Educational Project Levinson Durbin

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Educational ProjectLevinson Durbin

• Kyungtae Han
• Oct.14, 2002

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Outline

• The Levinson-Durbin Recursion
• Demos using Matlab

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Motivation

• Several different modeling signals
• To find the solution to a set of linear equations
  
• is a Hermitian Toeplitz matrix.
• Use the structure of to efficiently solve for

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Pronys All Pole Equations

• All Pole Equation
• Modeling Error

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Pronys All Pole Equations

• Matrix form
• Matrix notation

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Approach

• Using induction method
• We would like to solve for given the solution
  to

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The induction

• Initial condition
• Assume we know the solution to
• lets form
• where

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The induction (continued)

• Modify normal equations where
• Flipping and using Hermitian property

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The induction (continued)

• Adding two equations
• Reflection Coefficient

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The induction (continued)

• We arrive at
• where
• Levinson order-update equation
• The updated error

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Matlab Demos

• Example 5.2.1

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Discussion

• Property 1. The reflection coefficients that are
  generated by the Levinson-Durbin recursion to
  solve the autocorrelation normal equations are
  bounded by one in magnitude,
• If an arbitrary sequence of numbers
  are used in the matrix

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Discussion(continued)

• The unit magnitude constraint on the reflection
  coefficients and the nonnegativivity of the
  sequence are tied to the positive
  definiteness of the matix