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Scalable user selection in FDD massive MIMO
EURASIP Journal on Wireless Communications and Networking volumeÂ 2021, ArticleÂ number:Â 193 (2021)
Abstract
User subset selection requires full downlink channel state information to realize effective multiuser beamforming in frequencydivision duplexing (FDD) massive multiinput multioutput (MIMO) systems. However, the channel estimation overhead scales with the number of users in FDD systems. In this paper, we propose a novel propagation domainbased user selection scheme, labeled as zeromeasurement selection, for FDD massive MIMO systems with the aim of reducing the channel estimation overhead that scales with the number of users. The key idea is to infer downlink user channel norm and interuser channel correlation from uplink channel in the propagation domain. In zeromeasurement selection, the basestation performs downlink user selection before any downlink channel estimation. As a result, the downlink channel estimation overhead for both user selection and beamforming is independent of the total number of users. Then, we evaluate zeromeasurement selection with both measured and simulated channels. The results show that zeromeasurement selection achieves up to 92.5% weighted sum rate of genieaided user selection on the average and scales well with both the number of basestation antennas and the number of users. We also employ simulated channels for further performance validation, and the numerical results yield similar observations as the experimental findings.
1 Introduction
Massive multiinput multioutput (MIMO) improves wireless communication, which is a key component of nextgeneration wireless networks [1,2,3,4,5,6,7,8,9,10,11,12,13]. A key challenge for operating massive MIMO in frequencydivision duplex mode is the large channel overhead in acquiring channel state information. However, a significant fraction of spectrum allocations worldwide are for frequencydivision duplexing (FDD) operation. Therefore, there is a significant demand to enable massive MIMO operation in FDD mode.
The scaling challenge of channel measurement in FDD massive MIMO systems is twofold. First is measuring a large number of channels per user. In [14], we proposed directional training, a scalable channel estimation method, that addressed the peruser channel measurement scalability challenge.
The second challenge is measuring the downlink channels for a large number of users to aid in user subset selection for massive MIMO. Note that the massive MIMObased system will likely support an order of magnitude more users per timefrequency slot compared to past generation systems. The stateoftheart user subset selection methods [15,16,17,18,19,20,21,22,23,24,25,26,27] require full downlink channel state information. Even with directional training, the downlink channel measurement overhead scales with the number of users. Meanwhile, the number of users considered for user selection can be much smaller than the number of selected users for downlink beamforming. Therefore, a scalable user selection scheme is needed for FDD massive MIMO. To achieve graceful scalability with the number of users, the challenge is to perform effective user selection before any downlink channel estimation.
In this paper, to reduce the large downlink channel estimation overhead that scales with the number of users in FDD massive MIMO systems, we develop a novel propagation domainbased user selection scheme, labeled as zeromeasurement selection. As the name suggests, zeromeasurement selection lets the basestation perform user selection without any downlink channel estimation. Zeromeasurement selection is driven by the experimental findings that the downlink interuser channel correlation and user channel norm can be reliably inferred from the uplink channel information in the propagation domain. Overall, our main contributions in this paper are as follows:

1
Based on the observed channel low dimensionality and partial reciprocity in the propagation domain, we investigate the potential of inferring downlink user channel norm and interuser channel correlation from uplink channel propagation domain information. For the downlink channel norm, we show that the inferred downlink channel norm from uplink channel measurements leads to only about 15% normalized error. For the downlink interuser channel correlation, we find out that the estimated interuser channel correlation based on uplink information in the propagation domain, including paths angles and amplitudes, brings in only about 10% normalized error.

2
Driven by experimental findings that both uplink inferred downlink interuser channel correlation and uplink inferred user channel norm provide good approximation with small errors, we develop zeromeasurement selection for FDD massive MIMO systems. The key idea of zeromeasurement selection is to select users based on inferred downlink interuser channel correlation and user channel norm from free uplink channel information in the propagation domain. Therefore, no downlink channel estimation overhead is incurred in the zeromeasurement selection. Zeromeasurement selection is developed in the wellknown proportionalfair form. There are three main steps in zeromeasurement selection. First, we extract uplink propagation domain dominant angles and amplitudes from uplink channel and construct downlink channel propagation domain for each user, respectively. Second, during each round of selection, we calculate usersâ€™ orthogonal component to selected users channel space. Third, we select the user with the largest estimated weighted rate based on the orthogonal component. The output of zeromeasurement selection is the selected user set that can be then used by downlink multiuser beamforming methods.

