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Given a continuous-time impulse response $h(t)$, bandlimited to $B$. The discrete-time $h[n]=h(n/(2B))$ is then a unique and perfect representation of $h(t)$ and a discrete-time system $h[n]$ is then a perfect representation of $h(t)$ (impulse variance).

However, $h[n]$ generally requires infinitely many terms. I am looking for a name of systems $h(t)$ where $h[n]$ has finite terms (that means $h(t)$ has finite support) or can arbitrarily well approximated by finite terms. Obviously the name for $h[n]$ (the discrete-time equivalent) is a FIR filter.

Is a system with fading memory the right term? If not, why not and what is the right term?

PS: FIR seems to be a terms almost exclusively used for discrete-time systems. Why does it not exist for for continuous-time systems?

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What you want is theoretically impossible: if $h(t) \neq 0$ only on a finite interval, then it is not bandlimited. Fortunately, all real-world signals and systems have the property that $H(f) \rightarrow 0$ as $|f| \rightarrow \infty$. This allows you to approximate $h(t)$ very closely with only a finite number of samples. For engineering purposes, this is often more than enough.

Note that the property above means that $h(t)$ has finite energy and is absolutely integrable (it is in $L_1$). For the math to work out, it is often also required that $h(t)$ is in $L_2$, or that its square is absolutely integrable. These are the terms that are commonly used to describe these kinds of signals and systems.

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  • $\begingroup$ Thanks! But what exactly does fading memory mean then? Could you describe this in a similar way? $\endgroup$
    – divB
    Oct 22, 2020 at 14:35
  • $\begingroup$ Similarly, do I get advantages with regards to modeling as $h[n]$ when $h(t)$ is not only in $L_1$ and $L_2$ (as you say above) but has also fading memory? (For some reason I do not find a concice definition of this term although I feel I heard it frequently. I found the paper "Fading Memory and the Problem of Approximating Nonlinear Operators with Volterra Series" by Boyd but it is hard for me to follow. $\endgroup$
    – divB
    Oct 22, 2020 at 14:50
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    $\begingroup$ "Fading memory" is a term applied to non-linear systems that can't be characterized solely by an impulse response. A fading-memory system "tends to forget past inputs asymptotically over time"; see for example ieeexplore.ieee.org/document/331544 You may be conflating two different theories: the impulse response is useful when the system is linear; Volterra series are useful for non-linear systems with memory, of which fading memory systems are a subset. $\endgroup$
    – MBaz
    Oct 22, 2020 at 14:52
  • $\begingroup$ From Boyd's paper: For LTI systems, fading memory is actually identical to having a convolution representation. $\endgroup$
    – divB
    Oct 22, 2020 at 16:25

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