Z Transform, Sampling and Definition,

Cuthbert Nyack

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.

A mathematical representation of the sampled signal is shown below. This is equivalent to modulating a train of delta functions by the analog signal. The delta function effectively "filters" out the values of the signal at times corresponding to the zeros in the argument of the delta function. This process is also referred to as "ideal" sampling since it results in sampled signals of "zero" width and whose spectrum is perfectly periodic.

The above equation is equivalent to the following since the delta function has the effect of making x(t) nonzero only at times t = kT.

Taking the Laplace transform of the sampled signal using the integral definition and the properties of the delta function results in the following

The Laplace transform has the Laplace variable s occuring in the exponent and can be awkward to handle. A much simpler expression results if the following substitutions are made

produces the definition of the Z Transform

If the sampling time T is fixed then the Z Transform can also be written

The final result is a polynomial in Z. The Z Transform plays a similar role in the processing of sampled signals as the Laplace transform does in the processing of continuous signals.
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 2ⁿ - 1.
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.

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