Finite Impulse Response Equalizer Research Articles

Throughput of Cellular Systems with Conferencing Mobiles and Cooperative Base Stations

This paper considers an enhancement to multicell processing for the uplink of a cellular system, whereby the mobile stations are allowed to exchange messages on orthogonal channels of fixed capacity (conferencing). Both conferencing among mobile stations in different cells and in the same cell (inter- and intracell conferencing, resp.) are studied. For both cases, it is shown that a rate-splitting transmission strategy, where part of the message is exchanged on the conferencing channels and then transmitted cooperatively to the base stations, is capacity achieving for sufficiently large conferencing capacity. In case of intercell conferencing, this strategy performs convolutional pre-equalization of the signal encoding the common messages in the spatial domain, where the number of taps of the finite-impulse response equalizer depends on the number of conferencing rounds. Analysis in the low signal-to-noise ratio regime and numerical results validate the advantages of conferencing as a complementary technology to multicell processing.

An FIR Equalizer Design for Nonlinear Channels Using Hybrid GA and LMI

In this paper, a finite impulse response (FIR) equalizer for nonlinear discrete-time channels is designed by employing a hybrid genetic algorithm (GA) and linear matrix inequality (LMI) approach from an H∞ perspective. The GA technique is utilized to linearize the nonlinear channel model, and the approximate error can be viewed as a state uncertainty. Then, the design of the FIR equalizer is transformed into LMIs, and the coefficients of the FIR equalizer can be obtained by solving an LMI optimization problem. Finally, numerical examples are included to illustrate the effectiveness of the proposed methodology.

A 2.5- to 3.5-Gb/s Adaptive FIR Equalizer With Continuous-Time Wide-Bandwidth Delay Line in 0.25-<tex>$muhbox m$</tex>CMOS

This paper presents an adaptive finite impulse response (FIR) equalizer with continuous-time wide-bandwidth delay line in CMOS 0.25-mum process for 2.5-Gb/s to 3.5-Gb/s data communications. To achieve wide bandwidth, fractionally spaced structure is used and an inverter with active-inductor load design is proposed as the delay cell of the tap delay line. Close loop adaptation of the fractionally spaced FIR equalizer is demonstrated using a low-power and area-efficient pulse extraction method as on-chip error detector. Measurement results show that the proposed adaptive equalizer achieves over 75% horizontal eye opening when the channel loss at the half-data-rate frequency varies from 4 dB to 21 dB at 2.5-Gb/s data rate. At 3.5-Gb/s data rate, the equalizer achieves 68% horizontal eye opening when the channel loss is about 9.3 dB at the half-data-rate frequency. The adaptive equalizer including the FIR filter and the error detector occupies 0.095 mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2 </sup> die area and dissipates 95 mW at 2.5-Gb/s data rate from 2.5-V voltage supply

Filter Bank precoding for FIR equalization in high-rate MIMO communications

In this paper, the problem of designing finite-impulse-response (FIR) equalizers for multiple-input multiple-output (MIMO) FIR channels is considered. It is shown that an arbitrary MIMO frequency-selective channel can be rendered FIR equalizable by a suitable filter bank (FB) precoding operation that introduces redundancy at the transmitter. The expression for the minimum redundancy required to ensure FIR invertibility is derived. The analysis is extended to the case of MIMO multicarrier modulation. Optimum zero-forcing (ZF) and minimum mean-squared error (MMSE) solutions for the FIR equalizer are derived. Simulation results are provided to demonstrate that the proposed scheme achieves better performance than the block-processing methods while supporting a higher data rate.

Iterative joint optimization of minimal transmit redundancy FIR zero-forcing precoder-equalizer system for MIMO-ISI channel

An iterative joint finite-impulse response (FIR) zero-forcing (ZF) precoder-equalizer optimization algorithm for multiple-input multiple-output intersymbol interference (MIMO-ISI) channel is proposed. The existing joint precoder-equalizer design algorithms for MIMO-ISI channels require a guard period, which is longer than or equal to the channel order to avoid the interblock interference (IBI). This longer guard period is a kind of unnecessary redundancy consuming the valuable channel bandwidth. Based on space-time-modulated codes (STMC), this paper proposes the first algorithm for jointly optimizing the FIR precoder and equalizer without the guard-period constraint. Hence, the precoder-equalizer pairs obtained can achieve minimal transmit redundancy ISI-free communications for complex-valued signals. This greatly enhances the spectral efficiency for wide-band communications. The proposed algorithm is performed in an iterative basis. Sufficient conditions and convergence analysis of this algorithm are presented. The resultant precoder and equalizer are proved to be a least-square (LS) optimal solution for each other. The simulation results show that substantial performance gain is obtained with the proposed joint optimization algorithm.

