DESIGN AND BOUNDING OF LARGE SIGNAL SETS OCCUPYING A GIVEN TIME-BANDWIDTH (TB) PRODUCT AND HAVING SPECIFIED CORRELATION PROPERTIES
Abstract
The investigation of this thesis is concerned with two related problems. The first problem is the synthesis of a design algorithm that realizes large, equal energy signal sets whose members all occupy the same time-bandwidth (TB) product and have good correlation properties. Secondly, one wishes to have upper bounds on the signal set size that is theoretically possible given the constraints of TB product and peak correlation function magnitude. This work is motivated by the area of spread-spectrum code-division multiple-access communication systems. The receivers implemented in these systems are typically a matched filter or the equivalent correlator structure. Hence, mutal interference at the output of a receiver due to other users on the channel at the same time is determined by the correlation properties of the signal set employed. Obviously, this interference must be controlled to maintain acceptable system performance. A signal set design algorithm is realized that allows one to generate an arbitrarily sized set of time-limited, equal energy signals whose members have their correlation function magnitudes bounded at all integral shifts of a selected time increment. To implement the algorithm, a pseudo-noise sequence and finite sized set of orthonormal basis functions must be specified. In its most general form, the algorithm is independent of these selections, making a precise calculation of the resultant bandwidth required by the signals impossible. Examples of the algorithm utilizing sinusoids as the basis functions are presented. This choice of basis functions enables one to make a straightforward calculation of TB product. Computational results indicate that the resultant signal set has good correlation properties for all time shifts in comparison to a known uniform lower bound. As the last step in the signal set design area, an extension of the original algorithm is presented. The significance of this extension is that very large sets of signals having good correlation properites result when using the extended algorithm as compared to the original. Hence, a more efficient use of TB product in relation to signal set size is possible. To solve the theoretical question of bounding signal set size under the constraints of TB product and peak correlation magnitude, a mathematical model is defined which makes the problem tractable. The method employed is to rigorously relate the continuous time problem to a similar sequence set problem, enabling one to utilize known sequence set results. During the course of the theoretical investigation, different measures of bandwidth are defined for periodic signals versus time-limited signals. The results indicate that being precise with these measures of bandwidth cause separate bounds to be realized for the periodic and time-limited cases. To conclude the development of bounding signal set size under the aforementioned constraints, design examples are developed. These examples indicate that the periodic bounds are relatively tight, while the tightness of the aperiodic (time-limited) bounds is sensitive to the choice of bandlimitedness in relation to TB product magnitude.
Degree
Ph.D.
Subject Area
Electrical engineering
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