Chapter 6' Structures for DiscreteTime Systems

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Chapter 6. Structures for Discrete-Time Systems
6.1 Block Diagram Representation of Linear
Constant-Coefficient Difference
Equations 6.2 Signal Flow Graph Representation of
Linear Constant-Coefficient Difference
Equations 6.3 Basic Structures for IIR
Systems 6.4 Transposed Forms 6.5 Basic Network
Structures for FIR Systems 6.6 Overview of
Finite-Precision Numerical Effects 6.7 The
Effects of Coefficient Quantization 6.8 Effects
of Round-off Noise in Digital Filters 6.9
Zero-Input Limit Cycles in Fixed-Point
Realizations of IIR Digital Filters
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• 1. Digital circuits vs. Analog circuits

DIGITAL
ANALOG
(1) Expression
(2) Circuit Elements

-
3
DIGITAL
ANALOG
(3) Time - Domain
Discrete time Continuous - time
(4) Transform - Domain
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DIGITAL
ANALOG
(5) Signal Flow Graph
Straightforward
Conversion via KCL, KVL and branch equations
( no delay-free loop allowed)
( no improper Source-sink )
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Examples
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? Masons rule for signal flow graph
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2. Circuit Structure for IIR System
(1) Direct Form
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(2) Cascade Form - Second order factored form -
pole-zero pairing
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(3) Parallel Form - Partial fraction expansion
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(4) Transposed Form - Reverse the flow of a
structure, then you will get the identical
transfer function
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3. Circuit Structures for FIR Systems
(1) Direct Form

-
)
1
(
N
n
x
Z-1
Z-1
Z-1
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(2) Cascade Form
(3) Parallel Form (Frequency Sampling)
p
2
-
1
1
N
-
j
-


kn
)
(

IDFT


)
(
)
(
e
W
W
k
H
n
h
N
N
N
N

0
k
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(4) Linear Phase FIR Structure
-

)
(
)
(
n
M
h
n
h


)
0
(
h
)
1
(
h
-
)
(
M
h
)
1
(
M
h

)
0
(
h
)
(
M
h
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(5) Linear Phase FIR Structure in Quad Form
Quad zeros factors
coefficients in
A
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? IIR Structure of special interest
B
coefficients in
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Observation
grid denser
grid uniform
top and
bottom
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4. Effects of Coefficient Quantization
(1) Effects of pole-clustering
- after quantization
- sensitivity of (in denominator) on H(z)
is larger than that of
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Let
Therefore, a small change in could cause a
large change in when is close to , (or
when poles are clustered), causing a large
change in H(z) or its frequency spectrum
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Illustration Effects of quantization on the
zeros of a 27th order polynomial P(z)
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3) The resulting second order sections should be
ordered according to the closeness of the poles
to the unit circle, either in increasing
closeness to the unit circle or in its reverse
order
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Illustration a 12th order IIR bandpass filter
- with the cutoff frequencies at 0.3pi,
0.4pi - stopband attenuation of -40dB
Cascaded Structure
Direct-form Structure
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(3) Grid granularity and Filter Structure
(A) Arrangement 1 direct-form implementation
(A)
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Illustration zero locations for direct-form
implementation
4 bit quantization
7 bit quantization
Put Figure 6.42 (a) below
Put Figure 6.42 (b) below
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(B) Arrangement 2 coupled-form implementation
(B)
( Note) the cross points are physically
realizable pole points -guantization moves a
pole from one crosspoint to another -if
poles are on the upper region take
arrangement 1 (structure (A)), otherwise,
take arrangement 2 (structure (B))
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Illustration zero locations for coupled-form
implementation
4 bit quantization
7 bit quantization
Put Figure 6.44 (a) below
Put Figure 6.44 (b) below
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5 Limit Cycle
Y(n) ay(n-1) - x(n) -gt y(n) Qay(n-1) - x(n)
yn
yn
xn
xn
a
a
z-1
z-1
Q