Let x(t)=e^-|t|. Compute X(jω) and determine the value of X(j1) by comparing with the options.

(2)/(1+ω^2), which equals 1 at ω=1

(2)/(1+ω^2), which equals 1 at ω=1 with a negative sign

(1)/(1+ω^2), which equals 12 at ω=1

(2)/(1-ω^2), which diverges at ω=1

Exercise · ZW-CONTINUOUS-FOURIER-TRANSFORM-010 · ★★★ Advanced

Let \(x(t)=e^{-|t|}\). Compute \(X(j\omega)\) and determine the value of \(X(j1)\) by comparing with the options.

Choose one option.

Show answer

D. \(\frac{2}{1+\omega^2}\), which equals \(1\) at \(\omega=1\)

Explanation

Splitting the integral into two pieces gives \(X(j\omega)=\frac{2}{1+\omega^2}\), which equals \(1\) at \(\omega=1\). The version with a negative sign is wrong because the transform is real and positive.

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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.

\[X(j\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}\,dt\]

Let \(x(t)=e^{-|t|}\). Compute \(X(j\omega)\) and determine the value of \(X(j1)\) by comparing with the options.

Explanation

Review the concept: Continuous Fourier transform

More practice problems: continuous fourier transform