I Introduction
Over the last few years, there has been a growing interest in leveraging the opening up of the spectrum in the millimeter wave band ( GHz) in realizing the emerging higher data rate demands of cellular systems [1, 2, 3, 4]. Communications in the millimeter wave band suffers from increased path loss exponents, higher shadow fading, blockage and penetration losses, etc., than sub GHz systems leading to a poorer link margin than legacy systems [5, 6, 7, 8, 9, 10]. However, by restricting attention to small cell coverage and by reaping the increased array gains from the use of large antenna arrays at both the basestation and user ends, significant rate improvements can be realized in practice.
Millimeter wave propagation is spatially sparse with few dominant clusters in the channel relative to the number of antennas [5, 6, 11, 12]. Spatial sparsity of the channel along with the use of large antenna arrays motivates a subset of physical layer beamforming schemes based on directional transmissions for signaling. In this context, there have been a number of studies on the design and performance analysis of directional beamforming/precoding structures for singleuser multiinput multioutput (MIMO) systems [13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. These works [16, 17, 18, 19] show that directional schemes are not only good from an implementation standpoint, but are also robust to phase changes across clusters and allow a smooth tradeoff between peak beamforming gain and initial user discovery latency. There has also been progress in generalizing such directional constructions for multiuser MIMO transmissions [22, 23, 24, 25].
In this context, while legacy systems use as many radio frequency (RF) chains^{1}^{1}1An RF chain includes (but is not limited to) analogtodigital converters (ADCs), digitaltoanalog converters (DACs), mixers, lownoise and power amplifiers (PAs), etc. as the number of antennas, their higher cost, energy consumption, area and weight at millimeter wave carrier frequencies has resulted in the popularity of hybrid beamforming systems [26, 27, 28, 29]. A hybrid beamforming system uses a smaller number of RF chains than the number of antennas, with the one extreme case of a single RF chain being called the analog/RF beamforming system and the other extreme of as many RF chains as the number of antennas being called the digital beamforming system. Spatial sparsity of millimeter wave channels ensures that having as many RF chains as the number of dominant clusters in the channel is sufficient to reap the full array gain possible over these channels.
A number of recent works have addressed hybrid beamforming for millimeter wave systems. The problem of finding the optimal precoder and combiner with a hybrid architecture is posed as a sparse reconstruction problem in [17], leading to algorithms and solutions based on basis pursuit methods. While the solutions achieve good performance in certain cases, to address the performance gap between the solution proposed in [17] and the unconstrained beamformer structure, an iterative scheme is proposed in [30, 31] relying on a hierarchical training codebook for adaptive estimation of millimeter wave channels. The authors in [30, 31] show that a few iterations of the scheme are sufficient to achieve nearoptimal performance. In [32]
, it is established that a hybrid architecture can approach the performance of a digital architecture as long as the number of RF chains is twice that of the datastreams. A heuristic algorithm with good performance is developed when this condition is not satisfied. A number of other works such as
[33, 34, 35, 36] have also explored iterative/algorithmic solutions for hybrid beamforming.A common theme that underlies most of these works is the assumption of phaseonly control in the RF/analog domain for the hybrid beamforming architecture. This assumption makes sense at the user end with a smaller number of antennas (relative to the basestation end), where operating the PAs below their peak rating across RF chains can lead to a substantially poor uplink performance. On the other hand, amplitude control (denoted as amplitude tapering in the antenna theory literature) is necessary at the basestation end with a large number of antennas for sidelobe management and mitigating outofband emissions. Further, given that the basestation is a network resource, simultaneous amplitude and phase control of the individual antennas across RF chains is feasible at millimeter wave basestations at a lowcomplexity^{2}^{2}2Any calibration complexity can be seen as a onetime effort at the unit level for a large array and defrayed as a low network cost. and cost [37, pp. 285289], [38, 39]. In particular, the millimeter wave experimental prototype demonstrated in [40] allows simultaneous amplitude and phase control. Thus, it is important to consider a hybrid architecture with these constraints. Further, given the directional nature of the channel, a solution should both inherit a directional structure and provide an intuitive description of the beam weights. For example, a black boxtype algorithmic solution that does not provide an intuitive description of the beam weights is less preferable over a solution that is constructed out of measurement reports obtained over an initial beam alignment phase with a directional structure for the sounding beams.
