Existence of Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE) PDF Download

A system, which has system function, can only be stable if all the poles lie inside the unit circle. First, we check whether the system is causal or not. If the system is Causal, then we go for its BIBO stability determination; where BIBO stability refers to the bounded input for bounded output condition.

This can be written as;

Existence of Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE)

The above equation shows the condition for existence of Z-transform.

However, the condition for existence of DTFT signal is

Existence of Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE)

Example 1

Let us try to find out the Z-transform of the signal, which is given as

Existence of Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE)

Solution − Here, for −(−2)nu(n) the ROC is Left sided and Z<2

For 3nu(n) ROC is right sided and Z>3

Hence, here Z-transform of the signal will not exist because there is no common region.

Example 2

Let us try to find out the Z-transform of the signal given by

Existence of Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE)

Solution − Here, for −2nu(−n−1) ROC of the signal is Left sided and Z<2

For signal (0.5)nu(n)ROC is right sided and Z>0.5

So, the common ROC being formed as 0.5<Z<2

Therefore, Z-transform can be written as;

Existence of Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE)

Example 3

Let us try to find out the Z-transform of the signal, which is given as x(n) = 2r(n)

Solution − r(n) is the ramp signal. So the signal can be written as;

Existence of Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE)

Existence of Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE)

Here, for the signal u(−n−1) and ROC Z<1 and for 2nu(n) with ROC is Z>2.

So, Z-transformation of the signal will not exist.

Z -Transform for Causal System

Causal system can be defined as h(n) = 0,n < 0. For causal system, ROC will be outside the circle in Z-plane.

Existence of Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE)

Expanding the above equation,

Existence of Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE)

For causal systems, expansion of Transfer Function does not include positive powers of Z. For causal system, order of numerator cannot exceed order of denominator. This can be written as-

Existence of Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE)

For stability of causal system, poles of Transfer function should be inside the unit circle in Z-plane.

Z-transform for Anti-causal System

Anti-causal system can be defined as h(n) = 0,n ≥ 0. For Anti causal system, poles of transfer function should lie outside unit circle in Z-plane. For anti-causal system, ROC will be inside the circle in Z-plane.

The document Existence of 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 Existence of 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 technique used in electrical engineering to analyze discrete-time signals and systems. It allows us to convert a discrete-time signal from the time domain into the frequency domain.
2. How is the Z-transform related to the Laplace transform?
Ans. The Z-transform is a discrete-time counterpart of the Laplace transform, which is used for continuous-time signals and systems. While the Laplace transform operates on continuous-time signals, the Z-transform operates on discrete-time signals.
3. What are the advantages of using the Z-transform in electrical engineering?
Ans. The Z-transform offers several advantages in electrical engineering. It provides a convenient way to analyze and design discrete-time systems, helps in the stability analysis of digital filters, allows for the calculation of frequency response, and enables the representation of discrete-time signals and systems in a compact mathematical form.
4. How is the Z-transform used in digital filter design?
Ans. The Z-transform is extensively used in digital filter design. It allows engineers to analyze the frequency response of the filter, determine stability conditions, design filters with desired characteristics, and analyze the behavior of the filter under different input signals.
5. Can the Z-transform be used for real-time signal processing?
Ans. Yes, the Z-transform can be used for real-time signal processing in electrical engineering. By applying the Z-transform to a discrete-time signal, it is possible to analyze and process the signal in the frequency domain in real-time. This allows for the implementation of various signal processing techniques, such as filtering, modulation, and noise reduction, in real-time systems.
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