Blind and semi-blind equalization based on the constant power criterion

This paper focuses on the constant power (CP) criterion for blind linear equalization of digital communication channels. This recently proposed criterion is specially designed for the extraction of q-ary phase shift keying (q-PSK) signals using finite impulse response equalizers. When zero-forcing equalizers exist, the CP cost function accepts exact analytic solutions that are unaffected by undesired local extrema and spare costly iterative optimization. A subspace-based method exploiting the Toeplitz-like structure of the solution space is put forward to recover the minimum-length equalizer impulse response from the overestimated-length solutions. The proposed method is more robust to the relative weights of the minimum-length equalizer taps than existing techniques. In less ideal scenarios where the analytic solutions are only approximate minimizers of the criterion, a gradient-descent algorithm is proposed to minimize the cost function. To reduce the detrimental effects of suboptimal equilibria and accelerate convergence, the iterative algorithm is initialized with the approximate closed-form solution, and an optimal step size is incorporated into its updating rule. This optimal step size, which globally minimizes the cost function along the search direction, can be computed algebraically. A semi-blind implementation, which is useful when training data are available, further reduces the impact of undesired local extrema and enhances the convergence characteristics (particularly the robustness to the equalizer initialization) of the iterative algorithm from just a few pilot symbols. All these beneficial features are demonstrated with an experimental study of the proposed CP-based methods in a variety of channels and simulation conditions.

Low-complexity search for optimal delay in linear FIR MMSE equalization

In this letter, an efficient algorithm to determine the optimal delay in linear finite impulse response equalizers based on the minimum mean-square-error criterion is proposed. The algorithm uses the Levinson-Durbin (L-D) recursion as a starting point to find the values of the mean-square error for equalizers with all nontrivial delays. Despite the exhaustive search approach, the complexity of the proposed algorithm is only doubled when compared to the calculation of the equalizer with one prescribed delay. Such increase in complexity may be fully justified in practice because it yields globally optimal equalizer's design.

A Kalman-Filter Approach to Equalization of CDMA Downlink Channels

An efficient method for equalization of downlink CDMA channels is presented. By describing the observed signal in terms of a state-space model, the method employs the Kalman filter (KF) to achieve an unbiased signal estimate satisfying the linear minimum mean-squared error (LMMSE) criterion. The state-space model is realized at the symbol and chip levels. With the symbol-level model, the KF is used to estimate the transmitted chips that correspond to each symbol interval; whereas at the chip level, the transmitted chips are estimated individually. The symbol-level KF has a built-in tracking capability that takes advantage of the a priori known scrambling sequence, which renders the transmitted signal nonstationary. The chip-level KF reduces the complexity of the symbol-level KF significantly by ignoring the nonstationarity introduced by scrambling. A simple method for further reducing the KF complexity is also presented. The computational complexity of the proposed technique is analyzed and compared with that of several linear approaches based on finite-impulse response (FIR) filtering. Simulations under realistic channel conditions are carried out which indicate that the KF-based approach is superior to FIR equalizers by- in error-rate performance.

Time-varying FIR equalization for doubly selective channels

We propose a time-varying (TV) finite impulse response (FIR) equalizer for doubly selective (time- and frequency-selective) channels. We use a basis expansion model (BEM) to approximate the doubly selective channel and to design the TV FIR equalizer. This allows us to turn a complicated equalization problem into an equivalent simpler equalization problem, containing only the BEM coefficients of both the doubly selective channel and the TV FIR equalizer. The minimum mean-square error (MMSE) as well as the zero-forcing (ZF) solutions are considered. Comparisons with the block linear equalizer (BLE) are made. The TV FIR equalization we propose here unifies and extends many previously proposed serial equalization approaches. In contrast to the BLE, the proposed TV FIR equalizer allows a flexible tradeoff between complexity and performance. Moreover, through computer simulations, we show that the performance of the proposed MMSE TV FIR equalizer comes close to the performance of the ZF and MMSE BLE, at a point where the design as well as the implementation complexity are much lower.

