It is theorem of any course on signals that a linear time invariant system, whether in discrete or continuous time, is BIBO stable if and only if its impulse response is summable. The fact that summability implies BIBO stability is easy to prove. In fact, it's on the wikipedia page of BIBO stability: http://en.wikipedia.org/wiki/BIBO_stability(adsbygoogle = window.adsbygoogle || []).push({});

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

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# BIBO Stable if and only if impulse response is summable

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