Relation to the z Transform

The Laplace transform is used to analyze continuous-time systems. Its discrete-time counterpart is the $z$ transform:

$$X_d(z) \isdef \sum_{n=0}^\infty x_d(nT) z^{-n}$$

If we define $z=e^{sT}$ , the $z$ transform becomes proportional to the Laplace transform of a sampled continuous-time signal:

$$X_d(e^{sT}) = \sum_{n=0}^\infty x_d(nT) e^{-snT}$$

As the sampling interval $T$ goes to zero, we have

$$\lim_{T\to 0} X_d(e^{sT})T = \lim_{\Delta t\to 0} \sum_{n=0}^\infty x_d(t_n) e^{-st_n} \Delta t = \int_{0}^\infty x_d(t) e^{-st} dt \isdef X(s)$$

where $t_n\isdef nT$ and $\Delta t \isdef t_{n+1} - t_n = T$ .

In summary,

$\parbox{0.8\textwidth}{the {\it z} transform\ (times the sampling interval $T$) of a discrete time signal $x_d(nT)$\ approaches, as $T\to0$, the Laplace transform\ of the underlying continuous-time signal $x_d(t)$.}$

Note that the $z$ plane and $s$ plane are generally related by

$$\zbox {z = e^{sT}.}$$

In particular, the discrete-time frequency axis $\omega_d \in(-\pi/T,\pi/T)$ and continuous-time frequency axis $\omega_a \in(-\infty,\infty)$ are related by

$$\zbox {e^{j\omega_d T} = e^{j\omega_a T}.}$$

For the mapping $z=e^{sT}$ from the $s$ plane to the $z$ plane to be invertible, it is necessary that $X(j\omega_a )$ be zero for all $\vert\omega_a \vert\geq \pi/T$ . If this is true, we say $x(t)$ is bandlimited to half the sampling rate. As is well known, this condition is necessary to prevent aliasing when sampling the continuous-time signal $x(t)$ at the rate $f_s=1/T$ to produce $x(nT)$ , $n=0,1,2,\ldots\,$ (see [84, Appendix G]).