Modems designed for the public switched telephone network (PSTN) have conventionally been based on modeling assumptions which view the PSTN connection as an essentially analog medium. However, as the PSTN evolves toward all digital transport and switching, and particularly as major traffic sources, such as Internet service providers, increasingly have direct digital connections to the PSTN, it is appropriate to revisit the model assumptions. Recently, several modem and chipset manufacturers have announced "56 K" modems based on an emerging system paradigm in which one user (a residential Internet subscriber) has an analog connection to the PSTN, and the other (an Internet service provider) has a digital one. ITU-T is expected to finalize details of a corresponding recommendation in 1998. With this configuration, modem designs based on signaling with the /spl mu/-law alphabet become feasible, and the conventional Shannon limit disappears as the quantization noise is avoided. Thus, the conventional Shannon limit of about 36 kb/s can be beaten, and it is possible to approach the digital transmission rate of 64 kb/s. Modems employing this general approach have become known as /spl mu/-law or pulse-code modulation (PCM) modems. In this paper we present a signaling technique and the sampling theory based on this technique, display the structure and operating principles of a PCM modem equalizer, and show how this equalizer problem can be cast in the language of multiinput-multioutput (MIMO) system theory.

In many digital communications systems, crosstalk, rather than additive noise, is the primary channel impairment. For such systems, it is known that the spectral support of the optimum transmitter is not, in general, restricted to a Nyquist set, in contrast to the case for the additive-noise channel. Nevertheless, the problem of determining the optimum transmitter shaping function for the crosstalk channel without the Nyquist restriction is a difficult one, and has so far remained unsolved. Motivated by current interest in the high-speed digital subscriber line (HDSL) and related crosstalk-dominated applications, we explore a subcase of this problem in which only a single interferer is present. When applied to HDSL-like systems with a single (or dominant) interferer, our analysis and numerical results confirm that wider-than-Nyquist transmitters provide a large performance advantage over Nyquist-limited transmitters. Several interesting and counter-intuitive results also arise. For example, PAM and QAM systems operating at the same spectral efficiency do not, in general, perform identically over the crosstalk channel, despite their essential equivalence in additive noise. We explain why this is so, and show that for channels qualitatively similar to the HDSL wire-pair, QAM has a significant advantage over PAM at high data rates. Finally, we show how the characteristics of HDSL-like channels can be exploited by optimizing the symbol rate.>

Frigo's recent method for approximately calculating intermodulation distortion products for nonlinear systems is clarified, and a derivation from first principle is provided.>

A channel coding approach called diversity coding is introduced for self-healing and fault-tolerance in digital communication networks for nearly instantaneous recovery from link failures. To achieve this goal, the problem of link failures is treated as an erasure channel problem. Implementation details of this technique in existing and future communication networks are discussed.>

Consider a zero-mean, stationary Gaussian process g(t), to which a large positive constant A has been added. Define a distortion process h/sub A/(t) as equal to g(t)+A when the latter is negative and equal to zero otherwise. The author calculates the power spectrum of the process h/sub A/(t) asymptotically as A becomes large. The results have application for estimating the nonlinear-distortion power in the recovered signal when many frequency-multiplexed subcarriers collectively modulate a laser's output power, as would be the case for CATV transmission over an optical fiber. The process h/sub A/(t) then models the nonlinear distortion caused by occasional clipping of the DC-biased laser input.>

The matched filter bound (MFB) is exactly evaluated for a channel modeled by the sum of two delayed and independently Rayleigh-fading beams. It is shown that, if the two beams have comparable average powers, and if the delay spread is moderate or large, then considerable gain is obtained from a diversity-like effect. That is, it is as if the two beams could be detected separately and their results combined. This is true even though significant interference would seem to be present between the beams.>

The authors investigate the efficacy of using two different cyclic redundancy check (CRC) codes in tandem to increase error-burst detecting capability. For a set of pairs of CRCs which are used in standards, it is found that the guaranteed detectable burst length is less than the sum of the individual guaranteed detectable burst lengths, but not much less. Thus strengthened CRC codes can readily be obtained using existing devices.>

