Electrical Engineering (EE) Exam  >  Electrical Engineering (EE) Notes  >  Signals and Systems  >  Inverse Laplace & Z-Transform

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

Inverse Laplace Transform:
We know that there is a one to one correspondence between the time domain signal x(t) and its Laplace Transform X(s). Obtaining the signal 'x(t)' when 'X(s)' is known is called Inverse Laplace Transform (ILT). For ready reference , LT and ILT pair is given below :

X(s) = LT { x(t) } Forward Transform

 x(t) = ILT { X(s) } The Inverse TransformS

ome of the methods available for obtaining 'x(t)' from 'X(s)' are :

 

  • The complex inversion formulae.
  • Partial Fractions.
  • Series method.
  • Method of differential equations


In general:


If the Laplace Transform of 'x(t)' is 'X(s)' then the Inverse Laplace Transform of 'X(s)' is given by: 

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

Now 'C' is any vertical line in the s-plane that is parallel to the imaginary axis. 

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

 

Relationship between Laplace Transform and Fourier Transform 

The Fourier Transform for Continuous Time signals is infact a special case of Laplace Transform. This fact and subsequent relation between LT and FT are explained below.


Now we know that Laplace Transform of a signal 'x'(t)' is given by:

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

The s-complex variable is given by Inverse Laplace & Z-Transform | Signals and Systems - Electrical Engineering (EE)

But we consider Inverse Laplace & Z-Transform | Signals and Systems - Electrical Engineering (EE)  and therefore 's' becomes completely imaginary. Thus we have Inverse Laplace & Z-Transform | Signals and Systems - Electrical Engineering (EE) . This means that we are only considering the vertical strip atInverse Laplace & Z-Transform | Signals and Systems - Electrical Engineering (EE)

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

 From the above discussion it is clear that the LT reduces to FT when the complex variable only consists of the imaginary part . Thus LT reduces to FT along the Inverse Laplace & Z-Transform | Signals and Systems - Electrical Engineering (EE)  (Imaginary axis).

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

 Review:

We saw that if the imaginary axis lies in the Region of Convergence of 'X(s)' and the Laplace Transform is evaluated along it.
The result is the Fourier Transform of 'x(t)'.


Relationship between inverse Laplace Transform and inverse Fourier Transform 

Similarly while evaluating the Inverse Laplace Transform of 'X(s)' if we take the line ' C ' to be the imaginary axis (provided it lies in the Region of Convergence ). This is shown below as:

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

 Thus above we notice that we get the Inverse Fourier Transform of 'X(f)' as expected.


This tells us that there is a close relationship between the Laplace Transform and the Fourier Transform. In fact the Laplace Transform is a generalization of the Fourier Transform, that is, the Fourier Transform is a special case of the Laplace Transform only. The Laplace Transform not only provides us with additional tools and insights for signals and systems which can be analyzed using the Fourier Transform but they also can be applied in many important contexts in which Fourier Transform is not applicable. For example , the Laplace Transform can be applied in the case of unstable signals like exponential signals growing with time but the Fourier Transform cannot be applied to such signals which do not have finite energy.

 

Inverse Z - Transform

We know that there is a one to one correspondence between a sequence x[n] and its ZT which is X[z].
Obtaining the sequence 'x[n]' when 'X[z]' is known is called Inverse Z - Transform.
For a ready reference , the ZT and IZT pair is given below.

X[z] = Z { x[n] } Forward Z - Transform

 x[n] = Z-1 { X[z] } Inverse Z - Transform

 

For a discrete variable signal x[n], if its z - Transform is X(z), then the inverse z - Transform of X(z) is given by

Inverse Laplace & Z-Transform | Signals and Systems - Electrical Engineering (EE)
where ' C ' is any closed contour which encircles the origin and lies ENTIRELY in the Region of Convergence.

 

Relationship between Z - Transform and Discrete Time Fourier Transform (DTFT)

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

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

Which is same as the Discrete Time Fourier Transform (DTFT) of x[n]. Thus
Inverse Laplace & Z-Transform | Signals and Systems - Electrical Engineering (EE)

 

Similarly , on making the same substitution in the inverse z - Transform of X(z); provided the substitution is valid , that is, |z|=1lies in the ROC.

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

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

Hence we conclude that the z -Transform is just an extension of the Discrete Time Fourier Transform. It can be applied to a broader class of signals than the DTFT, that is, there are many discrete variable signals for which the DTFT does not converge but the z-Transform does so we can study their properties using the z -Transform.


Examples:

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

The z - Transform of this sequence is

 Inverse Laplace & Z-Transform | Signals and Systems - Electrical Engineering (EE)
Also we observe that the DTFT of the sequence does not exist since the summation

Inverse Laplace & Z-Transform | Signals and Systems - Electrical Engineering (EE)'

diverges. This example confirms that in some cases the z - Transform may exist but the DTFT may not.

 

Conclusion: 

In this lecture you have learnt:

  • If the Laplace Transform of 'x(t)' is 'X(s)' , then the Inverse Laplace Transform of X(s) is given by

 Inverse Laplace & Z-Transform | Signals and Systems - Electrical Engineering (EE)
where 'C' is any vertical line in the s plane, that is, parallel to the imaginary axis.

  • Fourier Transform of 'x(t)' = Laplace Transform of 'x(t)' when s=jw i.e. if the imaginary axis lies in the Region of Convergence of 'X(s)' and the Laplace Transform is evaluated along it , then the result is the Fourier Transform of 'x(t)'.
The document Inverse Laplace & Z-Transform | 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 - Signals and Systems - Electrical Engineering (EE)

1. What is the Laplace transform?
Ans. The Laplace transform is an integral transform used to convert a function of time into a function of complex frequency. It is particularly useful in solving linear differential equations with constant coefficients.
2. How is the inverse Laplace transform calculated?
Ans. The inverse Laplace transform is calculated by using a table of Laplace transforms or by applying partial fraction decomposition and then using the inverse transform of each term.
3. What is the Z-transform?
Ans. The Z-transform is a mathematical transform used to convert a discrete-time signal into a complex frequency representation. It is commonly used in digital signal processing and control systems analysis.
4. How is the Z-transform related to the Laplace transform?
Ans. The Z-transform is similar to the Laplace transform, but it is applied to discrete-time signals instead of continuous-time signals. The Z-transform can be seen as a discrete-time analogue of the Laplace transform.
5. How can the Z-transform be used to analyze discrete-time systems?
Ans. The Z-transform can be used to analyze discrete-time systems by converting their difference equations into a transfer function in the Z-domain. This transfer function can then be analyzed using techniques such as pole-zero analysis and frequency response analysis.
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