Channel state information (CSI) at the base station (BS) is crucial to achieve beamforming and multiplexing gains in multiple-input multiple-output (MIMO) systems. State-of-the-art limited feedback schemes require feedback overhead that scales linearly with the number of BS antennas, which is prohibitive for $5$G massive MIMO. This work proposes novel limited feedback algorithms that lift this burden by exploiting the inherent sparsity in double directional (DD) MIMO channel representation using overcomplete dictionaries. These dictionaries are associated with angle of arrival (AoA) and angle of departure (AoD) that specifically account for antenna directivity patterns at both ends of the link. The proposed algorithms achieve satisfactory channel estimation accuracy using a small number of feedback bits, even when the number of transmit antennas at the BS is large -- making them ideal for $5$G massive MIMO. Judicious simulations reveal that they outperform a number of popular feedback schemes, and underscore the importance of using angle dictionaries matching the given antenna directivity patterns, as opposed to uniform dictionaries. The proposed algorithms are lightweight in terms of computation, especially on the user equipment side, making them ideal for actual deployment in $5$G systems.