Spectral Audio Signal Processing
TOC
- Introduction and Overview
- Fourier Transforms and Theorems
- Organization
- Overview
- Discrete Time Fourier Transform
- Fourier Transform (FT) and Inverse
- Fourier Theorems for the DTFT
- Continuous-Time Fourier Theorems
- Spectral Interpolation
- Spectrum Analysis Windows
- Rectangular Window
- Generalized Hamming Window Family
- Blackman-Harris Window Family
- Spectrum Analysis of an Oboe Tone
- Bartlett (“Triangular”) Window
- Poisson Window
- Hann-Poisson Window
- Slepian or DPSS Window
- Kaiser Window
- Dolph-Chebyshev Window
- Gaussian Window and Transform
- Optimized Windows
- Optimal Window Design by linprog
- Spectrum Analysis of Sinusoids
- Spectrum of a Sinusoid
- Spectrum of Sampled Complex Sinusoid
- Spectrum of a Windowed Sinusoid
- Effect of Windowing
- The Rectangular Window
- Resolving Sinusoids
- Sinusoidal Peak Interpolation
- Optimal Peak-Finding in the Spectrum
- Spectrum Analysis of Noise
- Introduction to Noise
- Spectral Characteristics of Noise
- 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
- Welch’s Method
- Welch’s Method with Windows
- Filtered White Noise
- Processing Gain
- The Panning Problem
- Time-Frequency Displays
- The Short-Time Fourier Transform
- Classic Spectrograms
- Audio Spectrograms
- Overlap-Add STFT Processing
- Convolution of Short Signals
- Dual of Constant Overlap-Add
- Poisson Summation Formula
- Frequency-Domain COLA Constraints
- 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
- Weighted Overlap Add
- Review of Zero Padding
- 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
- Computational Examples in Matlab
- The DFT Filter Bank
- The Running-Sum Lowpass Filter
- Modulation by a Complex Sinusoid
- Making a Bandpass Filter from a Lowpass Filter
- Uniform Running-Sum Filter Banks
- FBS Window Constraints for R=1
- Nyquist(N) Windows
- Duality of COLA and Nyquist Conditions
- Specific Windows
- The Nyquist Property on the Unit Circle
- Portnoff Windows
- Downsampled STFT Filter Banks
- Downsampled STFT Filter Bank
- Downsampling with Anti-Aliasing
- Constant-Overlap-Add (COLA) Cases
- STFT with Modifications
- FBS Fixed Modifications
- Time Varying Modifications in FBS
- STFT Summary and Conclusions
- Applications of the STFT
- Estimation from Sinusoidal Peaks
- Cross Synthesis
- Spectral Envelope Extraction
- Cepstral Windowing
- Linear Prediction Spectral Envelope
- Spectral Envelope Examples
- 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
- Sines + Noise + Transients Models
- Gaussian Windowed Chirps
- Modulated Gaussian Windowed Chirp
- Identifying Chirp Rate
- 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
- FFT Filter Banks
- Audio Filter Banks
- Basic Idea
- Inverse Transforming STFT Bin Groups
- Fast Octave Filter Banks
- Pointers to Demos
- Multirate Filter Banks
- Upsampling and Downsampling
- Upsampling (Stretch) Operator
- Downsampling (Decimation) Operator
- Filtering and Downsampling
- Polyphase Filtering
- Two-Channel Case
- N-Channel Polyphase Decomposition
- Type II Polyphase Decomposition
- Filtering and Downsampling, Revisited
- Multirate Noble Identities
- Critically Sampled PR Filter Banks
- Two-Channel Critically Sampled Filter Banks
- Amplitude-Complementary 2-Channel Filter Bank
- Haar Example
- Polyphase Decomposition of Haar Example
- Quadrature Mirror Filters (QMF)
- Linear Phase Quadrature Mirror Filter Banks
- Conjugate Quadrature Filters (CQF)
- Orthogonal Two-Channel Filter Banks
- Perfect Reconstruction Filter Banks
- Simple Examples of Perfect Reconstruction
- Sliding Polyphase Filter Bank
- Hopping Polyphase Filter Bank
- Sufficient Condition for Perfect Reconstruction (PR)
- Necessary and Sufficient Conditions for PR
- Polyphase View of the STFT
- Polyphase View of the Overlap-Add STFT
- Polyphase View of the Weighted-Overlap-Add STFT
- Paraunitary Filter Banks
- Filter Banks Equivalent to STFTs
- Polyphase Analysis of Portnoff STFT
- MPEG Filter Banks
- Pseudo-QMF Cosine Modulation Filter Bank
- Perfect Reconstruction Cosine Modulated Filter Banks
- MPEG Layer III Filter Bank
- Review of STFT Filterbanks
- STFT, Rectangular Window, No Overlap
- STFT, Rectangular Window, 50% Overlap
- STFT, Triangular Window, 50% Overlap
- STFT, Hamming Window, 75% Overlap
- STFT, Kaiser Window, Beta=10, 90 % Overlap
- Sliding FFT, Any Window, Zero-Padded by 5
- Wavelet Filter Banks
- Geometric Signal Theory
- Further Reading
- Summary and Conclusions
- Notation
- Frequency and Time
- Signal Notation
- Fourier Transform Notation
- More Fourier Theorems
- Continuous Fourier Theorems
- Relation of Smoothness to Roll-Off Rate
- Statistical Signal Processing
- Stochastic Processes
- Correlation Analysis
- Cross-Correlation
- Cross-Power Spectral Density
- Autocorrelation
- Sample Autocorrelation
- Power Spectral Density
- Sample Power Spectral Density
- White Noise
- Making White Noise with Dice
- Independent Implies Uncorrelated
- Estimator Variance
- Gaussian Function Properties
- Gaussian Window and Transform
- Gaussians Closed under Multiplication
- Product of Two Gaussian PDFs
- Gaussians Closed under Convolution
- Fitting a Gaussian to Data
- Infinite Flatness at Infinity
- Integral of a Complex Gaussian
- Gaussian Integral with Complex Offset
- Fourier Transform of Complex Gaussian
- Why Gaussian?
