(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the Fourier transform of the following signal JUST by using the FT table and the FT properties

x(t) = sin(t) -pi<=t<=pi

0 otherwise

NOTE: I am using CONVOLVED WITH as a substitute for * (the real convolution operator) because I cannot express multiplication in any other way that I know of).

2. Relevant equations

Frequency Convolution : x_{1}(t)*x_{2}(t) ==> (1/2pi)X1([itex]\omega[/itex]) CONVOLVED WITH X2([itex]\omega[/itex])

Fourier Transforms:

sin(ω0t) ===> (pi/j)[δ(ω-ω0)-δ(ω+ω0)]

rect(t/[itex]\tau[/itex]) ===> [itex]\tau[/itex]sinc([itex]\omega[/itex][itex]\tau[/itex]/2pi)

3. The attempt at a solution

I know that x(t) can be expressed as a sine multiplied with a rectangle function

x(t) = sin(t) * rect(t/2pi)

so by the frequency convolution property, X(ω)= (1/2pi)[[itex]\frac{pi}{j}[/itex][δ(ω-1)+δ(ω+1)] CONVOLVED WITH 2pi*sinc(ω)]

My problem is how to do the convolution. Is it easier than it looks? I don't know how the imaginary (j) factors into the convolution. Can someone help explain how to simplify X(ω)? Thank you

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# Homework Help: Convolution in Frequency Domain

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