Hello. I'm studying signals and systems on my own this summerand I'm trying to get a good grasp of the convolution. Ithink I understand it mathematically enough to do some problems,but I don't have a firm grasp by any means. I'm studying bothdiscrete and continuous time cases. Before I get to myquestions, here is how I understand things in my own words (if anyof this is off, please let me know):

Sifting Property

The value of a signal at a time t can be found by summing theproduct of the impulse response and the signal at all times. Since the impulse response will be 1 only at time t and 0everywhere else, you will "sift" out only that value of the signalat time t. (1 times the signal at time t will just be thesignal)

Impulse Response

Written as in my texts (Oppenheim and Schaum's), theimpulse response is just the output of a system at a time when the input is the unit impulse.

In other words, I just think of the impulse response as what youget out of a system if you send in a 1 at a certain time.

This is where I may be confused. I'll ask a question aboutthis below.

And finally:

The Convolution

If you know the impulse response of a system at a time t, you justhave to scale it by the value of to get the response of the system to. For the system response:

I sort of think of this as multiplying a unit area (ie, 1m2) by a scalar, C, to get an area of (C m2).(the unit impulse response is analogous to the unit area, and thescalar is analogous to the input ).

Actually... I think I may be more confused about theconvolution than I realize. If you have any good tips on howto think about it, please let me know.

Ok. Now my questions:

Question I: SiftingProperty

What good is the sifting property??? It seems to be circularin its logic! I mean, you are basically saying you can get if you know !! You're just going through theextra step of multiplying all values of by to "catch" the ... But that means you already had in the first place! So what theheck is the point?!?

Question II: The ImpulseResponse

Is a 1, or infinity at time ?? When it's under the integral, Iknow it is 1, since it has unit are. But when the impulseresponse is described, it seems to be the response to the impulse,with no integral involved. Here is how my Schaum's definesit:

where T is the LTI system.

And if it is infinity, how can a system respond to an infiniteinput? This, I think, is my biggest point of confusion, andmay be why I'm having trouble understanding the convolutionfully.

Question III: TheConvolution

Why would you have the response of a system to the unit impulse,but not have its response to the signal ? If you could get the impulseresponse, why not just get the response and forget about the convolutionaltogether?


Well, I think that sums up my confusion for now. :) I hope myquestions made sense! I will be thrilled if someone is niceenough to clear some of these issues up for me!

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