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Experimental Investigation of Frequency Domain Channel Extrapolation in Massive MIMO Systems for Zero-Feedback FDD

Estimating the downlink channel state information (CSI) in the frequency division duplex (FDD) massive multi-input multi-output (MIMO) systems by means of the downlink pilots and the feedback requires an extra overhead which scales linearly with the copious number of antennas at the massive MIMO base station. To address this issue, this paper investigates the feasibility of the zero-feedback FDD massive MIMO systems based on channel extrapolation. The extrapolation uses high resolution parameter estimation (HRPE), specifically the space-alternating generalized expectation-maximization (SAGE) algorithm. HRPE is applied to two different channel models, vector spatial signature (VSS) and direction of arrival (DOA) models, to extract parameters of multipath components in the uplink channel and use these parameters to extrapolate the downlink CSI. These methods are evaluated by channel measurements using two separate channel sounder setups (switched array and virtual array), in two different environments (outdoor and indoor), in three microwave bands (2.4-2.5, 3.325-3.675, and 5-7 GHz). Performance metrics we evaluate include mean squared error, beamforming efficiency, and spectral efficiency in multi-user MIMO scenarios. The results show that extrapolation performs best under the VSS model, without requiring antenna calibration, if the base station is in an open outdoor environment having line-of-sight paths to well separated users.


A. Motivation and Problem Statement
Massive multiple-input multiple-output (MIMO) systems utilize tens to hundreds of antennas at the base station (BS) to increase spectral and energy efficiency of wireless networks, making them a promising solution for the challenges of rapidly growing numbers of wireless devices and soaring data capacity requirements [2]- [4]. Massive MIMO systems are generally assumed to operate in the time division duplexing (TDD) mode, where both the uplink and the downlink share the same frequency band 1 [5], [6]. This allows to exploit channel reciprocity to attain the downlink channel state information (CSI) using only the uplink pilots 2 .
In the frequency division duplex (FDD) operation, the uplink and the downlink frequency bands are separated. Because the channel coherence bandwidth is almost always much smaller than the frequency spacing between the uplink and the downlink band, channel reciprocity cannot be directly exploited for the FDD operation. Therefore, extra overheads such as the downlink pilots and the feedback are necessary to attain the downlink CSI during the FDD operation. In addition, these overheads usually scale with the number of antennas at the BS rather than with the total number of the user equipment (UE) antennas, as is the case for the TDD. Consequently, overhead in the FDD might become prohibitive.
While such high resource demands from the FDD operation suggests massive MIMO systems to employ the optimal TDD operation, many legacy wireless networks still operate in the FDD frequency spectrum [7]. Therefore, the FDD massive MIMO systems can reduce costs from modifying hardware, frequency allocations, and/or network operation when updating current BSs with small number of antennas to massive MIMO BSs. For this reason, extrapolation of . In particular, in [8], [9], we investigated theoretically an approach in which channel extrapolation is done based on high resolution parameter estimation (HRPE) of the multipath components (MPCs). More precisely, the transfer function of a wireless channel is the complex sum of the contributions of the individual MPCs. A HRPE algorithm determines the parameters of each MPC such as complex amplitude, delay, azimuth angle of arrival, and elevation angle of arrival, from the uplink channel measurements. Based on these results, one can synthesize the channel in the downlink frequency band. Assuming that the parameters of each MPC in the downlink channel are the same as in the uplink channel, the downlink CSI can be estimated, without any downlink pilots or feedback to resolve resource issue in the FDD massive MIMO systems (see Fig. 1).

