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 comprising a precoder recommendation from the transceiver to the transceiver 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 that is the product of 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; and wherein there are 2 N T conversion precoders, each one comprising a block diagonal matrix having two or more columns with blocks on the matrix diagonal, each such block comprising a DFT-based antenna-subgroup precoder that corresponds to a subgroup of N T /2 of the N T transmit antenna ports at the transceiver and provides 2 N T different DFT based beams for the corresponding subgroup, and where all of the 2 N T different conversion precoders are reusable, together with one or more of the tuning precoders, for forming a set of 2 N T 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 that is the product of 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; and wherein there are 2 N T conversion precoders, each one comprising a block diagonal matrix having two or more columns with blocks on the matrix diagonal, each such block comprising a DFT-based antenna-subgroup precoder that corresponds to a subgroup of N T /2 of the N T transmit antenna ports at the transceiver and provides 2 N T different DFT based beams for the corresponding subgroup, and where all of the 2 N T different conversion precoders are reusable, together with one or more of the tuning precoders, for forming a set of 2 N T different overall precoders, wherein each overall precoder represents a size- N T DFT-based beam over the N T transmit antenna ports, wherein there are 2 N T conversion precoders, each one comprising a block diagonal matrix having two or more columns with blocks on the matrix diagonal, each such block comprising a DFT-based antenna-subgroup precoder that corresponds to a subgroup of N T /2 of the N T transmit antenna ports at the transceiver and provides 2 N T different DFT based beams for the corresponding subgroup, and where all of the 2 N T different conversion precoders are reusable, together with one or more of the tuning precoders, for forming a set of 2 N T 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┐.