Copyright ? 2001,S,K,Mitra 1
z-Transform
? The DTFT provides a frequency-domain
representation of discrete-time signals and LTI
discrete-time systems
? Because of the convergence condition,in many
cases,the DTFT of a sequence may not exist
? As a result,it is not possible to make use of such
frequency-domain characterization in these cases
Copyright ? 2001,S,K,Mitra 2
z-Transform
? A generalization of the DTFT defined by
leads to the z-transform
? z-transform may exist for many sequences
for which the DTFT does not exist
? Moreover,use of z-transform techniques
permits simple algebraic manipulations
??
?
???
???
n
njj enxeX ][)(
Copyright ? 2001,S,K,Mitra 3
z-Transform
? Consequently,z-transform has become an
important tool in the analysis and design of
digital filters
? For a given sequence g[n],its z-transform
G(z) is defined as
where is a complex
variable
?
?
???
??
n
nzngzG ][)(
R e ( ) I m ( )z z j z? ? ? ￡
Copyright ? 2001,S,K,Mitra 4
z-Transform
? If we let,then the z-transform
reduces to
? The above can be interpreted as the DTFT
of the modified sequence
? For r = 1 (i.e.,|z| = 1),z-transform reduces
to its DTFT,provided the latter exists
?? jerz
??
?
???
????
n
njnj erngerG ][)(
}][{ nrng ?
Copyright ? 2001,S,K,Mitra 5
z-Transform
? The contour |z| = 1 is a circle in the z-plane
of unity radius and is called the unit circle
? Like the DTFT,there are conditions on the
convergence of the infinite series
? For a given sequence,the set R of values of
z for which its z-transform converges is
called the region of convergence (ROC)
?
?
???
?
n
nzng ][
Copyright ? 2001,S,K,Mitra 6
z-Transform
? From our earlier discussion on the uniform
convergence of the DTFT,it follows that the
series
converges if is absolutely
summable,i.e.,if
??
?
???
????
n
njnj erngerG ][)(
}][{ nrng ?
???
?
???
?
n
nrng ][
Copyright ? 2001,S,K,Mitra 7
z-Transform
? In general,the ROC of a z-transform of a
sequence g[n] is an annular region of the z-
plane,
where
?? ?? gg RzR
???? ?? gg RR0
ROC
gR? gR?
Copyright ? 2001,S,K,Mitra 8
? The z-transform is a form of the Cauchy-
Laurent series and is an analytic function at
every point in the ROC
? Let f (z) denote an analytic (or holomorphic)
function over an annular region centered
at
Cauchy-Laurent Series
?
oz
Re( )z
Im( )z ?
oz
Copyright ? 2001,S,K,Mitra 9
? Then f (z) can be expressed as the bilateral series
being a closed and counterclockwise integration
contour contained in
Cauchy-Laurent Series
( 1 )
( ) ( )
1
( ) ( )
2
n
no
n
n
no
f z z z
f z z z dz
j
?
?
?
?
?
? ??
??
??
??
?
??where
? ?
Copyright ? 2001,S,K,Mitra 10
z-Transform
? Example - Determine the z-transform X(z)
of the causal sequence and its
ROC
? Now
? The above power series converges to
? ROC is the annular region |z| > |?|
][][ nnx n ???
? ??? ???
?
?
??
???
?
0
][)(
n
nn
n
nn zznzX
1f o r,
1
1)( 1
1 ?????
?
? zzzX
Copyright ? 2001,S,K,Mitra 11
z-Transform
? Example - The z-transform ?(z) of the unit
step sequence ?[n] can be obtained from
by setting ? = 1,
? ROC is the annular region
1f o r,
1
1)( 1
1 ?????
?
? zzzX
??? z1
1f o r
1
1 1
1 ???
?
? zzz,)(?
Copyright ? 2001,S,K,Mitra 12
z-Transform
? Note,The unit step sequence ?[n] is not
absolutely summable,and hence its DTFT
does not converge uniformly
? Example - Consider the anti-causal
sequence
]1[][ ?????? nny n
Copyright ? 2001,S,K,Mitra 13
z-Transform
? Its z-transform is given by
? ROC is the annular region
? ???? ???
?
?
??
???
?
1
1
)(
m
mm
n
nn zzzY
1f o r,
1
1 1
1 ?????
?
? zz
z
zzz
m
mm
1
1
0
1
1 ?
??
?
??
??
???? ????
??z
Copyright ? 2001,S,K,Mitra 14
z-Transform
? Note,The z-transforms of the two
sequences and are
identical even though the two parent
sequences are different
? Only way a unique sequence can be
associated with a z-transform is by
specifying its ROC
]1[ ????? nn][nn??
Copyright ? 2001,S,K,Mitra 15
z-Transform
? The DTFT of a sequence g[n]
converges uniformly if and only if the ROC
of the z-transform G(z) of g[n] includes the
unit circle
? The existence of the DTFT does not always
imply the existence of the z-transform
)( ?jeG
Copyright ? 2001,S,K,Mitra 16
z-Transform
? Example - The finite energy sequence
has a DTFT given by
which converges in the mean-square sense
sin[ ] sin c,c c c
LP
nnh n n
n
? ? ?
? ??
??? ? ? ? ? ? ?
????
?
?
?
?????
????
??
c
cj
LP eH,0
0,1
)(
Copyright ? 2001,S,K,Mitra 17
z-Transform
? However,does not have a z-transform
as it is not absolutely summable for any value
of r,i.e,
? Some commonly used z-transform pairs are
listed on the next slide
][nhLP
[] nLP
n
h n r
?
?
? ??
??? r?
Copyright ? 2001,S,K,Mitra 18
Table 3.8,Commonly Used z-
Transform Pairs
Copyright ? 2001,S,K,Mitra 19
Rational z-Transforms
? In the case of LTI discrete-time systems we
are concerned with in this course,all
pertinent z-transforms are rational functions
of
? That is,they are ratios of two polynomials
in,
1?z
1?z
N
N
N
N
M
M
M
M
zdzdzdd
zpzpzpp
zD
zPzG
???
?
?
???
?
?
????
??????
)(
)(
....
....
)(
)()(
1
1
1
10
1
1
1
10
Copyright ? 2001,S,K,Mitra 20
Rational z-Transforms
? The degree of the numerator polynomial
P(z) is M and the degree of the denominator
polynomial D(z) is N
? An alternate representation of a rational z-
transform is as a ratio of two polynomials in
z,
NN
NN
MM
MM
MN
dzdzdzd
pzpzpzpzzG
????
?????
?
?
?
?
?
1
1
10
1
1
10
....
....
)( )(
Copyright ? 2001,S,K,Mitra 21
Rational z-Transforms
? A rational z-transform can be alternately
written in factored form as
?
?
?
?
?
?
?
?
? N
M
zd
zp
zG
1
1
0
1
1
0
1
1
? ?
? ?
)(
)(
)(
?
