The $z$-Transform

• The $z$-transform is a mathematical tool that is extensively used to evaluate the properties of discrete-time systems such as digital filters. In particular, it is convenient for determining the stability of a system. It is the discrete-time equivalent of the Laplace transform, which is used for continuous-time systems.

• The unilateral $z$-Transform of a discrete-time signal $x[n]$ is given by:
  $$X(z) \ensuremath{\stackrel{\Delta}{=}}\sum_{n = 0}^{+\infty} x[n] z^{-n},$$
  where $z$ is a complex variable.

• The $z$-transform maps a discrete-time signal to a function of the complex variable $z$.

• A convenient property of the $z$-transform is given by the Shift Theorem,
  $$x[n - \Delta] \leftrightarrow z^{-\Delta} X(z),$$
  which says that a delay of $\Delta$ samples in the time domain corresponds to a multiplication by $z^{-\Delta}$ in the $z$ domain.

• Using the shift theorem, we can easily calculate the $z$-transform of a digital filter's difference equation. Given the following second-order difference equation,
  $$y[n] = b_0 x[n] + b_1 x[n-1] + b_2 x[n-2] - a_1 y[n-1] - a_2 y[n-2],$$
  the $z$-transform can immediately be written (assuming the system is linear)
  $$Y(z) = b_0 X(z) + b_1 z^{-1} X(z) + b_2 z^{-2} X(z) - a_1 z^{-1} Y(z) - a_2 z^{-2} Y(z).$$

• From this expression, we can determine the transfer function, $H(z) = Y(z) / X(z)$, of the filter:
  $$H(z) = \frac{Y(z)}{X(z)} = \frac{b_0 + b_1 z^{-1} + b_2 z^{-2}}{1 + a_1 z^{-1} + a_2 z^{-2}}.$$

• The transfer function of a system is evaluated in the complex $z$-plane, as illustrated below:

  Figure 5: The complex $z$-plane.

  

• The $z$-transform is a more general version of the Discrete-Time Fourier Transform, which itself can be viewed as the limiting form of the DFT when its length $N$ is allowed to approach infinity.

• We can determine the frequency response of a system from its $z$-transform by setting $z = e^{j \omega T_s}$, where $\omega$ is in radians per second and $T_s$ is the sample period. In the complex $z$-plane, this is equivalent to evaluating the $z$-transform on the “unit circle” defined by $z = e^{j \omega T_s}$.