Convex Optimization in
Sinusoidal Modeling for
Audio Signal Processing
Michelle Daniels
PhD Student, University of California, San Diego
Outline
Introduction to sinusoidal modeling
Existing approach
Proposed optimization post

processing
Testing and results
Conclusions
Future work
2
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)
3
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
1
,
cos
1
( )
K
n k k k n
k
a n
n N
w f
y e
4
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
5
Sinusoidal Analysis Example
6
Sinusoidal Analysis Example
7
Sinusoidal Analysis Example
8
Sinusoidal Analysis Example
9
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
10
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:
1
ˆ
cos(
, 1
)
K
n k k k
k
a n
n N
w f
y
ˆ
e y
y
,,2
min  subject to
ˆ
 0, 1
a k
a
k K
w f
y
y
11
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
,,2
min  subject to
ˆ
 0, 1
a k
a
k K
w f
y
y
12
Convex Optimization Problem
2 2
2
min  , , ,
N K K N
x
R R
x y A x y
R
A
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
K
N N N N N
w w w w w w
w w w w w w
w w w w
A
2
) cos(( 1) )
K
N
w w
2 2 1
and tan
2
k K
k k k K k
k
a
f
x
x x
x
Magnitude and phase recovered as:
Classic least

squares problem:
13
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
14
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 2
max
0, 1
k k K
a k K
x x
max max
, 1
k
a a k K
x
15
Final Formulation
16
Quadratic Program:
Magnitude and phase recovered from
x
as:
2
max max
min   subject to , 1
x
k
a a k K
Ax y x
2 2 1
and tan
2
k K
k k k K k
k
a
f
x
x x
x
Test Signals
17
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
18
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
19
Test Signal 2: Sudden Onset
A single sinusoid starts half

way through an analysis
frame (the first half is silence)
20
Results 2: Sudden Onset
21
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
22
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.
23
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)
24
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
25
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.
26
Thanks for your attention!
For further information:
http://ccrma.stanford.edu/~danielsm/ifors2011.html
27
THE END
28
Convex Reformulation
Define:
Change of variables:
Define:
29
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)
30
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.
31
Results: Overlapping Sinusoids
The optimization is able to compensate for some of the errors in
initial magnitude and phase estimation, resulting in a lower MSE.
32
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