Sinusoidal Modeling for

yakzephyrAI and Robotics

Nov 24, 2013 (3 years and 6 months ago)

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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