Sampling

The signal $f(t)$ is a function on the real line $\mathbb{R}$. When we compute with a computer, usually we can't know everything about the function, nor can we store it all on the computer. So we just evaluate the signal at a sequence of times $t_n$, and define a vector

$$\mathbf{x} = (\ldots, x_{-2}, x_{-1}, x_0, x_1, x_2, \ldots)$$

with components $x_n = f(t_n),$ for all $n\in \mathbb{Z}$.

For reasons that have mostly to do with engineering technology, we usually take the time samples $t_n$ to be uniformly spaced. That is, we have a sequence of numbers separated by a uniform step $\Delta t$, and write $t_0 = 0, t_1 = \Delta t, t_2 = 2\Delta t, \ldots$, and so the vector $\mathbf{x}$ has components $$x_n = f(n\Delta t), \qquad \mbox{ for all } n\in \mathbb{Z}.$$

Although the vector $\mathbf{x}$ is infinitely long, that is it has infinitely many components, it is important to realize that most of the component $x_n$ are just zero. Why? Because we can't measure back to time minus infinity, or forward to plus infinity. So at some point we stop measuring, and can just assume everything else is zero.

The vector $\mathbf{x}$ is called a sampled signal. $\Delta t$ is called the sampling interval. $1/\Delta t$ is called the sampling rate.