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f00lishroy
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Homework Statement
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).
Homework Equations
Frequency Convolution : x1(t)*x2(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)
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