Main Contributions: With this backdrop, this work addresses these two fundamental issues in hybrid beamformer design. It is assumed that the basestation trains all the users in the cell with a cellspecific codebook of beamforming vectors over an initial beam alignment phase. Each user makes an estimate^{3}^{3}3In a practical implementation such as the Third Generation Partnership Project New Radio (3GPP 5GNR) design, is typically assumed both in terms of measurements and reporting [41]. The received is estimated as the received power of a beamformed link (corresponding to the beam pair under consideration) using a certain reference symbol resource. This metric is typically known as the reference symbol received power (RSRP) of the link. of the top (where ) beams over this phase and reports the beam indices to be used by the basestation as well as the measured/received signaltonoise ratios (s). The simplest implementation at the basestation uses only the best beam information for beam steering or zeroforcing as in [23, 24], with other beams serving as fall back options.
In contrast to this approach, we propose to reconstruct or estimate a rank approximation of the channel matrix between the basestation and the user (at the basestation end). To realize this reconstruction, we envision the additional feedback of the phase of the received signal estimate of the top beams over the beam alignment phase and the crosscorrelation information of the top beams at the user end with the beam used for multiuser reception. With this novel construction, the basestation can remain agnostic of the user’s top beams in precoder design. In terms of overhead, in 3GPP 5GNR, these quantities can be fed back over the physical uplink control channel (PUCCH) with a TypeII feedback scheme [41, Sec. 8.2.1.6.3, pp. 2426]; see Sec. VC for a detailed study that demonstrates this feedback overhead to be marginal. Leveraging the rank channel approximation, we propose the use of a zeroforcing structure that is then quantized to meet the RF precoding constraints (amplitude and phase control) at the basestation end for simultaneous transmissions.
To benchmark and compare the performance of the proposed scheme, we establish two upper bounds for the sum rate. This is a fundamentally difficult problem given the nonconvex dependence of the sum rate on the beamforming vectors [42, 43, 44]. The first bound is based on an intuitive parsing and understanding of the zeroforcing structure. The second bound is based on an alternating optimization of the beamformercombiner pair with signaltoleakage and noise ratio () [45] and signaltointerference and noise ratio () as optimization metrics. Numerical studies show that the proposed scheme performs significantly better than a naïve beam steering solution even for an initial beam alignment codebook of poor resolution. Further, the proposed scheme is comparable with the established upper bounds provided the beam alignment codebook resolution is moderatetogood. Thus, our work establishes the utility and efficacy of the proposed feedback techniques as well as opens up avenues for further investigation of such approaches in hybrid beamforming with millimeter wave systems.
Organization: This paper is organized as follows. Sec. II develops the system setup and explains the RF precoder architectural constraints adopted in this work. In Sec. III, we provide a background of the initial beam alignment phase and the feedback mechanism necessary for the multiuser beamforming envisioned in this work. Sec. IV generates two upper bounds on the sum rate to benchmark the performance of the proposed scheme. Sec. V performs a number of numerical studies to understand the performance of the proposed scheme relative to a naïve beam steering solution as well as to the upper bounds developed in Sec. IV. Concluding remarks are provided in Sec. VI.
Notations: Lower and uppercase bold symbols are used to denote vectors and matrices, respectively. The th entry of a vector and the th entry of a matrix are denoted by and , respectively. The regular matrix transpose and complex conjugate Hermitian transpose operations of a matrix are denoted by and , respectively. The twonorm of a vector is denoted as with , and
standing for the set of reals, complex numbers and the complex normal random variable, respectively.
