On a periodic training sequence in DFE to reduce the steady-stateerror probability

A periodic transmission of a short resetting sequence used to “fill up” the decision-feedback equalizer feedback filter memory with correct symbols is proposed to reduce the propagation error. Two strategies are considered: full and partial reset, where the entire memory and some part of it is reset, respectively. For any given steady-state bit-error rate, the period of the resetting sequence is found by use of the Markov chain limit theorem. It is shown, via an illustrative example, that for low signal-to-noise ratio (SNR), the resetting strategy outperforms Bose-Chaudhuri-Hocquenghen (BCH) code with the same code rate. Moreover, a joint resetting and error correction coding strategy is considered, which is better than BCH code for low and medium SNR     

Iterative design and detection of a DFE in the frequency domain

Error-propagation phenomena and computational complexity of the filters' design are important drawbacks of existing decision-feedback equalizers (DFE) for dispersive channels. In this paper, we propose a new iterative block DFE (IBDFE) which operates iteratively on blocks of the received signal. Indeed, a suitable data-transmission format must be used to allow an efficient implementation of the equalizer in the frequency domain, by means of the discrete Fourier transform. Two design methods are considered. In the first method, hard detected data are used as input of the feedback, and filters are designed according to the correlation between detected and transmitted data. In the second method, the feedback signal is directly designed from soft detection of the equalized signal at the previous iteration. Estimators of the parameters involved in the IBDFE design are also derived. From performance simulations on a wireless dispersive fading channel, we observed that the IBDFE outperforms existing DFEs. Moreover, the IBDFE exhibits a reduction of the computational complexity when compared against existing schemes, both in signal processing and in filter design.     

On the equivalence of multivariate polynomial matrices

The equivalence of system is an important concept in multidimensional (\(n\)D) system, which is closely related to equivalence of multivariate polynomial matrices. This paper mainly investigates the equivalence of some \(n\)D polynomial matrices, several new results and conditions on the reduction by equivalence of a given \(n\)D polynomial matrix to its Smith form are obtained.     

On the equivalence between SOVA and max-log-MAP decodings

It is shown that after a proper simple modification, the soft-output Viterbi algorithm (SOVA) proposed by Hagenauer and Hoeher (1989) becomes equivalent to the max-log-maximum a posteriori (MAP) decoding algorithm. Consequently, this modified SOVA allows to implement the max-log-MAP decoding algorithm by simply adjusting the conventional Viterbi algorithm. Hence, it provides an attractive solution to achieve low-complexity near-optimum soft-input soft-output decoding     

On the equivalence of PDA algorithm and SIC-MMSE algorithm

In this letter, by employing Gaussian distribution to approximate the probability density function （pdf） of the extrinsic information at the output of the multiuser detector as a function of the pdf of the input extrinsic messages, it is concluded that the Probabilistic Data Association （PDA） algorithm is equivalent to the Soft Interference Cancellation plus Minimum Mean Square Error algorithm （SIC-MMSE）.     

On the equivalence of set-theoretic and maxent MAP estimation

We establish an equivalence between two conceptually different methods of signal estimation under modeling uncertainty, viz., set-theoretic (ST) estimation and maximum entropy (maxent) MAP estimation. The first method assumes constraints on the signal to be estimated, and the second assumes constraints on a probability distribution for the signal. We provide broad conditions under which these two estimation paradigms produce the same signal estimate. We also show how the maxent formalism can be used to provide solutions to three important problems: how to select sizes of constraint sets in ST estimation (the analysis highlights the role of shrinkage); how to choose the values of parameters in regularized restoration when using multiple regularization functionals; how to trade off model complexity and goodness of fit in a model selection problem.     

On the analytical equivalence of electromagnetic fields solutionsfrom a known source distribution

An analytical proof of the equivalence of solutions from monopole and dipole techniques is presented for electric and magnetic fields by a simple lightning model. The proof is in the context of the lightning problem as stated by M. Rubinstein and M.A. Uman (1989). The expressions for the electric and magnetic fields are simplified considerably     

Theoretical and Measured Performance of a DFE Modem on a Fading Multipath Channel

A decision-feedback equalizer (DFE) is the basis of a recent development of a quadruple diversity troposcatter modem which can operate up to a data rate of 12.6 Mbit/s in a 99% bandwidth of 15 MHz. In this paper a theoretical approach is developed for the calculation of average bit error rate (ABER), including the effects of intersymbol interference due to multipath and the finiteness of the transversal filters used to realize the DFE. By omitting the intersymbol interference effect, the calculation provides a lower bound which can be used to assess the intersymbol interference penalty for a particular DFE structure. The paper includes calculations of a DFE configuration which has a three tap forward filter with tap spacing equal to one-half a symbol interval. Measured performance results from fading channel simulator tests of a three tap forward filter DFE are presented for data rates from 1.5 to 12.6 Mbit/s and for a wide range of multipath statistical conditions. The results for this DFE configuration show (1) excellent agreement between calculated and measured ABER, (2) a small intersymbol interference penalty when the 2σ multipath spread is less than approximnately one-half the data symbol interval, and (3) successful operation at values of multipath spread up to twice the data symbol interval. In a sequel to this paper, the results of a field test of the DFE modem are presented. These live links test results are consistent with both the calculated and simulator measured data presented here.     