An error-control-based approach, called diversity coding, that provides nearly instantaneous self-healing digital communication networks is presented. This is achieved by constructing an error-correcting code across logically independent channels and by treating link failures within the framework of an erasure channel model. Diversity coding is more efficient than previous approaches to self-healing communication networks since it is nearly instantaneous, is transparent to the end user, minimizes the required extra capacity, and does not need rerouting, resynchronization, or a feedback channel. It is applicable to both circuit-switched and packet-switched networks and to a wide variety of network topologies. Diversity coding can be extended to provide protection from short-duration environmental disruptions, such as multipath fading in radio networks and polarization dispersion in fiber-optic networks, or, in conjunction with previous error detection schemes, to provide forward error correction for random and burst errors.>

Lasers that are used for coherent optical communication exhibit phase instabilities which can be modeled as a Brownian motion. The effect that such a disturbance has on limiter-discriminator detection of FM signals received in additive Gaussian noise is investigated. Although a new theory must be developed to calculate FM click rates (and prove them finite), no effects are found which would preclude the use of such a detector with Brownian phase noise. In particular, for binary FM transmission at rate R, discriminator detection will result in 1 dB of degradation or less if B/sub pn//R>

A description is given of a methodology for designing baseband line codes with prescribed spectral nulls in the transmitted spectrum. These codes have the property that the transmitted power is adjustable (with a concomitant change in spectral shape, i.e. null width) and can be made arbitrarily close to the innovations power, while keeping the minimum distance between signal points (or sequences) constant. The essential design step requires the spectral factorization of a certain trigonometric polynomial. The line code that results can easily be used in conjunction with a large class of trellis-coded modulation schemes. Specific baseband codes are constructed using a representation of the general theory that involves a dither variable, which is used to create integer symbols and to minimize the size of the symbol alphabet. Emphasis is on the design of line codes with a double null at DC using the symbol alphabet (+or-1, +or-3).>

A method is described for modifying classical N-dimensional trellis codes to provide baseband codes that combine a spectral null at DC with significant coding gain. The information rate of the classical code is decreased by one bit, and this extra redundancy is used to keep the running digital sum bounded. Equivalently, if the rate is held constant, then twice as many signal points are needed, causing a power penalty of 6/N dB. Baseband trellis codes are presented for several information rates together with complete spectral plots and performance comparisons. A method of constructing baseband codes with multiple spectral nulls is also described.>

The authors reconsider the problem of determining the minimum distance between output sequences of an ideal band-limiting channel that are generated by uncoded binary sequences transmitted at a rate exceeding the Nyquist rate. For signaling rates up to about 25% faster than the Nyquist rate, it is shown that the minimum distance does not drop below the value which it would have in the ideal case wherein there is not intersymbol interference. Mathematically, the problem is to decide if the best L/sup 2/ Fourier approximation to the constant 1 on the interval (- sigma pi , sigma pi ), 00, with coefficients restricted to be =1 or =0, occurs when all coefficients are zero. This is shown to be optimal for 0.802...>

A trellis code is a "sliding window" method of encoding a binary data stream as a sequence of signal points in Rn. When a trellis code is used to encode data at the rate ofkbits/channel symbol, each channel input depends not only on the most recent block ofkbits to enter the encoder, but will also depend on a set of ν bits preceding this block. The ν bits determine the state of the encoder and the most recent block ofkbits generates the channel symbol conditional on the encoder state. The performance of a trellis code depends on a suitably defined minimum distance property of that code. This paper obtains upper bounds on this minimum distance that are better than any previously known.

A trellis code is a "sliding window" method of encoding a binary data stream as a sequence of real or complex numbers that are input to a noisy transmission channel. Ungerboeck has constructed simple trellis codes that provide the same noise immunity as is given by increasing the power of uncoded transmission by factors ranging from two to four. His method is to specify an underlying convolutional code and a rule (mapping by set partitioning) that maps the output of this code onto a fixed signal constallation. A new description of a trellis code is given that combines these two steps into one. The new description is analytic rather than graphical. Many practical codes can be described very simply, and strict bounds on performance can be obtained. A method for differential encoding trellis codes is presented that was suggested by the authors' representation.

The advantages with regard to noise enhancement of certain suboptimum nonlinear techniques for "equalization" of linear distortion on channels used for digital data transmission have recently received considerable attention. These nonlinear schemes fall into two classes: decision feedback and nonlinear precoding. Yet each of these methods has an associated peculiarity which has impeded attempts to give firm bounds on performance. Thus, for decision feedback error propagation has caused the analytical pains, while for nonlinear precoding of anL-level alphabet the unknown increase in transmitter power has been culpable. The former problem has recently received successful attention by one of the present authors and some colleagues. Here we address the concomitant problem for nonlinear precoding and its extension to quadrature amplitude modulation (QAM) and show that the transmitted normalized powerPsatisfies(L^{2} - 1)/3 \leq P \leq (L^{2} - 1)/3 + 1.

