Sampling theorem; aliasing cause; Nyquist

Sampling theorem

Sampling theorem: A continuous-time signal $x(t)$ that is band-limited to $\omega_M$ (that is, $X(j\omega)=0$ for $|\omega|>\omega_M$) is uniquely determined by its samples $x(nT)$ if the sampling frequency satisfies

$$\omega_s = \frac{2\pi}{T} > 2\omega_M \quad\text{or}\quad f_s > 2f_M$$

The minimum rate $2f_M$ is called the Nyquist rate, and $T_N = 1/(2f_M)$ is the Nyquist interval.

Aliasing effect and its cause

Aliasing is the appearance of a high-frequency component as a false lower frequency after sampling.

Cause: sampling repeats the spectrum at every multiple of $\omega_s$: $X_p(j\omega) = \frac1T\sum_k X(j(\omega - k\omega_s))$. If $\omega_s < 2\omega_M$ (under-sampling), or if the signal is not band-limited, neighbouring copies overlap. Frequencies above $f_s/2$ fold back into $0$ to $f_s/2$ and cannot be separated. Example: 700 Hz sampled at 1 kHz looks like 300 Hz.

Nyquist rate and interval of $x(t) = 5\cos(150\pi t) + 10\sin(350\pi t) + \cos(150\pi t)$

The first and last terms have the same frequency, so $x(t) = 6\cos(150\pi t) + 10\sin(350\pi t)$.

$f_M = 175$ Hz.

$$\begin{aligned} f_N &= 2 f_M = 350\ \text{Hz} \\ T_N &= \frac{1}{350} = 2.857\ \text{ms} \end{aligned}$$

Answer: Nyquist rate = 350 Hz; Nyquist interval = 1/350 s ≈ 2.86 ms.