Dealing with Acoustic Noise Part 1: Spectral Estimation

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Dealing with Acoustic Noise Part 1 Spectral
Estimation

• Mark Hasegawa-Johnson
• University of Illinois
• Lectures at CLSP WS06
• July 20, 2006

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Noise from the Perspective of the Brainstem

• Something happened!! (VAD)
• What happened?? (Recognition)

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Noise from the Perspective of the Brainstem

• Something happened!! (VAD)
• Which pixels belong to the new event (auditory
  scene analysis)?
• What are their amplitudes (spectral estimation)?
• What happened?? (Recognition)

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A Speech Recognition Model (Mixture Gaussian)

• Problem find the most probable class (Cn) given
  measurements of a function (f(S)) of the speech
  signal (S). For example, f(S) might be the PLP
  coefficients.
• Solution Choose n such that p(Cn,S)gtp(Cm,S) for
  n?m.
• What is p(Cn,S)? The most effective current
  computational model is the mixture Gaussian the
  weighted sum of exp(-m-f(S)W2), where
  xW2xTWx. 2W is called the precision matrix.

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How Should the Model React to Additive Noise?

• Suppose that we only have a noisy measurement, X
• ... where V is independent noise. Then Cn should
  maximize

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Answer By Computing fMMSE
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Definition of fMMSE
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Classical Estimators Maximum Likelihood
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Classical Estimators Maximum A Posteriori
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Classical Estimators Minimum Mean Squared Error
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Functions of Random Variables

• Since fMMSE(S)?f(SMMSE), it is necessary to find
  the probability density function of f(S)
  directly. Fortunately, the PDF of f(S) can
  always be computed from the PDF of S, as follows

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PDF of Speech Time Domain

• Speech samples sn are often modeled as
  Gaussian, because Gaussian PDFs are easy to
  manipulate.
• In fact, noise tends to be Gaussian, but
• Speech PDF is actually a mixture of Gaussian
  small-amplitude samples (the noise bits?) and
  Laplacian high-amplitude samples (the actual
  speech bits?)

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PDF of Speech Filter Outputs, e.g., STFT

• STFT is just a filter with complex-valued filter
  coefficients
• Central Limit Theorem says Sk should be a 2D
  Gaussian, if the window is infinitely long

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Err. Is it REALLY Gaussian?
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If we ignore the previous slide, and pretend that
Sk is a complex Gaussian, then it is possible to
analytically derive the PDFs of Sk2, Sk, and
phase of Sk. They are
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Classical Spectral Estimation Assumptions

• One signal is ongoing (call it the noise), so
  its lN is known (averaged over time prior to
  voice activity detection).
• The MMSE estimate of the other signal, S,
  combines two types of information
• a priori knowledge, lS ESk2
• a priori SNR xk lS / lN
• Maximum likelihood estimator, SML Xk2-lN
• Maximum likelihood SNR g k Xk2 / lN

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Classical Spectral Estimation ResultsWiener
Filter(Norbert Wiener, 1949)
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Classical Spectral Estimation ResultsMMSE
Spectral Amplitude Estimate(Ephraim and Malah,
1984)
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Classical Spectral Estimation ResultsMMSE Log
Amplitude Estimate (Ephraim and Malah, 1985)
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How does it sound?
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How does it sound?

• MVDR Beamformer eliminates high-frequency noise,
  MMSE-logSA eliminates low-frequency noise
• MMSE-logSA adds reverberation at low frequencies
  reverberation seems to not effect speech
  recognition accuracy

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What about, oh, say PLP?
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Loudness Spectrum

• Perceptual LPC (PLP) begins by computing an
  estimate of the perceptual loudness spectrum.
• Step 1 filter the signal using complex-valued
  Bark-scale critical-band filters hkm
• Step 2 compress the amplitudes with a
  nonlinearity

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MMSE Estimate of the Perceptual Loudness Spectrum

• gPLP requires numerical integration (of u1/3e-u)
• Numerical integration is a lot cheaper than it
  used to be (e.g., via lookup table).

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A Conservative Computational Auditory Model
x(t)
Auditory Nerve Carries The Loudness Spectrum
Xk2/3 (Fletcher Hermansky)
PLP Perceptual Formant Extraction (Hermansky)
Tandem Features Perceptual Magnet Effect (Niyog
i)
Average Background Loudness lN
Change Detection
Variable Threshold Synapses Amplitude
Compression (Ghitza, 1986)
MMSE Estimator of New Event E Sk2/3 X
Basilar Membrane Mechanical Filterbank (von
Bekesy)
Speech Feature Extraction
Auditory Scene Analysis
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Change Detection (VAD)

• Is there something new in the signal (hypothesis
  H1), or not (hypothesis H0)?
• p1p(H1) a priori, p0p(H0) a priori
• Solution compute the log likelihood ratio
• which has a simple form, in terms of gk