It is known that, when using a multi-tone FSK scheme in a wideband fading channel to achieve the wideband capacity limit, the codeword probability of error decays very slowly with bandwidth. In this paper, we consider a modified multi-tone FSK scheme which employs a feedback link. We show that, a small amount of feedback improves the error performance significantly

In the wideband limit, certain types of "flash" signaling, such as flash frequency-shift keying (FSK), achieve the capacity of multipath fading channels. It is not clear, however, whether these asymptotic results translate into insights for practical fading channels in the finite-bandwidth, power-limited regime. It is known that, for flash FSK, the size of the input alphabet grows slowly with increasing bandwidth, leading to very high-peak power per tone. Thus, for flash FSK, the codeword probability of error decays very slowly with bandwidth and feasible rates approach the wideband capacity limit extremely slowly. Without contradicting the above results, our results in this paper point to a more optimistic outlook, from the point of view of error exponents and achievable rates, for the applicability of flash techniques in practical scenarios. We consider multitone FSK (MFSK), which has the same asymptotic capacity-achieving property as flash FSK in the wideband limit, but allows a larger input alphabet size with the same bandwidth. First, we show, using an error exponent approach, that multitone FSK allows lower peak power per tone than flash FSK. Next, we present the capacity of single-tone and two-tone FSK schemes with hard-decision detection at finite bandwidths. For typical channel parameters, the capacities are close to the wideband capacity limit.

Flash signalling (with vanishing duty cycle) frequency shift keying (FSK) is known to be a capacity-achieving modulation for multipath 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 multitone FSK, which we have shown in our previous work that can also achieve the infinite-bandwidth capacity limit for multipath 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, multitone 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 multitone 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 multitone FSK in different situations and conclude that FSK is the preferable scheme when average power is the biting constraint, and multitone 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.