Convex Optimization in Sinusoidal Modeling for Audio Signal ProcessingMichelle DanielsPhD Student, University of California, San Diego

Outline Introduction to sinusoidal modeling Existing approach Proposed optimization post-processing Testing and results Conclusions Future work

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Analysis of Audio Signals Audio signals have rapid variations

Speech Music Environmental sounds

Assume minimal change over short segments (frames) Analyze on a frame-by-frame basis

Constant-length frames (46ms) Frames typically overlap

Any audio signal can be represented as a sum of sinusoids (deterministic components) and noise (stochastic components)

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Sinusoidal Modeling of Audio Signals Given a signal y of length N, represent as K component sinusoids plus noise e:

y and e are N-dimensional vectors Each sinusoid has frequency (w), magnitude (a), and phase (f) parameters K is determined during the analysis process Higher-resolution frequencies than DFT bins, no harmonic relationship required Model, encode, and/or process these components independently Applications:

Effects processing (time-scale modification, pitch shifting) Audio compression Feature extraction for machine listening Auditory scene analysis

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, cos 1( )K

n k k k nk

a n n Nw f

y e

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Estimation Algorithm Using frequency domain analysis (e.g. FFT), iterate up to K

times, until residual signal is small and/or has a flat spectrum: Identify the highest-magnitude sinusoid in the signal Estimate its frequency w Given w, estimate its magnitude a and phase f Reconstruct the sinusoid Subtract the reconstructed sinusoid to produce a residual

signal After all sinusoids have been removed, the final residual

contains only noise

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Sinusoidal Analysis Example6

Sinusoidal Analysis Example7

Sinusoidal Analysis Example8

Sinusoidal Analysis Example9

Estimation Challenges Energy in any DFT bin can come from:

Multiple sinusoids with similar frequency Both sinusoids and noise

Interference from other sinusoids and/or noise results in inaccurate estimates

Incorrect estimation of a single sinusoid corrupts the residual signal and affects all subsequent estimates

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Possible Solution Optimize frequency, magnitude, and phase to

minimize the energy in the residual signal The original parameter estimates are initial

estimates for the optimization Sinusoidal approximation: Residual: Optimization problem:

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ˆ cos( , 1)K

n k k kk

a n n Nw f

yˆ e y y

, , 2min || subject to ˆ || 0, 1a ka k Kw f yy

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Is it Convex?

Want convexity so the problem is practical to solve Not a convex optimization problem because each element of ŷ

is a sum of cosine functions of w and f Want convex function inside of the 2-norm instead With fixed frequencies, can reformulate optimization of

magnitudes and phases as convex problem Fix frequencies to initial estimates

, , 2min || subject to ˆ || 0, 1a ka k Kw f yy

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Convex Optimization Problem

2 22min || || , , , N K K N

x R Rx y A x y RA

1 2 1 2

1 2 1 2

1 2 1

sin(0) sin(0) sin(0) cos(0) cos(0) cos(0)sin( ) sin( ) sin( ) cos( ) cos( ) cos( )

sin(2 ) sin(2 ) sin(2 ) cos(2 ) cos(2 ) cos(2 )

sin(( 1) ) sin(( 1) ) sin(( 1) ) cos(( 1) ) cos(( 1)

K K

K K

KN N N N N

w w w w w ww w w w w w

w w w w

A

2 ) cos(( 1) )KNw w

2 2 1 and tan 2

k Kk k k K k

k

a f

xx xx

Magnitude and phase recovered as:

Classic least-squares problem:

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Related Work Petre Stoica, Hongbin Li, and Jian Li. “Amplitude estimation

of sinusoidal signals: Survey, new results, and an application”, 2000. Mentions least-squares as one approach to estimate amplitude

of complex exponentials No discussion of phase estimation

Hing-Cheung So. “On linear least squares approach for phase estimation of real sinusoidal signals”, 2005. Focuses on phase estimation Theoretical analysis

Not applied specifically to audio signals

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Constraints Analytic least-squares solution frequently results in

unrealistic magnitude values This is possibly the result of errors in frequency estimates

Constraints on magnitudes were required Ideal constraint: Relaxed constraint: Result is a constrained least squares problem that

can be solved using a generic quadratic program (QP) solver

2 2max0 , 1 k k K a k K x x

max max , 1 ka a k K x

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Final Formulation16

Quadratic Program:

Magnitude and phase recovered from x as:

2 max maxmin || || subject to , 1 x ka a k K Ax y x

2 2 1 and tan 2

k Kk k k K k

k

a f

xx xx

Test Signals17

Model test signals that reproduce challenging aspects of real-world signals

Reconstruct signal based on original model parameters and optimized parameters

Compare both reconstructions to original test signal and to each other

Test Signal 1: Overlapping Sinusoids Signal consists of two sinusoids close in frequency There is no additive noise, so the residual (the

noise component of the model) should be zero

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Results 1: Overlapping Sinusoids Without optimization, there is significant energy left in the residual (very

audible) With optimization, the residual power at individual frequencies is reduced

by as much as 50dB (now barely audible) The improvement with optimization generally decreases as the frequency

separation is increased

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Test Signal 2: Sudden Onset A single sinusoid starts half-way through

an analysis frame (the first half is silence)

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Results 2: Sudden Onset21

Original:MSE* = 2.76x10-5

Optimized:MSE* = 4.13x10-6

*MSE = Mean Squared Error

Test Signal 3: Chirp A single sinusoid with constant

magnitude and continuously-increasing frequency

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Results 3: Chirp Non-optimized peak magnitudes are close to constant between

consecutive frames Optimized peak magnitudes vary significantly from frame to frame The optimization produces peak parameters that do not reflect the

underlying real-world phenomenon.

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Conclusions Problem can be formulated using convex programming For several classic challenging signals, optimization

produces a more accurate model Constraints are necessary to ensure parameter

estimates reflect possible real-world phenomena Final formulation is quadratic program Parameters obtained via optimization may still not

represent the underlying real-world phenomenon as well as the original analysis (i.e. chirp)

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Future Work Explore robust optimization techniques to

compensate for errors in frequency estimates Integrate optimization into original analysis

instead of a post-processing stage Experiment with more real-world signals Further investigate constraints The ultimate goal: three-way joint optimization of

frequency, magnitude, and phase

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References M. Grant and S. Boyd. CVX: Matlab software for disciplined convex programming,

version 1.21. http://cvxr.com/cvx, May 2010. R. McAulay and T. Quatieri. Speech analysis/synthesis based on a sinusoidal

representation. IEEE Transactions on Acoustics, Speech, and Signal Processing, 34(4):744-754, Aug 1986.

Xavier Serra. A System for Sound Analysis/Transformation/Synthesis Based on a Deterministic Plus Stochastic Decomposition. PhD thesis, Stanford University, 1989.

Kevin M. Short and Ricardo A. Garcia. Accurate low-frequency magnitude and phase estimation in the presence of DC and near-DC aliasing. In Proceedings of the 121st Convention of the Audio Engineering Society, 2006.

Kevin M. Short and Ricardo A. Garcia. Signal analysis using the complex spectral phase evolution (CSPE) method. In Proceedings of the 120th Convention of the Audio Engineering Society, 2006.

Hing-Cheung So. On linear least squares approach for phase estimation of real sinusoidal signals. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, E88-A(12):3654-3657, December 2005.

Petre Stoica, Hongbin Li, and Jian Li. Amplitude estimation of sinusoidal signals: Survey, new results, and an application. IEEE Transactions on Signal Processing, 48(2):338-352, 2000.

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Thanks for your attention!

For further information:

http://ccrma.stanford.edu/~danielsm/ifors2011.html

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THE END28

Convex ReformulationDefine:

Change of variables:

Define:

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Test Signal: Sinusoid in noise A single sinusoid with stationary frequency

and corrupted by additive white Gaussian noise

Noise is present at all frequencies, including that of the sinusoid, corrupting magnitude and phase estimates

Test repeated using different variances for the noise (varying signal-to-noise ratios)

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Results: Sinusoid in noise• Without optimization, the sinusoid’s magnitude is over-

estimated and the noise’s energy is under-estimated• The optimization gives residual energy slightly closer to the

true noise energy.

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Results: Overlapping SinusoidsThe optimization is able to compensate for some of the errors in initial magnitude and phase estimation, resulting in a lower MSE.

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