A Relation Between Network Computation and Functional Index Coding Problems

In contrast to the network coding problem wherein the sinks in a network demand subsets of the source messages, in a network computation problem the sinks demand functions of the source messages. Similarly, in the functional index coding problem, the side information and demands of the clients include disjoint sets of functions of the information messages held by the transmitter instead of disjoint subsets of the messages, as is the case in the conventional index coding problem. It is known that any network coding problem can be transformed into an index coding problem and vice versa. In this work, we establish a similar relationship between network computation problems and a class of functional index coding problems, viz., those in which only the demands of the clients include functions of messages. We show that any network computation problem can be converted into a functional index coding problem wherein some clients demand functions of messages and vice versa. We prove that a solution for a network computation problem exists if and only if a functional index code (of a specific length determined by the network computation problem) for a suitably constructed functional index coding problem exists. And, that a functional index coding problem admits a solution of a specified length if and only if a suitably constructed network computation problem admits a solution.Comment: 3 figures, 7 tables and 9 page

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

On Precoding for Constant K-User MIMO Gaussian Interference Channel with Finite Constellation Inputs

This paper considers linear precoding for constant channel-coefficient $K$-User MIMO Gaussian Interference Channel (MIMO GIC) where each transmitter-$i$ (Tx-$i$), requires to send $d_i$ independent complex symbols per channel use that take values from fixed finite constellations with uniform distribution, to receiver-$i$ (Rx-$i$) for $i=1,2,\cdots,K$. We define the maximum rate achieved by Tx-$i$ using any linear precoder, when the interference channel-coefficients are zero, as the signal to noise ratio (SNR) tends to infinity to be the Constellation Constrained Saturation Capacity (CCSC) for Tx-$i$. We derive a high SNR approximation for the rate achieved by Tx-$i$ when interference is treated as noise and this rate is given by the mutual information between Tx-$i$ and Rx-$i$, denoted as $I[X_i;Y_i]$. A set of necessary and sufficient conditions on the precoders under which $I[X_i;Y_i]$ tends to CCSC for Tx-$i$ is derived. Interestingly, the precoders designed for interference alignment (IA) satisfy these necessary and sufficient conditions. Further, we propose gradient-ascent based algorithms to optimize the sum-rate achieved by precoding with finite constellation inputs and treating interference as noise. Simulation study using the proposed algorithms for a 3-user MIMO GIC with two antennas at each node with $d_i=1$ for all $i$, and with BPSK and QPSK inputs, show more than 0.1 bits/sec/Hz gain in the ergodic sum-rate over that yielded by precoders obtained from some known IA algorithms, at moderate SNRs.Comment: 15 pages, 9 figure

Square Complex Orthogonal Designs with Low PAPR and Signaling Complexity

Space-Time Block Codes from square complex orthogonal designs (SCOD) have been extensively studied and most of the existing SCODs contain large number of zero. The zeros in the designs result in high peak-to-average power ratio (PAPR) and also impose a severe constraint on hardware implementation of the code when turning off some of the transmitting antennas whenever a zero is transmitted. Recently, rate 1/2 SCODs with no zero entry have been reported for 8 transmit antennas. In this paper, SCODs with no zero entry for $2^a$ transmit antennas whenever $a+1$ is a power of 2, are constructed which includes the 8 transmit antennas case as a special case. More generally, for arbitrary values of $a$, explicit construction of $2^a\times 2^a$ rate $\frac{a+1}{2^a}$ SCODs with the ratio of number of zero entries to the total number of entries equal to $1-\frac{a+1}{2^a}2^{\lfloor log_2(\frac{2^a}{a+1}) \rfloor}$ is reported, whereas for standard known constructions, the ratio is $1-\frac{a+1}{2^a}$. The codes presented do not result in increased signaling complexity. Simulation results show that the codes constructed in this paper outperform the codes using the standard construction under peak power constraint while performing the same under average power constraint.Comment: Accepted for publication in IEEE Transactions on Wireless Communication. 10 pages, 6 figure