Poularikas A. D. “Two-Dimensional Z-Transform”
The Handbook of Formulas and Tables for Signal Processing.
Ed. Alexander D. Poularikas
Boca Raton: CRC Press LLC, 1999
8
Two-Dimensional
Z-Transform
8.1 The Z-Transform
8.2 Properties of the Z-Transform
8.3 Inverse Z-Transform
8.4 System Function
8.5 Stability Theorems
References
Appendix 1
Examples
8.1 The Z-Transform
8.1.1 Definition
∞
X ( z1 , z2 ) =
∞
∑ ∑ x (n , n ) z
1
2
− n1
1
z2− n2
n1 =−∞ n2 =−∞
8.1.2 Relationship to Discrete-Time Fourier Transform
∞
X ( z1 , z2 )
jω
jω
z1 = e 1 , z2 = e 2
=
∞
∑ ∑ x (n , n ) e
1
− jω1n1
2
e − jω 2n2 = X (ω 1 , ω 2 )
n1 =−∞ n2 =−∞
evaluated at z1 = e jω1 and z2 = e jω 2 .
8.1.3 Region of Convergence (ROC)
Points (z1,z2) for which
∑∑
n1
x(n1 , n2 ) z1
n2
are located in the ROC. This implies that
X ( z1 , z2 ) < ∞
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− n1
z2
− n2
<∞
If (z01,z02) point lies in the ROC, then all points (z1,z2) that satisfy
z1 ≥ z01 ,
z2 ≥ z02
also lie in the ROC.
For the first quadrant sequences, the boundary of the ROC must have nonpositive slope.
8.1.4 Sequences with Finite Support
M1
X ( z1 , z2 ) =
M2
∑ ∑ x (n , n ) z
1
2
− n1
1
z2− n2
n1 = N1 n2 = N2
The Z-transform converges for all finite values of z1 and z2, except possibly for z1 = 0 and z2 = 0.
8.1.5 Sequences with support on a wedge
If a sequence has support shown in Figure 8.1a, then its ROC is shown in Figure 8.1b (see Dudgeon
and Mersereau, 1984). If the point (z01,z02) belongs to the ROC, then
ln z1 ≥ ln z01
a
and
ln z2 ≥ Lln z1 + {ln z02 − Lln z01 }
b
ln|z2|
n2
Region of
support
slop = L
Region of
convergence
n1 = -Ln2
n1
ln|z1|
(z01, z02)
FIGURE 8.1
8.1.6 Sequences with Support on a Half-Plane
The boundary of the region of convergence is constrained to be a single-valued function of z1 (or
ln z1 ) (see Dudgeon and Mersereau, 1984).
8.1.7 Sequences with Support Everywhere
• x(n1,n2) = e − n1 −n2 converges for all values of (z1,z2)
n
n
• x(n1,n2) = 2 1 2 2 will not converge for any value of (z1,z2)
• Often the z-transform of a sequence with support everywhere will converge in a region of finite
area.
• A sequence with support everywhere can be split into four quadrant sequences, e. g.,
2
2
x(n1 , n2 ) = x1 (n1 , n2 ) + x 2 (n1 , n2 ) + x3 (n1 , n2 ) + x 4 (n1 , n2 )
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where
 x(n1, n2 )
 1 x (n , n )
2
1 2
x1 (n1, n2 ) =  1
x
n
(
1 , n2 )
4
0
for
for
for
for
n1 > 0,
n1 = 0,
n1 = n2
n1 < 0
n2 > 0
n2 > 0 or n1 > 0, n2 = 0
=0
or n2 < 0
Similarly are defined x2(n1,n2), x3(n1,n2), and x4(n1,n2), . The region of convergence of x(n1,n2) is the
intersection of the region of convergence of the four z-transforms of the four quadrants.
8.1.8 ROCs of Different Supports
Figure 8.2 shows the support of the function and its ROC (Lim, 1990).
8.2 Properties of the Z-Transform
8.2.1 Properties of the Z-Transform
TABLE 8.1 Properties of the 2-D Z-Transform
1.
x (n1 , n2 ) ←→ X ( z1 , z 2 ),
ROC : Rx
y(n1 , n2 ) ←→ Y ( z1 , z 2 ),
ROC : Ry
Linearity
ax (n1 , n2 ) + by(n1 , n2 ) ←→ aX ( z1 , z 2 ) + bY ( z1 , z 2 ),
2.
