1 Introduction
With the coming age of massive multipleinput multipleoutput (MIMO), the performance of wireless networks can be significantly boosted by multiantenna techniques like beamforming and precoding. Most multiantenna methods rely on accurate channel state information (CSI) at the base station (BS). In frequency division duplex (FDD) systems, due to the lack of reciprocity between uplink (UL) and downlink (DL) channels, the estimated DL CSI at the user equipment (UE) needs to be fed back to the BS. However, both the DL CSI estimation and feedback are computationconsuming and resourcedemanding. Specifically, with the increasing number of antennas and more stringent requirements of CSI accuracy, the overhead of CSI feedback proliferates. When more timefrequency resource is used for CSI feedback, the resource for the data transmission has to be decreased. As a result, the spectrum efficiency (SE) of data transmission is degraded, and it calls for highly efficient CSI sensing techniques.
The DL CSI estimated at a user equipment (UE) in practical cellular systems is usually quantized with a predefined codebook. Then a corresponding precoding matrix indicator (PMI) is fed back to the BS. The choice of codebook requires balancing the feedback overhead and the accuracy of CSI returned. Two kinds of codebooks are specified by the 3GPP [1]
– the lowresolution TypeI codebook that has been widely applied in 4G systems and the TypeII codebook that is added to 5G to enhance the CSI accuracy. The TypeI codebook characterizes the CSI with a single beam and conveys the coarse information about the DL CSI. The TypeII codebook depicts the CSI with multiple discrete Fourier transform (DFT) rays and keeps the amplitude and phase information. Therefore, the CSI conveyed by the TypeII codebook is more accurate and enhances the data transmission performance effectively. Systemlevel performance evaluation
[2, 3] has demonstrated that up to 30% performance improvement can be achieved by the TypeII codebook based beamforming over its counterpart on the TypeI codebook.Though the TypeII codebook based on CSI feedback brings excellent performance enhancement, it has not entirely replaced the TypeI codebook. First, the accuracy of CSI improved with the TypeII codebook is at the price of a much higher overhead. Because of that, the TypeII codebook currently supports up to rank2 transmission only, while up to rank8 transmission is enabled by the TypeI codebook. On the other hand, the TypeI codebook has been extensively used in 4G systems for many years. With the fact that a considerable number of UEs from the 4G age are still widely in use, and most of them only support the TypeI codebook, it can be anticipated that they will coexist with more advanced UEs like 5G cellphones supporting the TypeII codebook for a long time, resulting in a heterogeneous network as illustrated in Fig. 1.
In this paper, we consider such heterogeneous networks. Our goal is to improve the accuracy of CSI reconstruction at the BS for these UEs that can afford the TypeI codebook only. In particular, we aim to utilize the PMI and channel quality indicator (CQI) of the TypeI codebook fed back from the target UE (TU) to recover its CSI with high accuracy. This CSI sensing task can be formulated as a phase retrieval (PR) problem via appropriate training signal design [4, 5, 6]. However, like the conventional PR, which requires many measurements for accurate recovery, these methods require many feedbacks from the TU to recover the CSI well. To effectively reduce the feedback overhead, in this paper, we exploit the lowrank structure of the channel model and the PMI feedback to propose a novel dimension reduced and constrained PR (CPR) formulation, together with new precoding designs and efficient CSI recovery algorithms.
1.1 Related Works
There have been many efforts to reduce the feedback overhead in FDD systems in the literature. Conventionally, compressive sensing (CS) based methods were widely used to reduce the CSI feedback overhead [7, 8, 9, 10, 11]. The spatial correlation between closelypacked antennas was utilized, and an adaptive channel feedback protocol based on CS was proposed in [7] to reduce the CSI feedback load, with 2D discrete cosine transform and KarhunenLoeve transform used as bases. The work [10] considered the CSI estimation in FDD multiuser massive MIMO systems. A joint channel sparsity model was proposed, wherein the sparsity is divided into individual joint sparsity and distributed joint sparsity by different types of scattering. An algorithm based on orthogonal matching pursuit (OMP) was designed to reduce the training and feedback overhead for the CSI acquisition. Similar ideas were adopted in [11] and a sparse Bayesian learning framework was proposed for the joint channel estimation and user grouping. Although the designs in these works are shown to reduce the overhead effectively, they all have assumed that the UE can feedback the estimated CSI ideally without rate constraints and quantization.