Ii System Setup
We consider a cellular downlink scenario with a single basestation serving potential users. The basestation and each user are assumed to be equipped with planar arrays of dimensions antennas and antennas, respectively. At both ends, the interantenna element spacing is where is the wavelength of propagation. With and , the basestation and each user are assumed to have and RF chains, respectively.
For the channel between the basestation and the th user (where ), we assume an extended geometric propagation model over clusters/paths [46, 6]
(1) 
In (1), , and denote the complex gain, the array steering vector at the user end corresponding to the angle of arrival (AoA) in azimuth/zenith, and the array steering vector at the basestation corresponding to the angle of departure (AoD) in azimuth/zenith, respectively. The cluster gains are assumed to be independent and identically distributed (i.i.d.) standard complex Gaussian random variables: . The normalization of the channel ensures that .
In terms of the system model, we focus on the narrowband aspects and assume that the basestation serves users simultaneously with data along RF chains. The basestation precodes datastreams for the th user with the symbol vector using the digital/baseband precoder which is then upconverted to the carrier frequency by the use of the RF precoder . This results in the following system equation at the th user
(2) 
where is the preprecoding and is the white Gaussian noise vector added at the th user. We assume that are i.i.d. complex Gaussian random vectors with and .
At the th user, we assume that is processed (downconverted) with an userspecific RF combiner followed by a userspecific digital combiner to produce an estimate of as follows
(3)  
(4) 
The achievable rate (in nats/s/Hz) at the th user when treating multiuser interference as noise is given as
(5) 
where denotes the interference and noise covariance matrix
(6) 
The traditional use of finiterate feedback has been to convey the index of a precoder matrix from an appropriatelydesigned codebook of precoders to assist with adaptive transmissions to improve [47, 48]. More generally, feedback from users can also be used to aid in scheduling, channel estimation and advanced/noncodebook based precoder design. In this work, as we will see later in Sec. III, we assume that each user feeds back its top beam indices, an estimate of the received and signal phase, and crosscorrelation of the top receive beams to assist with the design of a noncodebook based multiuser precoder structure. In terms of precoder constraints, we make the assumption that .
For the RF precoder, we assume that the amplitude and phase of each entry in are controlled by a finite precision gain controller and phase shifter, respectively. In other words, the amplitude and phase come from a set of and quantization levels
(7) 
where . Prior work on hybrid beamforming such as [17, 30, 31, 32] etc., assume that the RF precoder can only be controlled by a phase shifter. However, such constraining assumptions are not reflective of practical implementations [38, 39, 40], where an independent gain controller can be used in every RF chain for every antenna. With these structural constraints on the precoder, the transmit power constraint is captured by
(8) 
We are interested in the design of RF and digital precoders with the sum rate, , being the metric to maximize. In general, we only need the constraints and . However, the considered sum rate optimization with such an assumption is quite complicated. To overcome this complexity, we consider a simple usecase in this work.
Iii MultiUser Beamformer Design
We are interested in the practicallymotivated setting where each user is equipped with only one RF chain and the basestation transmits one datastream to each user that is simultaneously scheduled. In this scenario, (for all ) and . The system decoding model in (2) and (4) reduce to
(9)  
(10) 
where and , and the second equation follows assuming^{4}^{4}4A simple realization of the hybrid precoding architecture is achieved by setting and the desired for the th user is set as the th column of . The desired is such that and meets the quantization constraints in (7). In a practical implementation, could be primarily used for subband precoding and in the narrowband context of this work, would reflect such an implementationdriven model. and . The power constraint is equivalent to and reduces to
(11) 
The focus of this section is to first develop an advanced feedback mechanism and a systematic design of the multiuser beamforming structure based on a directional representation of the channel. This structure allows the basestation to combat multiuser interference in simultaneous transmissions.
