Patent Publication Number: US-10312983-B2

Title: Precoding a transmission from a one-dimensional antenna array that includes co-polarized antenna elements aligned in the array&#39;s only spatial dimension

Description:
PRIORITY 
     This nonprovisional application is a U.S. National Stage Filing under 35 U.S.C. § 371 of International Patent Application Serial No. PCT/SE2015/050935 filed Sep. 4, 2015, and entitled “Precoding a Transmission from a One-Dimensional Antenna Array that Includes Co-Polarized Antenna Elements Aligned in the Array&#39;s Only Spatial Dimension.” 
     TECHNICAL FIELD 
     The present application relates generally to transmission precoding, and relates specifically to precoding a transmission from a one-dimensional antenna array that includes co-polarized antenna elements aligned in the array&#39;s only spatial dimension. 
     BACKGROUND 
     Precoding a transmission from an antenna array involves applying a set of complex weights to the signals that are to be transmitted from the array&#39;s antenna elements, so as to independently control the signals&#39; phase and/or amplitude. This set of complex weights is referred to as a “precoder”. The transmitting node conventionally chooses the precoder to match the current channel conditions on the link to the receiving node, with the aim of maximizing the link capacity or quality. If multiple data streams are simultaneously transmitted from the array&#39;s antenna elements using spatial multiplexing, the transmitting node also typically chooses the precoder with the aim of orthogonalizing the channel and reducing inter-stream interference at the receiving node. 
     In closed-loop operation, the transmitting node selects the precoder based on channel state information (CSI) fed back from the receiving node that characterizes the current channel conditions. The transmitting node in this regard transmits a reference signal from each antenna element to the receiving node, and the receiving node sends back CSI based on measurement of those reference signals. Transmission of the reference signals and feedback of the CSI contribute significant overhead to precoding schemes. For example, these reference signals and CSI feedback consume a significant amount of transmission resources (e.g., time-frequency resource elements in Long Term Evolution, LTE, embodiments). 
     Known approaches reduce overhead attributable to reference signal transmission by dedicating a reference signal for CSI measurement. LTE Release 10, for example, introduces a CSI Reference Signal (CSI-RS) specifically designed for CSI measurement. Unlike the cell-specific common reference signal (CRS) in previous LTE release, the CSI-RS is not used for demodulation of user data and is not precoded. Because the density requirements for data demodulation are not as stringent for CSI measurement, the CSI-RS can be relatively sparse in time and frequency, thereby reducing the number of transmission resources required for transmitting the CSI-RS. 
     Known approaches reduce overhead attributable to CSI feedback by limiting the usable precoders to a fixed set of precoders, i.e., a codebook. Each precoder in the codebook is assigned a unique index that is known to both the transmitting node and the receiving node. The receiving node determines the “best” precoder from the codebook, and feeds back the index of that precoder (often referred to as a “precoding matrix indicator”, PMI) to the transmitting node as a recommendation (which the transmitting node may or may not follow). Feeding back only an index, in conjunction with other CSI such as the recommended number of data streams (i.e., transmission rank) for spatial multiplexing, reduces the number of transmission resources required for transporting that CSI. This approach therefore reduces CSI feedback overhead considerably as compared to explicitly feeding back complex valued elements of a measured effective channel. 
     Despite this, overhead from closed-loop precoding remains problematic as antenna array technology advances towards more and more antenna elements. This antenna element escalation stems not only from increases to the number of elements in the traditional one-dimensional antenna array, but also from adoption of two-dimensional antenna arrays that enable beamforming in both the vertical and horizontal spatial dimension. Furthermore, although a codebook reduces CSI overhead, the effective channel quantization inherent in the codebook has heretofore limited the codebook&#39;s ability to flexibly adapt to different propagation environments. 
     SUMMARY 
     Codebook-based precoding according to teachings herein uses a multi-granular precoder that targets the spatial dimension of a one-dimensional antenna array at different granularities. A multi-granular precoder in this regard factorizes by way of a Kronecker Product into a coarse-granularity precoder and a finer-granularity precoder. Some embodiments exploit the multi-granular nature of such precoding to reduce the overhead associated with precoding. For example, some embodiments reduce the amount of transmission resources needed for transmitting reference signals, reduce the amount of transmission resources needed for transmitting CSI feedback, and/or reduce the computational complexity required to determine the CSI to feed back. Still other embodiments additionally or alternatively adapt the precoding codebook(s) to different propagation environments. 
     In particular, embodiments herein include a method for precoding a transmission from a one-dimensional antenna array that includes co-polarized antenna elements aligned in the array&#39;s only spatial dimension. The method is performed by a transmitting radio node for precoding the transmission to a receiving radio node. The method comprises precoding the transmission from each of different subarrays of the antenna elements using a coarse-granularity precoder. This precoder is factorizable from a multi-granular precoder targeting the array&#39;s spatial dimension at different granularities, so as to virtualize the subarrays as different auxiliary elements. The method also includes precoding the transmission from the different auxiliary elements using a finer-granularity precoder that is also factorizable from the multi-granular precoder. The multi-granular precoder comprises the Kronecker Product of the coarse-granularity precoder and the finer-granularity precoder. The coarse granularity precoder and the finer-granularity precoder are represented within one or more codebooks used for said precoding. 
     In at least some embodiments, the transmission comprises user data or a reference signal dedicated to the receiving radio node. 
     Alternatively or additionally, the transmitting radio node transmits a full-elements reference signal from the antenna elements without precoding. In one embodiment, the transmitting radio node also precodes transmission of an auxiliary-elements reference signal (e.g., at a later time) from the different subarrays of the antenna elements using respective coarse-granularity precoders that are factorizable from a multi-granular precoder targeting the array&#39;s spatial dimension at different granularities, so as to virtualize the subarrays as the different auxiliary elements. Finally, the transmitting radio node transmits the precoded, auxiliary-elements reference signal to the receiving radio node. 
     In this case, the full-elements and auxiliary-elements reference signals may be common reference signals transmitted from the antenna array to multiple receiving radio nodes. 
     In any event, the transmitting radio node in some embodiments transmits the precoded, auxiliary-elements reference signal more often than transmitting the full-elements reference signal. Alternatively, the transmitting radio node interlaces the precoded, auxiliary-elements reference signal with the full-elements reference signal in time. 
     In one or more embodiments, the transmitting radio node also receives from the receiving radio node, at different times, a complete recommendation that recommends both a coarse-granularity precoder and a finer-granularity precoder, and a partial recommendation that recommends only a finer-granularity precoder. In this case, precoding uses both a coarse-granularity precoder from the complete recommendation and a finer-granularity precoder from the partial recommendation. 
     In one such embodiment, the transmitting radio node receives a partial recommendation more often than receiving a complete recommendation. 
     Alternatively or additionally, the transmitting radio node may configure the receiving radio node to restrict precoders from which the receiving radio node selects, for recommending to the transmitting radio node, to a subset of precoders in a codebook that correspond to one or more coarse-granularity precoders, by transmitting codebook subset restriction signaling to the receiving radio node indicating those one or more coarse-granularity precoders. 
     In at least some embodiments, the transmitting radio node, for each of multiple possible coarse-granularity precoders, precodes transmission of a cell association reference signal from one or more of the different subarrays using that coarse granularity precoder, as factorizable from a multi-granular precoder targeting the array&#39;s spatial dimension at different granularities, so as to virtualize the subarrays as the different auxiliary elements. The transmitting radio node further transmits the precoded, cell association reference signal from one or more of the different subarrays for cell selection by the receiving radio node. 
     Embodiments herein also include a method for receiving a transmission from a one-dimensional antenna array that includes co-polarized antenna elements aligned in the array&#39;s spatial dimension. The antenna array is associated with a transmitting radio node. The method is performed by a receiving radio node. The method comprises receiving a first reference signal transmitted from the antenna array. Based on measurement of the first reference signal, the method entails generating a first type of recommendation that recommends either: (i) a multi-granular precoder in a multi-granular codebook targeting the array&#39;s spatial dimension at different granularities, each multi-granular precoder in the codebook comprising the Kronecker Product of a coarse-granularity precoder and a finer-granularity precoder; or (ii) a coarse-granularity precoder in a coarse-granularity codebook and a finer-granularity precoder in a finer-granularity codebook, the combination of which corresponds to a multi-granular precoder targeting the array&#39;s spatial dimension at different granularities. In either case, though, the method includes transmitting the first type of recommendation to the transmitting radio node. 
