Most real signals are analog and in order to utilise the processing power of modern digital processors it is necessary to convert these analog signals into some form which can be stored and processed by digital devices. The standard method is to sample the signal periodically and digitize it with an A to D converter using a standard number of bits 8, 16 etc. Digital signal processing is primarily concerned with the processing of these sampled signals. The diagram below illustrates the situation. The blue line shows the analog signal while the red lines shows the samples arising from periodic sampling at intervals T.

In the above equations x(kT) and x(k) represents a number arising from the sampling and digitizing process. For 8 bit quantization x(k) would have integer values from 0 to 255 (or -127 to +127) and for n bit quantization it would be from 0 to 2

Since the Laplace variable s is complex then the variable z is also complex and X(z) is a complex function having real and imaginary parts or magnitude and phase.

Because of the increasing occurence of digital signal processing and the fact that some come into dsp without analog signal processing(considered to be mathematically difficult by some), then the above equation can be used as a definition of the Z Transform without reference to the Laplace transform.

The above definition of X(z) uses only positive values of k and is sometimes referred to as the one sided definition of the Z Transform. It is also possible to have a 2 sided definition of the transform as in the following equation. The mathematical differences between the 2 definitions mainly relate to regions of convergence. Here negative values of k will be freely included when convergence is assured.

© 2000 Cuthbert A. Nyack.