A method in a wireless communication transceiver, wherein another transceiver precodes transmissions to the transceiver based at least in part on the transceiver sending channel state information to the other transceiver that includes precoder information and wherein the method is characterized by: selecting entries from one or more codebooks as a selected conversion precoder and a selected tuning precoder, or as a selected overall precoder corresponding to a selected conversion precoder and a selected tuning precoder; and transmitting indications of the selected entries as said precoder information included in the channel state information; wherein the one or more codebooks include entries comprising N T Q different conversion precoders, N T being the number of transmit antenna ports and Q being an integer value, and entries comprising a number of corresponding tuning precoders, or include entries comprising a plurality of overall precoders, with each overall precoder comprising the product of a conversion precoder arid a tuning precoder; and wherein each said conversion precoder out of said N T Q different entries comprises a block diagonal matrix in which each block comprises a DFT-based antenna-subgroup precoder that corresponds to a subgroup of N T transmit antenna ports at the transceiver and provides N T Q different DFT based beams for the corresponding subgroup, where the N T Q different conversion precoders, together with one or more of the tuning precoders, correspond to a set of N T Q different overall precoders, wherein each overall precoder represents a size- N T DFT-based beam over the N T transmit antenna ports, wherein said one or more codebooks includes conversion and tuning precoders (32, 34) or corresponding overall precoders for multiple transmission ranks, and wherein for transmission rank r>2: the tuning precoder has 2┌ r /2┐ rows and r columns, where r is the transmission rank, wherein all non-zero elements are constant modulus, every column has exactly two non-zero elements, and every row has exactly two non-zero elements, and two columns having non-zero elements in the same two rows are orthogonal to each other, and the conversion precoder has 2┌ r /2┐ columns, and if row m in a tuning precoder column in said tuning precoder has a non-zero element, so does row m + ┌ r /2┐. selecting entries from one or more codebooks as a selected conversion precoder and a selected tuning precoder, or as a selected overall precoder corresponding to a selected conversion precoder and a selected tuning precoder; and transmitting indications of the selected entries as said precoder information included in the channel state information; wherein the one or more codebooks include entries comprising N T Q different conversion precoders, N T being the number of transmit antenna ports and Q being an integer value, and entries comprising a number of corresponding tuning precoders, or include entries comprising a plurality of overall precoders, with each overall precoder comprising the product of a conversion precoder arid a tuning precoder; and wherein each said conversion precoder out of said N T Q different entries comprises a block diagonal matrix in which each block comprises a DFT-based antenna-subgroup precoder that corresponds to a subgroup of N T transmit antenna ports at the transceiver and provides N T Q different DFT based beams for the corresponding subgroup, where the N T Q different conversion precoders, together with one or more of the tuning precoders, correspond to a set of N T Q different overall precoders, wherein each overall precoder represents a size- N T DFT-based beam over the N T transmit antenna ports, wherein said one or more codebooks includes conversion and tuning precoders (32, 34) or corresponding overall precoders for multiple transmission ranks, and wherein for transmission rank r>2: the tuning precoder has 2┌ r /2┐ rows and r columns, where r is the transmission rank, wherein all non-zero elements are constant modulus, every column has exactly two non-zero elements, and every row has exactly two non-zero elements, and two columns having non-zero elements in the same two rows are orthogonal to each other, and the conversion precoder has 2┌ r /2┐ columns, and if row m in a tuning precoder column in said tuning precoder has a non-zero element, so does row m + ┌ r /2┐. the tuning precoder has 2┌ r /2┐ rows and r columns, where r is the transmission rank, wherein all non-zero elements are constant modulus, every column has exactly two non-zero elements, and every row has exactly two non-zero elements, and two columns having non-zero elements in the same two rows are orthogonal to each other, and the conversion precoder has 2┌ r /2┐ columns, and if row m in a tuning precoder column in said tuning precoder has a non-zero element, so does row m + ┌ r /2┐.