# Transmissibility

June 2, 2021

Back to: Fundamentals of Signal Processing

A transfer function, H, can uncover a resonance or anti-resonance in a system. Transmissibility T_{yx}(f) of a signal pair x and y is as follows, where S_{yy} is the PSD of y and S_{xx} is the PSD of x.

(1)

One should consider using transmissibility if they are not concerned with the statistical relationship between x and y or if there is not a statistical relationship at all. It is also helpful for determining a value between Ĥ_{1} and Ĥ_{2}, such as Ĥ_{v}.

### When not to Use Transmissibility

Transmissibility is not the best option if the goal is complex-valued system response data. Unlike all other transfer functions, transmissibility is real-valued and not complex-valued.

- T
_{yx}(f) ≜ Ŝ_{yy}(f) / Ŝ_{xx}(f) (real ÷ real = real) - Ĥ
_{1}(f) ≜ Ŝ_{yy}(f) / Ŝ_{yx}(f) (real ÷ complex = complex)

One should also not use transmissibility if they are concerned with the statistical relationship between x and y. Unlike all other transfer functions, transmissibility does not directly consider cross-statistical properties between x and y, it only uses the auto-statistical properties of x with x and y with y.

In particular, T_{yx} does not use the cross-spectral density of x and y; rather, it only uses the auto-spectral density (PSD) of x and PSD of y.

### Insights

- For a linear time invariant (LTI) system with frequency response H(f):

(2)

In the case of an LTI system, the transmissibility of x and y is equivalent to the magnitude of the frequency response |H(f)|.

- T
_{yx}is the geometric mean of Ĥ_{1}and Ĥ_{2}because T_{yx}is, in some sense, between Ĥ_{1}and Ĥ_{2}.

(3)

If there is noise on both input and output, then T_{yx} is an alternative to Ĥ_{v}, which is also between Ĥ_{1} and Ĥ_{2}.