The fifteen articles in this special issue are devoted to new equalization techniques for wireless communications, including new the latest theories and applications.

The specific requirements of a design for terrestrial mobile multimedia multicast physical layer are described. The technical characteristics of the FLOtrade physical layer are described in the context of the identified requirements. Simulation results are presented

Coherent spread-spectrum communications with transmitted reference-based, also called pilot-based, channel estimation is considered for code-division multiple-access (CDMA) communications over fading channels. Both noncontiguous time-division multiplexed reference symbol-based and continuous code-division multiplexed channel-based schemes are described. Assuming mean square error or optimal robust channel estimation, we derive the optimal receiver structure with a maximum-likelihood convolutional decoder. In addition, the Bhattacharyya bound and the cutoff rate of the reference-assisted coherent communications are derived. These analytical results are used for evaluating system performance and for selecting parameters such as coding rate and the data to reference power ratio to optimize system performance. Simulation results are given showing that the reverse link performance in a CDMA system can be significantly improved by using the reference-assisted coherent communication instead of noncoherent reception of orthogonally coded signal.

Optimal pilot assisted estimation of communication channels is considered for coherent cellular and PCS CDMA reverse link communications. Both pilot symbol and pilot channel based schemes are described and the optimal estimators for these two schemes are analyzed. Relative mean square estimation error (RMSEE) and optimal power allocation between data and pilot signals are derived based on the analysis. Finally, simulation results are given to show the reverse link performance can be significantly improved by using the pilot assisted coherent communication instead of non-coherent schemes for the CDMA reverse link.

In this paper, we developed a systematic frequency domain approach to analyze adaptive tracking algorithms for fast time-varying channels. The analysis is performed with the help of two new concepts, a tracking filter and a tracking error filter, which are used to calculate the mean square identification error (MSIE). First, we analyze existing algorithms, the least mean squares (LMS) algorithm, the exponential windowed recursive least squares (EW-RLS) algorithm and the rectangular windowed recursive least squares (RW-RLS) algorithm. The equivalence of the three algorithms is demonstrated by employing the frequency domain method. A unified expression for the MSIE of all three algorithms is derived. Secondly, we use the frequency domain analysis method to develop an optimal windowed recursive least squares (OW-RLS) algorithm. We derive the expression for the MSIE of an arbitrary windowed RLS algorithm and optimize the window shape to minimize the MSIE. Compared with an exponential window having an optimized forgetting factor, an optimal window results in a significant improvement in the h MSIE. Thirdly, we propose two types of robust windows, the average robust window and the minimax robust window. The RLS algorithms designed with these windows have near-optimal performance, but do not require detailed statistics of the channel.>

For CDMA (code-division multiple-access) wireless communications systems, orthogonal signaling with noncoherent detection has been proposed when a reference phase is not available. In this paper, orthogonal signaling using Walsh signals is considered. Performance of the channel using error-correcting coding together with 1) bit-by-bit interleaving or 2) Walsh word-by-word interleaving is investigated. Simulation results show that for several types of fading and non-fading channels with practical amounts of interleaving and with practical power control, the second technique with complete soft decision decoding that has added complexity in the decoder gives significant reduction in the required signal to interference ratio (SIR) over the first technique.>

The matched filter bound (MFB) for the two-path channel derived in Mazo (1991) is extended to Rayleigh fading channels with an arbitrary number of time-discrete paths. General results on antenna diversity reception and communication over correlated-path fading channels are presented. Examples of the MFB for several practically useful channels are given.>

A echo canceller fast training scheme for data-driven Nyquist in-band echo cancellers is presented. This scheme simultaneously estimates the desired near and far echo canceller coefficients by sending a special periodic training sequence and correlating a segment of the sequence with the real echo samples. The requirements and the generation of the training sequence are discussed. It is shown that the fast training method can also provide the parameters for the fast initialization of the phase roll compensator in the echo canceller, and thus complete the fast training of the entire echo canceller. Compared to the conventional LMS training and other training methods, this scheme provides accurate estimates of the echo canceller coefficients in a significantly reduced training time.>