On the equivalence of blind equalizers based on MRE and subspaceintersections

Two classes of algorithms for multichannel blind equalization are the mutually referenced equalizer (MRE) method by Gesbert et al. (see ibid. vol.45, p.2307-17, 1997) and the subspace intersection (SSI) method by van der Veen et al. (see ibid., vol. 45, no.1, p.173-90, 1997). Although these methods seem, at first sight, unrelated, we show that certain variants of the SSI and the MRE methods both optimize a new blind criterion, which is referred to as maximum coherence and, thus, are equivalent     

On the equivalence of linear MMSE chip-level equalizer and generalized RAKE

Several linear equalization schemes have been proposed for the downlink of direct-sequence code-division multiple-access systems. Previous numerical performance evaluation of such algorithms showed the equivalence of the linear minimum mean-square error chip-level equalizer and a slightly modified version of the generalized RAKE. In this letter, we establish this equivalence analytically.     

Performance analysis of a single-user MISO ultra-wideband time reversal system with DFE

This paper presents an analysis of the performance of a baseband multiple-input single-output (MISO) time reversal ultra-wideband system (TR-UWB) incorporating a symbol spaced decision feedback equalizer (DFE). A semi-analytical performance analysis based on a Gaussian approach is considered, which matched well with simulation results, even for the DFE case. The channel model adopted is based on the IEEE 802.15.3a model, considering correlated shadowing across antenna elements. In order to provide a more realistic analysis, channel estimation errors are considered for the design of the TR filter. A guideline for the choice of equalizer length is provided. The results show that the system’s performance improves with an increase in the number of transmit antennas and when a symbol spaced equalizer is used with a relatively small number of taps compared to the number of resolvable paths in the channel impulse response. Moreover, it is possible to conclude that due to the time reversal scheme, the error propagation in the DFE does not play a role in the system’s performance.     

On the equivalence between Berlekamp's and Euclid's algorithms (Corresp.)

It is shown that Berlekamp's iterative algorithm can be derived from a normalized version of Euclid's extended algorithm. Simple proofs of the results given recently by Cheng are also presented.     

On the equivalence of Kramer's and Shannon's sampling theorems (Corresp.)

The question of Campbell concerning the possibility of applying Shannon's sampling theorem to the same functions that can be sampled by Kramer's generalized sampling theorem is investigated and illustrated for more functions which are solutions of second-order differential equations with singular coefficients. A few definitions are introduced and some theorems are proved to show, in a more precise way, that under certain conditions the two sampling theorems are equivalent.     

On the equivalence of the operator and kernel methods for jointdistributions of arbitrary variables

Generalizing the concept of time-frequency representations, Cohen (see Englewood Cliffs, NJ: Prentice-Hall, 1995) has proposed a method, based on operator correspondence rules, for generating joint distributions of arbitrary variables. As an alternative to considering all such rules, which is a practical impossibility in general, Cohen has proposed the kernel method in which different distributions are generated from a fixed rule via an arbitrary kernel. We derive a simple but rather stringent necessary condition, on the underlying operators, for the kernel method (with the kernel functionally independent of the variables) to generate all bilinear distributions. Of the specific pairs of variables that have been studied, essentially only time and frequency satisfy the condition; in particular, the important variables of time and scale do not. The results warrant further study for a systematic characterization of bilinear distributions in Cohen's method     

On simultaneous signal estimation and parameter identificationusing a generalized likelihood approach

A common approach to blind deconvolution of Bernoulli-Gaussian processes consists of performing both signal restoration and hyperparameter identification through maximization of a single generalized likelihood criterion. It is shown on a simple example that the resulting hyperparameter estimates may not converge toward any meaningful value. Therefore, other more reliable approaches should be adopted whenever possible     

On the equivalence of generalized concatenated codes andgeneralized error location codes

We show that the generator matrix of a generalized concatenated code (GCC code) of order L consists of L submatrices, where the lth submatrix is the Kronecker product of the generator matrices of the lth inner code and the lth outer code. In a similar way we show that the parity-check matrix of a generalized error location code (GEL code) of order L consists of L submatrices, where the lth submatrix is the Gronecker product of the parity-check matrices of the lth inner code and the lth outer code. Then we use these defining matrices to show that for any GCC code there exists an equivalent GEL code and vice versa     

Comparison of DFE and MLSE Receiver Performance on HF Channels

Data communication at rates near or above 2 kbits/s on 3 kHz-baadwidth HF radio channels is subject to impairment from severe linear dispersion, rapid channel time variation, and severe fading. In this investigation, recorded 2.4 kbit/s QPSK signals received from HF channels were processed to extract a time-varying estimate of the channel impulse response. From the estimated channel impulse responses, performance-related parameters were computed for ideal matched filter reception, maximum-likelihood sequence-estimation (MLSE), and decision feedback equalization (DFE). The results indicated that the simpler DFE receiver suffered only a small theoretical performance degradation relative to the more complex MLSE receiver. Other HF channel impulse response statistics were also obtained to shed light on equalization and filter adaptation techniques.     

On the equivalence of geometrical-optics reflected fields and the stationary-phase solution of the associated radiation integrals

This tutorial shows how the geometrical-optics expression for the field reflected from a surface can be derived directly from the general first-order stationary-phase solution of the associated radiation integral. It will improve the understanding of how these two techniques are related to one another. The material presented is ideally suited for post-graduate courses on high-frequency electromagnetics techniques.     

About the asymptotic performance of MMSE MIMO DFE for filter-bankbased multicarrier transmission

We investigate the asymptotic performance of a multi-input multi-output decision-feedback (DF) equalizer used to detect a multicarrier (MC) signal based on a filter-bank and transmitted over a linear dispersive channel. We derive the optimum DF structure for a minimum-mean square-error criterion. We basically show that with infinite length forward and feedback filters and at high signal-to-noise ratio, the geometrical mean of prediction errors does not depend on the paraunitary filter-bank used in the transmitter. Hence, it appears that uncoded filter-bank based MC transmission and single carrier transmission, both used over the same channel and with optimum DF, lead to the same achievable bit rate