Convolution
x (n1 , n2 ) ∗ y(n1 , n2 ) =
∑ ∑ x (n − k , n
1
k1
3.
ROC : at least Rx ∩ Ry
1
2
− k 2 ) y(k1 , k 2 ) ←→ X ( z1 , z 2 ) Y ( z1 , z 2 ),
ROC : Rx ∩ Ry
k2
Separable Signals
x (n1 , n2 ) = x1 (n1 ) x 2 (n2 ) ←→ X ( z1 , z 2 ) = X1 ( z1 ) X 2 ( z 2 ),
ROC : z1 ∈ ROC X1 ( z1 ) and
z 2 ∈ ROC X 2 ( z 2 )
4.
Shift Property
x (n1 ± m1 , n2 ± m2 ) ←→ X ( z1 , z 2 ) = z1± m1 z 2± m2 X ( z1 , z 2 ) ,
ROC : Rx with possible exceptions z1 = 0, ∞
5.
z 2 = 0, ∞
Differentiation Property
− n1 x (n1 , n2 ) ←→ z1
∂ X ( z1 , z 2 )
,
∂z1
ROC : Rx
− n2 x (n1 , n2 ) ←→ z 2
∂ X ( z1 , z 2 )
,
∂z 2
ROC : Rx
n1 n2 x (n1 , n2 ) ←→ z1 z 2
6.
and
∂ 2 X ( z1 , z 2 )
,
∂z1 ∂z 2
ROC : Rx
Modulation Property
w(n1 , n2 ) = a n1 b n2 x (n1 , n2 ) ←→ X ( a −1 z1 , b −1 z 2 ), ROC : W ( z1 , z 2 ) has the same as X ( z1 , z 2 ),
but scaled by a in z1 variable and by b in the z 2 variable.
7.
Conjugate Properties
x (n1 , n2 ) = complex function ←→ X ( z1 , z 2 )
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TABLE 8.1 Properties of the 2-D Z-Transform (continued)
x ∗ (n1 , n2 ) ←→ X ∗ ( z1∗ , z 2∗ )
[ X ( z1 , z 2 ) + X ∗ ( z1∗ , z 2∗ ) ]
Re{x (n1 , n2 )} ←→
1
2
Im{x (n1 , n2 )} ←→
1
2j
[ X ( z1 , z 2 ) − X ∗ ( z1∗ , z 2∗ ) ]
ROC : same as X ( z1 , z 2 )
8.
Reflection Properties
x (n1 , n2 ) ←→ X ( z1 , z 2 )
x ( − n1 , n2 ) ←→ X ( z1−1 , z 2 )
x (n1 , − n2 ) ←→ X ( z1 , z 2−1 )
x ( − n1 , − n2 ) ←→ X ( z1−1 , z 2−1 )
9.
ROC : z1−1 , z 2−1
2
 1 
x (n1 , n2 ) y(n1 , n2 ) ←→ 

 2π j 
10.
in Rx
Multiplication Property
 zi
∫ ∫ X υ
C2 C1
,
1
z2 
dυ 1 dυ 2
Y ( z1 , z 2 )
υ 2 
υ1 υ 2
Parseval’s Theorem
∞
∞
 1 
2
 1
∑ ∑ x(n , n ) y (n , n ) ←→  2π j  ∫ ∫ X(z , z ) Y  z
∗
1
2
1
∗
2
1
n1 =−∞n2 =−∞
2
C2 C1
∗
1
,
1  dz1 dz 2
z 2∗  z1 z 2
Contours must: closed, counter - clockwise, encircle the origin, lie totally within ROC.
11.
Initial Value Theorems
x (n1 , n2 ) = 0
n1 < 0 , n2 < 0
lim X ( z1 , z 2 ) =
z1 →∞
∑ x(0, n ) z
2
− n2
2
n2
lim X ( z1 , z 2 ) =
z2 →∞
∑ x (n , 0) z
− n1
1
1
n1
lim X ( z1 , z 2 ) = x (0, 0)
z1 →∞
z2 →∞
12.
