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 −

Introduction to Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE)

The above equation represents a two-sided Z-transform equation.

Generally, when a signal is Z-transformed, it can be represented as −

Introduction to Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE)

Or   Introduction to Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE)

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)1Entire 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)2Mod(Z)>Mod(a)
−anu(−n−1)aZ−1/(1−aZ−1)2Mod(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 −

Introduction to Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE)

ROC is the entire Z-plane excluding Z = 0, ∞, -∞

The document Introduction to Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE) is a part of the Electronics and Communication Engineering (ECE) Course Digital Signal Processing.
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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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