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Convex Optimization in Sinusoidal Modeling for Audio Signal Processing

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Convex Optimization in Sinusoidal Modeling forAudio Signal Processing

Michelle Daniels

PhD Student, University of California, San Diego

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

- 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)

- Given a signal y of length N, represent as Kcomponent 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

- 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

- 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

- 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:

- 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

Classic least-squares problem:

Magnitude and phase recovered as:

- PetreStoica, 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

- 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

- Quadratic Program:
- Magnitude and phase recovered from x as:

- 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

- 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

- 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

- A single sinusoid starts half-way through an analysis frame (the first half is silence)

Original:

MSE* = 2.76x10-5

Optimized:MSE* = 4.13x10-6

*MSE = Mean

Squared Error

- A single sinusoid with constant magnitude and continuously-increasing frequency

- 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.

- 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)

- 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

- 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.
- PetreStoica, 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.

For further information:

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

Define:

Change of variables:

Define:

- 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)

- 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.

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