Level crossing rate in terms of the characteristic function: a new approach for calculating the fading rate in diversity systems

The level crossing rate (LCR) of a random process conveys useful information about the underlying process, and is of interest in diverse engineering fields. In wireless communications, it is related to the system characteristics such as handoff, outage probability, fading rate, average duration of fades, velocity (or maximum Doppler shift) of the mobile, and the effect of diversity on fading. The LCR formula was originally derived by Rice in terms of the joint probability density function (pdf) of the underlying process and its time derivative. In this letter, we express the LCR in terms of the joint characteristic function (cf). This new formula is useful for many cases where the joint cf is simpler to derive than the associated joint pdf. As an application and for a direct-sequence code-division multiple-access system, the fading rate at the output of a RAKE receiver with either maximal ratio combiner or postdetection equal gain combiner, operating over a frequency-selective fading channel with different path statistics, is easily calculated using the new cf-based LCR formula.