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Just relax: Convex programming methods for subset selection and sparse approximation
, 2004
"... Subset selection and sparse approximation problems request a good approximation of an input signal using a linear combination of elementary signals, yet they stipulate that the approximation may only involve a few of the elementary signals. This class of problems arises throughout electrical enginee ..."
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Cited by 103 (5 self)
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Subset selection and sparse approximation problems request a good approximation of an input signal using a linear combination of elementary signals, yet they stipulate that the approximation may only involve a few of the elementary signals. This class of problems arises throughout electrical engineering, applied mathematics and statistics, but small theoretical progress has been made over the last fifty years. Subset selection and sparse approximation both admit natural convex relaxations, but the literature contains few results on the behavior of these relaxations for general input signals. This report demonstrates that the solution of the convex program frequently coincides with the solution of the original approximation problem. The proofs depend essentially on geometric properties of the ensemble of elementary signals. The results are powerful because sparse approximation problems are combinatorial, while convex programs can be solved in polynomial time with standard software. Comparable new results for a greedy algorithm, Orthogonal Matching Pursuit, are also stated. This report should have a major practical impact because the theory applies immediately to many realworld signal processing problems.
Designing Structured Tight Frames via an Alternating Projection Method
, 2003
"... Tight frames, also known as general WelchBoundEquality sequences, generalize orthonormal systems. Numerous applicationsincluding communications, coding and sparse approximationrequire finitedimensional tight frames that possess additional structural properties. This paper proposes an alterna ..."
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Cited by 87 (10 self)
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Tight frames, also known as general WelchBoundEquality sequences, generalize orthonormal systems. Numerous applicationsincluding communications, coding and sparse approximationrequire finitedimensional tight frames that possess additional structural properties. This paper proposes an alternating projection method that is versatile enough to solve a huge class of inverse eigenvalue problems, which includes the frame design problem. To apply this method, one only needs to solve a matrix nearness problem that arises naturally from the design specifications. Therefore, it is fast and easy to develop versions of the algorithm that target new design problems. Alternating projection will often succeed even if algebraic constructions are unavailable. To demonstrate
A framework for spectrally efficient noncoherent communication
, 2000
"... Abstract—This paper considers noncoherent communication over a frequencynonselective channel in which the timevarying channel gain is unknown a priori, but is approximately constant over a coherence interval. Unless the coherence interval is large, coherent communication, which requires explicit c ..."
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Cited by 32 (6 self)
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Abstract—This paper considers noncoherent communication over a frequencynonselective channel in which the timevarying channel gain is unknown a priori, but is approximately constant over a coherence interval. Unless the coherence interval is large, coherent communication, which requires explicit channel estimation and tracking prior to detection, incurs training overhead which may be excessive, especially for multipleantenna communication. In contrast, noncoherent detection may be viewed as a generalized likelihood ratio test (GLRT) which jointly estimates the channel and the data, and hence does not require separate training. The main results in this paper are as follows 1) We develop a “signal space ” criterion for signal and code design for noncoherent communication, in terms of the distances of signal points from the decision boundaries. 2) The noncoherent metric thus obtained is used to guide the design of signals for noncoherent communication that are based on amplitude/phase constellations. These are significantly more efficient than conventional differential phaseshift keying (PSK), especially at high signaltonoise ratio (SNR). Also, known results on the highSNR performance of multiplesymbol demodulation of differential PSK are easily inferred from the noncoherent metric. 3) The GLRT interpretation is used to obtain nearoptimal lowcomplexity implementations of noncoherent block demodulation. In particular, this gives an implementation of multiple symbol demodulation of differential PSK, which is of linear complexity (in the block length) and whose degradation from the exact, exponential complexity, implementation can be made as small as desired. Index Terms—Differential phaseshift keying (PSK), differential quadrature amplitude modulation (QAM), generalized likelihood ratio test (GLRT), noncoherent communication, noncoherent distance. I.
Noncoherent Communication in MultipleAntenna Systems: Receiver design and Codebook construction
"... We address the problem of spacetime codebook design for noncoherent communications in multipleantenna wireless systems. In contrast with other approaches, the channel matrix is modeled as an unknown deterministic parameter at both the receiver and the transmitter, and the Gaussian observation noi ..."
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Cited by 7 (2 self)
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We address the problem of spacetime codebook design for noncoherent communications in multipleantenna wireless systems. In contrast with other approaches, the channel matrix is modeled as an unknown deterministic parameter at both the receiver and the transmitter, and the Gaussian observation noise is allowed to have an arbitrary correlation structure, known by the transmitter and the receiver. In order to handle the unknown deterministic spacetime channel, a generalized likelihood ratio test (GLRT) receiver is considered. A new methodology for spacetime codebook design under this noncoherent setup is proposed. This optimizes the probability of error of the GLRT receiver’s detector in the high signaltonoise ratio (SNR) regime, thus solving a highdimensional nonlinear nonsmooth optimization problem in a twostep approach: (i) firstly, a convex SDP relaxation of the codebook design problem yields a rough estimate of the optimal codebook; (ii) this is then refined through a geodesic descent optimization algorithm that exploits the Riemannian geometry imposed by the power constraints on the spacetime codewords. The results obtained through computer simulations illustrate the advantages of our method. For the specific case of spatiotemporal white observation noise, our codebook constructions replicate the performance of stateofart known solutions. The main point here is that our methodology permits to extend the codebook construction to any given correlated noise environment. The simulation results show the good performance of these
GoldbergCoxeter Construction for 3 and 4valent Plane Graphs
, 2004
"... We consider the GoldbergCoxeter construction GC k,l (G 0 ) (a generalization of a simplicial subdivision of the dodecahedron considered in [Gold37] and [Cox71]), which produces a plane graph from any 3 or 4valent plane graph for integer parameters k, l.Azigzag in a plane graph is a circuit of ed ..."
