UNIT 05 Z-TRANSFORM ANALYSIS | Basic Signal and Systems -AKTU

In this section, we will examine UNIT 05 Z-TRANSFORM ANALYSIS | Basic Signal and Systems -AKTU, subject “Electrical and Electronics Engineering”. I hope this article is useful to you as you prepare for your AKTU exams.

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Q1. Explain unilateral and bilateral Z-transform.

Ans. 1. For non-negative values of the time index (n ≥  0), the unilateral or one-sided Z-transform is assessed. This Z-transform variant is suitable for causal and LTI system-related issues.  

2. The unilateral Z-transform of a signal x[n] is defined as,

which depends only on x[n] for n >0.  

3. For a discrete-time signal x[n], the bilateral Z-transform is defined as 

4 The bilateral and unilateral Z-transform are equivalent if x[n] =0 for n < 0. 

Q2. What is region of convergence ?Also write the properties of ROC for Z-transform?  

Ans. A. ROC: ROC is defined as the set of values of z in the z-plane for which the magnitude of X(z) is finite. 

B. Properties of ROC for Z-transform: 

1. The ROC is a ring in the z-plane centred at the origin. 

2. The ROC cannot contain any poles. 

3. The ROC must be connected region. 

4. The ROC is the entirety of the z-plane, with the possible exception of z = 0 or z = o, if z(n) has a limited duration sequence, 

5. If x(n) is right-sided sequence, then the ROC exist outward from the outermost finite pole to z = ∞.

6. If x(n) is a left-sided sequence, then the ROC extends inward from the innermost pole to z = 0.

7. If x(n) is two sided sequence,then the ROC will consist a ring in the z plane, bounded on the interior and exterior by a pole and consistent with property (2). 

Example: Let X(z) = 1/ (z -1)  

Q3. Determine the Z-transform of 

Ans. i. We know that

ii. We know that

Q4. Find the Z-transform of the following sequences:

i. x₁[n]= {1, 2,3, 4, 5, 0, 7}

ii. x₂[n]= {1, 2, 3, 4, 5, 0, 7} 

                             ↑

Ans. i. Given: x₁[n] = {1, 2, 3, 4, 5, 0, 7}  

         To Find: X₁(z).

ii. Given:  x₂[n]= {1, 2, 3, 4, 5, 0, 7} 

                                          ↑

To Find: X₂(z) 

Q5. Find the inverse Z-transform of the following function: 

Ans. 

Q6. State and prove initial and final value theorem for Z-transform.

Ans. A. Initial value theorem: The initial value theorem of Z-transform states that, for a causal signal x [n]

          Proof:  

1. For a causal signal

2. Taking the limit z → ∞  on both sides, we have

3. The initial value of x[n] can be determined from X(z) using this theorem without the need to do its inverse Z-transform. 

B. Final value Theorem: The final value theorem of Z-transform states that, for a causal signal.

and if X(z) has no poles outside the unit circle, and it has no double or higher order poles on the unit circle centered at the origin of the z-plane, then 

Proof:

1. For a causal signal

5. This theorem enables us to find the steady-state value of x[n], i.e. x(∞) without taking the inverse Z-transform of X(z).

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