Electrical Engineering (EE) Exam > Electrical Engineering (EE) Notes > Signals and Systems > Z-Transform & Region of Convergence
Z-Transform & Region of Convergence | Signals and Systems - Electrical Engineering (EE) PDF Download
Objectives:
Scope of this lecture:
We have already seen the implementation of Fourier Transform and Laplace Transform for the study of Continuous Time (C.T.) signals and systems. Now our interest lies in frequency domain analysis and design of Discrete Time (D.T.) signals and systems. The ZTransform provides a valuable technique for frequency domain analysis of D.T. signals and design of DT-LTI systems. Further ZTransform offers an extremely convenient and compact way to describe digital signals and processors. Numerical problems are presented for a better understanding of the relevant concepts involved.
- We shall look at the definition of Z-transform .
- The need to consider Region of Convergence (ROC) with suitable illustrations
- The nature of ROC's in both Laplace Transform and Z-transform Domains..
Z-transform
The response of a linear time-invariant system with impulse response h[n] to a complex exponential input of the form can be represented in the following way :
In the complex z-plane , we take a circle with unit radius centered at the origin.
H(w) is periodic with period with respect to ' w ' .
When we replace z by ejω ,we get periodicity of 
in the form of a circle.
Nature of Region of Convergence
Laplace Transform:
The ROC of the Laplace transform X(s) of a two-sided signal lies between two vertical lines in the s-plane.
depend only on real part of s. For a right-sided signal 
and the corresponding ROC is referred to as right-half plane. Similarly for a left-sided signal 
This ROC is referred to as left-half plane. When x(t) is two-sided i.e; of infinite extent for both t > 0 and t < 0 ; both and are finite and the ROC thus turns out to be a vertical strip in the s-plane.
Z-transform:
The ROC of X(z) of a two sided signal consists of a ring in the z-plane centered about the origin.
depend only on magnitude of z. As in the case of Laplace transform for
a right-sided sequence and 
for aleft -sided sequence. If x[n] is two-sided ;the ROC will consist of a ring with both and finite and non-zero.
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FAQs on Z-Transform & Region of Convergence - Signals and Systems - Electrical Engineering (EE)
| 1. What is the Z-transform and how is it related to the region of convergence? |  |
The Z-transform is a mathematical tool used in digital signal processing to convert discrete-time signals into the Z-domain. It is analogous to the Laplace transform used in continuous-time signal processing. The Z-transform of a discrete-time signal is represented by the equation X(z) = Σ[x(n) * z^(-n)], where x(n) is the discrete-time signal and z is the complex variable.
The region of convergence (ROC) is a set of complex numbers in the Z-plane for which the Z-transform converges. It defines the range of values of z for which the Z-transform is valid. The ROC can be inside, outside, or on the unit circle in the Z-plane. The choice of ROC determines the causality and stability properties of the system represented by the Z-transform.
| 2. Why is the region of convergence important in Z-transform analysis? | |
The region of convergence (ROC) is crucial in Z-transform analysis as it provides information about the stability and causality of the system.
1. Stability: The ROC determines whether a system is stable or not. If the ROC includes the unit circle in the Z-plane, the system is stable. On the other hand, if the ROC lies outside the unit circle, the system is unstable.
2. Causality: The ROC also indicates the causality of the system. If the ROC includes the exterior of a certain radius in the Z-plane, the system is causal, meaning the output depends only on the current and past inputs. If the ROC includes the interior of a certain radius, the system is non-causal, meaning the output depends on future inputs as well.
By analyzing the ROC, engineers can determine the stability and causality properties of the system and make informed decisions for system design and analysis.
| 3. What are the different types of region of convergence in Z-transform analysis? | |
In Z-transform analysis, the region of convergence (ROC) can be classified into three main types:
1. Inside the unit circle: This type of ROC includes all the points inside the unit circle in the Z-plane. It represents stable and causal systems. The boundedness of the Z-transform within the unit circle ensures the stability of the system.
2. Outside the unit circle: This type of ROC includes all the points outside the unit circle. It represents unstable systems. The unboundedness of the Z-transform outside the unit circle indicates the instability of the system.
3. On the unit circle: This type of ROC includes points on the unit circle. It represents marginal stability. The Z-transform is bounded on the unit circle but not inside or outside it, indicating a borderline stability condition.
The type of ROC determines the stability and causality properties of the system and plays a vital role in system analysis and design.
| 4. How does the region of convergence affect the frequency response of a system? | |
The region of convergence (ROC) influences the frequency response of a system represented by the Z-transform. The frequency response describes how the system alters the input signal's frequency components.
The ROC determines the range of frequencies for which the system's frequency response is valid. For frequencies within the ROC, the frequency response can be obtained by evaluating the Z-transform on the unit circle. If the ROC includes the unit circle, the frequency response is well-defined and stable. However, if the ROC is outside the unit circle, the frequency response is undefined, indicating an unstable system.
In summary, the ROC restricts the valid range of frequencies for which the frequency response can be calculated, providing insights into the system's stability and behavior.
| 5. Can the region of convergence change for different signals in the same system? | |
Yes, the region of convergence (ROC) can vary for different signals within the same system. The ROC is signal-dependent and can change based on the characteristics of the input signal.
For a given system, the ROC represents the set of complex numbers for which the Z-transform converges. Different input signals may lead to different convergence properties, resulting in varying ROCs.
It is possible for a system to have multiple ROCs corresponding to different input signals. These ROCs can have different shapes and locations in the Z-plane, indicating different stability and causality properties for each input signal.
In practice, engineers analyze the ROC for various input signals to understand the system's behavior under different conditions and ensure its desired performance.
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