Moment space upper and lower error bounds for digital systems with intersymbol interference

A method for the evaluation of upper and lower bounds to the error probability of a linear pulse-amplitude modulation (PAM) system with bounded intersymbol interference and additive Gaussian noise is obtained via an isomorphism theorem from the theory of moment spaces. These upper and lower bounds are seen to be equivalent to upper and lower envelopes of some compact convex body generated from a set of kernel functions. Depending on the selection of these kernels and their corresponding moments, different classes of bounds are obtained. In this paper, upper and lower bounds that depend on the absolute moment of the intersymbol interference random variable, the second moment, the fourth moment, and an "exponential moment" are found by analytical, graphical, or iterative approaches. We study in detail the exponential moment case and obtain a family of new upper and a family of new lower bounds. Within each family, expressions for these bounds are given explicitly as a function of an arbitrary real-valued parameter. For two channels of interest, upper and lower bounds are evaluated and compared. Results indicate these bounds to be tight and useful.     

A residual intersymbol interference error bound for truncated-state Viterbi detectors

The Viterbi algorithm has been proposed for the symbol-by-symbol detection of discrete pulse amplitude modulated signals transmitted over noisy time-dispersive channels. The resulting detector has considerable complexity, requiring the storage of2^{m}state variables for binary signaling. A truncated-state detector assumes the channel memorymis much less than it really is and as such neglects some of the channel's intersymbol interference. An upper bound is prescribed for the detector's bit error probability due to this neglected interference in the case of binary signaling. This interference degrades the detector's performance in the same functional manner that intersymbol interference degrades the performance of the standard quantizing detector. This degradation can be considerably reduced for troublesome channels by optimally choosing the state variables for truncated-state detection.     

Semidefinite programming bounds on the probability of error of binary communication systems with inexactly known intersymbol interference

We consider the problem of evaluating the probability of error of binary communication systems in the presence of additive noise and intersymbol interferences whose statistics are inexactly known due to the estimation errors of the channel coefficients. We present a new method using semidefinite programming to evaluate tight bounds on the error probability based on the upper and lower bounds on the moments of those interferences. Numerical results are provided and compared with a previously published technique.     

On multiple interference error bounds

This letter presents upper and lower bounds for the error rate for (effectively) PAM signalling in the presence of thermal noise, cochannel interference, and intersymbol interference. To accomplish this, two known bounds and one new lower bound are used. For a class of examples from the literature, these simple bounds were found to provide an error-rate estimate accurate to 1 db in system SNR. Computational experiments indicate that this level of accuracy can be achieved when the system's eye is open by at least a factor of two.     

The intersymbol interference channel: lower bounds on capacity andchannel precoding loss

The discrete-time additive Gaussian intersymbol interference (ISI) channel with i.i.d. (not necessarily Gaussian) input signals is considered. Several new and old lower bounds on the capacity are derived in a unified manner by assuming different front-end receiver filters, in particular the sampled whitened matched filter (SWMF) and the minimum mean-squared error-decision feedback equalizer (MMSE-DFE) filter. The features of the bounds are demonstrated and compared in several examples with binary and quaternary input signals. It is also shown that the effect of an ideal post-cursor or tail cancellation, in an information-preserving context, depends primarily on the front-end filter. While, as is well known, ideal post-cursor cancellation at the output of the SWMF decreases the information, the opposite trend is seen when an MMSE-DFE front filter is considered. This observation reflects the basic theoretical obstacles in precoding, i.e., ideal post-cursor cancellation in the presence of a pre-cursor. It is used to assess the inherent loss (in terms of information rates as compared to the rates achievable with the hypothetical ideal post-cursor cancellation) associated with any post-cursor cancellation technique such as precoding, DFE, or other variants, when operating in synergy with the MMSE-DFE front-end filter. The effect of the front-end filter on an ideally interleaved, precoded coded-modulation system is also addressed     

A simple approximation to intersymbol interference error rate for low SNR (Corresp.)

An intersymbol interference bound is derived for multilevel and pseudoternary digital systems. The error bound is expressed in simple closed form. Monte Carlo trials are used to compare the accuracy of the bound with others recently published. The optimum matching point for the exponential bound is also determined.     

Moments of intersymbol interference

A new derivation of a recursion relation for the moments associated with intersymbol interference in multilevel PAM transmission is presented. This simple derivation uses certain properties of the cumulant generating function.     

