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89
Robust Uncertainty Principles: Exact Signal Reconstruction From Highly Incomplete Frequency Information
, 2006
"... This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discretetime signal and a randomly chosen set of frequencies. Is it possible to reconstruct from the partial knowledge of its Fourier coefficients on the set? A typical result of this pa ..."
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Cited by 1304 (42 self)
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This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discretetime signal and a randomly chosen set of frequencies. Is it possible to reconstruct from the partial knowledge of its Fourier coefficients on the set? A typical result of this paper is as follows. Suppose that is a superposition of spikes @ Aa @ A @ A obeying @�� � A I for some constant H. We do not know the locations of the spikes nor their amplitudes. Then with probability at least I @ A, can be reconstructed exactly as the solution to the I minimization problem I aH @ A s.t. ” @ Aa ” @ A for all
Nonlinear solution of linear inverse problems by waveletvaguelette decomposition
, 1992
"... We describe the WaveletVaguelette Decomposition (WVD) of a linear inverse problem. It is a substitute for the singular value decomposition (SVD) of an inverse problem, and it exists for a class of special inverse problems of homogeneous type { such asnumerical di erentiation, inversion of Abeltype ..."
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Cited by 182 (12 self)
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We describe the WaveletVaguelette Decomposition (WVD) of a linear inverse problem. It is a substitute for the singular value decomposition (SVD) of an inverse problem, and it exists for a class of special inverse problems of homogeneous type { such asnumerical di erentiation, inversion of Abeltype transforms, certain convolution transforms, and the Radon Transform. We propose to solve illposed linear inverse problems by nonlinearly \shrinking" the WVD coe cients of the noisy, indirect data. Our approach o ers signi cant advantages over traditional SVD inversion in the case of recovering spatially inhomogeneous objects. We suppose that observations are contaminated by white noise and that the object is an unknown element of a Besov space. We prove that nonlinear WVD shrinkage can be tuned to attain the minimax rate of convergence, for L 2 loss, over the entire Besov scale. The important case of Besov spaces Bp;q, p <2, which model spatial inhomogeneity, is included. In comparison, linear procedures { SVD included { cannot attain optimal rates of convergence over such classes in the case p<2. For example, our methods achieve faster rates of convergence, for objects known to lie in the Bump Algebra or in Bounded Variation, than any linear procedure.
Quantitative Robust Uncertainty Principles and Optimally Sparse Decompositions
, 2004
"... In this paper, we develop a robust uncertainty principle for finite signals in C N which states that for nearly all choices T, Ω ⊂ {0,..., N − 1} such that T  + Ω  ≍ (log N) −1/2 · N, there is no signal f supported on T whose discrete Fourier transform ˆ f is supported on Ω. In fact, we can mak ..."
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Cited by 119 (12 self)
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In this paper, we develop a robust uncertainty principle for finite signals in C N which states that for nearly all choices T, Ω ⊂ {0,..., N − 1} such that T  + Ω  ≍ (log N) −1/2 · N, there is no signal f supported on T whose discrete Fourier transform ˆ f is supported on Ω. In fact, we can make the above uncertainty principle quantitative in the sense that if f is supported on T, then only a small percentage of the energy (less than half, say) of ˆ f is concentrated on Ω. As an application of this robust uncertainty principle (QRUP), we consider the problem of decomposing a signal into a sparse superposition of spikes and complex sinusoids f(s) = � α1(t)δ(s − t) + � α2(ω)e i2πωs/N / √ N. t∈T We show that if a generic signal f has a decomposition (α1, α2) using spike and frequency locations in T and Ω respectively, and obeying ω∈Ω T  + Ω  ≤ Const · (log N) −1/2 · N, then (α1, α2) is the unique sparsest possible decomposition (all other decompositions have more nonzero terms). In addition, if T  + Ω  ≤ Const · (log N) −1 · N, then the sparsest (α1, α2) can be found by solving a convex optimization problem. Underlying our results is a new probabilistic approach which insists on finding the correct uncertainty relation or the optimally sparse solution for nearly all subsets but not necessarily all of them, and allows to considerably sharpen previously known results [9, 10]. In fact, we show that the fraction of sets (T, Ω) for which the above properties do not hold can be upper bounded by quantities like N −α for large values of α. The QRUP (and the application to finding sparse representations) can be extended to general pairs of orthogonal bases Φ1, Φ2 of C N. For nearly all choices Γ1, Γ2 ⊂ {0,..., N − 1} obeying Γ1  + Γ2  ≍ µ(Φ1, Φ2) −2 · (log N) −m, where m ≤ 6, there is no signal f such that Φ1f is supported on Γ1 and Φ2f is supported on Γ2 where µ(Φ1, Φ2) is the mutual coherence between Φ1 and Φ2.
