Table 1 Computational complexity

Full-digital BD

\({{\mathcal {O}}}\left( {{K^2}N_t^2{N_r}} \right)\)

Spatially sparse

\({{\mathcal {O}}}({\left( {N_{RF}^t} \right) ^4} + {N_t}{\left( {N_{RF}^t} \right) ^3} + {N_t}{\left( {N_{RF}^t} \right) ^2}K{N_s}\) \(+ N_{RF}^t{N_t}K{N_s})\)

MMSE

\({{\mathcal {O}}}({K^3}{\left( {N_{RF}^r} \right) ^4} + {K^3}{\left( {N_{RF}^r} \right) ^3}{N_r} + {K^3}{\left( {N_{RF}^r} \right) ^2}{N_r}{N_s}\) \(+ {K^2}N_{RF}^r{N_r}{N_s})\)

Limited feedback

\({{\mathcal {O}}}({K^3}{\left( {N_{RF}^t} \right) ^3} + K{N_t}{N_r}N_{RF}^r + K{N_t}N_{RF}^rN_{RF}^t)\)

Hybrid BD

\({{\mathcal {O}}}({K^4}{\left( {N_{RF}^r} \right) ^3} + K{\left( {N_{RF}^r} \right) ^3} + {K^2}{\left( {N_{RF}^r} \right) ^2}N_{RF}^t + {K^2}{N_t}{N_r}N_{RF}^r\) \(+ K{N_t}N_{RF}^tN_{RF}^r + {K^2}N_{RF}^tN_{RF}^r{N_s})\)

Proposed hybrid precoding

\({{\mathcal {O}}}({K^4}{\left( {N_{RF}^r} \right) ^3} + K{\left( {N_{RF}^r} \right) ^3} + {K^2}{\left( {N_{RF}^r} \right) ^2}{N_t} + {K^2}{N_t}{N_r}N_{RF}^r\) \(+ KN_t^2N_{RF}^r + {K^2}N_{RF}^r{N_t}{N_s})\)