A comparative study of two blind FIR equalizers

The paper investigates blind finite-impulse-response (FIR) equalization schemes for quadrature amplitude modulation signalling. We compare a bootstrap maximum a posteriori probability (MAP) equalizer with a recently introduced concurrent constant modulus algorithm (CMA) and decision directed (DD) equalizer (CMA+DD). Both equalizers are known to outperform the CMA considerably in the situations where carrier recovery is performed and the signal constellation is known. The concurrent CMA+DD equalizer has a complexity that is slightly more than twice of the CMA, and the bootstrap MAP equalizer has computational requirements that are only slightly more complex than the CMA. Simulation results indicate that the bootstrap MAP blind FIR equalizer has a faster convergence rate and better steady-state performance than the concurrent CMA+DD blind FIR equalizer, but tuning of the former is more complicated than the latter.

FIR H∞ equalization

We approach finite impulse response (FIR) equalization problem from an H ∞ perspective. First, we formulate the calculation of the optimal H ∞ performance for a given equalization setting as a semidefinite programming (SDP) problem. H ∞ criterion provides a set of FIR equalizers with different optimality properties. Among these, we formulate the calculation of risk-sensitive or minimum entropy FIR filter as the constrained analytic centring problem and mixed H 2/ H ∞ problem as another SDP. We provide examples to illustrate the procedures we described.

Frequency-domain design method for linear-phase and minimum-phase FIR equalizers

First the author shows that generally, the attenuation of equalizers should be optimized rather than their magnitude responses. Then he presents an algorithm for the design of linear-phase finite-impulse response (FIR) filters with Chebyshev approximation of desired attenuation functions. The design is quite similar to the Parks-McClellan algorithm, with two decisive deviations. The first concerns the choice of a weighting function, and secondly, the Remez algorithm is modified in order to be able to approximate also nontrivial attenuation functions. Finally, the design is extended to minimum-phase FIR filters, which can also equalize the phase response of minimum-phase systems. It is shown here that phase optimization is a consequence of attenuation optimization.

Adaptive blind source separation and equalization for multiple-input/multiple-output systems

In this paper, we investigate adaptive blind source separation and equalization for multiple-input/multiple-output (MIMO) systems. We first analyze the convergence of the constant modulus algorithm (CMA) used in MIMO systems (MIMO-CMA). Our analysis reveals that the MIMO-CMA equalizer is able to recover one of the input signals, remove the intersymbol interference (ISI), and suppress the other input signals. Furthermore, for the MIMO finite impulse response (FIR) systems satisfying certain conditions, the MIMO-CMA FIR equalizers are able to perfectly recover one of the system inputs regardless of the initial settings. We then propose a novel algorithm for blind source separation and equalization for MIMO systems. Our theoretical analysis proves that the new blind algorithm is able to recover all system inputs simultaneously regardless of the initial settings. Finally, computer simulation examples are presented to confirm our analysis and illustrate the effectiveness of blind source separation and equalization for MIMO systems.

Fast computation of channel-estimate based equalizers in packet data transmission

Computationally efficient procedures are introduced for the real-time calculation of finite-impulse-response (FIR) equalizers for packet-based data transmission applications, such as wireless data networks. In such packet data applications, the FIR equalizer filters are computed indirectly by first estimating the channel pulse response from a known training pattern embedded in each packet and then computing the equalizer for use in the recovery of the remaining unknown data in the packet. We find that a minimum mean-square-error decision-feedback equalizer (MMSE-DFE) with a finite-length constraint on its feedforward and feedback filters can be very efficiently computed from this pulse response. We combine a recent theory of finite-spectral factorization for the MMSE-DFE with the theory of structured matrices to derive these efficient procedures for computing the equalizer settings. The introduced method is much more computationally efficient than direct computation by matrix inversion or the use of popular gradient or least-squares algorithms over the duration of the packet.

