digital signal processing_ lecuter-3_ z transform
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Digital Signal Processing Lecture-2 Spring 2010Z Transform Chap.-3
Hassan Bhatti
Hassan Bhatti, DSP, Spring 2011
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z-Transform
The z-transform is the most general concept for the transformation of discrete-time series.
The Laplace transform is the more general concept for the transformation of continuous time processes.
For example, the Laplace transform allows you to transform a differential equation, and its corresponding initial and boundary value problems, into a space in which the equation can be solved by ordinary algebra.
The switching of spaces to transform calculus problems into algebraic operations on transforms is called operational calculus. The Laplace and z transforms are the most important methods for this purpose.
Hassan Bhatti, DSP, Spring 2010
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The Transforms
The Laplace transform of a function f(t):
0
)()( dtetfsF st
The one-sided z-transform of a function x(n):
0
)()(n
nznxzX
The two-sided z-transform of a function x(n):
n
nznxzX )()(
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Relationship to Fourier Transform
Note that expressing the complex variable z in polar form reveals the relationship to the Fourier transform:
n
nii
n
nini
n
n
ii
enxXeX
rifandernxreX
orrenxreX
)()()(
,1,)()(
,))(()(
which is the Fourier transform of x(n).
Hassan Bhatti, DSP, Spring 2010
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Relation ship of Transformations
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Region of Convergence
The z-transform of x(n) can be viewed as the Fourier transform of x(n) multiplied by an exponential sequence r-n, and the z-transform may converge even when the Fourier transform does not.
By redefining convergence, it is possible that the Fourier transform may converge when the z-transform does not.
For the Fourier transform to converge, the sequence must have finite energy, or:
n
nrnx )(
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Convergence, continued
n
nznxzX )()(
The power series for the z-transform is called a Laurent series:
The Laurent series, and therefore the z-transform, represents an analytic function at every point inside the region of convergence, and therefore the z-transform and all its derivatives must be continuous functions of z inside the region of convergence.
In general, the Laurent series will converge in an annular region of the z-plane.
Hassan Bhatti, DSP, Spring 2010
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Some Special Functions
First we introduce the Dirac delta function (or unit sample function):
0,1
0,0)(
n
nn
This allows an arbitrary sequence x(n) or continuous-time function f(t) to be expressed as:
dttxxftf
knkxnxk
)()()(
)()()(
or
0,1
0,0)(
t
tt
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Convolution, Unit Step
These are referred to as discrete-time or continuous-time convolution, and are denoted by:
)(*)()(
)(*)()(
ttftf
nnxnx
We also introduce the unit step function:
0,0
0,1)(or
0,0
0,1)(
t
ttu
n
nnu
Note also:
k
knu )()(
Hassan Bhatti, DSP, Spring 2010
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Poles and Zeros
When X(z) is a rational function, i.e., a ration of polynomials in z, then:
1. The roots of the numerator polynomial are referred to as the zeros of X(z), and
2. The roots of the denominator polynomial are referred to as the poles of X(z).
Note that no poles of X(z) can occur within the region of convergence since the z-transform does not converge at a pole.
Furthermore, the region of convergence is bounded by poles.
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Example
)()( nuanx n
The z-transform is given by:
0
1)()()(n
n
n
nn azznuazX
Which converges to:
azforazz
azzX
11
1)(
Clearly, X(z) has a zero at z = 0 and a pole at z = a.
a
Region of convergence
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Convergence of Finite Sequences
Suppose that only a finite number of sequence values are nonzero, so that:
2
1
)()(n
nn
nznxzX
Where n1 and n2 are finite integers. Convergence requires
.)( 21 nnnfornx
So that finite-length sequences have a region of convergence that is at least 0 < |z| < , and may include either z = 0 or z = .
