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In this paper, the average sum rate of two-way amplify-and-forward (AF) half-duplex relaying system is analyzed. To this end, we first derive the harmonic mean of two independent gamma distributed random variables which have the same shape parameter but different scale parameters. By deriving tight upper and lower bounds for the average sum rate of two-way relaying, we verify that two-way relaying can significantly recover the spectrum efficiency loss of one-way relaying. We also extend the two-way AF half-duplex relaying to the case where source and destination terminals both transmit Alamouti's orthogonal space time block code (OSTBC) utilizing two antennas and the relay has only one antenna. By deriving both upper and lower bounds for the average sum rate as well as an upper bound for the pairwise error probability (PEP) for the proposed two-way OSTBC scheme, we show that the average sum rate is further improved compared to the single antenna case and a diversity order of two is also achieved. Furthermore, optimal power allocations under a global power constraint for two-way relaying with single antenna and the proposed two-way OSTBC scheme are derived analytically.Index Terms-Two-way relaying, amplify-and-forward, harmonic mean, pairwise error probability, space time block code.

We consider the scenario in which multiple sensors send spatially correlated data to a fusion center (FC) via independent Rayleigh-fading channels with additive noise. Assuming that the sensor data is sparse in some basis, we show that the recovery of this sparse signal can be formulated as a compressive sensing (CS) problem. To model the scenario in which the sensors operate with intermittently available energy that is harvested from the environment, we propose that each sensor transmits independently with some probability, and adapts the transmit power to its harvested energy. Due to the probabilistic transmissions, the elements of the equivalent sensing matrix are not Gaussian. Besides, since the sensors have different energy harvesting rates and different sensor-to-FC distances, the FC has different receive signal-to-noise ratios (SNRs) for each sensor. This is referred to as the inhomogeneity of SNRs. Thus, the elements of the sensing matrix are also not identically distributed. For this unconventional setting, we provide theoretical guarantees on the number of measurements for reliable and computationally efficient recovery, by showing that the sensing matrix satisfies the restricted isometry property (RIP), under reasonable conditions. We then compute an achievable system delay under an allowable mean-squared-error (MSE). Furthermore, using techniques from large deviations theory, we analyze the impact of inhomogeneity of SNRs on the so-called k-restricted eigenvalues, which governs the number of measurements required for the RIP to hold. We conclude that the number of measurements required for the RIP is not sensitive to the inhomogeneity of SNRs, when the number of sensors n is large and the sparsity of the sensor data (signal) k grows slower than the square root of n. Our analysis is corroborated by extensive numerical results.

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