A game-theoretic framework based on smart pricing in power grids that incorporates heterogeneous user preferences and renewable power uncertainty is considered. The system operator adopts an adaptive pricing policy that depends on total consumption and renewable generation. The pricing policy sets up a non-cooperative game of incomplete information among users with heterogeneous preferences. Selfish, altruistic and welfare maximizing user behavior models are proposed. Information exchange models in which users only have private information, communicate or receive broadcasted information are considered. For each pair of behavior and information exchange models, rational consumption strategy is characterized. Numerical analyses reveal that communication is beneficial for the expected aggregate payoff while it does not affect the expected net revenue of the system operator. Moreover, the additional information to the users helps reduce the variance of total consumption among runs increasing the accuracy of demand predictions.

We consider a demand side management model in which the power provider adopts an adaptive pricing strategy that depends on fluctuations in renewable sources and consumption behavior of customers with heterogeneous marginal utilities in the smart grid. Given the adaptive pricing strategy, we formulate the power consumption behavior of customers as a repeated noncooperative game with incomplete information. We provide an explicit characterization of unique Bayesian Nash equilibrium strategy in terms of individual marginal utilities. The rational behavior is also characterized in a communication scheme where smart meters exchange consumption levels with neighboring meters. A local algorithm that computes equilibrium consumption and propagates beliefs is presented when the network is known. Simulation results show that communication is beneficial for welfare and that power provider can lower the peak-to-average ratio of total consumption by adjusting its target profit ratio.

A polar coding scheme for bit-interleaved coded modulation (BICM) is proposed. Code construction is performed by extending Arikan's heuristic to parallel channels. A lower-complexity and numerically robust implementation of list decoding with cyclic redundancy check is adopted. The performance of the design methods used for the construction of proposed bit-interleaved polar coded modulation (BIPCM) system is evaluated over 16-ary quadrature-amplitude modulation with different mapping schemes. It is shown that the proposed BIPCM architecture provides significant performance advantages over the BICM schemes implemented with other well-known codes for moderate block lengths.

A distributed system which comprises independent sensors and a fusion center is considered. Optimum quantization of sensor observations by Chernoff Information (CI) and deflection criterion (DC) for detection purpose is studied. Chernoff information of overall system is derived, and locally optimum quantization of sensor observations is proved to achieve global optimality. A relationship between DC and CI of quantized observations is established. A novel iterative algorithm which works similar to Lloyd-Max algorithm is proposed to find optimum quantization intervals. Results indicate that optimum 2-bit quantization gives detection performance that is close to what is achievable by collecting soft reports from sensors.

In this paper, we study demand response management and power scheduling in a smart grid with renewable energy sources represented by a wind farm. Total social welfare, defined in terms of consumer utility and cost of energy production (both scheduled and renewable), is maximized while the probability of energy shortfall due to the uncertainty in renewable production is limited by an upper bound. We find the outage-limiting optimal conventional energy production and consumption schedules, and investigate the impact of the statistical relationship between the turbines in a wind farm.

We introduce the design of a class of code sequences {en(m), n = 1, 2,... } with memory level m = 1,2,... based on the channel polarization idea, where {en(1)} coincides with the polar codes presented by Arıkan in [1]. The new codes achieve the symmetric capacity of arbitrary binary-input discrete memoryless channels. We derive bounds on the polarization performance as scaled with m. We show that {en(m)} offers monotonically decreasing encoding and decoding complexity with growing m.

In this paper, a channel precoding technique is proposed for M-ary continuous phase modulation (CPM) for multipath downlink channels. The proposed pre-equalizer, based on the approximation of CPM signals by complex exponential functions, applies modulo-arithmetic operations to ensure the stability as in Tomlinson-Harashima method. Spectrally efficient precoded signals are obtained with small envelope variations by employing a transmit antenna selection technique. Because the equalization functions are transferred to the base station, the complexity in the mobile terminals is reduced significantly. The simulation results depict the power spectral density of the transmitted signals and the bit-error rate. An upper bound on the error performance is computed and the relation between the number of transmit antennas and the spectral efficiency is derived.

In this paper, energy detectors are developed for wideband and ultra-wideband (UWB) pulse position modulation (PPM). Exact bit error probability (BEP) formulas are derived under different assumptions about the channel. More specifically, we present an expression for the instantaneous BEP for a specific channel realization, as well as an expression for the average BEP for a channel with independent and identically distributed zero-mean Gaussian coefficients. Simulation results corroborate the precision of the formulas.

