Nonlinear Sampling

2 Saturation in CCD sensors Dynamic range correction Optical devices High power amplifiers s(-t) Memoryless nonlinear distortion t=n

3 Easy or Hard? Memoryless nonlinear M, but ideal samples: Easy Standard setup Generalized samples but M=I: Easy Assuming => Oblique projection: Memoryless nonlinear M, generalized samples: Hard but M(A) usually not a subspace …

4 Perfect Reconstruction Setting: m(t) is invertible with bounded derivative y(t) is lies in a subspace A Uniqueness same as in linear case! Proof: Based on extended frame perturbation theory and geometrical ideas If and m(t) is invertible and smooth enough then y(t) can be recovered exactly If and m(t) is invertible and smooth enough then y(t) can be recovered exactly Theorem (uniqueness):

5 Main idea: 1. Minimize error in samples where 2. From uniqueness if Perfect reconstruction global minimum of Difficulties: 1. Nonlinear, nonconvex problem 2. Defined over an infinite space Optimization Based Approach Under the previous conditions any stationary point of is unique and globally optimal Under the previous conditions any stationary point of is unique and globally optimal Theorem : Only have to trap a stationary point!

6 Transform the problem into a series of linear problems: 1.Initial guess y 0 2.Linearization: Replace m(t) by its derivative around y 0 3.Solve linear problem and update solution y n y n+1 Algorithm: Linearization Algorithm converges to true input error in samples correction solving linear problem

7 Simulation Example Optical sampling system: optical modulator ADC

8 Simulation Initialization with First iteration: Third iteration:

9 Course Summary (So Far …) Crash course on linear algebra Subspace sampling (sampling of nonbandlimited signals, interpolation methods) Minimax recovery techniques Constrained reconstruction: minimax and consistent methods Nonlinear sampling And yet to come … Sampling random signals Sampling sparse signals

10 Summary BandlimitedIdeal point-wiseIdeal interpolation Subspace priors Smoothness priors Sparsity priors General linear sampling Non-linear distortions Minimax approach with simple kernels Signal Model Sampling Reconstruction

11 Our Point-Of-View Sampling can be viewed in a broader sense of projection onto any subspace 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 Perfect recovery of non-bandlimited signals after LPF …