Let \( x(t)=e^{-t}u(t) \) with \( X(\omega)=\frac{1}{1+j\omega} \). Compute the transform of \( y(t)=t\,x(t-1) \) using the shift and frequency-differentiation properties.
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C. \( e^{-j\omega}\left[\frac{1}{(1+j\omega)^2}+\frac{1}{1+j\omega}\right] \)
First \( x(t-1) \) has transform \( e^{-j\omega}\frac{1}{1+j\omega} \). Then multiplying by \( t \) in time is equivalent to \( j\,\frac{d}{d\omega} \), which produces a sum of two terms. Omitting the derivative term is a common error.
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.