ABSTRACT
When analyzing linear systems, one of the problems we often encounter is that of solving linear, constantcoefficient differential equations. A tool used for solving such equations is the Laplace transform. At the same time, to aid the analysis of linear systems, we extensively use Fourier-domain methods. With the advent of digital computers, it has become increasingly necessary to deal with discrete-time signals, or, sequences. These signals can be either obtained by sampling a continuous-time signal, or they could be inherently discrete. To analyze linear discrete-time systems, one needs a discrete-time counterpart of the Laplace transform (LT). Such a counterpart is found in the
z
transform, which similarly to the LT, can be used to solve linear constant-coefficient difference equations. In other words, instead of solving these equations directly, we transform them into a set of algebraic equations first, and then solve in this transformed domain. On the other hand, the
z
-transform can be seen as a generalization of the discretetime Fourier transform (FT)
(5.1)
This expression does not always converge, and thus, it is useful to have a representation which will exist for these nonconvergent instances. Furthermore, the use of the
z
-transform offers considerable notational simplifications. It also allows us to use the extensive body of work on complex variables to aid in analyzing discrete-time systems.
