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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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Copyright (C) 2005 Gner Arslan
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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Copyright (C) 2005 Gner Arslan
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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Copyright (C) 2005 Gner Arslan
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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Copyright (C) 2005 Gner Arslan
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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