Help with impulse response and convolution?

[lizm]

hey, i did my undergrad in music and english and now am lost beyond reckoning with the impulse response stuff in my postgrad. Could anyone help with these exercises? They're examples from my lecture notes and have no idea how my lecturer got the answers. Please help?

Ex.1) Show that x(t)*h(t) = h(t)*x(t)
EH? ANY IDEAS?

Ex.2) The unit response of a continuous system is h(t)=3e ^-2t-5e <sup>-4t/sup].
If the input is modeled as x(t)= [delta](t)-2[delta](t-1)+[delta](t-2.5), find the value of the output at time=1.5s.
Ans: y(t)=0.717 at t=1.5s.

Ex.3) An electrical system has unit-impulse response h(t)= 3te -4t . If a unit step function u(t) is applied to the system, use the convolution integral to determine the value of the output after 0.25s.
Ans: 4.95 x 10 -2 V
What's the unit step function for a start?

Ex.4) Evaluate the integral [integral infinity to minus infinity] f ₁ (t)f ₂ (t)dt
Where f ₁ = 2sin (2000[pie]t) and f ₂ (t) = [delta] (t-0.25 x 10 -3 )
Ans: 2
How do they get a pure whole number out of all those symbols?</sup>

 

[turin]

1) The * denotes the binary operator of convolution, for which your job is to show that the operation is commutative. It is defined by the convolution integral:

h(t)*x(t) = integral _{from 0 to t} { dτ h(t-τ) x(τ) }.

Let Y(s) = H(s)X(s), where the capitalization (not to mention the s argument) indicates that these functions are Laplace transforms of the original signals. Multiplication is commutative, so this expression makes it trivial to declare that Y(s) = X(s)H(s). Then, it is left to show that the Laplace transform of h(t)*x(t) is equal to the product of the Laplace transforms H(s)X(s).

The definition of the convolution operation gives you an integral expression for y(t). Substitute this into the Laplace integral. Then, you can allow the convolution integral to range to infinity by introducing a unit step function, u(t-τ). At this point, the integrations can be swapped, x(τ) can be taken outside of the inner integral, and the inner integral can be recognized as a Laplace transform with a delay: H(s)e^-sτ.



2) Use Laplace transforms. (I'm assuming that the unit response is an impulse response.) Laplace transform to H(s) and X(s). Then y(t) is the inverse transform of the product H(s)X(s).



3) The unit step function is

u(x) =
0 if x < 0
1 if x > 0.

x may, itself, be a dependent variable that is a function of the independent variable (or variable of integration). In this case, x = t. The effect is to merely reset the limits of integration.



4) δ(x) = 0 if x /= 0
integral { dt δ(t-t') f(t) } = f(t')

where the limits of integration include the value t'. This is not something that you have to prove; it is the definition of δ(t-t').

 

[lizm]

still lost!

okay, i haven't any background in maths at all and thanks for replying to my post. Could you really simpilfy it down to dummy maths?

 

[turin]

THERE IS A "SYSTEMS ENGINEERING" BOARD ON THESE FORUMS FROM WHICH YOU MAY GET SOME GOOD SUPPLEMENTAL ASSISTANCE.

Could you really simpilfy it down to dummy maths?

I'll try, but I'm doubtful. I'm not optimistic about demonstrating any of this without some integral calculus. I hope that's dummy enough for you. If not, I wonder what you are doing in such a class. I don't mean to offend, but, maybe you're out of sync with your program; maybe you've missed some prereqs. Oh well, I guess that doesn't help you now at the end of the semester (crunch time) so we'll see what I (as well as anyone else who would care to jump in at any time) can do. It would help, though, if you would be more specific with your background and what you don't understand in the problems. If I just go by what you stated in your initial post, I am shocked that you even considered such a course. If you're just one of those ambitious people, then more power to you, and I'll try to help you as best I can.

For #1:
a. Do you know what convolution is?
b. Do you know what a Laplace transform is?
c. Do you know what integration is?
d. Do you know what "commutive" means, and do you know of any examples of non-commutative combination (i.e. why it's a priori non-trivial)?

I don't know what else I can say to explain this problem.

For #2:
a. Do you know what an LTI causal system is?

