Lecture Digital signal processing: Lecture 6 - Zheng-Hua Tan

Lecture 6 - System structures for implementation presents the following content: Block diagram representation of computational structures, signal flow graph description, basic structures for IIR systems, transposed forms, basic structures for FIR systems.

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

Fourier-domain

representation

System analysis

MM1

MM2

MM6

Sampling and reconstruction MM5

System structure

System

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3

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

)

1 1

az

z b b z

[]

n u a b n u a b n

]1[][]

1[]

The input-output transformation x[n] Æ y[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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5

Part I: Block diagram representation

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[]

[]

n h n x n

y

n u a b n u a b n

=

−+

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7

Basic elements

Implementation based on the recurrence formula

derived from difference equation requires

y

8

Example of block diagram representation

A second-order difference equation

2 2

1 1

0

1)

b z

H

][]2[]

1[]

Demonstrates the

complexity, the steps,

the amount of resources

required

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M

k

k k

z a

z b z

−

k k N

][]

[]

[

]

[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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M

k

k k

M

k

k k N

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

[

][]

[

][]

[]

[

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

[

][][]

N

k k

k n w b n

y

n x k n w a n

w

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13

Minimum delay implementation

One big difference btw the two implementations

concerns the number of delay elements

),

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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15

An example

Direct form I and direct form II implementation

2 1

1

9.05

.11

21)

z z

H

16

Part II: Signal flow graph description

Block diagram representation of

Transposed forms

Basic structures for FIR systems

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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 ofincoming branches), while in the diagram a special symbol is usedfor adders and a node

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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 Æ linear 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

)

(

) ( )

(

) ( ) (

)

(

) ( )

(

) ( ) (

)

(

4 2

3 1

4

2

3

1 2

4

1

z W z

W

z

Y

z W

z

W

z X z W

z

W

z W

z

W

z X z W

)) ( ) ( ( ) (

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

] [ ]

1 [ ]

)

(

1 1

n u n

α

αα

α

? is system the

real, is

Causal!

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Part III: Basic structures for IIR systems

Transposed forms

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M

k

k k

z a

z b z

−

k k N

][]

[]

M

k

k k

z a

z b z

−

k k N

][]

[]

[

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Example

2 1

2 1 125 0 752

.

0

1

2 1

)

−

− +

−

+ +

=

z z

z z z

2 1

1

* 1 1

1 1

1

* 1 1

1

0

) 1

)(

1 ( ) 1 (

) 1

)(

1 ( ) 1

( 1

)

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

= N s

k k k

k k

k

z a z a

z b z b b z

H

2 2 1 1 0

1 )

(

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

) 25 0 1 ( ) 5 0 1 (

) 1 ( ) 1 ( 125

0 752

.

0

1

2 1

)

1 1

2 1

= +

−

+ +

=

z z

z z z

− +

1 10

) 1

)(

1 (

) 1 (

1 )

(

N

k k k

N

k k k N

k

k k

z d z d

z e B

z c

A z

C z H

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

1 1

2 2

11

1)

−+

[

]

[

a n

x

n

y

n x

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

−

=

az z

H

32

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

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

,0

, ,1,0 ,]

n

y

0

][]

[

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k M

n

z b z b b z

n h z

H

1

2 2

1 1 0 0

)(

][)

(

36

Linear-phase FIR systems

Generalized linear-phase system

Causal FIR systems have 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 ],[][

n y

0

][]

[

constantsreal

areand

offunction real

ais )

(

)()(

β

α

ωω

β ωα ω

ω

j

j j j j

e

A

e e A e

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Linear-phase FIR systems

]2/[

]/[])[

][](

[

]

[

][]

[

if

][

]2/[

]/[][

]

[

][][]

2/[

]/[][

]

[

][][

]

[

integereven

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

k

M

M k M

k

M

k

−+

+

−+

−

=

−+

Linear phase FIR systems

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

]2/[

]/[])[

][](

[]

+

−+

−

= ∑−

=

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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

40

Summary

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Course at a glance

Discrete-time signals and systems

Fourier-domain

representation

DFT/FFT

System analysis

System structure System

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