3
To evaluate zeromeasurement selection, we use both measured and simulated channels. Our measurement databased results show that zeromeasurement selection achieves up to 92.5% of the weighted sum rate of genieaided user selection on average when a 64antenna basestation selects 8 users out of 100 users for downlink beamforming. We then examine the performance scalability with both the number of basestation antennas and the number of users for zeromeasurement selection with both measured channels and 3GPPbased simulated channels. The numerical results show that zeromeasurement selection scales well, from average 90% weighted sum rate when a 64antenna basestation selects 8 users out of 100 users for downlink beamforming, to average 97% weighted sum rate when a 256antenna basestation selects 32 users out of 100 users for downlink beamforming; all comparisons are made with respect to the genieaided user selection.
The outline for the rest of the paper is as follows: SectionÂ 2.1 formulates the research problem for user selection. SectionÂ 2.2 illustrates the experimental findings that downlink interuser channel correlation and user channel norm can be effectively inferred from free uplink channel information in the propagation domain. SectionÂ 2.4 describes the proposed zeromeasurement selection. SectionÂ 3 provides experimental and numerical evaluation of zeromeasurement selection. Finally, SectionÂ 4 concludes this paper.
2 Methods
2.1 Problem formulation
We consider the singlecell FDD massive MIMO system, where an Mantenna basestation serves N singleantenna^{Footnote 1} users indexed on the set \(\{1,2, \ldots , N\}\). During each time slot, the basestation selects \(K < N\) users and performs downlink multiuser beamforming for data transmission to the selected K users. The selected user set index is denoted as \({\mathcal {S}} = \{s_1, s_2, \ldots , s_K\}\), where \({\mathcal {S}} \subset \{1,2,\ldots ,N\}\). For the selected K users, with downlink channels denoted as \({\mathbf {H}}_{{\mathcal {S}}} \in {\mathbb {C}}^{K \times M}\), the received signal at usersside \({\mathbf {y}}_{{\mathcal {S}}} \in {\mathbb {C}}^K\) can be modeled as:
where \({\mathbf {x}}_{{\mathcal {S}}} \in {\mathbb {C}}^{K}\) is the transmitted signals vector, \({\mathbf {W}}_{{\mathcal {S}}} \in {\mathbb {C}}^{M \times K}\) is the beamforming weights matrix based on zeroforcing beamforming, and \({\mathbf {n}}_{{\mathcal {S}}} \in {\mathbb {C}}^K\) is the additive noise with elements that follow standard complex Gaussian distribution.
When selecting users to be part of set \({\mathcal {S}}\) for downlink beamforming, different optimization goals can be considered, e.g., maximizing sum rate or minimizing delay. One wellknown fair user selection scheme is proportionalfair user selection, which maximizes the following weighted sum rate to find the best user set \({\mathcal {S}}\)
where \(\mu _{s_k} \left( t \right)\) is the weight of user \(s_k\) in time slot t, which is known at the basestation and can be determined by average throughput in the previous time window [15] or data queue length [16]. \(R_{s_k} \left( t \right)\) is the rate of user \(s_k\) in time slot t. To find the user set that maximizes the weighted sum rate in the above equation prior to downlink data transmission, downlink channel state information (CSI) of all the N users at the basestation side is necessary to estimate the weighted rate for each user during selection. However, to obtain downlink CSI, the overhead scales with the number of users N, which can be very large (e.g., 100s of users are common in a single cell). To reduce the downlink channel estimation overhead and make user selection scale with the number of users, we seek to perform downlink user selection without any downlink channel training. One possible direction is to investigate the feasibility of inferring partial downlink channel information from available information at the basestation side. Towards that end, we ask two questions:

Q1
What partial downlink channel information can we infer from uplink channel estimates, considering that full channel reciprocity does not hold in FDD mode?