I appologize for my hastyness in my previous response. You do NOT need Laplace transforms for this one, just the idea of delay. The impulse response tells you what would come out if you put in a δ(t) (δ(t) is the quasi-mathematical way of saying "unit impulse function with no delay"). Assuming the system is LTI, you just add the three impulse's corresponding responses with the appropriate delays. The delays manifest as a subtraction in the time argument. That is, everywhere that "t" shows up in h(t), you subtract τ, if the input δ has been delayed by τ. In other words:

if an input of δ(t) gives you an output of h(t),

then an input of δ(t-τ) gives you an output of h(t-τ).

This principle is known as "causality."

For #3:
Again, you need to know the convolution integral and of course how to integrate. Otherwise, we will have to first go over these two issues.

This problem is asking you to convolve (preform a convolution between) the h(t) that they gave you with the unit step function, u(t) (which I defined for you in my previous post). You should recognize that this is what they are asking you to do as a fundamental part of systems theory. I will restate the issue:

Assume that we have an LTI system (this is the default assumption). Since it is LTI, its behavior for any given input can be completely characterized by its impulse response. That is to say, if we input a unit impulse function into the input of an LTI system, the output of the system, being the response to this impulse (or, "impulse response" for short) can determine the output that the system would give under the influence of any input by convolution. (see attachment for a graphical depiction.)

After that, it is just a "plug and chug" together with realizing that the unit step simply changes one of the limits of integration. I suspect that the "limits of integration" issue is probably half of the point of the problem, and you should be able to figure it out if you think about it, so I don't want to be more specific (unless you need help with the anti-derivation part or the meaning of convolution). You can tell me what you think it does, and I will verify, but I don't want to just give you the answer.

For #4:
a. Do you know what the sine function is?
b. Do you know what integration is?

I gave this definition for the Dirac delta function (that's the δ(t-t') function) in the previous post. Maybe it would help you to think of it as a selector function. It trivializes the integration by selecting the value of the rest of the integrand out at the zero point of the delta function (at t = t').

 

[lizm]

by the way, I'm doing a postgrad in music technology and it's not my choice to do this module. it's part of it. We were not told about this at the start of the year. We don't touch on the laplace transform but have touched on the Fourier series and DFT. Maybe I'm in the wrong place asking for help. I haven't missed any of the classes.

 

[lizm]

i know what the sine function is and stuff. But we don't use LTI and I don't know what it is. I know what convolution and integration are but i don't know what communitive is.
I have basic higher level maths from my secondary school and that isn't helping me at all. And people in the class who have an undergrad in maths are lost. The whole module is extremely difficult but we need it as part of the systems processing that we use for music signal processing.

 

[turin]

I will try to fill in some more holes. In the meantime, you could find the list of prereqs for this class and post them. It might help. I have almost no idea how you should be approaching this material. Just out of curiosity, where did you learn about convolution? One thing that makes me a little nervous is our possible overlapping termonology.

Do you know anything about linear algebra, specifically matrices?

Commutative:
Assume we have some objects (for instance, you can think of your sound signals as objects), A and B. Then, let's imagine that there is some way to combine these objects to get another object, C, of the same type (i.e. another sound signal). One way to combine would be to add them, in the elementary school sense:

A + B = C

This does not depend on the order of the combination. Therefore, we could just as legitimately write:

B + A = C

We call this virture commutativity, and we say that the operation of addition is commutative for A and B. (This is true if A, B, and C are sound signals.)

Applying this idea to your first problem I hope is now obvious (in meaning). A, B, and C are sound signals. (well, I suppose the problem doesn't specify this, but I'm trying to put it into a relevant context. I can't see any problem with assuming that they are sound signals, as long as we also assume that their amplitudes are not large enough to induce any non-linearities.) The thing is, the combination in your first problem is no where near as simple as this elementary school combination of addition, and the commutativity is not at all obvious by cursory inspection. We could try a bunch of different signals and test the combination between them to see if it's commutative. In fact, this procedure works in the case of addition of two numbers, because, all we have to do is show it for any two numbers, and then recognize that they are all linearly related to each other. Unfortunately, this wouldn't tell us anything about convolution of sound signals, because, no matter how many different signals we demonstrate, there will always be infinity more signals. So, we have to show it using the trickes that I mentioned in the first post.

