Introduction to Z-Transform along with exercise Tasks

1. CS-419 Digital Signal Processing
Lab#5
CISE Department, NEDUET Karachi

2. Lab#05
Computation of Z – transform

3. Z – TRANSFORM
 There are numerous systems that are described by difference
equations rather than differential equations.
 In continuous systems, inputs and outputs are related by
differential equations and Laplace transform techniques are
used to solve those differential equations.
 In sampled systems, inputs and
 outputs are related by difference equations and Z-transform
techniques are used to solve those differential equations.
 Z-transform plays the role in sampled systems and represents
systems with
 transfer functions while Laplace transform plays the role in
continuous systems and represent systems with transfer
functions.
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4. Z – TRANSFORM
Z-transform is discrete in nature or we can say
that Z-transform operates in discrete time
domain.
With the help of Z-transform we can check the
stability of the systems such as filters, speech
processing systems, etc.
The Z-transform, like many integral
transforms, can be defined as either a one-
sided or two-sided transform.
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5. Z – TRANSFORM
 The bilateral or two-sided Z-transform of a discrete-
time signal x[n] is the function X(z) defined as:
𝑿 𝒛 = 𝒁 𝒙 𝒏 = 𝒏=−∞
∞
𝒙 𝒏 𝒛−𝒏
(5.1)
where, n is an integer and z is, in general, a complex number
𝒛 = 𝑨𝒆𝒋𝝋 = 𝑨 𝒄𝒐𝒔𝝋 + 𝒋𝒔𝒊𝒏𝝋 (5.2)
Where, A is the magnitude of z, and φ is the complex argument (also referred to
as angle or phase) in radians.
 Alternatively, in cases where x[n] is defined only for n
≥ 0, the single-sided or unilateral Z transform is
defined as:
𝑿 𝒛 = 𝒁 𝒙 𝒏 = 𝒏=𝟎
∞
𝒙 𝒏 𝒛−𝒏
(5.3)
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6. STABILITY CRITERIA FOR Z – TRANSFORM
 Once the poles and zeros have been found for a given
Z-Transform, they can be plotted onto the ZPlane.
 The Z-plane is a complex plane with an imaginary and
real axis referring to the complex-valued variable z.
 The position on the complex plane is given by r℮^(jθ)
and the angle from the positive, real axis around the
plane which is denoted by θ.
 When mapping poles and zeros onto the plane, poles
are denoted by an "x" and zeros by an "o".
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7. STABILITY CRITERIA FOR Z – TRANSFORM 7

8. Frequency response and stability of the system
 If the poles lie inside the unit circle the system is said to be
stable.
 If the poles lies on the unit circle the system is said to be
marginally stable.
 If the poles lies outside the unit circle the system will be
unstable.
 The placement of poles and zeros in the unit circle
provides us the frequency response as well as the
stability of the system.
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9. Z – Transform using MATLAB
 Following are the MATLAB commands which are related to the Z
transform.
» freqz used for calculating / displaying frequency response
» impz used for calculating / displaying impulse response
» zplane plots the zeros and poles with unit circle
» tf2zp finds zeros, poles and gain from H = B/A
» zp2tf Transforms from zero, poles , gain back to t = B/A
» residuez Finds residues, poles, direct terms of partial
fraction
» poly convert roots to polynomial
» roots computes roots of a polynomial
» conv used for multiplying 2 polynomials A & B.
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10. ZPLANE command
ZPLANE (Z-plane zero-pole plot):
ZPLANE (Z,P) plots the zeros Z and poles P (in column
vectors)with the unit circle for reference. Each zero is
represented with an 'o' and each pole with an 'x' on the
plot. Multiple zeros and poles are indicated by the
multiplicity number shown to the upper right of the zero
or pole.
 Example:
z = [1 2]; % coefficient of zeros
p = [1 0.9]; % coefficient of poles
zplane (z, p);
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11. Exercise
 Task # 1: Find out the zeros and poles of the following transfer
functions by using ‘roots’ command and also plot them by
using ‘pzmap’ command.
1) 𝐻 𝑧 =
𝑧+1
𝑧2
−0.9𝑧+0.81
2) 𝐻 𝑧 =
2𝑧2
+5𝑧+ 12
𝑧2
+2𝑧+10
 Task # 2: Use the ‘freqz’ function to evaluate the frequency
response of a Z–Transform shown of task 1. Where -20 ≤ ω ≤ 20
is the frequency vector in radians/seconds. (Use “linespace”
function to generate a vector with 200 samples).
Hint: H = freqz (b, a, w)
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12. Exercise
Task # 3: For above task, also plot the magnitude
response (in dB) and phase response (in degree) by
using the subplot command.
Task # 4: Write a MATLAB program to find poles
and zeros of the system given by:
y(n) = x(n) + 2x(n-1) - 0.9y(n-1)
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