 Research
 Open Access
 Published:
Analysis of scalable channel estimation in FDD massive MIMO
EURASIP Journal on Wireless Communications and Networking volume 2023, Article number: 29 (2023)
Abstract
One of the key ideas for reducing downlink channel acquisition overhead for FDD massive MIMO systems is to exploit a combination of two assumptions: (i) the dimension of channel models in propagation domain may be much smaller than the nextgeneration basestation array sizes (e.g., 64 or more antennas), and (ii) uplink and downlink channels may share the same lowdimensional propagation domain. Our channel measurements demonstrate that the two assumptions may not always hold, thereby impacting the predicted performance of methods that rely on the above assumptions. In this paper, we analyze the error in modeling the downlink channel using uplink measurements, caused by the mismatch from the above two assumptions. We investigate how modeling error varies with basestation array size and provide both numerical and experimental results. We observe that modeling error increases with the number of basestation antennas, and channels with larger angular spreads have larger modeling error. Utilizing our modeling error analysis, we then investigate the resulting beamforming performance rate loss. Accordingly, we observe that the rate loss increases with the number of basestation antennas, and channels with larger angular spreads suffer from higher rate loss.
1 Introduction
Massive multiinput and multioutput (MIMO) [1,2,3,4,5] uses many antennas at the basestation to improve communication in diverse ways [6,7,8,9,10,11,12,13]. However, due to the large number of antennas, the key challenge in enabling FDD^{Footnote 1} massive MIMO is that the downlink channel acquisition overhead scales with the basestation array size. Several recent works have proposed methods to exploit lower dimensionality of the channel with propagation domain channel characterizations to address this channel estimation challenge [14,15,16,17,18,19,20,21,22,23,24,25]. The proposed schemes leverage two key assumptions: (i) the cardinality of the alternate channel characterizations is much smaller than and independent of the number of basestation antennas, and (ii) uplink and downlink channels share the same lowdimensional propagation characteristics. With the two assumptions, the downlink channel can be parameterized by a few coefficients and an estimate of these coefficients can be obtained from uplink channel.
However, our recent channel measurementbased work [26] demonstrated that the two assumptions may not hold exactly. Our two main findings based on measured channels were: (i) dominant angles do not capture all the channel power, even though a significant fraction can be captured, and (ii) uplink and downlink dominant angles are not exactly the same, even though the angle correlation can be high. In short, the two assumptions can be good assumptions but not guaranteed to hold unilaterally. In the same paper [26], we had also proposed a directional training scheme, where the basestation trains downlink channel with estimated uplink dominant angles and, in essence, relies on approximation and reciprocity of channel in the propagation domain. However, the mismatch between the assumed channel model and the actual channel model can lead to downlink beamforming performance loss. Therefore, to quantify the possible performance gaps of training downlink channel in the propagation domain using uplink channel measurements, it is important to revisit the two key assumptions on FDD massive MIMO channels and quantify the loss in performance due to each of these assumptions.
In this paper, we focus on the fundamentals to analyze the modeling error in approximating downlink channel with uplink dominant angles in the propagation domain for FDD massive MIMO channels. With the proposed error analysis, we aim to answer two questions:

Q1
How does the modeling error scale with the number of basestation antennas?

Q2
How does the modeling error vary in different propagation environments?
Answering Question 1 is important to quantify the performance scalability in the largearray regime, which will have implications as the number of antennas is scaled in upcoming generations of massive MIMO systems. Similarly, answering Question 2 is important to understand the performance dependency on different channel scenarios. In particular, we will show that the channel angle spread is the important channel parameter that impacts modeling error.
Based on the modeling error analysis, we investigate the impact of downlink beamforming performance in training downlink channel with estimated uplink dominant angles. We quantify the downlink beamforming rate loss and study scalability with the basestation array size and dependency on the channel angle spread for the rate loss. Overall, our main contributions in this paper are as follows:

1
We define the modeling error to quantify the normalized error of approximating downlink channel with uplink dominant angles. There are two factors that affect modeling error. First is that dominant angles do not capture all the channel power; we label this error as approximation error. Second is that uplink and downlink dominant angles are not exactly the same; we label this as mismatch error. The two errors contribute additively toward the overall modeling error.

2
We employ the 3GPP spatial channel model [27] to investigate scalability with the basestation array size and dependency on the channel angle spread for modeling error. Using the channel model, we conduct extensive numerical simulations to examine modeling error in the finite array regime. We observe that modeling error increases with the number of basestation antennas, and a larger channel angle spread yields a larger modeling error.

3
We further validate our numerical observations utilizing our measured channels. The main finding is that our experimental results match the observations from the numerical results. We find that modeling error increases with the number of basestation antennas, from 2% as the average modeling error when the basestation is equipped with 4 antennas, to 28% as the average modeling error when the basestation is equipped with 64 antennas for nonlineofsight channels. The other observation is that larger angle spread in angle of arrivals leads to larger modeling error, with average modeling error as 28% for nonlineofsight channels, compared to the average modeling error as 13% for lineofsight channels when the basestation is equipped with 64 antennas.

