2.2. Aliasing#

The previous section introduced uniform sampling, which allows us to represent a continuous signal \(x(t)\) by a discrete sequence of sample values \(x[n]\).

In this section, we’ll see that this idea comes with some restrictions.

2.2.1. What is aliasing?#

Aliasing is the name we give to the phenomenon when two distinct continuous signals \(x_1(t)\) and \(x_2(t)\) produce the same sequence of sample values \(x[n]\) when sampled at a fixed rate \(f_s\). More specifically, we usually think of aliasing in terms of pure (sinusoidal) tones \(x(t) = A \cdot \cos\left(2\pi \cdot f \cdot t + \phi\right)\).

Theorem 2.1 (Aliasing)

Given a sampling rate \(f_s\), two frequencies \(f\) and \(f'\) are aliases of each other if for some integer \(k\),

(2.3)#\[f' = f + k \cdot f_s.\]

If sampled at rate \(f_s\), two waves \(x\) (at frequency \(f\)) and \(y\) (at frequency \(f'\))

\[\begin{align*} x[n] &= A \cdot \cos\left(2\pi \cdot f \cdot \frac{n}{f_s} + \phi\right)\\ y[n] &= A \cdot \cos\left(2\pi \cdot f' \cdot \frac{n}{f_s} + \phi\right) \end{align*}\]

will produce identical samples: \(x[n] = y[n]\) for all \(n = 0, 1, 2 \dots\).

Or, in words, frequency \(f'\) is \(f\) plus some whole number multiples of the sampling rate \(f_s\). Equation (2.3) is known as the aliasing equation, and it tells us how to find all aliasing frequencies for a given \(f\) and sampling rate.

Fig. 2.3 illustrates this effect: for any \(f\) that we choose, once a sampling rate \(f_s\) is chosen, there are infinitely many frequencies that produce an identical sequence of samples.