Complex correction of data acquisition channels using FIR equalizer filters

The common conviction about FIR (finite impulse response) digital filters is that the number of necessary taps, to reach the same performance as provided by IIR (infinite impulse response) digital filters, is usually too large. Moreover, the standard FIR filter design algorithm (Remez exchange) allows the design of linear-phase filters only. Therefore, IIR filters are often preferred over FIR ones, without any further investigations. This paper presents a case study of complex (amplitude and phase) equalization of the passband of a commercial anti-aliasing filter. The novelty is the usage of an FIR filter for this purpose (or an FIR one, combined with a low-order IIR filter), and a thorough discussion of the special design aspects. It turns out that for the given anti-aliasing filter (a Cauer filter of order 11) an FIR filter of length 60...100 can perform as well as a 26/26 (numerator order/denominator order) IIR one. The properties are even better if a low-order IIR filter is used in combination with an FIR one (orders, e.g., 1/1+40/0). Because of the absence of stability problems and the ease of implementation, the use of FIR filters is suggested. >

Ghost cancellation of analog TV signals: with applications to IDTV, EDTV, and HDTV

Techniques that can cancel ghosts in received analog TV (for improved-definition TV, extended-definition TV, and high-definition TV) signals are presented. The fact that there are short periods of time without the analog signal (the horizontal flyback interval between the lines) is utilized to periodically cleanse a finite impulse response (FIR) or an infinite impulse response (IIR) equalizer. This line-by-line processing (cleansing) overcomes the limitation of standard equalizers to allow for 40-50 dB of suppression of ghosts, even with nulls in the spectrum, as long as the ghost delay is less than the period of time without the analog signal. Furthermore, by using time inversion in combination with line-by-line processing, the stability problem of the conventional IIR equalizer can be eliminated. It is shown that it may be possible to implement this IIR equalizer on a single digital integrated circuit. Alternatively, an FIR equalizer can be used; although it requires multiple chips (i.e. more taps), it can acquire and adapt to the ghosted channel more rapidly than an IIR equalizer. >

FIR Adaptive Identification L∞ Bounds on Frequency Response Error

Using only input and noisy output measurements of a linear system, an adaptive finite impulse response (FIR) model may be fitted by means of on-line adaptive interpolation of the system's frequency response. The frequency response values are estimated using sharp notch filtering of the signals together with decoupled sets of two parameter (or single complex gain) identifiers, the FIR model is then constructed by Lagrange interpolation or, in the equispaced sample case, by discrete Fourier transform.The aim of this identification method is to explore jointly the L∞ frequency response error bounds of interpolation methods and the recent results of Ljung concerning frequency response modelling through filter selection. This allows us to investigate the design issues in the development of an adaptive FIR identification method to achieve L∞ accurate modelling. The method is applicable to both stable and unstable systems and an example is carried through for the FIR equalization (inversion) of a nominimum phase communications channel.

An adaptive Kalman equalizer: Structure and performance

The development of an adaptive infinite impulse response (IIR) linear equalizer is described. Using discrete time Wiener filtering theory, a closed form for the optimum mean-square error IIR filter is derived. A performance comparison using both minimum and non-minimum phase channels indicates the complexity/performance advantages inherent in the IIR system compared to an optimum finite impulse response (FIR) solution. The minimum phase spectral factorization, which is an integral part of the derivation of the IIR equalizer, may be circumvented through the use of a Kalman equalizer such as that originally proposed by Lawrence and Kaufman. The structure is made adaptive by using a system identification algorithm operating in parallel with a Kalman equalizer. In common with Luvison and Pirani, a least mean squares (LMS) algorithm was chosen for the system identification because the input to the channel is white and hence the LMS algorithm will produce consistent predictable results with little added complexity. A new technique is introduced which both estimates the variance of channel noise and compensates the Kalman filter for errors in the estimate of the channel impulse response. Computer simulation results show that the convergence performance of this new adaptive IIR filter is roughly equivalent to an FIR equalizer which is trained using a recursive least squares algorithm. However, the order of the new filter is always lower than the FIR filter.