Recently, with the popularity of deep learning and its efficacy validated in areas like image processing and natural language processing, many research works explored deep learning (DL) methods to reduce the CSI feedback and recovery overhead. In
[12], a CsiNet based on the autoencoder was first proposed for the CSI compression and recovery. It improves the quality of CSI recovery effectively over the CSbased methods and without making any sparsity assumption. Inspired by this, the autoencoderlikewise CSI feedback and recovery mechanisms were further investigated under more practical system settings like quantization, coding, and modulation to be considered [13, 14, 15, 16, 17, 18]. Besides the consideration of the practical setting in the CSI feedback, other existing works tried to improve the compression efficiency via exploiting the temporal correlation of CSI [19, 20, 21]. Among these designs, the recurrent NN (RNN) and longshorttemporalmemory (LSTM) structures were adopted to exploit the temporal and frequency correlation of channels.Although many schemes based on NN have been proposed to reduce the CSI feedback overhead, the feedback mechanism they rely on is not directly applicable to existing cellular systems. Moreover, most of these schemes only exploit temporal and frequency correlation. A few works have considered the spatial consistency, but are still limited on a small scale, i.e., the correlation of the CSI on different antennas of the same UE terminal [22, 23].
It is worth pointing out that there exist several works [4, 5, 6] that share the same goal as our current work and target improving the CSI recovery at the BS with lowresolution feedback. The work [4]
designed a binary CSI feedback scheme, where the receiver returns a binary indicator to inform whether the received signaltonoise ratio (SNR) is more significant than a preset threshold. A cutting planebased algorithm was proposed to learn the channel covariance matrix from these feedbacks. While the CSI accuracy can be improved from multiple coarse feedbacks, the scheme in
[4] cannot be applied to existing cellular systems with the PMI and CQI feedback mechanism. Comparatively, the designs in [5] and [6] also made use of the feedback PMI and CQI of the TypeI codebook. Still, the corresponding problem formulation and proposed schemes are different from our design. For instance, a volume minimization problem was formulated in [5] and a cutting plane method like that in [4] was proposed to search the channel covariance matrix. The work [6] formulated a 2norm regularized PR problem for CSI sensing, and an alternating minimization algorithm was devised. However, both methods in [5, 6] still take a large number of feedbacks (e.g., more than rounds of feedbacks in [5]) to achieve comparable performance as the TypeII codebook. Besides, they neither exploit the spatial consistency nor consider the precoding design.1.2 Contributions
In this paper, like [5, 6], we aim to design a DL CSI sensing scheme to enhance the CSI recovery accuracy of UEs that can afford the lowresolution TypeI codebook only. The objective is to effectively reduce the feedback overhead of the TU while achieving a sensing performance as good as that of the highresolution (TypeII) codebook in a computationally efficient manner.
To reduce the feedback overhead, we propose exploring the wireless channel’s lowrank structure, which enables us to project the original highdimensional CSI onto a lowdimensional space so that the BS only needs to recover a small number of unknown parameters from the TU feedback. Since in the PR the required number of measurements is proportional to the number of parameters, this strategy can effectively decrease the number of TU feedbacks for accurate CSI sensing. While such dimension reduction is possible, how to construct the associated basis matrix at the BS is unknown. To address it, we exploit the spatial consistency of the CSI among geographically nearby UEs. Specifically, as nearby UEs’ channels experience similar reflectors and scatters, they share similar structures as the TU’s channel. It is noted that experiments have widely validated the spatial consistency of the CSI [24, 25, 26] and it is observed in many studies [27, 28]. Recently, the spatial consistency feature was also emphasized in standards and channel models [29, 30]. Based on the spatial consistency, we assume that the TU’s CSI approximately lies in the same space as the CSI of nearby reference UEs (RUs). Therefore, as shown in Fig. 1, it is possible to construct the basis for the TU’s CSI from the TypeII PMI feedback of RUs.
