Training over sparse multipath channels in the low SNR regime

Training over sparse multipath noisy channels is explored. The energy allocation and the optimal shape of training signals that enable communications over unknown channels are characterized as a function of the channels´ statistics. The performance of training is evaluated by the reduction of the mean square error of the channel estimate and by the decrease in the the mutual information due to the uncertainty of the channel. The performance of low dimensional training signal is compared to the performance of a full dimensional one. Especially, The trade-off between the number of required measurements (signal dimensions) and the energy allocation is calculated, and it is proven that if the signal to noise ratio of the received training signal is low, reducing the number of channel measurements using compressed sensing is as efficient as training over the entire frequency band.