B. Contributions
1) In this paper, we apply the framework of [9] to extensive measurement results and evaluate the performance. The selected HRPE algorithm is the space-alternating generalized expectation-maximization (SAGE) algorithm, used in two variations based on the vector spatial signature (VSS) model [10] and the direction of arrival (DOA) model [11], forming separate sets of parameters for MPCs. 4 2) Multiple channel measurements have been conducted using two massive MIMO channel sounder setups (switched array and virtual array), in two different types of environment (outdoor and indoor), in three different microwave bands (2.4-2.5, 3.325-3.675, and 5-7 GHz) to assess the performance of the proposed channel extrapolation method without being biased toward certain measurement settings. The "measured channel (considered as ground truth channel in this paper) at the downlink frequency band" was compared with "estimated channel at the downlink frequency band", constructed by channel extrapolation using parameters of MPCs obtained purely from "measured channel at the uplink frequency band". This uplink frequency band in the FDD massive MIMO is also called a training band in channel extrapolation, which is a frequency band used for attaining parameters of MPCs to estimate channels at the other frequency bands, such as the downlink frequency band in the FDD massive MIMO.
3) Three different metrics assess the evaluated "accuracy" of the estimated downlink channel in comparison to the ground truth downlink channel, and the expected "performance" of the FDD massive MIMO systems using the estimated downlink channel. These metrics include mean squared error (MSE) of the channel estimate, beamforming efficiency, and spectral efficiency in multi-user MIMO scenarios. Overall, these metrics investigate the feasibility of the zero-feedback FDD massive MIMO systems using channel extrapolation based on HRPE algorithms. This paper makes several assumptions and acknowledges limitations in the following: 1) Implementing the proposed channel extrapolation method for the actual FDD massive MIMO systems in practice could be challenging due to computational complexity of HRPE algorithms. Rather, the emphasis in this paper is on the performance evaluation and the feasibility study of high-accuracy algorithms. There are several ways to speed up the MPC extraction using HRPE algorithm, but accuracy will be traded off by reducing the total number of MPCs, number of iterations, resolutions of parameters, etc. The performance resulting from such trade-offs is not studied in this paper.
2) While the switched array setup captures the channel within several milliseconds, a virtual array takes several seconds or minutes to record the channel. Therefore, virtual arrays were used for indoor scenarios with static environment while a switched array was used for outdoor scenarios. For the multi-user studies, we postulate that measurements with a single UE at different times is equivalent to measurement with multiple UEs at the same time, which is true only if the environment is completely static. Yet, for outdoor measurements, movement of scattering objects during the time it took to move physically the antenna from one location to the next might have occurred. We tried to minimize the effects of moving environmental objects, but quantitative assessment of the residual effects is not possible.
C. Literature Review 1) Theoretical studies of the FDD massive MIMO: There are several previous works that studied the feasibility of the FDD massive MIMO with reduced overhead. Among the suggested approaches are compressive sensing (CS) [12], using long term channel statistics and previous signals in a closed-loop manner [13], or a combination thereof [14]. There were also other potential solutions for the FDD massive MIMO systems to reduce or remove extra overheads, which utilize spatial correlation between multiple users [15], [16], spatial basis expansion model based on array theory [17], reciprocity based on reverse training [18], small number of dominant angles-of-departure [19], and deep learning [20]- [22].
There were also studies like this paper which used HRPE algorithms. One study used MUSIC and ESPIRIT to model channels in disjoint frequency bands [23]. In [24] and [25], the authors show that high resolution algorithms/MPC extractions are more suited than compressive sensing to exploit the FDD channel reciprocity. However, these papers did not verify the proposed performance empirically in real world channels.
2) Empirical Studies of the FDD massive MIMO: There were also several measurement-based studies which analyzed performance of the FDD massive MIMO systems (summarized in Table   I). Ref. [6] showed that the reciprocity-based TDD massive MIMO performs better than the feedback-based FDD massive MIMO with predetermined grid of beams, especially in non-lineof-sight (NLOS) cases, based on channel measurements at 2.6 GHz carrier frequency and 50 MHz bandwidth. In [26], in order to reduce the overheads, authors used the downlink training and  As mentioned in section 1-B, this paper is different from other papers as HRPE algorithm based on the VSS and the DOA models are used on the uplink channel to estimate the downlink channel without the extra overheads. We showed a theoretical analysis based on the DOA model in [8], [9], and initial outdoor measurement results in our conference paper [1]. The current paper expands our previous results by 1) comparing with another channel model (VSS) for the HRPE evaluation and extrapolation that has the advantage of not requiring pattern calibration, 2) using measurement settings including more locations and different (virtual array) setups, and

II. VSS AND DOA CHANNEL MODELS
In this work, we consider parametric channel models to 1) estimate the path parameters using an HRPE algorithm to get complete descriptions of the channels and 2) extrapolate CSI to the other frequency bands using the parameters obtained from the training band. The overall inputs and outputs of HRPE algorithms depending on two different models are shown in Fig. 2, which will be explained in the following subsections. 8