- Central Limit Theorem
- Iterated Convolutions
- Binomial Distribution
- Boltzmann Energy Distribution in Potential Well
- Gaussian Probability Density Function
- Maximum Entropy Property
- Entropy of a Probability Distribution
- Example: Random Bit String
- Maximum Entropy Distributions
- Gaussian Moments
- Gaussian Mean
- Gaussian Variance
- Higher Order Moments Revisited
- Moment Theorem
- Gaussian Characteristic Function
- Gaussian Central Moments
- Sums of Gaussian Random Variables
- Uncertainty Principle
- History of the Gaussian Function
- FIR Digital Filter Design
- The Ideal Lowpass Filter
- Least-Squares Impulse Response Design
- Examples
- Frequency-Sampling FIR Design
- The Window Method
- Matlab Support for the Window Method
- Online Demonstration of the Window Method
- Bandpass Filter Design Example
- Hilbert Transform Design Example
- Choice of Kaiser Window
- Windowing an “Ideal” Impulse Response
- Hilbert Transformer by the Window Method
- Comparison to Optimal Chebyshev FIR Filter
- Comparison to use of the hilbert function
- Postlude on Hilbert Transform Theory
- Generalized Window Method
- Minimum Phase Filter Design
- Minimum Phase and Causal Cepstra
- Optimal FIR Digital Filter Design
- Filter Specifications
- Ideal Lowpass Filter Revisited
- Optimal Least-Squares Filters
- Optimal Chebyshev Filters
- Lp norms
- Least-Squares Linear-Phase FIR Filter Design
- Chebyshev Optimal Linear Phase filter Design
- More General Real FIR Filters
- Complex FIR Filter Design
- Arbitrary FIR Magnitude and Phase Specification
- Second-Order Cone Problems (SOCP)
- Nonlinear Optimization in Octave
- Bilinear Audio Frequency Warping
- Introduction
- Bilinear Transform Frequency Warping
- “Allpass” Frequency Warping
- Section Outline
- The Bark Frequency Scale
- The Bilinear Transform
- Optimal Bark Warping
- Equivalent Rectangular Bandwidth
- Directions for Improvements
- Summary
- Simple Approximations to Various Frequency Warpings
- Examples in Matlab and Octave
- Matlab for Spectrum Analysis Windows
- Interpolating Spectral Peaks
- Matlab for Computing Spectrograms
- Matlab for Unwrapping Spectral Phase
- Non-Parametric Frequency Warping
- Fundamental Frequency Estimation
- Spectral Audio Modeling History
- Daniel Bernoulli’s Modal Decomposition
- The Telharmonium
- Early Additive Synthesis in Film Making
- The Hammond Organ
- Dudley’s Channel Vocoder
- Voder
- Phase Vocoder
- FFT Implementation of the Phase Vocoder
- Phase Vocoder Applications
- Additive Synthesis
- Oscillator-Bank Resynthesis
- Inverse FFT Synthesis
- Chirplet Synthesis
- Nonparametric Spectral Peak Modeling
- Efficient Specialized Methods
- Further Reading, Additive Synthesis
- Frequency Modulation (FM) Synthesis
- Phase Vocoder Sinusoidal Modeling
- Computing Vocoder Parameters
- Further Reading on Vocoders
- Speech Synthesis Examples
- Spectral Modeling Synthesis
- Perceptual audio coding
- Future Prospects
- The PARSHL Program
- Choice of Hop Size
- Filling the FFT Input Buffer (Step 2)
- Peak Detection (Steps 3 and 4)
- Peak Matching (Step 5)
- Parameter Modifications (Step 6)
- Synthesis (Step 7)
- Magnitude-only Analysis/Synthesis
- Preprocessing
- Applications
- Conclusions
- Acknowledgments
- Software Listing
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