Inputting Power Spectral Density into System Transfer Function to obtain Output PSD

I have come across the relation where and refer to the output and input spectral densities respectively, and is the system transfer function.

If I have already defined the system transfer function in the complex frequency domain using the following (for example)

s = tf('s');
H = (1+s*T2)/(1+s*(T2 + T1));

How would I go about inputting an arbitrary Power Spectral Density in this transfer function with the goal of observing the output spectral density? I am assuming the above relation is still valid in the complex frequency space but I may be wrong.

Any help would be greatly appreciated!

Answer 1

[Mathieu NOE]

hello

if your input PSD is a numerical vector defined for a frequency vector freq, you simply have to do dot products like this :

PSD_input = .... vector
freq = .... vector

s = j*2*pi*freq; % complex pulsation vector

% compute transfer function modulus 
tf_mod = abs((1+s*T2)./(1+s*(T2 + T1));

% PSD out 
PSD_out = (tf_mod.^2).*PSD_input; % that's all folks !

[RPS19]

Thanks Mathieu! Just to be clear, since I'm very new to all of this, the frequency vector would just be describing a range of frequencies that I am trying to investigate and the PSD_input vector would be of the same dimension describing the power at each of those frequencies?

[Mathieu NOE]

exactly !!