Pitch Estimation and Tracking with Harmonic Emphasis on ... · Bayesian Pitch Estimator Motivation...

Aalborg Universitet

Pitch Estimation and Tracking with Harmonic Emphasis On The Acoustic Spectrum

Karimian-Azari, Sam; Mohammadiha, Nasser; Jensen, Jesper Rindom; Christensen, MadsGræsbøllPublished in:I E E E International Conference on Acoustics, Speech and Signal Processing. Proceedings

DOI (link to publication from Publisher):10.1109/ICASSP.2015.7178788

Publication date:2015

Document VersionEarly version, also known as pre-print

Link to publication from Aalborg University

Citation for published version (APA):Karimian-Azari, S., Mohammadiha, N., Jensen, J. R., & Christensen, M. G. (2015). Pitch Estimation andTracking with Harmonic Emphasis On The Acoustic Spectrum. I E E E International Conference on Acoustics,Speech and Signal Processing. Proceedings, 4330-4334. https://doi.org/10.1109/ICASSP.2015.7178788

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https://doi.org/10.1109/ICASSP.2015.7178788

https://vbn.aau.dk/en/publications/a5ddf69b-88fa-4fca-b0b4-225c17dfe074

https://doi.org/10.1109/ICASSP.2015.7178788

Pitch Estimation and Tracking withHarmonic Emphasis on the Acoustic Spectrum

April 23, 2015

Sam Karimian-Azari1, Nasser Mohammadiha2,Jesper R. Jensen1, and Mads G. Christensen1

[email protected]

1Audio Analysis Lab, AD:MT, Aalborg University2University of Oldenburg

19

Pitch Estimation andTracking

Sam Karimian-Azariet al.

IntroductionHarmonic Signal Model

Noisy Signal Approx.

ML Pitch Estimation

Proposed MethodMotivation-Hypothesis

1- Discrete state-space:HMM

2- Continuous state-space:Kalman Filter

Numerical Results

Conclusion

Audio Analysis Lab, AD:MT,Aalborg University, Denmark

Agenda

I IntroductionI Noisy Harmonic Signal ApproximationI ML Pitch Estimate from UFE

I Bayesian MethodsI MotivationI HMMI Kalman Filter

I Numerical ResultsI Conclusion

19



Introduction2 Harmonic Signal Model


ML Pitch Estimation




Numerical Results

Conclusion



19





ML Pitch Estimation




Numerical Results

Conclusion



19





ML Pitch Estimation




Numerical Results

Conclusion



Harmonic Signal Model:

s(n) =

L(n)∑l=1

αl e j (ωl (n) n+ϕl ), (1)

where ωl (n) = lω0(n) for l = 1, . . . ,L(n),

L(n) : number of sinusoidsαl : real magnitudesω0 : fundamental frequencyϕl : phases of harmonics

50 100 150 200 250−2

0

2

4

6

n

s(n)

0 π/4 π/2 3π/4 π0

0.5

1

ω

|S(ω

)|

19




6 Noisy Signal Approx.

ML Pitch Estimation




Numerical Results

Conclusion


Signal ModelAdditive noise

The observed signal can be written as a sum of a desiredsignal s(n) and a noise signal v(n), i.e.,

x(n) =s(n) + v(n) (2)

=L∑

l=1

αl e j (ωl n+ϕl ) + v(n).

19





ML Pitch Estimation




Numerical Results

Conclusion


Signal ModelPhase noise

At a high narrowband SNR, the harmonic frequency ωl isperturbed with a real-valued phase-noise [S.Tretter 1985],which has a normal distribution with zero mean and thevariance

E∆ω2l (n) =

σ2

2α2l

(3)

We can approximate x(n) =∑L

l=1 αl e j (ωl n+ϕl ) + v(n) like

x(n) ≈L∑

l=1

αl e j (ωl n+∆ωl (n)+ϕl ) (4)

19





ML Pitch Estimation




Numerical Results

Conclusion


Signal ModelUnconstrained frequency estimates (UFE)

Unconstrained frequency estimates (UFE) of the constrainedfrequencies:

Ω(n) =[ω1(n), ω2(n), . . . , ωL(n)

]T (5)= dL(n)ω0(n) + ∆Ω(n), (6)

where

dL(n) =[

1,2, . . . , L(n)]T (7)

∆Ω(n), =[

∆ω1(n), ∆ω2(n), . . . , ∆ωL(n)]T, (8)

and

R∆Ω(n) = E∆Ω(n)∆ΩT(n) (9)

=σ2

2diag

1α2

1,

1α2

2, . . . ,

1α2

L

.

19





9 ML Pitch Estimation




Numerical Results

Conclusion


Max. Likelihood (ML) Pitch Estimator

For the time-frame x(n) =[

x(n), x(n−1), . . . , x(n−M−1)]T ,

the PDF of the UFE is

P(Ω(n)|ω0(n)) ∼ N (dL(n)ω0(n),R∆Ω(n)). (10)

The ML pitch estimator:

ω0(n) = arg maxω0(n)

log P(Ω(n)|ω0(n)) (11)

=[

dTL (n)R−1

∆Ω(n)dL(n)]−1

dTL (n)R−1

∆Ω(n)Ω(n) (12)

19





ML Pitch Estimation

Proposed Method10 Motivation-Hypothesis



Numerical Results

Conclusion


Bayesian Pitch EstimatorMotivation

I The ML Estimators are statistically efficient, e.g., thenon-linear least-squares (NLS), and the weighted leastsquares (WLS) [H.Li, et al. 2000], but the minimumvariance is limited by the number of samples.

