CISC 457, Lecture 14 - Impulse Trains

In the next few lectures, we'll address a very important question:

How many samples are required to accurately reconstruct $f(t)$ when we convert it to the frequency domain, then back to the spatial domain?

We'll first need some background first on "impulse trains".

Impulses

The Dirac delta is

$\delta(t) = \begin{cases} \infty \;\:\quad \text{if}\ t = 0 \\ 0 \qquad \text{otherwise} \end{cases}$

but has the property that

$\displaystyle \int_{-\infty}^\infty \delta(x)\ dx = 1$

and is drawn as

The Dirac delta can be used to sample a function at the origin, inside an integral:

$\displaystyle \int f(t)\ \delta(t)\ dt = f(0)$

The Dirac delta can be shifted to sample a function at any point, inside an integral:

$\displaystyle \int f(t)\ \delta(t-t_0)\ dt = f(t_0)$

Impulse Trains

"Trains" of impulses are used to sample a continuous function at a regular interval, $T$.

$\displaystyle s_T(t) = \sum_{i=-\infty}^\infty \delta( t - i T )$

Samples can be taken at each impulse:

$\begin{array}{rcl} (f \times s_T)(t) & = & f(t)\ s_T(t) \\ & = & \displaystyle \sum_{i \in \mathbb{Z}} f(iT)\ \delta(t-iT) \qquad \text{since this is non-zero at only one position} \\ & = & \begin{cases}f(t) \quad \text{if}\ t = iT\ \text{for some}\ i \in \mathbb{Z} \\ 0 \qquad \text{otherwise} \end{cases} \end{array}$

(note that this is a product of two functions, not the integral of the product)