Warped Burgs Method for ARModeling

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Warped Burgs Method for AR-Modeling

• Kari Roth Ismo Kauppinen
• Laboratory of Optics and Spectroscopy
• Department of Physics
• University of Turku, Finland
• Paulo Esquef Vesa Välimäki
• Laboratory of Acoustics and Audio Signal
  Processing
• Helsinki University of Technology, Finland

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Outline

• AR-Modeling
• Burgs Method
• Frequency Warping
• Warped Burgs Method
• Evaluation
• Applications
• Conclusions

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AR-Modeling

• Signal is modeled as an auto-regressive process
• The actual modeling is done by estimating the
  AR-coefficients ak
• This is done by minimizing the energy of the
  prediction error

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Burgs Method

• Burgs method uses a lattice structure for
  prediction error filter
• Reflection coefficients km are chosen so that the
  energies of each intermediate residual f (m) and
  b(m) are minimized
• Reflection coefficients are transformed into
  AR-coefficients by Levinson-Durbin recursion
• Step by step procedure that can be terminated at
  any stage
• The model order doesnt have to be known in
  advance

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Frequency Warping

• Achieved by a bilinear mapping in z-plane
• In filter structures a this is achieved by
  replacing unit delay by a first-order allpass
  filter
• The phase response determines the warped
  frequency scale

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Frequency Warped Burgs Method

• Same lattice structure as in conventional Burgs
  method
• Unit delays are replaced with D(z)

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Computational Costs

• Warped Burgs method is computationally about 33
  more expensive than conventional Burgs method
• When using warped AR-modeling most of the
  increase in computation comes from warped filter
  structures needed

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Comparison of the number of multiplications
needed for warped Burgs method and conventional
Burgs method
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Evaluation

• Example 2048 samples of guitar tone sampled at
  44.1kHz. Modeled using conventional Burgs
  method, warped Yule-Walker method, and warped
  Burgs method.
• Modeling setup l 0.723, p 50
• Frequency warping shifts the emphasis to the
  lower frequency range
• Warped Burgs method gives a model that has poles
  closer to unit circle and the peaks in the
  frequency response are sharper

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EvaluationInformation Extraction Capability

• Of the three methods Warped Burgs method
  provides a model with least pronounced peaks in
  the residual spectrum
• Warped Burgs method has captured more
  information about the signal to the model

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Extrapolation performance

• Signal segment is extrapolated by exciting the
  model with zero padded residual.
• Signal 2000 samples of guitar tone sampled at
  44.1kHz
• Employed setup warping factor l 0.5, model
  order p 50

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Applications

• Extrapolation
• Impulse response modeling
• Coding and compression
• Aki Härmä has used warped Burgs method for
  low-delay audio coding
• Any other application of AR-modeling...

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Conclusions

• Frequency warping and Burgs method can easily be
  combined
• Good qualities of both methods are preserved when
  used together
• Warped Burgs method provides more accurate
  models in a perceptual sense
• Good tool for high accuracy audio signal modeling

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Notice

• Some typos managed to stay unnoticed in the
  paper
• Equations 1 and 2 yn-p should be yn-m
• Figures 1 and 2 M should be p for consistency
  with Eqs. 1 and 2
• Corrected version of the paper can be downloaded
  from http//users.utu.fi/karrot/Pubs/WASPAA03_Rot
  h.pdf