I am a bit confused with convolution in the context of DSP.

The definition I have for convolution is as follows: if delta of n is applied to a digital system then the output will be the impulse response of the system h[n]. The impulse response can be determined by looking at the original function and replacing x[n] by delta[n]. Once we have the impulse function that is then convolved with x[n]. i.e. y[n]=x[n]*h[n].
My confusion is this is it the same x[n] that is convolved with h[n] that was used to create h[n]. Or was h[n] created using a separate x[n] to the one convolved with h[n].
Jag.

 

A delta function convolved with h[n] will result in h[n]

δ[n] * h[n]= h[n]

h[n] is determined by inputting the delta function to the system.

The result y[n] is the signal convolved with the transfer function h[n]

y[x] = x[x] * h[n]

 

The impulse response can be determined by looking at the original function and replacing x[n] by delta[n].

Once we have the impulse function that is then convolved with x[n].

My confusion is this is it the same x[n] that is convolved with h[n] that was used to create h[n]. Or was h[n] created using a separate x[n] to the one convolved with h[n].

The answer is right there in your own post. It specifically states that it is NOT the same x[n]. That to find h[n], the original x[n] is REPLACED by a delta function. After you have that, then the h[n] is convolved with the original x[n].

The h[n] describes the system and is independent of the signal fed into it.

Also, hopefully they are going about this material so that you understand WHY the convolution works and WHY a delta function input produces the system impulse response.

 

to both of you.

WBahn: Thanks for clearing that up. I know how convolution operation is carried out for LTI systems mathematically (using convolution sum). The impulse response for a system is found and than convolved with the signal function. I know digital filters can be developed using this method. There is an on line DSP book which shows this.

 

You know the mechanics of how to carry out a convolution, but do you understand WHY convolution works? Why does it yield the correct answer? Why do we use the impulse response and not the step response?

 

WBahn, As far as I understand this the step response (if u[n] is applied to a system the output is s[n]) is reserved for analogue processing. The Impulse (delta) function convolved with an input signal (x[n]) is equal to the input . As any arbitary function can be represented this way

\(^{\infty}_{k=\infty}\)\(\sum x[n] \delta[n-k] = x[n]\)

Therefore: \(x[n]= x[n] *\delta[n]\)

When the impulse response is determined for the input signal that is convolved with the input leading to the output y[n].

\(\sum x[n] \delta[n-k]\) \(\rightarrow\) \(\sum x[n] h[n-k]\)


Convolution in this form only applies to LTI systems, all those conditions are shown the sum itself. If the system is to be causal then the limits are 0 to n.

That is what I understand so far.

 

Your first sum should start at -∞ (simple typo).

When the impulse response is determined for the input signal that is convolved with the input leading to the output y[n].

The impulse response is determined for the system, not the input signal.

Most of you answer is correct, but it also has the flavor of being regurgitated from a textbook. My questions is whether YOU understand WHY convlolution (and not some other equation) works. Do you understant WHY it only works for LTI systems? In other words, WHY wouldn't it work if the system were nonlinear or time-invariant?

Can you explain, without using the word convolution until the very end when you say, "We call this process that we have developed 'convolution'," what convolution is in a manner in which someone that is about to learn it, but has never heard of it, will be able to go, "Oh, makes perfect sense."?

The idea is to start with things the person is already familiar with and use those to move to a point where you can say, "That's all there is to it. We do this often enough that we've given it the name 'convolution' and here is how we express it mathematically."

If you can't do that, then spend some time trying to figure out how you would do that as best you can and post that. Then I'll take my shot at it as we can compare the two.