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— Given an m × N matrix Φ, with m < N , the system of equations Φx = y is typically underdetermined and has infinitely many solutions. Various forms of optimization can extract a " best " solution. One of the oldest is to select the one with minimal 2 norm. It has been shown that in many applications a better choice is the minimal

We analyze mathematically the effect of quantization error in the circuit implementation of Analog to Digital (A/D) converters such as Pulse Code Modulation (PCM) and Sigma Delta Modulation (Σ∆). Σ∆ modulation, which is based on oversampling the signal, has a self correction for quantization error that PCM does not have, and that we believe to be a major… (More)

—Sigma–delta modulation, a widely used method of analog-to-digital (A/D) signal conversion, is known to be robust to hardware imperfections, i.e., bit streams generated by slightly imprecise hardware components can be decoded comparably well. We formulate a model for robustness and give a rigorous analysis for single-loop sigma–delta modulation applied to… (More)

This paper proposes a novel Nyquist-rate analog-to-digital (A/D) conversion algorithm which achieves exponential accuracy in the bit-rate despite using imperfect components. The proposed algorithm is based on a robust implementation of a beta-encoder with β = φ = (1 + √ 5)/2, the golden ratio. It was previously shown that beta-encoders can be implemented in… (More)

Quantization of compressed sensing measurements is typically justified by the robust recovery results of Candès, Romberg and Tao, and of Donoho. These results guarantee that if a uniform quantizer of step size δ is used to quantize m measurements y = Φx of a k-sparse signal x ∈ R N , where Φ satisfies the restricted isometry property, then the approximate… (More)

Sigma-Delta modulation is a popular method for analog-to-digital conversion of bandlimited signals that employs coarse quantization coupled with oversampling. The standard mathematical model for the error analysis of the method measures the performance of a given scheme by the rate at which the associated reconstruction error decays as a function of the… (More)

Quantization of compressed sensing measurements is typically justified by the robust recovery results of Candès, Romberg and Tao, and of Donoho. These results guarantee that if a uniform quantizer of step size δ is used to quantize m measurements y = Φx of a k-sparse signal x ∈ R N , where Φ satisfies the restricted isometry property, then the approximate… (More)

—Recent results make it clear that the compressed sensing paradigm can be used effectively for dimension reduction. On the other hand, the literature on quantization of compressed sensing measurements is relatively sparse, and mainly focuses on pulse-code-modulation (PCM) type schemes where each measurement is quantized independently using a uniform… (More)