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Introduction to Z-Transform Notes | Study Digital Signal Processing - Electrical Engineering (EE)

Document Description: Introduction to Z-Transform for Electrical Engineering (EE) 2022 is part of Digital Signal Processing preparation. The notes and questions for Introduction to Z-Transform have been prepared according to the Electrical Engineering (EE) exam syllabus. Information about Introduction to Z-Transform covers topics like and Introduction to Z-Transform Example, for Electrical Engineering (EE) 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Introduction to Z-Transform.

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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 Notes | Study Digital Signal Processing - Electrical Engineering (EE)

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 Notes | Study Digital Signal Processing - Electrical Engineering (EE)

Or   Introduction to Z-Transform Notes | Study Digital Signal Processing - Electrical Engineering (EE)

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 Notes | Study Digital Signal Processing - Electrical Engineering (EE)

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

The document Introduction to Z-Transform Notes | Study Digital Signal Processing - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Digital Signal Processing.
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