SPECTRAL AUDIO SIGNAL PROCESSING

• Preface
  • Book Series Overview
  • Acknowledgments
  
  
• Spectrum Analysis of Sinusoids
  • Spectrum of a Sinusoid
  • Spectrum of a Sampled Sinusoid
  • Spectrum of a Windowed Sinusoid
  • Effect of Windowing
  • The Rectangular Window
    • Rectangular Window Side-Lobes
    • Frequency Resolution
      • Two Cosines (``In-Phase'' Case)
      • One Sine and One Cosine ``Phase Quadrature'' Case
  • Main-Lobe Bandwidth
    • Other Definitions of Main Lobe Width
  • Choosing Window Length to Resolve Sinusoids
    • Periodic Signals
    • Tighter Bounds for Minimum Window Length
  
  
• Cont./Discrete Fourier Transforms
  • Discrete Time Fourier Transform
  • Fourier Transform (FT) and Inverse
    • Existence of the Fourier Transform
  • Fourier Theorems for the DTFT
    • Linearity of the DTFT
    • Time Reversal
    • Symmetry of the DTFT for Real Signals
      • Real Even (or Odd) Signals
    • Shift Theorem
    • Convolution Theorem
    • Correlation Theorem
    • Autocorrelation
    • Power Theorem
    • Stretch Operator
    • Repeat (Scaling) Operator
    • Stretch/Repeat (Scaling) Theorem
    • Downsampling and Aliasing
      • Proof of Aliasing Theorem
    • Differentiation Theorem Dual
  • Continuous Fourier Theorems
    • Radians versus Cycles
    • Differentiation Theorem
    • Differentiation Theorem Dual
    • Scaling Theorem
    • Shift Theorem
    • Convolution Theorem
    • Flip Theorems
    • Power Theorem
    • The Continuous-Time Impulse
    • Gaussian Pulse
    • Rectangular Pulse
    • Sinc Impulse
    • Impulse Trains
    • Poisson Summation Formula
    • Sampling Theory
    • The Uncertainty Principle
      • Second Moments
      • Time-Limited Signals
      • Time-Bandwidth Products are Unbounded Above
  • Relation of Smoothness to Roll-Off Rate
  • Exercises
  
  
• Spectrum Analysis Windows
  • Rectangular Window
    • Definition ( odd):
    • Transform:
    • Properties:
  • Generalized Hamming Window Family
    • Hann or Hanning or Raised Cosine
    • Matlab for the Hann Window
    • Summary of Hann window properties:
    • Hamming Window
    • Matlab for the Hamming Window
    • Summary of Generalized Hamming Windows
    • Definition:
    • Transform:
    • Common Properties
    • Rectangular window properties:
    • Hann window properties:
    • Hamming window properties:
    • The MLT Sine Window
      • Properties:
  • Blackman-Harris Window Family
    • Window Definition:
    • Window Transform:
    • Blackman Window Family
    • Classic Blackman
    • Matlab for the Classic Blackman Window
    • Three-Term Blackman-Harris Window
    • Frequency-Domain Blackman-Harris
    • Power-of-Cosine Window Family
      • Definition:
      • Properties:
      • Special Cases:
  • Bartlett (``Triangular'') Window
    • Definition:
    • Transform:
    • Properties:
    • Matlab for the Bartlett Window:
  • Poisson Window
    • Definition:
  • Hann-Poisson Window
    • Definition:
    • Matlab for the Hann-Poisson Window
  • Slepian or DPSS Window
    • Matlab for the DPSS Window
  • Kaiser Window
    • Definition:
    • Window transform:
    • Kaiser Window Beta Parameter
    • Kaiser Windows and Transforms
    • Minimum Frequency Separation vs. Window Length
    • Kaiser and DPSS Windows Compared
  • Dolph-Chebyshev Window
    • Matlab for the Dolph-Chebyshev Window
    • Example Chebyshev Windows and Transforms
    • Dolph-Chebyshev and Hamming Windows Compared
    • Dolph-Chebyshev Window Theory
      • Chebyshev Polynomials
      • Dolph-Chebyshev Window Definition
      • Dolph-Chebyshev Window Main-Lobe Width
      • Dolph-Chebyshev Window Length Computation
  • Gaussian Window and Transform
    • Matlab for the Gaussian Window
    • Gaussian Window and Transform
    • Exact Discrete Gaussian Window
  • Optimized Windows
    • Optimal Windows for Audio Coding
    • Conclusion
  • Exercises
  • Lab Assignments
  
