Impulse sampling and its equation

Impulse sampling is the ideal mathematical model of sampling in which the sampler output is a train of impulses at $t = kT$, each with strength (area) equal to the signal value at that instant. Real samplers give narrow pulses, but when the pulse width is very small compared with $T$ (and a hold follows) they can be modelled this way.

Diagram

 x(t) ---->o/o----> x*(t)
          T (sampler)

 x(t)            x*(t)
   /‾‾\            |  |
  /    \          ||  ||
 /      \__      |||  |||_
 ---------> t    -+-+-+-+-+--> t
                 0 T 2T 3T

Derivation

The ideal sampler multiplies the input by a unit impulse train:

$$\delta_T(t) = \sum_{k=0}^{\infty} \delta(t-kT)$$ $$\begin{aligned} x^*(t) &= x(t)\,\delta_T(t) = \sum_{k=0}^{\infty} x(t)\,\delta(t-kT) \\ &= \sum_{k=0}^{\infty} x(kT)\,\delta(t-kT) \end{aligned}$$

(using $x(t)\delta(t-kT) = x(kT)\delta(t-kT)$, with $x(t)=0$ for $t<0$).

Taking the Laplace transform, $\mathcal{L}[\delta(t-kT)] = e^{-kTs}$:

$$X^*(s) = \sum_{k=0}^{\infty} x(kT)\,e^{-kTs}$$

Putting $z = e^{Ts}$ gives the z-transform:

$$X(z) = X^*(s)\Big|_{s=\frac1T\ln z} = \sum_{k=0}^{\infty} x(kT)\,z^{-k}$$

Frequency-domain form

Expanding $\delta_T(t)$ as a Fourier series gives

$$X^*(s) = \frac{1}{T}\sum_{n=-\infty}^{\infty} X(s + jn\omega_s), \qquad \omega_s = \frac{2\pi}{T}$$

so the spectrum of the sampled signal is the original spectrum repeated every $\omega_s$. This leads to the sampling theorem $\omega_s > 2\omega_m$.