Optimal placement of training for frequency-selective block-fading channels

The problem of placing training symbols optimally for orthogonal frequency-division multiplexing (OFDM) and single-carrier systems is considered. The channel is assumed to be quasi-static with a finite impulse response of length ( + 1) samples. Under the assumptions that neither the transmitter nor the receiver knows the channel, and that the receiver forms a minimum mean square error (MMSE) channel estimate based on training symbols only, training is optimized by maximizing a tight lower bound on the ergodic training-based independent and identically distributed (i.i.d.) capacity. For OFDM systems, it is shown that the lower bound is maximized by placing the known symbols periodically in frequency. For single-carrier systems, under the assumption that the training symbols are placed in clusters of length (2 + 1), it is shown that the lower bound is maximized by a family of placement schemes called QPP, where QPP stands for quasi-periodic placement. These placement schemes are formed by grouping the known symbols into as many clusters as possible and then placing these clusters periodically in the packet. For both OFDM and single-carrier systems, the optimum energy tradeoff between training and data is also obtained.