To achieve a promising CSI sensing performance, we utilize the constructed lowrank CSI basis to design the training precoding matrix. In particular, we propose a hybrid precoding structure that is a product of a Gaussian random matrix and the projection matrix for the lowrank CSI subspace. The Gaussian random matrix is used for improving the sensing diversity in the PR
[31, 6]. In contrast, the projection matrix can align the training signal with the channel and thereby enhances the received SNR at the TU. Compared to the current work in [31, 6] which uses only Gaussian random precoders, the proposed hybrid precoder can further boost the CSI sensing performance. Moreover, we utilize the PMI fed back from the TU to characterize the feasible region that the CSI lies in, which leads to a novel CPR formulation.The formulated CPR problem is intricate to solve since it is not only nonconvex in the objective function but also involves a large number of nonconvex quadratic constraints. For example, the problem could have thousands of constraints due to the TypeI codebook size (e.g., more than codewords for the codebook with 32 CSI ports [1]). To address the challenges, we propose a twostage strategy. First, we propose a simple algorithm to identify redundant constraints and construct a ‘minimal’ effective constraint set (MECS) for the CPR problem. Second, we adopt the smoothed gradient descent ascent (SGDA) algorithm in [32] to solve the Lagrange dual problem of the CPR problem. Our technical contributions are summarized as follows.

Based on three novel ingredients, we propose in this paper a new CSI sensing scheme for achieving superior CSI recovery performance at the expense of considerably reduced feedback overhead for UEs that can afford the TypeI codebook only. In particular, firstly, we exploit the spatial consistency of wireless channels to perform dimension reduction for the PR problem. Secondly, a hybrid precoder is proposed to enhance the CSI sensing performance, and thirdly a constrained PR problem is formulated based on the PMI feedback. By using the primaldual (PD) optimization based algorithm in [33] (which we refer to as PDEVD in the manuscript), we show that the proposed scheme significantly outperforms the existing works in [5, 6] and can recover accurate CSI with only a handful number of feedbacks.

To solve the challenging CPR problem efficiently, we propose the twostage MECSSGDA algorithm, which removes redundant constraints from the CPR problem first, followed by solving the simplified problem by the SGDA algorithm. It will be shown that the MECSSGDA algorithm can save at least a magnitude of time in solving the CPR problem compared to the PDEVD algorithm.

The proposed CSI sensing scheme is further extended to frequency selective fading channels and multicarrier systems. Rather than naively applying the previous CSI sensing scheme to each of the subcarriers, we consider the channel correlation over different subcarriers and propose a new multicarrier CSI sensing scheme that can provide even better performance.
We remark that compared to the conference paper [33], the current paper presents the new MECSSGDA algorithm and the extension to multicarrier systems. In addition, comprehensive simulation results based on the DeepMIMO and QuaDriGa datasets are presented to demonstrate the efficacy of the proposed methods.
Synopsis: The system model and problem formulation are introduced in Section 2. In Section 3, the designs for dimension reduction, precoding and CPR modeling are elaborated in details. The algorithm design for the CPR problem is presented in Section 4 and the extension to the multicarrier systems is given in Section 5. Simulation results are given in Section 6. Finally, conclusions are drawn in Section 7.
2 System Model and Problem Formulation
As shown in Fig. 2, we consider a heterogeneous wireless network with one BS communicating with multiple UEs. The BS is equipped with antennas. Each UE is equipped with a single antenna. Here we assume flat fading channels, and latter extend the work to the frequencyselective fading channels in Section 5. It is assumed that the TU can only return the coarse CSI via the PMI of TypeI codewords. At the same time, the nearby RUs are capable of producing the highaccuracy CSI via the PMI of TypeII codewords. In the work, by integrating the sensing data returned to the BS, the spatial consistency of the CSI among nearby users is exploited to improve the CSI sensing accuracy for the TU.
The procedure of DL training and feedback between the BS and the TU is described as follows. Assume that antenna ports are used for CSI sensing. Denote the DL pilot signal at the time slot as . In the DL CSI estimation, the BS applies the precoding matrix . Assume that the (flat fading) channel is quasistatic and is unchanged during the sensing period. The signal received by the TU at time slot is expressed as
(1) 
where is the additive white Gaussian noise (AWGN). At the TU, without the knowledge of , only the effective CSI can be estimated as
(2) 
where is the error of channel estimation. Because returning the attained to the BS directly is too resource consuming, the TU in practice quantizes through the TypeI codebook
(3) 
and selects a PMI that is the index of the best matched codeword as the feedback message. Specifically, the PMI is determined by
(4) 
The CQI is also computed as
(5) 
and both PMI and CQI are sent to the BS.
