algorithmic s-z transformations for continuous-time to discrete- time filter conversion d. biolek,...
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ALGORITHMIC S-Z ALGORITHMIC S-Z TRANSFORMATIONS FOR TRANSFORMATIONS FOR CONTINUOUS-TIME TO CONTINUOUS-TIME TO DISCRETE-TIME FILTER DISCRETE-TIME FILTER CONVERSIONCONVERSION
D. Biolek, V. Biolkova
Brno University of TechnologyCzech Republic
http://www.vabo.cz/stranky/biolek
Generalized Pascal matrix: Using this matrix, the coefficients of transfer functions of the continuous-time and discrete-time linear circuits can be converted on the assumption that both circuits are related by a general first-order s-z transformation. Effective numerical procedure of computing all matrix elements for arbitrary first-order s-z transformation.
IntroductionIntroduction
approximate
z = f(s)
exact
sTez
K(s)X(s) Y(s)
j Im{s}
Re{s}0
K(z)X(z) Y(z)
j Im{z}
Re{z}0
s-z
S-Z TransformationsS-Z Transformations
S-Z TransformationsS-Z Transformations
,
0
0
M
i
ii
M
i
ii
zs
s
ssfz
,
0
0
M
i
ii
M
i
ii
sz
z
zzfs
j Im{s}
Re{s}0
j Im{z}
Re{z}0 1
s-z
10
0
1
0
0
M
ii
M
ii
S-Z TransformationsS-Z Transformations
M- order of s-z transformation (normally M=1)First-order linear s-z transformation:
N
k
kk
N
k
kk
DT
zb
zazK
0
0
N
k
kk
N
k
kk
CT
sd
scsK
0
0s-z
One-to-one linear correspondence
{ak} {ck} and {bk} {dk}
S-Z TransformationsS-Z Transformationslinear first-orderlinear first-order
BILINEAR (BL)1
1
z
zas
sa
saz
FT
a 22
BACKWARD DIFFERENCE (BD)z
zbs
1
sb
bz
F
Tb
1
FORWARD DIFFERENCE (FD) 1 zbsb
sz 1 F
Tb
1
PARAMETRIC BD-BLrz
z
T
rs
11
sTr
srTrz
1
1
….. 1,0r
S-Z TransformationsS-Z Transformationslinear first-orderlinear first-order
GENERAL First-Order (GFO)
BL
BD
BD-BL
wsu
vsuz
wzv
zus
1
FD
1,1, wvau
1,0, wvbu
0,1, wvbu
u, v, w
1,,1
wrvT
ru
…..
Preview GFO
Preview BD-BL
wzv
zus
1
S-Z TransformationsS-Z Transformationslinear first-orderlinear first-order
N
k
kk
N
k
kk
DT
zb
zazK
0
0
N
k
kk
N
k
kk
CT
sd
scsK
0
0
s-z
{ak} {ck} (and {bk} {dk}):
TNT
N ccccaaaa 210210 , CA
TNNN
NNN ucucucuc 22
110
~C
MAC ~
Generalized Pascal matrixGeneralized Pascal matrix
.,1,0,
,
0
Nik
wvM nknk
nki nk
in
iN
S-Z TransformationsS-Z TransformationsGeneralized Pascal matrix (GPM)Generalized Pascal matrix (GPM)
NN wv
wv
wv
wv
N
N
N
N
NN
NN
NN
33
22
11
33
22
11
11111
M
S-Z TransformationsS-Z TransformationsGeneralized Pascal matrix (GPM)Generalized Pascal matrix (GPM)
111111
531135
10222210
10222210
531135
111111
BL 1,1, wvau
Example: N=5
S-Z TransformationsS-Z TransformationsGeneralized Pascal matrix (GPM)Generalized Pascal matrix (GPM)
Example: N=5
BD 1,0, wvbu
100000
510000
1041000
1063100
543210
111111
S-Z TransformationsS-Z TransformationsGeneralized Pascal matrix (GPM)Generalized Pascal matrix (GPM)
Example: N=5
FD 0,1, wvbu
000001
000015
0001410
0013610
012345
111111
S-Z TransformationsS-Z TransformationsGeneralized Pascal matrix (GPM)Generalized Pascal matrix (GPM)
Example: N=5
BD-BL 1,,1
wrvT
ru
1
541322345
1064361634610
1046633616410
542332415
111111
2345
232434
232323
2222
rrrrr
rrrrrrrr
rrrrrrrrr
rrrrrrrr
rrrrr
S-Z TransformationsS-Z TransformationsAlgorithmic compilation of GPMAlgorithmic compilation of GPM
.,1,0,
,
0
Nik
wvwvM kknknk
nki k
i
k
iN
nk
i
n
iN
Ni
wDFTvDFTMDFT kkkii k
i
k
iN
,..,1,0
,.
M
… vector of DFT coefficients of i-th column of the GPM.
S-Z TransformationsS-Z TransformationsAlgorithmic compilation of GPMAlgorithmic compilation of GPM
1. i = 0.
2. Compiling vectors ai and bi of a size 1 x (N+1) according to the rule
N
iN
i wN
iw
iw
iw
ibv
N
iNv
iNv
iNv
iNa ,,
2,
1,
0,,,
2,
1,
0210210
3. Completing both vectors by as much zeros as needed to reach their length to integer power of two.
New vectors are obtained.'' , ii ba
S-Z TransformationsS-Z TransformationsAlgorithmic compilation of GPMAlgorithmic compilation of GPM
4. Performing FFT of vectors :, ''ii ba
5. Multiplying vectors above element per element and performing inverse FFT:
'''' , iiii bFFTBaFFTA
'''iiki BAIFFTM
6. First N+1 elements give the i-th column of the GPM.
7. i = i+1; if i<=N then go to the step 2 else end.
S-Z TransformationsS-Z TransformationsAlgorithmic compilation of GPM - exampleAlgorithmic compilation of GPM - example
Example: Parametric BD-BL, r=0.5, N=5.
Calculation of GPM in MATLAB:
1. i = 0;
2. ai = [1 2.5 2.5 1.25 0.3125 0.03125]; bi = [1 0 0 0 0 0];
3. = [1 2.5 2.5 1.25 0.3125 0.03125 0 0];'ia'ib = [1 0 0 0 0 0 0 0];
S-Z TransformationsS-Z TransformationsAlgorithmic compilation of GPM - exampleAlgorithmic compilation of GPM - example
4.
5-6.
)( ''ii aFFTA
7.59375 1.54929-5.12955i -1.1875-1.28125i -0.17429-0.12955i 0.03125 0.17429+0.12955i -1.1875+1.28125i 1.54929+5.12955i
)( ''ii bFFTB
1 1 1 1 1 1 1 1
1 2.5 2.5 1.25 0.3125 0.03125 0 0
)( '''iiki BAIFFTM % i =0
kiM
ConclusionsConclusions
• The proposed algorithm generates all elements of GPM of arbitrary first-order s-z transformation.
• The described approach is based on the FFT algorithm, which ensures the computation efficiency.
• The structure of this procedure is not dependent on the type of s-z transformation.
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