TimeVariant Channel Estimation Using Discrete Prolate Spheroidal Sequences
 IEEE Trans. Signal Processing
, 2005
"... We propose and analyze a lowcomplexity channel estimator for a multiuser multicarrier code division multiple access (MCCDMA) downlink in a timevariant frequencyselective channel. MCCDMA is based on orthogonal frequency division multiplexing (OFDM). The timevariant channel is estimated individu ..."
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Cited by 62 (21 self)
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We propose and analyze a lowcomplexity channel estimator for a multiuser multicarrier code division multiple access (MCCDMA) downlink in a timevariant frequencyselective channel. MCCDMA is based on orthogonal frequency division multiplexing (OFDM). The timevariant channel is estimated individually for every flatfading subcarrier, assuming small intercarrier interference. The temporal variation of every subcarrier over the duration of a data block is upper bounded by the Doppler bandwidth determined by the maximum velocity of the users. Slepian showed that timelimited snapshots of bandlimited sequences span a lowdimensional subspace. This subspace is also spanned by discrete prolate spheroidal (DPS) sequences. We expand the timevariant subcarrier coefficients in terms of orthogonal DPS sequences we call Slepian basis expansion. This enables a timevariant channel description that avoids the frequency leakage effect of the Fourier basis expansion. The square bias of the Slepian basis expansion per subcarrier is three magnitudes smaller than the square bias of the Fourier basis expansion. We show simulation results for a fully loaded MCCDMA downlink with classic linear minimum mean square error multiuser detection. The users are moving with 19.4 m/s. Using the Slepian basis expansion channel estimator and a pilot ratio of only 2%, we achieve a bit error rate performance as with perfect channel knowledge.
Noncommutative geometry, quantum fields and motives
 Colloquium Publications, Vol.55, American Mathematical Society
, 2008
"... ..."
Edge Direction Preserving Image Zooming: A Mathematical and Numerical Analysis
 SIAM Journal on Numerical Analysis
, 2000
"... We focus in this paper on some reconstruction/restoration methods which aim is to improve the resolution of digital images. The main point is here to study the ability of such methods to preserve 1D structures. Indeed such structures are important since they are often carried by the image "edges". W ..."
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Cited by 38 (5 self)
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We focus in this paper on some reconstruction/restoration methods which aim is to improve the resolution of digital images. The main point is here to study the ability of such methods to preserve 1D structures. Indeed such structures are important since they are often carried by the image "edges". We first focus on linear methods, give a general framework to design them and show that the preservation of 1D structures pleads in favor of the cancellation of the periodization of the image spectrum. More precisely, we show that preserving 1D structures implies the linear methods to be written as a convolution of the "sinc interpolation". As a consequence, we can not cope linearly with Gibbs effects, sharpness of the results and the preservation of the 1D structure. Secondly, we study variational nonlinear methods and in particular the one based on total variation. We show that this latter permits to avoid these shortcomings. We also prove the existence and consistency of an approximate sol...
Degrees of freedom in multipleantenna channels: A signal space approach
 IEEE Trans. Inf. Theory
, 2005
"... We consider multipleantenna systems that are limited by the area and geometry of antenna arrays. Given these physical constraints, we determine the limit to the number of spatial degrees of freedom available and find that the commonly used statistical multiinput multioutput model is inadequate. A ..."
