digital signal processing, fall 2006 - aalborg universitetkom.aau.dk/~zt/courses/dsp/mm6/digital...

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Digital Signal Processing, VI, Zheng-Hua Tan, 20061

Digital Signal Processing, Fall 2006

Zheng-Hua Tan

Department of Electronic Systems Aalborg University, Denmark

[email protected]

Lecture 6: System structures for implementation


Course at a glance

Discrete-time signals and systems

Fourier-domain representation

DFT/FFT

Systemanalysis

Filter design

z-transform

MM1

MM2

MM9, MM10MM3

MM6

MM4

MM7, MM8

Sampling andreconstruction MM5

Systemstructure

System

2


System implementation

LTI systems with rational system function e.g.

Impulse response

Linear constant-coefficient difference equation

Three equivalent representations!How to implement, i.e. convert to an algorithm or

structure?

|||| ,1

)( 1

110 az

azzbbzH >

−+

= −

−

]1[][][ 110 −+= − nuabnuabnh nn

]1[][]1[][ 10 −+=−− nxbnxbnayny



The input-output transformation x[n] y[n] can be computed in different ways – each way is called an implementation

An implementation is a specific description of its internal computational structureThe choice of an implementation concerns with

computational requirementsmemory requirements,effects of finite-precision,and so on

3


Part I: Block diagram representation

Block diagram representation of computational structuresSignal flow graph descriptionBasic structures for IIR systems Transposed formsBasic structures for FIR systems



Impulse response

is infinite-duration, impossible to implement in this way.

However, linear constant-coefficient difference equation provides a means for recursive computation of the output

][*][][]1[][][ 1

10

nhnxnynuabnuabnh nn

=−+= −

]1[][]1[][]1[][]1[][

10

10

−++−=−+=−−

nxbnxbnaynynxbnxbnayny

4


Basic elements

Implementation based on the recurrence formula derived from difference equation requires

addersmultipliersmemory for storing delayed sequence values

Fig. 6.1

]1[][]1[][ 10 −++−= nxbnxbnayny


Example of block diagram representation

A second-order difference equation

22

11

0

1)( −− −−=

zazabzH

][]2[]1[][ 021 nxbnyanyany +−+−=

Demonstrates thecomplexity, the steps, the amount of resourcesrequired.

5


General Nth-order difference equation

∑

∑

=

−

=

−

−= N

k

kk

M

k

kk

za

zbzH

1

0

1)(

∑∑==

−+−=M

kk

N

kk knxbknyany

01

][][][

][nv

A cascade of two systems!X[n] v[n], v[n] y[n]


Rearrangement of block diagram

A block diagram can be rearranged in many wayswithout changing overal function, e.g. by reversingthe order of the two cascaded systems.

6


System function decomposition

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−⎟⎠

⎞⎜⎝

⎛==

⎟⎠

⎞⎜⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−==

−=

∑∑

∑∑∑

∑

=

−=

−

=

−

=

−

=

−

=

−

N

k

kk

M

k

kk

M

k

kkN

k

kk

N

k

kk

M

k

kk

zazbzHzH

zbza

zHzH za

zbzH

1

021

0

1

12

1

0

1

1)()(

1

1)()(1

)(

][nv

][nw


In the time domain

⎪⎪⎩

⎪⎪⎨

⎧

+−=

−=

−+−=

∑

∑

∑∑

=

=

==

][][][

][][

][][][

1

0

01

nvknyany

knxbnv

knxbknyany

N

kk

M

kk

M

kk

N

kk

⎪⎪⎩

⎪⎪⎨

⎧

−=

+−=

∑

∑

=

=

][][

][][][

0

1M

kk

N

kk

knwbny

nxknwanw

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Minimum delay implementation

One big difference btw the two implementationsconcerns the number of delay elements

),max( MNMN +


Direct form I and II

Direct form I as shown in Fig. 6.3A direct realization of the difference equation

Direct form II or canonic direct form as shown in Fig. 6.5

There is a direct link between the system function (difference equation) and the block diagram

8


An example

Direct form I and direct form II implementation

21

1

9.05.1121)( −−

−

+−+

=zz

zzH


Part II: Signal flow graph description


9


Signal flow graph (SFG)

As an alternative to block diagrams with a few notational differences.A network of directed branches connecting nodes.

variable


Signal flow graph Nodes in SFG representboth branching points and adders (dependingon the number ofincoming branches), while in the diagram a special symbol is usedfor adders and a node has only one incomingbranch.

SGF is simpler to draw.

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From flow graph to system function

Fig. 6.12

Not a direct form, cannot obtain H(z) by inspection.But can write an equation for each node

involve feedback, difficult to solveBy z-transform linear equations

][][][]1[][

][][][][][

][][][

42

34

23

12

41

nwnwnynwnw

nxnwnwnwnw

nxnwnw

+=−=+=

=−=

α

]1[][ 34 −= nwnw



⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

+==

+==

−=

−

)()()()()(

)()()()()(

)()()(

42

31

4

23

12

41

zWzWzYzWzzW

zXzWzWzWzW

zXzWzWα

⎪⎪⎩

⎪⎪⎨

⎧

+=+=

−=

−

)()()())()(()(

))()(()(

42

21

4

42

zWzWzYzXzWzzW

zXzWzW α

][]1[][1

)(

)(1

)(

11

1

1

1

1

nununhz

zzH

zXz

zzY

nn +−

−

−

−

−

−−=−−

=

⎟⎟⎠

⎞⎜⎜⎝

⎛

−−

=

αααα

αα

? is system thereal, is If α All-pass

Causal!