The performance of impulse radio is investigated for ultra-wideband (UWB) channels and orthogonal M-ary pulse position modulation (PPM) in terms of the maximum achievable information rate, where the outputs of the M-ary correlation receiver implemented through the rake structure with conventional maximal ratio combining are subject to either soft or hard decisions. The major difference observed between the soft- and hard-output systems is that at low values of the mutual information the minimum information bit energy to noise ratio required for reliable communication increases with the constellation size for the hard-output systems. Furthermore, it is demonstrated that independent of the type of the output, achievable rates are considerably low at medium-to-large signal-to-noise ratio (SNR) values in UWB channels compared to those in additive white Gaussian noise channels, as the frequency-selectivity of the UWB channel destroys the orthogonality of the transmitted PPM signals. Interference from other users and constellation size put additional constraints on the achievable rates at high SNR.

In this paper, two novel iterative chip-level synchronization methods, one blind and the other data-aided, are proposed for chip-interleaved time-hopping (TH) ultra-wideband signals that are subject to intersymbol interference (ISI) and multiple access interference (MAI). Both algorithms are soft-input, soft-output devices which receive the a posteriori probability of each chip from the minimum mean-square error (MMSE) filter that suppresses both ISI and MAI. Using this soft information and the structure of TH codes, the blind synchronizer produces the probabilities of time delays which are fed into the channel decoder. In addition to the knowledge of TH codes and the soft information coming from the MMSE filter, the data-aided synchronizer uses a training sequence in the generation of the probabilities of time delays. Although the channel decoder for chip-interleaved systems is rather simple, here it is redesigned to incorporate the soft information. The resulting receivers perform as well as iterative equalization with perfect timing information, and catch the single-user performance at moderate signal-to-noise ratios.

In this paper, the performance of a time-division duplex ultra wideband system that has a transmitter/receiver pair of rake combining structures is optimized, where the total number of rake fingers to be deployed at the transmitter and the receiver is fixed. It is shown that there exists an optimum distribution of fingers between the two structures for systems with intersymbol interference, which maximizes the signal-to-interference plus noise ratio. Depending on the total number of rake fingers and/or post-rake fingers, i.e., those at the receiver, the optimum placement of post-rake fingers changes as the simulation results demonstrate.

In this paper, a doubly-iterative linear receiver, equipped with a soft-information aided frequency domain minimum mean-squared error (MMSE) equalizer, is proposed for the combined equalization and decoding of coded continuous phase modulation (CPM) signals over long multipath fading channels. In the proposed receiver architecture, the front-end frequency domain equalizer (FDE) is followed by the soft-input, softoutput (SISO) CPM demodulator and channel decoder modules. The receiver employs double turbo processing by performing back-end demodulation/decoding iterations per each equalization iteration to improve the a priori information for the front-end FDE. As presented by the computational complexity analysis and simulations, this process provides not only a significant reduction in the overall computational complexity, but also a performance improvement over the previously proposed iterative and noniterative MMSE receivers.

In this letter, an orthogonal space-time (ST) block coding (STBC) technique for continuous phase modulation (CPM) is proposed and combined with a frequencydomain equalizer (FDE) to obtain high performance gains over frequency-selective channels. The new method maintains the constant envelope and phase continuity of the CPM waveforms perfectly by using appropriate tail symbols during the addition of cyclic prefix and interblock transitions and, therefore, has no negative impact on the spectral efficiency. Because the STcoded CPM blocks remain orthogonal, FDE is applied as in the case of single antenna transmissions without any increase in the computational load. Simulations corroborate the performance improvement offered by STBC while also preserving the desired bandwidth efficiency of CPM.

For target tracking applications, wireless sensor nodes provide accurate information since they can be deployed and operated near the phenomenon. These sensing devices have the opportunity of collaboration among themselves to improve the target localization and tracking accuracies. An energy-efficient collaborative target tracking paradigm is developed for wireless sensor networks (WSNs). A mutual-information-based sensor selection (MISS) algorithm is adopted for participation in the fusion process. MISS allows the sensor nodes with the highest mutual information about the target state to transmit data so that the energy consumption is reduced while the desired target position estimation accuracy is met. In addition, a novel approach to energy savings in WSNs is devised in the information-controlled transmission power (ICTP) adjustment, where nodes with more information use higher transmission powers than those that are less informative to share their target state information with the neighboring nodes. Simulations demonstrate the performance gains offered by MISS and ICTP in terms of power consumption and target localization accuracy.

