Page 175 - Electrical Engineering Dictionary

2-D polynomial matrices A(z 1 ,z 2 ), B(z 1 , and
z 2 ) are called factor right (left) coprime if h
det B ˆ n 2 +ˆa n 2 −1 B ˆ n 2 −1 + ···
their greatest common right (left) divisor
is a unimodular matrix U(z 1 ,z 2 ) (nonzero i
ˆ
+ˆa 1 B +ˆa 0 I m 2 6= 0
detU(z 1 ,z 2 ) ∈ R).
2-D polynomial matrices A ∈ R p×m or
[z 1 ,z 2 ], B ∈ R p×m [z 1 ,z 2 ] (p+q ≥ m ≥ 1) ¯ ¯ m 1 −1
are called zero right coprime if there exists a det A ¯ m 1 + b m 1 −1 A + ···

pair (z 1 ,z 2 ) which is a zero of all m × m ¯ ¯ ¯ 6= 0
h i + b 1 A + b 0 I n 1
minors of the matrix A . The minor co-
B and
primeness of two 2-D polynomial matrices h
ˆ
implies their factor coprimeness. det A ˆ m 2 + b m 2 −1 A ˆ m 2 −1 + ···
i
ˆ ˆ
ˆ
coprimeness of 2-D polynomials a math- + b 1 A + b 0 I n 2 6= 0
ematical relationship of interest in control
systems. where
A 2-D polynomial  0 1 0 ... 0 
 0 0 1 ... 0 
n 2 n 1  
X X ¯
i j A =  ... ... ... ... ... 
p (z 1 ,z 2 ) = a i (z 1 ) z = a j (z 2 ) z  
2 1  0 0 0 ... 1 
i=0 j=0
−¯a 0 −¯a 1 −¯a 2 ... −¯a n 1 −1
is called primitive if a i (z 1 ), i = 1, 2,...,n 2  
0 1 0 ... 0
and a j (z 2 ), j = 1, 2,...,n 1 are coprime
 0 0 1 ... 0 
(have not a common factor). Two primitive  
ˆ
A =  ... ... ... ... ... 
2-D polynomials are called factor coprime if  
 0 0 0 ... 1 
their greatest common divisor is a constant.
The primitive 2-D polynomials −ˆa 0 −ˆa 1 −ˆa 2 ... −ˆa n 2 −1
 
0 1 0 ... 0
(z 2 ) ¯a (z 1 ,z 2 )
a (z 1 ,z 2 ) = a n 1  0 0 1 ... 0 
 
¯
(z 1 ) ˆa (z 1 ,z 2 ) B =  ... ... ... ... ... 
= a n 2
 
 0 0 0 ... 1 
¯
(z 2 ) b (z 1 ,z 2 )
b (z 1 ,z 2 ) = b m 1
¯ ¯ ¯ ¯
−b 0 −b 1 −b 2 ... −b m 1 −1
ˆ
(z 1 ) b (z 1 ,z 2 )
= b m 2  
0 1 0 ... 0
¯ a (z 1 ,z 2 ) = z n 1 +¯a n 1 −1 z n 1 −1 + ···  0
1 1 0 1 ... 0 
 
ˆ
+¯a 1 z 1 +¯a 0 (¯a i =¯a i (z 2 )) B =  ... ... ... ,,, ... 
 
n 2 −1  0 0 0 ... 1
n 2
ˆ a (z 1 ,z 2 ) = z +ˆa n 2 −1 z + ··· 
2 2
ˆ
ˆ
ˆ
ˆ
−b 0 −b 1 −b 2 ... −b m 2 −1
+ˆa 1 z 2 +ˆa 0 ˆ a i =ˆa i (z 1 )
¯
b (z 1 ,z 2 ) = z m 1 + b m 1 −1 z m 1 −1 + ···
¯
1 1
coprocessor a processor that is connected
¯ ¯ ¯ ¯
+ b 1 z 1 + b 0 b i = b i (z 2 ) to a main processor and operates concur-
b (z 1 ,z 2 ) = z m 2 + b m 2 −1 z m 2 −1 + ··· rently with the main processor, although un-
ˆ
ˆ
2 2
der the control of the main processor. Copro-
ˆ ˆ ˆ ˆ
+ b 1 z 2 + b 0 b i = b i (z 1 ) cessors are usually special-purpose process-
ing units, such as floating point, array, DSP,
are factor coprime if and only if or graphics data processors.

det B ¯ n 1 +¯a n 1 −1 B ¯ n 1 −1 + ···
copy-back in cache systems an opera-

¯
+¯a 1 B +¯a 0 I m 1 6= 0 tion that is the same as write-back—a write
c
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