# Characterizing a non-LTI system

How should we characterize a non-LTI system? For example we have: $$y[n]=x[3n]+x[2n]+x[n]$$ which is clearly not LTI. Also, the impulse response will be $$h[n]=3\delta[n]$$ and if we take the DTFT of this system I believe $$H(e^{j\omega})=3$$ which is not very intuitive. How can we characterize this system?

• A non-LTI system doesn't have an LTI impulse response, so what you conject can't be right. – Marcus Müller Apr 6 at 7:10
• You can't characterize a function about which you know literally nothing. You need to first set up a mathematical model of your system, and then you can deduct methods to deduct the parameters of that model. – Marcus Müller Apr 6 at 7:11
• @MarcusMüller Thanks for the reply. The only thing I know about this system is that y[n]=x[3n]+x[2n]+x[n]. How can I characterize this system? Is the time-domain equation the only way I can represent this system? – Alireza Apr 13 at 21:30
• It's certainly the most useful. You can do whatever math you want to that equation, and call it a representation, as long as the math is correct. But you'll usually not end up with anything useful. – Marcus Müller Apr 14 at 7:38