2.
1 The Z Transform
2.1.1 DEFINITION OF THE Z-TRANSFORM
The z-transform of a discrete-time signal x(n) is defined by
where z = rejω is a complex variable. The values of z for which the sum converges define
a region in the z-plane referred to as the region of convergence (ROC). Notationally, if
x(n) has a z-transform X(z), we write
CONVERGENCE CONDITION:
Furthermore, the Region of convergence ROC is determined by the range of values of r
for which the
Condition is satisfied.
Because the z-transform is a function of a complex variable, it is convenient to describe it
using the complex z-plane. With
the axes of the z-plane are the real and imaginary parts of z as illustrated in Fig. below,
and the contour corresponding to 1zl = 1 is a circle of unit radius referred to as the unit
circle. The z-transform evaluated on the unit circle corresponds to the DTFT,
More specifically, evaluating X(z) at points around the unit circle, beginning at z = 1(ω =
0), through z = j (ω = ∏/2), to z = - 1(ω=-∏), we obtain the values of X(ejω) for 0≤ω≤∏.
Fig. The unit circle in the complex z-plane
2.1.2 PROPERTIES OF ROC
1. ROC of X(Z) consists of a ring in the z-plane centered about the origin
2. The ROC cannot contain any poles.
3. If x[n] is finite duration, then the ROC is the entire z-plane except possibly at z=0
and/or z=∞. The point z = ∞ will be included if x(n) = 0 for n < 0, and the point z
= 0 will be included if x(n) = 0 for n > 0.
4. If x[n] is left sided, then the ROC is in the region in the z-plane inside the
innermost pole. (i.e inside the circle of radius equal to the smallest magnitude of
the poles of X(Z) other than any at z=0). Particularly if x[n] is anticausal (ie.left
sided and equal to zero for n>0)then the ROC also includes z=0.
5. If x[n] is right sided, then the ROC is in the region in the z-plane outside the
outermost pole. (i.e outside the circle of radius equal to the largest magnitude of
the poles of X(Z)). Particularly if x[n] is causal (ie.right sided and equal to zero for
n<0)then the ROC also includes z=∞.
6. If x[n] is an infinite duration, two sided sequence the ROC will consists of a ring
in the z-plane, bounded on the interior and exterior by a pole, not contain any
poles.
Problem 1. Find the z-transform of the sequence x(n) = αnu(n) and find ROC.
Problem 2.
Problem 3.
2.1.3 IMPORTANT PROPERTIES OF THE z-TRANSFORM
2. Time shifting
3. Scaling in z-domain (or) Frequency shifting
4. Time reversal
5. Differentiation in z-domain
Proof: By differentiating the z-transform definition on both sides
7. Convolution
8. Multiplication
Problems using z-transform properties
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
2.1.4 THE INVERSE Z-TRANSFORM
The inverse z-transform is formally given by,
Where the integral is a contour integral over a closed path C that encloses the origin and
lies within the region of convergence of X(z). For simplicity, C can be taken as a circle in
the ROC of X(z) in the z-plane. There are two methods that are often used for the
evaluation of the inverse z-transform. Two possible approaches are described below.
1. Partial Fraction Expansion
2. Power Series (or) Long division method
1. Partial Fraction Expansion
Problem 12
Problem 13
Problem 14
2. Power series expansion:
Problem 15
Problem 16
2.1.5 THE ONE-SIDED Z-TRANSFORM
2.1.6 Analysis of linear time invariant system using z-transform
1. System function
Consider an LTI system with input x(n) and output y(n). The system output
can be obtained by
y(n)=x(n)*h(n)
Taking z-transform on both the sides,
Y(z)=X(z)H(z)
The system function or transfer function is defined by,
H(z)=Y(z)/X(z)
Problem 17
2. Response of system with rational system function
Natural Response(zero-input response)
The natural response is the system output when the input is zero.
Note:In above table use ω instead of Ω
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
