# What is the relationship of S-parameters to frequency response?

I have some S-parameters. I'm reading about frequency response and the impulse response and step response.

If I have a set of S-parameters, then is the frequency response for a port (say output port 2, input port 1) simply $$S_{21}$$ for all the frequencies in my S-parameter file?

Let's use an example of a Wilkinson power divider that is used to take an input and split the power in half between two branches.

The ideal S-parameter matrix then looks like this

$$S = \begin{bmatrix} s_{11} & s_{12} & s_{13} \\ s_{21} & s_{22} & s_{23} \\ s_{31} & s_{32} & s_{33} \\ \end{bmatrix} = \begin{bmatrix} 0 & \frac{-j}{\sqrt{2}} & \frac{-j}{\sqrt{2}} \\ \frac{-j}{\sqrt{2}} & 0 & 0 \\ \frac{-j}{\sqrt{2}} & 0 & 0 \\ \end{bmatrix}$$

Practically speaking, you are interested if the device is doing its job. In this case, you want to see if the power divider is indeed splitting the input power in half for the frequencies of interest. In a real divider, there will be deviations on how it performs over frequency with the hope that the best performance is achieved for the bandwidth of interest. To see this, you can plot the S-parameters which are a type of frequency response.

Take this example of a Wilkinson power divider designed to operate at 2.5 GHz

"Frequency response" is a general term for a particular output you want to see over some frequency range. As you suggest, you can plot the S-parameter values over their frequencies, which is a type of "frequency response".

• Thank you for your response and the additional information/illustrations! – catLivingWithCuriosity Jun 18 '20 at 19:53
• @catLivingWithCuriosity If the answer suffices please mark it as accepted to help the site. – Envidia Jun 18 '20 at 20:08