This brings us to the Laplace transform. Forget about the name (for the purposes of this problem; perhaps you will never need to know). This is just a particular kind of integral. It is not needed that you know what the Laplace transform is, or even what the Laplace transform integral is. Familiarity only helps one to recognize (and compactify) the steps in proof. Let us see if it is not too complicated here to lay that proof out without mentioning the Laplace transform (any more than I already have).

I think that you might have a slightly different definition of the convolution integral than I gave in my first post. The limits of integration that you use may go from -infinity to +infinity instead of from 0 to t. I will assume that this is the case. I don't think it makes any difference, but maybe it will simplify the proof.

The quick method that takes the Fourier transform for granted:
Recognize that convolution in the time domain becomes multiplication in the frequency domain. That is:

F{x*h(t)}[ω] = F{x(t)}[ω]F{h(t)}[ω]

Then, recognize that multiplication is commutative. So:

F{x(t)}[ω]F{h(t)}[ω] = F{h(t)}[ω]F{x(t)}[ω]

By swapping x(t) and h(t) in the original formula:

F{h*x(t)}[ω] = F{h(t)}[ω]F{x(t)}[ω]

Then, we have from the commutativity of multiplication that:

F{x*h(t)}[ω] = F{h*x(t)}[ω]

Rearranging:

F{x*h(t)}[ω] - F{h*x(t)}[ω] = 0

Since the Fourier transform is linear:

F{x*h(t) - h*x(t)}[ω] = 0

Taking the inverse Fourier transform:

F^-1{F{x*h(t) - h*x(t)}[ω]}[t] = F^-1{0}[t]

The inverse transform of a transform is, by definition, what was originally transformed in the first place. The Fourier transform of 0 is 0. Therefore:

x*h(t) - h*x(t) = 0

If this proof is inadequate (i.e., you are not allowed to take the Fourier transform for granted), then we can talk about the more fundamental proof (which will simply consist of showing that the Fourier transform of convolution is equal to the product of the transforms, and will just add a step or two).

As far as the other three problems go, I don't know what more I could say.

 

[zak2386]

ouch

goodness, those music tech people are cruel, throwing you in a serious calculus based class with no background, that sucks. good luck to you. Turin, question, i am getting transfer functions and impulse functions a bit confused. If i am understand correctly (i think not) the transfer function, H(s), it is the output over the input of the function in frequency space. but then the impulse response, h(t), is the time space function assuming all initial conditions are zero?

 

[turin]

zak2386,
I think you have it right. But don't forget that the transfer function and impulse response (function) are intimately related. At this point it doesn't seem like lizm is even going to use the full blown frequency domain in the analysis (which seems to defeat the purpose, but this may be a weed-out class). The reason one goes to the frequency domain is to make convolution less convoluded (yuk, yuk ...).

 

[zak2386]

ok, thank you...and boooo on the joke, but you you might like this one, it's equally cheesy: there were 2 cats sitting on a roof, which one fell off? the one with the smaller mu. (hehehe )

 

[enigma]

I don't know how they'd actually stick you in that class for a music related field, even for graduate work.

Controls was and is the hardest class I have taken to date, and I'm in aerospace engineering. Ask me how much I remember of it...

 

[turin]

I don't know how they'd actually stick you in that class for a music related field, even for graduate work.

Controls was and is the hardest class I have taken to date, and I'm in aerospace engineering. Ask me how much I remember of it...

The stuff that lizm is asking about seems to be "pre-control-system" stuff. What gets me is that they don't seem to be focusing on the Laplace transform. You honestly can't think of a more difficult class than "controls" (what I think I probably refer to as "control systems")? Any class I ever took that dealt with non-linear systems always seemed to me more difficult than control systems, though, I certainly didn't consider my control systems class a walk in the park.

 

[enigma]

Well, it's off topic, but I haven't had anything with non-linear systems yet.

Control systems was certainly harder than my Orbital Dynamics, Navigation & Guidance, or Propulsion & Power classes.

 

[turin]

I wish I would've gone into AE. Just the names of the classes themselves sound exciting.

It seems we have lost lizm.

 

[zak2386]

I am likewise aerospace engineering, and i just finished taking dynamics, which for sophomore level classes covers basic control systems. Someof the classes are really awesome, i loved vehical preformance and design, it was my favorite class this year. Turin, what did you major in ME?

 

[zak2386]

If you actually know who wrote the Akin's Laws (which are very funny, and sadly so very true) you should suggest the addition of the KISS rule. Keep It Simple Stupid. :)