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Performance analysis of multichannel lattice equalization in coherent underwater communications

J.P. Gomes
ISR - Instituto Superior Tecnico, 1000 Lisboa, Portugal
A. Silva
and S.M.Jesus ,
SiPLAB-FCT, Universidade do Algarve, 8005-139 Faro, Portugal

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Ref.: Proc. OCEANS MTS/IEEE ' 2007, (ISBN:), p., Vancouver, Canada,  2007. (to appear)

Abstract : This paper presents experimental results for equalization of single-carrier transmissions using multichannel RLS lattice filters under fixed-point arithmetic. The performance of the lattice algorithm is illustrated and compared with other RLS algorithms using real data from the MREA'04 and MAKAI'05 ocean experiments. Recently, the implementation of signal processing algorithms in dedicated hardware has gained new momentum due to the availability of (1) large-capacity FPGAs that incorporate hardware multipliers and other useful features for mathematical computation, and (2) tools that can automatically generate reasonably-efficient HDL code from block diagrams or other high-level descriptions. Such power-efficent and potentially highly-parallel solutions are certainly appealing for equalization of underwater channels, where the computational complexity increases with the square of the number of coefficients, which is often large (long impulse responses, multiple receivers). However, the input covariance matrix tends to be ill-conditioned when fractional sampling or multiple receivers are used, especially with high filter orders, and in turn this leads to high dynamical ranges in the internal variables of several adaptive filtering algorithms, including plain RLS. Accomodating such numerical excursions in fixed-point arithmetic requires large word lengths, and hence more FPGA real estate and power consumption.
A new multichannel lattice algorithm for RLS filtering was proposed in [1]. The algorithm is based on a modular decomposition approach that enables unequal channel orders to be specified and leads to a filter that is structured as an array of interconnected scalar units which operate similarly to those found in conventional single-channel lattice filters. This avoids cumbersome matrix operations that are difficult to implement in hardware and not easily parallelizable. For a least-squares problem with L channels and M adjustable coefficients per channel the filter complexity is proportional to ML2, rather than (ML)2 as in plain RLS. Each scalar unit performs an input-output mapping independently of all the remaining ones, so that a fully-parallel implementation may be obtained by adding pipeline registers between units.
By exploiting a known link with Kalman filtering theory, the update recursions for individual units are written in array form, i.e., forward or backward (2-by-2) prearrays are built from each unit's internal variables, a Givens rotation is applied to create a postarray where a specific element is zeroed-out, and time-updated variables are read from other entries in the array. CORDIC algorithms enable Givens rotations to be iteratively implemented in hardware using only simple add/shift operations. The array formulation is known to exhibit excellent numerical properties in the single-channel case, and these were found to carry over to the proposed multichannel lattice. In simple simulated scenarios the array-based lattice exhibited none of the numerical instability problems that affect other fast RLS algorithms even for 10-bit fixed-point precision. Its behavior was found to be consistent in the sense that the output mean-square error (MSE) progressively increased as the numerical precision decreased. Moreover, for any given precision its MSE was lower than that obtained with other multichannel lattice and plain RLS algorithms.

This work examines the fixed-point performance of the array-based multichannel lattice under more challenging conditions, using data from two underwater communication experiments:
The MREA'04 (Maritime Rapid Environmental Assessment) sea trial was conducted in the continental shelf off the west coast of Portugal in April 2004, in an area to the north of the Setúbal Canyon. The acoustic source was suspended at a depth of about 60 m, and the receiver was an 8-element drifting array with hydrophones placed at depths 10, 15, 55, 60, 65, 70, 75, 80 m. During a period of approximately 90 minutes modulated data were transmitted over a range of about 2 Km, using a carrier frequency of 3600 Hz, symbol rates of 200 or 400 baud, and both 2-PSK and 4-PSK constellations. The transmitter was being towed throughout much of the experiment at speeds of up to 2 m/s, thus inducing significant Doppler scaling in received waveforms.
The MAKAI'05 experiment took place off the island of Kauai, Hawaii, in September/October 2005, in an area that is part of the Pacific Missile Range Facility (PMRF). MAKAI'05 was specifically planned to support the High-Frequency initiative, whose goal is to gain a better understanding of acoustic propagation at frequencies on the order of tens of kHz. A large number of teams from various countries and institutions were involved, each focused on a specific set of objectives related to its equipment and scientific goals. The data examined in this paper pertains to an experiment designed by the University of Algarve, Portugal, in which 2-PSK data were transmitted at 2000 baud (3 kHz bandwidth) around a carrier frequency of 10 kHz. Both moored and towed sources were used, and source-receiver ranges were on the order of 1 to 4 Km. The drifting receiver array was similar to the one used in MREA'04, but only 5 hydrophones were functional.

Both linear (FSE) and decision-feeback (DFE) equalizer architectures are considered in this work. Contrary to the usual transversal DFE, where the outputs of the feedforward and feedback filters can be individually accessed, all channels (fractionally-sampled multichannel acoustic data and previously-decoded symbols) are merged in a lattice DFE. This excludes some widely-used algorithms for carrier and symbol synchronization in DFE based on the MMSE criterion, and a few alternative options, MMSE-based or not, are discussed.

As in [1], it is found that the differences in performance between RLS variants are small under floating-point arithmetic. However, only the array-based lattice provides acceptable performance for 12-bit fixed-point precision. Plain RLS performs very poorly even for 16-bit precision.

[1] J. Gomes, V. Barroso, "Array-Based QR-RLS Multichannel Lattice Filtering," Submitted to IEEE Transactions on Signal Processing, April 2007.

ACKNOWLEDGMENT: this work was partially supported by FCT projects NUACE - POSI/CPS/47824/2002 and RADAR - POCTI/CTA/47719/2002.

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