algorithmic s-z transformations for continuous-time to discrete- time filter conversion d. biolek,...

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


,

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


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


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


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


111111

531135

10222210

10222210

531135

111111

BL 1,1, wvau

Example: N=5


Example: N=5

BD 1,0, wvbu

100000

510000

1041000

1063100

543210

111111


Example: N=5

FD 0,1, wvbu

000001

000015

0001410

0013610

012345

111111


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.


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


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.

algorithmic s-z transformations for continuous-time to discrete- time filter conversion d. biolek,...

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