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This 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.

Introduction

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.

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 importantcharacteristics 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 ofthe 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 belowshows the results of factoring the above rational function, [link] .

f x x 2 x 2 2 x 3 x 1

Thus, the roots of the rational function are as follows:

Roots of the numerator are: -2 2

Roots of the denominator are: -3 1

In order to understand rational functions, it is essential to know and understand the roots that make up the rationalfunction.

Discontinuities

Because we are dealing with division of two polynomials, we must be aware of the values of the variable that will causethe denominator of our fraction to be zero. When this happens, the rational function becomes undefined, i.e. we have a discontinuity in thefunction. Because we have already solved for our roots, it is very easy to see when this occurs. When the variable inthe denominator equals any of the roots of the denominator, the function becomes undefined.

Continuing to look at our rational function above, [link] , we can see that the function will have discontinuities at the followingpoints: x -3 1

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In respect to the Cartesian plane, we say that the discontinuities are the values along the x-axis where thefunction 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.

Intercepts

The x-intercept is defined as the point(s) where f x , i.e. the output of the rational functions, equals zero. Because we have alreadyfound the roots of the equation this process is very simple. From algebra, we know that the output will be zero wheneverthe 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 ofrepresenting the z-transform. Because of this, we can use the properties above, especially those of the roots, in order toreveal 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:

X z b 0 b 1 z -1 b M z M a 0 a 1 z -1 a N z N
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 thez-transform. The equation above, [link] , can be expressed in factored form just as was done for the simple rational function above, see [link] . Thus, we can easily find the roots of the numerator and denominator of thez-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 usefulways. 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 .

Questions & Answers

how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
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ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
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I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
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The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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Azam
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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
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