Negatively Correlated Branches in Frequency Diversity Systems to Overcome Multipath Fading

In multipath environments in which there is a compact concentration of the values of time-delay-difference (TDD) between signals arriving over each of the indirect paths and the direct path, it is possible, by suitably choosing frequency differences, to design a frequency diversity system in which the diversity branches are negatively correlated. This results in a reduced probability of simultaneous deep fades on all branches over that obtained with branches designed to be independent. In the first part of the paper the magnitude of the negative correlation is shown to depend on the number of indirect reflectors contributing to the total indirect signal and that it is maximum when the indirect signal arises from a single reflector. Also shown is how the magnitude of the negative correlation can be maximized using the probability density function of the TDD to establish the frequency assignment. This is a precursor to the second part of the paper in which an investigation of the improvement of the error probability by a judicious choice of frequency differences is reported. The error probability as a function of frequency differences is found for the specific case of noncoherent frequency-shift keying (FSK) in a specular multipath channel using square-law combining. A general result for
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th order diversity is obtained but a detailed study is made for two and three order diversity only. Results are presented for uniformly distributed TDD's with various ranges and mean values, the parameters being suggested by the physical conditions encountered when low flying aircraft are the multipath source or are themselves trying to communicate. Results for other densities of TDD, the normal and the gamma, are cited. For the second order case, it is found that an advantageous frequency spacing can be determined in terms of the average TDD. For the third order case, two frequency differences are available as design parameters. Results are presented based on a rule for selecting frequency differences similar to that used in the second order case and substantial improvements are found. Another rule investigated that gave even better results is cited. Finally, error probability comparisons are made with systems having no multipath, nondiversity systems with multipath, and with diversity systems in multipath with independent branches. For the conditions assumed, and for reasonable levels of transmitted signal power and receiver noise, order of magnitude improvements are found to be obtained over independent branches.