# Transfer Function

June 2, 2021

Back to: Fundamentals of Signal Processing

A system is often characterized by a transfer function H(f). In general, H(f) is not known but can be estimated. A transfer function estimate, Ĥ(f), is an estimate of the true transfer function; that is, an estimate helps to identify a system and its properties.

A transfer function H(f) of a system with an input x and output y is a ratio where X(f) is the Fourier transform of x and Y(f) is the Fourier transform of y.

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The transfer function helps to explain the transfer from the input X(f) to the output Y(f). That is, Y(f) = H(f)X(f).

If H is linear-time-invariant (LTI), then a single tone of frequency (f) at the input results in a single tone of the same frequency at the output.

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Mathematically, this means that sinusoids are eigenvectors of the linear operator H.

### Mathematical Details

It is not possible to know the true H(f) because, in general, systems are not linear, change over time (not time-invariant), or have measurement noise.

However, it is possible to estimate H. There are several ways to do so. Currently, ObserVIEW offers four estimators that are based on three other estimators.

- Ŝ
_{yx}(f): estimate of the cross-spectral density (CSD) of y and x. - Ŝ
_{xx}(f): estimate of the power spectral density (PSD) of x. - Ŝ
_{yy}(f): estimate of the power spectral density (PSD) of y.

The following equations are the exact definitions of the transfer functions.

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### Features

#### Transfer Function Estimator Ĥ_{1}

Ĥ_{1} is the least-squares optimal estimate when the input measurement noise is zero. In this case, using Ĥ1 is not just a rule of thumb but mathematically ideal.

When the system is also linear-time invariant (LTI), and the data is statistically stable (wide-sense stationary or WSS), then Ĥ_{1} is not just some estimate of the true H but is equal to H.

#### Transfer Function Estimator Ĥ_{2}

Ĥ_{2} is likely to be the optimal estimate when the output measurement noise is zero, but the input is noise is not zero. If there is no output measurement noise, H is LTI, and x is WSS, then Ĥ_{2} is equal to H.

In the previous case, Ĥ_{1} is biased relative to the true H. Particularly, Ĥ_{1} = H/(1 + S_{uu}/S_{xx}), where S_{uu} is the PSD of in the input noise and S_{xx} is the PSD of the measured signal. That is, Ĥ_{1} is biased and under-estimates H.

Similarly, when the output measurement noise is zero, but the input is noise is not zero, Ĥ_{2} is biased relative to the true H. Particularly, Ĥ_{2} = H(1 + S_{vv}/S_{yy}), where S_{vv} is the PSD of in the output noise and S_{yy} is the PSD of the measured output signal. Ĥ_{2} is biased and overestimates H.

If the input and output measurement noises are both zero, the best selection is Ĥ_{1} because it is the least-squares optimal estimate when the input noise is zero. If input noise = output noise = 0, the system is LTI, and the data is WSS, then Ĥ_{1} = Ĥ_{2} = H. Moreover, the ordinary coherence γ^{2}(f) = 1.

#### Transfer Function Estimator Ĥ_{v}

Again, if the input and output measurement noises are both zero, the best selection is Ĥ_{1}. Note that Ĥ_{1} and Ĥ_{2} are extremes, as Ĥ_{1} is ideal when input noise = 0 and Ĥ_{2} for when output noise = 0. Ĥ_{v} falls somewhere in between Ĥ_{1} ≤ Ĥ_{v} ≤ Ĥ_{2}.

As a generalization, Ĥ_{1}, Ĥ_{2}, and Ĥ_{v} are all special cases of the more general scaled transfer function estimator Ĥ_{s}(f) with scaling factor s, where 0 ≤ s ≤ ∞.

#### Transfer Function Estimator T_{yx}

T_{yx} is the magnitude of the geometric mean of Ĥ_{1} and Ĥ_{2}.

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