Loading in 2 Seconds...

Convex Optimization in Sinusoidal Modeling for Audio Signal Processing

Loading in 2 Seconds...

- 197 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about ' Convex Optimization in Sinusoidal Modeling for Audio Signal Processing' - astrid

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

### Convex Optimization in Sinusoidal Modeling forAudio 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

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)

Sinusoidal Modeling of Audio Signals

- 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

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

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

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:

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

Related Work

- 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

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

Final Formulation

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

Test Signals

- 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

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

Test Signal 2: Sudden Onset

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

Test Signal 3: Chirp

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

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.

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)

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

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.
- 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.

Thanks for your attention!

For further information:

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

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)

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.

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.

Download Presentation

Connecting to Server..