Evaluating integrals with delta function

nhoratiu
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Hi there!
I have a problem with one of the questions given to us in the signals and systems course. If anyone could help me I would greatly appreciate it!

Homework Statement



integral(from -infinity to +infinity) of u0(t) * cos(t) dt

u(t) is a step function.

Homework Equations



Why is this equal to 1 and not to sin(t), for t>0 ?

The Attempt at a Solution



I thought this is equal to sin(t) but the answer manual says it's equal to 1 because the whole thing transforms into a delta function. How does it do that?
 
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You need to define u_0.
 
u0(t) is a step function. its "0" from negative infinity to t=0 and "1" from t=0 to positive infinity.
 
So, I guess u0 is the same as u. And, I also guess, "*" stands for the convolution?
 
hey, the * is for multiplication. Overall we have:

integral from negative infinity to positive infinity of ( u(t) times cos(t) dt)

the solution manual says it's equal to 1. I thought it was equal to sin(t).
 
I would, formally (which is, strictly speaking, illegal), do it like that:

\int_{-\infty}^{\infty}u(x)\cos(x)dx=\int_{-\infty}^{\infty}u(x)\sin'(x)dx=-\int_{-\infty}^{\infty}u'(x)\sin(x)dx=-\int_{-\infty}^{\infty}\delta(x)\sin(x)dx=-\sin(0)=0

But I do not know what your course is based on, therefore I can't help more. And probably my result is not helping you anyway.
 
When you integrate cos x, you get sin x, but the problem is evaluating the result at the upper limit since sin x doesn't converge in the limit as x→∞. To get around this problem, you can use the convergence factor e-λx and take the limit as λ→0+:

\int_{-\infty}^\infty u(x)\cos(x)\,dx = \lim_{\lambda\to 0^+}\int_{-\infty}^\infty u(x)\cos(x)e^{-\lambda x}\,dx
 
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