A Geometrical Study of Transmitted Reference Communication Systems

Digital communication systems utilizing wide-band carriers require a coherent reference for optimal data processing. This reference may be either locally generated or transmitted simultaneously with the data. The latter system completely avoids the synchronization problem of locally generated reference systems but is degraded by the fact that two receivers are required, excess system dimension must be used, and the transmitter power is diluted by having to transmit the reference. This report is a study of a general transmitted reference (TR) system, with the primary objective of finding bounds on these restrictions. It is based upon geometrical considerations using the fact that a transmitted reference system can be described by vectors in, and linear transformations upon, an -dimensional Euclidean space. In this way it can be shown, for example, that the dimension of the transmitting channel must be greater than that of the space of carrier signals for unique operation. This increase in dimension need be only slight in the noiseless case, but an advantage is gained by increasing the dimension in noisy operation. An equivalent expression for output SNR is obtained in terms of norms of signal vectors and a comparison is made with that of a locally generated reference (LGR) system. It is shown that a TR system always operates with at least a 3-dB poorer SNR than that of an ideal LGR system, with the bound increasing to 6 dB if the transmitter is power limited, and gets progressively worse as the transmitter power to noisepower ratio decreases below 0 dB. Some curves are plotted to show these relationships quantitatively.