Note7 s2z Transform

Dr. Hongjiang Song, Arizona State UniversityDr. Hongjiang Song, Arizona State University1

EEE598D: Analog Filter & SignalEEE598D: Analog Filter & SignalProcessing CircuitsProcessing Circuits

Instructor:Dr. Hongjiang Song

Department of Electrical EngineeringArizona State University


Thursday February 7, 2002

Today: s- to z- Transformations• Standard s-z- Transformation• Matched s- z- Transformation• Numerical s-z- Transformations


S- To Z- Transformations

• Approximation of differential equation byappropriate difference equation

• Why?– Take advantages of continuous-time system design

techniques– Signal processing systems are often specified in

continuous-time domain– Even though they may be realized in sampled-data

systems.


General Approaches

• Standard (or Impulse Invariant) Approach• Numerical Approximation Approaches


Standard S-Z- Transformation

HA(s)

H (z)

Sampling

Sampling

δ(t)

y1(nT)

y2(nT)

,,,2,1,0)()( 21 == nnTynTyIf

Then H(z) is the standard s-z- transform (or equivalent) of HA(s)


Standard S-Z- Transformation

• For standard s-z- transform, we have

• Or for multiple poles :

• Standard s-z- transform is also called impulseinvariant transform

11)()( −−−=⇒

+=

zeAzH

psAsH pTA

)1

1()!1(

)1()()(

)( 11

11

−−−

−−

−−−

=⇒+

=zedp

dn

AzHpsAsH pTn

nn

nA


Matched-z Transformation

1

1

11

−−

−−

−⇒+

−⇒+

zezszeps

zT

pT

• It is closely related to standard s-z-transform• It simply maps the s-domain pole and zero

to z-domain by:

∏

∏

∏

∏

=

−−

=

−−

=

=

−

−=⇒

+

+= N

k

Tp

M

k

Tz

N

kk

M

kk

A

ze

zeKzH

ps

zsCsH

k

k

1

1

1

1

1

1

)1(

)1()(

)(

)()(


Forward Euler Transformation

• Forward Euler Integration Rule

tnT (n+1)T

f(t)f(nT)

)()1(

)(

)(

)())1(()(

)(1)(

)()(

)1(

zFzTzY

TnTf

nTyTnydttf

sFs

sY

dttfty

Tn

nTt

t

−≈⇒

⋅≈

−+=

=⇒

=

∫

∫

+

=error

(1)

(2)


Forward Euler Transformation


• An approximation of integration

sTzorzzT

s+=

−= −

−

11

11

1


Backward Euler Transformation

• Backward Euler Integration Rule

tnT (n+1)T

f(t) f(nT)

)()1(

)(

))1((

)())1(()(

)(1)(

)()(

)1(

zFzTzzY

TTnf

nTyTnydttf

sFs

sY

dttfty

Tn

nTt

t

−≈⇒

⋅+≈

−+=

=⇒

=

∫

∫

+

=

error

(3)

(4)


Backward Euler Transformation

• Backward Euler Integration Rule


sTzor

zT

s −=

−= − 1

11

111


Bilinear Transformation

• Bilinear Integration Rule

tnT (n+1)T

f(t)f(nT)

)()1(2)1()(

2)())1((

)())1(()(

)(1)(

)()(

)1(

zFzzTzY

TnTfTnf

nTyTnydttf

sFs

sY

dttfty

Tn

nTt

t

−+

≈⇒

⋅++

≈

−+=

=⇒

=

∫

∫

+

=error

(5)

(6)

f((n+1)T)


Bilinear Transformation



21

21

11

21

1

1

sT

sT

zorzzT

s −

+=

−+

= −

−


LDI Transformation

• LDI (midpoint) Integration Rule

tnT (n+1)T

f(t)f((n+1/2)T)

)()1(2)()(

))2/1((

)())1(()(

)(1)(

)()(

2/1

)1(

zFzzTzY

TTnf

nTyTnydttf

sFs

sY

dttfty

Tn

nTt

t

−≈⇒

⋅+≈

−+=

=⇒

=

∫

∫

+

=error

(5)

(6)


LDI Transformation


))2

(12

(1

1 21

2/1

sTsTzorz

zTs

+±=−

= −

−


Multi-Step Integration Transformation

• Multi-Step Integration Rule

tnT (n+1)T

f(t)

)()1(

)(

/))/1((

)())1(()(

)(1)()()(

0

0

/

00

)1(

zFcz

zcTzzY

cTTmknfc

nTyTnydttf

sFs

sYdttfty

m

km

m

k

mkm

m

km

m

km

Tn

nTt

t

∑

∑

∑∑

∫

∫

=

=

==

+

=

−≈⇒

⋅−+≈

−+=

=⇒=

error

(7)

(8)



• We have

• Let (slower clock) we have:• Multi-Step Transformation:

)1(1

10

/

−=

−

−≈∑

z

zcT

s

m

k

mkm

mTT ⇒

m

mm

m

kk

zzczcc

c

mTs −

−−

=

−+++

=

∑ 1...1 1

10

0



• Special cases:– Simpson’s Rule:

• m =2, c0 = c2 =1/3, c1 = 1/4

– Tick’s Rule:• m =2, c0 = c2 =0.3584, c1 = 1.2832

– ...


ψ Transformation

• Although z=eiωT appears quite natural. It isoften more convenient to use another complexvariable as intermediate step

sT

sT

ezsTor

ezzz

−−

−−−

−

=+−

==

=+−

=

ψψ

ψ

ψ

11),

2tanh(

,11

1

11

1

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Note7 s2z Transform