Digital Signal Processing, Fall 2006

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Digital Signal Processing, VI, Zheng-Hua Tan, 2006 1

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

System analysis MM1

MM2

MM6 Sampling and

reconstruction MM5

System structure System

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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

1 1

0 z a

az z b z b

H >

−

= + ₋⁻

] 1 [ ]

[ ]

[n =b₀a u n +b₁a ⁻¹u n−

h ⁿ ⁿ

] 1 [ ] [ ]

1 [ ]

[n −ay n− =b₀x n +b₁x n− y

System implementation

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

An implementation is a specific description of its internal computational structure

The choice of an implementation concerns with

computational requirements

memory requirements,

effects of finite-precision,

and so on

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Part I: Block diagram representation

Block diagram representation of computational structures

Signal flow graph description

Basic structures for IIR systems

Transposed forms

Basic structures for FIR systems

System implementation

Impulse response

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

However, linear constant-coefficient difference ]

[

* ] [ ] [

] 1 [ ]

[ ]

[ ₀ ₁ ¹

n h n x n y

n u a b n u a b n

h ⁿ ⁿ

=

− +

= ⁻

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Basic elements

Implementation based on the recurrence formula derived from difference equation requires

adders

multipliers

memory for storing delayed sequence values

Fig. 6.1

] 1 [ ] [ ] 1 [ ]

[n =ay n− +b₀x n +b₁x n− y

Example of block diagram representation

A second-order difference equation

2 2 1 1

0

) 1

( ₋ ₋

−

= −

z a z a z b

H

] [ ] 2 [ ]

1 [ ]

[n a₁y n a₂y n b₀x n

y = − + − +

Demonstrates the complexity, the steps, the amount of resources required.

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General Nth-order difference equation

∑

=

−

=

−

= _N

k

k k M

k

k k

z a

z b z

H

1 0

1 ) (

∑

= =

− +

−

= ^M

k k N

k

ky n k b x n k

a n

y

0 1

] [ ]

[ ]

[

] [n v

A cascade of two systems!

X[n]Æv[n], v[n]Æy[n]

Rearrangement of block diagram

A block diagram can be rearranged in many ways without changing overal function, e.g. by reversing the order of the two cascaded systems.

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System function decomposition

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

⎟⎠

⎜ ⎞

⎝

=⎛

=

⎟⎠

⎜ ⎞

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

=

−

=

∑ ∑

∑

=

= −

−

=

−

=

−

=

−

=

−

N

k

k k M

k

k k

M

k

k N k

k

k k N

k

k k M

k

k k

z a z

b z

H z H

z b z

a z

H z H z

a z b z

H

1 0

2 1

0 1

1 2

1 0

1 ) 1

( ) (

1 ) 1 ( ) ( 1

) (

] [n v

] [n w

In the time domain

⎪⎪

⎩

⎪⎪⎨

⎧

+

−

=

−

=

− +

−

=

∑

=

] [ ] [ ]

[

] [ ]

[

] [ ]

[ ]

[

1 0

0 1

n v k n y a n

y

k n x b n

v

k n x b k

n y a n

y

N

k k M

k k

M

k k N

k k

⎪⎪

⎩

⎪⎪⎨

⎧

−

=

+

−

=

∑

=

] [ ]

[

] [ ] [ ]

[

0 1 M

k k N

k k

k n w b n

y

n x k n w a n

w

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Minimum delay implementation

One big difference btw the two implementations concerns the number of delay elements

) , max(N M

M N +

Direct form I and II

Direct form I as shown in Fig. 6.3

A 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

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An example

Direct form I and direct form II implementation

2 1

1

9 . 0 5

. 1 1

2 ) 1

( ₋ ₋

−

+

−

= +

z z

z z H

Part II: Signal flow graph description

Transposed forms

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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 represent both branching points and adders (depending on the number of

incoming branches), while in the diagram a special symbol is used for adders and a node has only one incoming

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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 solve

By z-transform Ælinear equations

] [ ] [ ] [

] 1 [ ] [

] [ ] [ ] [

] [ ] [

] [ ] [ ] [

4 2

3 4

2 3

1 2

4 1

n w n w n y

n w n w

n x n w n w

n w n w

n x n w n w

+

=

−

=

+

=

−

= α

] 1 [ ]

[ ₃

4 n =w n−

w

From flow graph to system function

⎪⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

+

=

+

=

−

=

−

) ( ) ( ) (

) ( )

(

) ( ) ( ) (

) ( )

(

) ( ) ( ) (

4 2

3 1 4

2 3

1 2

4 1

z W z W z Y

z W z z W

z X z W z W

z W z W

z X z W z W

α

⎪⎪

⎩

⎪⎪

⎨

⎧

+

=

+

=

−

=

−

) ( ) ( ) (

)) ( ) ( ( ) (

4 2

2 1 4

4 2

z W z W z Y

z X z W z z W

z X z W z

W α

] [ ]

1 [ ]

[ ) 1 (

) 1 (

) (

1 1

n u n

u n

h

z z z

H

z z X z z

Y

n

n− +

−

=

−

= −

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

−

= −

α α

α α α

α

? is system the

real, is

Ifα All-pass

Causal!

