Sampling requirement; Nyquist of product

Sampling and the required sampling frequency

Sampling is the process of converting a continuous-time signal $x(t)$ into a discrete-time signal $x[n] = x(nT)$ by taking its values at equally spaced instants $t = nT$. $T$ is the sampling interval and $f_s = 1/T$ the sampling frequency.

Requirement: for a signal band-limited to $f_M$, the sampling frequency must be more than twice the highest frequency:

$$f_s > 2 f_M$$

$2f_M$ is the Nyquist rate. If this holds, $x(t)$ can be reconstructed exactly from its samples by an ideal low-pass filter; if not, aliasing occurs.

Nyquist rate of $x(t) = \frac{1}{2\pi}\cos(100\pi t)\cos(300\pi t)$

The product must first be written as a sum, using $\cos A\cos B = \tfrac12[\cos(A-B) + \cos(A+B)]$:

$$\begin{aligned} x(t) &= \frac{1}{2\pi}\cdot\frac{1}{2}\big[\cos(300\pi t - 100\pi t) + \cos(300\pi t + 100\pi t)\big] \\ &= \frac{1}{4\pi}\big[\cos(200\pi t) + \cos(400\pi t)\big] \end{aligned}$$

$f_M = 200$ Hz, so

$$f_N = 2 f_M = 400\ \text{Hz}$$

Answer: Nyquist rate = 400 Hz (400 samples/s, $\omega_N = 800\pi$ rad/s).