Second-Order Spectral Statistics for the Power Gain of Wideband Wireless Channels

We derive closed-form expressions for the second-order statistics of the power gain (as a function of frequency) of wideband microwave indoor channels. We obtain our results within a framework that is general enough to be compatible with several popular channel models, such as the Saleh-Valenzuela (SV) channel model and those proposed by the IEEE 802.15.3a and IEEE 802.15.4a task groups. As in all these models, our channel description is based upon clusters and rays with (possibly mixed) Poisson arrivals and random amplitudes. Our results consist of closed-form expressions for the second-order statistics of the channel power frequency response, where statistical averages involve expectations over ray amplitudes and arrival times. These expressions reveal that the autocovariance of the spectral power gain between any two frequencies decreases and tends to zero as the difference between these frequencies tends to infinity if and only if the cluster arrival rate goes to infinity. They also show that the variance-to-squared-mean ratio of the narrow-band power gain exhibits exactly the same behavior with respect to the center frequency. We then use these results to obtain closed-form expressions for the variance and the second-order moment of the aggregate channel power gain over any given interval of frequencies. This allows us to express the channel spectral diversity as a function of model parameters and bandwidth. In addition, we illustrate how these equations allow one to devise automatic cluster identification algorithms which, from empirical estimates of the second-order spectral statistics of the channel power gain, can confirm or deny the existence of clusters in a given scenario.