We consider the problem of estimating the distance, or range, between two locations by measuring the phase of a sinusoidal signal transmitted between the locations. This method is only capable of unambiguously measuring range within an interval of length equal to the wavelength of the signal. To address this problem signals of multiple different wavelengths can be transmitted. The range can then be measured within an interval of length equal to the least common multiple of these wavelengths. Estimation of the range requires solution of a problem from computational number theory called the closest lattice point problem. Algorithms to solve this problem require a basis for this lattice. Constructing a basis is non-trivial and an explicit construction has only been given in the case that the wavelengths can be scaled to pairwise relatively prime integers. In this paper we present an explicit construction of a basis without this assumption on the wavelengths. This is important because the accuracy of the range estimator depends upon the wavelengths. Simulations indicate that significant improvement in accuracy can be achieved by using wavelengths that cannot be scaled to pairwise relatively prime integers.
This study presents the symbol error probability (SEP) analysis of relay-multiplexing scheme in relay-assisted cellular systems such as the IEEE 802.16j and upcoming fifth generation (5G) systems. The availability of multiple relay paths in these systems motivates two relay configurations: (i) relay diversity and (ii) relay multiplexing. Available works in the literature have focused on the relay diversity methods. In this study, we explore the relay-multiplexing alternative whereby relay stations (RSs) act as independent data pipes for transmitting multiple independent data streams from the base station (BS) to the mobile station (MS). We examine the case when the BS is equipped with massive multiple-input multiple-output (MIMO) antennas, transmitting independent data streams to several MS simultaneously via different relay paths. We derive analytical expressions for the SEP performance of the proposed scheme, and also complement the analysis with simulations. The results show that parallel relaying of independent data streams via different RSs has acceptable SEP performance while boosting the capacity of the system linearly with the number of parallel RSs available in the system. The results also show that when this relaying approach is combined with the emerging massive MIMO techniques, a tremendous boost in data rate can be achieved.
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