Iiia Initial Beam Alignment
Enabling multiuser transmissions in practice is critically dependent on an initial beam acquisition process (commonly known as the beam alignment phase). In a practical implementation such as 3GPP 5GNR, beam alignment corresponds to a beam sweep over a block of secondary synchronization (SS) signals transmitted over multiple ports/RF chains. The use of multiple directional beams over multiple ports results in a composite beam pattern at the basestation end (as seen from the user side). The composite pattern can lead to uncertainty in the direction of the strongest path between the basestation and the user. This directional ambiguity is subsequently resolved with a beam refinement over the individual constituent beams that make the composite beam on separate resource elements. Beam refinement allows identification and ambiguity resolution of the constituent beams.
Such a “post directional ambiguity resolved” beam alignment process is modeled by assuming that the basestation is equipped with an element codebook
(12) 
and the th user is equipped with an element userspecific codebook
(13) 
A typical design methodology for is a hierarchical design with different sets of beams that tradeoff peak array gain at the cost of initial beam acquisition latency. For example, at least from the 3GPP 5GNR perspective, the designs of and are intended to be implementationspecific at the basestation and user ends, respectively. Nevertheless, overarching design guidelines for beam broadening are provided in [14, 19, 49, 50]. In particular, a broadened beam can be generated by an optimal cophasing of a number of array steering vectors in appropriately chosen directions. Both the number of such vectors as well as their steering directions can be optimized to produce a broadened beam. It must also be pointed out that most of the beam broadening works have some variations in terms of design principles and these variations themselves do not affect the flavor of results reported in this paper.
In the beam alignment phase, the top beam indices at the basestation and each user that maximize an estimate of the received are learned. In particular, the received corresponding to the th beam index pair at the th user is given as
(14) 
Let the beam pair indices at the th user be arranged in nonincreasing order of the received and let the top beam pair indices be denoted as
(15) 
With the simplified notation of
(16) 
we have . With the initial beam alignment methodology as described above, we now leverage the top beam information learned at the th user to estimate the channel matrix and to design at the basestation end.
IiiB Channel Reconstruction and Beamformer Design
A typical use of the feedback information at the basestation is to select the top/best beam indices for all the users and to leverage this information to construct a multiuser transmission scheme. Such an approach is adopted in [24], where multiuser beam designs leveraging only the top beam pair index, , and intended to serve different objectives are proposed: i) greedily (from each user’s perspective) steering a beam to the best direction for that user (called the beam steering scheme), ii) using the information collated from different users to combat interference to other simultaneously scheduled users via a zeroforcing solution (called the zeroforcing scheme), and iii) for leveraging both the beam steering and interference management objectives via a generalized eigenvector optimization (called the generalized eigenvector scheme). If the beam pair is blocked or fades, the th user requests the basestation to switch to the beam index and it switches to the beam with index (and so on) [10].
In this work, we propose to generalize the structures in [24] by leveraging all the top beam pair indices fed back from each user. In this direction, the basestation intends to reconstruct or estimate a rank approximation of (a scaled version of) the channel matrix corresponding to the th user as follows
(17) 
where and are defined as estimates of the array steering vectors and , respectively. Given the channel model structure in (1), (17) is simplified by estimating and by and , respectively, where
(18) 
for some choice of . In the above description, denotes an appropriatelydefined bit quantization operation^{5}^{5}5A bit quantization operation is precisely specified if disjoint intervals that exactly and entirely span the range of the quantity and a representative/quantized value from each interval are specified. of the quantity under consideration. However, estimating as in (17) is not complete until we have an estimate for and . The quantity can be estimated by the user with the same reference symbol resource (or pilot symbol) transmitted during the beam training phase with no additional training overhead. Therefore, we define as the bit quantization of the phase of an estimate of the pilot symbol
(19) 
for some choice of . The noise term captures the additive noise in the initial beam alignment process corresponding to the top beam pairs.
For , we note that the basestation not only needs the beam indices that are useful for the user side, but also the useful part of the user’s codebook () since the basestation is typically unaware of it. To avoid this unnecessary complexity and feedback given the proprietary structure of , we assume that the th user uses a multiuser reception beam . In the simplest manifestation, could be the best training beam learned in the beam alignment phase, . However, a more sophisticated choice for is not precluded. For example, an iterative choice that maximizes the (instead of the ) could be considered for .
