Dirac comb

Article summary

s spaced at intervals of T In mathematics, a Dirac comb (also known as sha function, impulse train or sampling function) is a periodic generalized function with the formula \operatorname{\text{Ш}}_{T}(t) := \sum_{k=-\infty}^{\infty} \delta(t - k T) for some given period . Here is a real variable and the sum extends over all integers . The Dirac delta function \delta and the Dirac comb are tempered distributions. The graph of the function resembles a comb (with the \delta s as the comb's 'teeth'), hence its name and the use of the comb-like Cyrillic letter sha (Ш) to denote the function. The symbol , where the period is omitted, represents a Dirac comb of unit period: \operatorname{\text{Ш}}(t) := \operatorname{\text{Ш}}_{1}(t) = \sum_{k=-\infty}^{\infty} \delta(t - k) This implies \operatorname{\text{Ш}}_{T}(t) = \frac{1}{T}\operatorname{\text{Ш}}\!\left({t}/{T}\right). Because the Dirac comb function is periodic, it can be represented as a Fourier series based on the Dirichlet kernel: \operatorname{\text{Ш}}_{T}(t) = \frac{1}{T}\sum_{n=-\infty}^{\infty} e^{i 2 \pi n {t}/{T}}. The Dirac comb function allows one to represent both continuous and discrete phenomena, such as sampling and aliasing, in a single framework of continuous Fourier analysis on tempered distributions, without any reference to Fourier series. The Fourier transform of a Dirac comb is another Dirac comb.…

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