Q2
Can the partial knowledge (Q1) be used effectively for user selection by the base station?
We first answer QuestionÂ 1 in SectionÂ 2.2 and investigate the effectiveness of inferring downlink interuser channel correlation and user channel norm from uplink channel information in the propagation domain based on measured channel dataset. Then, to answer QuestionÂ 2, we propose a scalable user selection scheme that lets the base station perform select users based on the inferred downlink interuser channel correlation and user channel norm from uplink.
2.2 Experimental findings on estimating downlink channel properties from uplink
In this section, we answer the first question posed above, on what partial downlink channel information can be inferred from uplink channel information? The key information required for effective user selection includes user channel norm and interuser channel correlation. Therefore, we investigate the possibility of inferring downlink user channel norm and interuser channel correlation from uplink channel information, such that no additional downlink channel estimation is needed for user selection.
2.3 FDD massive MIMO channels in propagation domain
In propagation domain, full downlink channel of nth user can be represented with multiple paths as
where the channel consists of \(L_n\) paths, and the lth path has complex coefficient \(\beta _{nl}\) and angle of departure with elevation as \(\theta _{nl}\) and azimuth as \(\varphi _{nl}\). The array response vector \({\mathbf {a}}_f \left( \theta ,\varphi \right)\) of an Mantenna uniform plane array consisting of \(M_r\) rows and \(M_c\) columns is defined as:
where f is the signal frequency, \(\uplambda\) is the signal wavelength, d is the antenna spacing, \(\theta\) is the elevation angle and \(\varphi\) is the azimuth angle.
Similarly, full uplink channel can also be characterized in propagation domain. While uplink and downlink full channel reciprocity does not hold in FDD mode, in our previous work [14], we found out that uplink and downlink channels share approximately the same lowdimensional propagation space. Thus, in this paper, we examine the potential of inferring downlink interuser channel correlation and user channel norm from uplink channel propagation domain information. Here we employ our measured FDD massive MIMO channels [14], with all the details explained therein. Overall, the channel dataset includes FDD massive MIMO channels corresponding to 21 nonlineofsight and 4 lineofsight user locations, and the basestation is equipped with an 8row 8column uniform plane array. For each basestation and user pair, two 20MHz widebands channels, each with 52 OFDM subcarriers and separated by about 72 MHz, are measured across around 5000 time frames.
2.3.1 Inferring downlink channel in propagation domain
Starting fromÂ (3), full downlink channel in propagation domain can be further extended as:
where downlink channel can be approximated with \(L_{\mathrm {D}n}\) dominant paths with largest power, and the lth dominant path has complex coefficient \(\beta _{\mathrm {D}nl}\) and angle of departure with elevation as \(\theta _{\mathrm {D}nl}\) and azimuth as \(\varphi _{\mathrm {D}nl}\). \({\mathbf {e}}_{\mathrm {a}n}\) denotes the corresponding dominant path approximation error. Also, using estimated \({\hat{L}}_{\mathrm {D}n}\) paths with complex coefficients \({\hat{\beta }}_{\mathrm {D}nl}\) and angle of departures with elevation as \({\hat{\theta }}_{\mathrm {D}nl}\) and azimuth as \({\hat{\varphi }}_{\mathrm {D}nl}\) to construct downlink channel will result in an additional estimation error \({\mathbf {e}}_{\mathrm {m}n}\).
Next, to estimate downlink channel in propagation domain, we need to first estimate dominant angles \(\left( {\hat{\theta }}_{\mathrm {D}nl},{\hat{\varphi }}_{\mathrm {D}nl}\right) , l=1,2,\ldots ,{\hat{L}}_{\mathrm {D}n}\) and corresponding complex coefficient \({\hat{\beta }}_{\mathrm {D}nl}, l=1,2,\ldots ,{\hat{L}}_{\mathrm {D}n}\). Using our results in [14], we extract dominant angles and corresponding amplitudes from the uplink channel estimates. However, the phase information is not known, so full downlink channel state information cannot be inferred from the uplink estimates. But we observed that estimated dominant angle vectors of one user are approximately orthogonal with each other. Therefore, even without phase information, we aim to use the estimated dominant angles and corresponding amplitudes from uplink channel to infer peruser channel norm and interuser channel correlation instead.
Using estimated dominant angles and corresponding amplitudes from uplink channel, we denote estimated downlink dominant angle vectors as:
where \(\left( {\hat{\theta }}_{\mathrm {U}nl},{\hat{\varphi }}_{\mathrm {U}nl}\right) , l=1,2,\ldots ,{\hat{L}}_{\mathrm {U}n}\) are estimated dominant angles from uplink channel employing the wellknown MUSIC estimator with spatial smoothing [28], and \({\hat{\beta }}_{\mathrm {U}nl}, l=1,2,\ldots ,{\hat{L}}_{\mathrm {U}n}\) are the corresponding estimated amplitudes obtained using leastsquare estimator, with details shown in [14].
We aim to estimate downlink user channel norm and interuser channel correlation from estimated angle space vectors \(\widehat{{\mathbf {C}}_n}\). To evaluate the effectiveness of estimating downlink user channel norm and interuser channel correlation from uplink channel, we employ our measured FDD massive MIMO channels [14]. Comparing downlink channel \({\mathbf {h}}_n\) and estimated dominant angle vectors \(\widehat{{\mathbf {C}}_n}\) from uplink channel, as shown in Fig.Â 1, there are three sources of error that can affect the estimation performance:

1
Dominant Paths Approximation Error: The approximate error \({\mathbf {e}}_{\mathrm {a}n}\) will lead to the performance gap between full downlink \({\mathbf {h}}_n\) and approximated downlink channel with downlink dominant angles as shown in Eq.Â 5.

2
Missing Phase Information Error: Without the phase information, we cannot obtain full downlink channel from estimated dominant angle vectors. Thus, the missing phase information will also affect estimation accuracy.

3
Dominant Angle Vectors Estimation Error: Downlink dominant angles and amplitudes are estimated from uplink channel; as a result, the dominant angle vectors estimation error will also degrade the estimation performance.
2.3.2 Downlink user channel norm estimation
User channel norm is one of the key information required for effective user selection. Thus, first, we evaluate the effectiveness of estimating downlink user channel norm from estimated angle space vectors set \(\widehat{{\mathbf {C}}_n}\), which is shown in Eq.Â 6. We estimate downlink user channel norm of nth user as follows
where \(\Vert .\Vert _2\) denotes \(l_2\)norm and \(\Vert .\Vert _{\mathrm {F}}\) Frobenius norm. The approximation is based on the observation that estimated dominant angle vectors of one user are approximately orthogonal with each other, i.e., \(\left {\mathbf {a}}\left( {\hat{\theta }}_{\mathrm {U}ni},{\hat{\varphi }}_{\mathrm {U}ni}\right) ^H {\mathbf {a}}\left( {\hat{\theta }}_{\mathrm {U}nj},{\hat{\varphi }}_{\mathrm {U}nj}\right) \right \approx 0, \forall i \ne j\). And the resulting error by the approximation is characterized as Missing Phase Information Error explained above. The cumulative distribution function of estimated downlink user channel norm is shown in Fig.Â 2.
Finding 1  Estimated Downlink Channel Norms From Uplink Leads to Small Error: From Fig.Â 2, we observe that utilizing estimated angle vectors from uplink channel, including dominant angles and amplitudes, to estimate downlink user channel norm, resulting in small error ranging from 0 to 15%.
Explanation for Finding 1: To understand the gap between uplink inferred downlink user channel norm and the actual one, we consider two more ways to estimate downlink user channel norm, including utilizing approximated downlink channel with downlink dominant angles and utilizing downlink dominant angles and amplitudes, to understand the aforementioned three different effects. First, approximation error effect causes error up to 15% error. Second, missing phase information effect leads to negligible error. Third, uplink/downlink angles and amplitudes mismatch effect results in up to 5% error. Thus, dominant paths approximation error effect is the dominant one that causes the estimation error. Also, channel norm inferred from dominant angles and amplitudes tends to be smaller due to the fact that dominant angles only capture part of downlink channel power.
2.3.3 Downlink interuser channel correlation estimation
Interuser channel correlation is the other one of the key information required for effective user selection. Thus, we evaluate the effectiveness of estimating downlink interuser channel correlation from estimated angle space vectors set \(\widehat{{\mathbf {C}}_n}\) in Eq.Â 6.
Consider users n and p. We denote their downlink channels as \({\mathbf {h}}_n\) and \({\mathbf {h}}_p\) , respectively. The downlink interuser channel correlation is formulated as the inner product of their channel vectors
As the channel correlation can directly affect multiuser beamforming rate, it is one of the key information required for effective user selection.
Then, we aim to estimate the downlink interuser channel correlation utilizing uplink dominant angles and amplitudes defined in Eq.Â 6. The estimated downlink interuser channel correlation is formulated as:
The cumulative distribution function of estimated downlink interuser channel correlation is shown in Fig.Â 3.
Finding 2  Estimated Downlink Interuser Channel Correlation From Uplink Leads to Small Error: From Fig.Â 3, we observe that utilizing uplink propagation domain information, including dominant angles and amplitudes, to estimate downlink interuser channel correlation brings in error range from 0 to 0.3 in terms of channel correlation estimation error.
Explanation for Finding 2: Missing phase information error is the dominant contribution to interuser channel correlation estimation error, causing up to 0.3 correlation gap. This is mainly due to the fact that channel vectorconstructed space is the subspace of dominant angles response vectorsconstructed space.
2.4 Proposed scheme: zeromeasurement selection
In this section, we answer the second question  Can the basestation perform effective user selection based on partial downlink channel information that is inferred from uplink only? Inspired by the finding that downlink interuser channel correlation and user channel norm can be effectively inferred from free uplink channel information in the propagation domain, we propose a scalable user selection scheme, labeled as zeromeasurement selection for FDD massive MIMO systems. As the name suggests, zeromeasurement selection lets the basestation select users for downlink multiuser beamforming before any downlink channel estimation and thus avoid the channel estimation overhead that scales with the number of users. We first provide an overview of the proposed zeromeasurement selection scheme, followed by details of each step therein.
