Abstract.Recently the problem of determining the best, in the least-squares sense, rank-1 approximation to a higher-order tensor was studied and an iterative method that extends the wellknown power method for matrices was proposed for its solution. This higher-order power method is also proposed for the special but important class of supersymmetric tensors, with no change. A simplified version, adapted to the special structure of the supersymmetric problem, is deemed unreliable, as its convergence is not guaranteed. The aim of this paper is to show that a symmetric version of the above method converges under assumptions of convexity (or concavity) for the functional induced by the tensor in question, assumptions that are very often satisfied in practical applications. The use of this version entails significant savings in computational complexity as compared to the unconstrained higher-order power method. Furthermore, a novel method for initializing the iterative process is developed which has been observed to yield an estimate that lies closer to the global optimum than the initialization suggested before. Moreover, its proximity to the global optimum is a priori quantifiable. In the course of the analysis, some important properties that the supersymmetry of a tensor implies for its square matrix unfolding are also studied.Key words. supersymmetric tensors, rank-1 approximation, higher-order power method, higherorder singular value decomposition AMS subject classifications. 15A18, 15A57, 15A69PII. S0895479801387413 Introduction.A tensor of order N is an N -way array, i.e., its entries are accessed via N indices.1 For example, a scalar is a tensor of order 0, a vector is a tensor of order 1, and a matrix is a second-order tensor. Tensors find applications in such diverse fields as physics, signal processing, data analysis, chemometrics, and psychology [4].The notion of rank can also be defined for tensors of order higher than 2. The way this is done is via an extension of the well-known expansion of a matrix in a sum of rank-1 terms. Thus, the rank, R, of an N th-order tensor T is the minimum number of rank-1 tensors that sum up to T . A rank-1 tensor of order N is given by the generalized outer product of N vectors, u (i) , i = 1, 2, . . . , N, i.e., its (i 1 , i 2 , . . . , i N ) entry

Filter bank-based multicarrier (FBMC) systems based on offset quadrature amplitude modulation (FBMC/OQAM) have recently attracted increased interest (in applications including DVB-T, cognitive radio, and powerline communications) due to their enhanced flexibility, higher spectral efficiency, and better spectral containment compared to conventional OFDM. FBMC/OQAM suffers, however, from an imaginary inter-carrier/inter-symbol interference that complicates signal processing tasks such as channel estimation. Most of the methods reported thus far in the literature rely on the assumption of (almost) flat subchannels to more easily tackle this problem, with the aim of addressing it in a way similar to OFDM. However, this assumption may be often quite inaccurate, due to the high frequency selectivity of the channel and/or the small number of subcarriers employed to cope with frequency dispersion in fast fading environments. In such cases, severe error floors are exhibited at medium to high signal-to-noise ratio (SNR) values, that cancel the advantage of this modulation over OFDM. Moreover, the existing methods provide estimates of the subchannel responses, most commonly in the frequency domain. The goal of this paper is to revisit this problem through an alternative formulation that focuses on the estimation of the channel impulse response itself and makes no assumption on the degree of frequency selectivity of the subchannels. The possible gains in estimation performance offered by such an approach are investigated through the design of optimal (in the mean squared error sense) preambles, of both the full and sparse types, and of the smallest possible duration of only one pilot FBMC symbol. Existing preamble designs for flat subchannels are then shown to result as special cases. The case of longer preambles, consisting of two consecutive pilot FBMC symbols, is also analyzed. Simulation results are presented, for both mildly and highly frequency selective channels, that demonstrate the significant improvements in performance offered by the proposed approach over both OFDM and the optimal flat subchannel-based FBMC/OQAM method. Most notably, no error floors appear anymore over a quite wide range of SNR values.

In this paper, preamble-based least squares (LS) channel estimation in OFDM systems of the QAM and offset QAM (OQAM) types is considered, in both the frequency and the time domains. The construction of optimal (in the mean squared error (MSE) sense) preambles is investigated, for both the cases of full (all tones carrying pilot symbols) and sparse (a subset of pilot tones, surrounded by nulls or data) preambles. The two OFDM systems are compared for the same transmit power, which, for cyclic prefix (CP) based OFDM/QAM, also includes the power spent for CP transmission. OFDM/OQAM, with a sparse preamble consisting of equipowered and equispaced pilots embedded in zeros, turns out to perform at least as well as CP-OFDM. Simulations results are presented that verify the analysis. Index TermsChannel estimation, cyclic prefix (CP), discrete Fourier transform (DFT), least squares (LS), mean squared error (MSE), orthogonal frequency division multiplexing (OFDM), quadrature amplitude modulation (QAM), offset QAM (OQAM), pilots, preamble.

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