The performance of noncoherent orthogonal M-FSK in the presence of timing and frequency errors

Practical M-FSK systems experience a combination of time and frequency offsets (errors). This paper assesses the deleterious effect ofthese offsets, first individually and then combined, on the average hit error probability performance ofthe system. Exact expressions for these various error prohability performances are derived and evaluated numerically for system parameters of interest. Also presented are upper boundsonaverage symbol error probability for the case of frequency error alone which are useful in assessing the absolute and relative performance of the system. Both continuous and discontinuous phase M-FSK cases are considered when timing error is present, the latter being much less robust to this type of offset. 1 .O INTRODUCTION Noncoherent orthogonal M-ary frequency-shift-keying (M-FSK) is a simple and robust form of digital communication when the transmission channel is such that fast reliable carrier recovery is difficult or impractical to achieve and thc bandwidth requirements are not overly stringent. Most studies of this modulatioddemodulation technique for the additive white Gaussian noise (AWGN) channel have focused on the error probability performance when the receiver is assumed to be perfectly time and frequency synchronized. That is, the receiver is assumed to have perfect knowledge of the instants of time at which the modulation can change state and also perfect knowledge of the received camer frequency. In practical systems, such perfect knowledge is never available and thus the receiver must dcrive this information from the received signal imbedded in the AWGN. Since the estimatcs of the time epoch and received carrier frequency derived at the receiver are, in general, random variables (because of the presence of the AWGN), there will exist an error between these estimates and their true values. This lack of perfect time and frequency synchronization gives rise to a degradation in error probability performance relative to that corresponding to the ideal case where perfect knowlcdgc of time and frequency is assumed known. The purpose of this paper is to evaluate this performance degradation, first by treating the two sources of degradation separately, and then by considering their simultaneous effect. In particular, we shall present exact cxprcssions for thc symbol and bit error probability performances of noncoherent orthogonal M-FSK conditioned on the presence of time and frequency errors. These expressions involve integrals of Marcum-Q functions and, as such, their numerical evaluation is cumbcrsomc. Thus, for the case of frequency error only, we present various upper bounds on error probability performances that. because of …