?
?
?
?
??
?
?
? N
M
MN
zd
zp
z
10
10
? ?
? ?
)(
)()(
?
?
Copyright ? 2001,S,K,Mitra 22
Rational z-Transforms
? At a root of the numerator polynomial
,and as a result,these values of z
are known as the zeros of G(z)
? At a root of the denominator
polynomial,and as a result,
these values of z are known as the poles of
G(z)
z ?? l
z ?? l
??)( ??G
0?)( ??G
Copyright ? 2001,S,K,Mitra 23
Rational z-Transforms
? Consider
? Note G(z) has M finite zeros and N finite
poles
? If N > M there are additional zeros at
z = 0 (the origin in the z-plane)
? If N < M there are additional poles at
z = 0
MN ?
NM ?
?
?
?
??
?
?
? N
M
MN
zd
zp
zzG
10
10
? ?
? ?
)(
)(
)( )(
?
?
Copyright ? 2001,S,K,Mitra 24
Rational z-Transforms
? Example - The z-transform
has a zero at z = 0 and a pole at z = 1
1f o r
1
1
1 ??? ? zzz,)(?
Copyright ? 2001,S,K,Mitra 25
Rational z-Transforms
? A physical interpretation of the concepts of
poles and zeros can be given by plotting the
log-magnitude as shown on
next slide for
)(l o g zG1020
21
21
640801
882421
??
??
??
???
zz
zzzG
..
..)(
Copyright ? 2001,S,K,Mitra 26
Rational z-Transforms
poles
zeros
Copyright ? 2001,S,K,Mitra 27
Rational z-Transforms
? Observe that the magnitude plot exhibits
very large peaks around the points
which are the poles of
G(z)
? It also exhibits very narrow and deep wells
around the location of the zeros at
6 9 2 8040,,jz ??
2121,,jz ??
Copyright ? 2001,S,K,Mitra 28
ROC of a Rational
z-Transform
? ROC of a z-transform is an important
concept
? Without the knowledge of the ROC,there is
no unique relationship between a sequence
and its z-transform
? Hence,the z-transform must always be
specified with its ROC
Copyright ? 2001,S,K,Mitra 29
ROC of a Rational
z-Transform
? Moreover,if the ROC of a z-transform
includes the unit circle,the DTFT of the
sequence is obtained by simply evaluating
the z-transform on the unit circle
? There is a relationship between the ROC of
the z-transform of the impulse response of a
causal LTI discrete-time system and its
BIBO stability
Copyright ? 2001,S,K,Mitra 30
ROC of a Rational
z-Transform
? The ROC of a rational z-transform is
bounded by the locations of its poles
? To understand the relationship between the
poles and the ROC,it is instructive to
examine the pole-zero plot of a z-transform
? Consider again the pole-zero plot of the z-
transform ?(z)
Copyright ? 2001,S,K,Mitra 31
ROC of a Rational
z-Transform
? In this plot,the ROC,shown as the shaded
area,is the region of the z-plane just outside
the circle centered at the origin and going
through the pole at z = 1
Copyright ? 2001,S,K,Mitra 32
ROC of a Rational
z-Transform
? Example - The z-transform H(z) of the sequence
is given by
? Here the ROC is just outside the circle going
through the point 60,??z
][)6.0(][ nnh n ???
,
.
)( 1
601
1
??? zzH
60.?z
Copyright ? 2001,S,K,Mitra 33
ROC of a Rational
z-Transform
? A sequence can be one of the following
types,finite-length,right-sided,left-sided
and two-sided
? In general,the ROC depends on the type of
the sequence of interest
Copyright ? 2001,S,K,Mitra 34
ROC of a Rational
z-Transform
? Example - Consider a finite-length
sequence g[n] defined for,
where M and N are non-negative integers
and
? Its z-transform is given by
NnM ???
??][ ng
0
[]
( ) [ ]
NM N M nN
n n
N
nM
g n M z
G z g n z
z
? ??
? ?
??
?
?? ??
Copyright ? 2001,S,K,Mitra 35
ROC of a Rational
z-Transform
? Note,G(z) has M poles at and N poles
at z = 0
? As can be seen from the expression for
G(z),the z-transform of a finite-length
bounded sequence converges everywhere in
the z-plane except possibly at z = 0 and/or at
??z
??z
Copyright ? 2001,S,K,Mitra 36
ROC of a Rational
z-Transform
? Example - A right-sided sequence with
nonzero sample values for is
sometimes called a causal sequence
? Consider a causal sequence
? Its z-transform is given by
?
?
?
??
0
11 ][)(
n
nznuzU
0?n
][1 nu
Copyright ? 2001,S,K,Mitra 37
ROC of a Rational
z-Transform
? It can be shown that converges
exterior to a circle,including the
point
? On the other hand,a right-sided sequence
with nonzero sample values only for
with M nonnegative has a z-transform
with M poles at
? The ROC of is exterior to a circle
,excluding the point
)(1 zU
1Rz ?
2Rz ?
??z
??z
??z
)(2 zU
][2 nu
Mn ??
)(2 zU
Copyright ? 2001,S,K,Mitra 38
ROC of a Rational
z-Transform
? Example - A left-sided sequence with
nonzero sample values for is
sometimes called a anti-causal sequence
? Consider an anti-causal sequence
? Its z-transform is given by
0?n
][1 nv
?
???
??
0
11 ][)(
n
nznvzV
Copyright ? 2001,S,K,Mitra 39
ROC of a Rational
z-Transform
? It can be shown that converges
interior to a circle,including the
point z = 0
? On the other hand,a left-sided sequence
with nonzero sample values only for
with N nonnegative has a z-transform
with N poles at z = 0
? The ROC of is interior to a circle
,excluding the point z = 0
Nn ?
)(1 zV
3Rz ?
)(2 zV
)(2 zV
4Rz ?
Copyright ? 2001,S,K,Mitra 40
ROC of a Rational
z-Transform
? Example - The z-transform of a two-sided
sequence w[n] can be expressed as
? The first term on the RHS,,
can be interpreted as the z-transform of a
right-sided sequence and it thus converges
exterior to the circle
???
?
???
?
?
?
?
?
???
? ???
1
0
][][][)(
n
n
n
n
n
n znwznwznwzW
5Rz ?
? ?? ?0 ][n nznw
Copyright ? 2001,S,K,Mitra 41
ROC of a Rational
z-Transform
? The second term on the RHS,,
can be interpreted as the z-transform of a left-
sided sequence and it thus converges interior
to the circle
? If,there is an overlapping ROC
given by
? If,there is no overlap and the
z-transform does not exist
? ? ??? ?1 ][n nznw
6Rz ?
65 RR ?
65 RR ?
65 RzR ??
Copyright ? 2001,S,K,Mitra 42
ROC of a Rational
z-Transform
? Example - Consider the two-sided sequence
where ? can be either real or complex
? Its z-transform is given by
? The first term on the RHS converges for
,whereas the second term converges
for
nnu ??][
???
