# Can I assume system is LTI when given by DTFT of impulse response

I'm having hard time to grasp it probably because i don't fully understand it.
I understand that when a system is given by $h(t)$ (in general $h(t-\tau)$) i can assume that it is a LTI system.
So i guess my gap of knowledge is that im not sure if it is possible to FT or DTFT impulse response of a non LTI system $h(t;\tau)$. I assume not because the transform depends on the time variable but here i have a 2 variables function and im not really sure how the transforms behave under these conditions.
So like the title says, if a system is given by $H(e^{j\theta})$ can i assume that it is a LTI system? and why is that?

• By convention, if a system has an impulse response, it's almost always assumed to be LTI. Furthermore, I would argue that if you're talking about a frequency response $H\left(e^{j\theta}\right)$, you're almost certainly talking about an LTI system. One can imagine what a system with a time-varying impulse response might look like, but a frequency response for a non-LTI system is a bit nonsensical. – Jason R Jul 19 '17 at 17:13
• Yes, thats also what my intuition says. I'm trying to cover this subject from all angles because every time I get stucked on this subject i realize i don't understand it deeply enough to get these conclusions and feel confident with them – Mark Elishaev Jul 19 '17 at 17:23

A 1D LTI system is completely characterized by the function $h(t)=T\{\delta(t)\}$ which is denoted as the impulse response of the system. Given an LTI system with impulse response $h(t)$ you can then find its frequency response as $H(j\omega)=\mathcal{F}\{h(t)\}$ where $\mathcal{F}$ stands for continuous time Fourier transform.
When a system is not LTI but, for example, linear time-varying, then it's characterization is possible with the 2D function $h(t,\tau)=T\{\delta(t-\tau)\}$ which is the response of the system to an impulse applied at time $t=\tau$
When such is the case, the Fourier transform can be aplied in the following sense, treating $\tau$ as fixed (or as a parameter) the Fourier transform $\mathcal{F}\{h(t)_{\tau}\} = H_{\tau}(j\omega)$. So in effect you have to consider 1D CTFT for each $\tau$. Whether this is a useful representation or not, or there are other more useful representations for such systems, requires an indepth analysis of time-varying systems.
• Oh I see, so as I understand it, it is possible to have a frequency response to a linear time-varying system but it depends on the time shifting parameter $\tau$. For different $\tau$ I'll get different frequency response. Thank you. – Mark Elishaev Jul 20 '17 at 4:57
• Rather than having a conventional unique and fixed frequency response which would completely characterize an LTI system, what you have is a plain Fourier transform of a (parametrized) function $h_{\tau}(t)$. And the collection of those transforms is an equivalent definition of the time-varying impulse response. – Fat32 Jul 20 '17 at 10:22