We then note that the estimated , defined as,
(20) 
is only dependent on in the form of . Building on this fact, each user generates , defined as,
(21) 
It then quantizes the amplitude and phase of for some choice of and and feeds them back
(22) 
For both and , without loss in generality, relative phases with respect to and (that is, and ) can be reported.
The mappings between the quantities of interest and the approximated quantities as well as the feedback overhead needed from each user to implement the proposed scheme are described in Table I. While the feedback overhead increases linearly with (the rank of the channel approximation), there are diminishing returns in terms of channel representation accuracy since the clusters captured in are subdominant as increases (and are eventually limited by ). Thus, it is useful to select to tradeoff these two conflicting objectives.
Quantity of Interest  Approximated Quantity  Feedback Overhead 

Array steering vector at basestation end ()  Basestation beam indices ()  
Gain of cluster coefficient ()  Received in beam alignment ()  
Phase of cluster coefficient ()  Estimated phase in beam alignment ()  
Array steering vector at user end ()  Amplitude of codebook correlation ()  
Phase of codebook correlation () 
Following the above discussion, the th user feeds back the matrix , defined as
(23) 
and the basestation approximates as follows
(24) 
In other words, is represented as a linear combination of the top beams as estimated from in the initial beam alignment phase. The weights in this linear combination correspond to the relative strengths of the clusters as distinguished by the codebook resolution (at both ends).
The basestation uses the channel matrix constructed for each user based on its feedback information () and generates a good beamformer structure, illustrated in the next result, for use in multiuser transmissions.
Proposition 1.
The zeroforcing beamformer structure is one where for every user that is simultaneously scheduled, the beam nulls the multiuser interference in with as given in (20). The beams in the zeroforcing structure are the unitnorm column vectors of the matrix , where is the matrix given as
(25) 
Iv Upper Bounds for
We are interested in benchmarking the performance of the zeroforcing structure against an upper bound on . The goal of optimizing over with perfect channel state information is a nonconvex optimization problem [42, 43, 44] that appears to be complicated. In this context, an alternate formulation based on the signaltoleakage and noise ratio metric [45] that simultaneously maximizes the array gain seen by the th user, , and minimizes the interfering array gain seen by the other users, is relevant. Since these objectives are in some sense conflicting and can be weighed differently, we consider the composite metric
(26) 
for an appropriate set of weighting factors with .
Iva Upper Bound Motivated by the Zeroforcing Structure
Building on Prop. 1, we now develop an upper bound for motivated by the zeroforcing structure. In this direction, we consider a signaltoleakagetype metric equivalent of (26) based on the estimated channel matrix
(27) 
for an appropriate set of weighting factors with .
Proposition 2.
Assuming that and are known at the basestation, the choice of that maximizes is given by the generalized eigenvector structure
(28) 
Proof.
See Appendix C. ∎
Several remarks are in order at this stage.

In the case where are set to zero for all (that is, the focus is not on interference management), the solution in (28) reduces to
(29) This is not surprising, and the basestation greedily steers a beam along the weighted set of top beams from for the th user. In other words, the basestation generates a set of transmit weights that are matched to the transmit angular spread of the channel as identified by the resolution of .

In the case where except if or (for a specific ), it can be seen that reduces to
(30) In other words, the specific design of in (30) removes a certain component of the beam corresponding to the th user from the beam corresponding to the th user.

In the general case, while it gets much harder to simplify in (28), it can be seen that has the structure
(31) for some complex scalars . In other words, the optimal is in the span of with the weights that make the linear combination being a complicated function of as well as .

The above observations are not entirely surprising given the KarhunenLoève interpretation of the eigenspace of the channel(s) [51, 52, 11] and utilizing an expansion of on this basis. Such an expansion is also consistent with Prop. 1 which shows that in the pure interference management case ( for all ), is given as
(32) where the matrix .