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From flow graph to system function

Fig. 6.13

Fig. 6.12

Different implementations, different amounts of computational resources

Part III: Basic structures for IIR systems

Transposed forms

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Direct form I

∑

=

−

=

−

= _N

k

k k M

k

k k

z a

z b z

H

1 0

1 ) (

∑

= =

− +

−

= ^M

k k N

k

ky n k b x n k

a n

y

0 1

] [ ]

[ ]

[

Direct form II

∑

=

−

=

−

= _N

k

k k M

k

k k

z a

z b z

H

1 0

1 ) (

∑

= =

− +

−

= ^M

k k N

k

ky n k b x n k

a n

y

0 1

] [ ]

[ ]

[

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Example

2 1

125 . 0 752 . 0 1

2 ) 1

( ₋ ₋

−

+

−

+

= +

z z

z z z

H

Cascade form

Factor the numerator and denominator polynomials

∏

∑

=

−

=

−

=

−

=

−

=

−

=

−

=

−

=

2 1

1

* 1 1

1 1

1

* 1 1

1

1 0

) 1

)(

1 ( ) 1 (

) 1

)(

1 ( ) 1

( 1

)

( _N

k

k k

N

k

k M

k

k k

M

k

k N

k k k M

k k k

z d z

d z

c

z g z

g z

f A

z a

z b z

H

∏

= − −

−

+

= ^N^s +

k k k

k k

k

z a z a

z b z b z b

H

1 2

2 1 1

2 2 1 1 0

) 1 (

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An example: from 2

^nd

-order to 1st-order cascade

) 25 . 0 1 ( ) 5 . 0 1 (

) 1 ( ) 1 ( 125

. 0 752 . 0 1

2 ) 1

( ₁ ₁

1 1

2 1

−

⋅

−

+

⋅

= + +

−

+

= +

z z

z z z

H

Parallel form by partial fraction expansion

∑

= − −

−

= −

=

−

− + −

+ −

=

2

1

* 1

1

1 1

0

) 1

)(

1 (

) 1 (

N

k k

k N

k k

k N

k k k

z d z d

z e B

z c z A

C z H

k k p

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Feedback in IIR systems

Feedback loop: a closed path Necessary but not sufficient condition for IIR system (Feedback introduced poles could be cancelled by zeros)

All loops must contain at least one unit delay element

1 1

2 2

1 1 ) 1

( ₋ ⁻

− = +

−

= − az

az z z a

H ]

[n u aⁿ

) 1 /(

] [ ] [

] [ ] [ ] [

a n x n y

n x n ay n y

−

=

+

=

Part IV: Transposed forms

Signal flow graph description

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)

1 1

) 1

( ₋

= − z az H

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

Signal flow graph description

Transposed forms

Direct form

So far, system function has both poles and zeros.

FIR systems as a special case.

Causal FIR system function has only zeros (except for poles as z=0)

⎩⎨

⎧ =

= 0, otherwise ,..., 1 , 0 ] ,

[ b n M

n

h ⁿ

∑

=

−

= ^M

k

kx n k b

n y

0

] [ ]

[

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Cascade form

Factoring the polynomial system function

∑ ∏

=

−

=

− = + +

= ^M^s

k

k k

k M

n

n b b z b z

z n h z

H

1

2 2 1 1 0 0

) (

] [ )

(

Linear-phase FIR systems

Generalized linear-phase system

Causal FIR systemshave generalized linear-phase if h[n] satisfies the symmetry condition

M n

n h n M h

M n

n h n M h

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

or

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

=

−

=

−

=

−

∑

=

−

= ^M

k

kx n k b

n y

0

] [ ]

[ constants

real are and

, of function real

a is ) (

) ( ) (

β α

ω ω

β ωα ω

ω j

j j j j

e A

e e A e

H = ⁻ ⁺

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Linear-phase FIR systems

] 2 / [

] / [ ]) [

] [ ](

[ ]

[

] [ ] [

if

] [

] 2 / [

] / [ ] [ ] [

] [ ] [ ]

2 / [

] / [ ] [ ] [

] [ ] [ ]

[

integer even

an is if

1 2 /

0

1 2 /

0 1

2 /

0

1 2 / 1

2 /

0 0

M n x M k h k M n x k n x k h n

y

n h n M h

k M n x k M h M

n x M k h k n x k h

k n x k h M

n x M k h k n x k h

k n x k h n

y M

M

k

M

k M

k

M

M k M

k M

k

− +

+

− +

−

=

−

+

−

− +

−

=

− +

−

=

−

=

∑

−

=

−

=

−

=

+

=

−

=

Linear phase FIR systems

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

] 2 / [

] / [ ]) [

] [ ](

[ ]

[

1 2 /

0

M n x M k h k M n x k n x k h n

y

M

k

− +

+

− +

−

=

∑

⁻

=

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Discussions

Implementation of FIR and IIR systems

Use signal block diagram flow graph representation to show the computational structures

Although 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

Transposed forms

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Course at a glance

Discrete-time signals and systems

Fourier-domain representation

DFT/FFT

System analysis

Filter design

z-transform MM1

MM2

MM9, MM10 MM3

MM6

MM4

MM7, MM8 Sampling and

reconstruction MM5

System structure System