If x(t) has transform X(ω) , what is the transform of (dx(t))/(dt) when x(t)=e^-tu(t) ?

Exercise · ZW-FOURIER-TRANSFORM-PROPERTIES-007 · ★★☆ Intermediate

If \( x(t) \) has transform \( X(\omega) \), what is the transform of \( \frac{dx(t)}{dt} \) when \( x(t)=e^{-t}u(t) \)?

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A. \( \frac{j\omega}{1+j\omega} \)

Explanation

With \( X(\omega)=\frac{1}{1+j\omega} \), differentiation in time equals multiplication by \( j\omega \), giving \( \frac{j\omega}{1+j\omega} \). The property already includes the impulse at \( t=0 \). Dividing by \( j\omega \) corresponds to integration, not differentiation.

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Review the concept: Fourier transform properties

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.

\[x(t-t_0) \leftrightarrow e^{-j\omega t_0}X(\omega)\]

If \( x(t) \) has transform \( X(\omega) \), what is the transform of \( \frac{dx(t)}{dt} \) when \( x(t)=e^{-t}u(t) \)?

Explanation

Review the concept: Fourier transform properties

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