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Cited by 34 (4 self)
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We consider multipleantenna systems that are limited by the area and geometry of antenna arrays. Given these physical constraints, we determine the limit to the number of spatial degrees of freedom available and find that the commonly used statistical multiinput multioutput model is inadequate. Antenna theory is applied to take into account the area and geometry constraints, and define the spatial signal space so as to interpret experimental channel measurements in an arrayindependent but manageable description of the physical environment. Based on these modeling strategies, we show that for a spherical array of effective aperture A in a physical environment of angular spread Ω  in solid angle, the number of spatial degrees of freedom is AΩ  for unpolarized antennas and 2AΩ  for polarized antennas. Together with the 2WT degrees of freedom for a system of bandwidth W transmitting in an interval T, the total degrees of freedom of a multipleantenna channel is therefore 4WTAΩ. 1
On the Bit Error Rate of Lightwave Systems with Optical Amplifiers
 J. Lightwave Technol
, 1991
"... We revisit the problem of evaluating the performances of communication systems with optical amplifiers and a wideband optical filter. We compute the exact probability of error and the optimal threshold and compare them with those predicted by Gaussian approximations for ASK, FSK or DPSK modulations, ..."
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Cited by 24 (0 self)
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We revisit the problem of evaluating the performances of communication systems with optical amplifiers and a wideband optical filter. We compute the exact probability of error and the optimal threshold and compare them with those predicted by Gaussian approximations for ASK, FSK or DPSK modulations, both for ideal photodetectors and for the case where shot noise is significant. 1 Introduction We consider the problem of receiving modulated signals that have passed through optical amplifiers. An optical amplifier with power gain G can be modeled as a linear optical field amplifier together with a source of white Gaussian noise over the bandwidth of interest. The noise has a two sided spectral density N 0 =2, with N 0 = N sp h(G \Gamma 1) [1]. There h is Planck's constant, is the frequency of interest, and N sp is a factor that takes various imperfections into account, its value is ideally 1. In the case of amplitude shift keying (ASK) the signal is either 0 or an optical pulse of dura...
Feedback capacity of stationary Gaussian channels
"... The capacity of stationary additive Gaussian noise channels with feedback is characterized as the solution to a variational problem. Toward this end, it is proved that the optimal feedback coding scheme is stationary. When specialized to the firstorder autoregressive movingaverage noise spectrum, ..."
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Cited by 22 (4 self)
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The capacity of stationary additive Gaussian noise channels with feedback is characterized as the solution to a variational problem. Toward this end, it is proved that the optimal feedback coding scheme is stationary. When specialized to the firstorder autoregressive movingaverage noise spectrum, this variational characterization yields a closedform expression for the feedback capacity. In particular, this result shows that the celebrated Schalkwijk–Kailath coding scheme achieves the feedback capacity for the firstorder autoregressive movingaverage Gaussian channel, resolving a longstanding open problem studied by Butman, Schalkwijk– Tiernan, Wolfowitz, Ozarow, Ordentlich, Yang–Kavčić–Tatikonda, and others. 1 Introduction and
The Generalized Information Network Analysis Methodology for Distributed Satellite Systems
 Doctor of Science Thesis, MIT
, 1998
"... A systematic analysis methodology for distributed satellite systems is developed that is generalizable and can be applied to any satellite mission in communications, sensing or navigation. The primary enabler is that almost all satellite applications involve the collection and dissemination of info ..."
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Cited by 16 (0 self)
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A systematic analysis methodology for distributed satellite systems is developed that is generalizable and can be applied to any satellite mission in communications, sensing or navigation. The primary enabler is that almost all satellite applications involve the collection and dissemination of information and can thus be treated as modular information processing networks. This generalization allows the adoption of the mathematics for information network ow, leading to a logical classi cation scheme for satellite systems. The bene ts and issues that are characteristic of each system class are identi ed, in terms of their capability, performance and cost. The quantitative analysis methodology speci es measurable, unambiguous metrics for the cost, capability, performance and adaptability. The Capabilities are characterized by four quality of service parameters that relate to the isolation, rate, integrity and availability of the information transferred between origindestination pairs within a market. Performance is the probability of satisfying the user's requirements for these parameters. The Cost per Function metric is the average cost incurred to provide satisfactory service to a single user, and Adapatability metrics are sensitivity indicators. Validation of the methodology is provided by a comprehensive quantitative analysis of the