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Z-Transform of sin/cos
ikeu(k) 1-i ze-11
U(z)
2ee
)cos(ku(k)ikik
Time Domain Z-Transform
2iee
)sin(ku(k)ikik
-ikeu(k)1-i- ze-1
1U(z)
21
1
2121
1
1111
1i1-i
zz2cos1zcos1
)z(sin)zcos(1zcos1
)/2zisinzcos1
1zisinzcos1
1(
)/2ze1
1ze-1
1(U(z)
21-
-1
zz2cos-1zsin
U(z)
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Exponentially Modulated sin/cos
2)(ae)(ae
)cos(ka(k)ukiki
kexpcos
2i)(ae)(ae
)sin(ka(k)ukiki
kexpsin
221-
-1
zazcos2a-1zsina
U(z)
0 2 4 6 8 10 12 14 16 18-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
u(k)=c os(k*pi/6)*0.9k
221-
-1
zazcos2a-1zsina
U(z)
A damped oscillating signal – a typical output of a second order system
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Inverse z-Transform
The inverse z-transform can be derived by using Cauchy’s integral theorem. Start with the z-transform
n
nznxzX )()(
Multiply both sides by zk-1 and integrate with a contour integral for which the contour of integration encloses the origin and lies entirely within the region of convergence of X(z):
transform.-z inverse the is)()(21
21
)(
)(21
)(21
1
1
11
nxdzzzXi
dzzi
nx
dzznxi
dzzzXi
C
k
n C
kn
C n
kn
C
k
Hassan Bhatti, DSP, Spring 2010
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Z-Transform/Inverse Z-Transform
LTI: yimpuse(k)=0.3k-1u (k)=0.7k y (k)?
0 5 10 150
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
*(convolution)
10.7z11
Z
=
·(multiplication)
= )0.7z)(10.3z(1z
11
1
0 5 10 150
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Z-1
1
-1
0.3z1z
Z
Transfer Function
0 2 4 6 8 10 12 14 16 180
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
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Inverse Z-Transform
Table Lookup – if the Z-Transform looks familiar, look it up in the Z-Transform table!
Long Division Partial Fraction Expansion
u(k) U(z)ZZ-1?
(k)2u(k)3uu(k) rampstep
(z)2U(z)3U
)z(12z
z13
U(z)
rampstep
21
1
1
Z-1?
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Long Division
Sort both nominator and denominator with descending order of z first
21
1
z2z1z3
U(z)
u(0)=3, u(1)=5, u(2)=7, u(3)=9, …, guess: u(k)=3ustep(k)+2uramp(k)
Hassan Bhatti, DSP, Spring 2010
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Partial Fraction Expansion
Many Z-transforms of interest can be expressed as division of polynomials of z
m
0j
jj
n
0i
ii
zb
zaU(z)
May be trickier:complex rootduplicate root
)p)...(zp)(zp(zb
zb...zbzbb
m21m
mm
2210
m
1j j
j0 pz
ccU(z)
,1kjdexpp p(k)u
j
m
1jdexppimpulse0 (k)u(k)ucu(k)
j
where k>0
1j
1
zp11
z
Hassan Bhatti, DSP, Spring 2010
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Real Unique Poles
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Repeat Real Poles(1)
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Repeat Real Poles(2)
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Complex Poles(1)
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Complex Poles(2)
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An Example
86zz1414z3z
U(z) 2
2
4z
c2z
ccU(z) 21
0
0k,4c2c
0k,cu(k) 1k
21k
1
0
(z-2)(z-4)
U1(z)=c0 Z-1 u1(k)=c0*uimpulse(k)
Z-1
Z-1 u2(k)=c1*2k-1, k>02z
c(z)U 1
2
4zc
(z)U 23
u2(k)=c2*4k-1, k>0
c0? c1? c2?
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Get The Constants!
86zz1414z3z
U(z) 2
2
4z
c2z
ccU(z) 21
0
(z-2)(z-4)
,4z
c2z
ccU(z) 21
0
,z ,cU(z) 0 3
86zz1414z3z
limc 2
2
z0
,c2z4)(zc
4)c(z4)U(z)-(zK(z) 21
0
3|2z
1414z3zcK(4) 4z
2
2
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Partial Fraction Expansion – cont’d
U(z)limcz0
m
0j
jj
n
0i
ii
zb
zaU(z)
m
1j j
j0 pz
ccU(z)
How to get c0 and cj’s ?
)U(z)p-(z(z)U jp j
)(pUc jpj j
define
Hassan Bhatti, DSP, Spring 2010
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An Example – Complete Solution
386zz1414z3z
limU(z)limc 2
2
zz0
4-z1414z3z
86zz1414z3z
2)(z(z)U
2
2
2
2
2-z1414z3z
86zz1414z3z
4)(z(z)U
2
2
2
4
86zz1414z3z
U(z) 2
2
4z
c2z
ccU(z) 21
0
14-2
1421423(2)Uc
2
21
32-4
1441443(4)Uc
2
42
4z3
2z1
3U(z)
0k,432
0k3,u(k) 1k1k
Hassan Bhatti, DSP, Spring 2010
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Solving Difference Equations
m)u(kb...1)u(kbn)y(ka...1)y(kay(k) m1n1
U(z)zb...U(z)zbY(z)za...Y(z)zaY(z) mm
11
nn
11
U(z)za...za1
zb...zbY(z) n
n1
1
mm
11
...y(k)
Z
Z-1Transfer Function
Hassan Bhatti, DSP, Spring 2010
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A Difference Equation Example
Exponentially Weighted Moving Average
y(k)=cy(k-1)+(1-c)u(k-1)
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Solve it!
