This paper studies a class of stochastic and time-varying Gaussian intersymbol interference (ISI) channels. The probability law for the$i^{th}$channel tap during time slot$t$is supported over an interval of centre$c_{i}$and radius$r_{i}$. The transmitter and the receiver only know the centres$c_{i}$and the radii$r_{i}$. The joint distribution for the array of channel taps and their realizations are unknown to both the transmitter and the receiver. A lower bound (achievability result) is presented on the channel capacity which results in an upper bound on the capacity loss compared to when all radii are zeros. The lower bound on the channel capacity saturates at a positive value as the maximum average input power$P$increases beyond what is referred to as the saturation power$P_{sat}$. Roughly speaking,$P_{sat}$is inversely proportional to the sum of the squares of the radii$r_{i}$. It is also verified that in the presence of channel state information at the receiver, if the so-called central channel frequency response is everywhere nonzero, then the aforementioned capacity loss is bounded from above by a constant that does not depend on$P$. A partial converse result is provided in a scenario where different channel taps vary independently of each other and each channel tap process is stationary with a finite differential entropy rate. It is shown that for every sequence of codebooks with vanishing probability of error, if the size of each symbol in every codeword is bounded away from zero by a constant that is proportional to$\sqrt {P}$, then the rate of that sequence of codebooks does not scale with$P$. Tools in matrix analysis such as matrix norms and Weyl’s inequality on perturbation of eigenvalues of symmetric matrices are used in order to analyze the probability of error. A result in the paper that may find other applications is a new and tight upper bound on the size of a Gaussian typical set.

This paper presents an outage analysis for a two-user parallel Gaussian interference channel consisting of two sub-channels. Each sub-channel is modeled as a two-user Gaussian interference channel with quasi-static and flat fading. Both users employ single-layer Gaussian code-books and maintain a statistical correlation ρ between the signals transmitted over the underlying sub-channels. When joint decoding (JD) is performed at the receivers, setting ρ = 0 minimizes the outage probability, regardless of the value of the signal-to-noise ratio (SNR). It is shown, however, that if the receivers treat interference as noise (TIN) or cancel interference (CI), the value of optimum ρ approaches 1 as SNR goes to infinity. Motivated by these observations, we let ρ = 0 under JD and ρ = 1 under TIN and CI and compute the outage probability in finite SNR, assuming that the direct and crossover channel coefficients are independent zero-mean complex Gaussian random variables with possibly different variances. In the asymptote of large SNR and assuming the transmission rate per user is r log snr, it is shown that the outage probability scales like snr-(1-r)under both TIN and CI, while it vanishes at least as fast as snr-min{2-r,4(1-r)}log snr under JD. This paper is concluded by extending some of the results to a two-user parallel Gaussian interference channel with an arbitrary number of sub-channels.

A sequence of lower and upper bounds is derived on the differential entropy of a Gaussian mixture where the Gaussian components only differ in mean values. As the sequence index increases, the computational complexity of the bounds increases; however, the gap between the lower and upper bounds becomes vanishingly small. We address the applications of these bounds in several communication scenarios where the transmitters utilize Pulse Amplitude Modulation (PAM) constellations to transmit data.

We address outage analysis for a two-user parallel Gaussian interference channel consisting of two sub-channels. Each sub-channel is modelled by a two-user Gaussian interference channel with quasi-static and flat fading. Both users employ single-layer Gaussian codebooks and maintain a statistical correlation ρ between the signals transmitted over the underlying sub-channels. If the receivers treat interference as noise (TIN) or cancel interference (CI), the value of ρ minimizing the outage probability approaches 1 as the signal-to-noise ratio (SNR) approaches infinity, while ρ = 0 is optimum under joint decoding (JD) regardless of the value of SNR. Motivated by these observations, we let ρ = 1 under TIN and CI and ρ = 0 under JD and compute the outage probability in finite SNR assuming the direct and crossover channel coefficients are independent zero-mean complex Gaussian random variables with possibly different variances. In the asymptote of large SNR and assuming the transmission rate per user is r log snr, we show that the outage probability scales as snr-(1-r)under both TIN and CI, while it vanishes at least as fast as snr-min{2-r;4(1-r)}log snr under JD.

