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If the signal x(t) = sin(t)/(πt) * sin(t)/(πt) with * denoting the convolution operation, then x(t) is equal to

1. sin(t)/(πt)
2. sin(2t)/(2πt)
3. 2 sin(t)/(πt)
4. (sin(t)/(πt))²

Correct answer: sin(t)/(πt)

Solution

sin(t)/(pi t) is an ideal lowpass (sinc) whose Fourier transform is a unit-height rectangle over |w|<1. Convolution in time multiplies the transforms, and rect*rect=rect, so the result is the same sinc: sin(t)/(pi t). Stored answer is wrong.

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