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  • Filters and LTI systems
    [notes]
    • Filters and LTI systems
    • discrete-time Fourier transforms
    • z-Transforms
    • Linear-phase filters
  • Continuous-domain signal processing
    [notes]
    • Continuous-domain LTI systems and convolution
    • Support, compactness, and FIR/IIR filters
    • Fourier and Laplace transforms
    • The Dirac impulse and its derivatives
  • Finite-energy signals and sampling
    [notes]
    • Finite-energy signals, inner products, and norms in ℓ2 and L2
    • Parseval’s and Plancherel’s identities
    • Ideal sampling and the Fourier transform of the Dirac comb
    • Spectral periodization and aliasing
  • Sampling and oversampling
    [notes]
    • The sampling theorem and sinc reconstruction
    • Oversampling and reconstruction from noisy samples
    • Error analysis using Poisson summation and Parseval’s identity
    • Mean-squared error reduction through oversampling
  • Gibbs phenomena and the need for localization
    [notes]
    • Fourier series and orthogonal projection in L2[0,1]
    • Low-pass filtering and convolution with the Dirichlet kernel
    • The Gibbs phenomenon and Fejér averaging
    • Localized approximation, cell averages, and Haar wavelets
  • Generalized sampling and multiresolution representations
    [notes]
    • Generalized sampling and piecewise-constant reconstruction
    • Local averages, resolution, and unit-energy normalization
    • Haar scaling functions, wavelets, and two-scale equations
    • Approximation and detail spaces, orthogonal direct sums, and multiresolution analysis
  • Multiresolution analysis and wavelet filtering
    [notes]
    • Multiresolution analysis and orthogonal detail spaces
    • Wavelet bases, direct-sum decompositions, and signal synthesis
    • Two-scale coefficients from orthogonal projections
    • Haar wavelet analysis through filtering and downsampling
  • Wavelet processing and perfect-reconstruction filter banks
    [notes]
    • The wavelet processing pipeline: acquisition, DWT, coefficient processing, and reconstruction
    • Inverse DWT through upsampling and synthesis filtering
    • Two-channel perfect-reconstruction conditions and alias cancellation
    • Haar filter banks and verification of perfect reconstruction
  • Designing two-channel perfect-reconstruction filter banks
    [notes]
    • Perfect-reconstruction conditions and synthesis-filter design
    • FIR filter banks through product-filter factorization
    • Polyphase decomposition and interpolation filters
    • Linear-phase half-band filters and complementary frequency responses
  • Orthogonal filter banks
    [notes]
    • Orthogonal filter banks, orthonormal expansions, and energy preservation
    • Double-shift orthogonality and completeness
    • Power complementarity, low-pass normalization, and even/odd tap sums
    • Conjugate mirror filters and construction of orthogonal perfect-reconstruction banks
  • Wavelets from filters
    [notes]
    • The cascade algorithm and scaling functions as fixed points
    • Fourier-domain refinement and the infinite-product formula
    • The fundamental theorem of wavelet analysis and construction of orthogonal MRAs
    • Recovering the Haar scaling function and wavelet from their filters