Rational Functions and the Z-Transformhttp://cnx.org/contenthttp://cnx.org/content/m10593/latest/m10593Rational Functions and the Z-Transform2.82002/05/222010/07/22 15:26:09.581 GMT-5MariyahPoonawalaMariyah Poonawalamariyah@rice.eduMichaelHaagMichael Haagmjhaag@gmail.comPrashantSinghPrashant Singhprash@ece.rice.edumjhaagmariyah prashmjhaagfunctionpolynomialrationalrational functionrational functionsz-transformMathematics and StatisticsThis module will introduce rational functions and describe some of their properties. In particular, it will discuss how rational functions relate to the z-transform and provide a useful tool for characterizing LTI systems.enIntroduction
When dealing with operations on polynomials, the term rational
function is a simple way to describe a particular
relationship between two polynomials.
rational function
For any two polynomials, A and B, their quotient is called a
rational function.
Below is a simple example of a basic rational function,
fx.
Note that the numerator and denominator can be polynomials of
any order, but the rational function is undefined when the
denominator equals zero.
fxx242x2x3
If you have begun to study the
Z-transform, you
should have noticed by now they are all rational functions.
Below we will look at some of the properties of rational
functions and how they can be used to reveal important
characteristics about a z-transform, and thus a signal or LTI system.
Properties of Rational Functions
In order to see what makes rational functions special, let us
look at some of their basic properties and characteristics.
If you are familiar with rational functions and basic
algebraic properties, skip to the next section to see how
rational functions are useful when dealing with the z-transform.
Roots
To understand many of the following characteristics of a
rational function, one must begin by finding the roots of
the rational function. In order to do this, let us factor
both of the polynomials so that the roots can be easily determined.
Like all polynomials, the roots will provide us with
information on many key properties. The function below
shows the results of factoring the above rational function,
.
fxx2x22x3x1
Thus, the roots of the rational function are as follows:
Roots of the numerator are:
-22
Roots of the denominator are:
-31
In order to understand rational functions, it is essential
to know and understand the roots that make up the rational
function.
Discontinuities
Because we are dealing with division of two polynomials, we
must be aware of the values of the variable that will cause
the denominator of our fraction to be zero. When this
happens, the rational function becomes undefined,
i.e. we have a discontinuity in the
function. Because we have already solved for our roots, it
is very easy to see when this occurs. When the variable in
the denominator equals any of the roots of the denominator,
the function becomes undefined.
Continuing to look at our rational function above,
, we can see that
the function will have discontinuities at the following
points:
x-31In respect to the Cartesian plane, we say that the
discontinuities are the values along the x-axis where the
function is undefined. These discontinuities often appear
as vertical asymptotes on the graph to
represent the values where the function is undefined.
Domain
Using the roots that we found above, the domain
of the rational function can be easily defined.
domain
The group, or set, of values that are defined by a given
function.
Using the rational function above,
, the domain
can be defined as any real number
x where
x does not equal 1 or
negative 3. Written out mathematical, we get the following:
xx-3x1Intercepts
The x-intercept is defined as the point(s) where
fx, i.e. the output of the
rational functions, equals zero. Because we have already
found the roots of the equation this process is very simple.
From algebra, we know that the output will be zero whenever
the numerator of the rational function is equal to zero.
Therefore, the function will have an x-intercept wherever
x equals one of the roots of
the numerator.
The y-intercept occurs whenever
x equals zero. This can be
found by setting all the values of
x equal to zero and solving
the rational function.
Rational Functions and the Z-Transform
As we have stated above, all z-transforms can be written as
rational functions, which have become the most common way of
representing the z-transform. Because of this, we can use the
properties above, especially those of the roots, in order to
reveal certain characteristics about the signal or LTI system
described by the z-transform.
Below is the general form of
the z-transform written as a rational function:
Xzb0b1z-1…bMzMa0a1z-1…aNzN
If you have already looked at the module about
Understanding Pole/Zero Plots and
the Z-transform, you should see how the roots of the
rational function play an important role in understanding the
z-transform. The equation above, , can be expressed in factored form just as
was done for the simple rational function above, see . Thus, we can easily
find the roots of the numerator and denominator of the
z-transform. The following two relationships become apparent:
Relationship of Roots to Poles and Zeros
The roots of the numerator in the rational function will be the
zeros of the z-transform
The roots of the denominator in the rational function will be the
poles of the z-transform
Conclusion
Once we have used our knowledge of rational functions to find its
roots, we can manipulate a z-transform in a number of useful
ways. We can apply this knowledge to representing an LTI
system graphically through a Pole/Zero Plot, or to analyze and design a
digital filter through Filter Design from the Z-Transform.