- #1
Jncik
- 103
- 0
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 hk[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
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 hk[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
Last edited by a moderator: