Degrees of freedom in fading channels with memory: Achievability through nonlinear decoding

In noncoherent communication systems, neither the transmitter nor the receiver knows the realization of channel fading coefficients. To overcome this limitation and facilitate reliable communication, the transmitter sends fixed values on some predetermined training symbols. The maximum number of symbols that can be used for communication in a block of time is the degrees of freedom (DOF) of the system. Given DOF equals to D, reliable communication (in high SNR regime) implies existence of a D dimensional subspace of the input signal space which is recoverable by the receiver with probability one. The training symbols specify this D dimensional subspace. Thus the minimum number of necessary training symbols determines the DOF of the system. We formalize this rationale and name it Dimension Counting Argument. We then use this argument to prove a lower bound on the DOF of fading channels with memory. We study the Multiple-Input Multiple-Output (MIMO) systems with n_t transmit antennas and n_r receive antennas where fading coefficients in blocks of length T have a temporal correlation matrix with rank Q. We prove that in these systems, n_t* transmit antennas (n_t* = min[n_t, ⌊T/Q⌋]) should be used to achieve min [n_r(T - n_t*Q), n_t*(T - n_t*)] DOF per block of time. We show that the geometric interpretation of the fading channel with memory is equivalent to a nonlinear mapping over manifolds of different dimensions. The inherent nonlinearity of the model manifests itself in the necessary nonlinearity of the proposed decoding algorithms.