State Space Modelling of Multiple Sinusoids in Noise

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State Space Modelling of Multiple Sinusoids in
Noise

• Svante Stadler
• Sound and Image Processing Lab
• EE, KTH

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State Space Modelling

• this course the model is known
• Real world model is often unknown, must be
  estimated
• Speech and Audio model is estimated from data
  only
• Estimation must be constrained (No Free Lunch
  etc)
• Assumption signal sinusoids noise

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Harmonic Estimation

• The clean signal can be expressed as
• Our goal is to minimize
• which leads to ML estimates if noise is gaussian.
• PROBLEM Nonlinear estimation, multiple minima,
  must search through parameter space

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Harmonic Estimation

• Trick first estimate a linear model, then
  estimate parameters from it
• First done by Prony (1795!)
• Linear models
• Prediction polynomial
• Power spectrum
• Covariance Matrix
• State space Model
• Frequencies, Phases and Amplitudes are estimated
  from the Linear Model
• Most work done is the 70s and 80s

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Pisarenkos Method

• Eigenvectors of Covariance Matrix are unaffected
  by additive white noise
• All Eigenvalues are shifted up by noise variance
• The smallest eigenvalue determines the noise
  variance and the corresponding eigenvector is the
  prediction polynomial for the clean signal
• PRO Simple, accurate, fast
• CON Model order must be known, sensitive to
  non-whiteness of noise

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MUltiple SIgnal Classification (MUSIC)

• Extension of Pisarenko approach
• Makes use of redundancy when Covariance Matrix is
  larger than p1

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Tufts and Kumaresan (TK)

• Least Squares Linear Prediction is sensitive to
  noise
• Apply SVD to data matrix and reduce rank to p

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State Space Form
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State Space Form
Linear prediction is identical to estimating
first row
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State Space Form

• Hankel matrix Y is factorized to estimate Model

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State Space Form

• When noise is present, F will not be in canonical
  form

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State Space Form

• PRO
• Low parameter sensitivity
• Yields amplitude and phase estimates along with
  frequency
• Does not need large order
• Synthesis filter is stable
• CON
• Matrix decompositions are computationally complex
• Methods become complicated in some cases

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Estimation Error
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Adaptive Estimation

• Sinusoids may change in frequency and amplitude
• State Space Model can be updated using
  decomposition of data matrix, qgtp
• No Forgetting Factor needed!

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Efficient Methods

• A filterbank can be used to separate harmonics
  (e.g. Chung and Fang, 1997)
• One harmonic can be tracked at a time, then
  subtracted (e.g. Prandoni and Vetterli, 1998)
• FFT can be used to find autocovariance
  (Wiener-Khinchin)

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References

• Pisarenko, V. F. The retrieval of harmonics from
  a covariance function, Geophysics, J. Roy.
  Astron. Soc., vol. 33, pp. 347-366, 1973.
• Schmidt, R.O, Multiple Emitter Location and
  Signal Parameter Estimation, IEEE Trans.
  Antennas Propagation, Vol. AP-34 (March 1986),
  pp.276-280.
• D. W. Tufts and R. Kumaresan, Estimation of
  frequencies of multiple sinusoids Making linear
  prediction work like maximum likelihood, Proc.
  IEEE, vol. 70, pp. 975-989, Sept. 1982.
• S. Y. Kung, K.S. Arun, and D. V. BhaskarRao,
  State-space singular value decomposition based
  methods for the harmonic retrieval problem, J.
  Opt. Soc. Amer., vol. 73, pp. 1799-1811, Dec.
  1983.
• D. B. Rao and K. Arun, Model based processing of
  signals A state space approach, Proc. IEEE,
  vol. 80, pp. 283--309, Feb. 1992.
• K. S. Arun and M. Aung, Detection and tracking
  of superimposed non-stationary harmonics, Fifth
  ASSP Workshop on Spectrum Estimation and
  Modeling, pp. 173-177, Oct. 1990.
• P. Prandoni and M. Vetterli, An FIR cascade
  structure for adaptive linear prediction, IEEE
  Transactions on Signal Processing, vol. 44, pp.
  16-31, Jan. 1998
• Y. Chung and W. Fang, An efficient approach for
  the harmonic retrieval problem via Haar Wavelet
  Transform, IEEE Signal Processing Letters, vol.
  4, No. 12, pp. 331-334, Dec. 1997