When and
is an identity matrix
, the codewordis simply an estimate of the DL CSI and can be directly used as the beamforming vector for data transmission. However, with a limited size, the TypeI codebook specified in
[1] only conveys the coarse information about the CSI and the performance is not satisfactory in general. While the more sophisticated TypeII codeword can return the more accurate CSI, it needs much larger overhead. As depicted in the introduction section, not all the UEs can afford the TypeII codebook. Therefore, it raises an interesting question: Can a BS achieve a CSI estimate as good as the TypeII codeword using only the TypeI codeword feedbacks from the TU?In [6], the authors showed that this is possible by solving the following PR problem
(6) 
According to theoretical results on the phase retrieval in [34, 35], it typically requires measurements to retrieve the complex vector accurately. Unfortunately, the resultant feedback overhead is too heavy even for a medium size of antenna ports at the BS, e.g., . Therefore, it calls for more effective design with light feedback.
3 Proposed CSI Sensing Scheme
This section delineates the design of our proposed CSI sensing scheme. The goal is to achieve the CSI sensing performance as good as the TypeII codebook feedback while using only PMI/CQI of the TypeI codebook. Given the formulation in (6) as a beginning, we propose three ingredients to achieve the desired goal, namely parameter dimension reduction via spatial consistency, hybrid precoder design, and a new constrained PR formulation.
3.1 Dimension Reduction
As mentioned before, the number of measurements required to recover the CSI in PR (6) is proportional to that of parameters . In view of this, we try to utilize the lowrank structure of wireless channels to reduce the dimension of parameters. The flat fading DL channel vector can be expressed as [36]
(7) 
which is composed of propagation paths. Each path is composed by a complex channel gain and the array response vector , where and correspond to the azimuth and elevation angles of departure (AoD), respectively.
While is typically large, because of the limited local scattering effect in the propagation, most of the energy of the channel is dominated by a small number of propagation paths. Hence it is sufficient to have a good approximation of the CSI by only considering these dominant paths. Specifically, by taking the first dominant path components, (7) can be approximated as
(8) 
where
(9) 
and . Substituting (8) into (6), the CSI sensing problem is updated as
(10) 
Compared with (6), as long as , the CSI sensing problem (10) can reduce the required rounds of feedback for achieving a good recovery performance. However, two fundamental questions regarding problem (10) are 1) the construction of the basis matrix and 2) the proper design of the precoder for improved performance. Next we show how they can be achieved by utilizing the spatial consistency of wireless channels.
3.2 Basis Construction by Spatial Consistency
The basis matrix in (9
) can be constructed by interpolating the associated path parameters of the TU from RUs’ path parameters. The interpolation explicitly utilizes the spatial consistency. However, this construction approach requires the estimation of RUs’ path parameters and welldesigned path aligning algorithms before interpolation. The computational cost due to parameter estimation and path alignment at the BS cannot be neglected.
To avoid the problem, the basis matrix is constructed by implicitly exploiting the spatial consistency instead. Specifically, given RUs near the TU, denote their corresponding CSI as . Here we assume all UEs’ locations are available at the BS so that the RUs can be selected from the neighborhood of the TU. In practice, the locations of UEs can be attained by the global positioning system (GPS), 5G NR positioning [37] or the combination of them, and reported to the BS. The CSI of RUs can be attained, for example, from the historical data stored at the BS or TypeII feedbacks. By the spatial consistency, the AoDs of RUs should be close to those of the TU. In view of this, we assume that
(11) 
where
(12) 
Therefore, can be approximately constructed based on
. Specifically, consider the singular value decomposition (SVD) of
as(13) 
with
being the ordered singular values, and the left and right singular vectors are
and , respectively. One may simply take(14) 
as an approximation of the basis of . Correspondingly, (8) is replaced by
(15) 
and the CSI sensing problem (10) is updated as
(16) 
It is worth noting that when only the TypeII codeword rather than of RU is available, in (12) can be replaced by
(17) 
from the RUs.
