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Unformatted text preview: ted versions of the blockdiagonal matrices
3 (ZA) = ZAZ T Z 2 AZ 2T Z 3 AZ 3T Z 3 = A(1) A(2) A(3) Z 3 .
A(1) A(2) A(3) This notional simpliﬁcation leads to another notational simpliﬁcation
A[n] := A(1) A(2) . . . A(n) ,
such that we can rewrite the expression for (ZA)n as
n (ZA) = A(1) A(2) . . . A(n) Z n = A[n] Z n
A[n] Finally, after all these notional modiﬁcations we can write out the series expansion
−1 (1 − ZA) = 1 + A[1] Z 1 + A[2] Z 2 + A[3] Z 3 + . . . , (7) which looks structurally very similar to the conventional form of the geometric series, in spite of the
elements in the series being blockdiagonal matrices. 5 Lecture 3 In the following we will use this modiﬁed von Neuman expansion (7) to further move on with producing
the Toeplitz operator T . The ﬁrst term in the expansion then looks like
A[1] Z 1 = A(1) Z 1 = = .. .
A−1 A0
A1 .. = .. . ..
. . . .. · .. . 0
1 0
1 0
1 ..
.. . .
0
A− 1 0
A0 0
A1 0
..
. .. . Following this recipe we can write up the second term of the series expansion more explicitely as
A[2] Z 2 = A(1) A(2) Z 2 = = = .. .
A−1 A0
A1
.. .. . ..
. · A− 1 A− 2 .. A−2 .
A0 A−1 A1 A0 0
0
A0 A−1 0
0
A1 A0 0
0
..
. A−1 .. 0
..
. .. . . A0 .. 0
0
1 0
0
1 0
0
..
. . ... .. .
· .. . . ... .. .
· .. . . .. . A−1 A−2
= = .. . ..
. 0
0
1 0
0
1 0
0
..
. .. . ..
. . 6 Lecture 3 Adding up the ﬁrst three terms of the series expansion produces the lowertriangular tridiagonal matrix ..
. .. . 1 . .. A1
1 [1] 1
[2] 2
1+A Z +A Z =
.
A2 A1
A2
1 A3 A2
A3
1 .. .
A4 A3 A4 ..
..
.
.
Just to make the construction principle more visible I add the forth term
matrix ..
.
. ..
1 ..
.
A1
1 1 + A[1] Z 1 + A[2] Z 2 + A[3] Z 3 = . .
.
A2 A1
A2
1 .. . A3 A2 A1
A3 A2
A3 0
A4 A3 A2 A4 A3 ..
..
.
.
Taking this intermediate result we can ﬁnally generate the Toeplitz ..
. ..
.
..
1 . . ..
C0
A1
1 .
C1
. .
A2 A1
A2
1 C2 . ..
..
A3 A2 A1
A3 A2
A3
. A4 A3 A2 A1 A4 A3 A2 A4 A3 ..
..
.
. of the series to produce the 1
A4
..
. 1
..
. .. . . operator according to (4) 1
A4
..
. 1
..
. .. . .. . .. . 0
B −1 0
B0 0
B1 Example: Finite Matrix
We now want to use a simple example to demonstrate the inner workings of the statespace computations.
To this end, let us consider the example of the ﬁnite dimensional 4 × 4 (nonToeplitz) matrix 1 1/2 1
,
T =
(8) 1/6 1/3
1
1/24 1/12 1/4 1
which we will interpret as a transfer operator of a simple timevarying system. The matrix corresponds
to a timevarying system since the matrix does not exhibit Toeplitz structure and because the matrix is 0
..
. .. . . 7 Lecture 3 ﬁnite dimensional. Furthermore, T is lowertriangular, indicating the system to be causal. Here T is a
4 × 4 matrix, however, real systems of interest are signiﬁcantly larger, typical of the order n ∼ 10, 000. In
the following discussion the term signal is used interchangeably with vector, likewise the term system is
used for matrix. Similar to our discussion about timeinvariant systems, we strive to represent the matrix
T in terms of the linear fractional map
−1 T = D + C (1 − ZA) ZB ....
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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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