?
???
?
?
?
?
?
???
? ??????
1
0
)(
n
nn
n
nn
n
nn zzzzU
??z
??z
Copyright ? 2001,S,K,Mitra 43
ROC of a Rational
z-Transform
? There is no overlap between these two
regions
? Hence,the z-transform of does
not exist!
nnu ??][
Copyright ? 2001,S,K,Mitra 44
ROC of a Rational
z-Transform
? The ROC of a rational z-transform cannot
contain any poles and is bounded by the
poles
? As an example,assume that a rational z-
transform X(z) has two simple poles at z = ?
and z = b with
? There are three possible ROCs associated
with X(z)
b??
Copyright ? 2001,S,K,Mitra 45
ROC of a Rational
z-Transform
Right-sided
Left-sided
Two-sided
Copyright ? 2001,S,K,Mitra 46
ROC of a Rational
z-Transform
? In general,if the rational z-transform has N
poles with R distinct magnitudes,then it has
ROCs
? Thus,there are distinct sequences with
the same z-transform
? Hence,a rational z-transform with a
specified ROC has a unique sequence as its
inverse z-transform
1?R
1?R
Copyright ? 2001,S,K,Mitra 47
ROC of a Rational
z-Transform
? The ROC of a rational z-transform can be
easily determined using MATLAB
determines the zeros,poles,and the gain
constant of a rational z-transform with the
numerator coefficients specified by the
vector num and the denominator
coefficients specified by the vector den
[z,p,k] = tf2zp(num,den)
Copyright ? 2001,S,K,Mitra 48
ROC of a Rational
z-Transform
[num,den] = zp2tf(z,p,k) implements the
reverse process
? The factored form of the z-transform can be
obtained using sos = zp2sos(z,p,k)
? The above statement computes the
coefficients of each second-order factor
given as an matrix sos 6?L
Copyright ? 2001,S,K,Mitra 49
ROC of a Rational
z-Transform
where
?
?
?
?
?
?
?
?
?
?
?
LLLLLL aaabbb
aaabbb
aaabbb
s o s
210210
221202221202
121101211101
??????
?
?
??
??
??
??? L
k kkk
kkk
zazaa
zbzbbzG
1
2
2
1
10
2
2
1
10)(
Copyright ? 2001,S,K,Mitra 50
ROC of a Rational
z-Transform
? The pole-zero plot is determined using the
function zplane
? The z-transform can be either described in
terms of its zeros and poles,
zplane(zeros,poles)
or,it can be described in terms of its
numerator and denominator coefficients,
zplane(num,den)
Copyright ? 2001,S,K,Mitra 51
ROC of a Rational
z-Transform
? Example - The pole-zero plot of
obtained using MATLAB is shown below
pole?? zeroo ?
12181533
325644162)(
234
234
????
?????
zzzz
zzzzzG
-4 -3 -2 -1 0 1
-2
-1
0
1
2
R e a l P a r t
I
m
a
gi
na
r
y P
a
r
t
Copyright ? 2001,S,K,Mitra 52
Inverse z-Transform
? General Expression,Recall that,for,
the z-transform G(z) given by
is merely the DTFT of the modified sequence
? Accordingly,the inverse DTFT is thus given
by
?? jerz
???? ? ??? ???? ??? ? n njnn n erngzngzG ][][)(
nrng ?][
1[ ] ( )
2
n j j ng n r G re e d? ??
? ??
?
?? ?
Copyright ? 2001,S,K,Mitra 53
Inverse z-Transform
? By making a change of variable,
the previous equation can be converted into
a contour integral given by
where is a counterclockwise contour of
integration defined by |z| = r
?? jerz
C?
dzzzGjng
C
n?
?? ?
? 1)(
2
1][
Copyright ? 2001,S,K,Mitra 54
z-Transform
()Xz[]xn
z-Transform,analysis equation
Inverse z-Transform,synthesis equation
time
domain z-domain
( ) [ ] n
n
X z x n z
?
?
? ??
? ?
11[ ] ( )
2
n
C
x n X z z d z
j?
?
?
? ??
Copyright ? 2001,S,K,Mitra 55
Inverse z-Transform
? But the integral remains unchanged when it
is replaced with any contour C encircling
the point z = 0 in the ROC of G(z)
? The contour integral can be evaluated using
the Cauchy’s residue theorem resulting in
? The above equation needs to be evaluated at
all values of n and is not pursued here
1
|[ ] Re s ( ( ) ) i
i
n
zz
z in s id e C
g n G z z ? ?? ?
Copyright ? 2001,S,K,Mitra 56
Inverse Transform by
Partial-Fraction Expansion
? A rational z-transform G(z) with a causal
inverse transform g[n] has a ROC that is
exterior to a circle
? Here it is more convenient to express G(z)
in a partial-fraction expansion form and
then determine g[n] by summing the inverse
transform of the individual simpler terms in
the expansion
Copyright ? 2001,S,K,Mitra 57
Inverse Transform by
Partial-Fraction Expansion
? A rational G(z) can be expressed as
? If then G(z) can be re-expressed as
where the degree of is less than N
?
?
?
?
?
?
?? N
i
i
i
M
i
i
i
zD
zP
zd
zp
zG
0
0
)(
)()(
)(
)()(
zD
zPNM zzG 1
0
?
?
?
? ??
?
?
??
NM ?
)(zP1
Copyright ? 2001,S,K,Mitra 58
Inverse Transform by
Partial-Fraction Expansion
? The rational function is called a
proper fraction
? Example - Consider
? By long division we arrive at
)(/)( zDzP1
21
321
20801
3050802
??
???
??
????
zz
zzzzG
..
...)(
21
1
1
20801
12555153
??
?
?
??
?????
zz
zzzG
..
....)(
Copyright ? 2001,S,K,Mitra 59
Inverse Transform by
Partial-Fraction Expansion
? Simple Poles,In most practical cases,the
rational z-transform of interest G(z) is a
proper fraction with simple poles
? Let the poles of G(z) be at,
? A partial-fraction expansion of G(z) is then
of the form
kz ?? Nk ??1
? ??
?
?
???
?
??
??
? ?
N
z
zG
1 11
)(
? ?
?
Copyright ? 2001,S,K,Mitra 60
Inverse Transform by
Partial-Fraction Expansion
? The constants in the partial-fraction
expansion are called the residues and are
given by
? Each term of the sum in partial-fraction
expansion has a ROC given by
and,thus,has an inverse transform of the
form
??
??? ??
????? zzGz )()1( 1
???z
][)( nn ??? ??
Copyright ? 2001,S,K,Mitra 61
Inverse Transform by
Partial-Fraction Expansion
? Therefore,the inverse transform g[n] of
G(z) is given by
? Note,The above approach with a slight
modification can also be used to determine
the inverse of a rational z-transform of a
noncausal sequence
? ????
?