The performance of impulse radio is investigated for frequency-selective ultra-wideband (UWB) channels and orthogonal M-ary pulse position modulation (PPM) in terms of the maximum achievable data rate, where the outputs of the M-ary correlation receiver implemented through the rake structure with conventional maximal ratio combining undergo either hard or soft decisions. The major difference observed between the soft and hard decision output systems is that at low code rates increasing the constellation size, M, requires an increase in the minimum signal-to-noise ratio (SNR) per communicated bit for the system with hard decisions. It is demonstrated that independent of the type of decision making, achievable rates in UWB channels are low compared to those in additive white Gaussian noise channels, as the frequency-selectivity of the UWB channel destroys the orthogonality of the transmitted PPM signals. The interference from the other users and the constellation size put additional constraints on the achievable data rates at high values of SNR.

The network lifetime measures for wireless sensor networks are mostly based on the percentage of live sensors. However, the lifetime definition must be in accordance with the primary functionality of the network. In surveillance applications, the network must be regarded as dead if the required security level cannot be provisioned. In this paper, we analyze the temporal resilience of deployment quality in surveillance wireless sensor networks and propose several lifetime definitions. It is concluded that having just the necessary number of sensors to satisfy the initial deployment quality requirement is the best strategy to favor the network lifetime.

Ultra-wideband (UWB) communication is envisaged to be deployed in indoor environments, where the noise distribution is decidedly non-Gaussian. A critical challenge for impulse radio UWB is the synchronization of nanosecond-long pulses. In this paper, we propose two acquisition schemes which are robust to uncertainties in the noise distribution and which operate directly on samples taken at symbol-rate. The only channel state information that the algorithms need can be obtained by estimation of an aggregate channel gain, avoiding channel-related complications. Following Huber's M-estimates for the Gaussian mixture noise model, the proposed robust acquisition systems outperform their traditional counterparts designed according to the Gaussian noise assumption. Necessary modifications for operation under multiple access interference are also introduced. Theoretical and simulation-based performance evaluations that reflect the asymptotic variance, normalized mean-square error and the bit error rates demonstrate the gains offered by the robust procedures.

A fully-distributed collaborative multi-target tracking framework that eliminates the need for a central data associaton or a central coordinating node for wireless sensor networks is defined. Details of the distributed data association architecture, which is more feasible than the ones relying on a coordinating entity, is described. It is shown that for target tracking applications, the collaboration improves the target localization performance of the distributed data collecting devices. In order to reduce the communication energy exhausted for collaboration, the performance of the collaboration logic manager is examined. Simulation results show that collaborating about a single target information is a rational decision. The problem of deciding which target information to collaborate among the detected targets arises. A mutual information based metric is shown to be a good candidate for deciding on the target which the sensor will collaborate about with the network.

Ultra-wideband (UWB) communications is envisaged to be deployed in indoor environments, where the noise distribution is decidedly non-Gaussian. A critical challenge for UWB is the synchronization of nanosecond-long pulses. In this paper, we propose a signal acquisition scheme that is robust to uncertainties in the noise distribution. Following Huber's M-estimates for the Gaussian mixture noise model, the successive sampling-based robust acquisition system outperforms Gaussian-optimal maximum likelihood approach. We present performance evaluation based on the normalized mean-square error and the bit-error rate.

Ultra-wideband (UWB) communications is envisaged to be deployed in indoor environments, where the noise distribution is decidedly non-Gaussian. A critical challenge for UWB is the synchronization of nanosecond-long pulses. In this paper, we propose an averaged template-based signal acquisition scheme that is robust to uncertainties in the noise distribution. Following Huber's M-estimates for the Gaussian mixture noise model, the averaged template-based robust acquisition system outperforms its traditional counterparts which are designed with Gaussian noise in mind. The authors present performance evaluation based on the asymptotic variance, normalized mean-square error, as well as the bit error rate.

