View
218
Download
1
Category
Preview:
DESCRIPTION
DSP-II p. 3 Version Lecture-3 Filter Realization Lecture 3 : Filter Realizations FIR Filter Realizations IIR Filter Realizations PS: We will assume real-valued filter coefficients
Citation preview
p. 1DSP-II
Digital Signal Processing II
Lecture 3: Filter Realization
Marc MoonenDept. E.E./ESAT, K.U.Leuven
marc.moonen@esat.kuleuven.behomes.esat.kuleuven.ac.be/~moonen/
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 2
PART-I : Filter Design/Realization
• Step-1 : define filter specs (pass-band, stop-band, optimization criterion,…)
• Step-2 : derive optimal transfer funcion FIR or IIR • Step-3 : filter realization (block scheme/flow graph)
direct form realizations, lattice realizations,… • Step-4 : filter implementation (software/hardware)
accuracy issues, …
question: implemented filter = designed filter ?
Lecture-2
Lecture-3
Lecture-4
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 3
Lecture 3 : Filter Realizations
• FIR Filter Realizations
• IIR Filter Realizations
PS: We will assume real-valued filter coefficients
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 4
FIR Filter Realizations
FIR Filter Realization =Construct (realize) LTI system (with delay elements,
adders and multipliers), such that I/O behavior is given by..
Several possibilities exist…1. Direct form2. Transposed direct form 3. Lattice (LPC lattice)4. Lossless lattice5. Frequency-domain realization: see Part-2
][....]1[.][.][ 10 Nkubkubkubky N
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 5
FIR Filter Realizations
1. Direct formu[k]
u[k-4]u[k-3]u[k-2]u[k-1]
xbo
+
xb4
xb3
+
xb2
+
xb1
+y[k]
][....]1[.][.][ 10 Nkubkubkubky N
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 6
FIR Filter Realizations
2. Transposed direct form Starting point is direct form :
`Retiming’ = select subgraph (shaded) remove delay element on all inbound arrows add delay element on all outbound arrows
u[k]
u[k-4]u[k-3]u[k-2]u[k-1]
xbo
+
xb4
xb3
+
xb2
+
xb1
+y[k]
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 7
FIR Filter Realizations
`Retiming’ : results in...
u[k]
u[k-1]
xbo
+
xb1
+y[k]
u[k-3]u[k-2]
xb4
xb3
+
xb2
+
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 8
FIR Filter Realizations
`Retiming’ : repeated application results in...
i.e. `transposed direct form’
u[k]
xbo
+y[k]
xb1
+
xb2
+
xb3
+
xb4
(=different software/hardware, same i/o-behavior)
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 9
FIR Filter Realizations
3. Lattice form : Derived from combined realization of
with `flipped’ version of H(z)
Reversed (real-valued) coefficient vector results in...
i.e. - same magnitude response - different phase response
][....]1[.][.][~ :)(.)(~01
1 NkubkubkubkyzHzzH NNN
212)(...)(~).(~)(~
jjj ezezezzHzHzHzH
][....]1[.][.][ :)( 10 NkubkubkubkyzH N
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 10
FIR Filter Realizations
Hence starting point is…
u[k] u[k-1]
u[k-2]
xb1
+
xb2
+
x
+
x
+
b3
u[k-3]
xb3
+
b2 x
+
xbo
+
y[k]
b4x
+
u[k-4]
xb4b1 x bo
y[k]~
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 11
FIR Filter Realizations
With , this is rewritten as
20
10332
0
20222
0
3011,00
0123
32101
0
0
0123
3210
0
0
1' ,
1' ,
1' '
]1[]...[.''''''''
.0
01.
11
][]...[.''''00''''
11
][~][
bbbbbbbbbbb
Nkukubbbbbbbb
z
Nkukubbbb
bbbbkyky
T
T
TNkukubbbbbbbbbb
kyky
][]...[.][~][
01234
43210
1 , 00
40 bb
`sloppy notation’
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 12
FIR Filter RealizationsThis is equivalent to...
…Now repeat procedure for shaded graph (=same structure as the one we started from)
y[k]
u[k] u[k-3]
u[k-2]
xb’o
+
xb’1
+
x
+
x
+
b’3
u[k-2]
xb’2
+
b’2 x
+
b’3
xb’1 x b’o
+
+
x
xko
y[k]~
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 13
FIR Filter Realizations
Repeated application results in `lattice form’
u[k]
y[k]
+
+
x
xko
+
+
x
xk1
+
+
x
xk2
+
+
x
xk3
xbo
y[k]~
= different software/hardware, same i/o-behavior
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 14
FIR Filter Realizations
Lattice form : • Also known as `LPC Lattice’ (`linear predictive coding lattice’) Ki’s are so-called `reflection coefficients’ Every set of bi’s corresponds to a set of Ki’s, and vice versa.