Linear Mapping of Variables
x (n1 , n2 ) = y( m1 , m2 )
n1 = I m1 + J m2 ,
n2 = Km1 + Lm2
I , J , K , L are integers
IL − KJ ≠ 0
X ( z1 , z 2 ) = Y ( z z , z z )
I
1
K
2
J
1
L
2
ROC : ( z1I z 2K , z1J z 2L ) in Rx
8.3. Inverse Z-Transform
8.3.1 Inverse Z-Transform
 1 
x(n1 , n2 ) = 

 2π j 
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2
∫ ∫ X(z , z ) z
1
C2 C1
2
n1 −1 n2 −1
1
2
z
dz1 dz2
FIGURE 8.2
C1, C2 both in the ROC
C1 counter-clockwise encircling the origin in the z1 plane with z2 fixed
C2 counter-clockwise encircling the origin in the z2 plane with z1 fixed
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8.4 System Function
8.4.1 System Function
∑ ∑ a(k , k ) y(n − k , n
1
k1
2
1
1
2
− k2 ) =
k2
∑ ∑ b( k , k ) x ( n − k , n
1
k1
2
1
1
2
− k2 )
k2
a(k1 , k2 ), b(k1 , k2 ) ≡ finite − extent sequences
∑ ∑ a( k , k ) z
Y ( z1 , z2 )
1
k1
2
− k1
1
∑ ∑ b(k , k ) z
z2− k2 = X ( z1 , z2 )
1
k1
k2
2
− k1
1
z2− k2
k2
or
∑ ∑ b(k , k ) z
Y (z , z )
H(z , z ) =
=
X(z , z )
∑ ∑ a(k , k ) z
1
1
1
2
k1
2
− k1
1
z2− k2
2
− k1
1
− k2
2
k2
2
1
2
1
k1
z
=
B( z1 , z2 )
A( z1 , z2 )
k2
8.5 Stability Theorems
8.5.1 Theorem 8.5.1.1 (Shanks, 1972)
Let H(z1,z2) = 1/A(z1,z2) be a first quadrant recursive filter. This filter is stable if, and only if, A(z1,z2) ≠
0 for every point (z1,z2) such that z1 ≥ 1 or z2 ≥ 1.
8.5.2 Theorem 8.5.1.2 (Shanks, 1972)
Let H(z1,z2) = 1/A(z1,z2) be a first quadrant recursive filter. Then H(z1,z2) is stable if, and only if, the
following conditions are true:
a) A( z1 , z2 ) ≠ 0 , z1 ≥ 1, z2 = 1
b) A( z1 , z2 ) ≠ 0 , z1 = 1, z2 ≥ 1
8.5.3 Theorem 8.5.1.3 (Huang, 1972)
Let H(z1,z2) = 1/A(z1,z2) be a first-quadrant recursive filter. The filter is stable if, and only if, A(z1,z2)
satisfies the following two conditions:
a) A( z1 , z2 ) ≠ 0 , z1 ≥ 1, z2 = 1
b) A( a, z2 ) ≠ 0 , z2 ≥ 1 for any a such that a ≥ 1
8.5.4 Theorem 8.5.1.4 (DeCarlo, 1977; Strintzis, 1977)
Let H(z1,z2) = 1/A(z1,z2) be a first quadrant recursive filter. The filter is stable if, and only if, A(z1,z2)
satisfies the following three conditions:
a) A( z1 , z2 ) ≠ 0 , z1 = 1, z2 = 1
b) A( a, z2 ) ≠ 0 , for z2 ≥ 1 for any a such that a = 1
c) A( z1 , b) ≠ 0 , for z1 ≥ 1 for any b such that b = 1
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References
DeCarlo, R., J. Murray, and R. Saeks, Multivariate Nyquist theory, Int. J. Control, 25, 657-75, 1977.
Dudgeon, Dan E. and Russell M. Mersereau, Multidimensional Digital Signal Processing, Prentice-Hall,
Inc., Englewood Cliffs, NJ, 1984.
Huang, Thomas S., Stability of two-dimensional recursive filters, IEEE Trans. Audio and Electroacoustics, AU-20, 158-63, 1972.
Lim, Jae S., Two-Dimensional Signal and Image Processing, Prentice-Hall Inc., Englewood Cliffs, NJ,
1990.
