Unformatted text preview: The target of statespace realization for these problems is to determine eﬃcient computations, where the
eﬃciency of the computations is measured in terms of the number of multiplications (arithmetic cost)
and in terms of the number of registers (delays), which denotes the memory cost of a computation. As
discussed earlier, truly timeinvariant systems are associated with matrices, which
1. are inﬁnite dimensional
2. exhibit Toeplitz structure.
Checking Equation 8 reveals that the matrix T neither has Toeplitz structure nor is it inﬁnite dimensional.
Hence it must correspond to a linear timevarying system. However, we can still interpret the columns of
T as timevarying impulse responses. Direct Realization
We proceed to draw a direct realization for the given matrix T as a timevarying system. As a start we
redraw the statespace realization for one time step k as is shown in Figure 2. The signal ﬂow graph on Dk yk xk+1 Bk uk Z Ak
Ck uk xk Bk xk xk+1 Ak
Ck Z Dk yk
Abbildung 2: Redrawing and simpliﬁcation
righthand side of Figure 2 is the basic building block of a timevarying system. Based on this buildingblock, the system that realizes the transfer function in Equation 8 is shown in Figure 3. The signal ﬂow
inherently depicts causality by its unidirectional signal ﬂwo, i.e. all the arrows strictly point from top to
bottom and from left to right. Other properties that can be observed are that this simple system uses
6 registers and 6 (nontrivial) multipliers for the realization of our given matrix T (adders are typically
not accounted for such complexity estimates). We denote the realization matrix for an indivdual block 8 Lecture 3 u1 u3 u2 u4
x4 x3
x2
y1 Z Z
Z 1/2 y2 1/4 Z 1/3 Z 1/12 Z 1/6 1/24 y3 Abbildung 3: Direct implementation of the matrix T in Equation 8 y4 9 Lecture 3 at timeindec k by
Bk
Dk Ak
Ck Σk = . (9) For the direct realization shown in Figure 3 we can write down the individual realization matrices Σk by
inspection as
Σ1
Σ2 Σ3 Σ4 = ·1
·1 1
0
1/2 1
0
0
1/6 = = = 1/24 (10)
0
1
1 0
1
0
1/3 ·
1/12 (11) 0
0 1
1 (12) ·
1 1/4 , (13) which we can combine to specify the timevarying realization matrix
A
C Σ= B
D where the corresponding blockdiagonal matrices are given as [·] 1 [A1 ] 0 [A2 ]
=
10
A= [A3 ] 0 1 [ A4 ] 00 [B1 ] [B2 ] =
B= [B3 ] [B4 ] C= D= [C1 ] [C2 ]
[C3 ]
[C4 ]
[D1 ] = [D2 ]
[D3 ]
[D4 ] [1] 0
1 [·] = 0
1
1 [·] [·] [1/2]
1/6 1/3 [1]
[1]
[1]
[1] . 1/24 1/12 Here and in the following a ‘[·]’ as an entry represents a zerodimensional matrix. 1/4 10 Lecture 3 Alternative Realization
Theocratically speaking an inﬁnite number of realizations are possible for a given transfer operator T . In
reality only a few alternates will be of interest. For example, we consider the implementation as shown in
Figure 4 that realizes the same transfer function as in Equation 8, but enables this with 3 registers (half
the number as used by the straightforward realization of Figure 3) and 5 multiplications. Although the
simpliﬁcation might not look signiﬁcant at the moment, its worth considering t...
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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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