Nyquist-Shannon theorem

In signal theory, the Nyquist-Shannon theorem (or sampling theorem) states that for a bandlimited CT signal $x$ with bandwidth $B$, sampled with period:

$$T_s\le \frac{1}{2B}$$ $${\omega }_s=2\pi /T_S\le 2(2\pi B)$$

then we can exactly recover the original signal $x$ from the sampled signal $x_s$. What this means is that we need a sample rate at least twice the bandwidth of the signal to avoid aliasing.

For a non-bandlimited signal, we need to sample at twice the rate of the fastest (angular) frequency to avoid aliasing. The minimum sampling frequency ${\omega }_s$ which this satisfies is $4\pi B$, called the Nyquist frequency.

Recall from the page on bandlimited signals. We can take the CTFT of the sampled signal $x_s$ to generate a periodic spectrum. If we window it to recover the original spectrum (centred at 0, i.e., a low-pass filter), then we can take the inverse CTFT to obtain the original signal.

Computations

The fastest angular frequency mentioned above only applies for the superposition of signals with a certain frequency. If we have the product of multiple signals, we need to somehow convert to a sum — then we can take the fastest angular frequency.

When we get aliasing, our CTFT spectrum is distorted.