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Lecture 9: Structure for Discrete-Time SystemXILIANG LUO
2014/11
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Block DiagramAdder, Multiplier, Memory, Coefficient
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Example
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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 GraphA 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 branchesSink 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 FormReal coefs:
Combine pairsof real factors/complex conjugate pairs
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Structures for IIRCascade Form
2nd –order subsystem
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Structures for IIRParallel Form
Group real poles in pairs:
Partial fraction expansion:
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Structures for IIRParallel Form
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Feedback Loops
If a network has no loops, then the system function has only zeros andthe impulse response has finite duration!
Loops are necessary to generate infinitely long impulse responses!
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
input unit impulse, the outputis: 𝑎𝑛𝑢[𝑛]
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Transposed Form
Transposition:1. reverse direction of all branches2. keep branch gains same3. reverse input/output
For SISO, transposition givesthe same system function!
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Transposed FormTransposed direct form II:
poles firstzeros first
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Structures for FIRDirect Form
Tapped delay line
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Structures for FIRCascade Form
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Structures for FIRwith Linear PhaseImpulse 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
][)1( na i
][)1( nb i
][)( na i
][)( nb i1z
ik
ik
2-port flow graph
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Lattice Filters: FIR
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Lattice Filters: FIRInput to i-th nodes:
Recursive computation oftransfer functions!
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Lattice Filters: FIRTo obtain a direct recursive relationship for the coefficients, or theimpulse response, we use the following definition:
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Lattice Filters: FIRFrom k-parameters to FIR impulse response:
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Lattice Filters: FIRFrom FIR impulse response to k-parameters:
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Lattice Filters: FIRFrom FIR impulse response to k-parameters:
𝐴 (𝑧 )=1−0.9𝑧−1+0.64 𝑧− 2−0.576 𝑧− 3
𝛼1(3)=0.9
𝛼2(3)=−0.64
𝛼3(3)=0.576
𝛼1(2)=
𝛼1(3 )+𝑘3𝛼2
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1−𝑘32 =0.795
𝛼2(2)=−0.182
𝑘2=𝛼2(2 )=−0.182
𝑘3=0.576
𝛼1(1)=0.673
𝑘1=𝛼1(1)=0.673
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Lattice Filters: FIR
Direct Form
Lattice Form
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Lattice Filters: IIRInvert the computations in the following figure:
𝐻❑ (𝑧 )= 1𝐴(𝑧 )
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Lattice Filters: IIRDerive 𝐴(𝑖− 1) (𝑧 ) 𝐴(𝑖 ) (𝑧 )from
FIR:
IIR:
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Lattice Filters: IIRDerive 𝐴(𝑖− 1) (𝑧 ) 𝐴(𝑖 ) (𝑧 )from
][)1( na i
][)1( nb i
][)( na i
][)( nb i1z
ik
ik
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Lattice Filters: IIR
𝐻❑ (𝑧 )= 1𝐴(𝑧 )
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