     The method also entails receiving a second reference signal transmitted from the antenna array. Based on measurement of the second reference signal, the method involves generating a second type of recommendation that recommends a finer-granularity precoder factorizable from a multi-granular precoder. And the method also includes transmitting the second type of recommendation to the transmitting radio node. 
     Finally, the method comprises receiving from the antenna array a data transmission that is precoded based on the first and second types of recommendations. 
     In at least one such embodiment, the first reference signal is a full-elements reference signal transmitted from the antenna elements without precoding. The second reference signal may be an auxiliary-elements reference signal transmitted from different subarrays of the antenna elements using a coarse-granularity precoder that is factorizable from a multi-granular precoder targeting the array&#39;s spatial dimension at different granularities, so as to virtualize the subarrays as different auxiliary elements. In this case, the second type of recommendation exclusively recommends a finer-granularity precoder, without also recommending a coarse-granularity precoder. 
     In such embodiments, the method may entail receiving the precoded, auxiliary-elements reference signal more often than receiving the full-elements reference signal. Additionally or alternatively, the method may be further characterized by receiving the precoded, auxiliary-elements reference signal interlaced with the full-elements reference signal in time. 
     In some embodiments, the method is further characterized by transmitting the second type of recommendation to the transmitting radio node more often than transmitting the first type of recommendation to the transmitting radio node. 
     In any case, though, both the first and second reference signals may alternatively be full-elements reference signals transmitted from the antenna elements without precoding. In one such embodiment, the receiving radio node generates the second type of recommendation to exclusively recommend a finer-granularity precoder, without also recommending a coarse-granularity precoder. 
     Alternatively, both the first and second reference signals may be full-elements reference signals transmitted from the antenna elements without precoding. But the receiving radio node generates the second type of recommendation to recommend either (i) a multi-granular precoder in the multi-granular codebook, wherein the multi-granular precoder factors into the coarse-granularity precoder from the first recommendation or (ii) a coarse-granularity precoder in the coarse-granularity codebook and a finer-granularity precoder in a finer-granularity codebook, wherein the coarse-granularity precoder is the coarse-granularity precoder from the first type of recommendation. 
     Embodiments herein also include a method for receiving a transmission from a one-dimensional antenna array that includes co-polarized antenna elements aligned in the array&#39;s spatial dimension. The antenna array is associated with a transmitting radio node. The method is performed by a receiving radio node and is characterized by receiving codebook subset restriction signaling from the transmitting radio node that indicates one or more coarse-granularity precoders. Each coarse-granularity precoder is factorizable along with a finer-granularity precoder from a multi-granular precoder targeting the array&#39;s spatial dimension at different granularities. A multi-granular precoder comprises the Kronecker Product of a coarse-granularity precoder and a finer-granularity precoder. Based on this signaling, the method entails restricting precoders from which the receiving radio node selects for recommending to the transmitting radio node to a subset of precoders in a codebook that correspond to the one or more indicated coarse-granularity precoders. The method further includes transmitting to the transmitting radio node a recommended precoder that is selected according to the restricting, and receiving from the antenna array a data transmission that is precoded based on the recommended precoder. 
     In one or more embodiments, this method is further characterized by receiving a full-elements reference signal transmitted from the antenna elements without precoding. Based on measurement of the full-elements reference signal, the method includes selecting said recommended precoder as either: (i) a multi-granular precoder in a multi-granular codebook, from amongst a subset of multi-granular precoders in the codebook that factorize into any of the one or more coarse-granularity precoders indicated by the codebook subset restriction signaling; or (ii) a coarse-granularity precoder in a coarse-granularity codebook, from amongst the one or more coarse-granularity precoders indicated by the codebook subset restriction signaling. Regardless, the method further includes transmitting the recommendation to the transmitting radio node. 
     In any of the above embodiments, the coarse-granularity precoder and the finer-granularity precoder may be Discrete Fourier Transform, DFT, vectors, wherein the product of the DFT vectors&#39; lengths equals the number of the antenna elements aligned along the array&#39;s spatial dimension. 
     Embodiments herein further include corresponding apparatus, computer programs, and computer program products. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1A  is a block diagram of a transmitting radio node configured to precode a transmission from a one-dimensional antenna array according to one or more embodiments. 
         FIGS. 1B-1C  are block diagrams of different precoding codebooks according to one or more embodiments. 
         FIG. 1D  is a block diagram of a transmitting radio node with additional details regarding how to precode the transmission according to one or more embodiments. 
         FIGS. 2A-2E  are plots illustrating the transmit beams possible according to precoding herein. 
         FIG. 3  is a block diagram of a transmitting radio node configured to transmit reference signals according to one or more embodiments. 
         FIG. 4  is a logic flow diagram of interaction between a transmitting radio node and a receiving radio node, for reducing reference signal transmission overhead, according to one or more embodiments. 
         FIG. 5  is a logic flow diagram of interaction between a transmitting radio node and a receiving radio node herein, for reducing CSI feedback transmission overhead and/or CSI feedback computational complexity according to one or more embodiments. 
         FIG. 6  is a logic flow diagram of interaction between a transmitting radio node and a receiving radio node herein, for codebook subset restriction according to one or more embodiments. 
         FIG. 7  is a block diagram of overall precoding by the transmitting radio node according to one or more embodiments. 
         FIG. 8  is a block diagram of multi-granular precoding of a transmission from an 8-element array according to one or more embodiments. 
         FIG. 9  is a logic flow diagram of a method for precoding a transmission according to one or more embodiments. 
         FIG. 10  is a logic flow diagram of a method for receiving a transmission from an antenna array according to one or more embodiments. 
         FIG. 11  is a logic flow diagram of a method also for receiving a transmission from an antenna array according to one or more embodiments. 
         FIG. 12  is a block diagram of a transmitting radio node according to some embodiments. 
         FIG. 13  is a block diagram of a receiving radio node according to one or more embodiments. 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 1A  depicts a transmitting radio node  10 , referred to for convenience as transmitting node  10 . The transmitting node  10  (e.g., an enhanced Node B in Long Term Evolution, LTE, embodiments) performs transmissions from an associated antenna array  12 . The array  12  is a one-dimensional array in the spatial domain. The array  12  includes co-polarized antenna elements  14  aligned in the array&#39;s spatial dimension (e.g., in a horizontal dimension or a vertical dimension). As shown, for example, the array  12  includes eight co-polarized antenna elements  14  aligned in the vertical dimension. The array  12  may include other antenna elements as well, but these elements are not shown in  FIG. 1A  or considered in  FIG. 1A &#39;s description. 
     The transmitting node  10  is configured to transmit a transmission  16  from the antenna array  12  to a receiving radio node (not shown and hereinafter referred to simply as “receiving node”). The transmission  16  in some embodiments, for example, comprises user data and/or a reference signal dedicated to the receiving node (e.g., a UE-specific reference signal or a Demodulation Reference Signal in LTE embodiments). The transmitting node  10  is configured to precode this transmission  16 .  FIG. 1A  depicts the transmitting node  10  in this regard as including one or more multi-granular precoding units  18   a ,  18   b  respectively configured to perform precoding for one or more simultaneously transmitted information streams (i.e., layers)  20   a ,  20   b.    
     When more than one information stream  20   a ,  20   b  is transmitted (i.e., the transmission  16  is a multi-stream transmission), the precoded information streams  22   a ,  22   b  that are output from the multi-granular precoding units  18   a ,  18   b  and that are destined for transmission from the same antenna element  14  are combined with adder  24  and sent to the destination antenna element  14 . In at least some multi-stream embodiments, the transmitting node  50  performs the same precoding for each of the multiple streams  20   a ,  20   b . In one embodiment, though, the transmitting node  50  performs a fixed unitary rotation of the streams  20   a ,  20   b  (not shown) prior to precoding. 
     Irrespective of whether the transmission  16  is a single-stream or multi-stream transmission, the transmitting node  10  according to embodiments herein advantageously precodes the transmission  16  at multiple different levels of granularity (i.e., resolution) in the array&#39;s spatial dimension. Multi-granular precoding in this regard involves precoding the transmission  16  at a coarse level of granularity, as well as at a finer level of granularity. As described more fully below, coarse-granularity precoding forms virtual transmit beams that have a coarse granularity in the array&#39;s spatial dimension, while finer-granularity precoding forms transmit beams that have a finer granularity within the array&#39;s spatial dimension and that are bounded by the virtual transmit beams&#39; envelope. 
