Z Transform, Sampling
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
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© 2000 Cuthbert A. Nyack.