3.3 Hybrid Precoding Design
The precoding matrix at the BS is not only critical for the UE side channel estimation but also for the BS side CSI sensing. The design of is based on two important observations. First, should be random which provides diversity and can improve the PR performance. Specifically, to learn about from as many different directions as possible, the precoding matrix applied at different time instants should be as diverse as possible [4]. Second, from (2) should be matched with to improve the received SNR at the TU side.
Based on the above two observations, we design in a hybrid fashion, i.e.,
(18) 
where brings randomness in the sensing and is to improve the received SNR. The design of follows a similar idea in the PR literature, where the measurement vectors are usually sampled from a random Gaussian matrix [31, 38]. Thus, in our CSI sensing problem, is chosen to be a random Gaussion matrix, i.e.,
(19) 
where
is the standard deviation.
To improve the received SNR at the TU side, should concentrate on the main components of the CSI rather than the whole space. In other words, should match the space of , i.e., . Inspired by the idea, is chosen to be the projection operator on and is expressed as
(20) 
where is given in (14).
It is not difficult to verify by checking (18) and (2) that the hybrid design essentially projects the original channel to the estimated space of the principal components firstly and then applies the random Gaussian matrix . The hybrid structure of brings benefits to the reduction of feedback rounds , especially for the wireless environment of copious propagation paths. Its efficacy will be shown in Section 6.
3.4 CPR Formulation
As described in Section 2, the PMI fed back from the TU to the BS is selected by (4) with the TypeI codebook (3). As a result, the CSI to reconstruct should satisfy the following inequalities
(21) 
Notice that the inequalities in (21) actually specify a feasible region for the CSI . Therefore, incorporating them into problem (16) would facilitate the search of the true CSI and enhance the sensing performance. By incorporating the inequalities in (21), problem (16) is updated as
(22a)  
(22b) 
There are three main challenges to solve the CPR problem (22). First, the objective in (22) is nonconvex. Second, the constraints are also nonconvex and difficult to project onto. As a result, the projected gradient descent method popularly used in the literature for solving the PR problem are no longer suitable for problem (22). Last but not least, the number of constraints in (22b) is , which is very large due to the codebook size . For instance, for the TypeI codebook of CSI ports, as specified by 3GPP [1]. Given a small value of like , the total number of constraints in problem (22) can easily exceed one thousand, which poses a big challenge for efficient CSI recovery.
4 Efficient Algorithm Designs
In this section, two algorithms to solve the CPR problem (22) are developed. We first review the algorithm in [33]
that is based on the PD optimization and eigenvalue decomposition (EVD). Then, to overcome the challenge of a large number of constraints in (
22), we further propose a computationally efficient twostage method that first removes redundant constraints from (22) followed by solving the Lagrange dual problem of (22) via a simple firstorder algorithm.4.1 PDEVD Algorithm
By defining and introducing matrices and , problem (22) is equivalently expressed as
(23a)  
s.t.  (23b)  
(23c) 
One can observe that the objective and all the constraints in problem (23) are convex except (23c). One alternative to handling (23) is to adopt the wellknown semidefinite relaxation (SDR) technique by simply relaxing to (positive semidefinite). However, this approach has two drawbacks. First, the rankone condition in (23c) is not guaranteed. Moreover, the resultant relaxation problem would still be computationally expensive due to the huge number of constraints. To cope with these problems, we instead develop a customized algorithm for problem (23) based on the primaldual optimization.
By introducing dual variables , the Lagrangian of (23) can be written as
(24)  
with . Then the dual problem of (23) is given by
(25) 
An algorithm to solve (25) can be developed by solving the inner problem and the outer problem alternatingly until a predefined stopping condition is satisfied. In our design, the inner problem of (25) is further processed by the majorizationminimization method following the PRIME algorithm in [31] to obtain a closedform solution in each iteration, while the outer problem is solved by the projected gradient ascent method.
