Let \( x(t)=\frac{\sin(2t)}{\pi t} \) with transform \( X(\omega)=\mathrm{rect}(\omega/4) \). What is the transform of \( z(t)=x(t)\cos(3t) \)? Check whether the magnitude at \( \omega=0 \) matches that of \( X \).
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B. \( \frac{1}{2}[X(\omega-3)+X(\omega+3)] \), whose magnitude at \( \omega=0 \) is 0
Cosine modulation shifts \( X \) to \( \pm 3 \) with factor \( \frac{1}{2} \). Since \( X \) is a rect of width 4 centered at 0, both shifted copies fall outside its support at \( \omega=0 \), giving 0. Using direct frequency multiplication confuses modulation with a product.
Set of rules describing how the Fourier transform \(X(\omega)\) of a signal \(x(t)\) changes under operations such as shifting, scaling, differentiation, or convolution.