Exercise · ZW-CONTINUOUS-FOURIER-TRANSFORM-009 · ★★☆ Intermediate
What is the transform of \(x(t)=\frac{d}{dt}e^{-t}u(t)\)?
Choose one option.
Show answer
- A. \(j\omega\cdot\frac{1}{1+j\omega}\)
Explanation
Time differentiation corresponds to multiplying by \(j\omega\) in frequency, and the transform of \(e^{-t}u(t)\) is \(\frac{1}{1+j\omega}\). The negative sign corresponds to a derivative with the sign flipped.
Review the concept: Continuous Fourier transform
The continuous Fourier transform \(X(j\omega)\) is a frequency-domain representation of an aperiodic continuous-time signal \(x(t)\), assigning to each frequency \(\omega\) its spectral content.
\[X(j\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}\,dt\]
More practice problems: continuous fourier transform
- ZW-CONTINUOUS-FOURIER-TRANSFORM-001 By definition, what is the kernel of the forward continuous Fourier transform?
- ZW-CONTINUOUS-FOURIER-TRANSFORM-002 What factor multiplies the integral in the inverse Fourier transform?
- ZW-CONTINUOUS-FOURIER-TRANSFORM-003 What is the Fourier transform of δ(t)?
- ZW-CONTINUOUS-FOURIER-TRANSFORM-004 If x(t) is real, what symmetry does |X(jω)| satisfy?
- ZW-CONTINUOUS-FOURIER-TRANSFORM-005 What is the value of X(j0) for integrable x(t)?
- ZW-CONTINUOUS-FOURIER-TRANSFORM-006 Compute the transform of x(t)=e^-atu(t) with a>0.
- ZW-CONTINUOUS-FOURIER-TRANSFORM-007 If X(jω) is the transform of x(t), what is the transform of x(t-2)?
- ZW-CONTINUOUS-FOURIER-TRANSFORM-008 Compute the transform of the convolution y(t)=x_1(t)*x_2(t) if X_1(jω)=(1)/(1+jω) and X_2(jω)=(2)/(…
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