June 2, 2021

A transfer function, H, can uncover a resonance or anti-resonance in a system. Transmissibility Tyx(f) of a signal pair x and y is as follows, where Syy is the PSD of y and Sxx is the PSD of x.

(1)   \begin{equation*} T_{yx}(f)=\sqrt{\frac{S_{yy}(f)}{S_{xx}(f)}} \end{equation*}

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 Ĥ1 and Ĥ2, such as Ĥv.

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.

  • Tyx(f) ≜ Ŝyy(f) / Ŝxx(f) (real ÷ real = real)
  • Ĥ1(f) ≜ Ŝyy(f) / Ŝyx(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, Tyx 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.


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

(2)   \begin{equation*} S_{yy}(f)=|H(f)|^2 S_{xx}(f) \end{equation*}

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

  1. Tyx is the geometric mean of Ĥ1 and Ĥ2 because Tyx is, in some sense, between Ĥ1 and Ĥ2.

(3)   \begin{equation*} T_{yx}=\sqrt{(\hat{H}_{1}\hat{H}_{2})} \end{equation*}

If there is noise on both input and output, then Tyx is an alternative to Ĥv, which is also between Ĥ1 and Ĥ2.