     The transmitting node  10  uses one or more codebooks  26 , e.g., stored in memory  28 , for performing this multi-granular precoding. As shown in  FIG. 1B , the codebook(s)  26  in some embodiments include both a coarse-granularity codebook C c  of different possible coarse-granularity precoders X c  as well as a finer-granularity codebook C f  of different possible finer-granularity precoders X f . Each coarse-granularity precoder X c  targets the array&#39;s spatial dimension at a coarse level of granularity, whereas each finer-granularity precoder X f  targets the array&#39;s spatial dimension at a finer level of granularity. As shown in  FIG. 10 , the codebook(s)  26  alternatively or additionally include a multi-granularity codebook C mg  of different possible multi-granular precoders X mg . Each multi-granular precoder X mg  targets the array&#39;s spatial dimension at different granularities. Each multi-granular precoder X mg  in this regard factorizes into a coarse-granularity precoder X c  and a finer-granularity precoder X f  Specifically, each multi-granular precoder X mg  is formed as the Kronecker Product of a coarse-granularity precoder X c  and a finer-granularity precoders X f  associated the finer level of granularity; that is, X mg =X f ⊗X c , where ⊗ represents the Kronecker Product. By way of its factorized structure, a multi-granular precoder X mg  represents a certain combination of a coarse-granularity precoder X c  and a finer-granularity precoder X f . Accordingly, separate application of precoders X c , X f  from the coarse and finer granularity codebooks C c , C f  is equivalent to application of a corresponding precoder X mg  from the multi-granular codebook C mg . 
     Regardless of the number or type of codebooks  26  employed,  FIG. 1A  shows that each of the transmitting node&#39;s multi-granular precoding units  18   a ,  18   b  is configured with different coarse-granularity precoding units  30  as well as a finer-granularity precoding unit  32  for performing multi-granular precoding for a respective one of the one or more information streams  20   a ,  20   b . The coarse-granularity precoding units  30  respectively precode the transmission  16  from each of different subarrays  34   a  and  34   b  of the antenna elements  14  using the coarse-granularity precoder X c  Referring briefly to  FIG. 1D , for example, the coarse-granularity precoder X c  is a 4×1 precoding vector applied via multipliers  36  such that the first entry X c ( 1 ) weights the transmission from the topmost antenna element  14  in each subarray  34   a ,  34   b , the second entry X c ( 2 ) weights the transmission from the second antenna element  14  in each subarray  34   a ,  34   b , the third entry X c ( 3 ) weights the transmission from the third antenna element  14  in each subarray  34   a ,  34   b , and the fourth entry X c ( 4 ) weights the transmission from the bottommost antenna element  14  in each subarray  34   a ,  34   b . Irrespective of the particular composition of these coarse-granularity precoders X c , though, precoding the transmission  16  in this way virtualizes the subarrays  34   a  and  34   b  as different respective “auxiliary” elements  38   a  and  38   b . That is, the coarse-granularity precoding virtualizes together the antenna elements  14  within subarray  34   a , and virtualizes together the antenna elements  14  within subarray  34   b , so that the antenna elements  14  effectively appear as a fewer number of auxiliary elements  38   a  and  38   b.    
     The finer-granularity precoding unit  32  precodes the transmission from these different auxiliary elements  38   a  and  38   b  using a finer-granularity precoder X f . As shown in  FIG. 1D , for example, a finer-granularity precoder X f  constituting a 2×1 precoding vector is applied via multipliers  40  such that the first entry X f ( 1 ) weights the transmission from the topmost auxiliary element  38   a , and thereby each antenna element  14  within subarray  34   a . And the second entry X f ( 2 ) weights the transmission from the bottommost auxiliary element  38   b , and thereby each antenna element  14  within subarray  34   b . In at least some embodiments, for instance, this means the auxiliary elements  38   a ,  38   b , which now will have a certain beamforming pattern formed by the virtualization X c , are virtualized by the precoder X f  so as to produce the final beamforming pattern. That is, X c  creates the shape of the auxiliary elements  38   a ,  38   b  and X f  creates a narrower beam within the beam pattern of the auxiliary elements  38   a ,  38   b.    
       FIGS. 2A-2E  help visualize the granular nature of this precoding approach by illustrating the different transmit beams realizable from exemplary codebook(s)  26 . In this example, the transmitter  10  is configured to precode the transmission from the two four-element subarrays  34   a ,  34   b  in  FIG. 1A  with a coarse-granularity precoder X c  constituting a 4×1 Discrete Fourier Transform (DFT) vector, and to precode the transmission from the two resulting auxiliary elements  38   a ,  38   b  with a finer-granularity precoder X f  constituting a 2×1 DFT precoding vector. 
       FIGS. 2A and 2B  show two different patterns of virtual transmit beams  42 A,  42 B that are coarse in granularity with respect to the zenith angle they cover. These two different patterns of virtual transmit beams  42 A,  42 B are realizable from precoding the transmission from each subarray  34   a ,  34   b  with two different possible coarse-granularity precoders X c . That is,  FIG. 2A  shows that selection of one precoder X c  from C c  (or a corresponding precoder X mg  from C mg ) forms one pattern of virtual beams  42 A, whereas  FIG. 2B  shows that selection of a different precoder X c  from C c  (or a corresponding precoder X mg  from C mg ) forms a different pattern of virtual beams  42 B. 
       FIGS. 2C and 2D  show different possible transmit beams  44 A,  44 B that are finer in granularity with respect to the zenith angle they cover (as compared to the coarse beams  42 A,  42 B). These different possible finer-granularity transmit beams  44 A,  44 B are realizable from precoding the transmission from the auxiliary elements  38   a ,  38   b  with different possible finer-granularity precoders X f . Finer-granularity transmit beams  44 A have amplitudes bounded by the envelope of the virtual transmit beams  42 A formed in  FIG. 2A , whereas finer-granularity transmit beams  44 B have amplitudes bounded by the envelope of the virtual transmit beams  42 B formed in  FIG. 2B . Accordingly, by selecting different combinations of precoders from C c  and C f , or by selecting different precoders from C mg , the transmitter  10  in some embodiments effectively controls both the transmit beam direction and the transmit beam amplitude. This contrasts with conventional precoding whereby a transmitter can only control the transmit beam direction, by selecting between different transmit beams  44 C like those shown in  FIG. 2E . 
     Some embodiments exploit multi-granular precoding herein and its associated Kronecker structure to reduce the amount of transmission resources needed for reference signal transmission and thereby reduce precoding overhead. Consider for example embodiments illustrated by  FIG. 3 . 
     As shown in  FIG. 3 , the transmitting node  10  transmits a full-elements reference signal RS fe  from the antenna elements  14  without precoding. This reference signal RS fe  is “full-elements” therefore in the sense that it may correspond to all available antenna elements  14 . In at least some embodiments, the transmitting node  10  does so by adding the full-elements reference signal RS fe  to the already precoded transmit signal via adders  46 . Specifically, the reference signal&#39;s first symbol RS fe ( 1 ) is added to the precoded transmission from the top-most antenna element  14 , the reference signal&#39;s second symbol RS fe ( 2 ) is added to the second-topmost antenna element  14 , and so on. In at least some embodiments, the full-elements reference signal RS fe  is mapped to different transmission resources (e.g., time-frequency resource elements in LTE) than the precoded transmission. 
     The transmitting node  10  also notably transmits a so-called auxiliary-elements reference signal RS ae  from the different subarrays  34   a ,  34   b  using a coarse-granularity precoder X c  that is factorizable from a multi-granular precoder X mg  targeting the array&#39;s spatial dimension at different granularities, so as to virtualize the subarrays  34   a ,  34   b  as the different auxiliary elements  38   a ,  38   b . In at least some embodiments, the transmitting node  10  does so by adding the auxiliary-elements reference signal RS ae  to the fine-grained precoded transmit signal via adders  48 . Specifically, the reference signal&#39;s first symbol RS ae ( 1 ) is added to the precoded transmission from the top auxiliary element  38   a , and the reference signal&#39;s second symbol RS ae ( 2 ) is added to the bottom auxiliary element  38   b . The transmitting node  10  transmits this precoded, auxiliary-elements reference signal RS ae  to the receiving radio node. 