Shanks, John L., Sven Treitel, and James H. Justice, Stability and synthesis of two-dimensional recursive
filters, IEEE Trans. Audio and Electroacoustics, AU-20, 115-28, 1972.
Strintzis, Michael G., Test of stability of multidimensional filters, IEEE Trans. Circuits and Systems,
CAS-24, 432-37, 1977.
Appendix 1
Examples
1.1
Two-Dimensional Z-transforms
Example 8.1
The Z-transform of x(n1 , n2 ) = a n1 b n2 u(n1 , n2 ) = a n1 b n2 u(n1 ) u(n2 ) is
∞
X ( z1 , z2 ) =
∑
a n1 u(n1 ) z1− n1
n1 =−∞
=
∞
∑
b n2 u(n2 ) z2− n2 =
n2 =−∞
∞
∑
a n1 z1− n1
n1 = 0
∞
∑b
n2
z2− n2
n2 = 0
1
1
1 − az1−1 1 − bz2−1
with region of convergence (ROC) az1−1 < 1 and bz2−1 < 1, or z1 > a and z2 > b (see Figure 8.3).
n2
n2
u(n1)
n2
u(n2)
×
=
n1
n1
|z2|
ROC
|b|
|a|
FIGURE 8.3
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u(n1, n2)
|z1|
n1
Example 8.2
n
The Z-transform of x(n1 , n2 ) = a 1 δ(n1 − n2 ) u(n1 , n2 ) is
∞
X ( z1 , z2 ) =
∞
∑∑
a n1 u(n1 ) u(n2 ) δ(n1 − n2 ) z1− n1 z2− n2 =
n1 =−∞ n2 =−∞
∞
=
∑a
n2
∞
∞
∑ ∑a
n1 = 0
z1− n2 z2− n2 =
n2 = 0
n1
δ(n1 − n2 ) z1− n1 z2− n2
n2 = 0
1
1 − az1−1 z2−1
From last summation the ROC is az1−1 z2−1 < 1 or a < z1 z2 equivalently
ln a < ln z1 + ln z2
The function and its ROC are shown in Figure 8.4.
n2
n2
u(n1, n2)
δ(n1- n2)
n2
×
x(n1, n2)
=
n1
n1
n1
ln|z2|
ROC
slope=-1
ln|z1|
FIGURE 8.4
Example 8.3
The Z-transform of x(n1 , n2 ) = a n1 u(n1 , n2 ) u(n1 − n2 ) is
∞
X ( z1, z2 ) =
∞
∑ ∑a
n1
u(n1, n2 ) u(n1 − n2 ) z1− n1 z2− n2
n1 =−∞ n2 =−∞
∞
=
∞
∑∑
n1 = 0
n2 = 0
∞
=
a n1 u(n1 − n2 ) z1− n1 z2− n2 =
∑a
n1
n1 = 0
ROC : z1 > a ,
=
( )
1 − z2−1
n1 +1
−1
2
1− z
z1 z2 > a
The function and its ROC are shown in Figure 8.5.
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=
∞
∑ ∑a
n1 = 0
− n1
1
z
∞
z1− n1 z2− n2
n2 = 0
1
1 − az1−1z2−1
(1 − az )(
−1
1
n1
)
|z2|
n2
|z1| = a
x(n1, n2)
ROC
|z1| |z2| = |a|
n1
|a|
|z1|
FIGURE 8.5
Example 8.4 (inverse integration)
If X(z1,z2) has a region of convergence in the 2-D unit surface,
 1 
x(n1 , n2 ) = 

 2π j 
2
 1 
=

 2π j 
2
=
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2π j
(2 π j ) 2
∫∫
z1n1 z2n2
dz dz
1 − 12 z1−1 − 14 z2−2 1 2
∫∫
( z2 − 14 ) −1 z1n1 z2n2
dz dz
z1 − [ 12 z2 /( z2 − 14 )] 1 2
C2 C1
C2 C1
∫
C2
( z2 − 14 ) −1
(n + n2 )!
( 12 ) n1 z2n1 n2
z dz = ( 12 ) n1 ( 14 ) n2 1
,
n1! n2 !
( z2 − 14 ) n1 2 2
n1 , n2 ≥ 0