Probability of error in PAM systems with intersymbol interference and additive noise

The effect of intersymbol interference and additive noise on the performance of a digital data transmission system is considered. The data sequence is assumed to be independent and equiprobable. The additive noise is independent of the signal but is not restricted to be Gaussian. A simple lower bound, an upper bound, and a simple approximation to the upper bound, on the probability of error are derived. The approximation to the upper bound is twice the lower bound; hence, either can be taken as an approximation to the actual error probability.     

Coding for intersymbol interference channels-combined coding andprecoding

This paper describes a new coding scheme for transmission over intersymbol interference (ISI) channels. This scheme, called ISI coding, combines trellis coding with precoding (used to combat ISI). Like the recently introduced precoding scheme of Laroia, Tretter, and Farvardin (LTF), the ISI coder makes it possible to achieve both shaping and coding gains over ISI channels. By combining coding and precoding, however, the ISI coder makes the “precoding loss” independent of the number of coset partitions used to generate the trellis code. At high rates (large signal-to-noise ratio (SNR)), this makes it possible to approach the Shannon capacity of an ISI channel. The V.34 (formerly V.fast) international modem standard for high-speed (up to 28.8 kb/s) communication over voice-band telephone lines uses the version of the ISI coder described in Section IV of this paper     

Binary intersymbol interference channels: Gallager codes, density evolution, and code performance bounds

We study the limits of performance of Gallager codes (low-density parity-check (LDPC) codes) over binary linear intersymbol interference (ISI) channels with additive white Gaussian noise (AWGN). Using the graph representations of the channel, the code, and the sum-product message-passing detector/decoder, we prove two error concentration theorems. Our proofs expand on previous work by handling complications introduced by the channel memory. We circumvent these problems by considering not just linear Gallager codes but also their cosets and by distinguishing between different types of message flow neighborhoods depending on the actual transmitted symbols. We compute the noise tolerance threshold using a suitably developed density evolution algorithm and verify, by simulation, that the thresholds represent accurate predictions of the performance of the iterative sum-product algorithm for finite (but large) block lengths. We also demonstrate that for high rates, the thresholds are very close to the theoretical limit of performance for Gallager codes over ISI channels. If C denotes the capacity of a binary ISI channel and if C/sub i.i.d./ denotes the maximal achievable mutual information rate when the channel inputs are independent and identically distributed (i.i.d.) binary random variables (C/sub i.i.d.//spl les/C), we prove that the maximum information rate achievable by the sum-product decoder of a Gallager (coset) code is upper-bounded by C/sub i.i.d./. The last topic investigated is the performance limit of the decoder if the trellis portion of the sum-product algorithm is executed only once; this demonstrates the potential for trading off the computational requirements and the performance of the decoder.     

Statistical detection for communication channels with intersymbol interference

The optimum detector under a fixed delay constraint D is derived for channels having memory and additive noise. The resulting receiver is recursive and does not grow with the message length. Its structure is presented for linear channels (AM and PSK) with known (or estimated) impulse response and noise statistics. The intersymbol interference is assumed to extend for L sampling periods and the receiver is allowed a "look-ahead" at D future received samples. Simulation results using actual channel characteristics show the detector to outperform a transversal equalizer even for relatively small values of D.     

Joint equalization and coding for intersymbol interference channels

We present a novel scheme that combines decision feedback equalization (DFE) with high-rate error-detection coding in an efficient manner. The proposed scheme is shown to considerably outperform the conventional practice on channels with high SNR, such as those encountered in twisted-pair telephony systems. In order to analyze the performance of our method, we introduce an approximate mathematical model taking into account the error propagation phenomenon. Based on this model, upper and lower bounds on the overall probability of error are developed. These show that a simple low-redundancy error-detecting code, when properly integrated with the equalizer, can make the overall probability of error several orders of magnitude lower than that obtained with the conventional DFE, or with a DFE followed by an error-correcting code. Computer simulations of the proposed method have been performed for several channels, including the so-called high-bit-rate digital subscriber line (HDSL) test-loop 4, which is known to have a considerable amount of intersymbol interference. For all these channels, our results show that a reduction in the probability of error by more than three orders of magnitude can be obtained using codes of rate 0.96 and above. This, in turn, translates into power savings (coding gain) of 2.5 to 3 dB     

Maximum likelihood sequence detection for intersymbol interference channels: A new upper bound on error probability

Maximum likelihood sequence detection of digital signaling subject to intersymbol interference and additive white Gaussian noise is considered. A new approach to the error probability analysis results in an upper bound which is tighter than the Forney bound. In addition, in the case of infinite-length intersymbol interference a locally convergent bound is shown to hold under fairly general assumptions on the signal autocorrelation.     

