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Exploiting channel sparsity at millimeter wave (mmWave) frequencies reducesthe high training overhead associated with the channel estimation stage.Compressive sensing (CS) channel estimation techniques usually adopt the(overcomplete) wavelet/Fourier transform matrix as a sparsifying dictionary.This may not be the best choice when considering non-uniform arrays, antennagain/phase errors, mutual coupling effects, etc. We propose two dictionarylearning (DL) algorithms to learn the best sparsifying dictionaries for channelmatrices from observations obtained with hybrid frequency-selective mmWavemultiple-input-multiple-output (MIMO) systems. First, we optimize the combineddictionary, i.e., the Kronecker product of transmit and receive dictionaries,as it is used in practice to sparsify the channel matrix. Second, consideringthe different array structures at the transmitter and receiver, we exploitseparable DL to find the best transmit and receive dictionaries. Once thechannel is expressed in terms of the optimized dictionaries, various CS-basedsparse recovery techniques can be applied for low overhead channel estimation.The proposed DL algorithms perform well under low SNR conditions inherent toany mmWave communication systems before the precoders/combiners can beoptimized. The effectiveness of the proposed DL algorithms has beencorroborated via numerical simulations with different system configurations,array geometries and hardware impairments.

Dictionary Learning for Channel Estimation in Hybrid Frequency-Selective mmWave MIMO Systems