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

In this chapter, we will understand the basic properties of Z-transforms.

Linearity

It states that when two or more individual discrete signals are multiplied by constants, their respective Z-transforms will also be multiplied by the same constants.

Mathematically,

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

Proof − We know that,

Properties of Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE)
Properties of Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE)      (Hence Proved)

Here, the ROC is Properties of Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE)

Time Shifting

Time shifting property depicts how the change in the time domain in the discrete signal will affect the Z-domain, which can be written as;

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

Proof

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

Let s = p-k

Properties of Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE)     (Hence Proved)

Here, ROC can be written as Z = 0 (p>0) or Z = ∞(p<0)

Example

U(n) and U(n-1) can be plotted as follows

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

Z-transformation of U(n) cab be written as;

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

Z-transformation of U(n-1) can be written as;

Properties of Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE)
So here Properties of Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE)    (Hence Proved)

Time Scaling

Time Scaling property tells us, what will be the Z-domain of the signal when the time is scaled in its discrete form, which can be written as;

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

Proof

Properties of Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE)      (Hence proved)

ROC: = Mod(ar1) < Mod(Z) < Mod(ar2) where Mod = Modulus

Example

Let us determine the Z-transformation of x(n) = ancos⁡ωn using Time scaling property.

Solution

We already know that the Z-transformation of the signal cos(ωn)cos⁡(ωn) is given by −

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

Now, applying Time scaling property, the Z-transformation of ancosωnancos⁡ωn can be written as;

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

Successive Differentiation

Successive Differentiation property shows that Z-transform will take place when we differentiate the discrete signal in time domain, with respect to time. This is shown as below.

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

Proof

Consider the LHS of the equation −   Properties of Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE)

Properties of Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE)     (Hence Proved)

ROC: R1< Mod (Z) <R2

Example

Let us find the Z-transform of a signal given by Properties of Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE)

By property we can write

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

Now, Z[n.y] can be found out by again applying the property,

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

Convolution

This depicts the change in Z-domain of the system when a convolution takes place in the discrete signal form, which can be written as −

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

Proof

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

Let n-k = l, then the above equation cab be written as −

Properties of Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE)   (Hence Proved)

 

ROC : ROC ⋂ ROC2

Example

Let us find the convolution given by two signals

x1(n) = {3,−2,2}     ...(eq. 1)

x2(n) = {2,0≤4 and 0 elsewhere}    ...(eq. 2)

Z-transformation of the first equation can be written as;

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

Z-transformation of the second signal can be written as;

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

So, the convolution of the above two signals is given by −

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

Taking the inverse Z-transformation we get,

x(n) = {6,2,6,6,6,0,4}

Initial Value Theorem

If x(n) is a causal sequence, which has its Z-transformation as X(z), then the initial value theorem can be written as;

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

Proof − We know that,

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

Expanding the above series, we get;

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

In the above case if Z → ∞ then Z−n→0 (Because n>0)

Therefore, we can say;

Properties of Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE)   (Hence Proved)

Final Value Theorem

Final Value Theorem states that if the Z-transform of a signal is represented as X(Z) and the poles are all inside the circle, then its final value is denoted as x(n) or X(∞) and can be written as −

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

Conditions

  • It is applicable only for causal systems.
  • X(Z)(1−Z−1) should have poles inside the unit circle in Z-plane.

Proof − We know that

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

Here, we can apply advanced property of one-sided Z-Transformation. So, the above equation can be re-written as;

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

Now putting z = 1 in the above equation, we can expand the above equation −

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

This can be formulated as;

Properties of Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE)        (Hence Proved)

Example

Let us find the Initial and Final value of x(n) whose signal is given by

X(Z) = 2 + 3Z−1 + 4Z−2

Solution − Let us first, find the initial value of the signal by applying the theorem

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

Now let us find the Final value of signal applying the theorem

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

Some other properties of Z-transform are listed below

Differentiation in Frequency

It gives the change in Z-domain of the signal, when its discrete signal is differentiated with respect to time.

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

Its ROC can be written as;

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

Example

Let us find the value of x(n) through Differentiation in frequency, whose discrete signal in Z-domain is given by  Properties of Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE)

By property, we can write that

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

Multiplication in Time

It gives the change in Z-domain of the signal when multiplication takes place at discrete signal level.

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

Conjugation in Time

This depicts the representation of conjugated discrete signal in Z-domain.

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

The document Properties 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 Properties 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 transform used in electrical engineering to convert discrete-time signals into complex frequency-domain representations. It is particularly useful in analyzing and designing digital filters.
2. What are the properties of the Z-transform?
Ans. The properties of the Z-transform in electrical engineering include linearity, time shifting, scaling, time reversal, convolution, and initial value theorem. These properties allow for easy manipulation and analysis of discrete-time signals.
3. How is the Z-transform related to the Laplace transform?
Ans. The Z-transform is closely related to the Laplace transform in electrical engineering. The Z-transform is a discrete-time equivalent of the continuous-time Laplace transform. By substituting the complex variable s in the Laplace transform with z, the Z-transform can be obtained.
4. What are the advantages of using the Z-transform in electrical engineering?
Ans. The Z-transform offers several advantages in electrical engineering. It allows for the analysis of discrete-time systems, such as digital filters, in the frequency domain. It provides a mathematical framework for studying stability, causality, and other system properties. Additionally, it enables the design of digital systems using transfer functions.
5. How is the inverse Z-transform calculated in electrical engineering?
Ans. The inverse Z-transform is used to convert a Z-transformed signal back into the time domain. The inverse Z-transform can be calculated using various methods, such as partial fraction expansion, power series expansion, or residue theorem. The choice of method depends on the complexity of the Z-transformed signal and the desired accuracy of the inverse transformation.
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