Michael B. Wakin
We are looking to recruit talented Ph.D. students with research interests in mathematical signal processing, data science, sensing systems, quantum information, and machine learning. Financial support may be available via research assistantships and/or departmental teaching assistantships. Prospective students are encouraged to email Prof. Wakin with the subject line "Wakin group".
This toolbox contains the MATLAB code necessary to reproduce the figures in the paper When Randomness Helps in Undersampling by Roel Snieder and Michael B. Wakin, published in SIAM Review, vol. 64, no. 4, pp. 1062-1080, November 2022. This code estimates various signals from randomly undersampled frequency spectra. Whereas uniform subsampling would introduce structural artifacts in the time series, random subsampling introduces a type of noise whose behavior we quantify in the paper.
Adaptive interference cancellation is rapidly becoming a necessity for our modern wireless communication systems, due to the proliferation of wireless devices that interfere with each other. To cancel interference, digital beamforming algorithms adaptively adjust the weight vector of the antenna array, and in turn its radiation pattern, to minimize interference while maximizing the desired signal power. While these algorithms are effective in ideal scenarios, they are sensitive to signal corruptions. In this work, we consider the case when the transmitter and receiver in a communication system cannot be synchronized, resulting in a carrier frequency offset that corrupts the signal. We present novel beamforming algorithms that are robust to signal corruptions arising from this time-variant carrier frequency offset. In particular, we bring in the Discrete Prolate Spheroidal Sequences (DPSS's) and propose two atomic-norm-minimization (ANM)-based methods in both 1D and 2D frameworks to design a weight vector that can be used to cancel interference when there exist unknown time-varying frequency drift in the pilot and interferer signals. Both algorithms do not assume a pilot signal is known. Noting that solving ANM optimization problems via semi-definite programs can be a computational burden, we also present a novel fast algorithm to approximately solve our 1D ANM optimization problem. Finally, we confirm the benefits of our proposed algorithms and show the advantages over existing approaches with a series of experiments.
This software will reproduce the figures in the paper Digital Beamforming Robust to Time-Varying Carrier Frequency Offset by S. Li, P. Nayeri, and M. B. Wakin.
The task of finding a sparse signal decomposition in an overcomplete dictionary is made more complicated when the signal undergoes an unknown modulation (or convolution in the complementary Fourier domain). Such simultaneous sparse recovery and blind demodulation problems appear in many applications including medical imaging, super resolution, self-calibration, etc. In this paper, we consider a more general sparse recovery and blind demodulation problem in which each atom comprising the signal undergoes a distinct modulation process. Under the assumption that the modulating waveforms live in a known common subspace, we employ the lifting technique and recast this problem as the recovery of a column-wise sparse matrix from structured linear measurements. In this framework, we accomplish sparse recovery and blind demodulation simultaneously by minimizing the induced atomic norm, which in this problem corresponds to the block L1 norm minimization.
This software will reproduce the figures in the paper Simultaneous Sparse Recovery and Blind Demodulation by Y. Xie, M. B. Wakin, and G. Tang.
The problem of estimating a sparse signal from low dimensional noisy observations arises in many applications, including super resolution, signal deconvolution, and radar imaging. In this paper, we consider a sparse signal model with non-stationary modulations, in which each dictionary atom contributing to the observations undergoes an unknown, distinct modulation. By applying the lifting technique, under the assumption that the modulating signals live in a common subspace, we recast this sparse recovery and non-stationary blind demodulation problem as the recovery of a column-wise sparse matrix from structured linear observations, and propose to solve it via block L1 -norm regularized quadratic minimization. Due to observation noise, the sparse signal and modulation process cannot be recovered exactly. Instead, we aim to recover the sparse support of the ground truth signal and bound the recovery errors of the signal's non-zero components and the modulation process. In particular, we derive sufficient conditions on the sample complexity and regularization parameter for exact support recovery and bound the recovery error on the support. Numerical simulations verify and support our theoretical findings, and we demonstrate the effectiveness of our model in the application of single molecule imaging.
This software will reproduce the figures in the paper Support Recovery for Sparse Signals With Unknown Non-Stationary Modulation by Y. Xie, M. B. Wakin, and G. Tang.