Space-frequency coded (SFC) orthogonal frequency-division multiple access (OFDMA) is considered in the presence of multitone (MT) interference. Analytical expressions for the bit-error probability (BEP) are derived for OFDMA with SFC in a frequency-selective fading environment. It is shown that SFC increases the resistance of OFDMA against the MT interference and reduces BEP considerably

In this paper, a doubly iterative receiver is proposed for joint turbo equalization, demodulation, and decoding of coded binary continuous-phase modulation (CPM) in multipath fading channels. The proposed receiver consists of three soft-input soft-output (SISO) blocks: a front-end soft-information-aided minimum mean square error (MMSE) equalizer followed by a CPM demodulator and a back-end channel decoder. The MMSE equalizer, combined with an a priori soft-interference canceler (SIC) and an a posteriori probability mapper, forms a SISO processor suitable for iterative processing that considers discrete-time CPM symbols which belong to a finite alphabet. The SISO CPM demodulator and the SISO channel decoder are both implemented by the a posteriori probability algorithm. The proposed doubly iterative receiver has a central demodulator coupled with both the front-end equalizer and the back-end channel decoder. A few back-end demodulation/decoding iterations are performed for each equalization iteration so as to improve the a priori information for the equalizer. As presented in the extrinsic information transfer (EXIT) chart analysis and simulation results for different multipath fading channels, this provides not only faster convergence to low bit error rates, but also lower computational complexity.

In this paper, a doubly-iterative receiver with a soft-information-aided frequency-domain minimum mean square error (MMSE) equalizer is proposed for coded continuous phase modulation (CPM). Equalization in the frequency-domain provides low computational complexity in long multipath fading channel responses. The front-end equalizer is followed by the soft- input soft-output (SISO) CPM and channel decoders at the back- end. The receiver exploits double turbo processing by employing back-end demodulation/decoding iterations per each equalization iteration to improve the a priori information for the front-end equalizer. Such processing results in high bit-error rate (BER) performance with less number of equalization iterations, as also shown in the simulation results.

The frequency-selective ultra-wideband (UWB) channels are subject to various types of electromagnetic noise, which is non-Gaussian (impulsive). The conventional linear receivers based on the additive white Gaussian noise assumption exhibit performance degradation. In this paper, a robust receiver using the M-estimation technique is introduced for multiple access UWB channels with impulsive noise, eliminating multiuser interference and impulsive noise at the same time, provided that accurate channel information about the users is available. Both the robust receiver and its simplified version are shown to outperform the linear receiver based on the Gaussian noise assumption.

In this paper, we propose a method to determine the quality of deployment in a surveillance wireless sensor network in an environment that contains obstacles. The maximum detection probability on the potential breach paths is used as the deployment quality measure. The watershed segmentation algorithm is applied on the graph formed by the iso-sensing curves to identify the possible breach paths. An algorithm is proposed to convert the watershed segmentation to an auxiliary graph which is then employed to determine the deployment quality measure.

In this paper, the convergence behavior of a doubly-iterative receiver, proposed for joint turbo equalization, modulation and decoding of coded binary continuous-phase modulation (CPM), is analyzed by using extrinsic information transfer (EXIT) charts. The doubly-iterative receiver consists of a central soft-input soft-output (SISO) CPM demodulator coupled with a front-end soft information aided minimum mean square error (MMSE) equalizer and a back-end SISO channel decoder. The EXIT chart analysis shows that performing a few back-end demodulation/decoding iterations per each equalization iteration improves the a priori information for the front-end equalizer, and provides faster convergence to low bit-error rates (BERs) with less computational complexity, as also illustrated in the BER simulations.

For target tracking applications, small wireless sensors provide accurate information since they can be deployed and operated near the phenomenon. These sensing devices have the opportunity of collaboration amongst themselves to improve the target localization and tracking accuracies. Distributed data fusion architecture provides a collaborative tracking framework. Due to the energy constraints of these small, sensing and wireless communicating devices, a common trend is to put some of them into a dormant state. In addition to a selective sensor activation strategy based on the maximum mutual information metric, in this paper, we devise the Information-Controlled Transmission Power (ICTP) adjustment in order to improve the energy savings. The essence of the proposed ICTP scheme for collaborative target tracking lies behind the idea that the sensors with more information use higher transmission powers than the sensors with less information in order to share their target state information with the neighboring sensors.

