- Spectrum of a Sinusoid
- Spectrum of a Sampled Sinusoid
- Spectrum of a Windowed Sinusoid
- Effect of Windowing
- The Rectangular Window
- Main-Lobe Bandwidth
- Choosing Window Length to Resolve Sinusoids
- Discrete Time Fourier Transform
- Fourier Transform (FT) and Inverse
- Fourier Theorems for the DTFT
- 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
- Relation of Smoothness to Roll-Off Rate
- Exercises
- Rectangular Window
- 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
- Blackman-Harris Window Family
- Bartlett (``Triangular'') Window
- Poisson Window
- Hann-Poisson Window
- Slepian or DPSS Window
- Kaiser Window
- Dolph-Chebyshev Window
- Gaussian Window and Transform
- Optimized Windows
- Exercises
- Lab Assignments
odd):- Ideal Spectral Interpolation
- Interpolating a DFT
- Zero Padding in the Time Domain
- Zero-Phase Zero Padding
- Example: Spectrum Analysis of an Oboe Tone
- Quadratic Peak Interpolation
- Bias of Parabolic Peak Interpolation
- Optimal Peak-Finding in the Spectrum
- Problems
- Lab Assignments
- 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
- Problems
- 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
- Applications in Perceptual Audio Compression
- Further Reading in Spectral Modeling
- Additive Synthesis Overview
- Additive Synthesis Analysis
- Sines + Noise Modeling
- Sines + Noise + Transients
- Additive Synthesis Implementations
- Exercises
- Lab Problem
- Convolution of Short Signals
- Convolving with Long Signals
- Dual of Constant Overlap-Add
- Overlap-Save Method
- Time Varying OLA Modifications
- Nonlinear Modifications
- Weighted Overlap Add
- Review of Zero Padding
- Exercises
- Lab Assignments
- Dual Views of the STFT
- STFT Filter Bank
- The DFT Filter Bank
- FBS Window Constraints for R=1
- Nyquist(N) Windows
- Duality of COLA and Nyquist Conditions
- Portnoff Windows
- Downsampled STFT Filter Banks
- STFT with Modifications
- STFT Summary and Conclusions
- Exercises
- Lab Assignments
- Cross Synthesis
- Spectral Envelope Extraction
- Vocoders
- PARSHL
- 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
- Sines + Noise + Transients Models
- Time Scale Modification
- Pitch Detection from Sinusoidal Peaks
- Demos
- Upsampling and Downsampling
- Polyphase Filtering
- Critically Sampled PR 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
- MPEG Filter Banks
- Review of STFT Filterbanks
- Further Reading
- The Ideal Lowpass Filter
- Least-Squares Impulse Response Design
- Frequency-Sampling FIR Design
- The Window Method
- Hilbert Transform Design Example
- Generalized Window Method
- Minimum Phase Filter Design
- Minimum Phase and Causal Cepstra
- The Hilbert Transform
- FIR Fractional Delay Filter Design
- 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
- Optimal FIR Filters: Arbitrary Magnitude and Phase
- Second-Order Cone Problems (SOCP)
- Optimal Window Design by linprog
- Linear Programming (LP)
- Matlab's LINPROG
- LP Formulation of Chebyshev Window Design
- Symmetric Window Constraint
- Positive Window Sample Constraint
- DC Constraint
- Sidelobe Specification
- Sidelobe Specification (cont.)
- LP Standard Form
- Remez Exchange Algorithm
- Monotonicity Constraint
- L-Infinity Norm of Derivative Objective
- L-One Norm of Derivative Objective
- Exercises
- Lab Assignments
Design, Cont'd - Gaussian Window and Transform
- Integral of a Complex Gaussian
- Gaussian Integral with Complex Offset
- Fourier Transform of Complex Gaussian
- Gaussian Windowed Chirps
- Identifying Chirp Rate
- Simulation Results
- Gaussians Closed under Multiplication
- Infinite Flatness at Infinity
- Gaussian Probability Density Function
- Maximum Entropy Property
- Uncertainty Principle
- Gaussian Moments
- Sums of Gaussian Random Variables
- Why Gaussian?
- History of the Gaussian Function
- References
- Introduction
- Frequency Warping via the Bilinear Transform
- ``Allpass'' Frequency Warping
- Section Outline
- The Bark Frequency Scale
- The Bilinear Transform
- Optimal Bark Warping
- Equivalent Rectangular Bandwidth
- Directions for Improvements
- Summary