On the other hand, from (24), we note that is itself a linear combination of the beams from . Thus, in (28) is a linear combination of beams from . In other words, the design of is equivalent to a search over scalar (complex) weights, where denotes the size of the initial beam alignment codebook at the basestation end.
With this interpretation, while Prop. 2 considers only the maximization of (not even the sum rate with ), we can consider the optimization of over from a class , defined as
(33) 
Theorem 1.
Assume that the same multiuser beams as in the zeroforcing scheme are used for reception at the th user. Let be defined as the solution to the search over the complex scalars
(34) 
With as above and
(35) 
we obtain an upper bound to the sum rate with the zeroforcing scheme. ∎
The proof is trivial following the structure of in the zeroforcing scheme in (32) and the definition of the class in (33). Since the structure in (35) is obtained as a search over scalar parameters, we call this upper bound a scalar optimizationbased upper bound. Further, while (35) is difficult to practically implement, it provides a benchmark to compare the realizable zeroforcing scheme of Prop. 1.
Another important consequence of (35) is that the coefficients of for either the zeroforcing or the upper bound are (in general) not of equal amplitude. Thus, has to be quantized for implementation to ensure that the RF beamforming constraints are satisfied. In particular, we compute with an appropriate quantization scheme as below
(36) 
and use them in transmissions for the th user. Good choices for will be discussed in Sec. VC.
IvB Bounding with an Alternating/Iterative Optimization
We now propose an iterative maximization algorithm to optimize over . In this approach, we first optimize the metric over (assuming is fixed), and then optimize the metric over (assuming is fixed). The algorithm is as follows:

Initialize randomly.

For , where is chosen according to a stopping criterion to determine convergence:

Compute with and for a (potential) upper bound.
Numerical studies show that for almost all channel realizations, the proposed algorithm converges in a small number of steps () to lead to a tolerable level of difference between successive iterates of . Further, while we are unable to theoretically establish that the proposed algorithm results in an upper bound to , numerical studies (see Sec. VD) suggest that it leads to an upper bound for almost all channel realizations.
V Numerical Studies
We now present numerical studies in a singlecell downlink framework to illustrate the advantages of the proposed beamforming solutions. The channel model from (1) is used to generate a channel matrix with
clusters, AoDs uniformly distributed in a
coverage area, and AoAs uniformly distributed in a coverage area for each of the users in the cell. The AoD spread captures a traditional threesector approach with a zenith coverage and the AoA spread corresponds to the assumption of the use of multiple subarrays [9] with the best subarray limited to a coverage. is justified from millimeter wave channel measurements reported in [9, 12]. The antenna dimensions assumed in these studies are and at the basestation end, and and at each user. We consider simultaneous transmissions from the basestation to out of the users in the cell.In terms of user scheduling, commonly used criteria include a round robin or a proportionate fair scheduler. On the other hand, a recently proposed directional scheduler [24] leverages the smaller beamwidths afforded by large antenna dimensions to schedule users with dominant clusters that are spatially wellseparated. In this work, the first of the users is scheduled randomly and the second user is chosen to ensure that . In other words, the considered scheduler implements a directional avoidance protocol with the dominant cluster in the channel of the first user separated spatially from the dominant cluster in the channel of the second user, as parsed by . With this scheduler, we now primarily focus on the beamforming aspects.
For the initial beam alignment codebooks, based on the beam broadening principles proposed in [19], Figs. 1(a)(d) illustrate the beam patterns in the azimuth plane for codebooks of sizes , , and , respectively, to cover the AoD space with a planar array at the basestation side. The optimization proposed in [19]
results in a discrete Fourier transform (DFT) codebook solution for
and . From Fig. 1, we observe that a beam codebook of small size (e.g., ) where each beam offers a broad directional coverage can reduce the acquisition latency at the cost of peak and/or worstcase array gain. On the other hand, a beam codebook of large size (e.g., ) where each beam can offer precision in terms of beamspace (and array gain) comes at the cost of acquisition latency. For the codebooks at the user end, two codebook sizes ( for a reduced acquisition latency and for performance improvement at the cost of acquisition latency) are considered with similar beam design principles as for the basestation side.(a)  (b) 
(a)  (b) 
At this stage, it is worth noting that a number of system parameters impact the performance of the proposed multiuser schemes such as: i) Granularity of and (initial beam alignment codebook sizes), ii) Coarseness of channel approximation (rank), iii) Finiterate feedback of channel reconstruction parameters, and iv) Quantization of the resulting multiuser beams.
