Complex decision problems present numerous modeling and analysis challenges. A number of approaches have been devised in an effort to meet these challenges; one of them is Influence Networks. In this paper, we present a new comprehensive approach to the formalism of Influence Networks that eliminates previously existing inconsistencies and incorporates temporal issues as well. We also use the approach to evaluate a specific decision problem.

We present and evaluate a class of signaling protocols for wireless digital networks. The presented signaling protocols are stable, they induce good delay characteristics and they possess resistance to feed back errors. We also evaluate a system deploying the IEEE 802.11 as a signaling protocol, instead. In the presence of relatively tight admission delay constraints in signaling, the latter is significantly inferior to our proposed signaling technique. As the above admission delay constraints diminish, the IEEE 802.11 MAC protocol breaks down, while our proposed signaling technique maintains its high performance characteristics.

We present, analyze, and numerically evaluate extended algorithms for detecting changes from an acting stochastic process to a number of possible alternatives. The algorithms are sequential, requiring minimal memory capacity and operational complexity, and they incorporate decision thresholds. The performance of the algorithms is controlled by the selection of the thresholds. An online learning algorithm adapts the thresholds dynamically, to attain prespecified error performance. Asymptotically, the first algorithmic extension detects the acting process correctly, in an expected stopping time sense. In addition, the probability of error induced by a reinitialization algorithmic extension converges asymptotically to zero, when the acting process changes infrequently (with order inversely proportional to the value of the decision thresholds). The presented algorithmic systems are quite powerful and their applications are numerous, ranging from industrial quality control, to identification of changes in patterns, to traffic and performance monitoring in high-speed networks.

We present, analyze, and numerically evaluate extended algorithms for detecting changes from an acting stochastic process to a number of possible alternatives. The algorithms are sequential, requiring minimal memory capacity and operational complexity, and they incorporate decision thresholds. The performance of the algorithms is controlled by the selection of the thresholds. Asymptotically, the first algorithmic extension detects the acting process correctly in an expected stopping time sense. In addition, the probability of error induced by a reinitialization algorithmic extension converges asymptotically to zero, when the acting process changes infrequently (with order inversely proportional to the value of the decision thresholds). The presented algorithmic systems are quite powerful and their applications are numerous, ranging from industrial quality control to traffic and performance monitoring in highspeed networks.

We consider stochastic neural networks, the objective of which is robust prediction for spatial control. We develop neural structures and operations, in which the representations of the environment are preprocessed and provided in quantized format to the prediction layer, and in which the response of each neuron is binary. We also identify the pertinent stochastic network parameters, and subsequently develop a supervised learning algorithm for them. The on-line learning algorithm is based an the Kullback-Leibler performance criterion, it induces backpropagation, and guarantees fast convergence to the prediction probabilities induced by the environment, with probability one.

Random multiple access (RMA) protocols comprise an important class for the accessing/signaling stage in multimedia ATM wireless networks. Within the RMA class, the ALOHA algorithm is most widely used at this point in time, while, as is well known, the latter algorithm becomes unstable, as the user population increases. In this paper, we consider the part-and-try algorithm, instead, for a system with different priorities. Based on this algorithm, we construct a mixed priority protocol whose total throughput-for all priorities combined-equals 0.487. In the special and important for ATM case of two, low, and high priorities, with respective Poisson intensities /spl lambda//sub L/ and /spl lambda//sub H/. We also present an analytical methodology which allows the computation of the average packet delays as functions of the rates /spl lambda//sub L/ and /spl lambda//sub H/.>

Two fundamental distributed decision network structures are considered: the first system consists of finite number of sensors, each collecting asymptotically many data, while the second one employs asymptotically many sensors, each collecting a single datum. For binary hypothesis testing, the Neyman-Pearson criterion is utilized and justified via information theoretic arguments. An asymptotic relative efficiency performance measure is used to establish tradeoffs between the two structures, by comparing the performance characteristics of the decentralized detection systems to their centralized counterparts.>

Hypothesis testing is a collective name for problems such as classification, detection, and pattern recognition. In this paper we propose two new classes of supervised learning algorithms for feedforward, binary-output neural network structures whose objective is hypothesis testing. All the algorithms are applications of stochastic approximation and are guaranteed to provide optimization with probability one. The first class of algorithms follows the Neyman-Pearson approach and maximizes the probability of detection, subject to a given false alarm constraint. These algorithms produce layer-by-layer optimal Neyman-Pearson designs. The second class of algorithms minimizes the probability of error and leads to layer-by-layer Bayes optimal designs. Deviating from the layer-by-layer optimization assumption, we propose more powerful learning techniques which unify, in some sense, the already existing algorithms. The proposed algorithms were implemented and tested on a simulated hypothesis testing problem. Backpropagation and perceptron learning were also included in the comparisons.

