I can't understand the discrete time unit impulse response and convolution

[Jncik]

hi, i have trouble in understanding the concepts of the impulse response

first of all, let's assume that we have a signal y[n] = x[n] which is time invariant and linear, hence if I understand correctly linear means that if for input

a*x1[n] we have an output a*y1[n]
b*x2[n] we have an output a*y2[n]

then for input a*x1[n] + b*x2[n] we will get an output a*y1[n] + b*y2[n]

as for the time invariance, suppose we have y[n] = x[n]

we do a left shift for x[n] by 2 and we have an output y1[n] = x[n+2]

the system would be time invariant if y[n+2] would give us the same output we get when we shift the input x[n+2] right? (please if i say something wrong correct me)

in this case the system is time invariant because y[n+2] = x[n+2] = y1[n]

OK now, let's go to the convolution stuff. I know that we can write a signal x[n] as a superposition of scaled versions of shifted unit impulses, i can understand this graphically and from the sum that we get

(1) [PLAIN]http://img69.imageshack.us/img69/8419/55080091.gif


Now in my book it has a page which is really important and I can't go on if I don't understand this page, so IM going to write it here

the RED will be what I don't understand

"The importance of the sifting property (1) lies in the fact that it represents x[n] as a superposition of scaled versions of a very simple set of elementary functions, namely, shifted unit impulses δ[n-k], each of which is nonzero(with value 1) at a single point in time specified by the corresponding value of k. The response of a linear system to x[n] will be the superposition of the scaled responses of the system to each of the shifted impulses.

"

ok wait a minute, what does "response" mean? the scaled responses of the system to each of the shifted impulses, I understand that we get the y[n] from these shifted impulses that get multiplied with x[n], but what kind of response does the system have to each of the shifted impulses? i don't get the meaning from this..

"More specifically, consider the response of a linear (but possibly time-varying) system to an arbitrary input x[n]. We can represent the input through (1) as a linear combination of shifted unit impulses. Let h_k[n] denote the response of the linear system to the shifted unit impulse δ[n-k]. Then from the superposition property for a linear system the response y[n] of the linear system to the input x[n] in (1) is simply the weighted linear combination of these basic responses. That is with the input x[n] to a linear system expressed in the form of (1) the output y[n] can be expressed as

[PLAIN]http://img101.imageshack.us/img101/6775/31522193.gif (2)

Thus, according to (2) if we know the response of a linear system to the set of shifted unit impulses, we can construct the response to an arbitrary input."

first red: can you please explain me what does h exactly mean? i don't understand it...
second red: how is this possible? i don't get it... can you please explain it to me with the simplest words you can find?

thanks in advance

 

[cepheid]

The response of a linear system to x[n] will be the superposition of the scaled responses of the system to each of the shifted impulses.


Let h_k[n] denote the response of the linear system to the shifted unit impulse δ[n-k].

Here is what the impulse response of a system means:

IF I send the system an input x[n] = δ[n], AND the output that results is a signal y[n] = h[n], then h[n] is called the "impulse response" of the system.

In other words, h[n] describes how the system responds (i.e. what output signal it produces) when you "hit" it with a spike (impulse) of unit height at t = 0. So the procedure would be: send a unit impulse into the system and measure the output signal that produces. The result is the impulse response of the system.

Note that although an impulse lasts for only one timestep, the response h[n] typically is a signal that goes on for quite a few timesteps (it takes the system a while to settle down after you give it that initial "kick" at t=0). h[n] may even be infinite in time e.g. if the system is mechanical -- like an oscillator with no friction, and you give it "kick" (impulse) at t = 0, it will continue to oscillate forever. Its response to an impulse will be a sinusoid of a certain period. Of course, this is a continuous-time example, but that's okay.

Thus, according to (2) if we know the response of a linear system to the set of shifted unit impulses, we can construct the response to an arbitrary input."

Okay, now: because of the time invariance of the system, if, instead of sending a unit impulse to the system at t = 0, I instead DELAY by one timestep, (i.e. I send δ[n-1] as an input), then the output will just be the time-delayed response: h[n-1].

We can represent ANY discrete time signal as a bunch of shifted and scaled unit impulses. Basically feeding in an input of x[n] is like sending a whole stream of impulses, one after the other, one for every timestep. Therefore, we can express x[n] as:

x[n] = ...+ x[-2]δ[n+2] + x[-1]δ[n+1] + x[0]δ[n-0] + x[1]δ[n-1] + x[2]δ[n-2] + ...

Where the sum goes to infinity in either direction. Each value of x[n] is just a unit impulse that has been shifted to the correct timestep and then scaled to the correct height (i.e. multiplied by the value that the function x is supposed to have at that timestep).

Due to linearity and time invariance, the response of the system to a bunch of shifted and scaled unit impulses will just be a bunch of shifted and scaled impulse responses (and yes, these do overlap with each other). Basically at time t = 0, you hit the system with x[0] producing an output x[0]h[n], but then at time t = 1, you hit the system with x[1], which produces output x[1]h[n-1], which overlaps with the previous output x[0]h[n] so that the result so far is the [STRIKE]product[/STRIKE] SUM of the two (EDIT: sorry for this mistake!). This continues at every time step so that the overall output is just:

y[n] = ...+ x[-2]h[n+2] + x[-1]h[n+1] + x[0]h[n-0] + x[1]h[n-1] + x[2]h[n-2] + ...

= ∑x[k]h[n-k] from k = -∞ to +∞

We call this type of sum a convolution sum. The result is that the output of a linear, time invariant system as a result of any input x[n] is just the convolution of that input with the impulse response h[n] of the system. If you know h[n] (how the system responds to a unit impulse input), then you know everything there is to know about the system.

 

[Jncik]

wow, thanks a lot cepheid! You explained it really great, made everything clear to me! :) :) :)