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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 )