N n
nng
1
][)(][
?
??
Copyright ? 2001,S,K,Mitra 62
Inverse Transform by
Partial-Fraction Expansion
? Example - Let the z-transform H(z) of a
causal sequence h[n] be given by
? A partial-fraction expansion of H(z) is then
of the form
).)(.().)(.(
)()(
11
1
601201
21
6020
2
??
?
??
??
??
??
zz
z
zz
zzzH
1
2
1
1
6.012.01
)( ??
?
??
?
??
zz
zH
Copyright ? 2001,S,K,Mitra 63
Inverse Transform by
Partial-Fraction Expansion
? Now
and
75.2
6.01
21)()2.01(
2.0
1
1
2.0
1
1 ??
?????
?
?
?
?
?
z
z z
zzHz
75.1
2.01
21)()6.01(
6.0
1
1
6.0
1
2 ???
?????
??
?
?
??
?
z
z z
zzHz
Copyright ? 2001,S,K,Mitra 64
Inverse Transform by
Partial-Fraction Expansion
? Hence
? The inverse transform of the above is
therefore given by
11 601
751
201
752
?? ???? zzzH,
.
.
.)(
][)6.0(75.1][)2.0(75.2][ nnnh nn ?????
Copyright ? 2001,S,K,Mitra 65
Inverse Transform by
Partial-Fraction Expansion
? Multiple Poles,If G(z) has multiple poles,
the partial-fraction expansion is of slightly
different form
? Let the pole at z = ? be of multiplicity L and
the remaining poles be simple and at
,
LN ?
???z LN ??? ?1
Copyright ? 2001,S,K,Mitra 66
Inverse Transform by
Partial-Fraction Expansion
? Then the partial-fraction expansion of G(z) is
of the form
where the constants are computed using
? The residues are calculated as before
?
??
???
??
??? ??
? ?
?
? ?
?
?
? L
i i
iLNNM
zz
zzG
1 11 10 )1(1
)(
? ?
?
?
?
?
i? ? ?
,)()1(
)()()!(
1 1
1 ??
?
??
?
? ??????? z
L
iL
iL
iLi zGzzd
d
iLLi ??1
??
Copyright ? 2001,S,K,Mitra 67
Partial-Fraction Expansion
Using MATLAB
[r,p,k]= residuez(num,den) develops the
partial-fraction expansion of a rational z-
transform with numerator and denominator
coefficients given by vectors num and den
? Vector r contains the residues
? Vector p contains the poles
? Vector k contains the constants ??
Copyright ? 2001,S,K,Mitra 68
Partial-Fraction Expansion
Using MATLAB
[num,den]=residuez(r,p,k) converts a z-
transform expressed in a partial-fraction
expansion form to its rational form
Copyright ? 2001,S,K,Mitra 69
Inverse z-Transform via Long
Division
? The z-transform G(z) of a causal sequence
{g[n]} can be expanded in a power series in
? In the series expansion,the coefficient
multiplying the term is then the n-th
sample g[n]
? For a rational z-transform expressed as a
ratio of polynomials in,the power series
expansion can be obtained by long division
1?z
1?z
nz?
Copyright ? 2001,S,K,Mitra 70
Inverse z-Transform via Long
Division
? Example - Consider
? Long division of the numerator by the
denominator yields
? As a result
21
1
12.04.01
21)(
??
?
??
??
zz
zzH
........)( ?????? ???? 4321 2224040520611 zzzzzH
02224040520611 ???? nnh },........{]}[{
?
Copyright ? 2001,S,K,Mitra 71
Inverse z-Transform Using
MATLAB
? The function impz can be used to find the
inverse of a rational z-transform G(z)
? The function computes the coefficients of
the power series expansion of G(z)
? The number of coefficients can either be
user specified or determined automatically
Copyright ? 2001,S,K,Mitra 72
Inverse z-Transform Using
MATLAB
>> num=[1 2];
>> den=[1 0.4 -0.12];
>> [h,t]=impz(num,den);
>> figure(1)
>> stem(t,h)
>> xlabel('n')
>> ylabel('h[n]')
h[n]=[1.0000 1.6000 -0.5200 0.4000 -0.2224 …]
Copyright ? 2001,S,K,Mitra 73
Table 3.9,z-Transform
Properties
Copyright ? 2001,S,K,Mitra 74
z-Transform Properties
? Example - Consider the two-sided sequence
? Let and with
X(z) and Y(z) denoting,respectively,their z-
transforms
? Now
and
]1[][][ ???b???? nnnv nn
][][ nnx n ??? ]1[[ ???b?? nny n
??
??
? ? z
z
zX,
1
1)(
1
b?
b?
? ? z
z
zY,
1
1)(
1
Copyright ? 2001,S,K,Mitra 75
z-Transform Properties
? Using the linearity property we arrive at
? The ROC of V(z) is given by the overlap
regions of and
? If,then there is an overlap and the
ROC is an annular region
? If,then there is no overlap and V(z)
does not exist
11 1
1
1
1
?? ?????? zzzYzXzV b?)()()(
??z b?z
b??
b??? z
b??
Copyright ? 2001,S,K,Mitra 76
z-Transform Properties
? Example - Determine the z-transform and
its ROC of the causal sequence
? We can express x[n] = v[n] + v*[n] where
? The z-transform of v[n] is given by
][)( c o s][ nnrnx on ???
][][][ 2121 nnernv nnjn o ????? ?
rz
zerz
zV
oj
???
?
??
??
?? ???,
1
1
1
1)(
12
1
12
1
Copyright ? 2001,S,K,Mitra 77
z-Transform Properties
? Using the conjugation property we obtain
the z-transform of v*[n] as
? Finally,using the linearity property we get
,
1
1
*1
1*)(*
12
1
12
1
???? ??????? zerzzV oj
*)(*)()( zVzVzX ??
???
?
???
?
?
?
?
? ????? 1121
1
1
1
1
zerzer oo jj
??z
Copyright ? 2001,S,K,Mitra 78
z-Transform Properties
? or,
? Example - Determine the z-transform Y(z)
and the ROC of the sequence
? We can write where
rz
zrzr
zrzX
o
o ?
???
???
??
?
,
)c o s2(1
)c o s(1)(
221
1
][)1(][ nnny n ????
][][][ nxnxnny ??
][][ nnx n ???
Copyright ? 2001,S,K,Mitra 79
z-Transform Properties
? Now,the z-transform X(z) of
is given by
? Using the differentiation property,we arrive
at the z-transform of as
][][ nnx n ???
][nxn
??
??
? ? z
z
zX,
1
1)(
1
??
??
???
?
?
z
z
z
dz
zXdz,
)1(
)(
1
1
Copyright ? 2001,S,K,Mitra 80
z-Transform Properties
? Using the linearity property we finally
obtain
21
1
1 )1(1
1)(
?
?
? ??
??
??
?
z
z
z
zY
??
??
