lecture12: Fractals, and wavelets + non-standard sampling
NB: not examinable

This lecture concerns a number of advanced topics: fractals and wavelets, and non-standard sampling. Note that this material is not examinable this year.

Topics covered

• Self-similarity
• Self-similarity: Koch Snowflake
• Self-similarity: IFS Fern
• Mandelbrot set I
• Mandelbrot set II
• Mandelbrot set III
• Statistical Self-similarity
• Ethernet traffic
• Statistical Self-similarity
• Example fGN: ($H=0.5$)
• Example fGN: ($H=0.75$)
• Example fGN: ($H=0.99$)
• Properties of Self-Similar Process %
• Asymptotic Statistical Self-similarity
• Long-range dependence
• LRD and SS
• LRD in the frequency domain
• Example fGN spectrum ($H=0.5$)
• Example fGN spectrum ($H=0.75$)
• Example fGN spectrum ($H=0.99$)
• 1/f noise
• Connection to Fractals
• fractional Gaussian Noise
• fractional Brownian Motion
• Wavelets: interpretation
• Dyadic grid
• Wavelet's as sub-band filters
• Wavelets and scaling
• Logscale diagram
• Wavelet estimator properties
• Shannon theorem
• Shannon interpolation
• Other sampling schemes
• Ordinate and Slope Sampling
• Interlaced sampling
• Implicit sampling
• Implicit sampling theory
• Irregular sampling
• Non-bandlimited signals
• Astronomical data
• Periodogram
• Lomb-Scargle Periodogram
• Lomb-Scargle Periodogram explained
• Nyquist limits
• Lomb-Scargle Periodogram examples
• Folded Plots
• Folded Plot example
• 2D irregular sampling: CGI jittering
• 2D possibilities: Hex grids
• Hexagonal grids
• Hexagonal Fourier Transform
• Generalization of L-S periodogram
• Sparse descriptions
• Sparse description example 1
• Sparse description example 2
• Sparse description example 3
• Sparse description example 4
• Basis pursuit
• Dictionary
• Sparse recovery
• Norms revisited
• Sparse recovery via $l^1$ norm
• Minimization of the $l^1$ norm
• Example
• Application
• Why does it work
• Relation to L-S periodogram