CHAPTER 6 Digital Filter Structures

CHAPTER 6 Digital Filter Structures

CHAPTER 6 Digital Filter Structures

Wang Weilian

[email protected]

School of Information Science and Technology

Yunnan University

2005-09 ~ 2006-01

云南大学课程：数字信号处理 Digital Filter Structures 2

OutlineOutline

• Block Diagram Representation

• Equivalent Structures

• Basic FIR Digital Filter Structures

• Basic IIR Digital Filter Structures

• FIR Cascaded Lattice Structures

• Summary


Block Diagram RepresentationBlock Diagram Representation

• Digital filters

• The difference equation

– A first-order causal LTI IIR digital filter:

• Knowing the initial condition y[-1] and the input x[n] for , we can computer y[n] for

[ ] [ ] [ ]k

y n h k x n k

1 1

[ ] [ ] [ ]N M

k kk k

y n d y n k p x n k

1 0 1[ ] [ 1] [ ] [ 1]y n d y n p x n p x n

1,0,1,2,n 0,1,2,n


Block Diagram RepresentationBlock Diagram Representation

• An LTI digital filter can be conveniently represented in block diagram form using the basic building blocks representing the unit delay, the multiplier, the adder and the pick-off nods.

– e.g. :1 0 1[ ] [ 1] [ ] [ 1]y n d y n p x n p x n

0[ ] px n

A first-order LTI digital filter

1z

1p

1z

1d

[ ]y n


Equivalent StructuresEquivalent Structures

• Two digital filter structures is equivalent if they have the same transfer function.

• There are a number of methods for the generation of equivalent structures:

– ( i ) Reverse all paths,

– ( ii ) Replace pick-off nodes by adders, and vice versa,

– ( iii ) Interchange the input and the output nodes.


Basic FIR Digital Filter StructuresBasic FIR Digital Filter Structures

• Direct forms

– In the time-domain:

– The transfer function:

– An FIR filter of order N requires N+1 multipliers and N two-input adders for implementation in general.

0

[ ] [ ] [ ]N

k

y n h k x n k

0 0

( ) [ ]N N

k ii

k i

H z h k z b z



• Direct Forms1z1z 1z

0b Nb1Nb 2b1bx[n]

y[n]h[0] h[1] h[2] h[N]h[N 1]

Direct form FIR structures

x n

1z 1z 1z

h 0 h 1 h N 2 h N 1 h N

y n



• Cascade Form

• 2L=N unit delays ， 2L+1=N+1 multipliers ， 2L=N adders

1 21 2

1

2

( ) [0] (1 ),

2, is evenwhere

( 1) 2, 0, is odd

K

k kk

k

H z h z z

N NK

N and N

11

21

1L

2L

12

22

x[n] y[n]

1z

1z

h[0]

1z 1z

1z 1z

Cascade form FIR structure



• Linear-Phase FIR Structures

– A linear-phase FIR filter of order N

• The symmetric impulse response:

• The antisymmetric impulse response:

[ ] [ ]h n h N n

[ ] [ ]h n h N n

Linear-phase FIR structure


Basic IIR Digital Filter StructuresBasic IIR Digital Filter Structures

• Direct Forms

– An Nth-order IIR digital filter transfer function requires 2N+1 multipliers and 2N two-input adders for implementation in general.

0 0

1 1

( )( )

( )1 1

M Mi i

i ii iN N

k kk k

k k

p z a zP z

H zD z

d z b z

( ) 1 1( ) ( ) ( )

( ) ( ) ( )

P zH z P z p z

D z D z D z



• Direct Forms

Direct form I Direct form It



• Cascade Realizations

– Various different cascade realizations of H(z) can be obtained by different pole-zero polynomial pairings.

– Usually, the polynomials are factored into a product of first-order and second-order polynomials.

• e.g. :

( ) ( ) ( )( )( )

( ) ( ) ( ) ( )i k k

k k kk k

P z P z A zP zH z or

D z D z D z B z

1 21 2

1 21 2

(1 )( ) [0] ,

(1 )k k

k k k

z zH z p

z z



• Cascade Realizations

Cascade realization of a IIR transfer function



• Parallel Realizations

– Making use of the partial-fraction expansion of the transfer function:

• The parallel form I:

• The parallel form II:

( )( )

( )k

kk

Q zH z

D z

10 1

0 1 21 2

( )1

k k

k k k

zH z

z z

1 21 2

0 1 21 2

( )1

k k

k k k

z zH z

z z



• Parallel Realizations

Parallel realization of a IIR transfer function


FIR Cascaded Lattice StructuresFIR Cascaded Lattice Structures

• Realization of Arbitrary FIR Transfer Functions

– All-Zeros filter:

– From fig 6.39:

1

( ) 1N

nN n

n

H z p z

11 0 1 0

11 1 0 0

( ) ( ) ( )

( ) ( ) ( )

X z X z k z X z

Y z k X z z X z

111 1

0

111 1

0

( )( ) 1

( )

( )( )

( )

X zH z k z

X z

Y zG z k z

X z


SummarySummary

• The realization of a causal digital transfer function, called structures, are usually represented in block diagram forms that are formed by interconnecting adders, multipliers, and unit delays.

• The structure can be analyzed to develop its input-output relationship either in the time-domain or in the transform-domain.

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CHAPTER 6 Digital Filter Structures