  
• Spectral Interpolation
  • Ideal Spectral Interpolation
  • Interpolating a DFT
  • Zero Padding in the Time Domain
    • Practical Zero Padding
      • Zero-Padding to the Next Higher Power of 2
      • Zero-Padding for Interpolating Spectral Displays
      • Zero-Padding for Interpolating Spectral Peaks
  • Zero-Phase Zero Padding
    • Matlab/Octave fftshift utility
      • Index Ranges for Zero-Phase Zero-Padding
      • Summary
  • Example: Spectrum Analysis of an Oboe Tone
  • Quadratic Peak Interpolation
    • Quadratic Interpolation of Spectral Peaks
    • Phase Interpolation at a Peak
    • Matlab for Parabolic Peak Interpolation
  • Bias of Parabolic Peak Interpolation
  • Optimal Peak-Finding in the Spectrum
    • Minimum Zero-Padding for High-Frequency Peaks
    • Minimum Zero-Padding for Low-Frequency Peaks
    • Matlab for Computing Minimum Zero-Padding Factors
    • Least Squares Sinusoidal Parameter Estimation
      • Sinusoidal Amplitude Estimation
      • Sinusoidal Amplitude and Phase Estimation
      • Sinusoidal Frequency Estimation
    • Maximum Likelihood Sinusoid Estimation
    • Likelihood Function
      • Multiple Sinusoids in Additive Gaussian White Noise
      • Non-White Noise
    • Generality of Maximum Likelihood Least Squares
  • Problems
  • Lab Assignments
  
  
• Spectrum Analysis of Noise
  • Introduction to Noise
    • Why Analyze Noise?
    • What is Noise?
  • Spectral Characteristics of Noise
  • White Noise
    • Testing for White Noise
  • Sample Autocorrelation
  • Sample Power Spectral Density
  • Biased Sample Autocorrelation
  • Smoothed Power Spectral Density
  • Cyclic Autocorrelation
  • Practical Bottom Line
  • Why an Impulse is Not White Noise
  • The Periodogram
    • Matlab for the Periodogram
  • Welch's Method
    • Welch Autocorrelation Estimate
    • Resolution versus Stability
  • Welch's Method with Windows
    • Matlab for Welch's Method
  • Filtered White Noise
    • Example: FIR-Filtered White Noise
    • Example: Synthesis of 1/F Noise (Pink Noise)
    • Example: Pink Noise Analysis
  • Processing Gain
  • The Panning Problem
  • Problems
  
  
• Time-Frequency Displays
  • The Short-Time Fourier Transform
    • Mathematical Definition of the STFT
    • Practical Computation of the STFT
    • Summary of STFT Computation Using the FFT
    • Two Dual Interpretations of the STFT
    • STFT in Matlab
  • Classic Spectrograms
    • Spectrogram of Speech
  • Audio Spectrograms
    • Auditory Filter Banks
    • Loudness Spectrogram
    • Loudness Spectrogram Examples
      • Multiresolution STFT
      • Excitation Pattern
      • Nonuniform Spectral Resampling
      • Specific Loudness
      • Spectrograms Compared
      • Instantaneous, Short-Term, and Long-Term Loudness
  • Summary
  
  
• Spectral Modeling Synthesis
  • History of Spectral Modeling
    • Early Additive Synthesis in Film Making
    • Vocoder
    • Phase Vocoder
    • Additive Synthesis in Computer Music
    • FFT Implementation of the Phase Vocoder
    • Short Time Fourier Analysis/Modification/Resynthesis
    • Sinusoidal Modeling Systems
      • Transient Models
      • Time-Frequency Reassignment
    • Applications in Perceptual Audio Compression
    • Further Reading in Spectral Modeling
  • Additive Synthesis Overview
  • Additive Synthesis Analysis
    • 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
  • Additive Synthesis Implementations
    • Additive Synthesis from Sinusoidal Peak Tracks
      • Oscillator-Bank Resynthesis
    • Inverse FFT Synthesis
    • Efficient Specialized Methods
      • Wavetable Synthesis
      • Group-Additive Synthesis
      • FM Synthesis
  • Exercises
  • Lab Problem
    • Matlab starter code
  
  
• Overlap-Add STFT Processing
  • Convolution of Short Signals
    • Cyclic FFT Convolution
    • Acyclic FFT Convolution
      • Acyclic Convolution in Matlab or Octave
      • Pictorial View of Acyclic Convolution
    • Acyclic FFT Convolution in Matlab or Octave
    • FFT versus Direct Convolution
      • Audio FIR Filters
      • Example 1: Low-Pass Filtering by FFT Convolution
      • Example 2: Time Domain Aliasing
  • Convolving with Long Signals
    • Overlap-Add Decomposition
    • COLA Examples
    • STFT of COLA Decomposition
    • Acyclic Convolution
    • Example of Overlap-Add Convolution
    • Overlap-Add FFT Processing Summary
    • The STFT as a Time-Frequency Distribution
      • Time-Frequency Parameters in the STFT
  • Dual of Constant Overlap-Add
    • Poisson Summation Formula
    • Frequency-Domain COLA Constraints
      • Strong COLA
    • PSF Dual and Graphical Equalizers
    • PSF and Weighted Overlap Add
    • Example COLA Windows for WOLA
  • Overlap-Save Method
  • Time Varying OLA Modifications
    • Time-Varying STFT Modifications
    • Length L FIR Frame Filters
  • Nonlinear Modifications
  • Weighted Overlap Add
    • WOLA Processing Steps
      • Choice of WOLA Window
  • Review of Zero Padding
  • Exercises
  • Lab Assignments
  
  
• Filter Bank View of the STFT
  • Dual Views of the STFT
    • Overlap-Add View of the STFT
    • Filter Bank View of the STFT
    • FBS and Perfect Reconstruction
  • STFT Filter Bank