2.4.1 Zeromeasurement selection overview
We develop zeromeasurement selection to achieve proportionalfair selection. As described in SectionÂ 2.1, we consider the singlecell FDD massive MIMO system, where an Mantenna basestation selects K out of N users in each time slot to perform multiuser beamforming. The goal of zeromeasurement selection is to solve:
The key to solving the above equation is to estimate the rate for each user during selection. When full downlink CSI of all the N users is available at the base station, optimal selection can be obtained using exhaustive search and search all possible user set, to find the one that maximizes the weighted sum rate. In [18], the authors proposed a low complexity greedy type scheme that selects users based on interuser channel correlation and user channel norm; here we label the scheme as fullchannel selection. However, as discussed in SectionÂ 2.1, CSI of all the N users is required and large downlink estimation overhead that scales with the number of users N O(N) incurs to obtain downlink CSI.
In contrast, zeromeasurement selection does not require any downlink channel estimation. Thus, zero overhead incurs during user selection and only O(K), where K is the number of selected users, incurs to obtain downlink CSI of the selected K users for the subsequent downlink beamforming usage. And the goal is zeromeasurement selection to achieve performance close to fullchannel selection.
Overall, there are three main steps in zeromeasurement selection, as shown in Fig.Â 4. First, extract usersâ€™ uplink propagation domain information, including dominant angles and amplitudes. Second, during each round of selection, calculate usersâ€™ orthogonal component to selected user channel space. Third, select the user with the larger weighted sum rate estimated from the orthogonal component. Next, we present the details of each step.
2.4.2 Zeromeasurement selection details
First, in step 1, extract uplink propagation domain information, including dominant angles and amplitudes. We assume that free estimated uplink CSI is available at the base station^{Footnote 2} and denote the uplink CSI as \(\hat{{\mathbf {h}}}_{\mathrm {U}n}, n = 1, 2, \ldots , N\). We employ the wellknown MUSIC estimator with spatial smoothing [28] to estimate dominant path angles and leastsquare estimator to estimate the corresponding dominant path amplitudes. All the estimation details are shown in SectionÂ 2.2. The estimated dominant angles of the nth user are denoted as \(\left( {\hat{\theta }}_{\mathrm {U}nl},{\hat{\varphi }}_{\mathrm {U}nl}\right) , l=1,2,\ldots ,{\hat{L}}_{\mathrm {U}n}\) and the estimated dominant path amplitudes are denoted as \(\Vert {\hat{\beta }}_{\mathrm {U}nl}\Vert , l=1,2,\ldots ,{\hat{L}}_{\mathrm {U}n}\), where \({\hat{L}}_{\mathrm {U}n}\) is the estimated number of uplink dominant paths. Then, we construct estimated downlink dominant angle vectors \(\widehat{{\mathbf {C}}}_n\) using definition in Eq.Â 6.
Second, in Step 2, calculate the channel orthogonal component to selected user channel space. The selected channel space is denoted as \(\hat{{\mathbf {S}}} \in {\mathbb {C}}^{M \times M}\) and initialized as \(\hat{{\mathbf {S}}} = {\mathbf {0}}_{M \times M}\). For the users to be selected, we calculate the channel orthogonal component of nth user as
Third, in Step 3, estimate user weighted rate and select the users with the largest estimated weighted rate during this round of selection. Based on the orthogonal propagation domain \(\widehat{{\mathbf {D}}}_n\), the estimated weighted rate of nth users is formulated as:
where P is the downlink transmission power; \(\Vert .\Vert\) stands for the Frobenius norm. The user with the largest weighted rate will be selected during the current round of selection. We denoted the selected user index as \(s_k\) for kth round of selection, where \(s_k \in {1, 2, \ldots , N}\). After selection, the selected user channel space will be updated by adding up the \(s_k\)th user propagation domain as
The key part of zeromeasurement selection is to estimate the weighted rate of each user in Step 3. Based on the findings that uplinkinferred downlink user channel norm and interuser channel correlation bring in small amount of estimation error; as a result, the estimated channel norm project to the complementary space of selected user channel space will be close to actual downlink channel norm as
where \({\mathbf {h}}_{\mathrm {D}n}\) is the downlink channel of nth user. Therefore, the estimated weighted sum rate \({\widehat{R}}_{\text {weighted},n}\) will be close to the actual one and zeromeasurement selection let the basestation select users based on the approximated weighted sum rate. Although there will still certain performance gap between zeromeasurement selection and fullchannel selection, which will be evaluated in the next section.
Combining the above three main steps, the zeromeasurement selection algorithm is summarized as follows:

Input:
Uplink CSIs of all the N users \(\hat{{\mathbf {h}}}_{\mathrm {U}n} \in {\mathbb {C}}^{M}, n = 1, 2, \ldots , N\).

Output:
Selected user set \({\mathcal {S}}\), where \({\mathcal {S}} = K\).

Step 1:
First extract uplink dominant path angles and amplitudes and construct estimated downlink dominant angle vectors \(\widehat{{\mathbf {C}}}_n, n=1,2,\ldots ,N\) for all the N user using definition in Eq.Â 6. Then initialize the remaining user set as \({\mathcal {T}} = \left\{ 1, 2, \ldots , N \right\}\), selected user set as \({\mathcal {S}} = \phi\), selected user channel space as \(\widehat{{\mathbf {S}}} = {\mathbf {0}}_{M \times M}\), loop index as \(i=1\).

Step 2:
For all users \(n \in {\mathcal {T}}\), calculate the orthogonal component \(\widehat{{\mathbf {D}}}_n\) as in Eq.Â 11.

Step 3:
Select user \(s_i\) that has the largest weighted rate as
$$\begin{aligned} s_i = \mathrm {arg} \, \underset{n \in {\mathcal {T}}}{ \mathrm {max} } \,\, \mu _n \log \left( 1 + \frac{P}{K} \Vert \widehat{{\mathbf {D}}}_n \Vert _{\mathrm {F}}^2 \right) , \end{aligned}$$(15)then update selected user set as \({\mathcal {S}} \leftarrow {\mathcal {S}} \cup \{ s_i \}\), selected user channel space as
$$\begin{aligned} \widehat{{\mathbf {S}}} \leftarrow \widehat{{\mathbf {S}}} + \widehat{{\mathbf {D}}}_{s_i}^H \left( \widehat{{\mathbf {D}}}_{s_i} \widehat{{\mathbf {D}}}_{s_i}^H \right) ^{1} \widehat{{\mathbf {D}}}_{s_i}, \end{aligned}$$(16)remaining user set as
$$\begin{aligned} {\mathcal {T}} = \left\{ n \in {\mathcal {T}}, k \ne s_i, \frac{\Vert \widehat{{\mathbf {C}}}_n \widehat{{\mathbf {D}}}_{\left( s\right) }^H \left( \widehat{{\mathbf {D}}}_{\left( s\right) } \widehat{{\mathbf {D}}}_{\left( s\right) }^H \right) ^{1} \widehat{{\mathbf {D}}}_{\left( s\right) } \Vert }{\Vert \widehat{{\mathbf {C}}}_n\Vert } < \alpha \right\} \end{aligned}$$(17)where \(\alpha\) is a small positive constant set to remove users with large correlation to the selected user, and loop index as \(i \leftarrow i + 1\). If \(i < K\), go to Step 2. Otherwise the algorithm is finished.
3 Results and discussion
In this section, we evaluate the performance of zeromeasurement selection. We first employ measured FDD massive MIMO channels to evaluate the performance of zeromeasurement selection by comparing to full channelbased selection and then examine the scalability with both the number of basestation antennas and the number of users for zeromeasurement selection. Then, for further performance validation, we employ the 3GPP spatial channel model to evaluate the performance and examine the scalability for zeromeasurement selection.
3.1 Experimental results
The details of measured channels are shown in [14]. Since the goal of downlink user selection is to maximize the weighted sum rate of K selected users, as shown in Eq.Â 10, we take the weighted sum rate of selected users as the key metric to evaluate the performance of zeromeasurement selection.
For comparison, we implement the fullchannel selection [18], which assumes perfect downlink CSIs of all the N users are available. To observe the performance gap between zeromeasurement selection and fullchannel selection, we consider the system where a 64antenna basestation selects \(K = 8\) users out of \(N = 100\) users for downlink. The cumulative distribution function of weighted sum rate based on 1000 cases is shown in Fig.Â 5. In each case, the 100 usersâ€™ uplink and downlink channels are randomly selected from the measured channel dataset and the weights are assumed i.i.d with U(0,Â 1] for simplicity. Then, we set different numbers of antennas and different numbers of users to evaluate the performance scalability for zeromeasurement selection.