1
1
1
1
1-1-
-1
1
1
0.8z11.2z
0.4z10.6z
0.8z1.2
0.4z0.6
0.8)0.4)(z(z0.6z
)0.8z-)(10.4z-(10.6z
U(z)0.4z1
0.6zY(z)
LTI: y(k)=0.4y(k-1)+0.6u(k-1)u (k)=0.8k y (k)?
U(z)0.6zY(z)0.4zY(z) 11 Z
10.8z11
U(z) Z
1-k1k 0.81.20.4-0.6y(k)
Z-1-1 0 1 2 3 4 5 6 7 8 9
0
0.2
0.4
0.6
0.8
1
-1 0 1 2 3 4 5 6 7 8 90
0.2
0.4
0.6
0.8
1
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Properties
z-transforms are linear:
The transform of a shifted sequence:
Multiplication:
But multiplication will affect the region of convergence and all the pole-zero locations will be scaled by a factor of a.
)()()()( zbYzaXnbynax Z
)()( 00 zXznnx nZ
)()( 1zaZnxan Z
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Convolution of Sequences
both. of econvergenc of regions the inside of values for)()()(
)()()(
let
)()(
)()()(
Then
)()()(
zzYzXzW
zzmykxzW
knm
zknykx
zknykxzW
knykxnw
k
k m
m
n
k n
n
n k
k
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More Definitions
Definition. Periodic. A sequence x(n) is periodic with period if and only if x(n) = x(n + ) for all n.
Definition. Shift invariant or time-invariant. Consider a sequence y(n) as the result of a transformation T of x(n). Another interpretation is that T is a system that responds to an input or stimulus x(n): y(n) = T[x(n)].
The transformation T is said to be shift-invariant or time-invariant if:
y(n) = T [x(n)] implies that y(n - k) = T [x(n – k)]
For all k. “Shift invariant” is the same thing as “time invariant” when n is time (t).
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k
k
k
kk
k
nhnxknhkxny
T
nhkxny
knTkxknkxTny
kn
knnh
).(*)()()()(
then , transform the of invariance time have weIf
).()()(
)()()()()(
:Then . at occurring
shock or spike"" a ),( to system the of response the be )( Let
This implies that the system can be completely characterized by its impulse response h(n). This obviously hinges on the stationarity of the series.
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Definition. Stable System. A system is stable if
k
kh )(
Which means that a bounded input will not yield an unbounded output.
Definition. Causal System. A causal system is one in which changes in output do not precede changes in input. In other words,
. for)()( then
for)()( If
021
021
nnnxTnxT
nnnxnx
Linear, shift-invariant systems are causal iff h(n) = 0 for n < 0.
Hassan Bhatti, DSP, Spring 2010
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.)()(
that so )()(Let
)()()(
Then. for )( let
is, That .sinusoidal be)( let)()()( Given
)(
nii
k
kii
k
kini
k
kni
ni
k
k
eeHny
ekheH
ekheekhny
nenx
nxnhkxny
Here H(ei) is called the frequency response of the system whose impulse response is h(n). Note that H(ei) is the Fourier transform of h(n).
Hassan Bhatti, DSP, Spring 2010
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We can generalize this state that:
. of function continuous a touniformly converges
and convergent absolutely is transform the then,)(
)(21
)(
)()(
n
nii
n
nii
nxIf
deeXnx
enxeX
This implies that the frequency response of a stable system always converges, and the Fourier transform exists.
These are the Fourier transform pair.
Hassan Bhatti, DSP, Spring 2010
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deekTxT
tx
ekTxeX
deeTXtx
deiXtx
TTiX
TeX
tiT
T k
Tkicc
k
Tkic
Ti
tiTiT
T
c
tiT
T
cc
cTi
)(2
)(
have we,)()( Since
.)(21
)(
:Combining
.)(21
)(
:transform Fourier time continuous the From
),(1
)(
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Z Transformation Pairs
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Z Transformation Pairs
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Properties of Z Transform, Linearity
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Time Shifting
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Multiplication by Exponential Sequence
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Time Reversal
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Convolution of Sequences
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Properties of Z Transformation
Hassan Bhatti, DSP, Spring 2010