The problem of binary hypothesis testing is revisited in the context of distributed detection with feedback. Two basic distributed structures with decision feedback are considered. The first structure is the fusion center network, with decision feedback connections from the fusion center element to each one of the subordinate decisionmakers. The second structure consists of a set of detectors that are fully interconnected via decision feedback. Both structures are optimized in the Neyman-Pearson sense by optimizing each decision-maker individually. Then, the time evolution of the power of the tests is derived. Definite conclusions regarding the gain induced by the feedback process and direct comparisons between the two structures and the optimal centralized scheme are obtained through asymptotic studies (that is, assuming the presence of asymptotically many local detectors). The behavior of these structures is also examined in the presence of variations in the statistical description of the hypotheses. Specific robust designs are proposed and the benefits from robust operations are established. Numerical results provide additional support to the theoretical arguments.>

A. Burrell et al. (1993; 1994) described an implementation efficient capacity allocation protocol for multimedia traffic (voice, video, high and low speed data). The protocol determines the capacity allocation of each traffic dynamically, and induces moving boundaries within prespecified and fixed channel frames. The channel frames are selected to satisfy the jittering constraints of all the traffic, such as voice, video and high versus low speed data. The high level protocol, in conjunction with the channel frame structure and the specifics of the transmission policy, may satisfy all traffic and QOSs and may simultaneously attain high channel utilization, for certain variability conditions of the traffic rates and at relatively low implementation cost. In the work presented, the system stability, channel utilization, traffic rejection rate and packet/cell delay are examined. Performance evaluation results are presented.>

Random multiple access (RMA) protocols comprise an important class for the accessing/signaling stage in multimedia ATM wireless networks. Within the RMA class, the ALOHA algorithm is most widely used, while, as is well known, the latter algorithm becomes unstable, as the user population increases. We consider the part-and-try algorithm, instead, for a system with different priorities. Based on this algorithm, we construct a mixed priority protocol whose total throughput for all priorities combined equals 0.487. In the special and important ATM case of two, low- and high-priorities, with respective Poisson intensities /spl lambda//sub L/ and /spl lambda//sub H/. We also present an analytical methodology which allows the computation of the average packet delays as the functions of the rates /spl lambda//sub L/ and /spl lambda//sub H/.

This paper considers the problem of Neyman-Pearson distributed detection. In distributed detection structures, a number of subordinate decision makers decide upon the active hypothesis based on their own data, and then transmit these decisions to one or more primary decision makers. Then the Neyman-Pearson performance criterion is deployed, the objective is to maximize the probability of detection (also known as power probability) induced by the primary decision makers, subject to a given false alarm constraint. In this formulation, the overall optimization problem reduces to the problem of threshold evaluation. This paper deals exactly with this issue. An on-line threshold learning algorithm is proposed that operates directly an data and requires-no explicit knowledge of the underlying probability distributions. The algorithm adapts recursively the pertinent threshold parameters in a way that minimizes the Kullback-Leibler distance between the observed and the desired output distribution. A formal convergence study is carried out and shows that, under some general conditions, the algorithm is strongly consistent; that is, the sequences of the produced threshold estimates converge to the optimal threshold values with probability 1. The rate of convergence is examined, and methods for controlling it are proposed. Simulation results are included and provide additional support to the theoretical arguments.>

Two feedforward neural structures intended for binary hypothesis testing are considered. The first structure, FFS1, is a tandem structure, while the second structure, FFS2, involves cumulative feedforward feedback. Both parametric and robust designs for the two structures are considered and analyzed in terms of induced false alarm and power probabilities. The inferiority of the FFS1 is rigorously proved in terms of the rate with which the induced power probability increases with respect to the number of the neural elements. Asymptotic results are presented, as well as numerical results, with emphasis on the Gaussian and location parameter nominal hypotheses model. Learning algorithms for the parameter involved in the robust network designs are discussed as well.>

