Question

A continuous-time signal \(x(t)\) is defined as \(x(t)=0\) for \(|t|>1\), and \(x(t)=1-|t|\) for \(|t|\le 1\). Let its Fourier transform be \(X(\omega)=\int_{-\infty}^{\infty} x(t)e^{-j\omega t}\,dt\). The maximum magnitude of \(X(\omega)\) is \(\underline{\hspace{2cm}}\).
 

Hint:

For even triangular signals, the maximum magnitude of the Fourier transform always occurs at \(\omega=0\).

Answer 1

Correct Answer: 1

At \(\omega = 0\):
\[ X(0) = \int_{-1}^{1} (1 - |t|)\,dt \] Compute area (triangle of height 1, base 2):
\[ X(0) = 1 \] Since Fourier magnitude decreases away from \(\omega=0\), the maximum occurs at zero frequency:
\[ \max |X(\omega)| = |X(0)| = 1 \] \[ \boxed{1} \]