z - transforms

UNIT - V

Z - TRANSFORMS AND DIFFERENCE EQUATIONS

Z-transforms - Elementary properties - Inverse Z - transform (using partial fraction and residues) - Convolution theorem - Formation of difference equations - Solution of difference equations using Z-transform.

Introduction

The Z-transform plays the same role for discrete systems as the Laplace transform does for continuous systems.

Applications of Z transform

1. Communication is one of the fields whose development is based on discrete analysis. Difference equations are also based on discrete system and their solutions and analysis are carried out by Z - transform.

2. In the system analysis area, the Z - transform converts convolutions to a product and difference equations to algebraic equations.

3. The stability of a discrete linear system can be determined by analyzing the transfer function H (z) given by the Z - transform.

4. Digital filters can be analyzed and designed using the Z - transform.

5. Digital control systems can be analyzed and designed using Z- transforms.

1. Z - transform, elementary properties of Z - transform

Z - transform

Definition: Z - transform (two-sided or bilateral)

Let {x (n)} be a sequence defined for all integers then its Z-transform is defined to be

where z is an arbitrary complex number.

Definition: Z - transform (one-sided or unilateral)

Let {x (n)} be a sequence defined for n = 0, 1, 2, ... and x (n) 0 for n < 0, then its Z- transform is defined to be

where z is an arbitrary complex number.

Definition: Z - transform for discrete values of t

If f (t) is a function defined for discrete values of t where t = nT, n = 0, 1, 2, 3, ... T being the sampling period, then Z - transform of f(t) is defined as

Note:  

1. Mostly we study one-sided Z transform

2. If ƒ (t) given, then replace 't’ by ‘n T'

3. The double braces {} are used for a sequence. 

Sometimes we use [ ] or ( ).

1. Problems based on Z-transform of some basic functions

Find the Z-transform of the following functions :

1. Prove that, Z [1] =

Solution: 

2. Prove that, Z [an] =

Solution: 

3. Prove that, Z (n) =

Solution: 

4. Prove that,

Solution : 

5. Prove that,

Solution :

6. Prove that,

Solution: 

7. Prove that,

Solution :

8. Find

Solution: 

9. Find the Z-transform of (n + 2)

Solution:

10. Find

Solution: 

2. Linearity property :

The Z - transform is linear

Proof: 

Find the Z-transform of the following functions :

1. Find Z (K)

Solution: 

2. Find Z [(-1)n]

Solution: 

3. Find Z [(-3)n]

Solution: 

4. Find

Solution :

5. Find Z [ean].

Solution :

6. Find Z [e-an]

Solution: 

7. Find Z [cos nθ] and Z [ sin nθ]

Solution :

Equating the real and imaginary parts, we get

8. Find Z [rncos nθ] and Z [rn sin nθ]

Solution:

Equating the real and imaginary parts, we get

9. Find Z (t).

Solution: 

10. Find Z [e-at]

Solution: 

11. Find Z [cos an]

Solution: 

12. Find Z [an-1]

Solution :

13. Find Z [cosh α n]

Solution :

14. Find Z [sinh 3n]

Solution :

III. Find the Z transform of the following:

1. Find

Solution :

2. Find

Solution :

3. Find

Solution :

4. Find

Solution : 

5. (a) Find

Solution :

5. (b) Find

Solution : 

6. Find

Solution :

7. Find

Solution :

8. Find Z [4.3n + 2 (−1)n]

Solution: 

9. Find

Solution :

10. Find the Z-transform of

Solution :

3. Differentiation in the Z-Domain

Proof :

Given: F(z) = Z [f(n)]

IV. Find the Z-transform of the following:

[Differentiation in the Z-Domain]

1. Find Z (n2)

Solution: 

2. Find Z (n3)

Solution: 

3. Find Z (nk)

Solution : 

Which is a recurrence formula.

4. Find Z [an2 + bn + c]

Solution : Z [an2 + bn + c]

= a Z [n2] + b Z [n] + c Z [1]

5. Find Z [n (n - 1)]

Solution: 

6. Find Z [nC2]

Solution :

7. Find Z [n cos nθ ]

Solution: 

8. Find Z [n (n − 1) (n − 2)]

Solution :

9. Given that F (z) = log (1 + az -1), for |z| > |a |, find f(n) and also find Z[n_f(n)].

Solution : Given : F(z) = log (1 + az-1)

10. Find the Z-transform of {nCk}.

Solution : 

This is the expansion of binomial theorem.

= (1+z-1)n

11. Find the Z-transform of {k+nCn}

Solution: 

12. Find the Z-transform of (n + 1) (n + 2).

Solution : 

4. First Shifting theorem [Frequency shifting]:

Damping rule

 [The geometric factor a-n when a < 1, damps the function un. Hence we use the name damping rule]

Proof: (i) Given: F(z) = Z {f(n)}

V. Problems based on first shifting theorem [Frequency shifting]

Find the Z-transform of the following:

1. Find Z [an n]

Solution : 

2. Find

Solution :

3. Find

Solution: 

4. Find Z  [an sin n θ]

Solution: 

5. Find Z [an cos n θ]

Solution: 

6. Find Z [an rn cos nθ]

Solution :

Aliter :

7. Find Z [(n – 1)an-1]

Solution :

8. Find Z [an cos nπ]

Solution: 

9. Find Z [an n3]

Solution: 

10. Find Z [a-nn2]

Solution: 

11. Find Z [2n n2]

Solution: 

12. Find Z [2n sinh 3n]

Solution: 

13. Find Z [an cosh α n]

Solution : 

14. (i) Find

Solution: 

5. Second shifting theorem [Time Shifting]

VI. Problems based on Time shifting

Find the Z-transform of the following:

1. Find

Solution :

2. Find Z [cos (n + 1) θ]

Solution :

3. Find Z [sin (n − 1)θ]

Solution :

6. Unit impulse sequence and unit step sequence.

Definition: Unit impulse sequence

The unit impulse sequence d (n) is defined as the sequence with values

(1) Z-Transfrom of unit impulse sequence is 1. i.e., Z [δ (n)] = 1

Proof : 

Definition: Unit step sequence

The unit step sequence u (n) has values

(2) Z-transform of unit step sequence i.e., Z (u (n)}=

Proof : 

(3) Z-transform of

Proof : 

VII. Find the L-transform of the following.

(based on unit impulse sequence and unit step sequence)

1. Find Z [δ (n − k)]

Solution: 

2. Find Z [an δ (n − k)]

Solution: 

3. Find Z [2n δ (n − 2)]

Solution: 

4. Find Z [3n δ (n − 1)]

Solution: 

5. Find Z [u (n - 1)]

Solution: 

6. Find

Solution : 

7. Find

Solution : 

7. Initial value theorem and final value theorem.

1. Initial value theorem

Proof : 

2. Final value theorem.

Proof :

VIII. Problems based on initial value theorem and final value theorem

1. If F (z)

Solution : By Initial value theorem

By Final value theorem we get

2. If evaluate u2 and u3.

Solution: 

By Initial value theorem :

3. If find the value of u0, u1 and u2.

Solution: 

By Initial value theorem :

8. Differentiation :

1. If

Solution : 

IX. Problems based on 

Find the Z-transform of (i) n an u (n) ; (ii) n (n − 1) an u (n)

Solution: 

Note: Put a = 1, we get