Error exponents for Multi-tone Frequency Shift Keying on wideband Rayleigh fading channels

Flash signalling (with vanishing duty cycle) Frequency Shift Keying (FSK) is known to be a capacity-achieving modulation for multi-path fading channels in the limit of infinite bandwidth. However, since capacity-achieving schemes using flash FSK build the richness of their codebooks in frequency, the data rates of such schemes increase slowly with bandwidth. We seek to establish schemes that exhibit better performance than flash FSK for large but finite bandwidth. We consider Multi-tone FSK, which we have shown in our previous work that can also achieve the infinite-bandwidth capacity limit for multi-path fading channels, but is more general than FSK. Given its increased generality vis-A-vis FSK, and its optimality from a capacity point of view, Multi-tone FSK is a good candidate for transmission over very wide bandwidth. In this paper, we discuss upper and lower bounds of error probabilities for the family of Multi-tone FSK in Rayleigh fading channels. We find that these two bounds coincide in the infinite bandwidth limit and are therefore asymptotically tight. We compare the error probabilities of FSK and Multi-tone FSK in different situations and conclude that FSK is the preferable scheme when average power is the biting constraint, and Multi-tone FSK may be preferable when peak power is a limiting factor. We also explore the relationship among capacity and parameters related to time efficiency and spectrum efficiency.