     In fact, in at least some embodiments, the full-elements and auxiliary-elements reference signals are common reference signals transmitted from the antenna array  12  to multiple receiving radio nodes. In LTE embodiments, for example, the full-elements and auxiliary-elements reference signals may be channel state information reference signals (CSI-RS) or cell-specific reference signals (CRS). In this case, therefore, the auxiliary-elements reference signal RS ae  differs from conventional common reference signals in that the auxiliary-elements reference signal RS ae  is precoded, even though it is a common reference signal. 
     In any event, some embodiments herein reduce reference signal overhead by transmitting the precoded, auxiliary-elements reference signal more often than transmitting the full-elements reference signal. And the transmitting node  10  configures transmission of the auxiliary-elements reference signal based on information obtained from prior transmission of the full-elements reference signal. Specifically, the transmitting node  10  determines a coarse-granularity precoder X c  based on feedback received responsive to transmission of the full-elements reference signal, and uses that coarse-granularity precoder X c  as a fixed virtualization on the antenna array  12  for transmission of the auxiliary-elements reference signal. That is, the transmitting node  10  effectively uses the full-elements reference signal to fix the virtual, coarse transmit beams over multiple transmissions of the auxiliary-elements reference signal, and uses the auxiliary-elements reference signal (which has a lower overhead than the full-elements reference signal) to form fine-grained transmit beams. 
     For example, in one embodiment, the transmitting node  10  transmits the full-elements reference signal at time instant 1, and transmits the auxiliary-elements reference signal at time instants 2, 3, 4, and 5. In doing so, the transmitting node  10  determines a coarse-granularity precoder X c  based on feedback received from transmission of the full-elements reference signal at time instant 1, and then to use that same coarse-granularity precoder X c  for precoding the transmission of the auxiliary-elements reference signal at time instants 2, 3, 4, and 5. The transmitting node  10  repeats this transmission pattern for future time instants. Hence, by transmitting the full-elements reference signal with lower periodicity than the auxiliary-elements reference signal, the amount of transmission resources required for reference signal transmission is reduced as compared to transmitting a full-elements reference signal at time instants 1, 2, 3, 4, and 5. 
     In yet other embodiments, the transmitting node  10  interlaces the precoded, auxiliary-elements reference signal with the full-elements reference signal in time. For example, the transmitting node  10  transmits the full-elements reference signal at time instant 1 and the auxiliary-elements reference signal at time instant 2, and repeats this pattern. Again, the transmitting node  10  configures transmission of the auxiliary-elements reference signal based on information obtained from prior transmission of the full-elements reference signal. 
     According to one approach, the transmitting node  10  actually re-configures coarse-granularity precoding of the auxiliary-elements reference signal based on the coarse-granularity precoder X c  obtained from prior transmission of the full-elements reference signal. In another approach, by contrast, the transmitting node  10  configures a set of multiple different auxiliary-elements reference signals that respectively correspond to different possible coarse-granularity precoders X c . The transmitting node  10  determines a coarse-granularity precoder X c  based on feedback from transmission of the full-elements reference signal, and then dynamically allocates to the receiving node the auxiliary-elements reference signal that corresponds to that coarse-granularity precoder X c . 
     In addition to lowering reference signal overhead, embodiments herein also increase the resulting quality of the channel estimates performed by the receiving node. Indeed, in at least some embodiments, the auxiliary-elements reference signal is beamformed using the virtualization from coarse-granularity precoding, resulting in a beamforming gain, e.g., of 10 log 10 (N c ), where N c  is the number of antenna elements virtualized by coarse-granularity precoding. This will increase the signal-to-interference-plus-noise (SINR) on the transmission resources (e.g., resource elements in LTE) of the auxiliary-elements reference signal, and lead to increased channel estimation quality. This will in turn lead to less link adaptation errors and increased system performance. 
     With these possible variations in mind,  FIG. 4  illustrates interaction between the transmitting node  10  and a receiving radio node  50  for reducing reference signal overhead according to some embodiments. As shown, the transmitting node  10  transmits a full-elements reference signal (RS)  52  from the antenna elements  14  without precoding (Step  52 ). The receiving node  50  estimates channel state information (CSI) from this full-elements RS (Step  54 ). In at least some embodiments, the receiving node  50  does so based on the multi-granular codebook C mg , or a combination of the coarse-granularity codebook C c  and the finer-granularity codebook C f , so as to apply both granularity levels. The receiving node  50  then reports the estimated CSI to the transmitting node  10  (Step  56 ). The CSI may include for instance an indicator (e.g., a precoding matrix index, PMI) for a recommended multi-granular precoder X mg , or a recommended combination of a coarse-granularity precoder X c  and a finer-granularity precoder X f . The transmitting node  10  determines a coarse-granularity precoder X c  and a finer-granularity precoder X f  based on the receiving node&#39;s recommendation (Step  58 ). 
     Using this determined coarse-granularity precoder X c , the transmitting node  10  transmits an auxiliary-elements reference signal (Step  60 ). That is, the auxiliary-elements reference signal is virtualized based on the coarse granularity level of the recommended precoder. The receiving node  50  estimates CSI from this auxiliary-elements RS (Step  62 ). The receiving node  50  does so based on the finer-granularity codebook C f ; that is, the receiving node  50  applies the finer-granularity level. The receiving node  50  then reports the estimated CSI to the transmitting node  10  (Step  64 ). The CSI may include for instance an indicator (e.g., PMI) for a recommended finer-granularity precoder X f . The transmitting node  10  determines a finer-granularity precoder X f  based on the receiving node&#39;s recommendation (Step  66 ). 
     The transmitting node  10  next precodes a data transmission, e.g., as described in  FIGS. 1A-1D , using the coarse-granularity precoder X c  determined in Step  56  and the finer-granularity precoder X f  determined in Step  66 . That is, the transmitting node  10  disregards the finer-granularity precoder X f  determined from the full-elements RS, in favor of the finer-granularity precoder X f  that was more recently determined from the auxiliary-elements RS with lower overhead. Finally, the transmitting node  10  transmits this precoded data transmission to the receiving node  50  (Step  70 ). 
     The above example with reference to  FIG. 4  illustrates certain embodiments herein whereby the transmitting node  10  receives a “complete” recommendation and a “partial” recommendation from the receiving node  50  at different times. In particular, the transmitting node  10  receives a complete recommendation in Step  56  by receiving a recommendation for both a coarse-granularity and a finer-granularity precoder. This recommendation may comprise for example either (i) an indicator (e.g., PMI) for a multi-granular precoder in the multi-granular codebook; or (ii) an indicator (e.g., PMI) for a coarse-granularity precoder in the coarse-granularity codebook and an indicator (e.g., PMI) for a finer-granularity precoder in the finer-granularity codebook. No matter its form, though, the recommendation is complete in the sense that it reflects each of the different levels of granularity. By contrast, the transmitting node  10  later receives a partial recommendation in Step  64  by receiving a recommendation for only a finer-granularity precoder (i.e., the recommendation does not reflect the coarse-granularity level). 
     Armed with these recommendations, the transmitting node  10  precodes the data transmission using both a coarse-granularity precoder from the complete recommendation as well as a finer-granularity precoder from the partial recommendation. The transmitting node  10  does so by basing its ultimate precoder selection on (i.e., considering) the precoders recommended by the complete and partial recommendations. In at least some embodiments, though, the transmitting node  10  is permitted to consider, but not necessarily follow, these recommendations. 
     In at least some embodiments alluded to above, the transmitting node  10  receives a partial recommendation more often than receiving a complete recommendation. These embodiments follow from the embodiments that transmit an auxiliary-elements reference signal more often than transmitting a full-elements reference signal. 
     With this in mind, other embodiments herein alternatively or additionally exploit multi-granular precoding to reduce the amount of transmission resources needed for transmitting CSI feedback and/or reduce the computational complexity required to determine the CSI to feed back. Consider for example embodiments illustrated by  FIG. 5 . 
     As shown in  FIG. 5 , the receiving node  50  receives a first reference signal transmitted from the antenna array  12  (Step  72 ). Based on measurement of the first reference signal, the receiving node  50  generates a first type of recommendation that recommends either a multi-granular precoder X mg , or a coarse-granularity precoder X c  and a finer-granularity precoder X f  the combination (i.e., Kronecker Product) of which corresponds to a multi-granular precoder (Step  74 ). This first type of recommendation may therefore be characterized as a complete recommendation according to some embodiments. In any event, the receiving node  50  transmits this first type of recommendation to the transmitting node  10  (Step  76 ). 
