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I've been trying to prove the converse, that is, if a system is BIBO stable, it is summable. I was able to prove it if the system is FIR:

Let H be an FIR LTI system that is BIBO stable. Then for any input signal x(t) bounded by M, i.e. |(x(t)| < M for all t, the response y(t) is bounded, i.e. |y(t)|<M'.

If the input is the impulse (which is clearly bounded), then output is the impulse response, h(t), which must also be bounded since we assumed BIBO stable.

So |h(t)| < M' for some M'. Since the system is FIR, h(t) has finite duration, so ## \int^{\infty}_{-\infty} |h(t)| dt ## exists. Hence the system is summable.

But is this true even if the system isn't FIR? If so, how might I prove this?

Thanks!

BiP