Electronics and Communication Engineering (ECE) Exam > Electronics and Communication Engineering (ECE) Notes > Digital Signal Processing > Introduction to Z-Transform
Introduction to Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE) PDF Download
Discrete Time Fourier Transform(DTFT) exists for energy and power signals. Z-transform also exists for neither energy nor Power (NENP) type signal, up to a certain extent only. The replacement z = ejw is used for Z-transform to DTFT conversion only for absolutely summable signal.
So, the Z-transform of the discrete time signal x(n) in a power series can be written as −
The above equation represents a two-sided Z-transform equation.
Generally, when a signal is Z-transformed, it can be represented as −
Or 
If it is a continuous time signal, then Z-transforms are not needed because Laplace transformations are used. However, Discrete time signals can be analyzed through Z-transforms only.
Region of Convergence
Region of Convergence is the range of complex variable Z in the Z-plane. The Z- transformation of the signal is finite or convergent. So, ROC represents those set of values of Z, for which X(Z) has a finite value.
Properties of ROC
- ROC does not include any pole.
- For right-sided signal, ROC will be outside the circle in Z-plane.
- For left sided signal, ROC will be inside the circle in Z-plane.
- For stability, ROC includes unit circle in Z-plane.
- For Both sided signal, ROC is a ring in Z-plane.
- For finite-duration signal, ROC is entire Z-plane.
The Z-transform is uniquely characterized by −
- Expression of X(Z)
- ROC of X(Z)
Signals and their ROC
| x(n) | X(Z) | ROC |
|---|---|---|
| δ(n) | 1 | Entire Z plane |
| U(n) | 1/(1−Z−1) | Mod(Z)>1 |
| anu(n) | 1/(1−aZ−1) | Mod(Z)>Mod(a) |
| −anu(−n−1) | 1/(1−aZ−1) | Mod(Z)<Mod(a) |
| nanu(n) | aZ−1/(1−aZ−1)2 | Mod(Z)>Mod(a) |
| −anu(−n−1) | aZ−1/(1−aZ−1)2 | Mod(Z)<Mod(a) |
| U(n)cosωn | (Z2−Zcosω)/(Z2−2Zcosω+1) | Mod(Z)>1 |
| U(n)sinωn | (Zsinω)/(Z2−2Zcosω+1) | Mod(Z)>1 |
Example
Let us find the Z-transform and the ROC of a signal given as x(n)={7,3,4,9,5}, where origin of the series is at 3.
Solution − Applying the formula we have −
ROC is the entire Z-plane excluding Z = 0, ∞, -∞
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FAQs on Introduction to Z-Transform - Digital Signal Processing - Electronics and Communication Engineering (ECE)
| 1. What is the Z-transform in electrical engineering? |  |
Ans. The Z-transform is a mathematical transformation technique used in electrical engineering to convert discrete-time signals into the Z-domain, similar to how the Laplace transform converts continuous-time signals into the frequency domain.
| 2. How is the Z-transform different from the Fourier transform? | |
Ans. The Z-transform and the Fourier transform are both used to analyze signals, but they differ in their applicability. The Fourier transform is used for continuous-time signals, while the Z-transform is used for discrete-time signals. The Z-transform provides information about the frequency content of a discrete-time signal, whereas the Fourier transform provides information about the frequency content of a continuous-time signal.
| 3. What are the advantages of using the Z-transform in electrical engineering? | |
Ans. The Z-transform offers several advantages in electrical engineering. It allows engineers to analyze and manipulate discrete-time signals using mathematical techniques, making it easier to design and analyze digital filters, control systems, and communication systems. The Z-transform also provides a bridge between the time domain and the frequency domain, enabling engineers to understand the behavior of discrete-time systems in terms of their frequency response.
| 4. How is the Z-transform applied in signal processing? | |
Ans. In signal processing, the Z-transform is used to analyze and process discrete-time signals. It allows engineers to represent discrete-time signals and systems in the Z-domain, where operations such as filtering, convolution, and frequency analysis can be performed using algebraic manipulation. The Z-transform is particularly useful in the design and analysis of digital filters, as it enables engineers to characterize the frequency response of the filter and adjust its parameters accordingly.
| 5. What are some common applications of the Z-transform in electrical engineering? | |
Ans. The Z-transform finds applications in various areas of electrical engineering. It is commonly used in digital signal processing for tasks such as signal filtering, noise removal, and spectral analysis. The Z-transform is also used in control systems engineering for designing and analyzing discrete-time control systems. Additionally, it is employed in communication systems for tasks such as channel equalization, signal detection, and error correction.
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