     The receiving node  50  also receives a second reference signal transmitted from the antenna array  12  (Step  78 ). Based on measurement of this second reference signal, the receiving node  50  generates a second type of recommendation that recommends a finer-granularity precoder X f (Step  80 ). As explained below, this second type of recommendation may be characterized as either a complete recommendation or a partial recommendation. Irrespective of its particular form, though, the receiving node transmits this second type of recommendation to the transmitting node  10  (Step  82 ). 
     Finally, the receiving node  50  receives from the antenna array  12  a data transmission that is precoded, e.g., as described in  FIGS. 1A-1D , based on the first and second types of recommendations (Step  84 ). 
     In at least some embodiments, as suggested above, the first reference signal is a full-elements reference signal transmitted from the antenna elements  14  without precoding, and the second reference signal is an auxiliary-elements reference signal transmitted from the different subarrayas  34   a ,  34   b  of antenna elements  14  using a coarse-granularity precoder. In this case, the second type of recommendation exclusively recommends a finer-granularity precoder, without also recommending a coarse-granularity precoder; that is, the second type of recommendation is a partial recommendation. In at least some embodiments, the auxiliary-elements reference signal is dedicated to the receiving node  50  (e.g., a demodulation reference signal, DM RS, in LTE). In other embodiments, the auxiliary-elements reference signal is a common reference signal transmitted from the antenna array  12  to multiple receiving nodes (e.g., a CRS in LTE). In these latter embodiments, the receiving node  50  decodes the precoded transmission of the auxiliary-elements reference signal using the coarse-granularity precoder. Regardless, these embodiments correspond to the embodiments illustrated in  FIG. 4 , from the perspective of the receiving node  50 . 
     In embodiments such as this, the receiving node  50  may also receive the precoded, auxiliary-elements reference signal more often than receiving the full-elements reference signal. Alternatively, the receiving node  50  may receive the precoded, auxiliary-elements reference signal interlaced with the full-elements reference signal in time. 
     In any event, the receiving node  50  in one or more of these and other embodiments transmits the second type of recommendation to the transmitting node  10  more often than transmitting the first type of recommendation to the transmitting node  10 . For example, the receiving node  50  may transmit the first type of recommendation at time instant 1, and transmit the second type of recommendation at time instants 2, 3, 4, and 5, e.g., based on constraining the precoders to use the last available coarse-granularity precoder from time instant 1. Hence, only a finer-granularity precoder is derived. Regardless, where the second type of recommendation exclusively recommends the finer-granularity precoder, not the coarse-granularity precoder, this reduces the amount of transmission resources needed for transmitting CSI feedback. Accordingly, in embodiments where the second reference signal is an auxiliary-elements reference signals transmitted more often than the first reference signal as a full-elements reference signal, transmission resource overhead from both reference signal transmission and CSI feedback is reduced. 
     Other embodiments, though, reduce the amount of transmission resources needed for transmitting CSI feedback, without necessarily reducing the amount of transmission resources needed for transmitting the reference signals. In these embodiments, contrary to those illustrated in  FIG. 4 , both the first and second reference signals are full-elements reference signals transmitted from the antenna elements  14  without precoding. No overhead reduction is achieved therefore from transmission of an auxiliary-elements reference signal. But the receiving node  50  generates the second type of recommendation to exclusively recommend a finer-granularity precoder, without also recommending a coarse-granularity precoder. Accordingly, although the receiving node  50  recommends a coarse-granularity precoder based on measurement of a full-elements reference signal in Step  76 , the receiving node  50  does not recommend a coarse-granularity precoder based on measurement of the other full-elements reference signal in Step  84 . That is, the first type of recommendation in this case amounts to a complete recommendation described above, whereas the second type of recommendations amounts to a partial recommendation described above. This approach thereby saves transmission resource overhead due to CSI feedback, since no feedback needs to be sent regarding a coarse-granularity precoder in the partial recommendation at Step  84 . Instead, the transmitting node  10  will precode the transmission in Step  86  based on the most recently recommended coarse-granularity precoder, e.g., as recommended in Step  76 . 
     In at least some embodiments, this approach also reduces the computational complexity required by the receiving node  50  to determine the CSI to feed back to the transmitting node  10 . Indeed, when generating the second type of recommendation as a partial recommendation, the receiving node  50  does not need to determine which coarse-granularity precoder to recommend to the transmitting node  10 . Rather, the receiving node  50  just needs to concern itself with recommending one or more finer-granularity precoders. 
     Still other embodiments reduce the computational complexity required by the receiving node  50  to determine the CSI to feed back to the transmitting node  10 , without reducing the amount of transmission resources needed for transmitting reference signals or CSI feedback. In these embodiments, again, both the first and second reference signals are full-elements reference signals transmitted from the antenna elements  14  without precoding. No overhead reduction is achieved therefore from transmission of an auxiliary-elements reference signal. Furthermore, the receiving node  50  generates the second type of recommendation to recommend either a multi-granular precoder in the multi-granular codebook or a coarse-granularity precoder in the coarse-granularity codebook and a finer-granularity precoder in a finer-granularity codebook. That is, the receiving node  50  generates both the first and second types of recommendations as complete recommendations, meaning that these embodiments do not reduce the amount of transmission resources required for sending the CSI feedback to the transmitting node  10 . 
     However, the receiving node  50  generates the second type of recommendation in a way that requires less computational complexity than that required to generate the first type of recommendation. First, the receiving node  50  refrains from re-evaluating which coarse-granularity precoder to recommend. Instead, the receiving node  50  simply recommends the same coarse-granularity precoder from (i.e., reflected by) a first type of recommendation (generated in Step  74 ). This effectively reduces the receiving node&#39;s precoder search space and thereby advantageously reduces computational complexity. 
     Still other embodiments herein additionally or alternatively exploit multi-granular precoding in order to adapt the precoding codebook(s)  26  to different propagation environments. Consider for instance the embodiments illustrated by  FIG. 6 , which do so using codebook subset restriction. 
     As shown in  FIG. 6 , the transmitting node  10  configures the receiving node  50  to restrict precoders from which the receiving node  50  selects, for recommending to the transmitting node  10 , to a certain subset of precoders. The receiving node  50  therefore cannot recommend any other precoder. This subset to which selection is restricted includes those precoders in a codebook  26  (e.g., C c  or C mg ) that correspond to one or more indicated coarse-granularity precoders. The transmitting node  10  does this by transmitting codebook subset restriction (CSR) signaling to the receiving node  50  indicating those one or more coarse-granularity precoders to which selection is restricted (Step  90 ). 
     For example, where the receiving node  50  selects from a coarse-granularity codebook C c , the signaling indicates different coarse-granularity precoders X c  in that codebook C c  from which the receiving node  50  is permitted to select. As another example, where the receiving node  50  selects from a multi-granular codebook C mg , the signaling indicates different coarse-granularity precoders X c  into which a multi-granular precoder X mg  selected from the codebook C mg  must factorize. That is, each multi-granular precoder X mg  in the subset to which the receiving node  50  is restricted to must factorize into (i.e., correspond to) any of the coarse-granularity precoders X c  indicated by the signaling. 
     In any event, based on this signaling, the receiving node  50  restricts precoders from which the receiving node  50  selects for recommending to the transmitting node  10  to the subset of precoders in a codebook  26  that corresponds to the one or more indicated coarse-granularity precoders (Step  92 ). The receiving node  50  then transmits to the transmitting node  10  a recommended precoder selected according to that restriction (Step  94 ). Finally, the receiving node  50  receives from the antenna array  12  a data transmission that is precoded based on the recommended precoder (Step  96 ). The transmitting node  10  in this regard may consider, but not necessarily follow, the receiving node&#39;s recommendation. 
     Because a coarse-granularity precoder defines an upper mask on the potential radiated power pattern from the transmitting node  10 , restricting codebook selection at this coarse level of granularity effectively controls how much power is radiated from the transmitting node  10  in different directions. The transmitting node  10  in at least some embodiments therefore chooses the coarse-granularity precoders that restrict precoder selection, in order to dynamically control the direction and amount of power radiated by the transmitting node  10 . This is thus an efficient way to adapt the codebook(s)  26  to a certain propagation environment. Indeed, the transmitting node  10  may prohibit the receiving node  50  from selecting or recommending certain precoders that generate harmful interference in certain directions. 
