Construction of Capacity-Achieving Lattice Codes: Polar Lattices

In this paper, we propose a new class of lattices constructed from polar codes, namely polar lattices, to achieve the capacity \frac{1}{2}\log(1+\SNR) of the additive white Gaussian-noise (AWGN) channel. Our construction follows the multilevel approach of Forney \textit{et al.}, where we construct a capacity-achieving polar code on each level. The component polar codes are shown to be naturally nested, thereby fulfilling the requirement of the multilevel lattice construction. We prove that polar lattices are \emph{AWGN-good}. Furthermore, using the technique of source polarization, we propose discrete Gaussian shaping over the polar lattice to satisfy the power constraint. Both the construction and shaping are explicit, and the overall complexity of encoding and decoding is $O(N\log N)$ for any fixed target error probability.Comment: full version of the paper to appear in IEEE Trans. Communication

The capacity region of broadcast channels with intersymbol interference and colored Gaussian noise

We derive the capacity region for a broadcast channel with intersymbol interference (ISI) and colored Gaussian noise under an input power constraint. The region is obtained by first defining a similar channel model, the circular broadcast channel, which can be decomposed into a set of parallel degraded broadcast channels. The capacity region for parallel degraded broadcast channels is known. We then show that the capacity region of the original broadcast channel equals that of the circular broadcast channel in the limit of infinite block length, and we obtain an explicit formula for the resulting capacity region. The coding strategy used to achieve each point on the convex hull of the capacity region uses superposition coding on some or all of the parallel channels and dedicated transmission on the others. The optimal power allocation for any point in the capacity region is obtained via a multilevel water-filling. We derive this optimal power allocation and the resulting capacity region for several broadcast channel models

CODING AND SCHEDULING IN ENERGY HARVESTING COMMUNICATION SYSTEMS

Wireless networks composed of energy harvesting devices will introduce several transformative changes in wireless networking: energy self-sufficient, energy self-sustaining, perpetual operation; and an ability to deploy wireless networks at hard-to-reach places such as remote rural areas, within the structures, and within the human body. Energy harvesting brings new dimensions to the wireless communication problem in the form of intermittency and randomness of available energy. In such systems, the communication mechanisms need to be designed by explicitly accounting for the energy harvesting constraints. In this dissertation, we investigate the effects of intermittency and randomness in the available energy for message transmission in energy harvesting communication systems. We use information theoretic and scheduling theoretic frameworks to determine the fundamental limits of communications with energy harvesting devices. We first investigate the information theoretic capacity of the single user Gaussian energy harvesting channel. In this problem, an energy harvesting transmitter with an unlimited sized battery communicates with a receiver over the classical AWGN channel. As energy arrives randomly and can be saved in the battery, codewords must obey cumulative stochastic energy constraints. We show that the capacity of the AWGN channel with such stochastic channel input constraints is equal to the capacity with an average power constraint equal to the average recharge rate. We provide two capacity achieving schemes: save-and-transmit and best-effort-transmit. In the save-and-transmit scheme, the transmitter collects energy in a saving phase of proper duration that guarantees that there will be no energy shortages during the transmission of code symbols. In the best-effort-transmit scheme, the transmission starts right away without an initial saving period, and the transmitter sends a code symbol if there is sufficient energy in the battery, and a zero symbol otherwise. Finally, we consider a system in which the average recharge rate is time-varying in a larger time scale and derive the optimal offline power policy that maximizes the average throughput, by using majorization theory. Next, we remove the battery from the model to understand the impact of stochasticity in the energy arrival on the communication rate. We consider the single user AWGN channel in the zero energy storage case. We observe that the energy arrival is a channel state and channel state information is available at the transmitter only. We determine the capacity in this case using Shannon strategies. We, then, extend the capacity analysis to an additive Gaussian multiple access channel where multiple users with energy harvesting transmitters of zero energy storage communicate with a single receiver. We investigate the achievable rate region under static and stochastic amplitude constraints on the users' channel inputs. Finally, we consider state amplification in a single user AWGN channel with an energy harvesting transmitter to analyze the trade-off between the objectives of decoding the message and estimating the energy arrival sequence. Next, we specialize in the finite battery regime in the energy harvesting channel. We focus on the case of side information available at the receiver side. We determine the capacity of an energy harvesting channel with an energy harvesting transmitter and battery state information available at the receiver side. This is an instance of a finite-state channel and the channel output feedback does not increase the capacity. We state the capacity as maximum directed mutual information from the input to the output and the battery state. We identify sufficient conditions for the channel to have stationary input distributions as optimal distributions. We also derive a single-letter capacity expression for this channel with battery state information at both sides and infinite-sized battery at the transmitter. Then, we determine the capacity when energy arrival side information is available at the receiver side. We first find an n-letter capacity expression and show that the optimal coding is based on only current battery state s_i. We, next, show that the capacity is expressed as maximum directed information between the input and the output and prove that the channel output feedback does not increase the capacity. Then, we consider security aspects of communication in energy harvesting systems. In particular, we focus on a wiretap channel with an energy harvesting transmitter where a legitimate pair of users wish to establish secure communication in the presence of an eavesdropper in a noisy channel. We characterize the rate-equivocation region of the Gaussian wiretap channel under static and stochastic amplitude constraints. First, we consider the Gaussian wiretap channel with a static amplitude constraint on the channel input. We prove that the entire rate-equivocation region of the Gaussian wiretap channel with an amplitude constraint is obtained by discrete input distributions with finite support. We also prove the optimality of discrete input distributions in the presence of an additional variance constraint. Next, we consider the Gaussian wiretap channel with an energy harvesting transmitter with zero energy storage. We prove that single-letter Shannon strategies span the entire rate-equivocation region and obtain numerically verifiable necessary and sufficient optimality conditions. In the remaining parts of this dissertation, we consider optimal transmission scheduling for energy harvesting transmitters. First, we consider the optimization of single user data transmission with an energy harvesting transmitter which has a limited battery capacity, communicating over a wireless fading channel. We consider two objectives: maximizing the throughput by a deadline, and minimizing the transmission completion time of the communication session. We optimize these objectives by controlling the time sequence of transmit powers subject to energy storage capacity and causality constraints. We, first, study optimal offline policies. We introduce a directional water-filling algorithm which provides a simple and concise interpretation of the necessary optimality conditions. We show the optimality of the directional water-filling algorithm for the throughput maximization problem. We solve the transmission completion time minimization problem by utilizing its equivalence to its throughput maximization counterpart. Next, we consider online policies. We use dynamic programming to solve for the optimal online policy that maximizes the average number of bits delivered by a deadline under stochastic fading and energy arrival processes with causal channel state feedback. We also propose near-optimal policies with reduced complexity, and numerically study their performances along with the performances of the offline and online optimal policies. Then, we consider a broadcast channel with an energy harvesting transmitter with a finite capacity battery and M receivers. We derive the optimal offline transmission policy that minimizes the time by which all of the data packets are delivered to their respective destinations. We obtain structural properties of the optimal transmission policy using a dual problem and determine the optimal total transmit power sequence by a directional water-filling algorithm. We show that there exist M-1 cut-off power levels such that each user is allocated the power between two corresponding consecutive cut-off power levels subject to the availability of the allocated total power level. Based on these properties, we propose an iterative algorithm that gives the globally optimal offline policy. Finally, we consider parallel and fading Gaussian broadcast channels with an energy harvesting transmitter. Under offline knowledge of energy arrival and channel fading variations, we characterize the transmission policies that achieve the boundary of the maximum departure region in a given interval. In the case of parallel broadcast channels, we show that the optimal total transmit power policy that achieves the boundary of the maximum departure region is the same as the optimal policy for the non-fading broadcast channel, which does not depend on the priorities of the users, and therefore is the same as the optimal policy for the non-fading scalar single user channel. The optimal total transmit power can be found by a directional water-filling algorithm while optimal splitting of the power among the parallel channels is performed in each epoch separately. In the case of fading broadcast channels, the optimal power allocation depends on the priorities of the users. We obtain a modified directional water-filling algorithm for fading broadcast channels to determine the optimal total transmit power allocation policy

