The Impulse Function & Sampling

The impulse function

Sampling — turning a continuous signal into a sequence of numbers — is easiest to describe with one tool: the unit impulse function, also called the delta function or the Dirac delta function, after the physicist Paul Dirac.

δ (t - a) = {0 \infty t \neq = a t = a, \int_{a - ε}^{a + ε} δ (t - a) d t = 1 (ε > 0)

It's zero everywhere except at $t = a$ , infinite there, and yet the area underneath it is exactly one. You can build it as the limit of an ever-narrower, ever-taller rectangular pulse of unit area:

δ (t - a) = Δ \to 0 lim δ_{Δ} (t - a)

The sifting property

Multiply any function that's continuous at $t = a$ by an impulse and integrate, and every value of that function except the one at $t = a$ is thrown away:

\int_{- \infty}^{\infty} f (t) δ (t - a) d t = f (a)

This is called the sifting property — the delta function sifts out a single sample of $f$ from the entire continuum. It's the mechanism that turns sampling from a hand-wavy idea ("just grab the value every so often") into an exact algebraic operation.

Note

Integrating an impulse train produces the unit step function $u (t - a)$ : 0 before $a$ , 1 from $a$ onward. Equivalently, the step function's derivative is the impulse — $δ (t - a)$ is the rate of change of a signal that jumps instantaneously. In discrete form, $u (t) = \sum_{k = 0}^{\infty} δ (t - k)$ .

Sampling as multiplication by an impulse train

Now the payoff. Take a continuous signal $x (t)$ and multiply it by a train of unit impulses spaced one unit apart. At every integer $t = n$ , the sifting property pulls out exactly the value $x (n)$ ; everywhere else, the product is zero. The result is the sampled signal:

x_{s} (t) = n = - \infty \sum \infty x (n) δ (t - n)

That's it — that's sampling. A continuous curve becomes a set of weighted impulses, one at each sample instant, each one scaled by the original signal's value there.

Interactive · Sampling

Samples per cycle

The impulse function in two dimensions

Since an image is a function of two variables, sampling an image needs a 2-D impulse function $δ (x, y)$ , with the same defining properties — zero everywhere except the origin, unit volume underneath it, and its own sifting property:

\iint_{- \infty}^{\infty} δ (x, y) d x d y = 1 \iint_{- \infty}^{\infty} f (x, y) δ (x - x_{0}, y - y_{0}) d x d y = f (x_{0}, y_{0})

Sliding the impulse to $(x_{0}, y_{0})$ and sifting against $f$ pulls out the pixel value at that exact location — the direct 2-D analogue of picking a single sample out of a continuous 1-D signal.

Check your understanding

Why does δ(t − a) inside the integral ∫ f(t)δ(t − a) dt behave like a 'pick out one value' operator?