Band-limited signals; aliasing; Nyquist rate

Band-limited signal

A signal $x(t)$ is band-limited if its Fourier transform is zero above some finite frequency $\omega_M$: $X(j\omega) = 0$ for $|\omega| > \omega_M$. Example: a sum of sinusoids, or speech after a 3.4 kHz low-pass filter.

Aliasing effect with example

When a band-limited signal is sampled at $f_s < 2f_M$, the shifted copies of its spectrum overlap. A component of frequency $f_0 > f_s/2$ then gives the same samples as a lower frequency $|f_0 - kf_s|$ and is mistaken for it. This is aliasing.

Example: $x_1(t) = \cos(2\pi\cdot 10t)$ and $x_2(t) = \cos(2\pi\cdot 50t)$, sampled at $f_s = 40$ Hz:

$$x_2[n] = \cos\!\left(2\pi\frac{50}{40}n\right) = \cos\!\left(2\pi n + \frac{2\pi n}{4}\right) = \cos\!\left(2\pi\frac{10}{40}n\right) = x_1[n]$$

The samples are identical, so 50 Hz is an alias of 10 Hz. Aliasing is avoided by sampling above the Nyquist rate and using an anti-aliasing filter.

Nyquist rate and interval of $x(t) = 3\sin(230\pi t) + 50\cos(225\pi t) + \sin(200\pi t)$

$f_M = 115$ Hz.

$$\begin{aligned} f_N &= 2f_M = 2 \times 115 = 230\ \text{Hz} \\ T_N &= \frac{1}{f_N} = \frac{1}{230} = 4.348\ \text{ms} \end{aligned}$$

Answer: Nyquist rate = 230 Hz; Nyquist interval = 1/230 s ≈ 4.35 ms.