? ? z
z
,
)1(
1
21
z-Transform
? The DTFT provides a frequency-domain
representation of discrete-time signals and LTI
discrete-time systems
? Because of the convergence condition,in many
cases,the DTFT of a sequence may not exist
? As a result,it is not possible to make use of such
frequency-domain characterization in these cases
Copyright ? 2001,S,K,Mitra 2
z-Transform
? A generalization of the DTFT defined by
leads to the z-transform
? z-transform may exist for many sequences
for which the DTFT does not exist
? Moreover,use of z-transform techniques
permits simple algebraic manipulations
??
?
???
???
n
njj enxeX ][)(
Copyright ? 2001,S,K,Mitra 3
z-Transform
? Consequently,z-transform has become an
important tool in the analysis and design of
digital filters
? For a given sequence g[n],its z-transform
G(z) is defined as
where is a complex
variable
?
?
???
??
n
nzngzG ][)(
R e ( ) I m ( )z z j z? ? ? ￡
Copyright ? 2001,S,K,Mitra 4
z-Transform
? If we let,then the z-transform
reduces to
? The above can be interpreted as the DTFT
of the modified sequence
? For r = 1 (i.e.,|z| = 1),z-transform reduces
to its DTFT,provided the latter exists
?? jerz
??
?
???
????
n
njnj erngerG ][)(
}][{ nrng ?
Copyright ? 2001,S,K,Mitra 5
z-Transform
? The contour |z| = 1 is a circle in the z-plane
of unity radius and is called the unit circle
? Like the DTFT,there are conditions on the
convergence of the infinite series
? For a given sequence,the set R of values of
z for which its z-transform converges is
called the region of convergence (ROC)
?
?
???
?
n
nzng ][
Copyright ? 2001,S,K,Mitra 6
z-Transform
? From our earlier discussion on the uniform
convergence of the DTFT,it follows that the
series
converges if is absolutely
summable,i.e.,if
??
?
???
????
n
njnj erngerG ][)(
}][{ nrng ?
???
?
???
?
n
nrng ][
Copyright ? 2001,S,K,Mitra 7
z-Transform
? In general,the ROC of a z-transform of a
sequence g[n] is an annular region of the z-
plane,
where
?? ?? gg RzR
???? ?? gg RR0
ROC
gR? gR?
Copyright ? 2001,S,K,Mitra 8
? The z-transform is a form of the Cauchy-
Laurent series and is an analytic function at
every point in the ROC
? Let f (z) denote an analytic (or holomorphic)
function over an annular region centered
at
Cauchy-Laurent Series
?
oz
Re( )z
Im( )z ?
oz
Copyright ? 2001,S,K,Mitra 9
? Then f (z) can be expressed as the bilateral series
being a closed and counterclockwise integration
contour contained in
Cauchy-Laurent Series
( 1 )
( ) ( )
1
( ) ( )
2
n
no
n
n
no
f z z z
f z z z dz
j
?
?
?
?
?
? ??
??
??
??
?
??where
? ?
Copyright ? 2001,S,K,Mitra 10
z-Transform
? Example - Determine the z-transform X(z)
of the causal sequence and its
ROC
? Now
? The above power series converges to
? ROC is the annular region |z| > |?|
][][ nnx n ???
? ??? ???
?
?
??
???
?
0
][)(
n
nn
n
nn zznzX
1f o r,
1
1)( 1
1 ?????
?
? zzzX
Copyright ? 2001,S,K,Mitra 11
z-Transform
? Example - The z-transform ?(z) of the unit
step sequence ?[n] can be obtained from
by setting ? = 1,
? ROC is the annular region
1f o r,
1
1)( 1
1 ?????
?
? zzzX
??? z1
1f o r
1
1 1
1 ???
?
? zzz,)(?
Copyright ? 2001,S,K,Mitra 12
z-Transform
? Note,The unit step sequence ?[n] is not
absolutely summable,and hence its DTFT
does not converge uniformly
? Example - Consider the anti-causal
sequence
]1[][ ?????? nny n
Copyright ? 2001,S,K,Mitra 13
z-Transform
? Its z-transform is given by
? ROC is the annular region
? ???? ???
?
?
??
???
?
1
1
)(
m
mm
n
nn zzzY
1f o r,
1
1 1
1 ?????
?
? zz
z
zzz
m
mm
1
1
0
1
1 ?
??
?
??
??
???? ????
??z
Copyright ? 2001,S,K,Mitra 14
z-Transform
? Note,The z-transforms of the two
sequences and are
identical even though the two parent
sequences are different
? Only way a unique sequence can be
associated with a z-transform is by
specifying its ROC
]1[ ????? nn][nn??
Copyright ? 2001,S,K,Mitra 15
z-Transform
? The DTFT of a sequence g[n]
converges uniformly if and only if the ROC
of the z-transform G(z) of g[n] includes the
unit circle
? The existence of the DTFT does not always
imply the existence of the z-transform
)( ?jeG
Copyright ? 2001,S,K,Mitra 16
z-Transform
? Example - The finite energy sequence
has a DTFT given by
which converges in the mean-square sense
sin[ ] sin c,c c c
LP
nnh n n
n
? ? ?
? ??
??? ? ? ? ? ? ?
????
?
?
?
?????
????
??
c
cj
LP eH,0
0,1
)(
Copyright ? 2001,S,K,Mitra 17
z-Transform
? However,does not have a z-transform
as it is not absolutely summable for any value
of r,i.e,
? Some commonly used z-transform pairs are
listed on the next slide
][nhLP
[] nLP
n
h n r
?
?
? ??
??? r?
Copyright ? 2001,S,K,Mitra 18
Table 3.8,Commonly Used z-
Transform Pairs
Copyright ? 2001,S,K,Mitra 19
Rational z-Transforms
? In the case of LTI discrete-time systems we
are concerned with in this course,all
pertinent z-transforms are rational functions
of
? That is,they are ratios of two polynomials
in,
1?z
1?z
N
N
N
N
M
M
M
M
zdzdzdd
zpzpzpp
zD
zPzG
???
?
?
???
?
?
????
??????
)(
)(
....
....
)(
)()(
1
1
1
10
1
1
1
10
Copyright ? 2001,S,K,Mitra 20
Rational z-Transforms
? The degree of the numerator polynomial
P(z) is M and the degree of the denominator
polynomial D(z) is N
? An alternate representation of a rational z-
transform is as a ratio of two polynomials in
z,
NN
NN
MM
MM
MN
dzdzdzd
pzpzpzpzzG
????
?????
?
?
?
?
?
1
1
10
1
1
10
....
....
)( )(
Copyright ? 2001,S,K,Mitra 21
Rational z-Transforms
? A rational z-transform can be alternately
written in factored form as
?
?
?
?
?
?
?
?
? N
M
zd
zp
zG
1
1
0
1
1
0
1
1
? ?
? ?
)(
)(
)(
?
?
?
?
?
??