This paper addresses the following question regarding Gaussian networks: Is there an alternative to decoding interference or treating interference as Gaussian noise? To state our result, we study a decentralized network of one primary user (PU) and one secondary user (SU) modeled by a two-user Gaussian interference channel. In one scenario, the primary transmitter is constellation-based and PUs codebook is constructed over its modulation signal set. Assuming SU is aware of the constellation points of PU, the interference plus noise at the secondary receiver is modeled by a mixed Gaussian process. We show that SU can achieve larger rates by matching its decoder to the actual interference plus noise compared with the case where the secondary receiver performs nearest neighbor decoding (NND). In another scenario, we assume that PU utilizes a predetermined point-to-point code. We ask if SU can utilize its knowledge about PUs codebook without decoding PUs codewords. The proposed strategy assumes each transmitted codeword of SU overlaps with infinitely many transmitted codewords of PU, referred to as the unequal codeword-length (UCL) strategy. The secondary receiver views PU as a virtual user that is constellation-based and its modulation signal set is the codebook of the actual PU. UCL is compared with other strategies, namely, interference cancellation (IC), joint decoding, and NND. It is shown that UCL can outperform both IC and NND simultaneously.

This paper studies a decentralized wireless network of asynchronous transmitter-receiver pairs with burst transmissions. Each receiver learns about the number of active users, channel coefficients, and mutual delays based on locally available measurements. The estimates for the mutual delays are not perfect, however, they are reliable enough to guarantee successful decoding. Two signalling schemes are addressed, namely, randomized masking (RM) and reduced cycle transmission (RCT). Under RM, the n symbols of a codeword are generated according to a Bernoulli-Gaussian distribution with activity factor 0 <; θ ≤ 1. This is in contrast to RCT where each codeword consists of ⌈θn⌉ Gaussian symbols followed by n-⌈θn⌉ zeros. Assuming the transmitters are unaware of the number of users, channel coefficients, and mutual delays, the probability of outage under RM is considerably lower compared with RCT if the signal-to-noise ratio (SNR) is sufficiently large. A generalized RCT scheme is also examined where the n - ⌈θn⌉ zero symbols are not necessarily located at the end of a codeword. In the asymptote of large SNR, the outage probability becomes vanishingly small under RM, however, it is bounded away from zero for generalized RCT regardless of the value of SNR.

This paper addresses a hybrid setup for cognitive radio based on Gaussian interference channel where the secondary user can use both interweave and underlay strategies. The primary user does not cooperate or adapt since it is using legacy hardware. We show that these assumptions lead to a non-convex achievable rate region for various types of hybrid interweave-underlay strategies and that the rate optimization problem for the secondary user is in general non-convex and non-smooth. We analyze the structure of this optimization problem to reduce it to a number of tractable subproblems in various interference regimes. Numerical simulations are also presented to give insight into the performance of the proposed schemes.

This paper addresses a two-user Gaussian multiple access channel (MAC) under peak power constraints at the transmitters. It is shown that generating the code-books of both users according to discrete distributions with a finite number of mass points achieves the largest weighted sum-rate in the network. This verifies that any point on the boundary of the capacity region of a two-user MAC under peak power constraints at both transmitters is achieved by discrete distributions with a finite number of mass points. Although the capacity-achieving distributions are not necessarily unique, it is verified that only discrete distributions with a finite number of mass points can achieve a point on the boundary of the capacity region. It is shown that there exist an infinite number of sum-rate-optimal points on the boundary of the capacity region. In contrast to the Gaussian MAC with average power constraints, we verify that time division (TD) cannot achieve any of the sum-rate-optimal points in the Gaussian MAC with peak power constraints. Using the so-called I-MMSE identity of Guo et al., the largest achievable sum-rate by orthogonal code division (OCD) is characterized where it is shown that Walsh-Hadamard spreading codes of length 2 are optimal. In the symmetric case where the peak power constraints at both transmitters are identical, we verify that OCD can achieve a sum-rate that is strictly larger than the highest sum-rate achieved by TD. Finally, it is demonstrated that there are values for the maximum peak power at the transmitters such that OCD can not achieve any of the sum-rate-optimal points on the boundary of the capacity region.

