# Does fading memory mean impulse response with finite support?

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?

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.

• Thanks! But what exactly does fading memory mean then? Could you describe this in a similar way?
– divB
Oct 22, 2020 at 14:35
• 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.
– divB
Oct 22, 2020 at 14:50
• "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.
– MBaz
Oct 22, 2020 at 14:52
• From Boyd's paper: For LTI systems, fading memory is actually identical to having a convolution representation.
– divB
Oct 22, 2020 at 16:25