Dirac comb: Information from Answers.com

A Dirac comb is an infinite series of Dirac delta functions spaced at intervals of T.

In mathematics, a Dirac comb (also known as an impulse train and sampling function in electrical engineering) is a periodic Schwartz distribution constructed from Dirac delta functions

$\Delta_T(t) \ \stackrel{\mathrm{def}}{=}\ \sum_{k=-\infty}^{\infty} \delta(t - k T)$

for some given period T. Some authors, notably Bracewell as well as some textbook authors in electrical engineering and circuit theory, refer to it as the Shah function (possibly because its graph resembles the shape of the Cyrillic letter sha Ш). Because the Dirac comb function is periodic, it can be represented as a Fourier series:

$\Delta_T(t) = \frac{1}{T}\sum_{n=-\infty}^{\infty} e^{i 2 \pi n t/T}.$

1 Scaling property
2 Fourier series
3 Fourier transform
4 Sampling and aliasing
5 See also
6 References

Scaling property

The scaling property follows directly from the properties of the Dirac delta function

$\sum_{k=-\infty}^{\infty} \delta(t - k T) = |\alpha|\cdot \sum_{k=-\infty}^{\infty} \delta\bigg(\alpha\cdot (t - k T)\bigg).$

Fourier series

It is clear that Δ_T(t) is periodic with period T. That is

$\Delta_T(t+T) = \Delta_T(t)\,$

for all t. The complex Fourier series for such a periodic function is

$\Delta_T(t) = \sum_{n=-\infty}^{+\infty} c_n e^{i 2 \pi n t/T} \$

where the Fourier coefficients, c_n are

$c_n\,$	$= \frac{1}{T} \int_{t_0}^{t_0 + T} \Delta_T(t) e^{-i 2 \pi n t/T}\, dt \quad ( -\infty < t_0 < +\infty ) \$
	$= \frac{1}{T} \int_{-T/2}^{T/2} \Delta_T(t) e^{-i 2 \pi n t/T}\, dt \$
	$= \frac{1}{T} \int_{-T/2}^{T/2} \delta(t) e^{-i 2 \pi n t/T}\, dt \$
	$= \frac{1}{T} e^{-i 2 \pi n \, 0/T} \$
	$= \frac{1}{T}. \$

All Fourier coefficients are 1/T resulting in

$\Delta_T(t) = \frac{1}{T}\sum_{n=-\infty}^{\infty} e^{i 2 \pi n t/T}$ .

Fourier transform

The Fourier transform of a Dirac comb is also a Dirac comb.

Unitary transform to ordinary frequency domain (Hz):

$\sum_{n=-\infty}^{\infty} \delta(t - n T) \quad \stackrel{\mathcal{F}}{\longleftrightarrow}\quad {1\over T}\sum_{k=-\infty}^{\infty} \delta \left( f - {k\over T} \right) \quad = \sum_{n=-\infty}^{\infty} e^{-i2\pi fnT}$

Unitary transform to angular frequency domain (radians/sec):

$\sum_{n=-\infty}^{\infty} \delta (t - n T) \quad \stackrel{\mathcal{F}}{\longleftrightarrow}\quad \frac{\sqrt{2\pi }}{T} \sum_{k=-\infty}^{\infty} \delta \left( \omega -k \frac{2\pi }{T}\right) \quad = \frac{1}{\sqrt{2\pi}}\sum_{n=-\infty}^{\infty} e^{-i\omega nT} \,$

Sampling and aliasing

Main article: Nyquist–Shannon sampling theorem

Reconstruction of a continuous signal from samples taken at sampling interval T is done by some sort of interpolation, such as the Whittaker–Shannon interpolation formula. Mathematically, that process is often modelled as the output of a lowpass filter whose input is a Dirac comb whose teeth have been weighted by the sample values. Such a comb is equivalent to the product of a comb and the original continuous signal. That mathematical abstraction is often described as "sampling" for purposes of intoducing the subjects of aliasing and the Nyquist-Shannon sampling theorem.

References

Bracewell, R.N. (1986), The Fourier Transform and Its Applications (revised ed.), McGraw-Hill ; 1st ed. 1965, 2nd ed. 1978.
Córdoba, A (1989), "Dirac combs", Letters in Mathematical Physics 17: 191-196

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Dirac comb

Contents

Scaling property

Fourier series

Fourier transform

Sampling and aliasing

See also

References