CHAPTER 6 Digital Filter Structures

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CHAPTER 6 Digital Filter Structures

• Wang Weilian
• wlwang_at_ynu.edu.cn
• School of Information Science and Technology
• Yunnan University
• 2005-09 2006-01

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Outline

• Block Diagram Representation
• Equivalent Structures
• Basic FIR Digital Filter Structures
• Basic IIR Digital Filter Structures
• FIR Cascaded Lattice Structures
• Summary

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Block Diagram Representation

• Digital filters
• The difference equation
• A first-order causal LTI IIR digital filter
• Knowing the initial condition y-1 and the input
  xn for , we can computer yn
  for

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Block Diagram Representation

• An LTI digital filter can be conveniently
  represented in block diagram form using the basic
  building blocks representing the unit delay, the
  multiplier, the adder and the pick-off nods.
• e.g.

A first-order LTI digital filter
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Equivalent Structures

• Two digital filter structures is equivalent if
  they have the same transfer function.
• There are a number of methods for the generation
  of equivalent structures
• ( i ) Reverse all paths,
• ( ii ) Replace pick-off nodes by adders, and vice
  versa,
• ( iii ) Interchange the input and the output
  nodes.

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Basic FIR Digital Filter Structures

• Direct forms
• In the time-domain
• The transfer function
• An FIR filter of order N requires N1 multipliers
  and N two-input adders for implementation in
  general.

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Basic FIR Digital Filter Structures

• Direct Forms

Direct form FIR structures
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Basic FIR Digital Filter Structures

• Cascade Form
• 2LN unit delays,2L1N1 multipliers,2LN adders

Cascade form FIR structure
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Basic FIR Digital Filter Structures

• Linear-Phase FIR Structures
• A linear-phase FIR filter of order N
• The symmetric impulse response
• The antisymmetric impulse response

Linear-phase FIR structure
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Basic IIR Digital Filter Structures

• Direct Forms
• An Nth-order IIR digital filter transfer function
  requires 2N1 multipliers and 2N two-input adders
  for implementation in general.

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Basic IIR Digital Filter Structures

• Direct Forms

Direct form I
Direct form It
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Basic IIR Digital Filter Structures

• Cascade Realizations
• Various different cascade realizations of H(z)
  can be obtained by different pole-zero polynomial
  pairings.
• Usually, the polynomials are factored into a
  product of first-order and second-order
  polynomials.
• e.g.

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Basic IIR Digital Filter Structures

• Cascade Realizations

Cascade realization of a IIR transfer function
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Basic IIR Digital Filter Structures

• Parallel Realizations
• Making use of the partial-fraction expansion of
  the transfer function
• The parallel form I
• The parallel form II

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Basic IIR Digital Filter Structures

• Parallel Realizations

Parallel realization of a IIR transfer function
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FIR Cascaded Lattice Structures

• Realization of Arbitrary FIR Transfer Functions
• All-Zeros filter
• From fig 6.39

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Summary

• The realization of a causal digital transfer
  function, called structures, are usually
  represented in block diagram forms that are
  formed by interconnecting adders, multipliers,
  and unit delays.
• The structure can be analyzed to develop its
  input-output relationship either in the
  time-domain or in the transform-domain.