In cellular radio, accessing requests are transmitted through signaling channels in the form of minipackets. The accessing requests are generated by a variety of mobile users that impose different constraints. The authors consider data and high-priority public-safety cellular users and isolate a single channel that is shared by the mixed user population. They devise and analyze a mixed protocol that satisfies the constraints of the high-priority public-safety users, while maintaining the data traffic with satisfactory delays. For the high-priority users, a well-defined and finite but bursty population and a number of different priorities are considered. The population of data users is assumed time-varying and not well-defined, and is modeled as limit Poisson and in the analysis of the protocol.>

The authors are concerned with the effective accommodation of mixed traffic (high and low priority) by the cellular ISDN system. They consider two mutually independent traffic streams sharing a single common channel for transmission. The two streams have different arrival and required transmission time characteristics, as well as different delay constraints. The authors present and analyze an algorithm for accommodating both traffics on the common transmission medium, so that the expected throughput of the system is high and the expected per message delays for both traffics are low. Numerical results are presented.>

The authors consider data networks carrying mixed traffic, partially low and partially high priority. For such networks, they propose a synchronous limited sensing random access algorithm per channel, which gives a delay advantage to the high-priority traffic. The algorithm basically consists of two dynamically coupled window algorithms, one for the high and one for the low-priority packets. Assuming limit Poisson user models for both the traffic classes, the authors optimize the windows of the mixed algorithm for: (a) throughput maximization under no specific delay constraints, and (b) throughput maximization subject to expected delay constraints for the high-priority traffic. For cases (a) and (b), they study the delay characteristics of the algorithm, for the two classes of data traffic. With appropriate selection of the two window sizes, the algorithm can attain relatively high throughput, while it simultaneously maintains low expected per high-priority packet delays, whose values are only slightly affected by the total traffic rate.>

The author considers a system where a single channel is shared by both high- and low-priority data. He assumes packet transmissions from both data categories and slotted channel. In addition, he assumes binary (collision versus noncollision) feedback per slot, and limited feedback sensing capabilities for all users in the system. He assumes that the high-priority data are generated by a well-defined finite-number user population, while he adopts the limit Poisson user model (infinitely many independent Bernoulli users) for the low-priority traffic. For this system, he proposes and analyzes a transmission algorithm which is a mixture of a deterministic tree search for the high-priority users, and a random-access algorithm for the low-priority traffic. The algorithm is stable for both traffic classes, it guarantees a strict upper bound on the delays of the high-priority packets, and induces good throughput-delay characteristics for the low-priority data.>

Consideration is given to data networks carrying mixed low- and high-priority traffic. For such networks, the authors propose a synchronous limited sensing random access algorithm per channel, which gives a delay advantage to the high-priority traffic. The algorithm basically consists of two dynamically coupled window algorithms, one for the high- and one for the low-priority packets. Assuming limit Poisson user models for both the traffic classes, the authors optimize the windows of the mixed algorithm for: (a) throughput maximization under no specific delay constraints and (b) throughput maximization subject to expected delay constraints for high-priority traffic. For both cases, the delay characteristics of the algorithm are studied for the two classes of data traffic. With appropriate selection of the two window sizes, the algorithm can attain relatively high throughput, while it simultaneously maintains low expected per high priority packet delays, whose values are only slightly affected by the total traffic rate.>

The authors propose and analyze a stable random access algorithm which is a modification of the two-cell algorithm proposed by M. Paterakis (1984). The authors assume ternary feedback broadcast per slot, and in the presence of capture, identification of the captured packet by all the users in the system. It is also assumed that a successfully received signal transmission cannot be distinguished from a capture in the presence of multiple transmission. The performance of the algorithm is compared to an appropriate modification of J.I Capetanakis' (1979) dynamic algorithm. The proposed algorithm, and in contrast to the latter, it can be easily modified to operate in limited feedback sensing environments.>

The authors consider the problem of random access communication over a time-slotted channel, with binary success/failure feedback. The feedback informs the users only whether or not there was a success (single transmission) in the previous slot. They propose and analyze a limited feedback-sensing algorithm (each user is required to observe the channel feedback, from the time he generates a packet to the time that this packet is successfully transmitted). The algorithm requires central control implemented by a central receiver. The limit Poisson user model is adopted. The algorithm achieves a throughput of 0.322 and induces low delays for relatively low input rates.>