Wireless multiple-access protocols using retransmission diversity resolve packet collisions through source separation techniques. Retransmissions are applied to achieve the necessary diversity level to determine the collision multiplicity blindly, whose proper detection is essential for good resolution performance. The so-called network-assisted diversity multiple access (NDMA) employs the rank test (RT) method for blind multiplicity detection, but static channel parameters throughout the retransmissions and the tight control of the transmitter phase are required. In this paper, a superior blind collision multiplicity-detection technique is introduced by combining the minimum description length criterion and a variation of the RT method. Collision resolution is accomplished through independent component analysis (ICA). Unlike blind NDMA,the new protocol, named ICA-NDMA, works in quasi-static fading channels, and does not need any phase control. Extensive performance analyses corroborate the performance advantages offered by ICA-NDMA.

A frequency-hopping (FH) space-frequency (SF)-coded clustered orthogonal frequency division multiple access (OFDMA) system is considered. It is shown that SF coding reduces the number of collisions in OFDMA with FH provided that the subcarrier clusters are formed randomly at each hopping instant. Analytical expressions for the collision rate, which is defined as the expected number of collisions per symbol, are derived. Bit error rate (BER) computations via simulations in a frequency-selective fading channel corroborate the theoretical findings for SF-coded FH-OFDMA.

A frequency hopping, space-frequency coded clustered orthogonal frequency-division multiple access (OFDMA) system is considered. It is shown analytically that SF coding reduces the number of collisions in OFDMA with random subcarrier hopping. Expressions for the expected number of symbol loss in a cluster collision are derived. Bit error rate computations via simulations in a frequency-selective fading channel corroborate the theoretical findings.

The sensing coverage area of surveillance wireless sensor networks, which is determined by applying the Neyman-Pearson detection rule, is considered. Using a graph model for the perimeter, the weakest breach path is found through Dijkstra's shortest path algorithm. The breach probability is a critical parameter for assessing the required number of sensor nodes and the surveillance performance of the network. Analysis indicates that for the Neyman-Pearson detection model, the most significant parameter is the false alarm rate, which is inversely proportional to the breach probability. Furthermore, the width of the field has a significant impact on the breach probability.

Frequency-shift keying (FSK) is a popular modulation scheme in power-limited communication links. This paper introduces space-time FSK (ST-FSK), which does not require any channel state information at the transmitter and the receiver, as in conventional noncoherent FSK. ST-FSK can be viewed as a special unitary ST modulation design. However, ST-FSK has a number of advantages over existing unitary ST modulation designs. ST-FSK is easier to design, and enjoys lower decoding complexity. Furthermore, ST-FSK guarantees full diversity. Finally, ST-FSK can be adopted in the digital as well as in the analog domain, and merges very naturally with frequency-hopping multiple access. As expected, all these advantages come at the cost of a decrease in spectral efficiency.

Location area (LA) planning plays an important role in cellular networks because of the tradeoff caused by paging and registration signalling. The upper boundary for the size of an LA is the service area of a mobile services switching center (MSC). In that extreme case, the cost of paging is at its maximum but no registration is needed. On the other hand, if each cell is an LA, the paging cost is minimal but the cost of registration is the largest. Between these extremes lie one or more partitions of the MSC service area that minimize the total cost of paging and registration. In this paper, we seek to determine the location areas in an optimum fashion. Cell to switch assignments are also determined to achieve the minimization of the network cost. For that purpose, we use the available network information to formulate a realistic optimization problem, and propose an algorithm based on simulated annealing (SA) for its solution. Then, we investigate the quality of the SA-based technique by comparing it to greedy search, random generation methods, and a heuristic algorithm.

Network-assisted diversity multiple access (NDMA) is a novel algorithm that resolves packet collisions deterministically by means of source separation techniques. The blind version of NDMA (B-NDMA) relies on the presence of a fading process that induces constant channel parameters over long slot times, as well as, the transmitter-receiver phase synchronization. In this paper, independent component analysis (ICA) is integrated into NDMA to provide blind collision resolution with high throughput/delay performance in a fast-fading channel environment and without phase control.

During the optimization of the Global System for Mobile Communications (GSM) network, the operations and maintenance center (OMC) measurements are of significant importance. Because of false predictions, the capacity planning of a cell may be done inaccurately. If the capacity of the cell is not adequate to handle all of the busy-hour requests, the requests are not granted a channel and users are blocked. Thus, when the blocking ratio is high, the cellular capacity should be replanned. However, the capacity planning should not rely on the raw OMC measurements since these are inflated by the redials and retrials of fresh calls, as well as the retrials of the hand-off call attempts. In this paper, we provide a simple-to-implement analytical model that computes the average number of retrials and redials per original call attempt using only the OMC statistics. The effective load is then determined through appropriate scaling. That way, unnecessary waste of channels to sustain the retrials and redials is avoided during the cellular capacity replanning phase.