4
To investigate the performance impact of modeling error, we also quantify the resulting downlink beamforming rate loss. Similar to modeling error, we provide both numerical and experimental results. From both numerical and experimental results, we observe that the rate loss increases with the number of basestation antennas, and more distributed power channel will bring in larger rate loss. Also, even though the rate loss increases with the number of basestation antennas, we find out resulted beamforming rate still increases with array size. So we conclude that beamforming based on scalable channel estimation schemes that exploit the propagation domain, e.g., directional training [26], still benefits from the array gain in FDD massive MIMO.
As a future extension, the proposed approach can be used to understand the impact of modeling error on other methods [14,15,16,17,18,19,20,21,22,23,24,25].
The rest of the paper is outlined as follows. Section 2 formulates the research problem and defines the model error to analyze channel estimation errors in FDD massive MIMO. Section 3 provides both numerical and experimental results of modeling error and evaluates the performance impact of modeling error. Finally, Sect. 4 concludes this paper.
2 Methods
2.1 System model
We consider a singlecell FDD massive MIMO system where an Mantenna basestation serves K singleantenna users through downlink beamforming. With downlink channel as \({\textbf{H}} \in {\mathbb {C}}^{K \times M}\) and beamforming weights matrix as \({\textbf{W}} \in {\mathbb {C}}^{M \times K}\), the received signals at the mobile user can be written as
where \({\textbf{x}} \in {\mathbb {C}}^{K}\) is the transmitted signals and \({\textbf{n}} \in {\mathbb {C}}^K\) is the additive noises that follow standard complex Gaussian distribution.
To design beamforming weights \({\textbf{W}}\) for effective downlink multiuser beamforming, e.g., conjugate beamforming or zeroforcing beamforming, the key step is to estimate downlink channel state information at the basestation side. Therefore, next, we first present details of FDD massive MIMO channels, including propagation domain channel model and measured channels, and then show potential errors of existing scalable channel estimation schemes in FDD massive MIMO.
2.2 FDD massive MIMO channels
2.2.1 Propagation domain channels
We adopt the popular geometrical raytracing approach and employ the 3GPP spatial channel model [27] to model FDD massive channels in the propagation domain. Note that the main propagation mechanisms include lineofsight, reflection, diffraction, and scattering of the transmitted electromagnetic signals. In FDD mode, even though uplink and downlink transmit at different frequency bands, the frequency gap between uplink and downlink is relatively small in many cases, e.g., less than 100 MHz in sub6 GHz FDD bands. As a result, with the proximity of the wavelengths, uplink and downlink can be approximated to undergo through the same propagation paths and have the same amplitude for each corresponding path. However, since the phase is very sensitive to wavelength difference, the phases are often modeled as uniform i.i.d. random, \(U[0, 2\pi )\).
Therefore, consider the system where the basestation is equipped with an M antennas uniform plane array, consisting of \(M_r\) rows and \(M_c\) columns. We use the geometrical raytracing approach as illustrated in Fig. 1. First, the downlink channel with frequency \(f_{\textrm{D}}\) between the Mantenna basestation and a singleantenna user can be modeled as
where the channel consists of I clusters and the ith cluster consists of \(J_i\) paths; the jth path of ith cluster has power \(g_{ij}\), independent phase \(\phi _{\textrm{D}ij} \sim U[0, 2\pi )\) and angle with elevation as \(\theta _{ij}\) and azimuth as \(\varphi _{ij}\). The array response vector \({\textbf{a}}_f \left( \theta ,\varphi \right)\) corresponding to the \(M_r\) rows and \(M_c\) columns uniform plane array is defined as
where f is the received signal frequency; \(\lambda\) is the signal wavelength; d is the antenna spacing; \(\theta\) is the elevation angle; and \(\varphi\) is the azimuth angle.
Then, accordingly, using the geometrical raytracing approach, the corresponding uplink channel operated at a different frequency band \(f_{\textrm{U}}\) of the same basestation user pair can be modeled as
For small frequency differences, uplink and downlink channel will have the same number of clusters I and same number of paths \(J_i\) in the ith cluster. The jth path of ith cluster has the same power \(g_{ij}\) and same angles \((\theta _{ij}\), \(\varphi _{ij})\) as downlink channel. The only different channel parameter is the phase component, with value i.i.d. in \(U[0, 2\pi ),\) respectively.
2.2.2 Measured channels
The measured FDD massive MIMO channels are presented in our previous work [26], with all the details explained therein. Overall, the channel dataset includes FDD massive MIMO channels corresponding to 21 nonlineofsight and 4 lineofsight user locations. For each user location, two 20 MHz wideband channels, each with 52 OFDM subcarriers and separated by about 72 MHz, are measured across around 5000 time frames. Further, the basestation is equipped with an 8row 8column uniform plane array, as shown in Fig. 2 with plots from [26]. Overall, the entire dataset includes 52 (subcarriers) × 52 (subcarriers) × 25 (user locations)= 67600 FDD massive MIMO uplink/downlink channel instance pairs, including 56784 nonlineofsight ones and 10816 lineofsight ones.
2.3 Scalable channel estimation in FDD massive MIMO
To investigate the potential error of downlink channel estimation, here we focus on the singleuser case only. Without loss of generality, the user downlink channel is denoted as \({\textbf{h}}_{\textrm{D}} \in {\mathbb {C}}^{M}\). Based on the propagation domain channel model, as shown in Eq. 2, downlink channel model can be rewritten as
where
denotes full propagation domain and
coefficients in the propagation domain.
In scalable channel estimation schemes that exploit channel lowdimensional domain, there are two main steps. First, estimate the lowdimensional propagation domain \(\hat{{\textbf{S}}} \in {\mathbb {C}}^{M \times L}, L < M\) for downlink training, where L is the number of training vectors in the estimated domain. Second, estimate domain coefficients \(\hat{{\textbf{b}}} \in {\mathbb {C}}^L\) via downlink training and uplink feedback. After that, the downlink channel can be reconstructed as
Previous works focus on domain coefficients estimation as in the second step only and assume perfect knowledge of the channel propagation domain. However, the error in propagation domain estimation \(\hat{{\textbf{S}}}\) also contributes to channel estimation error. Therefore, it is important to investigate the impact of propagation domain estimation error.
Since we aim to quantify the propagation domain modeling error—normalized error of approximating downlink channel with uplink dominant angles, next we ask and answer two questions on propagation domain estimation corresponding to the imperfectness of the two aforementioned assumptions that will affect the normalized error:

1
What will be the normalized error if utilizing dominant angles in the propagation domain instead of all angles to approximate the downlink channel?

2
What will be the extra normalized error if utilizing uplink dominant angles instead of downlink ones to approximate the downlink channel?
The answers to the questions depend on channel properties only. Therefore, to answer the questions, we seek to start from the fundamentals, i.e., FDD massive MIMO channels, to investigate the normalized error of approximating downlink channel with uplink dominant angles. We use a combination of a numerical and experimental approach. For the numerical approach, we employ the spatial channel model to formulate modeling error and examine the scalability with the basestation array size and dependency on the channel angle spread; for the experimental approach, we further validate the observations of modeling error based on measured channels.
2.4 Modeling error definition
In this section, we characterize the normalized error of approximating downlink channel with uplink dominant propagation domain, defined as modeling error, by answering the two questions brought up in Sect. 2.3. Each question corresponds to one source of error, with the first one denoted as approximation error and the second one denoted as mismatch error.
For singleuser case, the resulting rate with conjugate beamforming based on estimated channel \(\hat{{\textbf{h}}}_{\textrm{D}} \in {\mathbb {C}}^{M}\) is
where P denotes the downlink transmission power for the user. From Eq. (9), maximizing rate is same as minimizing the following normalized error
Note \(E \in [0,1]\), where \(E=0\) occurs when there is no channel estimation error and \(E=1\) occurs when the estimated channel is orthogonal to the actual channel, a worstcase scenario. Thus, smaller E is better.
We focus on propagation domain estimation and assume genieaided domain coefficients training. Consider estimated propagation domain \(\hat{{\textbf{S}}} \in {\mathbb {C}}^{M \times L}, L < M\), the genieaided estimated domain coefficients will be
based on leastsquare estimator, where \(\hat{{\textbf{S}}}^{\dag }\) stands for the pseudoinverse matrix of \(\hat{{\textbf{S}}}\). To evaluate estimated propagation domain \(\hat{{\textbf{S}}}\), based on Eq. 10, we consider the normalized error of estimated downlink channel utilizing the estimated propagation domain, which is formulated as
We aim to quantify modeling error—the normalized error of approximating downlink channel with uplink dominant propagation domain. To analyze modeling error, we need to obtain both downlink domain of low dimensionality and uplink one. In propagation domain, the low dimensionality part is constructed from array response vectors corresponding to dominant channel angles. Therefore, first, we try to construct downlink dominant propagation domain \({\textbf{S}}_{\textrm{d}} \in {\mathbb {C}}^{M \times L_d}, L_d \ll M\), where M is the number of antennas at the basestation and \(L_d\) is the number of dominant angles. The downlink dominant angle set is defined as
where \({\textbf{A}}_{\textrm{d}} = \begin{bmatrix} {\textbf{a}}\left( \theta _{1},\varphi _{1}\right)&\cdots&{\textbf{a}}\left( \theta _{L_d},\varphi _{L_d}\right) \end{bmatrix}\) and \({\textbf{A}}_{\textrm{d}}^{\dag }\) stands for the pseudoinverse matrix of \({\textbf{A}}_{\textrm{d}}\). We extract downlink dominant angles from full downlink CSI \({\textbf{h}}_{\textrm{D}} \in {\mathbb {C}}^{M}\) utilizing maximum likelihood estimator [26]. Then, the downlink dominant propagation domain based on dominant angles is constructed as
where \({\textbf{a}}\) is the array response vector defined in Eq. (3).
Second, to construct uplink dominant propagation domain \({\textbf{S}}_{\textrm{u}} \in {\mathbb {C}}^{M \times L_d}\), similarly, we extract uplink dominant angles \(\left\{ \left( \theta _{\textrm{u}1},\varphi _{\textrm{u}1} \right) , ..., \left( \theta _{\textrm{u}L_d},\varphi _{\textrm{u}L_d} \right) \right\}\) from full uplink CSI \({\textbf{h}}_{\textrm{U}} \in {\mathbb {C}}^{M}\). Then, the uplink dominant propagation domain is constructed as
Corresponding to the two questions brought up in Sect. 2.3, there are two factors that affect modeling error. First, to keep the low dimensionality of propagation domain under channel training overhead constraints, only \(L_d\) dominant anglesbased response vectors constructed propagation domain \({\textbf{S}}_{\textrm{d}} \in {\mathbb {C}}^{M \times L_d}\) is utilized for downlink channel approximation. As a result, there will still be certain channel estimation error due to the approximation; we denote this normalized channel estimation error as approximation error. Following the definition of normalized channel estimation error in Eq. 10, approximation error is formulated as
