1. Convex Optimization in Sinusoidal Modeling for Audio Signal Processing Michelle Daniels PhD Student, University of California, San Diego

2. Outline  Introduction to sinusoidal modeling  Existing approach  Proposed optimization post-processing  Testing and results  Conclusions  Future work 2

3. 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

4. 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  )  magnitude ( a ), and phase  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 4

5. 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   Given , estimate its magnitude a and phase   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

6. Sinusoidal Analysis Example 6

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10. 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

11. 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: 11

12. 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  and   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 12

13. Convex Optimization Problem Magnitude and phase recovered as: Classic least-squares problem: 13

14. 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

15. 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 15

16. Final Formulation 16  Quadratic Program:  Magnitude and phase recovered from x as:

17. 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

18. 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

19. 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

20. Test Signal 2: Sudden Onset  A single sinusoid starts half-way through an analysis frame (the first half is silence) 20

21. Results 2: Sudden Onset 21 Original: MSE* = 2.76x10 -5 Optimized: MSE* = 4.13x10 -6 *MSE = Mean Squared Error

22. Test Signal 3: Chirp  A single sinusoid with constant magnitude and continuously-increasing frequency 22

23. 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

24. 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

25. 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

26. 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

27. Thanks for your attention! For further information: http://ccrma.stanford.edu/~danielsm/ifors2011.html 27

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29. Convex Reformulation Define: Change of variables: Define: 29

30. 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

31. 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

32. 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