• Procedure for computing Ki’s from bi’s corresponds to the well-known `Schur-Cohn’ stability test (from control theory):
problem = for a given polynomial B(z), how do we find out if all the zeros of B(z) are stable (i.e. lie inside unit circle) ? solution = from bi’s, compute reflection coefficients Ki’s (following procedure on previous slides). Zeros are stable if and only if all reflection coefficients statisfy |Ki|<1
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 15
FIR Filter Realizations
Lattice form :
• Procedure breaks down if |Ki|=1 is encountered. Means at least one root of B(z) lies on or outside the unit circle (cfr Schur-Cohn). Then have to select other realization (direct form, lossless lattice, …) for B(z).
• Lattice form is often applied to `minimum phase’ filters, i.e. filter with only stable zeros (roots of B(z) strictly inside unit circle). Then design procedure never breaks down.
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 16
FIR Filter Realizations
4. Lossless lattice : Derived from combined realization of (possibly scaled) H(z)
with which is such that
(*)
PS : interpretation ?… (see next slide)
][.~~...]1[.
~~][.~~][~~ :)(
~~10 NkubkubkubkyzH N
][....]1[.][.][ :)( 10 NkubkubkubkyzH N
1)(~~).(
~~)().( 11 zHzHzHzH
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 17
FIR Filter RealizationsPS : Interpretation ? When evaluated on the unit circle, this formula is equivalent to (for filters with real-valued coefficients)
i.e. and are `power complementary’ (= form a 1-input/2-output `lossless’ system, see also below)
1)(~~)(
22
j
j
ezez
zHzH
)(~~ zH)(zH
2
)(~~ jeH
2)( jeH
1
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 18
FIR Filter RealizationsPS : How is computed ?
Note that if `a’ is a root of R(z), then `1/a’ is also a root of R(z), hence
Note that ai’s can be selected such that all of them lie inside the unit circle. Then is a minimum-phase FIR filter.
This is referred to as spectral factorization, =spectral factor.
)(~~ zH
)(
11 )().(1)(~~).(
~~
zR
zHzHzHzH
iii azazzHzH ))(()(
~~).(~~ 11
iiazzH )()(
~~ 1
)(~~ zH
)(~~ zH
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 19
FIR Filter Realizations
Hence starting point is…
u[k] u[k-1]
u[k-2]
xb1
+
xb2
+
x
+
x
+
u[k-3]
xb3
+
x
+
xbo
+
y[k]
x
+
u[k-4]
xb4
x
y[k]~
0
~~b 1
~~b 2
~~b 4
~~b3
~~b
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 20
FIR Filter Realizations
From (*) (page 16), it follows that
Hence there exists a theta_0 such that
T
T
Nkukubbbb
bbbbz
Nkukubbbb
bbbbkyky
]1[]...[.'
~~'~~'
~~'~~
''''.
001
.cossinsincos
][]...[.'
~~'~~'
~~'~~0
0''''cossinsincos
][~~][
3210
32101
00
00
3210
3210
00
00
TNkukubbbbb
bbbbbkyky
][]...[.~~~~~~~~~~][~~][
43210
43210
4
4
0
04040 ~~~~ 0
~~.~~.
b
b
b
bbbbb
`sloppy notation’
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 21
FIR Filter Realizations
This is equivalent to...
…Now repeat procedure for shaded graph (=same structure as the one we started from!)
(why?)
u[k] u[k-3]
u[k-2]
xb’o
+
xb’1
+
x
+
x
+
u[k-2]
xb’2
+
x
+
b’3
xx
y[k]
0cos
+
+
xx
xx
0sin
0cos
y[k]~~
0sin
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 22
FIR Filter Realizations
Repeated application results in `lossless lattice’
u[k]
x
3sin
2cos
+
+
xx
xx
2sin
1cos
+
+
xx
xx
1sin
3cos
+
+
xx
xx
3cos2cos1cos
2sin1sin 3sin
y[k]
0cos
+
+
xx
xx
0sin
0cos
y[k]~~
0sin
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 23
FIR Filter Realizations
Lossless lattice :
• also known as `paraunitary lattice’ (see also Lecture 6)
• each section is based on an orthogonal transformation, which preserves norm/energy/power
(= forms a 2-input/2-output `lossless’ system) hence overall structure preserves the input energy, i.e. is `lossless’ (see also next slides)
22
21
22
21
2
1
2
1 )()()()(.cossinsincos
OUTOUTININININ
OUTOUT
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 24
FIR Filter Realizations
• State-space description Example: 2nd-order system:
- R is `realization matrix’ (A-B-C-D-matrix) for 1-input/2-output system - Orthogonality of R implies `losslessness’ (see next slide)
2
][][2
][1
2
][~~][
]1[2
]1[1
2
1
11
11
00
00
2
1
hence ,2
1 if 100010001
.