Specifically, given in problem (25), the inner problem is
(26a)  
s.t.  (26b) 
To achieve efficient computation, the first term is further majorized at the point as
(27) 
with to be a step size parameter, where takes the largest eigenvalue of . Therefore, the majorization problem for (26) can be written as
(28a)  
s.t.  (28b) 
It is not difficult to show that problem (28) is equivalent to
(29) 
where
(30) 
and is a diagonal matrix whose diagonal entries are . Obviously, problem (29
) is a leading eigenvector problem admitting a closedform solution
(31) 
where and represents the leading eigenvalue and eigenvector of and can be computed via the power method efficiently.
Once the inner problem is solved, the dual variables in the outer problem of (25) can be updated via the projected dual ascent method, i.e.
(32) 
where is the step size. The steps of the overall algorithm are summarized in Algorithm 1.
Experiment results presented in Section 6 will show that the above PDEVD algorithm can provide promising CSI sensing performance with a significantly reduced feedback overhead. Nevertheless, it is also found that the computation of PDEVD is still heavy. One reason is that step 7 in PDEVD involves the computation of the leading eigenvector that is not computationally cheap. In addition, PDEVD needs to process all the constraints of problem (25), which is another bottleneck to achieve efficient solution. In view of this, we next develop a twostage approach to solve the CPR problem (22) via the MECS construction and the firstorder SGDA algorithm.
4.2 TwoStage MECSSGDA Algorithm
With the observation that many of the constraints in the CPR problem (22) may not be active at the optimum, finding the effective ones and removing those redundant ones from the problem would achieve complexity reduction. On the other hand, to avoid computing the leading eigenvector, we consider the firstorder methods to handle problem (22). This idea gives rise to a twostage method as described below.
MECS Construction: The task of constructing the MECS is to find the minimal number of constraints that effectively bound the true followed by eliminating those redundant constraints from problem (22). However, it is highly nontrivial to find all the effective constraints in (22b) theoretically. Instead, we resort to a suboptimal method. The construction process starts with an initial point as well as an empty MECS, and adding the unsatisfied constraints to the MECS gradually until a point feasible to all the constraints is found.
Specifically, initialize and the iteration index . Given an initial point , these unsatisfied constraints at constitute a set and the MECS is updated by
(33) 
Afterwards, is updated by solving the following feasibility problem
find  (34a)  
s.t.  (34b) 
where . The updates of and proceed alternatingly until a point meeting all the constraints in (22b) is found. The details of the MECS construction are given in Algorithm 2. Because the original CSI satisfies all the constraints in (22b) and the set will not shrink during the iterations, it is not difficult to verify that Algorithm 2 will terminate within a finite number of iterations.
The main step in Algorithm 2 is to solve problem (34), which is a nonconvex quadratically constrained problem. While the consensusADMM algorithm proposed in [39] can undertake the task, the approach involves eigendecomposition that is usually not computationally cheap. Therefore, problem (34) is handled in a different way in our work. Specifically, by introducing a continuous and nonconvex function
(35) 
problem (34) can be transformed into
(36) 
Thus, problem (36) becomes an unconstrained problem with a smooth objective function and can be efficiently solved by the gradient descent method or ADMMbased methods [40].
SGDA Algorithm: By Algorithm 2, a feasible point satisfying all constraints of problem (22) and a MECS can be obtained. Then, what remains is to solve problem (22) with the MECS:
(37a)  
s.t.  (37b) 
Rather than considering the PDEVD algorithm or other methods based on the convexconcave procedure (CCP) [41], we propose to employ firstorder methods to handle problem (37). In particular, inspired by the smoothed gradient descent ascent (SGDA) algorithm framework proposed in [32], we consider the Lagrangian dual problem of (37) as follows
(38) 
where
(39) 
is the Lagrangian function and are dual variables.
Note that (38) is a minmax problem which is concave w.r.t. but not necessarily convex w.r.t. . To overcome the nonconvexity, SGDA introduces an auxiliary variable and solves the following problem as
(40) 
with as a constant parameter and is large enough so that is convex w.r.t. . In SGDA, the gradient descent and ascent are conducted in the alternating fashion w.r.t. the primal variable and dual variables , while the auxiliary is updated via an averaging step. The details of SGDA for problem (38) are summarized in Algorithm 3, where are stepsize parameters. As seen, the algorithm involves only simple gradient descent/ascent updates and therefore are computationally cheap in each iteration. Conditions for the SGDA algorithm to converge to a stationary solution of problem (38) have been proved in [32].
