Bandlimited signal

In signal theory, a CT signal $x$ is bandlimited with a bandwidth $B>0$ if its CTFT spectrum $X$ satisfies $X(j\omega )=0\forall \left|\frac{\omega }{2\pi }\right|\ge B$, i.e., the spectrum is zero outside an interval $[-2\pi B,2\pi B]$. Many signals are in practice bandlimited or filtered such that they become bandlimited.

The CTFT spectrum of a sampled bandlimited signal is given by:

$$X_s(j\omega )=\frac{1}{T_s}\sum _{k=-\infty }^{\infty }X(j(\omega -k{\omega }_s))$$

What this means is that the spectrum of the sampled signal is a periodised version (with period ${\omega }_s$) of the original signal’s spectrum in the frequency domain.

If ${\omega }_s\ge 4\pi B$, then we have a relation given by:¹ What this means is that for a sampled signal $x_s(t)$ that is bandlimited, we can uniquely reconstruct the signal. We first take the CTFT to get the periodic spectrum, then we window it within a small interval (i.e., we apply a low-pass filter) to get a single peak centred at 0. Then we can take the inverse CTFT to recover the original signal.

Footnotes

1. From Prof Simpson-Porco’s lecture slides. ↩