(Blank Abstract)
Introduction
One of the most important concepts of DSP is to be able to
properly represent the input/output relationship to a given LTIsystem. A linear constant-coefficient
difference
equation (LCCDE) serves as a way to express just this
relationship in a discrete-time system. Writing the sequenceof inputs and outputs, which represent the characteristics of
the LTI system, as a difference equation help in understandingand manipulating a system.
difference equation
- An equation that shows the relationship between
consecutive values of a sequence and the differences amongthem. They are often rearranged as a recursive formula so
that a systems output can be computed from the inputsignal and past outputs.
As stated briefly in the definition above, a difference
equation is a very useful tool in describing and calculatingthe output of the system described by the formula for a given
sample
. The key property of
the difference equation is its ability to help easily find thetransform,
,
of a system. In the following two subsections, we will look atthe general form of the difference equation and the general
conversion to a z-transform directly from the differenceequation.
Difference equation
The general form of a linear, constant-coefficient
difference equation (LCCDE), is shown below:
We can also write the general form to easily express a
recursive output, which looks like this:
From this equation, note that
represents the outputs and
represents the inputs. The value of
represents the
order of the difference equation and
corresponds to the memory of the system being represented.Because this equation relies on past values of the output,
in order to compute a numerical solution, certain past outputs,referred to as the
initial conditions , must be known.
Using the above formula,
[link] , we can
easily generalize the
transfer function ,
, for any difference equation. Below are the steps
taken to convert any difference equation into its transferfunction,
i.e. z-transform. The first
step involves taking the
Fourier Transform of all the terms in
[link] . Then we use
the linearity property to pull the transform inside thesummation and the time-shifting property of the z-transform
to change the time-shifting terms to exponentials. Oncethis is done, we arrive at the following equation:
.
Conversion to frequency response
Once the z-transform has been calculated from the difference
equation, we can go one step further to define the frequencyresponse of the system, or filter, that is being
represented by the difference equation.
Remember that the reason we are dealing with these
formulas is to be able to aid us in filter design. A LCCDEis one of the easiest ways to represent FIR filters. By
being able to find the frequency response, we will be able tolook at the basic properties of any filter represented by a
simple LCCDE.
Below is the general formula for the frequency response of a
z-transform. The conversion is simple a matter of takingthe z-transform formula,
,
and replacing every instance of
with
.
Once you understand the derivation of this formula, look atthe module concerning
Filter Design
from the Z-Transform for a look into how all of these
ideas of the
Z-transform , Difference Equation, and
Pole/Zero Plots play a role in filter design.