?
?
? N
M
MN
zd
zp
z
10
10
? ?
? ?
)(
)()(
?
?
Copyright ? 2001,S,K,Mitra 22
Rational z-Transforms
? At a root of the numerator polynomial
,and as a result,these values of z
are known as the zeros of G(z)
? At a root of the denominator
polynomial,and as a result,
these values of z are known as the poles of
G(z)
z ?? l
z ?? l
??)( ??G
0?)( ??G
Copyright ? 2001,S,K,Mitra 23
Rational z-Transforms
? Consider
? Note G(z) has M finite zeros and N finite
poles
? If N > M there are additional zeros at
z = 0 (the origin in the z-plane)
? If N < M there are additional poles at
z = 0
MN ?
NM ?
?
?
?
??
?
?
? N
M
MN
zd
zp
zzG
10
10
? ?
? ?
)(
)(
)( )(
?
?
Copyright ? 2001,S,K,Mitra 24
Rational z-Transforms
? Example - The z-transform
has a zero at z = 0 and a pole at z = 1
1f o r
1
1
1 ??? ? zzz,)(?
Copyright ? 2001,S,K,Mitra 25
Rational z-Transforms
? A physical interpretation of the concepts of
poles and zeros can be given by plotting the
log-magnitude as shown on
next slide for
)(l o g zG1020
21
21
640801
882421
??
??
??
???
zz
zzzG
..
..)(
Copyright ? 2001,S,K,Mitra 26
Rational z-Transforms
poles
zeros
Copyright ? 2001,S,K,Mitra 27
Rational z-Transforms
? Observe that the magnitude plot exhibits
very large peaks around the points
which are the poles of
G(z)
? It also exhibits very narrow and deep wells
around the location of the zeros at
6 9 2 8040,,jz ??
2121,,jz ??
Copyright ? 2001,S,K,Mitra 28
ROC of a Rational
z-Transform
? ROC of a z-transform is an important
concept
? Without the knowledge of the ROC,there is
no unique relationship between a sequence
and its z-transform
? Hence,the z-transform must always be
specified with its ROC
Copyright ? 2001,S,K,Mitra 29
ROC of a Rational
z-Transform
? Moreover,if the ROC of a z-transform
includes the unit circle,the DTFT of the
sequence is obtained by simply evaluating
the z-transform on the unit circle
? There is a relationship between the ROC of
the z-transform of the impulse response of a
causal LTI discrete-time system and its
BIBO stability
Copyright ? 2001,S,K,Mitra 30
ROC of a Rational
z-Transform
? The ROC of a rational z-transform is
bounded by the locations of its poles
? To understand the relationship between the
poles and the ROC,it is instructive to
examine the pole-zero plot of a z-transform
? Consider again the pole-zero plot of the z-
transform ?(z)
Copyright ? 2001,S,K,Mitra 31
ROC of a Rational
z-Transform
? In this plot,the ROC,shown as the shaded
area,is the region of the z-plane just outside
the circle centered at the origin and going
through the pole at z = 1
Copyright ? 2001,S,K,Mitra 32
ROC of a Rational
z-Transform
? Example - The z-transform H(z) of the sequence
is given by
? Here the ROC is just outside the circle going
through the point 60,??z
][)6.0(][ nnh n ???
,
.
)( 1
601
1
??? zzH
60.?z
Copyright ? 2001,S,K,Mitra 33
ROC of a Rational
z-Transform
? A sequence can be one of the following
types,finite-length,right-sided,left-sided
and two-sided
? In general,the ROC depends on the type of
the sequence of interest
Copyright ? 2001,S,K,Mitra 34
ROC of a Rational
z-Transform
? Example - Consider a finite-length
sequence g[n] defined for,
where M and N are non-negative integers
and
? Its z-transform is given by
NnM ???
??][ ng
0
[]
( ) [ ]
NM N M nN
n n
N
nM
g n M z
G z g n z
z
? ??
? ?
??
?
?? ??
Copyright ? 2001,S,K,Mitra 35
ROC of a Rational
z-Transform
? Note,G(z) has M poles at and N poles
at z = 0
? As can be seen from the expression for
G(z),the z-transform of a finite-length
bounded sequence converges everywhere in
the z-plane except possibly at z = 0 and/or at
??z
??z
Copyright ? 2001,S,K,Mitra 36
ROC of a Rational
z-Transform
? Example - A right-sided sequence with
nonzero sample values for is
sometimes called a causal sequence
? Consider a causal sequence
? Its z-transform is given by
?
?
?
??
0
11 ][)(
n
nznuzU
0?n
][1 nu
Copyright ? 2001,S,K,Mitra 37
ROC of a Rational
z-Transform
? It can be shown that converges
exterior to a circle,including the
point
? On the other hand,a right-sided sequence
with nonzero sample values only for
with M nonnegative has a z-transform
with M poles at
? The ROC of is exterior to a circle
,excluding the point
)(1 zU
1Rz ?
2Rz ?
??z
??z
??z
)(2 zU
][2 nu
Mn ??
)(2 zU
Copyright ? 2001,S,K,Mitra 38
ROC of a Rational
z-Transform
? Example - A left-sided sequence with
nonzero sample values for is
sometimes called a anti-causal sequence
? Consider an anti-causal sequence
? Its z-transform is given by
0?n
][1 nv
?
???
??
0
11 ][)(
n
nznvzV
Copyright ? 2001,S,K,Mitra 39
ROC of a Rational
z-Transform
? It can be shown that converges
interior to a circle,including the
point z = 0
? On the other hand,a left-sided sequence
with nonzero sample values only for
with N nonnegative has a z-transform
with N poles at z = 0
? The ROC of is interior to a circle
,excluding the point z = 0
Nn ?
)(1 zV
3Rz ?
)(2 zV
)(2 zV
4Rz ?
Copyright ? 2001,S,K,Mitra 40
ROC of a Rational
z-Transform
? Example - The z-transform of a two-sided
sequence w[n] can be expressed as
? The first term on the RHS,,
can be interpreted as the z-transform of a
right-sided sequence and it thus converges
exterior to the circle
???
?
???
?
?
?
?
?
???
? ???
1
0
][][][)(
n
n
n
n
n
n znwznwznwzW
5Rz ?
? ?? ?0 ][n nznw
Copyright ? 2001,S,K,Mitra 41
ROC of a Rational
z-Transform
? The second term on the RHS,,
can be interpreted as the z-transform of a left-
sided sequence and it thus converges interior
to the circle
? If,there is an overlapping ROC
given by
? If,there is no overlap and the
z-transform does not exist
? ? ??? ?1 ][n nznw
6Rz ?
65 RR ?
65 RR ?
65 RzR ??
Copyright ? 2001,S,K,Mitra 42
ROC of a Rational
z-Transform
? Example - Consider the two-sided sequence
where ? can be either real or complex
? Its z-transform is given by
? The first term on the RHS converges for
,whereas the second term converges
for
nnu ??][
???
