Instructor: Yonina Eldar

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Modern Sampling Methods 049033

• Instructor Yonina Eldar
• Teaching Assistant Tomer Michaeli
• Spring 2009

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Sampling Analog Girl in a Digital World Judy
Gorman 99
Digital world
Analog world
Sampling A2D
Signal processing Denoising Image analysis
Reconstruction D2A
(Interpolation)
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Applications Sampling Rate Conversion

• Common audio standards
• 8 KHz (VOIP, wireless microphone, )
• 11.025 KHz (MPEG audio, )
• 16 KHz (VOIP, )
• 22.05 KHz (MPEG audio, )
• 32 KHz (miniDV, DVCAM, DAT, NICAM, )
• 44.1 KHz (CD, MP3, )
• 48 KHz (DVD, DAT, )

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Applications Image Transformations

• Lens distortion correction
• Image scaling

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Applications CT Scans
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Applications Spatial Superresolution
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Applications Temporal Superresolution
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Applications Temporal Superresolution
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Our Point-Of-View

• The field of sampling was traditionally
  associated with methods implemented either in the
  frequency domain, or in the time domain
• Sampling can be viewed in a broader sense of
  projection onto any subspace or union of
  subspaces
• Can choose the subspaces to yield interesting new
  possibilities (below Nyquist sampling of sparse
  signals, pointwise samples of non bandlimited
  signals, perfect compensation of nonlinear
  effects )

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Bandlimited Sampling Theorems

• Cauchy (1841)
• Whittaker (1915) - Shannon (1948)
• A. J. Jerri, The Shannon sampling theorem - its
  various extensions and applications A tutorial
  review, Proc. IEEE, pp. 1565-1595, Nov. 1977.

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Limitations of Shannons Theorem

• Towards more robust DSPs
• General inputs
• Nonideal sampling general pre-filters, nonlinear
  distortions
• Simple interpolation kernels

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Sampling Process Linear Distortion
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Sampling Process Nonlinear Distortion
Original Initial guess
Reconstructed signal

• Replace Fourier analysis by functional analysis,
  Hilbert space algebra, and convex optimization

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Sampling Process Noise

• Employ estimation techniques

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Signal Priors
x(t) bandlimited
x(t) piece-wise linear
Different priors lead to different reconstructions
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Signal Priors Subspace Priors

• Shift invariant subspace
• General subspace in a Hilbert space

Common in communication pulse amplitude
modulation (PAM)
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Beyond Bandlimited

• Two key ideas in bandlimited sampling
• Avoid aliasing
• Fourier domain analysis

Misleading concepts!

• Suppose that
  with
• Signal is clearly not bandlimited
• Aliasing in frequency and time
• Perfect reconstruction possible from samples

Aliasing is not the issue
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Signal Priors Smoothness Priors
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Signal Priors Stochastic Priors
Original Image
Bicubic Interpolation
Matern Interpolation
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Signal Priors Sparsity Priors

• Wavelet transform of images is commonly sparse
• STFT transform of speech signals is commonly
  sparse
• Fourier transform of radio signals is commonly
  sparse

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Reconstruction Constraints Unconstrained Schemes
Sampling
Reconstruction
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Reconstruction Constraints Predefined Kernel
Sampling
Reconstruction

• Minimax methods
• Consistency requirement

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Reconstruction Constraints Dense Grid
Interpolation

• To improve performance Increase reconstruction
  rate

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Reconstruction Constraints Dense Grid
Interpolation
Bicubic Interpolation
Second Order Approximation to Matern
Interpolation with K2
Optimal Dense Grid Matern Interpolation with K2
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Course Outline (Subject to change without further
notice)

• Motivating introduction after which you will all
  want to take this course (1 lesson)
• Crash course on linear algebra (basically no
  prior knowledge is assumed but strong interest in
  algebra is highly recommended) (3 lessons)
• Subspace sampling (sampling of nonbandlimited
  signals, interpolation methods) (2 lessons)
• Nonlinear sampling (1 lesson)
• Minimax recovery techniques (1 lesson)
• Constrained reconstruction minimax and
  consistent methods (2 lessons)
• Sampling sparse signals (1 lesson)
• Sampling random signals (1 lesson)