Dirac comb

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In mathematics, a Dirac comb (also known as sha function, impulse train or sampling function) is a periodic generalized function with the formula Ш T ⁡ ( t ) := ∑ k = − ∞ ∞ δ ( t − k T ) {\displaystyle \operatorname {\text{Ш}} _{T}(t):=\sum _{k=-\infty }^{\infty }\delta (t-kT)} for some given period ⁠ T {\displaystyle T} ⁠. Here ⁠ t {\displaystyle t} ⁠ is a real variable and the sum extends over all integers ⁠ k {\displaystyle k} ⁠. The Dirac delta function δ {\displaystyle \delta } and the Dirac comb are tempered distributions. The graph of the function resembles a comb (with the δ {\displaystyle \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 ⁠ Ш ( t ) {\displaystyle {\text{Ш}}(t)} ⁠, where the period ⁠ T {\displaystyle T} ⁠ is omitted, represents a Dirac comb of unit period: Ш ⁡ ( t ) := Ш 1 ⁡ ( t ) = ∑ k = − ∞ ∞ δ ( t − k ) {\displaystyle \operatorname {\text{Ш}} (t):=\operatorname {\text{Ш}} _{1}(t)=\sum _{k=-\infty }^{\infty }\delta (t-k)} This implies Ш T ⁡ ( t ) = 1 T Ш ( t / T ) . {\displaystyle \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: Ш T ⁡ ( t ) = 1 T ∑ n = − ∞ ∞ e i 2 π n t / T . {\displaystyle \operatorname {\text{Ш}} _{T}(t)={\frac {1}{T}}\sum _{n=-\infty }^{\infty }e^{i2\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...