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) = ancosωn 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 ancosωnancosωn 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 −
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.
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1. What is the Z-transform in electrical engineering? |
2. What are the properties of the Z-transform? |
3. How is the Z-transform related to the Laplace transform? |
4. What are the advantages of using the Z-transform in electrical engineering? |
5. How is the inverse Z-transform calculated in electrical engineering? |
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