Structures for Discrete-Time Systems

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Structures forDiscrete-Time Systems

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Content

• Introduction
• Block Diagram Representation
• Signal Flow Graph
• Basic Structure for IIR Systems
• Transposed Forms
• Basic Structure for FIR Systems
• Lattice Structures

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Structures forDiscrete-Time Systems

• Introduction

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Characterize an LTI System

• Impulse Response
• z-Transform
• Difference Equation

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Example
Noncomputable
Computable
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Basic Operations

• Addition
• Multiplication
• Delay

In fact, there are unlimited variety of
computational structures.
Computable
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Why Implement Using Different Structures?

• Finite-precision number representation of a
  digital computer.
• Truncation or rounding error.
• Modeling methods
• Block Diagram
• Signal Flow Graph

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Block Diagram Representation
Adder
Multiplier
Unit Delay
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Example
x(n)
y(n)
y(n?1)
y(n?2)
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Higher-Order Difference Equations
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Block Diagram Representation(Direct Form I)
v(n)
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Block Diagram Representation(Direct Form I)
v(n)
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Block Diagram Representation(Direct Form I)
v(n)
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Block Diagram Representation(Direct Form I)
Implementing zeros
Implementing poles
v(n)
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Block Diagram Representation(Direct Form I)
How many Adders? How many multipliers? How many
delays?
v(n)
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Block Diagram Representation (Direct Form II)
Assume M N
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Block Diagram Representation (Direct Form II)
Assume M N
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Block Diagram Representation (Direct Form II)
Implementing poles
Implementing zeros
Assume M N
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Block Diagram Representation (Direct Form II)
How many Adders? How many multipliers? How many
delays?
Assume M N
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Block Diagram Representation (Canonic Direct Form)
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Block Diagram Representation (Canonic Direct Form)
How many Adders? How many multipliers? How many
delays? max(M, N)
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Structures for Discrete-Time Systems

• Signal Flow Graph

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Nodes And Branches
Associated with each node is a variable or node
value.
wj(n)
wk(n)
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Nodes And Branches
Input wj(n)
Output A linear transformation of input, such as
constant gain and unit delay.
Brach (j, k)
Each branch has an input signal and an output
signal.
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More on Nodes
An internal node serves as a summer, i.e., its
value is the sum of outputs of all branches
entering the node.
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Source Nodes

• Nodes without entering branches

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Sink Nodes

• Nodes that have only entering branches

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Example
Source Node
Sink Node
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Block Diagram vs. Signal Flow Graph
w(n)
y(n)
x(n)
w1(n)
x(n)
y(n)
w2(n)
w3(n)
w4(n)
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Block Diagram vs. Signal Flow Graph
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Block Diagram vs. Signal Flow Graph
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Structures for Discrete-Time Systems

• Basic Structure for IIR Systems

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Criteria

• Reduce the number of constant multipliers
• Increase speed
• Reduce the number of delays
• Reduce the memory requirement
• Modularity VLSI design
• The effects of finite register length and
  finite-precision arithmetic.

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Basic Structures

• Direct Forms
• Cascade Form
• Parallel Form

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Direct Forms
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Direct Form I
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Direct Form I
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Direct Form II
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Direct Form II
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Example
0.75
2
Direct Form I
?0.125
2
0.75
Direct Form II
?0.125
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Cascade Form
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Cascade Form
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Cascade Form
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Cascade Form
x(n)
y(n)
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Another Cascade Form
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Parallel Form
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Parallel Form
Group Real Poles
Complex Poles
Real Poles
Poles at zero
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Parallel Form
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Parallel Form
y(n)
x(n)
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Example
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Example
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Structures for Discrete-Time Systems

• Transposed Forms

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Signal Flow Graph Transformation

• To transform signal graphs into different forms
  while leaving the overall system function between
  input and output unchanged.

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Transposition of Signal Flow Graph

• Reverse the directions of all arrows.
• Changes the roles of input and output.

x(n)
y(n)
x(n)
y(n)
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Transposition of Signal Flow Graph
Are there any relations between the two systems?
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Example
x(n)
y(n)
x(n)
y(n)
x(n)
y(n)
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Transposition of Signal Flow Graph

• Reverse the directions of all arrows.
• Changes the roles of input and output.

Detail proof see reference
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Structures for Discrete-Time Systems

• Basic Structure for FIR Systems

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FIR

• For causal FIR systems, the system function has
  only zeros.

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Direct Form
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Direct Form
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Direct Form
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Cascade Form
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Cascade Form
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Structures for Linear Phase Systems

• A generalized linear phase system satisfies

h(M?n) h(n) for n 0,1,,M
h(M?n) ?h(n) for n 0,1,,M
or
Type I
Type II
Type VI
Type III
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Type I
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Type I
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Type II, III and VI
Construct them in a similar manner by yourselves.
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Structures for Discrete-Time Systems

• Lattice Structures

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FIR Lattice
Consider x(n)?(n), one will see
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FIR Lattice
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FIR Lattice
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FIR Lattice
Define
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FIR Lattice
Show that
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FIR Lattice
FIR Lattice
i1
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FIR Lattice
FIR Lattice
i n Assumed true
Prove
i n1 also true.
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FIR Lattice
FIR Lattice

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FIR Lattice
FIR Lattice
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FIR Lattice
FIR Lattice
Given the lattice, to find A(z).
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FIR Lattice
FIR Lattice
Given A(z), to find the lattice.
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FIR Lattice
FIR Lattice
Given A(z), to find the lattice.
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Example
0.6728
0.7952
0.9
?0.1820
?0.64
0.576
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Example
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Inverse Filter
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All-Pole Filter
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All-Pole Filter
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All-Pole Filter
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All-Pole Filter
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Example
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Example
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Stability of All-Pole Filter
All zeros of A(z) have to lie within the unit
circle.
Necessary and sufficient conditions
All of k-parameters kis satisfy ki lt 1.
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Normalized Lattice
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Normalized Lattice
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Normalized Lattice
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Normalized Lattice
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Normalized Lattice
Three-Multiplier Form
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Normalized Lattice
Four-Multiplier, Kelly-Lochbaum Form
Four-Multiplier, Normalized Form
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Normalized Lattice
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Normalized Lattice
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Normalized Lattice
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Lattice Systems with Poles and Zeros
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Lattice Systems with Poles and Zeros
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Lattice Systems with Poles and Zeros
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Example
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Example
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Example