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Cited by 7 (4 self)
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We consider the GoldbergCoxeter construction GC k,l (G 0 ) (a generalization of a simplicial subdivision of the dodecahedron considered in [Gold37] and [Cox71]), which produces a plane graph from any 3 or 4valent plane graph for integer parameters k, l.Azigzag in a plane graph is a circuit of edges, such that any two, but no three, consecutive edges belong to the same face; a central circuit in a 4valent plane graph G is a circuit of edges, such that no two consecutive edges belong to the same face. We study the zigzag (or central circuit) structure of the resulting graph using the algebraic formalism of the moving group,the(k, l)product and a finite index subgroup of SL 2 (Z), whose elements preserve the above structure. We also study the intersection pattern of zigzags (or central circuits) of GC k,l (G 0 ) and consider its projections, obtained by removing all but one zigzags (or central circuits).
Construction of Equiangular Signatures for Synchronous CDMA Systems
, 2004
"... Welch bound equality (WBE) signature sequences maximize the uplink sum capacity in directspread synchronous code division multiple access (CDMA) systems. WBE sequences have a nice interference invariance property that typically holds only when the system is fully loaded and the signature set must b ..."
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Cited by 2 (0 self)
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Welch bound equality (WBE) signature sequences maximize the uplink sum capacity in directspread synchronous code division multiple access (CDMA) systems. WBE sequences have a nice interference invariance property that typically holds only when the system is fully loaded and the signature set must be redesigned and reassigned as the number of active users changes to maintain this property. An additional equiangular constraint on the signature set, however, maintains interference invariance. Finding such signatures requires imposing equiangular side constraints on an inverse eigenvalue problem. This paper presents an alternating projection algorithm that can design WBE sequences that satisfy equiangular side constraints. The proposed algorithm can be used to find Grassmannian frames as well as equiangular tight frames. Though one projection is onto a closed but non convex set, it is shown that this algorithm converges to a fixed point, and these fixed points are partially characterized.
2006 IEEE Ninth International Symposium on Spread Spectrum Techniques and Applications On The Construction of Noncoherent Space Time Codes from Highdimensional Spherical Codes
"... Abstract — This paper analyzes the performance of noncoherent space time codes obtained from a new class of spherical codes by mapping the surface of a higherdimensional halfsphere to the Grassmann manifold. The deployed spherical code contains a structure which can be exploited by the receiver, l ..."
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Abstract — This paper analyzes the performance of noncoherent space time codes obtained from a new class of spherical codes by mapping the surface of a higherdimensional halfsphere to the Grassmann manifold. The deployed spherical code contains a structure which can be exploited by the receiver, leading to a reasonable decoding complexity. Therefore, highdimensional coding spaces (Grassmann manifolds) can be used as a basis for the construction. This corresponds to codes with larger block lengths, as opposite to the usual construction of space time codes from smalldimensional spaces, such as the unitary group U(nT), where nT is the number of transmit antennas. The construction is flexible, meaning that codes of various dimensions and rates (spectral efficiency) may be constructed. Additionally, because of the properties of the deployed mapping, the structure of the spherical codes applies to the Grassmannian constellations as well. I.
Index Terms
, 2004
"... It is known that at high signal to noise ratio (SNR), or for large coherence interval (T), a constellations of unitary matrices can achieve the capacity of the noncoherent multipleantenna system in block Rayleigh flatfading channel. For a single transmit antenna system, a unitary constellation i ..."
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It is known that at high signal to noise ratio (SNR), or for large coherence interval (T), a constellations of unitary matrices can achieve the capacity of the noncoherent multipleantenna system in block Rayleigh flatfading channel. For a single transmit antenna system, a unitary constellation is simply a collection of Tdimensional unit vectors. Nevertheless, except for a few special cases, the optimal constellations are obtained only through exhaustive or random search, and their decoding complexity is exponential in the rate of the constellation and the length of the coherence interval, T. In this work, we propose a recursive construction method for realvalued single transmit antenna noncoherent constellations, in which a Tdimensional unitary constellation is constructed by using a number of (T − 1)dimensional unitary or spherical constellations as its equilatitude subsets. Comparison of the minimum distances achieved by the proposed constructions with the best known packings in G(T, 1) [1] shows that, for practical values of T, the recursive constellations are close to optimal. We also propose a simple lowcomplexity decoding algorithm for the singleantenna recursive constellations. The complexity of the proposed decoder is linear in the total number of the twodimensional constituent subsets, which is usually much smaller than the number of the constellation points. Nevertheless, the performance of the suboptimal decoder is similar to the optimal decoder. A comparison of the error rate performance of the recursive constellations with the complexvalued systematic designs of [2] shows that the proposed realvalued constellations have similar performance to the complexvalued systematic designs. The recursive designs also show a significant gain over the lowcomplexity PSK constellations of [3].