Optimum Binary FSK for Transmitted Reference Systems Over Rayleigh Fading Channels

Abstract

It is well known that in communicating over randomly time-varying channels, a receiver which performs a channel measurement can make a better decision than one that does not. Furthermore, if the channel characteristics vary relatively slowly in comparison to a large number of adjacent message intervals, a small portion of the transmittter energy can be devoted to channel measurement, and, in spite of the loss of energy in the information bearing portion of the signal, the resulting system performs better than one with no measurement. This paper shows that improved system performance from a channel measuring system occurs, even when the channel characteristics are fixed only during the present message interval. The randomly time-varying channel studied is that of a Rayleigh fading medium with independently fading mark and space channels whose fading is fixed over one baud interval but is independent from baud to baud. The transmission system is a modified frequency shift keying (FSK) system such that during a portion of a baud interval, the mark and space frequencies are always transmitted so as to act as reference signals. For this system, the following has been established: 1) optimum receiver configuration 2) optimum ratio α of information energy to total signal energy as a function of total available SNR for a single fading channel. 3) asymptotic optimum α for an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> -diversity channel 4) error probabilities for item 2 and asymptotic error probabilities for item S for α <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">opt</inf> as a function of total SNR. The asymptotic results show that by using reference techniques the order of diversity is effectively doubled.