Periodic impulse train

The periodic impulse train is an infinite sum of unit impulses spaced $T$ apart:

${\delta }_T(t)={\sum }_{n=-\infty }^{\infty }\delta (t-nT).$

So at $t=0,T,2T,\dots$ and $-T,-2T,\dots$ we have an impulse; elsewhere the signal is zero. The subscript on ${\delta }_T$ denotes the period. So ${\delta }_4(t)$ is an impulse train with period 4 — impulses at $t=\dots ,-8,-4,0,4,8,\dots$.

The impulse train is sometimes called the Dirac comb, especially in math contexts.

Shifting and scaling

By analogy with a single impulse, we can shift and scale an impulse train. The expression $16{\delta }_8(t-3)$ means a periodic impulse train with period $8$, shifted right by $3$, with each impulse of strength $16$ — so impulses at $t=\dots ,-5,3,11,19,\dots$, each of strength $16$.

Multiplication = ideal sampling

The primary use of the impulse train in this course is to model the act of sampling a continuous-time signal. Multiplying $x(t)$ by ${\delta }_{T_s}(t)$ uses the equivalence property of each impulse:

$x(t)\cdot {\delta }_{T_s}(t)={\sum }_{n}x(n{T}_s)\delta (t-n{T}_s).$

The product is a train of impulses at the sample times, each impulse with strength equal to the signal’s value at that sample time. This is impulse sampling — see Sampling for the full story.

Fourier transform

The Fourier transform of a periodic impulse train is another periodic impulse train:

${\delta }_T(t)\stackrel{\mathcal{F}}{\leftrightarrow }\frac{1}{T}{\delta }_{1/T}(f).$

Spacing $T$ in time, spacing $1/T$ in frequency. This pair is the foundation of sampling theory: when you multiply a signal by an impulse train in time, you convolve its spectrum with an impulse train in frequency, which produces shifted copies of the spectrum spaced $1/T$ apart. That’s aliasing if the copies overlap, exact reconstruction if they don’t.

Fourier series

The impulse train is periodic with period $T$, so it has a Fourier series. All its harmonic coefficients are equal to $1/T$:

$c_x[k]=\frac{1}{T}\text{ for all integers }k.$

So all harmonics are equally present — the impulse train has a flat line spectrum. This is the discrete-frequency analog of the white-spectrum-of-the-impulse result $\delta (t)\leftrightarrow 1$.