A. Channel Matrix for HRPE Algorithm
HRPE algorithms takes complex channel data as an input, producing parameter values for MPCs that are applied to channel models. The structure and processing of the measured channel data are described in this subsection (details of measurements are further discussed in Section IV).
First, raw channel data measured by the receiver of a channel sounder, H meas , with dimension M × K are selected, where M is the number of antennas at the BS and K is the number of frequency samples of the measurement. Subscript meas indicates that the data has not yet been compensated for the response of the RF system, h RF . 3 h RF can be measured by the back-to-back calibration of the sounder transmitter (TX) and receiver (RX), connected by a cable connection.
The compensated frequency response of the channel and the antennas only, H, is attained by where h RF,k −1 represents the inverse of the RF system frequency response at kth frequency index.
If the averaged power of H is expressed as µ H 2 = ||H || 2 F MK where || · || F is the Frobenius norm, then the normalized channel measurement matrixH = H µ H . AmongH with dimension of M × K, a subset which represents channel measurement data of the uplink frequency band,H u , is selected, with dimension M × K u .H u becomes the input for HRPE algorithm.
While there are many well-known HRPE algorithms such as MUSIC [33], ESPRIT [34], CLEAN [35], and RiMAX [36], SAGE was selected. The choice of HRPE algorithm was arbitrary, as the extrapolation results were dependent on channel models rather than types of HRPE algorithms during preliminary analysis. Two channel models (VSS and DOA) were used to attain different types and values of path parameters, which are discussed in the following subsections. A more detailed description of the SAGE algorithm are given in Appendix A.

B. Vector Spatial Signature (VSS) Model
The main difference between the VSS and the DOA models is that the VSS model does not require calibrated array pattern, A, as it does not estimate angles of arrival of MPCs. The only input for the VSS algorithm other than the normalized channel measurement data at the uplink frequency band,H u , is L, the number of MPCs (Fig. 2). L is arbitrarily chosen because the actual number of MPCs is not known. While using a very large number can improve the performance within the selected bandwidth for SAGE evaluations, it may lead to overfitting and channel extrapolation usually performs worse outside the training bandwidth during channel extrapolation [1].
The second parameter is called "vector spatial signature", represented byâ v .â v is a frequency independent complex vector with size M × 1 which according to [10], is "not explicit functions of angles of arrival, but instead abstractly represent the response of the array pattern for path v with delay τ v ". The extrapolation range is limited for the VSS model because in practice, most array patterns are frequency dependent. However, the estimation within the training band will be very accurate for massive MIMO systems because the number of estimated parameters in vector spatial signatures scales with M; more parameters can be adjusted to best estimate the channel response. Also, the VSS model does not suffer from calibration errors since it does not rely on calibration. The VSS channel model is overall expressed as: Once bothâ v and τ v are estimated for all L paths from the training frequency band, CSI at the desired frequency, f d , can be attained by setting f = f d in eq. (2), where f d will be within the downlink frequency band. The physical meaning of the VSS model is that only phase changes with frequency.
In total, 2M L + L real-valued parameters are estimated by the VSS model, where 2M L comes from real and imaginary values of the vector spatial signatures for all antennas for all paths, and L represents the total number of delays.

C. Direction of Arrival (DOA) Model
As the name indicates, the DOA model requires a frequency dependent calibrated antenna array pattern, A, to determine the directions of incoming MPCs. Aside from the topic of channel extrapolation, this model is useful when angular information of the channel is needed. Like the VSS model, the DOA model also requires normalized channel measurement data in the uplink frequency band,H u , and the total number of MPCs, L (Fig. 2).
A is attained differently for a switched array and a virtual array [1]. Virtual array calibration, however, omits switching, so a 3-D calibration data of a single antenna with dimension N az × N el × K will be created first. This 3-D calibration data of a single antenna will then be rotated or moved M times to form a 4-D calibration data of a virtual antenna array with selected geometrical shape and numbers. The impact of the rotation/movement is computed geometrically, not measured explicitly. The formulated calibration data, A, with the dimension N az × N el × M × K, are then used in the same way as the switched-array calibration data.
In [1], [9], we have already shown output parameters of MPCs attained by the SAGE algorithm and the DOA models. These parameters are summarized in Table II. There are 5L parameters to estimate for L paths, which are much less than the VSS model's 2M L + L parameters.
After parameters are estimated from the uplink frequency band, they are plugged into the channel model to estimate the channel frequency response at any selected frequency f . The where is between two measured frequency points during calibration. The physical meaning is that both the antenna response and the exponential term indicate phase change with frequency.

