Exercise · ZW-CONTINUOUS-FOURIER-TRANSFORM-005 · ★☆☆ Basic
What is the value of \(X(j0)\) for integrable \(x(t)\)?
Choose one option.
Show answer
- A. The area under \(x(t)\)
Explanation
Evaluating at \(\omega=0\) removes the exponential and leaves the integral of \(x(t)\), which is the area. Energy comes from Parseval's theorem, not from this value.
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-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)/(…
- ZW-CONTINUOUS-FOURIER-TRANSFORM-009 What is the transform of x(t)=(d)/(dt)e^-tu(t)?
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