Notice that Algorithm 3 is applied to solve problem (38) rather than the original problem (37). Due to the nonconvexity of problem (37), Algorithm 3 cannot guarantee to find a solution satisfying all the constraints in (37). It is necessary to emphasize that the goal here is not to meet all the constraints but to find a solution satisfying most of the constraints in order to improve the sensing performance of the unconstrained counterpart. Moreover, the tests in Section 6 showcase that the solution returned by Algorithm 3 indeed meets most of the constraints and achieves a noticeable improvement on the sensing performance.
5 Extension to Multicarrier Systems
So far, the proposed CSI sensing scheme considers only the flat fading channels. For frequency selective fading channels, we consider the multicarrier transmission system (e.g., OFDM) and extend the CSI sensing scheme to this scenario. Rather than applying it to each of the subcarriers, we explore the channel correlation between different subcarriers.
Assume that the multicarrier system has subcarriers. The DL channel in the th subcarrier can be expressed as [36]
(41) 
where and are the power, phase and delay of the th propagation path, respectively. Correspondingly, the CSI to sense is
(42) 
Since the number of propagation paths is usually much smaller than the number of subcarriers , we consider transforming into the antennadelay domain by
(43) 
where contains the first columns of the inverse discrete Fourier transform (IDFT) matrix of size . Similar to (8), can be further approximated by the first few dominant paths (say paths), like
(44) 
where the basis matrix and the coefficient matrix .
From the aforementioned derivations, one can observe that the number of the parameters to recover is reduced from to . The transformation from to
in essence utilizes the correlation of the CSI on the angular and frequency domains jointly, making it possible to further cut down the feedback overhead via sensing the dimension reduced
. Correspondingly, the extensions of the basis matrix construction, precoder design and CPR formulation to the multicarrier system are developed as follows.Basis matrix construction: Notice that is the basis matrix for the CSI in the delay domain. Denote the CSI of RUs as . We take their delaydomain counterparts by
(45) 
Similar to (13) based on the spatial consistency, by taking the left principal singular vectors of , we can obtain a basis matrix for , denoted by
Precoder design: Define as the precoding matrix for each subcarrier in the th round. Like (18), is constructed in a hybrid mode as
(46) 
where
is still sampled from the complex Gaussian distribution, i.e.,
. The construction of relies on the basis matrix for the subspace that lie in. Analogously, we use the CSI of RUs(47) 
and take the first principal left singular vectors of , denoted by , as the basis matrix. Then, the precoder is chosen to be
(48) 
in order to maximize the received SNR at each subcarrier .
CPR formulation: With the precoder over the th subcarrier, the effective CSI estimated at the TU is
(49) 
where is the channel estimation error. Like (4), the PMI from the TypeI codebook is selected via
(50) 
and the corresponding CQI is computed as
(51) 
As a result, similar to (21), the associated constraints indicated by the PMI can be expressed as
(52) 
With the above basis matrix , we approximately have ^{1}^{1}1In general, does not hold as is the truncated IDFT matrix and . The rationale behind the approximation is that the energy of the CSI mainly dominate parts of the temporal components because of the limited local scattering effect. That is, given as the CSI in the time domain with being the full IDFT matrix, the absolute values of the entries in are almost zero. Therefore, can be approximately recovered by .
(53) 
where is the th column vector of . The CPR problem for the multicarrier system is formulated as
(54a)  
s.t.  (54b)  
(54c) 
Notice that problem (54) can also be solved by the proposed twostage MECSSGDA algorithm.
Until now, the extension to the multicarrier system is based on the setting that the PMI and CQI are fed back per carrier. For more flexible resource utilization, the computation and feedback of PMI and CQI can be conducted over a group of subcarriers like physical resource block (PRB) rather than per carrier. For that scenario, we have the following remark.
Remark 1
Assume that the overall frequency band is divided into groups of carriers, and each group contains subcarriers. Denote the estimate of the effective channel over the th carrier within the th group as . The PMI in the th round for the th group can be selected by
(55) 
and the CQI can be computed as
(56) 
It is not difficult to extend our previous designs to the scenario, and our proposed algorithms are still applicable.