III. PERFORMANCE METRICS
Three different types of performance metrics are selected to assess the performance of the extrapolation algorithms: mean squared error (MSE) of the channel estimates, beamforming efficiency, and spectral efficiency for multi-user scenarios.

A. Mean Squared Error (MSE) of the Channel Estimates
MSE averages the squared magnitude of the differences between normalized complex channel response of a measured (ground truth) channel and an estimated channel at the selected frequency over all antennas: norm. If f lies within the training band (e.g. the uplink band in the FDD massive MIMO systems), then MSE will be a measure of the interpolation performance. If f lies outside the training band (e.g. the downlink band in the FDD massive MIMO systems), then MSE will be a measure of extrapolation performance. In the results in later section, MSE will be represented on a dB scale.

B. Beamforming Efficiency
Massive MIMO array can give a beamforming gain by combining constructive contributions of the many antenna elements within the array. One common way to optimize beamforming gain (in single-user case) is in the form of maximum-ratio combining, by using matched filtering based on estimated channel response. Beamforming efficiency indicates how beamforming gain with estimated CSI compares with beamforming gain based on perfect CSI, based on an assumption that the measured CSI is the perfect CSI. Beamforming gain (BG) with perfect CSI, estimated CSI, and uniform beamforming are expressed as: where † is the conjugate transpose operator. Therefore, using eq. (5) and (6), beamforming efficiency (BE) is: and this value ranges from 0 to 1 with 1 indicating full efficiency. As in the case of MSE, beamforming efficiency will be represented on a dB scale in later discussions.

C. Spectral Efficiency in Multi-User MIMO Systems
Earlier metrics were based on a single-user assumption. Spectral efficiency for every UE in a multi-user MIMO system can be determined by SIMO measurements at multiple locations.
Spectral efficiency of the UE n at the frequency f is represented as: where SINR(f) is a signal-to-interference-plus-noise ratio at a given frequency and the index of each UE, n, varies from 1 to N. The received signal by the n-th UE at frequency f during the downlink phase is: where T is a transpose operator, r (n) ( f ) is a complex value received by the UE at the selected where σ s (n) 2 and σ w (n) 2 are variances of the signal and the noise, assumed to be the same for all N UEs (i.e., all UEs are expected to experience the same transmit power from BS and same noise power).
The normalized precoding vector,G (n) ( f ), can be created as follows. First, represents an estimated complex channel frequency response between the BS antenna m and the UE n at the frequency f . Then, the precoding matrix for N UEs can be determined in two ways: maximum ratio (MR) and zero-forcing (ZF): where dimension of both matrices is N × M. G (n) ( f ) is a precoding vector for UE n with size . Finally, we normalize this precoding vector to get normalized precoding vector per UE,

IV. MEASUREMENT SETUPS AND SETTINGS
Measurement campaigns were conducted using two types of channel sounders in different environments. For both channel sounders, an omnidirectional antenna was used at the TX emulating a UE. In order to consider the measured channel as the reference "ground truth" channel to compare with the estimated channel, high SNR was necessary during measurement.
Therefore, transmitting power up to 40 dBm was used during outdoor measurements and up to 28 dBm during indoor measurements. On the BS side, the outdoor channel sounder setup used a switched array and the indoor channel sounder setup used a virtual array.

A. Outdoor Measurements with Switched Array
To measure channel characteristics of outdoor channels with short coherence time, a switched array with 64 antenna elements was used as a RX, emulating a massive MIMO base station (Fig. 3a).

