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,

Proof − We know that,

      (Hence Proved)

Here, the ROC is 

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;

Or 

Proof −

Let s = p-k

     (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

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

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

So here     (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;

Proof −

      (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 −

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

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.

Proof −

Consider the LHS of the equation −   

     (Hence Proved)

ROC: R1< Mod (Z) <R2

Example

Let us find the Z-transform of a signal given by

By property we can write

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

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 −

Proof −

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

   (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;

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

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

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;

Proof − We know that,

Expanding the above series, we get;

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

Therefore, we can say;

   (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 −

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

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

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

This can be formulated as;

        (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

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

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.

Its ROC can be written as;

Example

Let us find the value of x(n) through Differentiation in frequency, whose discrete signal in Z-domain is given by  

By property, we can write that

Multiplication in Time

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

Conjugation in Time

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