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Unformatted text preview: hat for large systems with
n ∼ 10, 000 and given that r << n, the complexity of implementation could be reduced to O(8 · r · n)
compared to O((1/2)n2 ) for a straightforward implementation. However, the question is how can we u1 u3 u2
x2 1/2 1/3 Z 1/3 y1 x3 u4 1/4
1/4 Z x4 Z y3 y2 y4 Abbildung 4: Simpliﬁed implementation of the matrix T in Equation 8
tell if implementation complexity reduction is possible for a given system, and how to determine the
minimalcomplexity implementation? To answer this, we take a closer look at the implementations. For
the system in Figure 4 we can read oﬀ the individual realization matrices
· 1/2
·1 Σ1 = Σ2 = 1/3
1 1/3
1 1/4
1 Σ3 = 1/4
1 Σ4 = ·
1 ·
1 , which we can decompose and reassemble to form the timevarying realization matrix
A
C Σ= B
D with the corresponding blockdiagonal matrices [·]
[ A1 ] [1/3]
[A2 ]
=
A = [A3 ]
[A4 ] B = [B1 ] [B2 ]
[B3 ]
[B4 ] = [1/4]
[·] [1/2] , [1/3]
[1/4]
[·] , (14) 11 Lecture 3 C = [C1 ]
[C2 ]
[C3 ]
[C4 ] [D1 ] D = [·] = = [D2 ]
[D3 ]
[D4 ] , [1]
[1] [1] [1] . [1]
[1]
[1] Realization of Inverse Operator
Based on the realization given in the previous section, we can determine a realization Γ, which implements
the inverse operator T −1 . We write the realization of the inverse operator as
ˆˆ
AB
ˆD
Cˆ Γ= We can determine
ˆ
Ak
Γk =
ˆk
C . the individual realization matrices Γk as
−
−
ˆ
Bk
Ak − Bk · Dk 1 · Ck Bk · Dk 1
=
−1
−1
ˆ
−Dk · Ck
Dk
Dk . (15) For the in system depicted in Equation 14, the inverse realization can be done by using Equation 15, and
is given as
Γ1 = ˆ
A= ˆ
B= ˆ
C= ˆ
D= ·
· 1/2
1 ˆ
[A1 ] ˆ
[B1 ] ˆ
[C1 ] ˆ
[D1 ] Γ2 = ˆ
[A2 ] ˆ
[B2 ] ˆ
[C2 ] ˆ
[D2 ] ˆ
[A3 ] ˆ
[B3 ] ˆ
[C3 ] ˆ
[D3 ] 0
−1 ˆ
[A4 ] ˆ
[B4 ] ˆ
[C4 ] 1/3
0 1/4
Γ3 =
1
−1 1 [·] [1/3]
= [1/4] = = Γ4 = [·] [1/2] [1/3]
[1/4]
[·] = [1]
[1]
[1] [1]
[1] [·] ·
−1 ·
1 (16) [1]
ˆ
[1]
[D4 ]
The realization is depicted in Figure 5. The inverse transfer function can be readoﬀ as 1
0
−1 −1/2 1
.
ˆ
ˆ
ˆ
ˆ
T −1 = D − C 1 − Z A
ZB = 0 −1/3
1
0
0
−1/4 1 (17) 12 Lecture 3 y1 1/3 1/2 Z y4
1/4 Z
−1 u1 y3 y2 Z
−1 u2 −1 u3 Abbildung 5: Inverse realization of the system u4 13 Lecture 3 Eﬃcient Methods for Solving Linear Systems of Equations
In a more general setting we consider the problem of solving the set of linear equations T · u = y , for the
variable u where here the matrix T is not assumed to have any special structure. The variable u takes
on the interpretation of an input signal applied to the input of a linear timevariant system, which we
describe by means of the matrix T . The signal y is the corresponding output signal computed by a linear
matrixvector multiplication. In many technical situations we may need to invert this map. We can solve
the inversion problem by determining the inverse of T , which we may compute using standard linear
algebra tools such as Gaussian elimination or similar. However, for a general matrix T this inversion
process costs O(n3 ) operations, if n denotes the size of the matrix n.
Similar to the strategy of using Fast Fourier techniques...
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This note was uploaded on 07/07/2013 for the course EI 2012 taught by Professor Tum during the Winter '12 term at TU München.
 Winter '12
 TUM

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