Exercise · ZW-CONTINUOUS-FOURIER-TRANSFORM-006 · ★★☆ Intermediate
Compute the transform of \(x(t)=e^{-at}u(t)\) with \(a>0\).
Choose one option.
Show answer
- D. \(\frac{1}{a+j\omega}\)
Explanation
Integrating from zero to infinity, \(\int_0^\infty e^{-(a+j\omega)t}dt=\frac{1}{a+j\omega}\). The option with a minus sign on \(j\omega\) comes from using the inverse kernel.
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
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- ZW-CONTINUOUS-FOURIER-TRANSFORM-005 What is the value of X(j0) for integrable x(t)?
- 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)/(…
- ZW-CONTINUOUS-FOURIER-TRANSFORM-009 What is the transform of x(t)=(d)/(dt)e^-tu(t)?
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