Channel Probing under a Power Budget

Consider a multi-channel system between a transmitter and receiver where each channel undergoes independent block fading. At the beginning of each fading block, a certain number of slots are used for sending probing signals (training period) allowing the receiver to learn some information about the fading gains. The receiver uses this information to figure out the best channel for transmission (which maximizes the expected SNR at the receiver) and conveys this choice to the transmitter through a delay free noiseless feedback link. One can then ask the following question: given a fixed power budget and number of slots for probing, how should these resources be divided among the channels so as to maximize the expected SNR at receiver? In this paper, we formulate this general problem and analyze it for some simple cases. For a single probing slot and channels with Gaussian fading gains and additive Gaussian noise, we show that the probing strategy varies from probing a single channel to probing all the channels depending on the probe power budget. Since the exact analytical characterization of the optimal probe power distribution is difficult to obtain, we only present partial analytical results and numerical simulations to illustrate the behavior of the optimal probe power distribution. For the multiple slots case, we show that if probing is done in parallel, then there is no additional benefit to be gained over the single slot case. The problem of sequential probing is very similar to the multi-armed bandit problem and is very difficult to analyze. Here we consider a heuristic scheme and numerically compare its performance to the single slot probing case.