     In general, therefore, the codebook(s)  26  herein do not necessarily maximize the expected SNR for a receiving node as is conventional; rather, the codebook(s)  26  include precoders that have certain properties that can be used in other ways to increase system performance. 
     Alternatively or additionally, codebook subset restriction signaling defined at a coarse level of granularity advantageously lowers the amount of transmission resources required for such signaling. For example, signaling a certain subset of multi-granular precoders to which selection shall be restricted requires fewer transmission resources when done by signaling indices for corresponding coarse-granularity precoders (rather than by signaling a greater number of indices for those multi-granular precoders themselves). 
     Regardless, note that the above codebook subset restriction signaling embodiments comport well with embodiments that employ a full-elements reference signal. In this regard, the receiving node  50  in at least some embodiments is configured to receive a full-elements reference signal transmitted from the antenna elements  14  without precoding. Based on measurement of the full-elements reference signal, the receiving node  50  in one embodiment selects the recommended precoder a multi-granular precoder X mg  in a multi-granular codebook C mg , from amongst a subset of multi-granular precoders in the codebook that factorize into any of the one or more coarse-granularity precoders indicated by the codebook subset restriction signaling. Alternatively, the receiving node  50  in another embodiment selects the recommended precoder as a coarse-granularity precoder X c  in a coarse-granularity codebook C c , from amongst the one or more coarse-granularity precoders indicated by the codebook subset restriction signaling. The receiving node  50  then transmits this recommendation to the transmitting node  10 . 
     Of course, although various figures herein illustrates multi-granular precoding with an antenna array  12  that has a certain number of antenna elements  14 , embodiments herein are equally extendable to arrays with a different number of antenna elements  14 . 
     Also, the array&#39;s spatial dimension as described herein may be any dimension in the spatial domain, whether horizontal, vertical, or otherwise. 
     Still further, the antenna array  12  herein may also include additional antenna elements that are spatially aligned with antenna elements  14  and with one another, but that are cross-polarized with elements  14 . In at least some embodiments, transmission from these cross-polarized elements proceeds in a like manner as that described above. 
     Furthermore, note that an antenna element as used herein is non-limiting in the sense that it can refer to any virtualization (e.g., linear mapping) of a transmitted signal to physical antenna elements. For example, groups of physical antenna elements may be fed the same signal, and hence share the same virtualized antenna port when observed at the receiver. Hence, the receiver cannot distinguish and measure the channel from each individual antenna element within the group of elements that are virtualized together. Accordingly, the terms “antenna element”, “antenna port” or simply “port” should be considered interchangeable herein, and may refer to either a physical element or port or a virtualized element or port. 
     Also note that the precoders herein may form all or just a part of an overall precoder applied to the transmitted signal.  FIG. 7  illustrates overall precoding according to at least some of these embodiments. 
     As shown in  FIG. 7 , a precoding unit  98  receives input data, e.g., information symbols to be transmitted, and it includes layer processing units  100  that are responsive to a rank control signal from a precoding controller  102 . Depending on the transmit rank in use, the input data is placed onto one or more spatial multiplexing layers and the corresponding symbol vector(s) s are input to a precoder  104 . 
     The information carrying symbol vector s is multiplied by an N T ×r precoder matrix W, which serves to distribute the transmit energy in a subspace of the N T  (corresponding to N T  antenna elements) dimensional vector space. The r symbols in s each correspond to a layer and r is referred to as the transmission rank. In this way, spatial multiplexing is achieved since multiple symbols can be transmitted simultaneously over the same time/frequency resource element (TFRE). The number of symbols r is typically adapted to suit the current channel properties. 
     In any case, the precoder  104  outputs precoded signals to additional processing  106  that processes the signals before providing them towards a number of antenna elements  108  associated with the antenna array  12 . In at least some embodiments, such as for OFDM-based transmission schemes like LTE, this additional processing  106  includes Inverse Fast Fourier Transform (IFFT) processing units. In other exemplary embodiments, such as those based on CDMA, the additional processing  106  involves multiplying the signals with spreading sequences. 
     One example of embodiments where the precoders herein form just part of the overall precoder W will now be described, as a concrete example. 
     In general, a factorized precoder structure may be used such that W=W 1 W 2 . In one embodiment, this overall precoder is tailored to a 2N-element 1D antenna array. The first precoder W 1  is a wideband precoder targeting long term channel characteristics and the second precoder W 2  is a frequency-selective precoder targeting short term channel characteristics/co-phasing between polarizations. A precoder matrix indicator (PMI) for each of the two precoders may be supplied by the receiving node, choosing each precoder from a limited set of available precoders (codebooks). The PMI reporting for each of the two precoders can be configured with different frequency granularity. Note that the labeling of W 1  as a wideband precoder and W 2  as a frequency-selective precoder merely describes the typical use case of the factorized precoder structure and should be considered as non-limiting. 
     The wideband precoder 
               W   1     =     [         X       0           0       X         ]           
in some embodiments has a block diagonal structure targeting a uniform 1D antenna array of N cross-polarized antennas (i.e. the number of antenna elements is 2N). With this structure, the same N×1 precoder X is applied to each of the two polarizations.
 
     In one or more embodiments, the precoder X constitutes a multi-granular precoder X mg  as described above. That is, the precoder X is constructed by means of a Kronecker Product between a coarse-granularity precoder X c  and a finer-granularity precoder X f  These precoders in this embodiment are vectors. In one embodiment, for example, the vectors are Discrete Fourier Transform (DFT) vectors. That is, the precoders are DFT-based precoders, implementing a Grid-of-Beams codebook, supplying the receiving node  50  with beams pointing in different directions. The DFT vectors may have entries such that the finer-granularity vector is described as 
                 X   f     =       [         1         e     j   ⁢           ⁢   2   ⁢   π   ⁢       1   ⁢           ⁢     l   f           N   f     ⁢     Q   f                 …         e     j   ⁢           ⁢   2   ⁢   π   ⁢         (       N   f     -   1     )     ⁢     l   f           N   f     ⁢     Q   f                   ]     T       ,       l   f     =   0     ,   …   ⁢           ,         N   f     ⁢     Q   f       -   1     ,         
where Q f  is an integer oversampling factor, controlling the number of beams available in the finer-granularity codebook and N f  corresponds to the length of the DFT vector. Similarly, the DFT vectors may have entries such that the coarse-granularity vector is described as
 
                 X   c     =       [         1         e     j   ⁢           ⁢   2   ⁢   π   ⁢       1   ⁢           ⁢     l   c           N   c     ⁢     Q   c                 …         e     j   ⁢           ⁢   2   ⁢   π   ⁢         (       N   c     -   1     )     ⁢     l   c           N   c     ⁢     Q   c                   ]     T       ,       l   c     =   0     ,   …   ⁢           ,         N   c     ⁢     Q   c       -   1     ,         
where Q c  is an integer oversampling factor, controlling the number of beams available in the coarse-granularity codebook and N c  corresponds to the length of the DFT vector. The total precoder X is created as X f ⊗X c  described herein. Regardless, the product of the DFT vectors&#39; lengths equals the number of the antenna elements aligned along the array&#39;s spatial dimension. That is, the vectors are created such that N f N c =N. This means that X will have length N, which in turn means that W 1  corresponds to 2N ports. Note that the associative property holds for Kronecker products; that is, (A⊗B)⊗C=A⊗(B⊗C), meaning that it is not necessary to specify the prioritization order of the binary Kronecker product operations.
 
     In another embodiment, the vectors are not constrained to have a DFT structure. The frequency-selective precoder W 2  may then, for example, for rank 1 be defined as 
                 W   2     =     [         1             e     j   ⁢           ⁢   ω             ]       ,         
where
 
               ω   =       2   ⁢   π   ⁢           ⁢   p     P       ,     p   =   0     ,   …   ⁢           ,       P   -     1   ⁢           ⁢   and   ⁢           ⁢   P       =   4.           
In this case, the resultant overall precoder becomes
 
             W   =         W   1     ⁢     W   2       =         [         X       0           0       X         ]     ⁡     [         1             e     j   ⁢           ⁢   ω             ]       =       [         X               e     j   ⁢           ⁢   ω       ⁢   X           ]     .               
Note that other approaches to constructing W 2  are envisioned herein as well.
