On the stability of sparse convolutions

Walk, Philipp; Jung, Peter; Pfander, Götz E.

doi:10.1016/j.acha.2015.08.002

Cited by 7 publications

(6 citation statements)

References 31 publications

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“…Bilinear compressive sensing problems have also been investigated in e.g. [16,38,28]. These papers focus on the issue of injectivity and the principle of identifiability.…”

Section: Related Work and Our Contributionsmentioning

confidence: 99%

Self-calibration and biconvex compressive sensing

2015

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The design of high-precision sensing devises becomes ever more difficult and expensive. At the same time, the need for precise calibration of these devices (ranging from tiny sensors to space telescopes) manifests itself as a major roadblock in many scientific and technological endeavors. To achieve optimal performance of advanced high-performance sensors one must carefully calibrate them, which is often difficult or even impossible to do in practice. In this work we bring together three seemingly unrelated concepts, namely Self-Calibration, Compressive Sensing, and Biconvex Optimization. The idea behind self-calibration is to equip a hardware device with a smart algorithm that can compensate automatically for the lack of calibration. We show how several self-calibration problems can be treated efficiently within the framework of biconvex compressive sensing via a new method called SparseLift. More specifically, we consider a linear system of equations y = DAx, where both x and the diagonal matrix D (which models the calibration error) are unknown. By "lifting" this biconvex inverse problem we arrive at a convex optimization problem. By exploiting sparsity in the signal model, we derive explicit theoretical guarantees under which both x and D can be recovered exactly, robustly, and numerically efficiently via linear programming. Applications in array calibration and wireless communications are discussed and numerical simulations are presented, confirming and complementing our theoretical analysis.

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“…Bilinear compressive sensing problems have also been investigated in e.g. [16,38,28]. These papers focus on the issue of injectivity and the principle of identifiability.…”

Section: Related Work and Our Contributionsmentioning

confidence: 99%

Self-calibration and biconvex compressive sensing

2015

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“…The main assertion of the theorem is: The smallest ℓ 2 −norm over all convolutions of s− and f −sparse normalized sequences can be determined solely in terms of s and f , where we used the fact that the sparse convolution can be represented by sparse vectors in n = ⌊2 2(s+ f −2) log(s+ f −2) ⌋ dimensions, due to an additive combinatoric result. An analytic lower bound for α, which decays exponentially in the sparsity, has been found very recently in (Walk, Jung, and Pfander, 2014). Although D n,min{s, f } is decreasing in n (since we extend the minimum to a larger set by increasing n) nothing seems to be known on the precise scaling in n. Nevertheless, since n depends solely on s and f it is sufficient to ensure that D n,min{s, f } is non-zero.…”

Section: The Rnmp For Sparse Convolutionsmentioning

confidence: 97%

Sparse Model Uncertainties in Compressed Sensing with Application to Convolutions and Sporadic Communication

Jung

Walk

2015

Compressed Sensing and Its Applications

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The success of the compressed sensing paradigm has shown that a substantial reduction in sampling and storage complexity can be achieved in certain linear and non-adaptive estimation problems. It is therefore an advisable strategy for noncoherent information retrieval in, for example, sporadic blind and semi-blind communication and sampling problems. But, the conventional model is not practical here since the compressible signals have to be estimated from samples taken solely on the output of an un-calibrated system which is unknown during measurement but often compressible. Conventionally, one has either to operate at suboptimal sampling rates or the recovery performance substantially suffers from the dominance of model mismatch. In this work we discuss such type of estimation problems and we focus on bilinear inverse problems. We link this problem to the recovery of low-rank and sparse matrices and establish stable low-dimensional embeddings of the uncalibrated receive signals whereby addressing also efficient communication-oriented methods like universal random demodulation. Exemplary, we investigate in more detail sparse convolutions serving as a basic communication channel model. In using some recent results from additive combinatorics we show that such type of signals can be efficiently low-rate sampled by semi-blind methods. Finally, we present a further application of these results in the field of phase retrieval from intensity Fourier measurements.

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“…Here, A x = T x T * x is the L × L autocorrelation matrix of x, which is the identity scaled by x 2 if L < K. Hence, each normalized Huffman sequence, generates an isometric operator T x having the best stability among all discrete-time LTI systems, as studied in [19].…”

Section: Timing Offset and Effective Delay Spread For Bmoczmentioning

confidence: 99%

MOCZ for Blind Short-Packet Communication: Basic Principles

Walk

Jung

Hassibi

2019

IEEE Trans. Wireless Commun.

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We will investigate practical aspects for a recently introduced blind (noncoherent) communication scheme, called modulation on conjugate-reciprocal zeros (MOCZ), which enables reliable transmission of sporadic and short-packets at ultra-low latency in unknown wireless multipath channels, which are static over the receive duration of one packet. Here the information is modulated on the zeros of the transmitted discrete-time baseband signal's z−transform. Due to ubiquitous impairments between transmitter and receiver clocks a carrier frequency offset (CFO) will occur after a down-conversion to the baseband, which results in a common rotation of the zeros. To identify fractional rotations of the base angle in the zero-pattern, we propose an oversampled direct zero testing decoder to identify the most likely one. Integer rotations correspond to cyclic shifts of the binary message, which we compensate by a cyclically permutable code (CPC). Additionally, the embedding of CPCs into cyclic codes, allows to exploit additive error correction which reduces the bit-error-rate tremendously. Furthermore, we use the trident structure in the signals autocorrelation to estimate timing offsets and the channels effective delay spread. We finally demonstrate how these impairment estimations can be largely improved by receive antenna diversity, which enables extreme bursty reliable communication at low latency and SNR.

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On the stability of sparse convolutions

Cited by 7 publications

References 31 publications

Self-calibration and biconvex compressive sensing

Self-calibration and biconvex compressive sensing

Sparse Model Uncertainties in Compressed Sensing with Application to Convolutions and Sporadic Communication

MOCZ for Blind Short-Packet Communication: Basic Principles

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