1. Dealing with Acoustic Noise Part 1: Spectral Estimation
Mark Hasegawa-Johnson
University of Illinois
Lectures at CLSP WS06
July 20, 2006

2. Noise from the Perspective of the Brainstem
Something happened!! (VAD)
What happened?? (Recognition)

3. 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)

4. 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)>p(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 |x|W2≡xTWx. 2W is called the “precision” matrix.

5. 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:

6. Answer: By Computing fMMSE

7. Definition of fMMSE

8. Classical Estimators: Maximum Likelihood

9. Classical Estimators: Maximum A Posteriori

10. Classical Estimators: Minimum Mean Squared Error

11. 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:

12. PDF of Speech: Time Domain
Speech samples s[n] 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?)

13. 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…

14. Err…. Is it REALLY Gaussian?

15. If we ignore the previous slide, and pretend that Sk is a complex Gaussian, then it is possible to analytically derive the PDFs of |Sk|2, |Sk|, and phase of Sk. They are:

16. 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 = E[|Sk|2]
a priori SNR: xk = lS / lN
Maximum likelihood estimator, SML = |Xk|2-lN
Maximum likelihood SNR: g k = |Xk|2 / lN

17. Classical Spectral Estimation Results: Wiener Filter (Norbert Wiener, 1949)

18. Classical Spectral Estimation Results: MMSE Spectral Amplitude Estimate (Ephraim and Malah, 1984)

19. Classical Spectral Estimation Results: MMSE Log Amplitude Estimate (Ephraim and Malah, 1985)

20. How does it sound?

21. 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

22. What about,
oh, say…
PLP?

23. 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 hk[m]:
Step 2: compress the amplitudes with a nonlinearity:

24. 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).

25. A Conservative Computational Auditory Model
x(t)
Auditory
Nerve Carries
The Loudness
Spectrum
~ |X[k]|2/3
(Fletcher;
Hermansky)
PLP ≈
Perceptual
Formant
Extraction
(Hermansky);
Tandem
Features
Magnet
Effect
(Niyogi)
Average
Background
Loudness lN
Change
Detection
Variable
Threshold
Synapses: Amplitude Compression
(Ghitza, 1986)
MMSE
Estimator of
“New Event”
E[ |S[k]|2/3 | X ]
Basilar
Membrane:
Mechanical
Filterbank
(von Bekesy)
Speech Feature
Extraction
Auditory
Scene Analysis

26. Change Detection (VAD)
Is there something new in the signal (hypothesis H1), or not (hypothesis H0)?
p1=p(H1) a priori, p0=p(H0) a priori
Solution: compute the log likelihood ratio…
… which has a simple form, in terms of gk: