Z Transform part 1.pdf
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Transcript of Z Transform part 1.pdf
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The z-transform
Part 1
Dr. Ali Hussein Muqaibel
http://faculty.kfupm.edu.sa/ee/muqaibel/
Dr. Ali Muqaibel
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The material to be covered in this lecture is
as follows:
Dr. Ali Muqaibel
Introduction to the z-transform
Definition of the z-transform
Derivation of the z-transform
Region of convergence for the transform
Examples.
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After finishing this lecture you should be
able to:
Dr. Ali Muqaibel
Find the z-transform for a given signal utilizing the z-
transform definition
Calculate the region of convergence for the transform
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Derivation of the z-Transform
Dr. Ali Muqaibel
The z-transform is the basic tool for the analysis and synthesisof discrete-time systems.
The z-transform is defined as follows:
The coefficient
denote the sample value and
denotes that the sample occurs sample periods afterthe 0 reference. Note that the lower limit of the summation can start from
zero if the signal is causal (Unilateral z-transform)
Rather than starting form the given definition for the z-transform, we may start from the continuous-time functionand derive the z-transform. This is done in the next slide.
nn
z x nT zX
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Derivation of the z-transform
Dr. Ali Muqaibel
The sampled signal may be written as
Since
0for all t except
at , can be replaced by. And Assuming 0 for 0. Then,
Taking Laplace transform yields Rearranging
By sifting property of the delta
function
sn
x t x t t nT
0
s s
n
sx t x tnT nT
0
0
t
n
ss
X s x nt t n eT dt
0
0
st
s
n
X s x nT t nT e dt
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Continue Derivation
Dr. Ali Muqaibel
Defining the complex variable z as the Laplace time-shift
operator becomes, We could have started from the last expression but it is good
to relate to the s-domain
In the -domain the left-half plane corresponds to 0 ismapped to || 1 in the z-plane which is the region insidethe unit circle.
j
1 Re[z]
Im[z]
S-planez-plane
Left-half
Unit
circle
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Region of Convergence (ROC) is converged for 0 (left-half of s-plane). This corresponds
to 1. This is the region inside the unit circle. is NOT converged for 0 (right-half of s-plane). This
corresponds to 1 which is the region outside the unit circle
The mapping of the Laplace variable into the z-plane through is illustrated in the figure.
j
1Re[z]
Im[z]
S-planez-plane
Left-half
Unit
circle
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The Z-Transform in Summary
Dr. Ali Muqaibel
The coefficient denotes the sampled value The square bracket is used to indicate discrete times.
denotes that the sample occurs
sample periods after
the 0 reference. is simply the -second time shift The parameter is simply shorthand notation for the Laplace
time shift operator
For instance, 30 denotes a sample, having value 30, whichoccurs 40 sample periods after the t=0 reference
Matlab has special tools for z-transform: ztrans, iztrans , pretty
Where
and
0
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Example 1:
Dr. Ali Muqaibel
Determine the z-transform for the
following signal
Solution:
We know that
hence
1
2
,
,
1
0
1
2
1
0,
1
,
,
n
n
x n n
n
otherwise
X zn
nx nz
1 0 1 2
1
1
2
2
1 0X 1 2
2
z n x x xx n z z z z z
X z z z z
x
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Example 2: Sampled Step Function
(Important Functions)
Dr. Ali Muqaibel
Consider a unit step sample sequence
defined by
Find the z-transform.
Solution
The sum converges absolutely to
outside the unit circle
t
u(t)
T 2 T 3 T 4 T 5 T
1
0
1 2 3 10
1( ) 1 ........... , 1
1
n
n
U z X z z z z z zz
0
n
n
X z z
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Sampled Dirac Delta Function (an other
important function)
Dr. Ali Muqaibel
The Dirac Delta Function is defined to be
For a delayed version of delta is defined as
Applying the definition of the z-transform
1 0[ ]
0 0
nn
n
1[ ]0
n kn kn k
Dirac
function( )t
0( ) ( ) (0) 1s T
k
X z t z
( ) 1X z
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The Unit Exponential Sequence
Dr. Ali Muqaibel
The unit exponential sequence is defined to be
Apply z-transform definition
,we get
, 0[ ]0 0
k
e kx kk
k
1
1
1[ ] 1
zX k e z z e
1
0 0
1
( ) ( )
1
( ) 1
k k k
k k
X z e z e z
z
X z e z z e
| |where z e
if k e then 11
( )1
zX z
kz z k
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Example 3 with Poles and Zeros
Dr. Ali Muqaibel
Determine thez-transform of the signal
0.5 Depict the ROC and the locations of poles and zeros of X(z) in the z-plane
Solution:
Substituting is the definition of the z-transform
0.5 0.5 .
This is a geometric series of infinite length in the ratio 0.5/z; the sumconverges, provided that . 1 or 0.5. Hence the z-transform is
Pole at , zero at . , ROC is the light blue region
10
0.5 1, 0.5
1 0.5
, 0.50.5
n
n
X z zz z
z zz
Im[z]
0.50
0 1 2
0 1 2
2
1 1 1...
2 2 2
1 11 ...
2 4
X z z z z
X zz z
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Self Test 1:
Dr. Ali Muqaibel
Find the z- transform of the following signal:
Hint :
Answer:
2 2
1
( ) 1X z z a
a z
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Self Test 2:
Dr. Ali Muqaibel
Determine the z-transform of the signal
1 0.5 Depict the ROC and the locations of poles and zeros of in the z-
plane Answer:
the sum converges, provided that 0.5 and 1.
1
0
0 0
0.5
0.51
n
n
n n
n
k
n k
X z zz
zz
1
1 11 , 0.5 1
1 0.5 1
2 1.5, 0.5 1
0.5 1
X z zz z
z zz
z z
Poles at z=0.5, 1, zeros at z=0, 0.75.ROC is the region in between
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Dr. Ali Muqaibel
1
1 11 , 0.5 1
1 0.5 1
2 1.5 , 0.5 10.5 1
X z zz z
z zz
z z
Poles at z=0.5, 1, zeros at z=0, 0.75. ROC is the region in between
Continue Self-test