(a)  (b) 
(c)  (d) 
Va Impact of Initial Beam Alignment Codebook
In the first study, we consider the relative performance of the zeroforcing scheme (proposed in Prop. 1) relative to a baseline beam steering scheme with different initial beam alignment codebooks. We assume that the system has infiniteprecision feedback of channel reconstruction parameters and infiniteprecision resolution in the quantization of multiuser beams. We also compare the performance of the proposed schemes with the zeroforcing scheme presented in [23, 24], where the system is assumed to be able to find perfectly aligned directional beams in the training phase. Fig. 2 illustrates this comparative performance with different choices of in approximating and different codebook sizes ( and ).
While it is intuitive that there should be diminishing performance as increases (since increasing beyond the channel rank is not expected to improve performance), whether this saturation in performance is observed with a lowrank channel approximation is dependent on the resolution of the codebooks. In particular, increasing when the codebook granularity is already poor (small and ) does not lead to any performance improvement than observed with (beam steering). On the other hand, with a high resolution for (large ), even a rank approximation appears to be sufficient to reap most of the performance improvement gains. This is because the performance of the baseline (beam steering) scheme is already quite good and significant relative improvement over it with increasing
has a lower likelihood unless the channel has a large number of similar gain clusters (a lowprobability event). When
is large and is small, the beam steering performance is poor and the channel can be better approximated with the higher codebook resolution of leading to a sustained performance improvement for even up to . For example, with or and , zeroforcing based on a rank channel approximation leads to around bps/Hz improvement at the median level.In terms of performance comparison, note that the scheme from [23, 24] assumes but infiniteprecision in terms of beam alignment (). Thus, it is not surprising that as and increase, the performance of the proposed schemes compare well with that of [23, 24]. For lower codebook resolutions, the proposed schemes overcome the codebook disadvantage by leveraging a better channel approximation as increases. These observations suggest that the optimal choice of the rank in approximating (which in turn determines the feedback overhead) depends not only on the rank of the true channel , but also on the codebook granularities. In general, a higher (and feedback overhead) is necessary if the codebook resolution is rich enough at the user end to allow the parsing of the channel better, but poor enough at the basestation end to allow a sustained performance improvement with increasing . In particular, we provide the following heuristic design guidelines based on our studies
(41) 
(a)  (b) 
(c)  (d) 
VB Quantizer Design
Towards the second study, we utilize different quantization functions to quantize the different parameters needed in channel reconstruction. For a phase term with a dynamic range of (e.g., and ), we use a uniform quantizer of the form
(42) 
where stands for a function that rounds off the underlying quantity to the nearest integer. For an amplitude term with a dynamic range of (e.g., ), we use a nonuniform quantizer of the form
(43) 
The reason for scaling with respect to in (43) instead of by is because we want the quantized set to include both and for proper crosscorrelation quantization. For example, in the typical case where the multiuser reception beam , we have and the use of a uniform amplitude quantizer will not allow the correct reproduction of this important quantity at the basestation end.
Quantization of the is performed on a dB scale rather than on a linear scale. This is intuitive since measurements have a wide dynamic range. The proposed quantizer is similar to quantizations considered in Fourth Generation (4G) systems. In particular, for a received term (in dB) with a theoretically unbounded range (e.g., ), we first cap to a maximum value of and quantize a spread of (in dB) with quantization levels (denoted as ) as follows:
(44) 
The quantization of is given as
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