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

[tex]x(t) = \int_{-\infty}^{+\infty}x(\tau)\delta(t-\tau)d\tau[/tex]

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

Written as [itex]h(t)[/itex] in my texts (Oppenheim and my Schaum's), the impulse response is just the output of a system at a time [itex]t_0[/itex] when the input is the unit impulse [itex]\delta(t-t_0)[/itex].

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

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

And finally:

If you know the impulse response of a system at a time t, you just have to scale it by the value of x(t) to get the response of the system to x(t). For the system y(t):

[tex]y(t) = \int_{-\infty}^{+\infty}x(\tau)h(t-\tau)d\tau[/tex]

I sort of think of this as multiplying a unit area (ie, 1 m^2) by a scalar, q, to get an area of (q m^2). (the unit impulse response is analogous to the unit area, and the scalar is analogous to the input [itex]x(\tau)[/itex]).

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

Ok. Now my questions:

What good is the sifting property??? It seems to be circular in its logic! I mean, you are basically saying you can get x([itex]t_0[/itex]) if you know x([itex]t_0[/itex])!! You're just going through the extra step of multiplying all values of x(t) by [itex]\delta(t-t_0)[/itex] to "catch" the x([itex]t_0[/itex])... But that means you already had x([itex]t_0[/itex]) in the first place! So what the heck is the point?!?

Is [itex]\delta(t-t_0)[/itex] a 1, or infinity at time [itex]t_0[/itex]?? When it's under the integral, I know it is 1, since it has unit are. But when the impulse response is described, it seems to be the response to the impulse, with no integral involved. Here is how my Schaum's defines it:

[tex]h(t) = \textbf{T}\{\delta(t)\} [/itex]

where T is the LTI system.

And if it is infinity, how can a system respond to an infinite input? This, I think, is my biggest point of confusion, and may be why I'm having trouble understanding the convolution fully.

Why would you have the response of a system to the unit impulse, but not have its response to the signal x(t)? If you could get the impulse response, why not just get the x(t) response and forget about the convolution altogether?

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

Thanks!

**Sifting Property**[tex]x(t) = \int_{-\infty}^{+\infty}x(\tau)\delta(t-\tau)d\tau[/tex]

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

**Impulse Response**Written as [itex]h(t)[/itex] in my texts (Oppenheim and my Schaum's), the impulse response is just the output of a system at a time [itex]t_0[/itex] when the input is the unit impulse [itex]\delta(t-t_0)[/itex].

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

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

And finally:

**The Convolution**If you know the impulse response of a system at a time t, you just have to scale it by the value of x(t) to get the response of the system to x(t). For the system y(t):

[tex]y(t) = \int_{-\infty}^{+\infty}x(\tau)h(t-\tau)d\tau[/tex]

I sort of think of this as multiplying a unit area (ie, 1 m^2) by a scalar, q, to get an area of (q m^2). (the unit impulse response is analogous to the unit area, and the scalar is analogous to the input [itex]x(\tau)[/itex]).

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

Ok. Now my questions:

**Question I: Sifting Property**What good is the sifting property??? It seems to be circular in its logic! I mean, you are basically saying you can get x([itex]t_0[/itex]) if you know x([itex]t_0[/itex])!! You're just going through the extra step of multiplying all values of x(t) by [itex]\delta(t-t_0)[/itex] to "catch" the x([itex]t_0[/itex])... But that means you already had x([itex]t_0[/itex]) in the first place! So what the heck is the point?!?

**Question II: The Impulse Response**Is [itex]\delta(t-t_0)[/itex] a 1, or infinity at time [itex]t_0[/itex]?? When it's under the integral, I know it is 1, since it has unit are. But when the impulse response is described, it seems to be the response to the impulse, with no integral involved. Here is how my Schaum's defines it:

[tex]h(t) = \textbf{T}\{\delta(t)\} [/itex]

where T is the LTI system.

And if it is infinity, how can a system respond to an infinite input? This, I think, is my biggest point of confusion, and may be why I'm having trouble understanding the convolution fully.

**Question III: The Convolution**Why would you have the response of a system to the unit impulse, but not have its response to the signal x(t)? If you could get the impulse response, why not just get the x(t) response and forget about the convolution altogether?

**Conclusion**Well, I think that sums up my confusion for now. I hope my questions made sense! I will be thrilled if someone is nice enough to clear some of these issues up for me!

Thanks!

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