Dirac Comb
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The Dirac comb (also called the Shah function or sampling function) is an infinite train of equally-spaced Dirac delta functions. It is fundamental to sampling theory and creating periodic functions.
Definition
The Dirac comb with unit spacing is:
$$ \text{comb}(x) = \mathrm{III}(x) = \sum_{n=-\infty}^{\infty} \delta(x - n) $$The symbol $\mathrm{III}$ (Shah, from the Cyrillic letter Ш) visually resembles the function’s graph.
For arbitrary spacing $T$:
$$ \text{comb}\left(\frac{x}{T}\right) = \sum_{n=-\infty}^{\infty} \delta\left(\frac{x}{T} - n\right) = T \sum_{n=-\infty}^{\infty} \delta(x - nT) $$This places delta functions at positions $x = 0, \pm T, \pm 2T, \ldots$
Fourier Transform
The Dirac comb is its own Fourier transform (up to scaling):
$$ \mathcal{F}\{\text{comb}(x)\} = \text{comb}(f) $$More generally:
$$ \mathcal{F}\left\{\text{comb}\left(\frac{x}{T}\right)\right\} = T \cdot \text{comb}(Tf) $$This self-reciprocal property is crucial for understanding sampling and periodicity in Fourier analysis.
Key Properties
| Operation | Result |
|---|---|
| Multiplication $f(x) \cdot \text{comb}(x/T)$ | Samples $f$ at intervals of $T$ |
| Convolution $f(x) * \text{comb}(x/T)$ | Replicates $f$ at intervals of $T$ (periodic extension) |
Creating Periodic Functions
Convolving any function $g(x)$ with a Dirac comb produces a periodic function:
$$ p(x) = g(x) * \text{comb}\left(\frac{x}{X}\right) = \sum_{n=-\infty}^{\infty} g(x - nX) $$The function $p(x)$ has period $X$, meaning $p(x + X) = p(x)$.
Two-Dimensional Case
For a 2D function:
$$ p(x, y) = g(x, y) * \left[\text{comb}\left(\frac{x}{X}\right) \text{comb}\left(\frac{y}{Y}\right)\right] $$creates a doubly-periodic function with:
- Period $X$ in the $x$ direction
- Period $Y$ in the $y$ direction
This tiles copies of $g$ on an $X \times Y$ lattice.
Applications in Optics
- Diffraction gratings: A grating can be modeled as a slit function convolved with a Dirac comb
- Sampling: The comb function models ideal sampling of continuous signals
- Discrete Fourier analysis: Periodic extension connects continuous and discrete Fourier transforms