An algorithm previously used for estimating parameters of the vocal tract with a digital filter is here reexamined from the point of view of equalization theory. The procedure of adjusting equalizer taps to minimize the mean-squared sample values at the output of the equalizer (while keeping a particular equalizer tap fixed at unit value) is shown to result in an equalizer that inverts the channel provided that 1) the channel is minimum phase, and 2) the equalizer is one-sided.

An upper bound on the error probability of a decision-feedback equalizer which takes into account the effect of error propagation is derived. The bound, which assumes independent data symbols and noise samples, is readily evaluated numerically for arbitrary tap gains and is valid for multilevel and nonequally likely data. One specific result for equally likely binary symbols is that if the worst case intersymbol interference when the firstJfeedback taps are Set to zero is less than the original signal voltage, then the error probability is multiplied by at most a factor of2^Jrelative to the error probability in the absence of decision errors at highS/Nratios. Numerical results are given for the special case of exponentially decreasing tap gains. These results demonstrate that the decision-feedback equalizer has a lower error probability than the linear zero-forcing equalizer when there is both a highS/Nratio and a fast roll-off of the feedback tap gains.

Automatic equalization for digital systems often has as its immediate goal the minimization of some cost function that depends explicitly on the values of intersymbol interference, and thus implicitly on the tap settings. For ease of implementation it is often desirable to slightly modify the conventional distortion measures. A theoretical comparison of several such criteria, some of which are of the peak distortion variety and some of which are of the mean-square class, is made. With one exception, the tap settings that minimize the modified mean-square criteria yield to first-order identical values for the conventional quadratic cost function. All the analogous peak distortion criteria result in zero forcing of sample values, provided that the binary eye is open. Some comments about automatic equalization that are of a tutorial nature are also made.

A model of nonlinear data wave regeneration without retiming is analyzed. Attention is focused on performance after regeneration and subsequent detection. The communication model consists of a pseudoternary signal that is regenerated at one data station, but not retimed prior to retransmission to another station. The signal regeneration is accomplished by a three-level slicer, and this nonlinear transformation degrades the overall system performance in the presence of noise. The extent of this degradation is our subject matter. The formulation and analysis are constructed so that one can put to effective use the fact that, for a certain Gaussian process, one can give definite expressions for the chance that its sample paths lie below a piecewise linear curve on an appropriately restricted time interval. We are able to use events of this type to give upper and lower bounds for the error rate. The bounds are compared with the error rate that would have been obtained had detection been accomplished before the nonlinear regeneration. We hope that the techniques employed here may be of use in other nonlinear communication problems.

The problem of determining the first-order probability density of a filtered version of hard-limitedRCnoise is considered. An integral equation for the density is derived by using Siegert's results on zero-crossing distributions forRCnoise. We also show how the integral equation can be obtained from a two-dimensional Fokker-Planck equation. The exact solution of the integral equation has not been found, but some general information is available. An iterative scheme for determining the moments has been found, and some information has been obtained regarding the behavior of the density as the variable approaches the ends of its range. One result of this portion of the investigation is a disproof of au expression for the density conjectured by Pawula and Tsai. Finally, curves for the density are obtained by solving the integral equation numerically.

We analyze the locking properties of a digital rate-locked loop that may be used to synchronize a slave oscillator with a master. Formulating the dynamics of this system by a nonlinear differentialintegral equation, we rigorously prove that in the absence of filtering in the loop the slave oscillator output waveform has periodic zero crossings occurring at exactly the same rate as those of the master oscillator. The main requirement for this is only that the quantization of the staircase comparator in the slave oscillator not be too coarse. A similar proof for a more general loop remains elusive.

A statistic often encountered in various estimation problems is the conditional ensemble average of the time derivative of a random signal given the signal. It turns out that for a very large class of random signals this statistic is equal to zero. This is a rather surprising result and as far as can be determined has not been precisely stated and rigorously proven. A precise statement and a rigorous proof of this theorem is the subject of this paper. Our result is the following. Lety(t)be a stationary random process possessing a mean-square derivative\dot{y}(t). Then the conditional ensemble averageE \{\dot{y}(t)\mid y(t) \}always vanishes.