General order-recursive least-squares estimation employing a sliding window is described. It is shown that time and order updates of any order-recursive sliding window least-squares algorithm can be obtained solely by 3*3 hyperbolic Householder transformations. Applying this general observation to the sliding window least-squares estimation of time-series signals results in a new algorithm: the hyperbolic Householder lattice (HHL) algorithm. This work broadens the range of applications of QR-decomposition (QRD)-based adaptive least-squares algorithms by allowing a sliding window formulation in addition to the known exponentially windowed form.>

One of the main problems in the proposed Northern American digital cellular system (IS-54) is the poor performance of the receiver for fast fading channels. The authors propose to use a novel joint data and channel estimation technique to improve performance. The basic idea of this method was described by N. Seshadri (1990) in the context of blind equalization. Simulation results indicate that this approach solves the problems of decision delay and divergence caused by error decision feedback in conventional maximum likelihood sequence estimation (MLSE). The specified IS-54 requirement of 19 dB at a BER of 3% and a speed of 100 km/h can be met with a margin of 8 dB. This result is 4.5 dB better than the conventional maximum likelihood receiver.>

A multichannel Givens lattice algorithm is presented, which has a modular structure and is suitable for parallel implementation. The basic processing module requires a 2*2 plane rotation (Givens rotation), which has superior numerical properties due to its circular nature. The application of the multichannel Givens lattice algorithm to channel equalization is also discussed.>

A method is proposed for estimating the impulse response of a complex system based on its complex input and only the real part of its output. A unique periodic white complex testing sequence is constructed, and it is then sent to the system to be estimated. An important feature of the testing sequence is that its real and imaginary parts are mutually orthogonal, i.e. their cross-correlation is zero. Due to this orthogonality, the effects of the system's real imaginary parts are ideally decoupled. The cross-correlation between the complex testing sequence and the system's real part output yields an accurate estimate of the system's complex impulse response. In applications where the unknown system's impulse response is sparse, it is possible to squeeze it and choose a considerably small period for the sequence to reduce complexity. An effective algorithm for this purpose is presented. The application of this new method to fast training of the Nyquist echo cancellers in data modems is described. It is shown that using these techniques, fast and accurate estimation can be achieved efficiently.>

An extension is presented of the work of V.M. Eyubogin (see ibid., vol.36, p.401-9, Apr. 1988) on decision-feedback equalization (DFE) applied to coded systems with interleaving. The authors study the adaptive implementation of the DFE using recursive-least-squares algorithms (RLS). System performance on time-dispersive channels with nulls as well as on channels with relatively small intersymbol interference is investigated. A reference insertion method is used to improve system performance, and a two-stage processing technique is adopted to use the more reliable decisions from the decoder in equalizer coefficient adaptation. Simulation results show that the RLS-implemented DFE systems with coded modulation can successfully combat severe channel distortion while maintaining the coding gain over corresponding uncoded systems.>

LS (least squares) lattice algorithms based on Givens rotation, called Givens-lattice algorithms, are presented. They are derived by exploiting the relationship between the Givens algorithms and the RMGS (recursive modified Gram-Schmidt) algorithm. It is shown that the Givens-lattice algorithms are suitable for LS estimation of time-series signals and are computationally more efficient than the recently derived fast QR algorithm for the same purpose. Systolic array implementation of Givens-lattice algorithms using the same basic processing cells in the systolic arrays for the original Givens algorithm is discussed. Computer simulation results are given.>

The use of an intersymbol interpolation method in training fractionally spaced equalizers (FSEs) is investigated. It is shown that the optimal interpolation filter depends on the amplitude frequency response of the transmitter filter and the channel. Using a nonoptimal interpolation filter increases the stead-state mean-square error of the FSE. An interpolated complex FSE (CFSE) using a stochastic gradient, or LMS, adaptive algorithm has very little advantage over an LMS CFSE with symbol-rate updating. However, an interpolated LMS phase-splitting FSE (PS-FSE) has a convergence speed that is twice as fast as a conventional PS-FSE. Special precautions for evaluating the performance of interpolated FSEs are discussed, and a novel evaluation scheme is proposed.>

