Exercise · ZW-CONTINUOUS-FOURIER-TRANSFORM-008 · ★★☆ Intermediate
Compute the transform of the convolution \(y(t)=x_1(t)*x_2(t)\) if \(X_1(j\omega)=\frac{1}{1+j\omega}\) and \(X_2(j\omega)=\frac{2}{2+j\omega}\).
Choose one option.
A \(\frac{2}{1+j\omega+2+j\omega}\) B \(\frac{2}{(1+j\omega)(2+j\omega)}\) C \(\frac{1}{1+j\omega}+\frac{2}{2+j\omega}\) D \(\frac{2}{(1-j\omega)(2-j\omega)}\) Show answer B. \(\frac{2}{(1+j\omega)(2+j\omega)}\)← Back to the question
Explanation Time-domain convolution becomes a product in frequency, so \(Y=X_1X_2\). The option with a sum confuses convolution with adding signals.
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