1 The z-transform

If you have studied the Laplace transform either in a Mathematics course for Engineers and Scientists or have applied it in, for example, an analog control course you may recall that

1. the Laplace transform definition involves an integral
2. applying the Laplace transform to certain ordinary differential equations turns them into simpler (algebraic) equations
3. use of the Laplace transform gives rise to the basic concept of the transfer function of a continuous (or analog) system.

The z-transform plays a similar role for discrete systems, i.e. ones where sequences are involved, to that played by the Laplace transform for systems where the basic variable $t$ is continuous. Specifically:

1. the z-transform definition involves a summation
2. the z-transform converts certain difference equations to algebraic equations
3. use of the z-transform gives rise to the concept of the transfer function of discrete (or digital) systems.

Notes:

1. The z-transform only involves the terms $y_n$ , $n=0,1,2,\dots$ of the sequence. Terms $y_{-1},y_{-2},\dots$ whether zero or non-zero, are not involved.
2. The infinite series in (1) must converge for $Y\left(z\right)$ to be defined as a precise function of $z$ . We shall discuss this point further with specific examples shortly.
3. The precise significance of the quantity (strictly the ‘variable’) $z$ need not concern us except to note that it is complex and, unlike $n$ , is continuous.

Less strictly one might write $\mathbb{Z}{y}_n=Y\left(z\right)$ . Some texts use the notation $y_n\leftrightarrow Y\left(z\right)$  to denote that (the sequence) $y_n$ and (the function) $Y\left(z\right)$ form a z-transform pair.

We shall also call $\left\{y_n\right\}$ the inverse z-transform of $Y\left(z\right)$ and write symbolically

$\left\{y_n\right\}={\mathbb{Z}}^{-1}Y\left(z\right)$ .