Lecture 9:
Structure for Discrete-Time
System
XILIANG LUO
2014/11
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Block Diagram
Adder, Multiplier, Memory, Coefficient
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Example
3
General Case
Direct Form 1
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Rearrangement
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Rearrangement
Zeros 1st
Poles 1st
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Canonic Form
Minimum number of delay elements:
max{M, N}
Direct Form 2
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Signal Flow Graph
A directed graph with each node being a variable or a node value.
The value at each node in a graph is the sum of the outputs of
all the branches entering the node.
Source node: no entering branches
Sink node: no outputs
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Signal Flow Graph
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Structures for IIR:
Direct Form
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Structures for IIR:
Direct Form
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Structures for IIR:
Cascade Form
Real coefs:
Combine pairs
of real factors/
complex conjugate pairs
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Structures for IIR
Cascade Form
2nd –order subsystem
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Structures for IIR
Parallel Form
Partial fraction expansion:
Group real poles in pairs:
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Structures for IIR
Parallel Form
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Feedback Loops
Loop: closed path starting at a node and returning to same node by
traversing branches in the direction allowed, which is defined by
the arrowheads
If a network has no loops, then the system function has only zeros and
the impulse response has finite duration!
Loops are necessary to generate infinitely long impulse responses!
input unit impulse, the output
is: 𝑎𝑛 𝑢[𝑛]
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Transposed Form
Transposition:
1. reverse direction of all branches
2. keep branch gains same
3. reverse input/output
For SISO, transposition gives
the same system function!
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Transposed Form
Transposed direct form II:
poles first
zeros first
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Structures for FIR
Direct Form
Tapped delay line
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Structures for FIR
Cascade Form
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Structures for FIR
with Linear Phase
Impulse response satisfies the following symmetry condition:
or
So, the number of coefficient multipliers can be essentially halved!
Type-1:
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Lattice Filters
a(i 1)[n ]
a (i )[n ]
 ki
2-port flow graph
 ki
b(i 1)[n ]
z
1
b (i )[n ]
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Lattice Filters: FIR
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Lattice Filters: FIR
Input to i-th nodes:
Recursive computation of
transfer functions!
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Lattice Filters: FIR
To obtain a direct recursive relationship for the coefficients, or the
impulse response, we use the following definition:
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Lattice Filters: FIR
From k-parameters to FIR impulse response:
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Lattice Filters: FIR
From FIR impulse response to k-parameters:
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Lattice Filters: FIR
From FIR impulse response to k-parameters:
𝐴 𝑧 = 1 − 0.9𝑧 −1 + 0.64𝑧 −2 − 0.576𝑧 −3
(3)
𝛼1 = 0.9
(3)
𝛼2
(3)
𝛼3
= −0.64
= 0.576
𝑘3 = 0.576
(3)
𝛼1 + 𝑘3 𝛼23
=
= 0.795
2
1 − 𝑘3
(2)
𝛼2 = −0.182
(2)
𝛼1
(2)
(1)
𝛼1 = 0.673
(1)
𝑘1 = 𝛼1 = 0.673
𝑘2 = 𝛼2 = −0.182
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Lattice Filters: FIR
Direct Form
Lattice Form
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Lattice Filters: IIR
Invert the computations in the following figure:
𝐻
1
𝑧 =
𝐴(𝑧)
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Lattice Filters: IIR
Derive 𝐴
𝑖−1
𝑧 from 𝐴
𝑖
𝑧
FIR:
IIR:
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Lattice Filters: IIR
Derive 𝐴
𝑖−1
𝑧 from 𝐴
𝑖
𝑧
a (i )[n ]
a(i 1)[n ]
ki
 ki
(i )
b [n ]
z 1
b(i 1)[n ]
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Lattice Filters: IIR
𝐻
1
𝑧 =
𝐴(𝑧)
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