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Properties of Laplace and Z-Transform

1) Linearity For Laplace:


If  Properties of Laplace & Z-Transform | Signals and Systems - Electrical Engineering (EE) with ROC R1 and Properties of Laplace & Z-Transform | Signals and Systems - Electrical Engineering (EE)with ROC R2,

 then

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

The ROC of X(s) is at least the intersection of R1 and R2,which could be empty,in which case x(t) has no Laplace transform.


For z-transform :

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

then

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

 

2) Differentiation in the time domain

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

This property follows by integration-by-parts. Specifically let

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

Then

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

and hence  Properties of Laplace & Z-Transform | Signals and Systems - Electrical Engineering (EE)

 

he ROC of sX(s) includes the ROC of X(s) and may be larger.


This property holds for z-transform as well.

 

3) Time Shift For Laplace transform:

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

For z-transform:

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

Because of  Properties of Laplace & Z-Transform | Signals and Systems - Electrical Engineering (EE) with ROC = R except for the possible addition or deletion of the origin or infinity

he multiplication by  Properties of Laplace & Z-Transform | Signals and Systems - Electrical Engineering (EE) for n0 > 0  poles will be introduced at z=0, which may cancel corresponding zeroes of X(z) at z=0. In this case the ROC  for Properties of Laplace & Z-Transform | Signals and Systems - Electrical Engineering (EE) equals the ROC of X(z) but with the origin deleted. Similarly, if n0 < 0 may get deleted.

 

 4) Time Scaling 

For Laplace transform:

If Properties of Laplace & Z-Transform | Signals and Systems - Electrical Engineering (EE) with ROC=R.

then

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

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

Where  Properties of Laplace & Z-Transform | Signals and Systems - Electrical Engineering (EE)

 A special case of time scaling is time reversal, when a= -1

 

For z transform: 

The continuous-time concept of time scaling does not directly extend to discrete time. However, the discrete time concept of time expansion i.e. of inserting a number of zeroes between successive values of a discrete time sequence can be defined. The new sequence can be defined as

x(k)[n] = x[n / k] if n is a multiple of k

= 0 if n is not a multiple of k

has k - 1 zeroes inserted between successive values of the original sequence. This is known as upsampling by k. If

x[n] ↔ X(z) with ROC = R

then x (k)[n] ↔ X(zk) with ROC = R1/k

i.e. z k R i.e. X(zk) = S x[n](zk)-n, −∞ < n < ∞

= S x[n]z-nk, −∞ < n < ∞

 

For Laplace transform
If
x(t) ↔ X(s) with ROC = R 

then 

eαtx(t) ↔ X(s - α) where Re(s - α)  ROC(X(.)) 

For z-transform

If x[n] ↔ X(z) with ROC = R 

then 

βn ↔ X(z/β) β ≠ 0 where zβ-1 ROC(X(.))

Consider

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

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

 

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

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

 Conclusion:

In this lecture you have learnt:


If Properties of Laplace & Z-Transform | Signals and Systems - Electrical Engineering (EE)with ROC = R then 


1. Properties of Laplace & Z-Transform | Signals and Systems - Electrical Engineering (EE)with ROC = R. 


2. Properties of Laplace & Z-Transform | Signals and Systems - Electrical Engineering (EE) with ROC = R.


3. Properties of Laplace & Z-Transform | Signals and Systems - Electrical Engineering (EE)

4. 4. eαtx(t) ↔ X(s - α) where Re(s - α)  ROC(X(.))

5. 5. If y(t) = (x*h)(t), Y(s) = H(s).X(s) where ROC of Y(s) = ROC(X)  ROC(H) 

 

If Properties of Laplace & Z-Transform | Signals and Systems - Electrical Engineering (EE) with ROC = R then 


1. Properties of Laplace & Z-Transform | Signals and Systems - Electrical Engineering (EE)with ROC = R except for the possible addition or deletion of infinity from ROC.


2. The continuous-time concept of time scaling does not directly extend to discrete time.Read upsampling for the reason.


3. Other properties of z-transform are similar to that of Laplace transform.

The document Properties of 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 Properties of Laplace & Z-Transform - Signals and Systems - Electrical Engineering (EE)

1. What is the Laplace transform?
Ans. The Laplace transform is an integral transform that converts a time-domain function into a complex frequency-domain function. It is often used to solve differential equations in the frequency domain.
2. What is the Z-transform?
Ans. The Z-transform is a mathematical transform that converts a discrete-time signal into a complex frequency-domain function. It is commonly used in digital signal processing to analyze and process discrete-time signals.
3. What are the properties of the Laplace transform?
Ans. The Laplace transform has several properties that make it a powerful tool in solving differential equations. Some of the key properties include linearity, time shift, scaling, differentiation, and integration.
4. What are the properties of the Z-transform?
Ans. Similar to the Laplace transform, the Z-transform also possesses important properties. Some of these properties include linearity, time shifting, scaling, convolution, and initial value theorem.
5. How are the Laplace and Z-transform related?
Ans. The Laplace transform and the Z-transform are related to each other through a mapping known as the bilinear transform. The bilinear transform allows for the conversion between continuous-time signals (Laplace domain) and discrete-time signals (Z domain). This relationship is especially useful in the design and analysis of digital filters.
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