where \({\textbf{S}}_{\textrm{d}} \in {\mathbb {C}}^{M \times L_d}\) is the downlink dominant propagation domain as illustrated in Eq. 14. As evident from the above equation, approximation error will be in the range from 0 to 1. When the downlink dominant propagation domain \({\textbf{S}}_{\textrm{d}} \in {\mathbb {C}}^{M \times L_d}\) gets closer to downlink channel in antenna domain, approximation error will decrease and get closer to 0.
Second, since downlink propagation domain information is not available before any downlink channel training, uplink channelinferred dominant propagation domain instead of actual downlink channel dominant propagation domain is utilized for downlink channel training. Consequently, there will be extra normalized channel estimation error due to the uplink and downlink dominant propagation domain mismatch; we denote this normalized channel estimation error as mismatch error, which is formulated as
where \({\textbf{S}}_{\textrm{u}} \in {\mathbb {C}}^{M \times L_d}\) is the uplink dominant propagation domain as illustrated in Eq. 15. As evident from the above equation, mismatch error will also be in the range from 0 to 1. When the uplink channelinferred dominant propagation domain \({\textbf{S}}_{\textrm{u}} \in {\mathbb {C}}^{M \times L_d}\) gets closer to actual downlink channel one \({\textbf{S}}_{\textrm{d}} \in {\mathbb {C}}^{M \times L_d}\), mismatch error will decrease and get closer to 0.
Combining approximation error and mismatch error, the total normalized error of approximating downlink channel with uplink dominant propagation domain \({\textbf{S}}_{\textrm{u}} \in {\mathbb {C}}^{M \times L_d}\), i.e., modeling error, is formulated as
As evident from the above equation, modeling error will be in the range from 0 to 1. When the uplink channelinferred dominant propagation domain \({\textbf{S}}_{\textrm{u}}\) gets closer to downlink channel in antenna domain, the modeling error will get closer to 1.
Here, we focus on modeling error in the propagation domain, while similar error analysis can be applied to other domainbased channel characterization. Also, we want to emphasize that modeling error is determined by FDD massive MIMO channel properties only and thus schemeindependent. But, modeling error is an important and necessary part to analyze and evaluate the performance of scalable FDD massive MIMO channel estimation schemes.
2.5 Performance impact
Modeling error quantifies the normalized estimation error of approximating downlink channel with uplink dominant angle response vectors. As expected, the modeling error will result in beamforming performance impact due to channel estimation error. To understand the performance impact of modeling error, here we first derive the beamforming rate loss corresponding to modeling error.
To quantify the beamforming performance impact of modeling error, as illustrated in Section 2.4, we focus on the singleuser case and evaluate singleuser beamforming achievable rate with conjugate beamforming. When the basestation has the perfect downlink CSI \({\textbf{h}}_{\textrm{D}}\) available, the achievable rate with conjugate beamforming will be
And utilizing approximated downlink channel with uplink dominant angles \({\textbf{h}}_{\textrm{uD}}\) defined as
where \({\textbf{S}}_{\textrm{u}}\) includes uplink dominant angle response vectors as shown in Eq. 15. With conjugate beamforming, the achievable rate will be
Then, the rate gap between beamforming based on perfect downlink CSI \({\textbf{h}}_{\textrm{D}}\) and beamforming based on approximated downlink channel \({\textbf{h}}_{\textrm{uD}}\), denoted as rate loss, is formulated as
As evident from the above equation, the modeling error will affect the rate loss and larger modeling error will lead to larger rate loss.
3 Results and discussion
3.1 Modeling error: numerical results
To find out how the modeling error varies with the basestation array size, we employ the spatial channel model to investigate modeling error for the case where the basestation is equipped with a finite number of antennas. We conduct numerical simulations to observe how the modeling error varies with the number of basestation antennas and the channel angle spread. To facilitate the simulations, we set propagation domain channel parameters as follows:

For each channel instance, both the number of clusters and the number of dominant angles for approximation are set as 4.

Uniform distribution for the central angle of each cluster with elevation \(\theta _i \sim U[0^o, 180^o]\) and azimuth \(\varphi _i \sim U[0^o, 180^o]\).

All clusters have the same channel angle spread \(\Delta\) for both azimuth and elevation. In one cluster, the number of path is determined by the angle spread \(\Delta\), with all path angles uniformly sampled in elevation range \(\left[ \varphi  \Delta / 2, \varphi + \Delta / 2 \right]\) and azimuth range \(\left[ \theta  \Delta / 2, \theta + \Delta / 2 \right]\) with angle density as \(1^o\).

All clusters have the same total power, with uniform power distribution across all the paths in each cluster: \(g_{pr} = \frac{g_{p}}{R_p}\).
We first examine the simulation results on modeling error, as shown in Fig. 3. We investigate both the scalability with basestation array size and the dependency on channel angle spread for modeling error. Then, we examine the decomposed modeling error components, including approximation error and mismatch error, with simulation results on average error, as presented in Fig. 5. We also investigate both the scalability with basestation array size and the dependency on channel angle spread for both approximation error and mismatch error. All the numerical results are based on 10, 000 simulated independent channel instances.
Observation 1—Modeling Error Increases with Both Basestation Array Size and Channel Angle Spread: In terms of the scalability with basestation array size, we observe that the average modeling error increases with the number of basestation antennas, from Fig. 3. For example, when the channel angle spread is 2degree, the average modeling error increases from 0.02 with 16 antennas, to 0.06 with 64 antennas, and to 0.65 with 10, 000 antennas. The same trend is also observed in Fig. 4 with the cumulative distribution function of modeling error, which shows that more basestation antennas lead to larger modeling error. In terms of the dependency on channel angle spread, we observe that larger angle spread will result in larger average modeling error, from Fig. 3. For example, when the basestation is equipped with 64 antennas, the average modeling error increases from 0 with 0degree angle spread, to 0.06 with 2degree angle spread and to 0.23 with 8degree angle spread. The same trend is also observed in Fig. 4 with the cumulative distribution function of modeling error, which shows that larger channel angle spread results in larger modeling error.
Explanation for Observation 1: Modeling error quantifies the normalized error of approximating downlink channel with a fixed number of uplink dominant angle response vectors. For the scalability with basestation array size, since the array beamwidth is inversely proportional to the number of basestation antennas, the captured relative channel power with a fixed number of dominant angle response vectors will get smaller when the basestation is equipped with more antennas. As a result, modeling error increases with the basestation array size. For the dependency on channel angle spread, since larger channel angle spread indicates more distributed channel power, a fixed number of dominant angle response vectors will capture less channel power when the channel angle spread gets larger. Consequently, modeling error increases with the channel angle spread.
Observation 2—Approximation Error Increases with Both Basestation Array Size and Channel Angle Spread: As shown in Fig. 5, the average approximation error increases with the number of basestation antennas. For example, when the channel angle spread is 2degree, the average approximation error increases from 0.01 with 16 antennas, to 0.02 with 64 antennas and to 0.6 with 10, 000 antennas. And larger angle spread will result in more approximation error, from 0.02 when angle spread is 2degree to 0.09 when angle spread is 8degree with 64antenna at the basestation.
Explanation for Observation 2 Approximation error quantifies the normalized error of approximating downlink channel with a fixed number of dominant angle response vectors instead of all angle response vectors. For both the scalability with basestation array size and the dependency on channel angle spread, the reasons are the same ones as in the explanation for Observation 1.
Observation 3—Mismatch Error First Increases Then Decreases with Basestation Array Size and Larger Channel Angle Spread Leads to More Mismatch Error As shown in Fig. 5, the average mismatch error first increases and then decreases with the number of basestation antennas. For example, when the channel angle spread is 8degree, the average mismatch error increases from 0.04 with 16 antennas, to 0.3 with 400 antennas, and then drops to 0.08 with 10, 000 antennas. Also, larger channel angle spread will result in larger mismatch error, from 0.04 when angle spread is 2degree to 0.14 when angle spread is 8degree with 64antenna at the basestation.
Explanation for Observation 3 Mismatch error quantifies the normalized error gap between approximated downlink channel with uplink dominant angle response vectors and approximated downlink channel with downlink dominant angle response vectors. For the scalability with the basestation array size, due to uplink/downlink dominant angles difference resulting from channel phase parameters difference, the captured normalized channel power by uplink dominant angle response vectors decreases at a larger rate than that by downlink dominant angle response vectors when the number of basestation antennas increases. As a result, mismatch error first increases and then decreases with the basestation array size. For the dependency on channel angle spread, the reason is the same one as in the explanation for Observation 1.
3.2 Modeling error: experimental results
To further validate the observations on modeling error from numerical results, here we investigate modeling error based on measured FDD massive MIMO channels as detailed in Sect. 2.2. We take the measured FDD massive MIMO channels as input to evaluate modeling error. Similar to numerical results, we examine the scalability with basestation array size and the dependency on channel angle spread for modeling error, and the results are shown in Figs. 6 and 7.
Finding 1—Modeling error Increases with Basestation Array Size and NonLineofsight Channels Yield Larger Modeling error Than Lineofsight Channels In terms of the scalability with basestation array size, we find out that the average modeling error increases with the number of basestation antennas from Fig. 6. For example, for nonlineofsight channels, the average modeling error increases from 0.02 with 4 antennas, to 0.18 with 16 antennas, and to 0.28 with 64 antennas. The same trend is also observed in Fig. 7 with the cumulative distribution function of modeling error, which shows that more basestation antennas always lead to smaller modeling error. When comparing lineofsight channels with nonlineofsight channels, we find out that nonlineofsight channels result in larger average modeling error than lineofsight channels from Fig. 6. For example, when the basestation is equipped with 64 antennas, the average modeling error of nonlineofsight channels is 0.28, which is larger than the average modeling error 0.13 of lineofsight channels. The same finding is also obtained from Fig. 7 with the cumulative distribution function of modeling error.
