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

Properties of Laplace and Z-Transform

1) Linearity For Laplace:

If   with ROC R1 and with ROC R2,

 then

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 :

then

2) Differentiation in the time domain

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

Then

and hence 

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:

For z-transform:

Because of   with ROC = R except for the possible addition or deletion of the origin or infinity

he multiplication by   for n_{0 > 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  equals the ROC of X(z) but with the origin deleted. Similarly, if n_{0 < 0} may get deleted.

 4) Time Scaling 

For Laplace transform:

If  with ROC=R.

then

Where 

 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 = R^1/k

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

= S x[n]z^-nk, −∞ < n < ∞

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

then 

e^αtx(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

 Conclusion:

In this lecture you have learnt:

If with ROC = R then 

1. with ROC = R. 

2.  with ROC = R.

3.

4. 4. e^αtx(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  with ROC = R then 

1. 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.