Finding 3  Zeromeasure Selection Performs Close to Fullchannel Selection: From Fig.Â 5, we can observe that zeromeasure selection achieves average 92.5% weighted sum rate of fullchannel selection one. While in terms of overhead, \(O(N=100)\) downlink channel estimation overhead incurs to obtain full downlink channel, while no channel estimation incurs for zeromeasure selection and only \(O(K=8)\) channel estimation overhead is needed to perform effective beamforming to selected K users.
Explanation for Finding 3: To understand the performance gap between zeromeasure selection and fullchannel selection, we also implement two other user selection schemes, one is user selected based on downlink dominant paths approximated channel, and the other one is user selection based on downlink dominant angle vectors. We find out that the main source of performance gap comes from the missing dominant path phase information (3.5%) and dominant angles with amplitudes mismatch effect (4%). For the missing dominant path phase information effect, without phase information, the estimated user weighted rate will be smaller than the actual one, as explained in the previous section. For the dominant angles with amplitudes mismatch effect, both inferred interuser channel correlation and user channel norm from uplink will have estimation error and thus affect zeromeasure selection performance.
Finding 4  Zeromeasure Selection Scales with Basestation Array Size: From Fig.Â 6, we can observe that the weighted sum rate based on zeromeasurement selection increases with the number of basestation antennas, from 7.25Â bps/Hz with 16 antennas to 22Â bps/Hz with 64 antennas. Compared to fullchannel selection, zeromeasurement selection achieves a larger relative weighted sum rate with more basestation antennas, from 59% with 16 antennas to about 93% with 64 antennas. Besides, the weighted sum rate gap between zeromeasurement selection and fullchannel selection decreases with the number of basestation antennas, from 5.05Â bps/Hz with 16antenna to 1.6Â bps/Hz with 64antenna. A similar trend is also shown in Fig.Â 8.
Explanation for Finding 4: The main source of zeromeasurement selection performance loss comes from the missing dominant path phase effect and dominant angles with amplitudesâ€™ mismatch effect, while both effects will have a smaller impact on zeromeasurement selection with more basestation antennas. As a result, the weighted sum rate gap between zeromeasurement selection and fullchannel selection decreases with the number of base station.
Finding 5  Zeromeasure Selection Scales with the Number of Selected Users: For the scalability with the number of selected users, from Fig.Â 7, we can observe that the weighted sum rate based on zeromeasurement selection increases with the number of selected users, from 8.1Â bps/Hz with 2 users to 21.8Â bps/Hz with 8Â users. While the weighted sum rate gap compared to fullchannel selection increases slightly with the number of selected users, from 0.6Â bps/Hz with 2 users to 1.7Â bps/Hz with 8 users. A similar trend is also shown in Fig.Â 8.
Explanation for Finding 5: Since both the missing path phase information effect and dominant angles with amplitudes mismatch effect exist for each selected user, the weighted sum rate gap between zeromeasurement selection and fullchannel selection increases with the number of selected users.
3.2 Numerical results
To further validate experimental findings on zeromeasurement selection in a larger array regime, we employ the 3GPP spatial channel model [29], to evaluate the performance of zeromeasurement selection. The system setup here is similar to experimental setup, and the results are shown in Figs.Â 9, 10, and 11.