Explanation for Finding 1: For the scalability with basestation array size, the finding matches numerical Observation 1. For the comparison between lineofsight channels and nonlineofsight channels, since nonlineofsight channels exhibit more distributed channel power over angles than lineofsight channels, a fixed number of dominant angles will capture less channel power for nonlineofsight channels. When comparing measured channels to simulated channels, lineofsight channels have a smaller channel angle spread than nonlineofsight channels. Therefore, for the dependency on channel angle spread, the experimental finding also matches numerical Observation 3.
3.3 Performance impact: numerical results
To understand the performance impact of modeling error, we first employ the spatial channel model to investigate rate loss. Here as illustrated in Sect. 2.4, we focus on the singleuser case and evaluate singleuser beamforming achievable rate with conjugate beamforming. We conduct numerical simulations to observe how the rate loss varies with the number of basestation antennas and the channel angle spread.
Observation 4—Rate Loss Increases with Both Basestation Array Size and Channel Angle Spread The simulation results on rate loss are shown in Fig. 8. In terms of the scalability with basestation array size, we observe that rate loss increases with the number of basestation antennas, from Fig. 8. For example, when the channel spread is 2 degrees, rate loss increases from 0.1 bps/Hz with 4 antennas, to 0.25 bps/Hz with 64 antennas, and to 3 bps/Hz with 10, 000 antennas. In terms of the dependency on channel angle spread, we observe that larger angle spread will result in larger rate loss, from Fig. 8. For example, when the basestation is equipped with 64 antennas, rate loss increases from 0 bps/Hz with 0degree angle spread, to 0.25 bps/Hz with 2degree angle spread, and to 1 bps/Hz with 8degree angle spread.
Explanation for Observation 4: We explain the observation based on the relationship between rate loss and modeling error as shown in Eq. 22. Since modeling error decreases with the basestation array size, as shown in Observation 1, accordingly, rate loss power decreases with the basestation array size. And since larger channel angle spread leads to smaller modeling error, accordingly, rate loss decreases with the channel angle spread.
3.3.1 Performance impact: experimental results
To further validate the observation of rate loss from numerical results, we also investigate the rate loss based on measured FDD massive MIMO channels. Again, we focus on the singleuser case and evaluate singleuser beamforming achievable rate with conjugate beamforming. Similar to numerical results, we examine the scalability with the basestation array size and the dependency on the channel angle spread for the rate loss.
Finding 2—Rate Loss Increases with Basestation Array Size and Lineofsight Channels Yield Smaller Rate Loss Than Nonlineofsight Channels The experimental results on rate loss are shown in Fig. 10. In terms of the scalability with basestation array size, we observe that rate loss increases with the number of basestation antennas, from Fig. 11. For example, for lineofsight channels, rate loss increases from 0.03 bps/Hz with 4 antennas to 0.4 bps/Hz with 64 antennas. When comparing lineofsight channels with nonlineofsight channels, we find out that lineofsight channels result in smaller rate loss than nonlineofsight channels from Fig. 6a. For example, when the basestation is equipped with 64 antennas, rate loss of lineofsight channels is 0.4 bps/Hz, which is smaller than rate loss 1 bps/Hz of nonlineofsight channels.
Explanation for Finding 2 For the scalability with basestation array size, the finding matches numerical Observation 4. For the comparison between lineofsight channels and nonlineofsight channels, since nonlineofsight channels exhibit more distributed channel power over angles than lineofsight channels, a fixed number of dominant angles will capture less channel power for nonlineofsight channels. When comparing measured channels with simulated channels, lineofsight channels have a smaller channel angle spread than nonlineofsight channels. Therefore, for the dependency on channel angle spread, the experimental finding also matches numerical Observation 4.
Combining the numerical observation with the experimental finding, we find out that even though modeling error leads to rate loss that increases with the number of basestation antennas, more basestation antennas still bring in more beamforming performance improvement with estimated downlink channel, as shown in both Fig. 9 and Fig. 11. Therefore, we can conclude that taking the modeling error into account, scalable channel estimation schemes that exploit the propagation domain, e.g., directional training in [26], can still benefit from the array gain and bring in beamforming performance improvement in the massive MIMO regime.
4 Conclusion
Inspired by the experimental findings on channel propagation domain properties, we first formulate modeling error to quantify the normalized estimation error of approximating downlink channel with uplink dominant angles. From our analysis and numerical results, we observe that modeling error increases with the number of basestation antennas, and more distributed power channels lead to larger modeling error. We also validate the observation by experimental results.
Then, we investigate the performance impact of modeling error and quantify the resulting downlink beamforming rate loss of approximating downlink channel with uplink dominant angles. From both numerical and experimental results, we observe that the rate loss increases with the number of basestation antennas, and more distributed power channels will result in larger rate loss. Also, even though the rate loss increases with the number of basestation antennas, we find out the beamforming rate still increases with array size. Based on the observation, we conclude that beamforming based on scalable channel estimation schemes that exploit the propagation domain, e.g., directional training in [26], benefits from the array gain in FDD massive MIMO.
In this paper, we mainly focus on channel in the propagation domain with angle response vectors characterization and investigate the corresponding modeling error, along with the beamforming performance impact. However, similar error analysis can be conducted for alternate channel characterizations. Also, our analysis will be an important part to help improve FDD channel estimation schemes and other related applications. A detailed example is to optimize time resources allocated to downlink channel training during channel estimation, where there is a tradeoff between training overhead and channel estimation error.
Availability of data and materials
Not applicable.
Notes
Frequencydivision Duplex
Abbreviations
 FDD:

Frequencydivision duplexing
 MIMO:

Multiinput multioutput
 CSI:

Channel state information
 OFDM:

Orthogonal frequencydivision multiplexing
 3GPP:

Thirdgeneration partnership project
References
T.L. Marzetta, Noncooperative cellular wireless with unlimited numbers of base station antennas. IEEE Trans. Wirel. Commun. 9(11), 3590–3600 (2010)
E.G. Larsson, O. Edfors, F. Tufvesson, T.L. Marzetta. Massive MIMO for next generation wireless systems. arXiv:1304.6690 (2013)
F. Rusek, D. Persson, B.K. Lau, E.G. Larsson, T.L. Marzetta, O. Edfors, F. Tufvesson, Scaling up MIMO: opportunities and challenges with very large arrays. IEEE Signal Process. Mag. 30(1), 40–60 (2013)
H.Q. Ngo, E.G. Larsson, T.L. Marzetta, Energy and spectral efficiency of very large multiuser MIMO systems. IEEE Trans. Commun. 61(4), 1436–1449 (2013)
C. Shepard, H. Yu, N. Anand, E. Li, T. Marzetta, R. Yang, L. Zhong. Argos: Practical manyantenna base stations, in Proceedings of the 18th Annual International Conference on Mobile Computing and Networking, pp. 53–64 (2012). ACM
C.S. Lee, M.C. Lee, C.J. Huang, T.S. Lee. Sectorization with beam pattern design using 3D beamforming techniques, in 2013 AsiaPacific Signal and Information Processing Association Annual Summit and Conference, pp. 1–5 (2013). IEEE
X. Cheng, B. Yu, L. Yang, J. Zhang, G. Liu, Y. Wu, L. Wan, Communicating in the real world: 3D MIMO. IEEE Wirel. Commun. 21(4), 136–144 (2014)
L. You, X. Gao, X.G. Xia, N. Ma, Y. Peng, Pilot reuse for massive MIMO transmission over spatially correlated rayleigh fading channels. IEEE Trans. Wirel. Commun. 14(6), 3352–3366 (2015)
H. Xie, F. Gao, S. Zhang, S. Jin, A unified transmission strategy for TDD/FDD massive MIMO systems with spatial basis expansion model. IEEE Trans. Veh. Technol. 66(4), 3170–3184 (2016)
P. Patcharamaneepakorn, S. Wu, C.X. Wang, M.M. Alwakeel, X. Ge, M. Di Renzo, Spectral, energy, and economic efficiency of 5G multicell massive MIMO systems with generalized spatial modulation. IEEE Trans. Veh. Technol. 65(12), 9715–9731 (2016)
N. Garcia, H. Wymeersch, E.G. Larsson, A.M. Haimovich, M. Coulon, Direct localization for massive MIMO. IEEE Trans. Signal Process. 65(10), 2475–2487 (2017)
X. Du, A. Sabharwal. Shared anglesofdeparture in massive MIMO channels: Correlation analysis and performance impact. submitted to IEEE Transactions on Wireless Communications (2019)
X. Du, Y. Sun, N. Shroff, A. Sabharwal, Balance queueing and retransmission: Latencyoptimal massive MIMO design. arXiv:1902.07676 (2019)
A. Adhikary, J. Nam, J.Y. Ahn, G. Caire, Joint spatial division and multiplexingthe largescale array regime. IEEE Trans. Inform. Theory 59(10), 6441–6463 (2013)
W. Shen, L. Dai, B. Shim, S. Mumtaz, Z. Wang, Joint CSIT acquisition based on lowrank matrix completion for FDD massive MIMO systems. IEEE Commun. Lett. 19(12), 2178–2181 (2015)