?
???
?
?
?
?
?
???
? ??????
1
0
)(
n
nn
n
nn
n
nn zzzzU
??z
??z
Copyright ? 2001,S,K,Mitra 43
ROC of a Rational
z-Transform
? There is no overlap between these two
regions
? Hence,the z-transform of does
not exist!
nnu ??][
Copyright ? 2001,S,K,Mitra 44
ROC of a Rational
z-Transform
? The ROC of a rational z-transform cannot
contain any poles and is bounded by the
poles
? As an example,assume that a rational z-
transform X(z) has two simple poles at z = ?
and z = b with
? There are three possible ROCs associated
with X(z)
b??
Copyright ? 2001,S,K,Mitra 45
ROC of a Rational
z-Transform
Right-sided
Left-sided
Two-sided
Copyright ? 2001,S,K,Mitra 46
ROC of a Rational
z-Transform
? In general,if the rational z-transform has N
poles with R distinct magnitudes,then it has
ROCs
? Thus,there are distinct sequences with
the same z-transform
? Hence,a rational z-transform with a
specified ROC has a unique sequence as its
inverse z-transform
1?R
1?R
Copyright ? 2001,S,K,Mitra 47
ROC of a Rational
z-Transform
? The ROC of a rational z-transform can be
easily determined using MATLAB
determines the zeros,poles,and the gain
constant of a rational z-transform with the
numerator coefficients specified by the
vector num and the denominator
coefficients specified by the vector den
[z,p,k] = tf2zp(num,den)
Copyright ? 2001,S,K,Mitra 48
ROC of a Rational
z-Transform
[num,den] = zp2tf(z,p,k) implements the
reverse process
? The factored form of the z-transform can be
obtained using sos = zp2sos(z,p,k)
? The above statement computes the
coefficients of each second-order factor
given as an matrix sos 6?L
Copyright ? 2001,S,K,Mitra 49
ROC of a Rational
z-Transform
where
?
?
?
?
?
?
?
?
?
?
?
LLLLLL aaabbb
aaabbb
aaabbb
s o s
210210
221202221202
121101211101
??????
?
?
??
??
??
??? L
k kkk
kkk
zazaa
zbzbbzG
1
2
2
1
10
2
2
1
10)(
Copyright ? 2001,S,K,Mitra 50
ROC of a Rational
z-Transform
? The pole-zero plot is determined using the
function zplane
? The z-transform can be either described in
terms of its zeros and poles,
zplane(zeros,poles)
or,it can be described in terms of its
numerator and denominator coefficients,
zplane(num,den)
Copyright ? 2001,S,K,Mitra 51
ROC of a Rational
z-Transform
? Example - The pole-zero plot of
obtained using MATLAB is shown below
pole?? zeroo ?
12181533
325644162)(
234
234
????
?????
zzzz
zzzzzG
-4 -3 -2 -1 0 1
-2
-1
0
1
2
R e a l P a r t
I
m
a
gi
na
r
y P
a
r
t
Copyright ? 2001,S,K,Mitra 52
Inverse z-Transform
? General Expression,Recall that,for,
the z-transform G(z) given by
is merely the DTFT of the modified sequence
? Accordingly,the inverse DTFT is thus given
by
?? jerz
???? ? ??? ???? ??? ? n njnn n erngzngzG ][][)(
nrng ?][
1[ ] ( )
2
n j j ng n r G re e d? ??
? ??
?
?? ?
Copyright ? 2001,S,K,Mitra 53
Inverse z-Transform
? By making a change of variable,
the previous equation can be converted into
a contour integral given by
where is a counterclockwise contour of
integration defined by |z| = r
?? jerz
C?
dzzzGjng
C
n?
?? ?
? 1)(
2
1][
Copyright ? 2001,S,K,Mitra 54
z-Transform
()Xz[]xn
z-Transform,analysis equation
Inverse z-Transform,synthesis equation
time
domain z-domain
( ) [ ] n
n
X z x n z
?
?
? ??
? ?
11[ ] ( )
2
n
C
x n X z z d z
j?
?
?
? ??
Copyright ? 2001,S,K,Mitra 55
Inverse z-Transform
? But the integral remains unchanged when it
is replaced with any contour C encircling
the point z = 0 in the ROC of G(z)
? The contour integral can be evaluated using
the Cauchy’s residue theorem resulting in
? The above equation needs to be evaluated at
all values of n and is not pursued here
1
|[ ] Re s ( ( ) ) i
i
n
zz
z in s id e C
g n G z z ? ?? ?
Copyright ? 2001,S,K,Mitra 56
Inverse Transform by
Partial-Fraction Expansion
? A rational z-transform G(z) with a causal
inverse transform g[n] has a ROC that is
exterior to a circle
? Here it is more convenient to express G(z)
in a partial-fraction expansion form and
then determine g[n] by summing the inverse
transform of the individual simpler terms in
the expansion
Copyright ? 2001,S,K,Mitra 57
Inverse Transform by
Partial-Fraction Expansion
? A rational G(z) can be expressed as
? If then G(z) can be re-expressed as
where the degree of is less than N
?
?
?
?
?
?
?? N
i
i
i
M
i
i
i
zD
zP
zd
zp
zG
0
0
)(
)()(
)(
)()(
zD
zPNM zzG 1
0
?
?
?
? ??
?
?
??
NM ?
)(zP1
Copyright ? 2001,S,K,Mitra 58
Inverse Transform by
Partial-Fraction Expansion
? The rational function is called a
proper fraction
? Example - Consider
? By long division we arrive at
)(/)( zDzP1
21
321
20801
3050802
??
???
??
????
zz
zzzzG
..
...)(
21
1
1
20801
12555153
??
?
?
??
?????
zz
zzzG
..
....)(
Copyright ? 2001,S,K,Mitra 59
Inverse Transform by
Partial-Fraction Expansion
? Simple Poles,In most practical cases,the
rational z-transform of interest G(z) is a
proper fraction with simple poles
? Let the poles of G(z) be at,
? A partial-fraction expansion of G(z) is then
of the form
kz ?? Nk ??1
? ??
?
?
???
?
??
??
? ?
N
z
zG
1 11
)(
? ?
?
Copyright ? 2001,S,K,Mitra 60
Inverse Transform by
Partial-Fraction Expansion
? The constants in the partial-fraction
expansion are called the residues and are
given by
? Each term of the sum in partial-fraction
expansion has a ROC given by
and,thus,has an inverse transform of the
form
??
??? ??
????? zzGz )()1( 1
???z
][)( nn ??? ??
Copyright ? 2001,S,K,Mitra 61
Inverse Transform by
Partial-Fraction Expansion
? Therefore,the inverse transform g[n] of
G(z) is given by
? Note,The above approach with a slight
modification can also be used to determine
the inverse of a rational z-transform of a
noncausal sequence
? ????
?
N n
nng
1
][)(][
?