B. Indoor Measurements with Virtual Array
For the indoor measurements, a vector network analyzer (VNA) based channel sounder with a rotating horn antenna forming a virtual massive MIMO array (Fig. 3b) on the RX side was selected. While indoor channel characteristics can also be measured using the real time channel sounder with switched array, we opted for a different measurement method both to provide data with different setups. Also, the use of a precision VNA as TX and RX can result in higher accuracy, though its use is only feasible for measurements over short distances. Each measurement was conducted two times, at two frequency bands (2.4-2.5 GHz and 5-7 GHz).
When repeating measurement two times at two separate frequency bands, several setup con- This effect is not taken into account by the MSE but well by the beamforming efficiency. Results of beamforming efficiency in Fig. 7 indeed provide alternative views on channel extrapolation.
Beamforming efficiencies were averaged per case (4 UEs), using the VSS model (Fig. 7a) and the DOA model (Fig. 7c). Also, the CDF of beamforming efficiencies for all 12 locations (raw values for all UEs, not averaged values), 105 MHz away from the training band (marked in pink in Fig. 7a and 7c), are shown in Fig. 7b and Fig. 7d, with additional results when L = 1. There are several points to be made. First, in general, the DOA model provides higher variance and more deviation from perfect CSI than the VSS model. This may be attributed to 1) imperfect calibration of antenna array and 2) not enough parameters to estimate per MPC. It is therefore more practical to use the VSS model for extrapolation than the DOA model because no calibration data is required, and the computing speed is quicker. Second, regardless of the model, using only one path was usually better (Fig. 7b and Fig. 7d). The outdoor measurements, even in the NLOS case, were better explained with a single path, than with multiple paths. Therefore, choosing many paths possibly resulted in overfitting during the extrapolation. Lastly, while every case outperforms uniform beamforming, the performance decreases in the order of the LOS case, the Each figure contains four plots. There are single-user performance with perfect CSI (which is also very close to multi-user ZF performance with perfect CSI -not shown on the figure), single-user performance with estimated/extrapolated CSI, multi-user ZF performance with estimated/extrapolated CSI, and multi-user MR performance with estimated/extrapolated CSI.
The results show that extrapolated single-user spectral efficiency matches closely with singleuser spectral efficiency based on perfect CSI, while the deviations become larger as we move to the OLOS and then to the NLOS cases. Also, despite using just two users, multi-user scenarios show large deviation from single-user scenario. This indicates that interference exists at the extrapolated frequencies even for well separated users, even though such interference does not exist within the training band. Lastly, ZF performs better than MR in a LOS scenario where the noise power is relatively smaller than the interference power, while they perform relatively similar in OLOS/NLOS scenarios where due to the decrease in signal strength the noise power dominates the SINR value. Overall, in terms of spectral efficiency, extrapolation works best for single-user scenario and when the user is in LOS.   algorithm. Yet, in Fig. 10, beamforming efficiency shows that channel extrapolation could be useful in the LOS cases. In Fig. 10a and 10c Beamforming efficiency outside the training band performs worse for indoors than outdoors.

B. 2.4-2.5 and 5-7 GHz Indoor
Unlike Fig. 7b and 7d which showed that most extrapolated channels provide -3 dB or greater beamforming efficiency 105 MHz away from the training band, less than 50 percent of the LOS cases provided greater than -3 dB beamforming efficiency in Fig. 10b and less than 70 percent in Fig. 10d. The figures also showed that one path provided higher beamforming efficiency. This is possibly due to many paths other than the LOS path contributing to the channel in the indoors environment, and the LOS path is best extrapolated. The fact that the large extrapolation range ( scattering environment, as theoretically foreseen in [9]. In terms of the model, the VSS model had better performance than the DOA model for the 2.4-2.5 GHz case, while both models had similar performances for the 5-6 GHz case. Lastly, Fig. 11 shows the spectral efficiency for UE3 served by BS1 (LOS) and BS2 (NLOS).
Only the results for the 2.4-2.5 GHz case using the VSS model was plotted due to space restriction and similarity of the DOA results. Just like in Fig. 8 σ w (n) 2 = 10 8 (80 dB) was used based on the assumption that the difference between the transmit power of a BS and a noise floor of a UE is around 85 dB (for example, 15 dBm transmit power from outdoor BS and -70 dBm noise floor of UE).
The results show that while the single-user performance using extrapolated channels is relatively small for the LOS case (differing by 2 bits/s/Hz at maximum), it does not perform as well as the LOS case in outdoor environment as in Fig. 8a. As expected, the extrapolated channel provides large loss in spectral efficiency for the NLOS case. For the NLOS case, single-user performance and the multi-user performance are rather similar in the extrapolated bands. While ZF provides better than MR for the LOS case, they are overlapping for the NLOS case. Overall, extrapolation is difficult in multi-user scenarios, especially in the NLOS case, while it performs relatively better for single-user scenarios in LOS case.