 
     In a variation, the wideband precoder is instead 
                 W   1     =     [             X   ~       (       l   f     ,     l   c       )           0           0           X   ~       (       l   f     ,     l   c       )             ]       ,         
where {tilde over (X)} (l     f     ,l     c     ) =[X (l     f     ,l     c     )  . . . X (l     f     +N     b     −1,l     c     ) ], l f =0, . . . , N f Q f −1, l c =0, . . . , N c Q c −1. In this case, {tilde over (X)} (l     f     ,l     c     )  is a multi-column matrix where each column corresponds to a precoder from the previously described DFT-based Kronecker codebook.
 
     Note that the example above is merely an illustrative example of how the wideband precoder in this embodiment may be constructed; in this case, by setting the columns of {tilde over (X)} (l     f     ,l     c     )  to be precoders of the Kronecker codebook with adjacent l f -indices. This should not be considered limiting. Rather, in the general case, the wideband precoder may be constructed by setting the columns to any of the precoders of the Kronecker codebook, not just precoders with adjacent l f -indices. 
     Regardless, W 2  may then be extended to be a tall matrix consisting of selection vectors which selects one of the precoders in {tilde over (X)} (l     f     ,l     c     )  (in addition to changing the phase between polarizations in some embodiments), consistent with how W 2  is defined for the 8TX codebook in the LTE Rel. 12 standard. 
     Consider a simple example where the antenna array is a vertical antenna array of cross-polarized antenna elements. In this example, the antenna array is described by the variables M v =8 and M p =2, where M v =8 indicates that the antenna array has eight antenna elements in the vertical dimension, and M v =2 indicates that the antenna array has two antenna elements in the polarization (non-spatial) dimension. The total number of antenna elements is thus M=M v M=16. In one embodiment herein, the multi-granular precoder may be defined with N f =2, and N c =4, yielding X (l     f     ,l     c     ) =X f ⊗X c . In at least some embodiments, X c  is a precoder that is configured for the vertical dimension of a 2D antenna array, and X f  is a precoder that is configured for the horizontal dimension of a 2D antenna array, but those precoders are nonetheless applied as described above to a 1D antenna array. Embodiments such as this may therefore be said to apply a higher-dimensioned Kronecker-structured codebook to a lower-dimensioned antenna array in order to exploit that reduction in dimensionality to subgroup antenna elements together in the spatial domain. In other words, the codebook is over-dimensioned in the spatial domain relative to the spatial dimension of the antenna array. In any event, for this example, X f  will have length 2 and X c  will have length 4 meaning that there will be in total 16 elements (8 elements per polarization). The codebook may be applied such that X (l     f     ,l     c     ) =X f ⊗X c , is connected to the 1D antenna array, with X( 1 ) corresponding to the topmost antenna element of the array, X( 2 ) corresponding to the second top most antenna elements, and so on, as shown in  FIG. 8 . In this way, the virtualization X c  is applied to subarrays in the 1D antenna array and a CSI-RS is used on the resulting antenna elements as illustrated with X 1 ( 1 ) and X 1 ( 2 ). Only one polarization is illustrated in  FIG. 8 , but the same would be done also in the other polarization. 
     In another embodiment, codebook subset restriction is used on a full-elements (16 elements) CSI-RS in order to reduce the number of possible X c  precoders. This may be beneficial in order to reduce the number of needed auxiliary-elements (4 elements) CSI-RSs in the previous embodiments. In another embodiment, Q c &lt;Q f  in order to create a relatively low number of possible X c  vectors. Hence, this will require a relatively low number of 4 ports CSI-RS. X f  will on the other hand have a finer granularity. 
     In another embodiment, the CSI-RS is defined such that the receiving node  50  dynamically switches between measuring on the described 16 elements CSI-RS, assuming the codebook X (l     f     ,l     c     ) =X f ⊗X c , and a 4 port CSI-RS corresponding with an associated codebook X (f) =X f . Hence the PMI reporting for the different precoders X c , X f  are configured with different time resolutions. 
     The codebook(s)  26  herein in at least some embodiments are parameterizable to (at least) tailor the codebook(s)  26  for different antenna array configurations of the transmitting node  10 . In one embodiment, for example, the one or more parameterized codebooks  26  define sets of different possible coarse-granularity precoders and finer-granularity precoders. The parameters defining the codebook(s)  26  may be signaled from the transmitting node  10  to the receiving node  50 . These parameters may be signaled from the transmitting node  10  to the receiving node  50  in the form of a length of the precoders. For example, the previously described codebook may be signaled by signaling the values of the DFT vector lengths, i.e., the parameters (Nf,Nc) In another embodiment also the corresponding oversampling factors (Q f ,Q c ) are signaled. 
     Regardless, the parameters of the parametrizable codebook(s)  26  are signaled to the receiving node  50 . The signaling may be conducted by e.g. Radio Resource Control (RRC), MAC header element or dynamically using physical downlink control channels. The receiving node  50  knows the general structure of the codebook(s)  26  that applies for the signaled parameters. Based on that and based on the signaled parameters, the receiving node  50  can determine the constituent precoders in the actual precoder codebook(s)  26 . 
     Embodiments herein are also applicable for precoding transmission of a cell association reference signal (e.g., a Discovery Reference Signal, DRS) from the array  12 . In one embodiment, the transmitting radio node  10 , for each of multiple possible coarse-granularity precoders, precodes transmission of a cell association reference signal from one or more of the different subarrays  34   a ,  34   b  using that coarse-granularity precoder, as factorizable from a multi-granular precoder targeting the array&#39;s spatial dimension at different granularities, so as to virtualize the subarrays  34   a ,  34   b  as the different auxiliary elements  38   a ,  38   b . The transmitting radio node  10  then transmits the precoded, cell association reference signals for cell selection by the receiving radio node  50 . 
     In some embodiments, the transmitting radio node  10  controls one of multiple cells available to serve the receiving radio node  50 . Other transmitting radio nodes also transmit cell association reference signals. The receiving radio node  50  compares the received power on the cell associated reference signals transmitted from different transmitting radio nodes. Based on this comparison, the receiving radio node  50  is allocated to a serving transmitting radio node or serving cell. 
     Transmitting a set of beamformed reference signals used for cell association in this way advantageously lowers downlink reference signal overhead for cell search in some embodiments. Indeed, by transmitting the cell association reference signals as described, the beams of the cell association reference signals only coarsely correspond to the set of beams of the precoders in the codebook(s)  26  used by the transmitting radio node  10 . As depicted in the example of  FIGS. 2A-2E , the set of beams in the codebook(s)  26  which have the same coarse-granularity precoder will lie within (a scaled version of) the envelope of the beam of the auxiliary element  38 , i.e., the beam pattern resulting from combining groups of the antenna elements  14  with the coarse-granularity virtualization X c . This means that setting the cell association reference signals to the beam patterns of the auxiliary elements  38   a ,  38   b , for each of the possible coarse-granularity precoders X c , causes the cell association reference signals&#39; received power to correspond to the received power the receiving radio node  50  would get with a beam from the codebook(s)  26  used by the transmitting radio node  10 . This way, the receiving radio node  50  is able to leverage the coarsely precoded cell association reference signals to choose the serving cell that is able to provide the receiving radio node  50  with the best possible beam for data transmission. Yet the overhead in transmitting this relatively smaller number of coarsely precoded cell association reference signals is lower than if every beam in the codebook(s) would have been sent as a cell association reference signal. 
     Note that although terminology from 3GPP LTE has been used in this disclosure to exemplify embodiments herein, this should not be seen as limiting the scope of the embodiments to only the aforementioned system. Other wireless systems, including WCDMA, WiMax, UMB and GSM, may also benefit from exploiting embodiments herein. 
     Note that the transmitting node  10  and receiving node  50  herein may correspond to any pair of nodes configured to transmit radio signals and otherwise interact in the way described. In one embodiment, though, the transmitting node  10  comprises a base station (e.g., an eNodeB in LTE) or a relay node, whereas the receiving node comprises a wireless communication device (e.g., a UE in LTE). 
     Terminology such as eNodeB and UE should be considering non-limiting and does in particular not imply a certain hierarchical relation between the two; in general “eNodeB” could be considered as device  1  and “UE” device  2 , and these two devices communicate with each other over some radio channel. Herein, we also focus on wireless transmissions in the downlink, but embodiments herein are equally applicable in the uplink. 