Location area (LA) planning plays an important role in cellular networks because of the trade-off caused by paging and registration signaling. The upper bound on the size of an LA is the service area of a mobile switching center (MSC). In that extreme case, the cost of paging is at its maximum, but no registration is needed. On the other hand, if each cell is an LA, the paging cost is minimal, but the registration cost is the largest. In general, the most important component of these costs is the load on the signaling resources. Between the extremes lie one or more partitions of the MSC service area that minimize the total cost of paging and registration. In this paper, we try to find an optimal method for determining the location areas. For that purpose, we use the available network information to formulate a realistic optimization problem. We propose an algorithm based on simulated annealing (SA) for the solution of the resulting problem. Then, we investigate the quality of the SA technique by comparing its results to greedy search and random generation methods.

We consider the packet reservation window multiple access protocol (PRWMA) that successfully combines voice and data traffic over a cellular network. In agreement with the mobile channel environment, the presence of capture is also modeled. PRWMA is modified to accommodate the advantages presented by the capture effect. The proposed protocol's capacity and average delay performance is studied under the power capture model through simulations. Experiments indicate that significant performance gains are offered by the new algorithm.

Robust single-user detection is employed in a DS/CDMA system in which the noise process contains impulsive components. The breakdown point is computed for a mixture noise model. Noise rather than interference is shown to be the primary obstacle in achieving good performance when the signal power is low. The bit error probability expressions are also derived under a Gaussian mixture. DS/CDMA employing a robust correlator receiver performs better than the conventional matched filter in impulsive noise environment.

GSM operators work with call data that include retrials and redials as well. Based on these inflated, virtual call attempt statistics, it appears that enormous demands are put on the available spectrum. In this paper, we develop simple models to estimate the expected number of retrials and redials per original call attempt, which is used to determine the effective load and the number of traffic channels required to meet the grade of service. It is shown that significantly fewer channels are capable of handling high traffic loads.

We propose a new packet reservation multiple access (PRMA) scheme for the joint transmission of voice and data traffic over a wireless medium. The collision resolution protocol within the system is based on a modification of the two-cell window random access algorithm, which has superior properties compared to the conventional slotted Aloha. The proposed algorithm, which we call packet reservation window multiple access (PRWMA), works in distinct modes for voice and data, and the user performs slightly different operations depending on the information type.

Ultrasonic range sensors are used to obtain the information required for collision-free navigation of a mobile robot in a semi-structured or unstructured environment. A set of range readings from a ring of sonars are correlated to acquire a 2D map of the robot's environment. The map is continuously enhanced via novel matching and update algorithms as new data are collected while the robot is in motion. The algorithm utilizes confidence measures that are directly obtained from the sonar's resolution or accuracy. There are no additional modeling assumptions and the approach is robust. The algorithms are experimentally tested on the Nomad 200 mobile robot and Nomad's Cognos Software Development Package.

The multiple access problem as characterized by infinite user population and a slotted-time channel is examined, and an algorithm that utilizes a control mini-slot is proposed. The stability region of the proposed algorithm is determined and compared to the random access algorithm with the highest known throughput. A break-even point is also given.

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.>

This paper presents several formulations for aggregating opinions about the outcome of a future event-e.g., opinions in the form of probabilities are aggregated into a single probability. The approach used to derive these aggregation formulas is maximum entropy inference, with the assumption that the opinions and the event being predicted are joint random variables. As part of the presentation, the novel, axiomatically developed aggregation formula due to Bordley is used as a foil for contrasting and comparing the results developed here. It is shown that Bordley's result is overly restrictive when looked at in terms of maximum entropy derivations that are clearly using more information. Initially, we suppose that the opinions given on the future event are in the form of odds; then we consider the case where these opinions are in the form of probabilities; and finally, we reach a general formulation in which they could be odds, probabilities or even binary, yes/no, predictions of a future event with interactions between experts. In regard to this last point, the use of Darroch and Ratcliff's method in maximum entropy aggregation is sketched.>