11



Fig. 6.13

Fig. 6.12

Different implementations, different amounts of computational resources


Part III: Basic structures for IIR systems

Block diagram representation of computational structuresSignal flow graph descriptionBasic structures for IIR systemsTransposed formsBasic structures for FIR systems

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Direct form I

∑

∑

=

−

=

−

−= N

k

kk

M

k

kk

za

zbzH

1

0

1)(

∑∑==

−+−=M

kk

N

kk knxbknyany

01

][][][


Direct form II

∑

∑

=

−

=

−

−= N

k

kk

M

k

kk

za

zbzH

1

0

1)(

∑∑==

−+−=M

kk

N

kk knxbknyany

01

][][][

13


Example

21

21

125.0752.0121)( −−

−−

+−++

=zz

zzzH


Cascade form

Factor the numerator and denominator polynomials

∏∏

∏∏

∑

∑

=

−−

=

−

=

−−

=

−

=

−

=

−

−−−

−−−=

−=

21

21

1

1*1

1

1

1

1*1

1

1

1

0

)1)(1()1(

)1)(1()1(

1)( N

kkk

N

kk

M

kkk

M

kk

N

k

kk

M

k

kk

zdzdzc

zgzgzfA

za

zbzH

∏=

−−

−−

−−++

=sN

k kk

kkk

zazazbzbb

zH1

22

11

22

110

1)(

14


An example: from 2nd-order to 1st-order cascade

)25.01( )5.01( )1( )1(

125.0752.0121)( 11

11

21

21

−−

−−

−−

−−

−⋅−+⋅+

=+−++

=zz

zzzz

zzzH


Parallel form by partial fraction expansion

∑

∑∑

=−−

−

=−

=

−

−−

−+

−+=

2

1

11*1

1

11

0

)1)(1()1(

1)(

N

k k

k

N

k k

kN

k

kk

zdzdzeB

zcA

zCzH

k

k

p

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Feedback in IIR systems

Feedback loop: a closed pathNecessary but not sufficient condition for IIR system (Feedback introduced polescould be cancelled by zeros)

All loops must contain at leastone unit delay element

11

22

11

1)( −−

−

+=−−

= azaz

zazH

][nuan

)1/(][][][][][

anxnynxnayny

−=+=


Part IV: Transposed forms


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Transposed form for a first-order system

Flow graph reversal or transposition also provides alternatives: reversing the directions of all branches and reversing the input and output

Resulting in same H(z)

111)( −−

=az

zH


Transposed direct form II and direct form II

The transposed direct form II implements the zeros first and then the poles, being important effect for finite-precision existing

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Part V: Basic structures for FIR systems



Direct form

So far, system function has both poles and zeros. FIR systems as a special case.Causal FIR system function has only zeros (exceptfor poles as z=0)

Form I and form II are the same.⎩⎨⎧ =

=otherwise ,0

,...,1,0 ,][

Mnbnh n∑

=

−=M

kk knxbny

0

][][

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Cascade form

Factoring the polynomial system function

∏∑=

−−

=

− ++==sM

kkkk

M

n

n zbzbbznhzH1

22

110

0

)(][)(


Linear-phase FIR systems

Generalized linear-phase system

Causal FIR systems have generalized linear-phaseif h[n] satisfies the symmetry condition

MnnhnMh

MnnhnMh

,...,1,0 ],[][ or

,...,1,0 ],[][

=−=−

==−

∑=

−=M

kk knxbny

0

][][

constants real are and , offunction real a is )(

)()(

βαωω

βωαωω

j

jjjj

eA

eeAeH +−=

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Linear-phase FIR systems

]2/[]/[])[][]([][

][][ if

][][]2/[]/[][][

][][]2/[]/[][][

][][][

integereven an is if

12/

0

12/

0

12/

0

12/

12/

0

0

MnxMkhkMnxknxkhny

nhnMh

kMnxkMhMnxMkhknxkh

knxkhMnxMkhknxkh

knxkhny

M

M

k

M

k

M

k

M

Mk

M

k

M

k

−++−+−=

=−

+−−+−+−=

−+−+−=

−=

∑

∑∑

∑∑

∑

−

=

−

=

−

=

+=

−

=

=


Linear phase FIR systems

M is an even integer and h[M-n]=h[n]

]2/[]/[])[][]([][12/

0MnxMkhkMnxknxkhny

M

k−++−+−= ∑

−

=

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Discussions

Implementation of FIR and IIR systemsUse signal block diagram flow graph representation to show the computational structuresAlthough two structures may have equivalent input-output charateristics for infinite-precision represenations of coefficients and variables, they may have dramatically different behaviour when the numerical precision is limited.


Summary


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Course at a glance

Discrete-time signals and systems

Fourier-domain representation

DFT/FFT

Systemanalysis

Filter design

z-transform

MM1

MM2

MM9, MM10MM3

MM6

MM4

MM7, MM8

Sampling andreconstruction MM5

Systemstructure

System

digital signal processing, fall 2006 - aalborg universitetkom.aau.dk/~zt/courses/dsp/mm6/digital...

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