The author investigates the convergence characteristics of the last mean square (LMS) and the recursive least squares (RLS) adaptive algorithms when the correlation matrix of the input signal does not have a full rank. It is shown that the initial convergence rate of the LMS algorithm is inversely proportional to the rank of correlation matrix, or equivalently, the number of nonzero eigenvalues. The same conclusion holds for the RLS algorithms if the minimum norm solution (MNS) is used in each iteration. A simple time-recursive method to obtain approximate MNSs in each iteration is presented and proven. The effect of additive noise is discussed.>

Investigation of the eigenstructure of the autocorrelation matrix of the received signal in a complex or real fractionally-spaced equalizer indicates that this matrix has a certain number of nontrivial eigenvalues. By using singular value decomposition, it is shown that the excess mean-square error in least-squares adaptive algorithms is proportional to the number of nontrivial eigenvalues, instead of the number of taps. This leads to methods for effectively reducing the misadjustment error.>

In practical applications of the LMS adaptive transversal filtering algorithm, a delay in the coefficient update is sometimes required. In this paper, the behavior of the Delayed LMS (DLMS) algorithm is studied. A stablility bound for the step size is derived. The relation between the step size and the convergence speed and the effect of the delay on the convergence behavior are also investigated. Computer simulation results are provided and compared with the analytical results.

The development of systolic array processors is very important for real-time adaptive array processing applications. In this paper a numerically robust algorithm, based on the modified Gram-Schmidt (MGS) method, is presented. The algorithm can be efficiently realized using a systolic architecture and is capable of handling both exponential and finite memory windows in order to cope with time-varying data. Finally, the relation of these algorithms to similar schemes based on the Given's transformation is investigated.

In this paper, we show that it is possible to approximate true division by dividing a number by a power-of-two in the LS lattice algorithm that is based on the new direct-coefficient-update form of the exact least squares (LS) lattice algorithm. The resulting pseudo-LS lattice algorithm is modestly slower in initial convergence than the true LS lattice algorithm, while it has a steady state error similar to the latter. The results are extended to a family of LS estimation algorithms covering a broader scope of applications.

New formulas are presented for direct updating of the reflection coefficients in the a priori and a posteriori forms of the least squares (LS) lattice algorithms. An analysis of the numerical characteristics of the new LS lattice forms is given and a comparison is made with the numerical characteristics of the conventional LS lattice algorithms. Computer simulation results are provided on the numerical accuracy of conventional and new LS lattice algorithms for several values of finite word length.

This paper presents two types of adaptive lattice decisionfeedback equalizers (DFE), the least squares (LS) lattice DFE and the gradient lattice DFE. Their performance has been investigated on both time-invariant and time-variant channels through computer simulations and compared to other kinds of equalizers. An analysis of the self-noise and tracking characteristics of the LS DFE and the DFE employing the Widrow-Hoff least mean square adaptive algorithm (LMS DFE) are also given. The analysis and simulation results show that the LS lattice DFE has the faster initial convergence rate, while the gradient lattice DFE is computationally more efficient. The main advantages of the lattice DFE's are their numerical stability, their computational efficiency, the flexibility to change their length, and their excellent capabilities for tracking rapidly time-variant channels.

This paper provides a quantitative analysis of the tracking characteristics of least squares algorithms. A comparison is made with the tracking performance of the LMS algorithm. Other algorithms that are similar to least squares algorithms, such as the gradient lattice algorithm and the Gram-Schmidt orthogonalization algorithm are also considered. Simulation results are provided to reinforce the analytical results and conclusions.

This paper investigates the numerical accuracy and the numerical stability of adaptive estimation algorithms. A feedback system model is provided to investigate the numerical stability of the algorithms. Simulation results on the comparison of numerical accuracy of LS adaptive algorithms are given. Two new methods are given that improve the numerical accuracy of the adaptive algorithms when a short word length is used.

A generalized least squares lattice algorithm (GLSLA), which is appropriate for multi-channel adaptive filtering and estimation, is presented. This algorithm has all the advantages of a conventional least-squares (LS) lattice algorithm, such as computational efficiency, numerical stability, and the property that successive stages are decoupled. Besides these advantages, it also has the flexibility that the number of stages in each channel may be different. This property does not exist in the conventional LS lattice algorithms. As an application, a decision feedback equalizer (DFE) with different numbers of feedforward and feedback stages is proposed and implemented using the GLSLA. Some results obtained from a computer simulation of the algorithm are also presented.