Two-User Gaussian Interference Channel with Finite Constellation Input and FDMA

In the two-user Gaussian Strong Interference Channel (GSIC) with finite constellation inputs, it is known that relative rotation between the constellations of the two users enlarges the Constellation Constrained (CC) capacity region. In this paper, a metric for finding the approximate angle of rotation (with negligibly small error) to maximally enlarge the CC capacity for the two-user GSIC is presented. In the case of Gaussian input alphabets with equal powers for both the users and the modulus of both the cross-channel gains being equal to unity, it is known that the FDMA rate curve touches the capacity curve of the GSIC. It is shown that, with unequal powers for both the users also, when the modulus of one of the cross-channel gains being equal to one and the modulus of the other cross-channel gain being greater than or equal to one, the FDMA rate curve touches the capacity curve of the GSIC. On the contrary, it is shown that, under finite constellation inputs, with both the users using the same constellation, the FDMA rate curve strictly lies within (never touches) the enlarged CC capacity region throughout the strong-interference regime. This means that using FDMA it is impossible to go close to the CC capacity. It is well known that for the Gaussian input alphabets, the FDMA inner-bound, at the optimum sum-rate point, is always better than the simultaneous-decoding inner-bound throughout the weak-interference regime. For a portion of the weak interference regime, it is shown that with identical finite constellation inputs for both the users, the simultaneous-decoding inner-bound, enlarged by relative rotation between the constellations, is strictly better than the FDMA inner-bound.Comment: 12 pages, 10 figure

The Noncoherent Rician Fading Channel -- Part I : Structure of the Capacity-Achieving Input

Transmission of information over a discrete-time memoryless Rician fading channel is considered where neither the receiver nor the transmitter knows the fading coefficients. First the structure of the capacity-achieving input signals is investigated when the input is constrained to have limited peakedness by imposing either a fourth moment or a peak constraint. When the input is subject to second and fourth moment limitations, it is shown that the capacity-achieving input amplitude distribution is discrete with a finite number of mass points in the low-power regime. A similar discrete structure for the optimal amplitude is proven over the entire SNR range when there is only a peak power constraint. The Rician fading with phase-noise channel model, where there is phase uncertainty in the specular component, is analyzed. For this model it is shown that, with only an average power constraint, the capacity-achieving input amplitude is discrete with a finite number of levels. For the classical average power limited Rician fading channel, it is proven that the optimal input amplitude distribution has bounded support.Comment: To appear in the IEEE Transactions on Wireless Communication