][][][
.
010cos.0sinsin.0cos
00
.
cossin00sincos0000100001
][~~][
]1[]1[
kukxkx
kykykxkx
T
R
RR
kukxkx
kykykxkx
~
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 25
FIR Filter Realizations
• Orthogonality of R implies `losslessness’ : assume initial internal state is x1[0], x2[0] (2nd order example) input sequence is u[0],u[1],u[2],…,u[L] corresponding output sequences are y[0],y[1],… and y[0],y[1],... then orthogonality implies that `energy’ in initial states + inputs = `energy’ in final states + outputs
• Equivalent transfer function based property is… i.e. H(z) is `embedded’ in 1-input/2-output lossless system (see page 17)
][~~...]0[
~~][...]0[]1[]1[
][...]1[]0[]0[]0[22222
221
22222
21
LyyLyyLxLx
Luuuxx
1)(~~)(
22
j
j
ezez
zHzH
~~~~
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 26
FIR Filter Realizations
PS : Relevance of lattice realizations : robustness example : o = original transfer function + = transfer function after 8-bit truncation of lattice filter parameters (angles theta) - = transfer function after 8-bit truncation of direct-form coefficients (bi’s)
PS : A 2x2 orthogonal transformation (rotation) may be implemented in hardware based on a so-called `CORDIC’ (`COR’ for CO-ordinate Rotation’) architecture, which is more efficient (and more robust) than a
multiply-add based implementation
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 27
IIR Filter Realizations
Construct LTI system such that I/O behavior is given by..
Several possibilities exist… 1. Direct form 2. Transposed direct form 3. Lattice-ladder form 4. Lossless lattice 5. Parallel and cascade realization 6. State space realization
NN
NN
zazazbzbb
zAzBzH
...1...
)()()( 1
1
110
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 28
IIR Filter Realizations
1. Direct form Starting point is… u[k]
xbo
xb4
xb3
xb2
xb1
+ +++y[k]
+ +++
x-a4
x-a3
x-a2
x-a1
)(1zA
)(zB
NN
NN
zazazbzbb
zAzBzH
...1...
)()()( 1
1
110
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 29
IIR Filter Realizationswhich is equivalent to...
PS : If all a_i=0 (hence H(z) is FIR), then this reduces to a direct form FIR
xbo
xb4
xb3
xb2
xb1
+ +++y[k]
u[k] + +++
x-a4
x-a3
x-a2
x-a1
Direct form for B(z)
)(1zA
)(zB
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 30
IIR Filter Realizations -State-space description:
-N delay elements (=minimum number =max of numerator and denominator polynomial order = max(p,q)) -2N+1 multipliers, 2N adders
u[k]
xbo
xb4
xb3
xb2
xb1
+ +++y[k]
+ +++
x-a4
x-a3
x-a2
x-a1
x1[k] x2[k] x3[k] x4[k]
][.
][][][][
.....][
][.
0001
][][][][
.
010000100001
]1[]1[]1[]1[
0
4
3
2
1
044033022011
4
3
2
13321
4
3
2
1
kub
kxkxkxkx
babbabbabbabky
ku
kxkxkxkxaaaa
kxkxkxkx
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 31
IIR Filter Realizations
2. Transposed direct form Starting point is…
+ +++
x-a4
x-a3
x-a2
x-a1
)(1zA
y[k]
u[k]
xbo
xb4
xb3
xb2
xb1
+ +++
)(zB
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 32
IIR Filter Realizations
which is equivalent to...
)(1zA
)(zB
x-a4
x-a3
x-a2
x-a1
y[k]
u[k]
xbo
xb4
xb3
xb2
xb1
+ +++
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 33
IIR Filter Realizations
Transposed direct form is obtained after retiming ...
PS : If all a_i=0, then this reduces to a transposed direct form FIR
Transposeddirect form B(z)
x-a4
x-a3
x-a2
x-a1
y[k]
u[k]
xbo
xb4
xb3
xb2
xb1
+ +++
)(zB
)(1zA
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 34
IIR Filter Realizations
- State-space description
i.e. direct forms state space matrices are `transpose’ of each other (which justifies the name `transposed dir.form’).