Outlier-resistant algorithms that detect a change from a given nominal stationary process to another such process are given. The nominal processes are assumed to be mutually independent and to satisfy some general regularity conditions. The outlier sequences are assumed to be independently and identically distributed and independent of the nominal processes. The proposed algorithms are sequential and consist of uniformly bounded steps. The asymptotic performance of the algorithms is analyzed, both in the absence and the presence of outliers. Breakdown points and influence functions are defined and analyzed. The algorithms are studied in more detail for Gaussian autoregressive nominal processes.>

A simple full-feedback-sensing window random access algorithm is proposed and analyzed. The throughput of the algorithm is 0.429; its delay and resistance to feedback channel errors are better than those induced by J.I. Capetanakis's (ibid., vol.IT-25, p.505-15, Sept. 1979) window algorithm. In addition, the simple operations of the algorithm allow for the analytical evaluation of the output-traffic interdeparture distribution.>

A simple full-feedback-sensing window random-access algorithm is proposed and analyzed in the presence of the Poisson user model. The throughput of the algorithm is 0.43. The proposed algorithm is better than J.I. Capetanaki's dynamic algorithm (1979) with respect to its delay and resistance to channel errors. The simple operations of the algorithm, along with its regenerative properties, facilitate the computation of the output traffic interdeparture distribution. The obtained distribution is useful in the evaluation of interacting systems which adopt the algorithm for their internal transmissions.>

Random multiple-access algorithms are used to control the accessing of a common communication channel by a large population of bursty channel users. For such algorithms, the induced transmission delay is a key performance measure. A systematic method for finding the delay characteristics of random multiple-access algorithms, whose delay process is regenerative, is presented. The method uses a powerful result from the theory of regenerative processes, in effect, to reduce the problem of determining the delay moments to the problem of solving denumerable dimensional systems of linear equations. Techniques for finding tight bounds on the solutions of such systems are presented. The "0.487" algorithm is used to exemplify the method.

We examine the relation between the finite and the infinite population models for a class of random access algorithms. The algorithms in the class are a combination of random access and reservation techniques, they are synchronous, and they are studied under the condition that each of the users can monitor the channel feedback continuously (full feedback sensing). For any finite number of independent and identical users in the system, and any i.i.d. arrival process per user, the algorithms are stable, provided that the total input rate is less than one. However, as the population size increases, the stability of an algorithm in the class is determined by its throughput in the presence of the infinite population model for all practical purposes.

We consider Poisson packet traffic accessing a single-slotted channel. We assume the existence of a ternary feedback per channel slot. We also adopt the limited feedback sensing model where each user senses the feedback only while he has a packet to transmit. For this model we develop a collision resolution algorithm with last come-first served characteristics. The algorithm attains the same throughput as Gallager's algorithm without the latter's full feedback sensing requirement. In addition, it is easy to implement, requires reasonable memory storage, induces uniformly good transmission delays, and is insensitive to feedback errors. In the presence of binary (collision versus noncollision) feedback the algorithm may attain a throughput of0.4493.

The problem of detecting a change from one given stationary and ergodic stochastic process to another such process is considered. It is assumed that both stochastic processes are processes with memory and that they are mutually independent. A sequential test is proposed and analyzed. It is proved that the proposed test is asymptotically optimal in a mathematically precise sense.

In this correspondence, we consider the random accessing of a single slotted channel by a large number of packet transmitting users, whose cumulative traffic is Poisson. We assume the existence of the same feedback as that of the MCRAI protocols of Georgiadis, and full channel sensing, and we develop collision resolution algorithms that utilize the absorption concept of Gallager. We observe the improvement in the throughputs induced by the absorption, as well as the improved delay characteristics. Finally, we draw some conclusions about the limitations of the absorption idea.

A slotted packet broadcast channel with an infinite user population is considered. A limited feedback sensing algorithm is proposed and analyzed for collision versus noncollision binary feedback. The algorithm bas maximum throughput equal to0.42(packets/slot), has uniformly good delay characteristics within its stability region, and is robust in the presence of feedback errors. A variation of the algorithm, for ternary feedback, attains maximum throughput0.425and bas uniformly good delay characteristics within its stability region. In contrast, the highest throughput limited feedback sensing algorithm existing for ternary feedback attains maximum throughput0.456, but induces relatively high delays for Poisson intensities below0.3.