This paper addresses a decentralized wireless networks of K separate transmitter-receiver pairs. Users treat each other as noise and there is no central controller to assign the resources to the users. Each user randomly spreads the symbols in its Gaussian codewords by the so-called signatures of spreading gain N. Any receiver is aware of the signatures of its affiliated transmitter, however, it is unaware of the signatures of other users. This makes the interference plus noise at each receiver be mixed Gaussian, and hence, there is no closed expression for the achievable rates of users. Invoking conditional entropy power inequality and a key upper bound on the differential entropy of a mixed Gaussian random vector, we develop a lower bound on the achievable rates of users. This lower bound has the same signal-to-noise ratio (SNR) scaling as that of the exact achievable rate. It is shown that the sum multiplexing gain (SMG) in the network can be made arbitrarily close to (K/N) for any finite values of K and N where K ≤ N. The effect of matched filtering is studied in the particular case where the signatures are constructed over a binary alphabet. It is established that the SMG of the network is larger than (1/2e) regardless of the value of K as long as N = 2 and the signatures are generated according to a proper nonuniform distribution. This paper is concluded by a section on signature design in the finite SNR regime. The main observation is that for any two different methods A and B of designing the signatures, if method A results in a larger achievable rate per user for sufficiently large SNR values, then construction B is likely to yield larger achievable rates for sufficiently small values of SNR. This behavior is attributed to the interplay between two critical factors, namely, the multiplexing gain per user and what we refer to as the interference entropy factor.

It is well-known that the so-called Shannon Achievable Region (SAR) in a collocated two-user Gaussian Two-Way Channel (GTWC) does not depend on the self-interference that is due to the leakage of the signal transmitted by each user at its own receiver. This is simply because each user can completely remove its self-interference. In this paper, we study a class of GTWCs where each user is unable to cancel the self interference due to random erasures at its receiver. The mixture of the intended signal for each user and its self-interference is erased independently from transmission slot to transmission slot. It is assumed that both users adopt PAM constellations for transmission purposes. Due to the fact that both users are unaware of the erasure pattern, the noise plus interference at each user is mixed Gaussian. To analyze this setup, a sequence of upper and lower bounds are developed on the differential entropy of a general mixed Gaussian random variable where it is shown that the upper and lower bounds meet as the sequence index increases. Utilizing such bounds, it is shown that the achievable rate for each user is monotonically increasing in terms of the level of self-interference and eventually saturates as self-interference grows to infinity. This saturation effect is justified analytically by showing that as self-interference increases, each user is enabled to extract the erasure pattern at its receiver. Treating the erasure pattern as side information, both users are able to cancel self-interference and decode the useful information at higher transmission rates.

This paper is a follow up to [1] on the two-user Gaussian Multiple Access Channel (MAC) with peak constraints at the transmitters. It is shown that there exist an infinite number of sum-rate-optimal points on the boundary of the capacity region. In contrast to the Gaussian MAC with power constraints, we verify that Time Division (TD) can not achieve any of the sum-rate-optimal points in the Gaussian MAC with peak constraints. Using the so-called I-MMSE identity of Guo et.al, the largest achievable sum-rate by Orthogonal Code Division (OCD) is characterized where it is shown that Walsh-Hadamard spreading codes of length 2 are optimal. In the symmetric case where the peak constraints at both transmitters are similar, we verify that OCD can achieve a sum-rate that is strictly larger than the highest sum-rate achieved by TD. Finally, it is demonstrated that there are values for the maximum peak at the transmitters such that OCD can not achieve any of the sum-rate-optimal points on the boundary of the capacity region.