][.
][][][][
.0001][
][.
.
.
..
][][][][
.
000100010001
]1[]1[]1[]1[
0
4
3
2
1
044
033
022
011
4
3
2
1
4
3
2
1
4
3
2
1
kub
kxkxkxkx
ky
ku
babbabbabbab
kxkxkxkx
aaaa
kxkxkxkx
u[k]
x-a4
x-a3
x-a2
x-a1
y[k]
xbo
xb4
xb3
xb2
xb1
+ +++x1[k] x2[k] x3[k] x4[k]
T
DCBA
DCBA
formdirect TransposedformDirect
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 35
IIR Filter Realizations
3. Lattice-ladder form Derived from combined realization of
with...
- numerator polynomial is denominator polynomial with reversed coefficient vector (see also page 9)
- hence is an `all-pass’ (=`SISO lossless’) filter :
NN
NN
zazazbzbb
zAzBzH
...1...
)()()( 1
1
110
NN
NNN
zazazzaa
zAzAzH
...1
.1...)()(~
)(~1
1
11
1)(~)(
)(~2
22
j
j
j
ez
ezez zA
zAzH
)(~ zH
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 36
IIR Filter RealizationsStarting point is…
u[k]
+ +
x xb1
+
b2 x
+
b0x b3 x b4x xa3 xa2 xa1 x1
xa1 xa2 xa3 xa4
+ + ++
+ + ++
y[k]y[k]~
a4
x[k]
-
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 37
IIR Filter Realizations
This is equivalent to…
+ +
x x
+
x xx x x x
+ + +
x’[k-1]
x
xu[k]
b0
y[k]
y[k]~
x
+
x
+
0cos
4.411.43
aaaaa
4.412.42
aaaaa
4.413.41
aaaaa
1
4.411.01aaabb
4.412.02aaabb
4.413.03aaabb
4.414.04aaabb
+
x
x x x
+ ++ -
4.413.41
aaaaa
4.412.42
aaaaa
4.411.43
aaaaa
0sin
0sin
0cos
040sin ka
(prove it!)
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 38
IIR Filter Realizations
Right-hand part has the same structure as what we started from, hence repeated application leads to `lattice-ladder form’
u[k]
y[k]y[k]~
x
x
x
x
+
x
+
0cos
+
x
0cos
0sin
0sin
b0 =los
sles
s se
ctio
n
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 39
IIR Filter Realizations
Lattice-Ladder form :
• Ki’s are `reflection coefficients’• Procedure for computing Ki’s (=sin(theta_i) !!) from ai’s again
corresponds to `Schur-Cohn’ stability test (see lecture-2): all zeros of A(z) are stable (i.e. lie inside unit circle) iff all reflection coefficients statisfy |Ki|<1 (i=1,…,N-1) (ps: procedure breaks down if |Ki|=1 is encountered)• Orthogonal transformations correspond to `lossless’ sections
see next slide…
22
21
22
21
2
1
2
1 )()()()(.cossinsincos
OUTOUTININININ
OUTOUT
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 40
IIR Filter Realizations• State-space description Example: 2nd-order system:
- R is `realization matrix’ (A-B-C-D-matrix) for u[k] -> y[k] (SISO) - R is orthogonal matrix (R^T.R=I), which implies `losslessness’ , i.e.
][][][
.][
and
][][][
.1000sincos0cossin
.sincos0cossin0
001
][~]1[]1[
2
1
012
2
1
11
11
00
002
1
kukxkx
ky
kukxkx
kykxkx
R
~
1)(~ 2
jezzH
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 41
IIR Filter Realizations
PS : `All-pass’ part (u[k]->y’[k]) is known as `Gray-Markel’ structure
PS : Note that the all-pass part corresponds to A(z) (i.e. N angles theta_i correspond to N coefficients a_i), while the ladder section corresponds to B(z). If all a_i=0 (hence H(z) is FIR !), then all theta_i=0, hence the all-pass part reduces to a delay line, and the lattice-ladder form reduces to a direct-form FIR.
x
x
x
+
0cos
+
x
0cos
0sin
0sin
in
out
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 42
IIR Filter Realizations
4. Lossless-lattice : Derived from combined realization of
with...
- numerator polynomial is C(z) is such that
i.e. and are `power complementary’ (p.17-18)
NN
NN
zazazbzbb
zAzBzH
...1...
