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Inverse Laplace & Z-Transform of Rational Functions | Signals and Systems - Electrical Engineering (EE) PDF Download

Inverse Laplace transform :: Rational functions

Consider an arbitrary rational polynomial in Laplace Transform

Inverse Laplace & Z-Transform of Rational Functions | Signals and Systems - Electrical Engineering (EE) 

Examples: 

1) Let us consider the function in s:


Inverse Laplace & Z-Transform of Rational Functions | Signals and Systems - Electrical Engineering (EE)


2) Let us consider an LTI system with system function:

Inverse Laplace & Z-Transform of Rational Functions | Signals and Systems - Electrical Engineering (EE)Inverse Laplace & Z-Transform of Rational Functions | Signals and Systems - Electrical Engineering (EE)

As the ROC has not been specified. there are several different ROCs and correspondingly, several different system impulses. Possible ROCs for the system with poles at s = -1and s = 2 and a zero at s = 1

Inverse Laplace & Z-Transform of Rational Functions | Signals and Systems - Electrical Engineering (EE)

 Inverse Laplace & Z-Transform of Rational Functions | Signals and Systems - Electrical Engineering (EE)

 

 Inverse Laplace & Z-Transform of Rational Functions | Signals and Systems - Electrical Engineering (EE)

Inverse Laplace & Z-Transform of Rational Functions | Signals and Systems - Electrical Engineering (EE)

 

Inverse Laplace & Z-Transform of Rational Functions | Signals and Systems - Electrical Engineering (EE)

Inverse Laplace & Z-Transform of Rational Functions | Signals and Systems - Electrical Engineering (EE)

Conclusions:


Properties of certain class of systems can be explained simply in terms of the locations of the poles. Particularly, consider a causal LTI system with a rational system function H(s). Since the system is causal, the ROC is to the right of the right most pole. Consequently, for this system to be stable (i.e. for the ROC to include the j-axis), the right most pole of H(s) must be to the left of the j-axis. i.e.

 

Inverse Laplace & Z-Transform of Rational Functions | Signals and Systems - Electrical Engineering (EE)

 

Inverse Z - transform: Consider an arbitrary rational z-transform:

 Inverse Laplace & Z-Transform of Rational Functions | Signals and Systems - Electrical Engineering (EE)

  Example 1: 

Consider the z transform

 Inverse Laplace & Z-Transform of Rational Functions | Signals and Systems - Electrical Engineering (EE) 

Example :

Consider the z transform

 Inverse Laplace & Z-Transform of Rational Functions | Signals and Systems - Electrical Engineering (EE)
There are two poles one at z=1/4 and at z=1/3. The partial fraction expansion, expressed in polynomials in 1/z, is

Inverse Laplace & Z-Transform of Rational Functions | Signals and Systems - Electrical Engineering (EE)


Thus, x[n] is the sum of 2 terms, one with z - transform 1/[1-(1/4z)] and the other with z - transform 2/[1-(1/3z)]. Thus,

Inverse Laplace & Z-Transform of Rational Functions | Signals and Systems - Electrical Engineering (EE)

As the ROC is not mentioned, we get different inverses for different possible ROCs. We do not discuss causality and stability as this may not be a system function. One possible inverse is worked out, the other two left as an exercise to the reader.

Inverse Laplace & Z-Transform of Rational Functions | Signals and Systems - Electrical Engineering (EE)

 

We can identify by inspection ,

Inverse Laplace & Z-Transform of Rational Functions | Signals and Systems - Electrical Engineering (EE)

  

Inverse Laplace & Z-Transform of Rational Functions | Signals and Systems - Electrical Engineering (EE)

 Conclusion:

In this lecture you have learnt:

  • if the system is causal then the ROC extends from the right most pole to infinity.
  • A system is stable if the ROC includes the imaginary axis and therefore the right most pole of 'H(s)' must be to the left of the imaginary axis
  • A causal system with a rational function 'H(s)' is stable if and only if all poles of H(s) lie in the left-half of the s-plane and must include the unit radius circle in the z-plane.

 

The document Inverse Laplace & Z-Transform of Rational Functions | Signals and Systems - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Signals and Systems.
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FAQs on Inverse Laplace & Z-Transform of Rational Functions - Signals and Systems - Electrical Engineering (EE)

1. What is the inverse Laplace transform?
Ans. The inverse Laplace transform is an operation that allows us to obtain the original function in the time domain from its Laplace transform. It essentially reverses the process of taking the Laplace transform.
2. How do you find the inverse Laplace transform of a rational function?
Ans. To find the inverse Laplace transform of a rational function, we first decompose it into partial fractions. Then, we use tables, properties, and techniques such as the method of residues or convolution to find the inverse Laplace transform of each partial fraction.
3. What is the z-transform?
Ans. The z-transform is a mathematical transform that converts a discrete-time signal into a complex function of a complex variable, usually denoted as z. It is used to analyze and process discrete-time signals and systems in areas such as digital signal processing.
4. How do you find the z-transform of a rational function?
Ans. To find the z-transform of a rational function, we represent the function as the ratio of two polynomials in z. Then, we use the properties of the z-transform, such as linearity, shifting, and scaling, along with the tables of common z-transform pairs, to determine the z-transform of each term in the rational function.
5. What is the relationship between the Laplace transform and the z-transform?
Ans. The Laplace transform and the z-transform are related to each other through a mapping. The z-transform is essentially a discrete-time version of the Laplace transform, where the complex variable s in the Laplace transform is replaced by the complex variable z in the z-transform. The z-transform can be seen as a special case of the Laplace transform when the Laplace variable s is replaced by z on the unit circle of the complex plane.
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