Last time, I explained transfer functions, but how exactly are they calculated in a typical FFT analysis system?
Let me write the definition formula again:
H (f) ＝ Y (f) / X (f) ---------------------- (1)
Here, X(f) and Y(f) are the Fourier transforms of the system's input and output signals x(t) and y(t). In actual calculations, equation (1) above is not used. Multiplying the numerator and denominator of the right-hand side of equation (1) by X(f)* (complex conjugate) gives:
H (f) ＝ Y (f) / X (f) ＝ X (f) * Y (f) / X (f)* X (f)
＝ Gxy (f) / Gxx (f) ------------------- (2)
This means that the transfer function H(f) is the ratio of the cross-spectrum Gxy(f) of the input and output of the system to the power spectrum Gxx(f) of the input, and is calculated by equation (2) in a typical FFT analyzer.
Furthermore, the actual spectral estimation is performed by taking M blocks of N-point FFT time window lengths from a sufficiently long sampled time-series signal and averaging them.
Gxx (f) = 1/M Σ (X (f) * X (f))---------------- (3)
Gxy (f) = 1/M Σ (X (f) * Y (f))---------------- (4)
(Σ: The subscript for addition is omitted.)
In other words, the estimated value of the transfer function can be obtained as the ratio of the estimated value of the cross spectrum (equation 4) to the estimated value of the power spectrum (equation 3).
The advantages or reasons for calculating in this way are:
- Using equation (1) directly, the average is unstable and the phases are scattered. Also, with random signals, the instantaneous spectrum X(f) may be zero, making calculation impossible in some cases.
- By using random signals as the input source and performing sufficient averaging, a nonlinear system can be approximated linearly.
- Even if the output signal contains noise unrelated to the signal source, averaging can reduce random errors.
These are some examples.
If we multiply the numerator and denominator of the right-hand side of equation (1) by Y(f)* instead of X(f)* (complex conjugate),
H (f) = Y (f) / X (f) = Y (f) * Y(f) / Y(f)* X(f)
＝ Gyy (f) / Gyx (f) ------------------ (5)
Therefore, it can also be calculated as the ratio of the output power spectrum to the input/output cross spectrum (complex conjugate).
The transfer function obtained from equation (2) is sometimes called H1, and the transfer function obtained from equation (5) is sometimes called H2. H2 estimation can reduce random errors by using sufficient averaging when there is a large amount of noise in the input.
(Excerpt from the email newsletter issued on September 19, 2003)
