filter realization

7/25/2019 Filter Realization

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Filter Realization


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Classification of Digital Systems - IIR

Systems are classified in to two types based on impulse R

h[n] of the systems

IIR Filters * FIR Filters

Representation of Filters using System Function

FIR systems are all zero systems and all poles of H(z) are at Z=0

IIR systems are all pole systems & H(z) is a rational polynomial

FIR and IIR filters can be represented as difference equati


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Convolution properties

Commutative property

H1[n] * H2[n] = H2[n] * H1[n]

Associative property

H1[n] * {H2[n] * H3[n]} = {H1[n] * H2[n]} * H3[n]

Distributive property

H1[n] * { H2[n] + H3[n] } = { H1[n] * H2[n] } + { H1[n] * H3[n]


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Copyright (C) 2005 Gner Arslan

Elements for Block Diagram Representation of a sys

LTI systems with rational systemfunction can be represented asLinear constant coefficient difference

equation. The implementation of difference

equations requires delayed values ofthe

input

output

intermediate results

The requirement of delayedelements implies need for storage

We also need means of

addition

multiplication


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Digital Filter Structures

M

k

k nxbnv0

][

Direct Form I implementation


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On the z-domain

or equivalently

)()()()(0

1 zXzbzXzHzVM

k

k

k


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Direct FormI Structure Block diagram


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Direct Form1 Modified Structure Block di


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By changing the order of H1 and H2, consider the equivalencethe z-domain:

where

Let

Then

)()()( 21 zHzHzH

M

k

k

kzbzH0

1 )(

)()()()(0

1 zWzbzWzHzYM

k

k

k


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In the time domain,

We have the following equivalence for implementation:

M

k

k knwbny0

][][

We assume

here


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Note that the exactly the same signal, w[k], is stored in thetwo chains of delay elements in the block diagram. Theimplementation can be further simplified as follows:

Direct Form II (orCanonic Direct Form)implementation


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Direct FormII Structure Block diag


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Direct FormII Structure Block diag


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Direct FormII Structure Modified Block d


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By using the direct form II implementation, the number ofdelay elements is reduced from (M+N) to max(M,N).

Example:

21

1

9.05.11

21)(

zz

zzH

Direct form I implementation Direct form II implementat


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C d F


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Cascade Form

If we factor the numerator and denominator polynomials,can express H(z) in the form:

where M=M1+2M2 and N=N1+2N2, gk and gk* are a compconjugate pair of zeros, and dk and dk* are a complex conjugpair of poles.

It is because that any N-th order real-coefficient polynomequation has n roots, and these roots are either real or compconjugate pairs.

1 2

1 2

1 1

111

1 1

111

)1)(1()1(

)1)(1()1(

)(N

k

N

k

kkk

M

k

M

k

kkk

zdzdzc

zgzgzf

zH

A general form is


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A general form is

where

and we assume that MN. The real poles and zeros have beecombined in pairs. If there are an odd number of zeros, one o

the coefficients b2k will be zero.It suggests that a difference equation can be implemented vthe following structure consisting of a cascade of second-order and first-order systems:

cascade form of implementation (with a direct form II realizatieach second-order subsystems)

2/)1( NNs

Example:


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Cascade structure: direct form I implementation

Cascade structure: direct form II implementation

Example:

)25.01)(5.01(

)1)(1(

125.075.01

21)(

11

11

21

21

zz

zz

zz

zzzH


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Parallel form Realization


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Parallel Form

If we represent H(z) by additions of low-order rational syste

where N=N1+2N2. If MN, then Np = MN.

Alternatively, we may group the real poles in pairs, so that

21

111

1

11

0 )1)(1(

)1(

1)(

N

k kk

kk

N

k k

k

N

k

k

kzdzd

zeB

zc

A

zCzH

p


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Example: consider still the same system


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Example: consider still the same system

21

1

21

21

125.075.01

878

125.075.01

21)(

zz

z

zz

zzzH


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Transposed Forms


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p

Linear signal flow graph property:

Transposing doesnt change the input-output relation

Transposing:

Reverse directions of all branches

Interchange input and output nodes Example:

Reverse directions of branches and interchange input and output

1az1

1zH

Example


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p

Transpose

Both have the same system function or difference equation

2nxb1nxbnxb2nya1nyany

21021


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Basic Structures for FIR Systems: Direct Form


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Special cases of IIR direct form structures

Transpose of direct form I gives direct form II

Both forms are equal for FIR systems

Tapped delay line


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Basic Structures for FIR Systems: Cascade Form


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Obtained by factoring the polynomial system function

M

0n

M

1k

2k2

1k1k0

nS

zbzbbznhzH


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l k i i l l


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Block Diagram Signal Flow G


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filter realization

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