     In some embodiments a non-limiting term UE is used. The UE herein can be any type of wireless device capable of communicating with a network node or another UE over radio signals. The UE may also be a radio communication device, target device, device to device (D2D) UE, machine type UE or UE capable of machine to machine communication (M2M), a sensor equipped with UE, iPAD, Tablet, mobile terminals, smart phone, laptop embedded equipped (LEE), laptop mounted equipment (LME), USB dongles, Customer Premises Equipment (CPE) etc. 
     Also in some embodiments generic terminology, “radio network node” or simply “network node (NW node)”, is used. It can be any kind of network node which may comprise of base station, radio base station, base transceiver station, base station controller, network controller, evolved Node B (eNB), Node B, Multi-cell/multicast Coordination Entity (MCE), relay node, access point, radio access point, Remote Radio Unit (RRU) Remote Radio Head (RRH), or even core network node, etc. 
     In view of the above modifications and variations, those skilled in the art will appreciate that a transmitting radio node  10  herein generally performs the method  110  shown in  FIG. 9  for precoding a transmission from a one-dimensional antenna array  12  that includes co-polarized antenna elements  14  aligned in the array&#39;s only spatial dimension. The method  110  includes precoding the transmission from each of different subarrays  34   a ,  34   b  of the antenna elements  14  using a coarse-granularity precoder that is factorizable from a multi-granular precoder targeting the array&#39;s spatial dimension at different granularities, so as to virtualize the subarrays  34   a ,  34   b  as different auxiliary elements  38   a ,  38   b  (Block  112 ). The method  110  also comprises precoding the transmission from the different auxiliary elements  38   a ,  38   b  using a finer-granularity precoder that is also factorizable from the multi-granular precoder (Block  114 ). The multi-granular precoder comprises the Kronecker Product of the coarse-granularity precoder and the finer-granularity precoder. The coarse granularity precoder and the finer-granularity precoder are represented within one or more codebooks used for the precoding. 
     Those skilled in the art will appreciate that a receiving radio node  50  herein generally performs the method  116  shown in  FIG. 10  for receiving a transmission from a one-dimensional antenna array  12  that includes co-polarized antenna elements  14  aligned in the array&#39;s only spatial dimension, wherein the antenna array  12  is associated with a transmitting radio node  10 . The method  116  comprises receiving a first reference signal transmitted from the antenna array  12  (Block  118 ). The method  116  also comprises, based on measurement of the first reference signal, generating a first type of recommendation (Block  120 ). This first type of recommendation recommends either (i) a multi-granular precoder in a multi-granular codebook targeting the array&#39;s spatial dimension at different granularities, each multi-granular precoder in the codebook comprising the Kronecker Product of a coarse-granularity precoder and a finer-granularity precoder; or (ii) a coarse-granularity precoder in a coarse-granularity codebook and a finer-granularity precoder in a finer-granularity codebook, the combination of which corresponds to a multi-granular precoder targeting the array&#39;s spatial dimension at different granularities. The method  116  then includes transmitting the first type of recommendation to the transmitting radio node (Block  122 ). The method  116  also entails receiving a second reference signal transmitted from the antenna array  12  (Block  124 ). The method involves, based on measurement of the second reference signal, generating a second type of recommendation that recommends a finer-granularity precoder factorizable from a multi-granular precoder (Block  126 ). The method  116  then comprises transmitting the second type of recommendation to the transmitting radio node  10  (Block  128 ). Finally, the method includes receiving from the antenna array a data transmission that is precoded based on the first and second types of recommendations (Block  130 ). 
     Embodiments herein also include a method  132  for receiving a transmission from a one-dimensional antenna array  12  that includes co-polarized antenna elements  14  aligned in the array&#39;s only spatial dimension, as shown in  FIG. 11 . The method  132  is performed by a receiving radio node  50 . The method  132  includes receiving codebook subset restriction signaling from the transmitting radio node that indicates one or more coarse-granularity precoders, each coarse-granularity precoder factorizable along with a finer-granularity precoder from a multi-granular precoder targeting the array&#39;s spatial dimension at different granularities (Block  134 ). A multi-granular precoder comprises the Kronecker Product of a coarse-granularity precoder and a finer-granularity precoder. Based on this signaling, the method  132  includes restricting precoders from which the receiving radio node  50  selects for recommending to the transmitting radio node  10  to a subset of precoders in a codebook  26  that correspond to the one or more indicated coarse-granularity precoders (Block  136 ). The method  132  also entails transmitting to the transmitting radio node  10  a recommended precoder that is selected according to the restricting (Block  138 ). Finally, the method  132  includes receiving from the antenna array  12  a data transmission that is precoded based on the recommended precoder (Block  140 ). 
       FIG. 12  illustrates an example transmitting radio node  10  (e.g., a base station) configured according to one or more embodiments herein. The transmitting radio node  10  comprises one or more communication interfaces  142  for communicating with the receiving radio node  50  via the antenna array  12 . The one or more communication interfaces may also interface with other nodes in a wireless communication network. For communicating with the receiving radio node  50 , though, the interface(s)  142  may include transceiver circuits that, for example, comprise transmitter circuits and receiver circuits that operate according to LTE or other known standards. The transmitting radio node  10  also comprises processing circuits  144 , which may comprise one or more processors, hardware circuits, firmware, or a combination thereof. Memory  146  may comprise one or more volatile and/or non-volatile memory devices. Program code for controlling operation of the transmitting node  10  is stored in a non-volatile memory, such as a read-only memory or flash memory. Temporary data generated during operation may be stored in random access memory. The program code stored in memory, when executed by the processing circuit(s), causes the processing circuit(s) to perform the methods shown above. 
       FIG. 12  illustrates the main functional components of the processing circuit(s)  144  according to one exemplary embodiment. The functional components include a coarse-granularity precoding unit  148  and a finer-granularity precoding unit  150 , e.g., as depicted in  FIGS. 1A-1D . In one embodiment, these units each comprise a programmable circuit that is configured by program code stored in memory to perform their respective functions. In other embodiments, one or more of the functional components may be implemented, in whole or in part, by hardware circuits. Regardless, the units are collectively configured to perform the method in  FIG. 9 . 
     Also in view of the above modifications and variations, those skilled in the art will appreciate that  FIG. 11  illustrates an example receiving radio node  50  configured according to one or more embodiments herein. The receiving radio node  50  comprises one or more communication interfaces  152  for communicating with the transmitting radio node  10  one or more antennas. The interface(s)  152  may include transceiver circuits that, for example, comprise transmitter circuits and receiver circuits that operate according to LTE or other known standards. The receiving radio node  50  also comprises processing circuits  154 , which may comprise one or more processors, hardware circuits, firmware, or a combination thereof. Memory  156  may comprise one or more volatile and/or non-volatile memory devices. Program code for controlling operation of the receiving radio node  50  is stored in a non-volatile memory, such as a read-only memory or flash memory. Temporary data generated during operation may be stored in random access memory. The program code stored in memory, when executed by the processing circuit(s), causes the processing circuit(s) to perform the methods shown above. 
       FIG. 13  illustrates the main functional components of the processing circuit(s)  154  according to different embodiments. In one exemplary embodiment, the functional components include a receiving unit  158 , a recommendation unit  160 , and a transmitting unit  162 . In one embodiment, these units each comprise a programmable circuit that is configured by program code stored in memory to perform their respective functions. In other embodiments, one or more of the functional components may be implemented, in whole or in part, by hardware circuits. Regardless, the units are collectively configured to perform the method in  FIG. 10 . In another embodiment, by contrast, the functional components include a receiving unit  164 , a codebook subset restricting unit  166 , and a transmitting unit  168 . Again, in one embodiment, these units each comprise a programmable circuit that is configured by program code stored in memory to perform their respective functions. In other embodiments, one or more of the functional components may be implemented, in whole or in part, by hardware circuits. Regardless, the units are collectively configured to perform the method in  FIG. 11 . 
     Embodiments herein also include a computer program comprising instructions which, when executed by at least one processor of a radio node  10 ,  15 , causes the radio node to carry out any of the methods herein. In one or more embodiments, a carrier containing the computer program is one of an electronic signal, optical signal, radio signal, or computer readable storage medium. 
     The present invention may, of course, be carried out in other ways than those specifically set forth herein without departing from essential characteristics of the invention. The present embodiments are to be considered in all respects as illustrative and not restrictive, and all changes coming within the meaning and equivalency range of the appended claims are intended to be embraced therein.