Applications of the STFT

This chapter briefly tours selected applications involving spectral audio signal processing, with associated examples in matlab. Related historical background and techniques appear in Appendix H. In this chapter, however, we relax historical considerations and focus primarily on efficient practical methods involving the Short-Time Fourier Transform (STFT) (defined and discussed in §6.1). The following applications are considered:

• fundamental frequency estimation from spectral peaks
• cross synthesis
• spectral envelope extraction by cepstral smoothing
• spectral envelope extraction by linear prediction
• sinusoidal modeling of audio signals
• sines+noise modeling
• sines+noise+transients modeling
• chirplet modeling
• time-scale modification
• frequency shifting
• audio FFT filter banks

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Subsections

• Estimation from Sinusoidal Peaks
  • Useful Preprocessing
  • Getting Closer to Maximum Likelihood
  • References on Estimation
  
  
• Cross Synthesis
• Spectral Envelope Extraction
  • Cepstral Windowing
  • Linear Prediction Spectral Envelope
    • Linear Prediction is Peak Sensitive
    • Linear Prediction Methods
    • Computation of Linear Prediction Coefficients
    • Linear Prediction Order Selection
    • Summary of LP Spectral Envelopes
  • Spectral Envelope Examples
    • Signal Synthesis
    • Spectral Envelope by the Cepstral Windowing Method
    • Spectral Envelope by Linear Prediction Method
    • Linear Prediction in Matlab and Octave
  
  
• Sinusoidal Modeling of Sound
• Additive Synthesis Overview
• Additive Synthesis Analysis
  • Following Spectral Peaks
  • Sinusoidal Peak Finding
  • Tracking Sinusoidal Peaks in a Sequence of FFTs
  
  
• Sines + Noise Modeling
  • Sines+Noise Analysis Procedure
  
  
• Sines + Noise + Transients
  • Why Model Transients Separately?
  • Transient models
  • Time Scale Modification
  
  
• Sines + Noise + Transients Models
• Gaussian Windowed Chirps
  • Modulated Gaussian Windowed Chirp
  • Identifying Chirp Rate
    • Chirplet Frequency-Rate Estimation
    • Simulation Results
  
  
• Time Scale Modification
  • TSM Using WOLA Phase Vocoder
  • Phase Continuation in a Time-Scaling Vocoder
  • More Recent Phase-Continuation Methods
  • Time-Scaling Phase Vocoder in Matlab
  • Example Vocoder Waveforms
  • References
  
  
• FFT Filter Banks
  • Audio Filter Banks
  • Basic Idea
    • Summing STFT Bins
  • Inverse Transforming STFT Bin Groups
    • Improving the Channel Filters
  • Fast Octave Filter Banks
    • Spectral Rotation of Real Signals
    • Improving the Octave Band Filters
    • Aliasing on Downsampling
    • Restricting Aliasing to Stopbands
    • Optimal Band Filters
  
  
• Pointers to Demos

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Next: Fundamental Frequency Estimation from Sinusoidal Peaks

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.