A decentralized network of one Primary User (PU) and several Secondary Users (SU) is studied. PU is licensed to exploit the resources, while the party of SUs intend to share the resources with PU. Each SU must guarantee to not disturb the performance of PU beyond a certain level, while maintaining a satisfactory quality of service for itself. It is proposed that each secondary transmitter adopts a Randomized Masking (RM) strategy with full average transmission power where it remains silent or transmits a symbol in its codeword independently from transmission slot to transmission slot. We consider a setup where the primary transmitter is unaware of channel coefficients, code-books of secondary users and the number of secondary users. SUs are anonymous to each other, i.e, they are unaware of each others' code-books, however, each SU is smart in the sense that it is aware of the code-book of PU, channel coefficients and the number of active SUs. Invoking the concept of ε-outage capacity, we define the (ε,ν)-admissible region as the set of masking probabilities for each SU such that the probability of outage for PU is maintained under a threshold \varepsilon in a case where PU sets its transmission rate at a fraction ν of its ε-outage capacity as if there were no SUs in the network. The masking probability of SUs is designed through maximizing the average (with respect to channel coefficients) achievable rate per SU over the (ε,ν)-admissible region. In our analysis, the primary receiver treats interference as noise, however, each secondary receiver has the option to decode and cancel the interference caused by PU, while treating the signals of other SUs as noise. In another approach, referred to as Continuous Transmission with Power Control (CTPC), each SU transmits continuously (no masking is applied), however, it adjusts its transmission power in order to yield the largest value for average achievable rate per SU. The schemes RM and CTPC are compared for different values of transmission power for each SU and PU and distance between different users. It is observed that neither of RM or CTPC always outperforms the other in various scenarios in terms of the underlying system parameters. A combination of RM and CTPC referred to as Randomized Masking with Power Control (RMPC) is also investigated where each SU controls both its probability of masking and average transmission power. It is demonstrated through simulations that RMPC can outperform both RM and CTPC.

This paper addresses a two-user Gaussian Multiple Access Channel (MAC) under peak constraints at the transmitters. It is shown that generating the code-books of both users according to discrete distributions achieves the largest sum-rate in the network. In other words, sum-capacity achieving input distributions for this channel are discrete with a finite number of mass points. We also demonstrate uniqueness of the input distributions which achieve rates at any of the corner points of the capacity region of the channel.

This paper addresses the following question regarding Gaussian networks: Is there an alternative to decoding interference or treating interference as Gaussian noise? By answering this question we aim to establish a benchmark for practical systems where multiuser decoding is not a common practice. To state our result, we study a decentralized network of one Primary User (PU) and one Secondary User (SU) modeled by a two-user Gaussian interference channel. The primary transmitter is constellation-based, i.e., PU is equipped with a modulator and its code-book is constructed over a modulation signal set. SU utilizes random Gaussian codewords with controlled transmission power that guarantees a certain level of Interference-to-Noise Ratio (INR) at the primary receiver. Both users are unaware of each other's code-book, however, SU is smart in the sense that it is aware of the constellation set of PU. While interference at the primary receiver is modeled as additive Gaussian noise, the secondary receiver can utilize the structure of PU's modulator as side information to decode its message without decoding the message of PU. The instantaneous realizations of symbols in a codeword transmitted by PU are unknown to both ends of SU's direct link, however, the sample space of such symbols is available to SU. This makes the interference plus noise at the secondary receiver be a mixed Gaussian process. Invoking entropy power inequality and an upper bound on the differential entropy of a mixed Gaussian vector, we develop an achievable rate for SU that is robust to the structure of PU's modulation signal set and only depends on its constellation size and the dimension of the euclidean space that the constellation points lie in. Moreover, we obtain an achievable rate for PU using Fano's inequality in conjunction with a Gallager-type upper bound on the probability of error in decoding constellation points at the primary receiver. The developed achievable rates for PU and SU enable us to show that the sum rate can be improved compared to a scenario where both users employ Gaussian codewords and treat each other as Gaussian noise.

In this paper, we consider a decentralized wireless communication network with a fixed numberuof frequency subbands to be shared amongNtransmitter-receiver pairs. It is assumed that the number of active users is a realization of a random variable with a given probability mass function. Moreover, users are unaware of each other's codebooks and hence, no multiuser detection is possible. We propose a randomized frequency hopping (FH) scheme in which each transmitter randomly hops over a subset ofusubbands from transmission slot to transmission slot. Assuming all users transmit Gaussian signals, the distribution of the noise plus interference is mixed Gaussian, which makes calculation of the mutual information between the transmitted and received signals of each user intractable. We derive lower and upper bounds on the mutual information of each user and demonstrate that, for large signal-to-noise ratio (SNR) values, the two bounds coincide. This observation enables us to compute the sum multiplexing gain of the system and obtain the optimum hopping strategy for maximizing this quantity. We compare the performance of the FH system to that of the frequency division (FD) system in terms of the following performance measures: average sum multiplexing gain (η(1)) and average minimum multiplexing gain per user (η(2)). We show that (depending on the probability mass function of the number of active users) the FH system can offer a significant improvement in terms of η(1)and η(2)(implying a more efficient usage of the spectrum). In the sequel, we consider a scenario where the transmitters are unaware of the number of active users in the network as well as the channel gains. Developing a new upper bound on the differential entropy of a mixed Gaussian random vector and using entropy power inequality, we obtain lower bounds on the maximum transmission rate per user to ensure a specified outage probability at a given SNR level. We demonstrate that the so-called outage capacity can be considerably higher in the FH scheme than in the FD scenario for reasonable distributions on the number of active users. This guarantees a higher spectral efficiency in FH compared to FD.