6 Simulation Results
In this section, the proposed CSI sensing scheme is evaluated extensively based on the DeepMIMO dataset [36] and the CSI generated by QuaDriGa platform^{2}^{2}2https://quadrigachannelmodel.de/. Two metrics are applied to evaluate the performance of the sensed CSI – the correlation
(57) 
and the normalized mean squared error with a common phase rotation () [31] defined as
(58) 
6.1 Performance on DeepMIMO Dataset
The test data is from the DL CSI from BS 4 to user within row in the DeepMIMO ‘O1’ scenario. The DL CSI consists of paths in that area, which is almost the most complicated propagation area covered by BS 4. To examine the performance gain of our proposed scheme clearly, samples are selected from the total samples in the area. Besides, other UE are selected to model a given dataset of RUs whose TypeII codewords are available. The other parameters in experiments are given in Table 1. For each TU, RUs with the smallest distance to the TU are selected from the RU dataset. The TypeII codewords of RUs are used to construct as (17). Unless otherwise specified, the number of CSI ports is set as . The value to generate the random in (19) is set to 1. For each TU, s are randomly generated and the performance of the sensed CSI is averaged over them. The presented performance is averaged over all TUs.
Parameter  Value 
BS height  6 meter 
UE height  1.5 meter 
Number of antennas at the BS,  
Number of antennas in the horizontal dimension,  8 
Number of antennas in the vertical dimension,  4 
Antenna spcaing  
Center carrier frequency (GHz)  3.5 
Oversampling rate of the TypeI, TypeII codebook 
6.1.1 Significance of dimension reduction
The basis matrix in (14) designed by exploiting the spatial consistency over the CSI of nearby UEs plays a critical role in the dimension reduction. To show that, two schemes are compared. The first one solves problem (6) to obtain without dimension reduction, while the second one solves problem (16) with . The basis matrix is constructed according to (17) in the second scheme. Both problem (6) and problem (16) are solved by the PRIMEpoweracce algorithm (PRIME for short) [31]. In the test, the BS only applies the random precoding .
Fig. 3 shows the averaged performance of the two schemes. ‘Prob. (6), GS’ stands for the scheme which solves problem (6) with , while ‘Prob. (16), GS’ stands for the scheme which solves problem (16) with firstly and then constructs . ‘GS’ represents the random precoding sampled from the complex Gaussian distribution. One can see that the performance of ‘Prob. (16), GS’ is elevated quickly with slightly increased , while that of ‘Prob. (6), GS’ is improved slowly. The correlation achieved by ‘Prob. (16), GS’ is always better than that of the TypeI codeword. However, ‘Prob. (6), GS’ needs about rounds of feedback to achieve a higher correlation than the TypeI codeword. Even with , there is still a gap on between the two schemes. The results are reasonable. As analyzed in [34, 35], the variable with a larger dimension in problem (6) requires more rounds of feedback to get a good sensing performance. Through our dimension reduction design, the dimension of is much smaller than the orginal , alleviating the requirement of feedback. In addition, it verifies our proposed basis matrix construction enables a good sensing recovery to .
6.1.2 Hybrid precoding design
The proposed hybrid precoding scheme is compared with the ‘GS’ precoding scheme. Both the two schemes solve problem (16).
Fig. 4 demonstrates that the ‘hybrid’ scheme outperforms the ‘GS’ scheme, under different . The reason behind is that the proposed in the hybrid precoding scheme can concentrate the measurements more on the major components of the CSI. Therefore, it alleviates the requirement of feedback to attain a good sensing performance in the wireless environment with copious propagation paths.
Besides, the impact of the ’s dimension is also tested. Fig. 5 compares the sensing performance of the proposed hybrid precoding scheme under different values of . It shows that the performance with a smaller value of is elevated more quickly. The scheme with achieves the best performance for . This is because it is easier to recover a lowdimensional in (16) with high accuracy under limited rounds of feedback. In comparison, the schemes with a larger dimension achieve the better performance when is larger, as they can better approximate the channel and characterize it when the number of feedbacks is sufficient to guarantee a good recovery of .