This software package contains algorithms for identifying spectral parameters (frequencies and damping ratios) using nuclear norm minimization, specifically in settings where only partial samples are available.
In particular, as detailed in the file Readme.txt, this software will reproduce the figures in the paper Recovery analysis of damped spectrally sparse signals and its relation to MUSIC by S. Li, H. Mansour, and M. B. Wakin.
This software package contains algorithms for performing modal analysis (joint sparse frequency estimation) using atomic norm minimization, specifically in settings where only partial samples or randomly compressed signal measurements are available.
In particular, as detailed in the file readme.txt, this software will reproduce the figures in the paper Atomic Norm Minimization for Modal Analysis from Random and Compressed Samples by S. Li, D. Yang, G. Tang, and M. B. Wakin.
This software package contains a collection of tools for implementing fast alorithms for working with the Slepian basis, also known as discrete prolate spheroidal sequences. See the included readme file for a detailed description of the contents and for usage instructions.
For further details, see the paper The Fast Slepian Transform by S. Karnik, Z. Zhu, M. B. Wakin, J. Romberg, and M. A. Davenport.
This code demonstrates a fast two-phase algorithm for super-resolution. Given the low-frequency part of the spectrum of a sequence of impulses, Phase I consists of a greedy algorithm that roughly estimates the impulse positions. These estimates are then refined by local optimization in Phase II. The backbone of our work is the fundamental work of Slepian et al. involving discrete prolate spheroidal wave functions and their unique properties.
The function TwoPhaseAlg.m (called with no input arguments) will reproduce Figure 1 in the manuscript "Greed is Super: A Fast Algorithm for Super-Resolution" by A. Eftekhari and M. B. Wakin.
The bulk of the Compressive sensing (CS) literature has focused on the case where the acquired signal has a sparse or compressible representation in an orthonormal basis. In practice, however, there are many signals that cannot be sparsely represented or approximated using an orthonormal basis, but that do have sparse representations in a redundant dictionary. Standard results in CS can sometimes be extended to handle this case provided that the dictionary is sufficiently incoherent or well-conditioned, but these approaches fail to address the case of a truly redundant or overcomplete dictionary.
This software package implements a variant of the iterative reconstruction algorithm CoSaMP for this more challenging setting. In contrast to prior approaches, the method is "signal-focused"; that is, it is oriented around recovering the signal rather than its dictionary coefficients.
For further details, see the paper Signal Space CoSaMP for Sparse Recovery with Redundant Dictionaries, by M.A. Davenport, D. Needell, and M.B. Wakin.
Compressive sensing (CS) has recently emerged as a framework for efficiently capturing signals that are sparse or compressible in an appropriate basis. While often motivated as an alternative to Nyquist-rate sampling, there remains a gap between the discrete, finite-dimensional CS framework and the problem of acquiring a continuous-time signal.
This software package provides a set of tools for bridging this gap through the use of Discrete Prolate Spheroidal Sequences (DPSS's), a collection of functions that trace back to the seminal work by Slepian, Landau, and Pollack on the effects of time-limiting and bandlimiting operations. DPSS's form a highly efficient basis for sampled bandlimited functions; by modulating and merging DPSS bases, we obtain a dictionary that offers high-quality sparse approximations for most sampled multiband signals. This multiband modulated DPSS dictionary can be readily incorporated into the CS framework.
For further details, see the paper Compressive Sensing of Analog Signals Using Discrete Prolate Spheroidal Sequences, by M.A. Davenport and M.B. Wakin.
DART contains all of the software necessary to reproduce the results presented in this paper. It can be downloaded here. Please e-mail markad-at-stanford-dot-edu if you find any bugs or have any questions.
This Matlab package computes the player advantage in blackjack using two strategies: (1) basic strategy and (2) Thorp's complete point count system for card counting. The analysis uses Markov chains, as described in the manuscript: M. B. Wakin and C. J. Rozell, A Markov Chain Analysis of Blackjack Strategy, 2004. Installation: Download and unzip code, then see the main Matlab script BJMCmainScript.m. This software is released under a Creative Commons license (Attribution 3.0 Unported).