We address a two-user decentralized interference channel with static non-frequency selective channel gains. Both users are unaware of each other's code-books and there is no central controller to manage the allocation of resources between the two users. As multiuser detection is not possible, the conventional scheme of transmitting a continuous stream of i.i.d. symbols from Gaussian codebooks by each transmitter (referred to as continuous transmission) results in excessive interference. To provide both users with a partially interference-free channel, we propose that each user randomly quits transmitting from transmission slot to transmission slot independently with a probability of 1 - ε; ε ∈ (0, 1). This is called the Randomized Masking (RM) protocol. Due to the on-off nature of transmissions, the noise plus interference process has a mixed distribution. As a result, the mutual information between the input and the output of the channels does not accept any closed form expression. Assuming each user transmits i.i.d. signals upon activation, the highest achievable rate by each user is denoted by CRM-I. We derive upper and lower bounds on CRM-Iwhere Entropy Power Inequality (EPI) and the extremal inequality of Liu and Viswanath are two important tools in this analysis. Using the proposed lower bound, we devise a distributed strategy to select the activity factor e. Note that this strategy includes the conventional continuous transmission by setting ε = 1. The main result of the paper states that there exist values of 0 ≤ αRM-Ias far as the Signal-to-Noise Ratio (SNR) is sufficiently large. Therefore, it is proved that transmitting i.i.d. signals in consecutive transmission slots is not optimum under the RM protocol.

This paper addresses a distributed signaling scheme to improve the multiplexing gain (MG) in a wireless decentralized network with a fixed number u > 1 of frequency sub-bands to be shared among K transmitter-reciever pairs. In a decentralized network, users are not aware of the code-books of each other. Hence, in the high SNR regime, interference highly degrades the achievable rates of users as canceling the interference is impossible. On the other hand, decentralized networks have no fixed underlying infrastructure, i.e., there is no central management to assign certain non-overlaping portions of the spectrum to different users. As such, choosing the same sub-band by different users may result in losing the data transmitted on this sub-band. These shortcomings motivate us to propose a decentralized scheme that enables all users to coexist fairly, while utilizing the spectrum efficiently. We introduce a distributed signaling scheme (using i.i.d. Gaussian code-books) called repetition-frequency hopping (RFH) where all users keep transmitting the same set of independent signals over different portion of the spectrum along a certain repetition frame. Due to the dynamic nature of interference, sensing the spectrum to locate the interference is practically not possible. This makes the interference plus noise probability density function (PDF) be mixed Gaussian. We obtain upper and lower bounds on the rates of users that coincide as SNR tends to infinity. This enables us to derive a general formula for the sum-rate multiplexing gain in the network. We show that it is possible to achieve higher multiplexing gains in such systems if the length of the repetition frame along the time-axis is large enough. In fact, in many cases, there is a certain amount of repetition that leads to the highest multiplexing gain per user.

We consider a two-user interference channel where the users are not synchronous meaning there exists a delay between their transmitted codes. Assuming no user is aware of the location of the interference burst on its code, no interference cancellation is performed, i.e., users treat each other as noise. By the same token, the interference is no longer Gaussian as a result of the ambiguity on the start of the interference burst. We propose a stationary channel model for this setup for which we are able to derive the achievable rates based on upper and lower bounds on the mutual information between the input and output of the channel. These bounds meet each other as the code length grows to infinity. We define the outage capacity for each user as the largest transmission rate such that the outage probability is ensured to be below a certain threshold. In case the users are sharing a certain number of frequency sub-bands, we propose to divide the spectrum among the users to maximize the outage capacity for each user. We demonstrate that depending on the probabilistic parameters of the delay model and the value of the outage threshold, there are cases where the best strategy is to assign both private and common frequency sub-bands to the users.

