In this Workbook we have developed the theory and some applications of the z-transform from first principles. We mentioned much earlier that the z-transform plays essentially the same role for discrete systems that the Laplace transform does for continuous systems. We now explore the precise link between these two transforms. A brief knowledge of Laplace transform will be assumed.
At first sight it is not obvious that there is a connection. The z-transform is a summation defined, for a sampled signal , as
while the Laplace transform written symbolically as is an integral , defined for a continuous time function as
Thus, for example, if
which has a (simple) pole at say.
As we have seen, sampling gives the sequence with z-transform
The z-transform has a pole when where
[Note the abuse of notations in writing both and here since in fact these are different functions.]
The continuous time function has Laplace transform
Firstly write down the pole of this function and its order:
has its pole at . The pole is second order.
Now obtain the z-transform of the sampled version of , locate the pole(s) of and state the order:
The ramp sequence has z-transform
(see Key Point 8)
This has a (second order) pole when .
We have seen in both the above examples a close link between the pole of the Laplace transform of and the pole of the z-transform of the sampled version of i.e.
where is the sample interval.
Multiple poles lead to similar results i.e. if has poles then has poles where .
The relation (1) between the poles is, in fact, an example of a more general relation between the values of and as we shall now investigate.
The unit impulse function can be defined informally as follows:
The rectangular pulse of width and height shown in Figure 21 encloses unit area and has Laplace transform
As becomes smaller becomes taller and narrower but still encloses unit area. The unit impulse function (sometimes called the Dirac delta function) can be defined as
The Laplace transform, say , of can be obtained correspondingly by letting in (2), i.e.
A shifted unit impulse is defined as as illustrated below.
Obtain the Laplace transform of this rectangular pulse and, by letting , obtain the Laplace transform of .
which reduces to the result (3)
These results (3) and (4) can be compared with the results
for discrete impulses of height 1.
Now consider a continuous function . Suppose, as usual, that this function is sampled at for
This sampled equivalent of , say can be defined as a sequence of equidistant impulses, the ‘strength’ of each impulse being the sample value i.e.
This function is a continuous-time signal i.e. is defined for all . Using (4) it has a Laplace transform
If, in this sum (5) we replace by we obtain the z-transform of the sequence of samples:
The Laplace transform
of a sampled function is equivalent to the z-transform of the sequence of sample values with .
This table can be compared with the table of the z-transforms of sequences on the following page.
|Radius of convergence|
Note: is such that the closed forms of (those listed in the above table) are valid for .
|unit step sequence|