)()()( 1
1
110
NN
NN
zazazczcc
zAzCzH
...1
....)()()(
~~1
1
110
)(~~ zH )(zH
)().()().()().(
1)(~~).(
~~)().(111
11
zAzAzCzCzBzB
zHzHzHzH
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 43
IIR Filter Realizations
`Lossless-lattice’ : similar derivation (but more complicated -hence skipped) leads to…
PS: Gray-Markel is a special case of this (ki’=0)
x
x
x
x
+
+
x
u[k]
y[k]y[k]~
x
x
x
+
x +
x=l
ossl
ess
sect
ion
'0
'0 sink
'0cos
'0cos
'0sin
0cos
0cos
00 sink
0sin
N~sin
N~cos
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 44
IIR Filter Realizations
• Orthogonal transformations correspond to `lossless sections
• State-space description with orthogonal realization matrix R (try it) implies overall lossless characteristic
• Note that 2N+1 filter coefficients (a_i’s and b_I’s) define 2N+1 rotation angles. If all a_i=0 (hence H(z) is FIR), rotation angles will be such that all feedback paths are cancelled (in a magic way)
Example: (draw it/try it!)
23
22
21
23
22
21
3
2
1
3
2
1
)()()()()()(
.'cos0'sin
010'sin0'cos
.1000cossin0sincos
OUTOUTOUTINININ
INININ
OUTOUTOUT
62,
21,
31
21
21)(
~~21
21)( '
1'00
11 kkkzzHzzH
1)(~~)(
22
j
j
ezez
zHzH
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 45
IIR Filter Realizations
5. Parallel & Cascade Realization: Parallel real. obtained from partial fraction decomposition, e.g. for simple poles:
– similar for the case of multiple poles– each term realized in, e.g., direct form– same number of delay elements, additions and multiplications (if
p=q) as direct forms– transmission zeros are realized iff signals from different sections
exactly cancel out. =Problem in finite word-length implementation
21
121
1
11
110 11)(
)()(N
i N
iN
i i
i
zzz
zc
zAzBzH
u[k]
y[k]
+
+
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 46
IIR Filter Realizations
Cascade realization obtained from pole-zero factorization of H(z) e.g. for N even:
– similar for N odd– each section realized in, e.g., direct form– second-order sections are called `bi-quads’– same number of delay elements, additions and multiplications as
direct forms
2/
121
21
0 ..1..1.
)()()(
N
i ii
ii
zzzzb
zAzBzH
u[k]
y[k]
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 47
IIR Filter Realizations
• Cascade realization is non-unique: - multiple ways of pairing poles and zeros - multiple ways of ordering sections in cascade
• ``Pairing procedure’’ : - pairing of little importance in high-precision (e.g. floating point implementation), but important in fixed-point implementation (with short word-lengths) - principle = pair poles and zeros to produce a frequency response for each section that is as flat as possible (i.e. ratio of max. to min. magnitude response close to unity) - obtained by pairing each pole to a zero as close to it as possible - procedure : start with pole pair nearest to the unit circle, and pair this to nearest complex zeros. remove pole-zero pair, and repeat, etc….
u[k]
y[k]
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 48
IIR Filter Realizations 6. State-space Realization :
• State-space description:
• State-space realization = realization such that all the elements of A,B,C,D are the multiplier coefficients in the structure
• Example: (transposed) direct form is NOT a state-space realization, cfr. elements (bi-bo.ai) in B or C
][.][.][][.][.]1[
kuDkxCkykuBkxAkx
+
+
CBy[k]
u[k]
D
A
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 49
IIR Filter Realizations
• Example: bi-quad `coupled realization’ - direct form realization of bi-quad may be unsatisfactory, e.g. for short word-lengths and poles near z=1 or z=-1 (see lecture-4) - alternative realization:
state space description :
][][][
.][
][.01
][][
.]1[]1[
2
1
2
1
2
1
kukxkx
ky
kukxkx
kxkx
21
21
..1..1)(
zzzzzH
ii
iii
poles : .j
+
+ +
-
y[k]
u[k]
DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 50
IIR Filter Realizations • State-space description/realization is non-unique: any non-singular matrix T transforms A,B,C,D into an alternative state-space
description/realization
with
• State-space realization with orthogonal realization matrix is referred to as `orthogonal filter’ :
Relevance : see lecture-4
][.~][~.~][
][.~][~.~]1[~
kuDkxCky
kuBkxAkx
][.][~100
..100
~~~~
1
1
kxTkx
TDCBAT
DCBA
IRRRR
DCBA
R
TT
..
Recommended