Problems based on Z-transform of some basic functions

I. problems based on Z-transform of some basic functions

Find the Z-transform of the following functions:

 

1. Prove that, Z [1] = z/-1, z > 1,

Solution: We know that, Zx (n)}z

 

Linearity property:

The Z-transform is linear

 (i.e.,) Z {ax (n) + by (n)} = a Z {x (n)} + b Z {y (n)}

 

II. Problems based on Z (1) = z/z-1 and Z [aⁿ] z /z-a if |z| > |a|

Find the Z-transform of the following functions:

 

1. Find Z (K)

 

2. Find Z [(-1)"]

 

3. Find Z [(-3)"]

 

4. Find Z[1/3ⁿ]

5. Find Z [e^an]

6. Find Z [e^-an]

7. Find Z [cos nθ] and Z d Z [sin nθ]

8. Find Z [rⁿcos nθ] and Z [rⁿ sinθ ]

9. Find Z (t).

10. Find Z [e^-at]

11. Find Z [cos an]

12. Find Z [a^n-1]

Solution :

We know that, Z [aⁿ]

13. Find Z [cosh a n]

14. Find Z [sinh 3n]

 

 

III. Find the Z transform of the following:

1. Find Z [cos n π/2]

2. Find Z [1/n (n + 1)]

3. Find Z [n-2/n (n + 1)]

4. Find 2 [sin(n π /2+a)]

5. (a) Find [sin² nπ /2]

5. (b) Find Z [sin² nπ /4]

6. Find Z [sin³ (nπ /6)]

 

Differentiation in the Z-Domain

 

IV. Find the Z-transform of the following:

[Differentiation in the Z-Domain]

 

1. Find Z (n²)

2. Find Z (n³)

3. Find Z (n^k)

Which is a recurrence formula.

4. Find Z [an² + bn + c]

Solution: Z [an² + bn + c]

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

Solution: Z [n (n − 1)] == Z [n²-n]

6. Find Z [nC₂]

8. Find Z [n (n - 1) (n = 2)]

 

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}.

 

11. Find the Z-transform of {k+n C₂}

 

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

Solution: Z[(n + 1) (n + 2) ] = Z [n2 + 2n + n + 2]

 

 

First Shifting theorem [Frequency shifting]:

Damping rule

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

 

 

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

Find the Z-transform of the following:

1. Find Z [a" n]

2 Find Z [aⁿ/n!]

3.Find [aⁿ/n]

4. Find Z [aⁿ sin nθ ]

5. Find Z [aⁿ cos nθ]

6. to Find Z [aⁿ rⁿ cos nθ]

7. Find Z [(n-1) a^n-1]

 

8. Find Z [an cos n л] 

9. Find Z [aⁿ n³]

10. Find Z[a-ⁿ n²]

11. Find Z [2ⁿ n²]

12. Find Z [2ⁿ sinh 3n]

13. Find Z [aⁿ cosh a n]

14. (i) Find Za" sin

 

Second shifting theorem [Time Shifting]

 

VI. Problems based on Time shifting

Find the Z-transform of the following:

1. Find Z [1/(n + 2)!]

2.Find Z [cos (n + 1) θ]

3. Find Z [sin (n-1)θ]

 

Unit impulse sequence and unit step sequence.

Definition: Unit impulse sequence

The unit impulse sequence 8 (n) is defined as the sequence with values

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

P

(2) Z-transform of unit step sequence i.e., Z (u (n)} = z/Z-1

Proof : We know that,

 

VII. Find the 2-transform of the following.

(based on unit impulse sequence and unit step sequence)

1. Find Z [δ (n-k)]

2. Find Z [^an δ (n − k)]

 

3. Find Z [2ⁿ δ (n-2)]

 

4. Find Z [3ⁿ δ (n − 1)]

 

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

 

 

 

 

Initial value theorem and final value theorem.

1. Initial value theorem

 

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

 

 

Differentiation:

 

IX. Problems based on