  Exploiting channel sparsity at millimeter wave (mmWave) frequencies reduces
the 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, antenna
gain/phase errors, mutual coupling effects, etc. We propose two dictionary
learning (DL) algorithms to learn the best sparsifying dictionaries for channel
matrices from observations obtained with hybrid frequency-selective mmWave
multiple-input-multiple-output (MIMO) systems. First, we optimize the combined
dictionary, i.e., the Kronecker product of transmit and receive dictionaries,
as it is used in practice to sparsify the channel matrix. Second, considering
the different array structures at the transmitter and receiver, we exploit
separable DL to find the best transmit and receive dictionaries. Once the
channel is expressed in terms of the optimized dictionaries, various CS-based
sparse recovery techniques can be applied for low overhead channel estimation.
The proposed DL algorithms perform well under low SNR conditions inherent to
any mmWave communication systems before the precoders/combiners can be
optimized. The effectiveness of the proposed DL algorithms has been
corroborated via numerical simulations with different system configurations,
array geometries and hardware impairments.