??
Copyright ? 2001,S,K,Mitra 62
Inverse Transform by
Partial-Fraction Expansion
? Example - Let the z-transform H(z) of a
causal sequence h[n] be given by
? A partial-fraction expansion of H(z) is then
of the form
).)(.().)(.(
)()(
11
1
601201
21
6020
2
??
?
??
??
??
??
zz
z
zz
zzzH
1
2
1
1
6.012.01
)( ??
?
??
?
??
zz
zH
Copyright ? 2001,S,K,Mitra 63
Inverse Transform by
Partial-Fraction Expansion
? Now
and
75.2
6.01
21)()2.01(
2.0
1
1
2.0
1
1 ??
?????
?
?
?
?
?
z
z z
zzHz
75.1
2.01
21)()6.01(
6.0
1
1
6.0
1
2 ???
?????
??
?
?
??
?
z
z z
zzHz
Copyright ? 2001,S,K,Mitra 64
Inverse Transform by
Partial-Fraction Expansion
? Hence
? The inverse transform of the above is
therefore given by
11 601
751
201
752
?? ???? zzzH,
.
.
.)(
][)6.0(75.1][)2.0(75.2][ nnnh nn ?????
Copyright ? 2001,S,K,Mitra 65
Inverse Transform by
Partial-Fraction Expansion
? Multiple Poles,If G(z) has multiple poles,
the partial-fraction expansion is of slightly
different form
? Let the pole at z = ? be of multiplicity L and
the remaining poles be simple and at
,
LN ?
???z LN ??? ?1
Copyright ? 2001,S,K,Mitra 66
Inverse Transform by
Partial-Fraction Expansion
? Then the partial-fraction expansion of G(z) is
of the form
where the constants are computed using
? The residues are calculated as before
?
??
???
??
??? ??
? ?
?
? ?
?
?
? L
i i
iLNNM
zz
zzG
1 11 10 )1(1
)(
? ?
?
?
?
?
i? ? ?
,)()1(
)()()!(
1 1
1 ??
?
??
?
? ??????? z
L
iL
iL
iLi zGzzd
d
iLLi ??1
??
Copyright ? 2001,S,K,Mitra 67
Partial-Fraction Expansion
Using MATLAB
[r,p,k]= residuez(num,den) develops the
partial-fraction expansion of a rational z-
transform with numerator and denominator
coefficients given by vectors num and den
? Vector r contains the residues
? Vector p contains the poles
? Vector k contains the constants ??
Copyright ? 2001,S,K,Mitra 68
Partial-Fraction Expansion
Using MATLAB
[num,den]=residuez(r,p,k) converts a z-
transform expressed in a partial-fraction
expansion form to its rational form
Copyright ? 2001,S,K,Mitra 69
Inverse z-Transform via Long
Division
? The z-transform G(z) of a causal sequence
{g[n]} can be expanded in a power series in
? In the series expansion,the coefficient
multiplying the term is then the n-th
sample g[n]
? For a rational z-transform expressed as a
ratio of polynomials in,the power series
expansion can be obtained by long division
1?z
1?z
nz?
Copyright ? 2001,S,K,Mitra 70
Inverse z-Transform via Long
Division
? Example - Consider
? Long division of the numerator by the
denominator yields
? As a result
21
1
12.04.01
21)(
??
?
??
??
zz
zzH
........)( ?????? ???? 4321 2224040520611 zzzzzH
02224040520611 ???? nnh },........{]}[{
?
Copyright ? 2001,S,K,Mitra 71
Inverse z-Transform Using
MATLAB
? The function impz can be used to find the
inverse of a rational z-transform G(z)
? The function computes the coefficients of
the power series expansion of G(z)
? The number of coefficients can either be
user specified or determined automatically
Copyright ? 2001,S,K,Mitra 72
Inverse z-Transform Using
MATLAB
>> num=[1 2];
>> den=[1 0.4 -0.12];
>> [h,t]=impz(num,den);
>> figure(1)
>> stem(t,h)
>> xlabel('n')
>> ylabel('h[n]')
h[n]=[1.0000 1.6000 -0.5200 0.4000 -0.2224 …]
Copyright ? 2001,S,K,Mitra 73
Table 3.9,z-Transform
Properties
Copyright ? 2001,S,K,Mitra 74
z-Transform Properties
? Example - Consider the two-sided sequence
? Let and with
X(z) and Y(z) denoting,respectively,their z-
transforms
? Now
and
]1[][][ ???b???? nnnv nn
][][ nnx n ??? ]1[[ ???b?? nny n
??
??
? ? z
z
zX,
1
1)(
1
b?
b?
? ? z
z
zY,
1
1)(
1
Copyright ? 2001,S,K,Mitra 75
z-Transform Properties
? Using the linearity property we arrive at
? The ROC of V(z) is given by the overlap
regions of and
? If,then there is an overlap and the
ROC is an annular region
? If,then there is no overlap and V(z)
does not exist
11 1
1
1
1
?? ?????? zzzYzXzV b?)()()(
??z b?z
b??
b??? z
b??
Copyright ? 2001,S,K,Mitra 76
z-Transform Properties
? Example - Determine the z-transform and
its ROC of the causal sequence
? We can express x[n] = v[n] + v*[n] where
? The z-transform of v[n] is given by
][)( c o s][ nnrnx on ???
][][][ 2121 nnernv nnjn o ????? ?
rz
zerz
zV
oj
???
?
??
??
?? ???,
1
1
1
1)(
12
1
12
1
Copyright ? 2001,S,K,Mitra 77
z-Transform Properties
? Using the conjugation property we obtain
the z-transform of v*[n] as
? Finally,using the linearity property we get
,
1
1
*1
1*)(*
12
1
12
1
???? ??????? zerzzV oj
*)(*)()( zVzVzX ??
???
?
???
?
?
?
?
? ????? 1121
1
1
1
1
zerzer oo jj
??z
Copyright ? 2001,S,K,Mitra 78
z-Transform Properties
? or,
? Example - Determine the z-transform Y(z)
and the ROC of the sequence
? We can write where
rz
zrzr
zrzX
o
o ?
???
???
??
?
,
)c o s2(1
)c o s(1)(
221
1
][)1(][ nnny n ????
][][][ nxnxnny ??
][][ nnx n ???
Copyright ? 2001,S,K,Mitra 79
z-Transform Properties
? Now,the z-transform X(z) of
is given by
? Using the differentiation property,we arrive
at the z-transform of as
][][ nnx n ???
][nxn
??
??
? ? z
z
zX,
1
1)(
1
??
??
???
?
?
z
z
z
dz
zXdz,
)1(
)(
1
1
Copyright ? 2001,S,K,Mitra 80
z-Transform Properties
? Using the linearity property we finally
obtain
21
1
1 )1(1
1)(
?
?
? ??
??
??
?
z
z
z
zY
??
??
? ? z
z
,
)1(
1
21
