Flowchart of transfer function estimation calculation using an FFT analyzer
The averaging process loop includes recording of 2-channel time signals and averaging of the spectra (power and cross). The transfer function is calculated as a post-processing calculation from the power spectra (for both channels) and the cross-spectrum estimation results. In addition to the transfer function, the coherence function γ2(f), which can be used to check the reliability of the transfer function, can also be calculated. The coherence function will be discussed in the next installment.
Furthermore, by performing an inverse transform (IFFT) on the calculated transfer function, the impulse response of the original system can also be obtained.
Finally, here's a summary.
- The output of a linear system is represented by the convolution integral of the system's impulse response and the input signal.
- The transfer function of a system can be defined as the ratio of the Fourier spectra of the input and output signals, and is generally a function of complex numbers.
- The actual transfer function estimation method in an FFT analyzer involves dividing the estimated input/output cross-spectrum by the estimated input power spectrum, and this method is called crossspectral...
This is called the "Ru method." - Transfer function estimation using the cross-spectral method has advantages such as minimizing random errors and allowing for linear approximation of nonlinear systems.
- As a post-processing step for spectral estimation using an FFT analyzer, you can calculate not only the transfer function but also the coherence function.
【keyword】
Laplace transform, impulse response, convolution integral, convolution theorem, transfer function, frequency response function, FRF, power spectrum, cross spectrum, cross spectrum method, least squares approximation, post-processing calculation, coherence function
[Reference materials]
- "Digital Fourier Analysis (2) - Advanced Level -" by Kenichi Kido, Corona Publishing Co., Ltd. (2007)
- "Signal Processing," by Iwao Morishita and Hidefumi Obata, Society of Instrument and Control Engineers (1982)
(Excerpt from the email newsletter issued on September 19, 2014)