I Consecutive pitch values are estimated independently

19





ML Pitch Estimation




Numerical Results

Conclusion


Bayesian Pitch EstimatorMotivation

I Pitch values are usually correlated in a sequence, i.e.,

P(ω0(n)|ω0(n−1), ω0(n−2), · · · ), (13)

that motivate Bayesian methods to minimize an errorincorporating prior distributions.

I State-of-the-art methods mostly track pitch estimates in asequential process without concerning noise statistics.

19





ML Pitch Estimation




Numerical Results

Conclusion


Bayesian Pitch EstimatorHypothesis

1- Jointly estimate and track pitch incorporating both theharmonic constraints and noise characteristics.2- Estimate the state ω0(n) through a series of noisyobservations:

P(ω0(n)|Ω(n), Ω(n−1), · · · ) (14)

3- Recursively update the prior distribution of the pitch value.

19





ML Pitch Estimation


13 1- Discrete state-space:HMM


Numerical Results

Conclusion


Bayesian Pitch EstimatorDiscrete state-space (HMM)

ω0(n) : Discrete random variable (Hidden states)

P(ω0(n)|ω0(n−1)) : Transition probability in a 1st-order Markov model,

i.e.,∑ω0(n)

P(ω0(n)|ω0(n−1)) = 1

ω0(n) = arg maxω0(n)

log P(ω0(n)|Ω(n), Ω(n−1), · · · ) (15)

= arg maxω0(n)

log P(Ω(n)|ω0(n)) + log P(ω0(n)|Ω(n−1), · · · ).

The priori distribution is defined recursively like

P(ω0(n)|Ω(n−1), Ω(n−2), · · · ) = (16)∑ω0(n−1)

P(ω0(n)|ω0(n−1))P(ω0(n−1)|Ω(n−1), · · · ),

where P(ω0(n−1)|Ω(n−1), · · · ) is the past estimate.

19





ML Pitch Estimation



14 2- Continuous state-space:Kalman Filter

Numerical Results

Conclusion


Bayesian Pitch Estimatorstate-space representation of the pitch continuity

Continuous state-space:

ω0(n) = ω0(n−1) + δ(n)

Ω(n) = dL(n)ω0(n) + ∆Ω(n),

where δ(n) ∼ N (0, σ2t ) and ∆Ω(n) ∼ N (0,R∆Ω(n)) are the

state evolution and observation noise, respectively.

19





ML Pitch Estimation




Numerical Results

Conclusion


Bayesian Pitch EstimatorContinuous state-space (Kalman filter)

First, a pitch estimate is predicted using the past estimates as

ω0(n|n−1) = ω0(n−1|n−1) (17)

with the variance

σ2K (n|n−1) = σ2

K (n−1|n−1) + σ2t . (18)

Second, the pitch estimate is updated with the error of

e(n) = Ω(n)− dL(n) ω0(n|n−1). (19)

Then, the predicted estimate is updated:

ω0(n|n) = ω0(n|n−1) + hK(n)e(n) (20)

hK(n) = σ2K (n|n−1)dT

L (n)[

ΠL(n)σ2K (n|n−1) + R∆Ω(n)

]−1, (21)

where ΠL(n) = dL(n)dTL (n), and update

σ2K (n|n) =

[1− hK(n)dL(n)

]σ2

K (n|n−1). (22)

19





ML Pitch Estimation




Numerical Results

Conclusion


Covariance MatrixEstimation

The ML estimator of the covariance matrix among N estimates:

R∆Ω(n) = E∆Ω(n)∆ΩT(n)

=1N

n∑i=n−N+1

∆Ω(i)∆ΩT(i), (23)

where ∆Ω(n) = Ω(n)− µ(n), and µ(n) = EΩ(n).

Exponential moving average:

µ(n) = λ Ω(n) + (1−λ) µ(n−1) (24)

The forgetting factor 0 < λ < 1 recursively updates thetime-varying mean value.

19





ML Pitch Estimation




17 Numerical Results

Conclusion


Numerical ResultsSynthetic signal

A linear chirp signal (r = 100 Hz/s) with L = 5 harmonics,random phases, and identical amplitudes during 0.1 s.

0 2 4 6 8 10 12 14 16 18 20

10−7

10−6

10−5

10−4

SNR [ dB ]

MSE

ω1

ω0,WLSω0,MLω0,HMMω0,KF

M = 80, ω0(1) = 400π/fs , fs = 8.0 kHz, σt =√

2πr/f2s , and for the HMM-based pitch estimator, the frequency rangeω ∈ [150, 280] × (2π/fs ) was discretized into Nd = 1000 samples.

19





ML Pitch Estimation




18 Numerical Results

Conclusion


Numerical ResultsReal signal

Speech signal + Car noise at SNR= 5 dB.Freq.[Hz]

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

500

1000

1500

2000

2500

3000

3500

4000

Time [ s ]

Freq.[Hz]

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1100

150

200

250

300

350

400

WLSMLKFHMM

The MAP order estimation [Djuric 1998], M = 240, λ = 0.9, and N = 150.

19





ML Pitch Estimation




Numerical Results

19 Conclusion


Conclusion

I For pitch estimation, we have formulated the ML estimatefrom the UFE.

I For pitch estimation and tracking, we have proposedHMM- and KF-based methods.

I Experimental results showed that both HMM- andKF-based methods outperform the corresponding ML pitchestimators.

I The KF-based method statistically performs better thanthe HMM-based method, while the it tracks pitch changesmore accurate than the KF-based method.

Thank you!

Pitch Estimation and Tracking with Harmonic Emphasis on ... · Bayesian Pitch Estimator Motivation...

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