In this paper the multiple-access problem in a packet network is considered. A finite number of independent, packet transmitting users is assumed. Protocols are developed to allocate the channel resources among the users. Part of the channel capacity is assigned to control information. This information is used to make probabilistic inferences of the queue lengths and delay characteristics of the users. The probabilistic inferences are used to develop dynamic scheduling protocols for data transmission. Protocols are developed for two different levels of control information. An analysis of the relationship between amount of control information and system performance is carried out, and a study of the frequency of transmission of control information is also provided.

In this paper, we consider the random accessing of a single slotted channel by a large number of packet-transmitting, bursty users. We assume that feedback broadcasting is available where some different information, in addition to the information assumed by the Capetanakis, Gallager, Massey, etc., models, is included in the feedback. In particular, we assume that the existence of energy detectors permits the broadcasting of the number of collided packets within each collision slot, whenever this number is below a certain limit. We first consider this limit to be infinity, and then a finite small number. For the model considered, we propose and analyze a collision resolution protocol (CRAI), whose implementation is simple. For Poisson input traffic and infinite number of energy detectors, we found that the CRAI is stable for input rates below 0.53237. For finite number of energy detectors, we propose a modified version of the CRAI (MCRAI). We found that the MCRAI reaches the throughput 0.53237, through the utilization of only about eight energy detectors. These protocols, like the ones introduced by Capetanakis, Gallager, Massey, etc., have good delay properties.

Sliding block encoders operating within a statistically contaminated source environment are considered. Stationary analog sources are studied using the rho-bar distance as the measure of contamination. Sufficient constructive conditions are developed for the design of sliding block encoders that guarantee rho-bar stability of their output when operating within a statistically contaminated source environment. The relationship of this stability to the robustness of the encoding scheme is discussed, where robustness is a local stability property applied on the output entropy and the average distortion.

We summarize some new frequency domain expressions of statistical distance measures between stationary vector Gaussian processes recently derived by the authors. Both time-discrete and time-continuous processes are treated. Some of the frequency domain distance measures have been empirically verified to be very useful speech recognition and speech analysis-synthesis.

The problem of sequential detection between Poisson processes is analyzed and resolved. Wald's results on sequential analysis are used, but a substantial modification is shown to be necessary due to the discontinuous nature of the likelihood ratio. It is shown that, in general, one can take into account the "excess over the boundary" so as to design sequential tests with a given error performance.

In the models used by Hampel as well as by Gray and the author for the qualitative analysis of robust parameter estimators, the Prohorov stability criterion was used. The Vasershtein distance is proposed here as an alternative stability measure. This measure, in contrast to the Prohorov criterion, implies performance stability of the estimators and leads to a constructive analysis for exponentially fast convergence.

Algorithms for the use of Origin-to-Destination (Endto-End) observations for the monitoring of direct network links (direct finals or primary high usage groups) were previously proposed and analyzed. In this paper a theoretical framework intended to shed insight on the potential for also monitoring non-direct network links is developed. In particular, a maximum likelihood based approach is applied to an idealized network model with fixed routing probability structure and point-to-point loads. This approach allows sufficient conditions for the identifiability (estimation) of a fixed set of per link call failure probabilities to be developed, indicating that identifiability problems can be resolved by the addition of observations on selected links from tandems. Because of the dynamically changing character of network loads and hence routing probabilities, the extension appears thus attractive to implement only for limited portions of the network. Through the proposed scheme, unnecessary complexity in the minicomputer network that realizes the quality control of the network links is avoided.

The automation of network service observing through message classification software permits a great increase in the rate at which observations can be made, thus allowing the monitoring of network links as well as switching nodes. This paper develops and analyzes a quality control algorithm which is suitable for monitoring the performance of direct (end-office to endoffice) links via high-volume service observing data. This algorithm processes a sample of successful as well as unsuccessful call attempts sequentially and in real time to quickly and effectively detect deteriorations in the performance of a link. Discussion of the performance qualities and the robustness of this algorithm is presented. The discussion is supported by quantitative results.

A three-level adaptive delta modulator for image encoding is presented. Its advantage is better tracking of signal levels. The characteristic constants of this modulator are designed to satisfy certain convergence and error properties. Behavioral curves of the modulator for different choices of the characteristic constants are included as well as the reconstructed versions of pictures that were encoded by this modulator.