First, from Fig.Â 9, we can observe that zeromeasure selection performs close to fullchannel selection, achieving an average 90% weighted sum rate with 64 antennas at the base station, which matches FindingÂ 3. For the scalability with the basestation array size, we can observe that the weighted sum rate based on zeromeasurement selection increases with the number of basestation antennas, from 12.4Â bps/Hz with 16 antennas to 41.6Â bps/Hz with 1024 antennas, and the weighted sum rate gap compared to fullchannel selection decreases with the number of basestation antennas, which matches FindingÂ 4 in a larger number of basestation antennas regime.
Second, for the scalability with the number of selected users, from Fig.Â 10, we can observe that the weighted sum rate based on zeromeasurement selection increases with the number of selected users, from 18.3Â bps/Hz with 4 users to 95.5Â bps/Hz with 32 users when the basestation is equipped with 256 antennas. While the weighted sum rate gap compared to fullchannel selection increases slightly with the number of selected users, from 1.3Â bps/Hz with 4 users to 2.5Â bps/Hz with 32 user, and the trend matches FindingÂ 5.
Third, we can observe the combined scalability with both the number of basestation antennas and the number of selected users from Fig.Â 11. In more detail, the ratio between the number of basestation antennas and the number of selected users is fixed as 8. The results show that, compared to fullchannel selection, zeromeasurement selection achieves average 90% weighted sum rate when a 64antenna base station selects 8 users out of 100 users for downlink beamforming. And the weighted sum rate percentage goes up to 97% when a 256antenna basestation selects 32 users out of 100 users for downlink beamforming.
4 Conclusions
To sum up, motivated by the experimental findings that both uplink inferred downlink interuser channel correlation and uplink inferred user channel norm provide good approximation with small errors, we develop zeromeasurement selection for FDD massive MIMO systems. Zeromeasurement selection utilizes the uplink channel in the propagation domain to estimate the weighted sum rate and selects users that maximize the weighted sum rate as in proportionalfair selection. Thus, no downlink channel estimation overhead incurs in zeromeasurement selection.
Then, we evaluate zeromeasurement selection with both measured and simulated FDD massive MIMO channels. The results show that zeromeasurement selection performs close to fullchannelbased user selection, with only 7.5% performance loss on average when a 64antenna selects 4 users out of 100 users for downlink beamforming. Also, zeromeasurement selection scales well with both the number of basestation antennas and the number of users, as shown in both experimental and numerical results.
In this paper, while we develop zeromeasurement selection in the proportionalfair form, the selection scheme can be easily extended to other forms of user selection.
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Notes
The scheme proposed here can be extended to multiantenna users scenario, but due to experiment data limitation, we mainly focus on singleantenna users scenario.
Uplink CSI can be obtained from uplink transmission and for a wideband system, uplink CSI of one OFDM subcarrier only is enough.
Abbreviations
 FDD:

Frequencydivision duplexing
 MIMO:

Multiinput multioutput
 ZMS:

Zeromeasurement selection
 CSI:

Channel state information
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The authors were partially supported by NSF Grant 1518916 and support from Qualcomm, Inc.
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Zhang, X., Sabharwal, A. Scalable user selection in FDD massive MIMO. J Wireless Com Network 2021, 193 (2021). https://doi.org/10.1186/s13638021020734
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DOI: https://doi.org/10.1186/s13638021020734