Optimum soft-output detection for channels with intersymbol interference

In contrast to the conventional Viterbi algorithm (VA) which generates hard-outputs, an optimum soft-output algorithm (OSA) is derived under the constraint of fixed decision delay for detection of M-ary digital signals in the presence of intersymbol interference and additive white Gaussian noise. The OSA, a new type of the conventional symbol-by-symbol maximum a posteriori probability algorithm, requires only a forward recursion and the number of variables to be stored and recursively updated increases linearly, rather than exponentially, with the decision delay. Then, with little performance degradation, a suboptimum soft-output algorithm (SSA) is obtained by simplifying the OSA. The main computations in the SSA, as in the VA, are the add-compare-select operations. Simulation results of a convolutional-coded communication system are presented that demonstrate the superiority of the OSA and the SSA over the conventional VA when they are used as detectors. When the decision delay of the detectors equals the channel memory, a significant performance improvement is achieved with only a small increase in computational complexity.<>     

Filters for data transmission with minimum intersymbol interference

Lowpass filters for data transmission with and without finite imaginary-axis transfer zeros are considered. The transfer-function poles adjust the equidistance between the rectangular pulse response zeros and the minimum intersymbol interference is automatically attained. The improvement in filter selectivity is provided by the imaginary-axis zeros.     

Random geometric series and intersymbol interference

An interesting and long-standing problem in probability theory is surveyed, and its applications to analyzing the effects of intersymbol interference in digital transmission systems are discussed. Old and new results are presented which enable one to obtain analytical forms for the probability density function of the intersymbol interference in special eases. The probability of error is then computed and compared with some popular upper bounds.     

Efficient sequence detection for intersymbol interference channelswith run-length constraints

This paper addresses the data detection problem of intersymbol interference (ISI) channels with a specific modulation code-constraint known as the (d, k) run-length-limited (RLL) constraint, a popular modulation code-constraint for data storage channels as well as certain communication channels. A computationally efficient sequence detection algorithm is proposed which yields a performance close to that of the maximum likelihood sequence detector when applied to such ISI channels. The proposed detector is derived as a high signal-to-noise ratio approximation to the delay-constrained optimum detector, one which minimizes the symbol error probability given a fixed decision-delay constraint. The proposed algorithm is essentially a fixed-delay tree search (FDTS) algorithm with systematic ambiguity checking and is closely related to existing finite-depth tree search algorithms. It is observed that long critical error events common in uncoded ISI channels are eliminated by the RLL constraint. Based on this observation, we show that for some important RLL constrained channels, the proposed FDTS algorithm yields the same minimum Euclidean distance between distinguishable channel output sequences as the unconstrained maximum likelihood sequence detector     

Intersymbol interference error bounds with application to ideal bandlimited signaling

An upper bound is derived for the probability of error of a digital communication system subject to intersymbol interference and Gaussian noise. The bound is applicable to multilevel as well as binary signals and to all types of intersymbol interference. The bound agrees with the exponential portion of a normal distribution in which the larger intersymbol interference components subtract from the signal amplitude, and the smaller components add to the noise power. The results are applied to the case of random binary signaling with sinx/xpulses. It is shown that such signals are not so sensitive to timing error as is commonly believed, nor does the total signal amplitude become very large with significant probability. However, the error probability does grow very rapidly as the system bandwidth is reduced below the Nyquist band.     

单载波频域均衡消除符号间干扰的研究

介绍了单载波频域均衡（SC-FDE）系统以及SC-FDE所常用的频域块最小均方（FBLMS）算法。为了进一步改进单载波频域均衡系统的性能，基于抽头泄漏的思想，通过增加抽头的泄漏进一步优化了FBLMS算法，并对基于优化算法的单载波频域均衡系统进行了计算机仿真模拟。其仿真结果表明：与常用算法相比较，该优化算法能有效提高均衡器的收敛速度和显著改善均衡器消除符号间干扰（ISI）的性能。