Convolution Example: Unit Step with Exponential Part 1

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An example of computing the continuous time convolution of a unit step function with an exponential function.

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Your videos are the only thing carrying me through signals and systems. Thank you, sir.

austinanderson

in general, ..
if we have x(t), then x(t-T) is same as x(t) but sifted to the right by the value of T. x(t+T) is same as x(t) but shifted to the left by the value of T.
Note that if you have done the reversal first, the shifting direction is also reversed. That is, x(T-t) is same as x(-t) but shifted to the RIGHT by the value of T.
x(-T-t) is same as x(-t) but shifted to the LEFT by the value of T.

laztom

@Jdonovanford The 1-etc comes from integrating e^{-a tau}; the integral is -1/a e^{-a tau}. Then you put in the upper limit of the integral (t) to get the e^{-at} term, and the lower limit of the integral (0) to get the e^{-a 0} = 1 term.

DarrylMorrell

11:32 -12:12
If it looks like step response of RC circuit why voltage across capacitor does not reach 1, but only 1/a? In real circuit voltage across capacitor approaches to voltage of step that is 1.

AlexAlex-fogt

When doing convolution, u(t-tau) is a function of tau for some fixed t. u(t-tau) is first flipped about the origin, then shifted to the right by tau. My video "Convolution-What's τ got to do with it?" explains this much better than this video does; it may help clear things up.

DarrylMorrell

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DarrylMorrell

Can anyone confirm that the unit step function in this example is shifted to the right when you add t in u(t-τ)? Because I think it should be shifted to the left, is it?

brngva

Thanks for adding this explanation. It helps clarify things nicely.

DarrylMorrell

Another question: when you do the integral at 8:29, I don't understand where the "one minus etc." is coming from. And also why the exponent of -e exp(-at) is t, and not tau. Thanks.

Jdonovanford

If you convolute two delta fuctions is the result = 1?
δ(t-3)*δ(t) = ?

Imbanz

u(-T+t), , , shoud be shift left of t right?

humanye

well "1/a (1-e^-at), t>=0". r u sure about that part is not missin something ? I guess thats valid when t starts from zero up to the some point. Am i wrong ?

onculkoparan