VI. CONCLUSION
To eliminate large overheads in the FDD massive MIMO systems, an experimental study of the feasibility of channel extrapolation using parameters of MPCs at a selected training frequency band attained by HRPE algorithm to estimate the channel frequency response at a desired frequency band was presented. Empirical data obtained by three massive MIMO channel measurement setups in outdoor and indoor environments were used for verification.
The results showed that the best extrapolation performance can be achieved when 1) a channel contains a dominant LOS path, 2) the BS is not surrounded by interacting objects (outdoor), and 3) UEs are well separated. One of the reasons why the extrapolation in the NLOS cases is difficult is because small-scale fading is less predictable. Another source of error may be simplicity of the channel model, which assumes MPCs to be planar waves with perfect vertical polarization.
Such model assumption will not hold when BS is too close to interacting objects. Therefore, the suggested extrapolation method will perform best in the FDD massive MIMO systems that follow conditions 1)-3). We conjecture that a particularly appealing case could be channels in stationary unmanned aerial vehicle (UAV) communication systems which usually involve high LOS probability, outdoor environment, well separated aerial vehicles, and long enough channel coherence time. The FDD massive MIMO systems for terrestrial applications involving mainly NLOS cases, however, may still require at least partial downlink pilot and feedback overheads or better extrapolation techniques in order to improve performance.
Another notable result was that extrapolation based on a simpler VSS model, which estimates an abstract antenna array pattern, performs better than extrapolation based on the DOA model, which uses measured calibration data of antenna array to find arriving directions of MPCs. This is somewhat counter-intuitive: more knowledge about the antenna array used during the measurement would be expected to provide better results, but the opposite occurs. One possible explanation is the sensitivity to calibration and model errors. If the goal of HRPE is reconstructing an estimated channel based on observation rather than finding angular characteristics of MPCs, the VSS might thus be a preferable algorithm.
Several topics arising from the current work will be considered in the future, including: 1) extrapolation performance dependence on number and/or geometry (planar or cylindrical) of antennas at BS and/or UEs, 2) extension and/or improvement of channel model including polarization parameter and/or spherical wave model, and 3) channel measurements between a drone and an antenna array to verify HRPE based channel extrapolation techniques for a realistic massive MIMO system dominated by LOS path.

APPENDIX A SAGE ALGORITHM
The SAGE algorithm, explained in [11], identifies "the estimated parameters describing MPCs in a channel",Ψ, maximizing the likelihood of "the observed complex channel frequency H(m, f k ;Ψ) is "the estimated complex channel frequency response" related toΨ = (ψ 1 , . . . ,ψ L ), where L is the total number of paths in a channel, through the VSS or the DOA models previously introduced in section II.
This optimization problem is challenging because of 1) the high-dimensionality scaling with L and 2) the non-linear dependence on the path parameters. The SAGE algorithm provides an efficient suboptimal solution to the problem of relying on an iterative approach. Fig. 12 is the SAGE algorithm flow chart. The estimates of "the ground truth parameters", Ψ, at iteration µ, are denoted byΨ(µ). Successive ordered cancellation is used to estimate initial parameters,Ψ(0), as explained in [11]. At each iteration, only the parameters corresponding to one path, e.g.,ψ v orψ d , are optimized while parameters for other paths keep their past value. Optimizing parameters per path reduces the search dimensions by a factor L. A set of L iterations, called an iteration cycle, is required to update the parameters for each of the L paths.
After an iteration cycle, each path is re-estimated based on the updated values of other paths.
Each iteration consists of two steps: 1) expectation step and 2) maximization step. The specific operations performed in the two steps for the VSS and the DOA algorithms are shown in Fig. 13. During the expectation step, the interference due to other paths is canceled from the measured channel response based on their current estimate. Then, during the maximization step, the parametersψ v orψ d are re-estimated. To further simplify the complexity of the problem, the optimization over the different parameters of each path is simplified into several one-dimensional searches optimizing each parameter at a time. The algorithm iterates until convergence or if a maximal number of iterations is achieved.