Convex Optimization in
Sinusoidal Modeling for
Audio Signal Processing
Michelle Daniels
PhD Student, University of California, San Diego
Outline
2
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Introduction to sinusoidal modeling
Existing approach
Proposed optimization post-processing
Testing and results
Conclusions
Future work
Analysis of Audio Signals
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Audio signals have rapid variations
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Assume minimal change over short segments (frames)
Analyze on a frame-by-frame basis
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Speech
Music
Environmental sounds
Constant-length frames (46ms)
Frames typically overlap
Any audio signal can be represented as a sum of sinusoids
(deterministic components) and noise (stochastic components)
Sinusoidal Modeling of Audio Signals
4

Given a signal y of length N, represent as K component sinusoids plus noise e:
K
y n   ak cos(wk n fk )  e n, 1  n  N
k 1
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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:
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Effects processing (time-scale modification, pitch shifting)
Audio compression
Feature extraction for machine listening
Auditory scene analysis
Estimation Algorithm
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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
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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
Sinusoidal Analysis Example
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Sinusoidal Analysis Example
7
Sinusoidal Analysis Example
8
Sinusoidal Analysis Example
9
Estimation Challenges
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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
Possible Solution
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Optimize frequency, magnitude, and phase to minimize
the energy in the residual signal
The original parameter estimates are initial estimates
for the optimization
K
yˆ n   ak cos(wk n fk ), 1  n  N
Sinusoidal approximation:
k 1
ˆ
e

y

y
Residual:
Optimization problem:
minw ,a,f || y  yˆ ||2 subject to ak  0, 1  k  K
Is it Convex?
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minw ,a,f || y  yˆ ||2 subject to ak  0, 1  k  K
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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
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Fix frequencies to initial estimates
Convex Optimization Problem
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Classic least-squares problem:
min x || Ax  y ||2 , A  R N 2 K , x  R 2 K , y  R N
sin(0)
sin(0)


sin(w2 )
 sin(w1 )
A   sin(2w1 )
sin(2w2 )


 sin(( N  1)w ) sin(( N  1)w )
1
2

sin(0)
cos(0)
cos(0)
sin(wK )
cos(w1 )
cos(w2 )
sin(2wK )
cos(2w1 )
cos(2w2 )
sin(( N  1)wK ) cos(( N  1)w1 ) cos(( N  1)w2 )
Magnitude and phase recovered as:


cos(wK ) 
cos(2wK ) 


cos(( N  1)wK ) 
cos(0)
x
ak  xk2  xk2  K and fk  tan 1  k  K
 xk
 

 2
Related Work
14

Petre Stoica, Hongbin Li, and Jian Li. “Amplitude estimation of
sinusoidal signals: Survey, new results, and an application”, 2000.
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Hing-Cheung So. “On linear least squares approach for phase
estimation of real sinusoidal signals”, 2005.
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Mentions least-squares as one approach to estimate amplitude of
complex exponentials
No discussion of phase estimation
Focuses on phase estimation
Theoretical analysis
Not applied specifically to audio signals
Constraints
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Analytic least-squares solution frequently results in
unrealistic magnitude values
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This is possibly the result of errors in frequency estimates
Constraints on magnitudes were required
Ideal constraint:
0 x x
 a , 1 k  K
Relaxed constraint: a  x  a , 1  k  K
Result is a constrained least squares problem that can
be solved using a generic quadratic program (QP)
solver
2
k
max
2
k K
k
max
max
Final Formulation
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
Quadratic Program:
min x || Ax  y ||2 subject to  amax  x k  amax , 1  k  K
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Magnitude and phase recovered from x as:
ak  x  x
2
k
2
k K
 xk  K
and fk  tan 
 xk
1
 

 2
Test Signals
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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
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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
Results 1: Overlapping Sinusoids
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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
Test Signal 2: Sudden Onset
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A single sinusoid starts half-way through an analysis
frame (the first half is silence)
Results 2: Sudden Onset
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Original:
MSE* =
2.76x10-5
Optimized:
MSE* =
4.13x10-6
*MSE = Mean
Squared Error
Test Signal 3: Chirp
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A single sinusoid with constant magnitude and
continuously-increasing frequency
Results 3: Chirp
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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.
Conclusions
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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)
Future Work
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
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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
References
26
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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.
Thanks for your attention!
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For further information:
http://ccrma.stanford.edu/~danielsm/ifors2011.html
THE END
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Convex Reformulation
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Define:
Change of variables:
Define:
Test Signal: Sinusoid in noise
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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)
Results: Sinusoid in noise
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•
•
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.
Results: Overlapping Sinusoids
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The optimization is able to compensate for some of the errors in
initial magnitude and phase estimation, resulting in a lower MSE.