Dirac comb

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In mathematics, a Dirac comb is a periodic Schwartz distribution constructed from Dirac delta functions

<math>\Delta_T(t) \ \stackrel{\mathrm{def}}{=}\ \sum_{k=-\infty}^{\infty} \delta(t - k T)</math>

for some given period T. Some authors, notably Bracewell, refer to it as the Shah function (probably 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:

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

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

[edit] Scaling property

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

<math>\sum_{k=-\infty}^{\infty} \delta(t - k T) = |\alpha|\cdot \sum_{k=-\infty}^{\infty} \delta\bigg(\alpha\cdot (t - k T)\bigg).</math>

[edit] Fourier series

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

<math> \Delta_T(t+T) = \Delta_T(t) \quad \forall t </math>.

The complex Fourier series for such a periodic function is

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

where the Fourier coefficients, c_n are

<math>c_n\,</math>	<math>= \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 ) \ </math>
	<math>= \frac{1}{T} \int_{-T/2}^{T/2} \Delta_T(t) e^{-i 2 \pi n t/T}\, dt \ </math>
	<math>= \frac{1}{T} \int_{-T/2}^{T/2} \delta(t) e^{-i 2 \pi n t/T}\, dt \ </math>
	<math>= \frac{1}{T} e^{-i 2 \pi n \, 0/T} \ </math>
	<math>= \frac{1}{T} \ </math>

All Fourier coefficients are 1/T resulting in

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

[edit] Fourier transform

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

Unitary transform to ordinary frequency domain (Hz):

<math>\sum_{n=-\infty}^{\infty} \delta(t - n T) \quad \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}</math>

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

<math>\sum_{n=-\infty}^{\infty} \delta (t - n T) \quad \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} \,</math>

[edit] Sampling and aliasing

Multiplication of a continuous signal by a Dirac comb is sometimes called an ideal sampler with sampling interval T. When used as an ideal sampler, it can be used to understand the effects of aliasing and as a proof of the Nyquist-Shannon sampling theorem.