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Room Impulse Response Estimation Using Higher-order Statistics from Music Signals

İzzet Özçelik 2 November 2011. Room Impulse Response Estimation Using Higher-order Statistics from Music Signals. Outline. Blind Deconvolution HOS: cumulants and polyspectra Symmetries in cumulants Examples on Bispectrum Music signal model Cumulant s of music model

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Room Impulse Response Estimation Using Higher-order Statistics from Music Signals

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  1. İzzet Özçelik 2 November 2011 Room Impulse Response Estimation Using Higher-order Statistics from Music Signals

  2. Outline • Blind Deconvolution • HOS:cumulants and polyspectra • Symmetries in cumulants • Examples on Bispectrum • Music signal model • Cumulants of music model • An algorithm for RIR estimation from music (BD) • Simulations

  3. distorting unknown system (room impulse response, communication channel ) deconvolving system ŝ s h w y source (any sound, speech, music. communication symbols) estimated source signal distorted signal Blind Deconvolution knowns:y and possibly statistical information about s unknowns:h and s • requirements: • h must be invertible i.e. no zeros on the unit circle. • samples of smust be independent and identically distributed.

  4. white noise y, observed signal h • Forming a linear set of equations in terms of h. kth-order statistics of y kth-order statistics of source which is a constant and known • h can be solved using iterative methods like kurtosis maximization or some information theoretic cost function measuring nonGaussianity.

  5. : i.i.drv for different r, deterministic constant in time Audio signals are correlated • Speech and music signals are not spectrally white. Music signal model Speechsignal model

  6. Cumulants • Cumulant is obtained from the cumulant generating function, which is the logarithm of the moment generating function. moment generating function cumulant generating function • Since its favourable properties it is one most commonly used tool of the probability theory in nongaussian and nonminimum-phase signal processing.

  7. More practical multivarite definitions • By higher-ordercumulants, cumulants of order greater than 2 is meant.

  8. Special cumulants • The 2nd, 3rd and 4th-order cumulants evaluated at zero-lags are given special names. variance skewness kurtosis

  9. (1) (2) (2) (1) (1) Symmetries in cumulants • If the rv is realthen there are some symmetries in the cumulants. • The symmetries can make calculations easier as well leading to somealgorithms, e.g. phase estimation method (by Matsuoka & Ulrych) in geophysics .

  10. Polyspectra • The Fourier transform of the higher-order cumulants are called polyspectra.The 2nd and 3rd order cumulants and their spectra are defined as: Power spectra Bispectra • The Fourier transform of 3rd order cumulant is called the bispectrum. The phase information is preserved in the bispectrum.

  11. Polyspectra: Symmetries in bipsectrum

  12. Trispectra • The Fourier transform of the 4th order cumulant is defined similarly and referred to as the trispectrum.

  13. Cumulants : Most Distinctive Properties • The higher-order (>2) cumulants of gaussian distributed signals are zero. • Higher-order (>2) cumulants preserve the phase information in a (nongaussian) signal (while 2nd-order statistics is phase blind). • Cumulants of i.i.d signals are multidimensional Kronecker delta functions. • Cumulants of independent rvs equal to the sum of their cumulants.

  14. Why do we need the cumulants? • Cumulants (of order>2) represent higher-order statistics which convey information on the phase of a noniminimum-phase system. • They measure the deviation from normality, since the cumulants (of order>2) of gaussian process are zero. • They allow estimation using data corrupted by (colored or white) gaussian noise. • They can detect the nonlinear effects, e.g. phase coupling between two harmonics. • They can measure statistical independence.

  15. Where are higher-order cumulants are used? • Blind parametric system identification. • Particularly useful in phase estimation for blind deconvolution, time-delay estimation, detecting and removing nonlinear effects. • The areas where cumulants are used : telecommunications, geophysics (seismic deconvolution), oceanography, economic time-series, image processing, source separation etc.

  16. Difficulties AssociatedWithCumulantBasedMethods • They may require long data to estimate with a reasonable variance. • A stable estimate might not be guaranteed all the time. • They are computationally more loaded.

  17. Examples Using Bispectrum • AR bispectrum estimation • Phase estimation • Magnitude estimation

  18. Model Based Approaches Using Cumulants • An AR model is assumed for the system excited by a NGWN. • Multiply each side by and then take expectation. • Third-order recursion equation:

  19. Bispectrum estimation for AR systems (parametric approach) • Using the recursive formula for • we get p+1 equations.

  20. Phase estimation (By Matsuoka&Ulrchy) (frequency domain approach) • Consider the system Bispectrum of x(k) With the restrictions

  21. Phase estimation Sample phase spectra and define From symmetry equations: Iterate with the assumption

  22. Magnitude Estimation From Bispectra

  23. Spectral Modeling Synthesis of Music Signals • SMS model: sinusoids+noise • Noise modelled as an AR process

  24. Removal of Sinusoids Linear Prediction Blind Deconvolution • Distorted music signal:

  25. Cumulants of music signal • Music model: • The amplitude and the frequency are constant. But, different assumptions can be put on the phase. i.i.d rv with zero mean deterministic constant i.i.d rv for different r

  26. : i.i.drv for different r, deterministic constant in time • Music model: • 3rd-order cumulant

  27. Applying the definition of the 3rd-order cumulant: • If we calculate 4th order cumulants, we observe that the deterministis part exits.

  28. After time-averaging • Bispectraof the distorted signal (in the form linear system excited by noise)

  29. Algorithm • Estimate the AR model of order p from estimated the cumulant sequence for whitening • Apply 3rd-order whitening using the ARmodel. • Estimate the truncated cumulant sequence using the whitened sequence. • Calculate • Estimate from

  30. Estimation of the Cumulant Sequence

  31. Example: 3rd-order cumulant of brass 3rd-order cumulant: very close to a 2-D impulse (negatively skewed).

  32. 3rd-order cumulant of brass (two different views) 3rd-order cumulant of brass after whitening (two different views) 3rd-order whitening with an AR(6) filter

  33. Magnitude Response Estimation input : noise room : Vimal’s listening room (the first 2000 samples)

  34. Room Impulse Response : 256 samples of Whittlahall(R1)16khz Input: Brass

  35. Room Impulse Response : 512 samples of Whittlahall(R1)16khz Input: Brass

  36. Conclusions • 3rd-order cumulants can be very usefull to estimate and extract certains features about a system. • We used them eliminate the sinusiodal part of music signals and approximate the RIR estimation to conventional blind deconvolution problem. • We observe a degree of succes in estimating the magnitude response of RIR. • RIR are vey long, music signal are not stationary.

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