Z. Gao, L. Dai, Z. Wang, S. Chen, Spatially common sparsity based adaptive channel estimation and feedback for FDD massive MIMO. IEEE Trans. Signal Process. 63(23), 6169–6183 (2015)
X. Zhang, J. Tadrous, E. Everett, F. Xue, A. Sabharwal, Angleofarrival based beamforming for FDD massive MIMO, in 2015 49th Asilomar Conference on Signals, Systems and Computers, pp. 704–708 (2015). IEEE
Z. Jiang, A.F. Molisch, G. Caire, Z. Niu, Achievable rates of FDD massive MIMO systems with spatial channel correlation. IEEE Trans. Wirel. Commun. 14(5), 2868–2882 (2015)
J. Fang, X. Li, H. Li, F. Gao, Lowrank covarianceassisted downlink training and channel estimation for FDD massive MIMO systems. IEEE Trans. Wirel. Commun. 16(3), 1935–1947 (2017)
D. Fan, F. Gao, G. Wang, Z. Zhong, A. Nallanathan, Angle domain signal processingaided channel estimation for indoor 60ghz TDD/FDD massive MIMO systems. IEEE J. Sel. Areas Commun. 35(9), 1948–1961 (2017)
H. Xie, F. Gao, S. Jin, J. Fang, Y.C. Liang, Channel estimation for TDD/FDD massive MIMO systems with channel covariance computing. IEEE Trans. Wirel. Commun. 17(6), 4206–4218 (2018)
M.B. Khalilsarai, S. Haghighatshoar, X. Yi, G. Caire, FDD massive MIMO via UL/DL channel covariance extrapolation and active channel sparsification. IEEE Trans. Wirel. Commun. 18(1), 121–135 (2018)
Y. Han, Q. Liu, C.K. Wen, M. Matthaiou, X. Ma, Tracking fdd massive mimo downlink channels by exploiting delay and angular reciprocity. IEEE J. Sel. Top. Signal Process. 13(5), 1062–1076 (2019)
F. Rottenberg, T. Choi, P. Luo, C.J. Zhang, A.F. Molisch, Performance analysis of channel extrapolation in fdd massive mimo systems. IEEE Trans. Wirel. Commun. 19(4), 2728–2741 (2020)
B. Banerjee, R.C. Elliott, W.A. Krzymień, H. Farmanbar, Towards fdd massive mimo: Downlink channel covariance matrix estimation using conditional generative adversarial networks, in 2021 IEEE 32nd Annual International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC), pp. 940–946 (2021). IEEE
X. Zhang, L. Zhong, A. Sabharwal, Directional training for FDD massive MIMO. IEEE Transactions on Wireless Communications (2018)
3GPP: Study on 3D channel model for LTE. TR 36.873, 3rd Generation Partnership Project (3GPP) (2015). http://www.3gpp.org/dynareport/36873.htm
Acknowledgements
Not applicable.
Funding
The authors were partially supported by NSF grant 1518916 and support from Qualcomm, Inc.
Author information
Authors and Affiliations
Contributions
The authors contributed equally. Both the authors have read and approved the final version of the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Zhang, X., Sabharwal, A. Analysis of scalable channel estimation in FDD massive MIMO. J Wireless Com Network 2023, 29 (2023). https://doi.org/10.1186/s1363802202199z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s1363802202199z
Keywords
 Massive MIMO
 FDD
 Channel estimation
 Propagation domain