The Performance of FSK Permutation Modulations in Fading Channels and Their Comparison Based on a General Method for the Comparison of M-ary Modulations

Abstract

A problem that arises in multipath fading channels is that of finding a modulation which enables one to obtain high data rates and at the same time does not lead to an excessively complicated system implementation or the requirement of excessive transmitter power. In many dispersive channels the product of time dispersion and Doppler spread is large, so that coherent modulation systems are not feasible. Examples of such overspread or nearly overspread channels are the West Ford channel, the endoionospheric channel when using aurora injection, and, in some instances, the moon relay channel. For such channels consideration must be given to the use of incoherent modulation systems. Consideration is given to the performance in fading channels of an incoherent modulation system which uses a subclass of permutation modulation called frequency-shift keying (FSK) permutation (PFSK) modulation. The channel is assumed to have the following characteristics. First, the (Gaussian) noise for each propagation mode is white and independent, with equal noise density spectrums; and second, the fadings for the different propagation modes are Rayleigh distributed and independent with equal variance. Extensive curves are given for the performance of various PFSK modulations in terms of the modulation signal-waveform error probabilities (often referred to as the word error probabilities) for the case of frequency-selective fading. Bounds on the modulation performance are given for the case of flat fading. Consideration is given to the problem of comparing the PFSK modulations with each other and with other available modulations to determine which is best for communicating a given <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</tex> -ary message sequence, with respect to the transmitter power and bandwidth required. Toward this end, a relatively simple method is presented for comparing modulations, which circumvents the problem of requiring detailed knowledge of the mapping from the original information data source (which, for greatest generality, is assumed to consist of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s L_{i}</tex> -state Markov chains, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i = 1,...,s,</tex> which are to be transmitted simultaneously) to the modulation waveforms. The method provides upper and lower bounds for the required receiver signal energy to noise density ratio per <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</tex> -ary symbol for a specified <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</tex> -ary error rate (or else, bounds for the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</tex> -ary error rates for a specified receiver signal energy to noise density ratio per <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</tex> -ary symbol). The upper bound is based on the performance of the modulation for the ensemble average of all possible mappings, or, equivalently, for a random mapping (analogous to random coding). The lower bound corresponds to the performance of the modulation for an optimum mapping, which is in effect a generalization of the Gray-code-type mapping used for phaseshift keying (PSK) modulation. (To obtain the lower bound on the modulation performance, the details of the optimum mapping need not be known; moreover, it remains to prove the existence of this optimum mapping.) The method is general and applicable to a wide class of modulations and channels. Applying the technique to FSK, FSK permutation, and amplitude-keyed (AK) carrier modulations yields results which conveniently show, for different <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L_{i}</tex> -ary source sequences, the cost in increased transmitter power required to achieve higher data rates with various modulations. The comparison data indicates that, in some instances, relative to the FSK modulation, one can obtain with PFSK modulations increased data rates for a given signal bandwidth without requiring an increased received signal energy to noise density ratio per <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</tex> -ary source symbol.As a consequence of these results, PFSK modulations should perhaps be of interest to the designer of fading communications channels not having large time-dispersion times Doppler-spread products, as well as to the designer of highly perturbed channels. Finally, a means is given for transmitting PFSK modulations in time-dispersive fading channels, which does not require the use of a linear power amplifier.