This paper provides a unified measure to design binary pseudo-random (PN) codes in a wireless decentralized network in which several transmitter-reciever pairs share the spectrum. In a decentralized network, users are not aware of the code-books of each other. Hence, in the high SNR regime, interference highly degrades the achievable rates of users as interference cancellation is impossible. On the other hand, decentralized networks have no fixed underlying infrastructure, i.e., there is no central management to assign ldquogoodrdquo PN codes with appropriate cross-correlation properties to different users. As such, choosing the same PN code by different users may result in losing the packets transmitted by these users. These shortcomings motivate us to propose a decentralized scheme that enables all users to coexist fairly, while utilizing the spectrum efficiently. We introduce a distributed signaling scheme (using i:i:d: Gaussian code-books) called Bernoulli-Direct-Sequence (BDS) where all users spread their signals by locally generated binary PN codes. Due to the dynamic nature of interference, sensing the spectrum to measure the interference is practically not possible. This makes the interference plus noise probability density function (PDF) be mixed Gaussian. We obtain upper and lower bounds on the rates of users that coincide as SNR tends to infinity. This enables us to derive a general formula for the sum-rate multiplexing gain in the network. Subsequently, we propose a general rule to design the PN codes in the sense of increasing the sum-rate multiplexing gain in the network. It is shown that depending on the number of active users in the system, there is a certain amount of spreading length that leads to the highest multiplexing gain per user. Several design examples are provided at the end.

In this paper, a downlink communication system, in which a base station (BS) equipped with M antennas communicates with N (N Gt 1) single-antenna users, in a Rician fading environment is considered. The asymptotic (in terms of the number of users) sum-rate capacity of the system, as well as the capacity-achieving strategies, are derived. The main results of the paper are as follows: i) in the region of K = o(log N), where K denotes the Rician factor, the sum-rate capacity scales as M log(1 + P/Meta), where P denotes the SNR and eta =Deltalog N/1+K, which is achieved by zero-forcing beam-forming (ZFBF) along with a low-complexity user selection algorithm that considers only the scattered component of the userspsila channels, ii) in the region K = omega(log N), in the case of co-located transmit antennas, the capacity scales as log(1+MP), which is achieved by time division multiple access (TDMA), iii) in the region K = omega(log N), in the case of isotropically-distributed specular components, the sum-rate capacity behaves as M log(1 + P), which is achieved by ZFBF, along with a user selection algorithm that considers only the specular component of the userspsila channels.

We consider a wireless communication network with a fixed number of frequency sub-bands to be shared among several transmitter-receiver pairs. In traditional frequency division (FD) systems, the available sub-bands are partitioned into disjoint clusters (frequency bands) and assigned to different users (each user transmits only in its own band). If the number of users sharing the spectrum is random, this technique may lead to inefficient spectrum utilization (a considerable fraction of the bands may remain empty most of the time). In addition, this approach inherently requires either a central network controller for frequency allocation, or cognitive radios which sense and occupy the empty bands in a dynamic fashion. These shortcomings motivate us to look for a decentralized scheme (without using cognitive radios) which allows the users to coexist, while utilizing the spectrum efficiently. We consider a frequency hopping (FH) scheme (with iid Gaussian code-books) where each user transmits over a selection of sub-bands and hops to another selection (with the same cardinality) from transmission to transmission. We derive lower and upper bounds on the achievable rate of each user and demonstrate that for large signal-to-noise ratio (SNR) values, the two bounds coincide. This observation enables us to compute the sum-rate multiplexing gain (SMG) of the system. Subsequently, we show how each user can regulate its rate to guarantee fairness while maximizing SMG. We compare the FH and FD systems in terms of the following performance measures: average sum-rate multiplexing gain (eta1), average multiplexing gain per user (eta2), the minimum multiplexing gain per user (eta3) and service capability. We show that (depending on the probability mass function of the number of active users), the FH system can offer a significant improvement in terms of eta1and eta2(implying a more efficient usage of the spectrum). It is also shown that 1/epsi les eta3(FH)/eta3(FD)les 1, i.e., the loss incurred in eta3is not more than 1/epsi . Finally, computation of the so-called service capability shows that in FH systems any number of users can coexist fairly, while the maximum number of users in FD system is limited by the number of available bands.