In this paper a high order error measure is used for the evaluation and design of nonlinear modulation schemes. This high order measure works as a deterrent against threshold effects, and through its use a simple, two-dimensional nonlinear modulation scheme is fully analyzed and efficiently designed.

A constructive approach to robust parameter estimation that carries over naturally to the nonparametric estimation is presented. Vagueness in previous notions of "robustness" has prevented such a connection from being made. To eliminate vagueness, robustness is defined in a precise mathematical way that leads to isolation of constructive analytical properties which characterize robust parameter estimators. The approach used in this paper is an extension of the qualitative approach introduced by Hampel.

In this correspondence the problem of sequential estimation is considered when the observation delays are, in general, costly. General theorems determining the "optimal" stopping time in this case are stated, and specific cases are worked out.

The choice of the proper evaluation measure is critical for the design of a good parameter estimator. The mean-square

The error performance of a digital FM system is studied in the presence of additive Gaussian noise. This system is a conventional one employing a voltage-controlled oscillator as the modulator and a limiter-discriminator followed by a low-pass filter as the demodulator. The notion of "clicks" introduced by Rice, and used by Mazo and Salz, is adopted and the performance of the above system is theoretically studied when the intersymbol interference caused by band limitation cannot be neglected. Special consideration has been given to binary frequency-shift keying (FSK) systems. For such systems, the probability of error is given in a closed form.

A three-element detector, consisting of anRCfilter with time constant1/a, a hard limiter, and finally anRCfilter with time constant1/b, is considered. The detector is stimulated by a nonzero-mean white Gaussian noise input, and the system output is observed. A Markov assumption on the hard limiter output leads to a closed-form expression for the output density, which is verified to be close to the actual output density of the detector by computer simulations.

It is known that under certain restrictions on the posterior density and assigned cost function, the Bayes estimate of a random parameter is the conditional mean. The restrictions on the cost function are that it must be a symmetric convex upward function of the difference between the parameter and the estimate. In this correspondence, asymmetrical cost functions of the following form are examined: \begin{equation} C(a, \hat{a})= \begin{cases} f_1(a- \hat{a}),& a \geq \hat{a} \\ f_2(\hat{a}- a),& af_1(\cdot), f_2(\cdot)are both twice-differentiable convex upward positive functions on[0, \infty]that intersect the origin. It is shown that for posterior densities satisfying a certain symmetry condition, the biased Bayes estimate is a generalized median. Furthermore, for linear polynomial functionsf_1(\cdot), f_2(\cdot), the unbiased Bayes estimate is shown to be the conditional mean.

Nonparametric tests have been extensively investigated asymptotically for small signals and large numbers of samples. More realistic, though, in many engineering applications is the small-sample large-signal case, which has had little study because of its complexity. The problem of testing the hypothesis of probability density symmetry about a positive versus a negative value is investigated. The efficiency of the optimum rank and the efficiency of the Wilcoxon rank-sum test are found with respect to the optimum parametric test for normally distributed independent samples with large signal-to-noise ratios. Specifically, it is found that for the same signal-to-noise ratio and probability of error the optimum rank test requires at most 4/3 of the number of samples (or equivalently, 4/3 higher signal-to-noise ratio for the same number of samples) of the optimum parametric test; the Wilcoxon nonparametric test requires at most a factor of2/\sqrt{2}more. Thus, the efficiency of the Wilcoxon nonparametric test is very close to that of the optimum rank test for normal alternatives, although neither are as close to the efficiency of the optimum parametric test as in the large-sample small-signal problem (where, as is well known, the asymptotic relative efficiencies are\pi/3and 1).

Suppose white noise is put into anRCfilter with time constant1/awhose output is hard limited and fed into a secondRCfilter with time constant1/b. The density of the outputy(t)of the system has been the subject of several investigations. Most recently, Pawula and Tsai made the conjecture that for Gaussian white noise they(t)density is given by a certain expression, a special case of which was derived earlier by Doyle, McFadden, and Marx forb/a = 2. In this paper an expression for thenth moment ofy(t)is found. From the expression for the fourth moment, it is proven that the Pawula-Tsai conjecture is not true in general. It is further found that if the output of the hard limiter were wide-sense Markov, the Pawula-Tsai conjecture would be true.