Correlationbased radio localization in an indoor environment
 Thomas Callaghan^{1},
 Nicolai Czink^{2}Email author,
 Francesco Mani^{3},
 Arogyaswami Paulraj^{4} and
 George Papanicolaou^{4}
DOI: 10.1186/168714992011135
© Callaghan et al; licensee Springer. 2011
Received: 1 December 2010
Accepted: 21 October 2011
Published: 21 October 2011
Abstract
We investigate the feasibility of using correlationbased methods for estimating the spatial location of distributed receiving nodes in an indoor environment. Our algorithms do not assume any knowledge regarding the transmitter locations or the transmitted signal, but do assume that there are ambient signal sources whose location and properties are, however, not known. The motivation for this kind of node localization is to avoid interaction between nodes. It is most useful in nonlineofsight propagation environments, where there is a lot of scattering. Correlationbased node localization is able to exploit an abundance of bandwidth of ambient signals, as well as the features of the scattering environment. The key idea is to compute pairwise cross correlations of the signals received at the nodes and use them to estimate the travel time between these nodes. By doing this for all pairs of receivers, we can construct an approximate map of their location using multidimensional scaling methods. We test this localization algorithm in a cubiclestyle office environment based on both raytracing simulations, and measurement data from a radio measurement campaign using the Stanford broadband channel sounder. Contrary to what is seen in other applications of crosscorrelation methods, the strongly scattering nature of the indoor environment complicates distance estimation. However, using statistical methods, the rich multipath environment can be turned partially into an advantage by enhancing ambient signal diversity and therefore making distance estimation more robust. The main result is that with our correlationbased statistical estimation procedure applied to the real data, assisted by multidimensional scaling, we were able to compute spatial antenna locations with an average error of about 2 m and pairwise distance estimates with an average error of 1.84 m. The theoretical resolution limit for the distance estimates is 1.25 m.
Keywords
indoor localization sensor networks signal correlations rich scattering multidimensional scaling1 Introduction
Indoor localization is a longstanding open problem in wireless communications [1], particularly in wireless sensor networks [2, 3]. Localization techniques in nonlineofsight indoor environments face two major challenges: (i) multipath from rich scattering makes it difficult to identify the direct path, limiting the use of distance estimation based on timedelayofarrival (TDOA) methods; (ii) the strongly changing propagation loss due to shadowing impairs distance estimation based on the received signal strength (RSS).
In both kinds of algorithms, TDOA and RSS, nodes can estimate their own location relative to several "anchor nodes" acting as transmitters. This is commonly done by estimating the distances to the anchor nodes and subsequently using triangulation for position estimation.
The estimation of the TDOA is done either by roundtrip time estimation [4], the transmission of specific training sequences [5], or simply by detecting the first peak of the received signal [6]. Ultrawide band communications are specifically suited for TDOA distance estimation because of the large available bandwidth [7].
Many publications discuss RSSbased distance estimation. The work presented in [8] provides a comprehensive overview of an actual implementation using WiFi hotspots in a selfconfiguring network.
Another technique described in [9] uses spatial signatures for localization. However, this requires multiple antennas at the nodes and a database of spatial locations. Moreover, this technique is limited to specific antenna requirements.
Correlationbased methods [10] have been widely used in the last few years in a variety of fields, including sensor networks. Some examples include estimation of the local propagation speed of surface seismic waves and even earthquake prediction [11]. The idea is to cross correlate seismic noise signals from seismographs deployed in a wide area so as to estimate the travel time of the seismic waves from one sensor to the other. Given the sensor locations, the wave speed can be estimated using travel time tomography.
Contribution
In this paper, we investigate the feasibility of passive, correlationbased indoor radio localization.
In contrast to previous works, our localization scheme only relies on ambient signals with wide bandwidth. Thus, no dedicated transmitters need to be deployed as long as the ambient signals from other wireless systems are sufficiently rich. In effect, the radio signals are unknown, the location of the sources is unknown, even the number of effective sources is unknown.
Even under these very stringent conditions, the distances between the receiving nodes can be estimated in a threestep procedure: (i) first, all nodes are receiving and recording ambient signals, (ii) the nodes communicate their received signals to a central entity or node, (iii) the central entity estimates the pairwise travel times, hence the distances, between all the nodes by cross correlating their received signals and identifying peaks in the crosscorrelation function. If the ambient signals have sufficient spatial diversity, then the peaks of the cross correlations provide a robust estimate of the distance between the two receiving antennas. By doing this for all pairs of receiving nodes, we construct an approximate map of their locations using weighted leastsquares methods, in particular multidimensional scaling (MDS) [12, 13].
This method suggests that there are several advantages for radio localization:

There is no communication overhead between nodes by active probing. Ranging is done without nodes cooperating or even communicating with each other. Nodes do not even know how many other peers are in their vicinity.

Only the central entity has the information from which to estimate the location of the nodes. The gains of cooperative localization (i.e., the pairwise distance estimates between peers) are achieved at the central entity, without having the nodes cooperate. This is advantageous for situations, where nodes do not want to reveal their location to other peers, as with active probing.

While the performance of TDOA ranging methods is inherently limited by the bandwidth of the (known) transmitted signals, correlationbased localization is only limited by the bandwidth of the (unknown) received signals, depending on communication or other wireless activities in the environment. Thus, correlationbased methods are not limited by scarce bandwidth allocations. Using wideband receivers, a much higher ranging resolution can be obtained by simply recording ambient signals from any occupied bands. By that, the performance improves with the employed bandwidth of the receivers.
To show the feasibility of this approach, we explore the performance of correlationbased radio localization in an indoor environment. To quantify it, we use (i) raytracing simulations and (ii) data from a recently conducted radio measurement campaign, using the RUSK Stanford multiantenna radio channel sounder with a center frequency of 2.45 GHz and bandwidth of 240 MHz [14].
The strongly scattering nature of the indoor environment makes the pairwise distance or travel time estimation challenging. However, in contrast to other localization methods, multipath from rich scattering is now both helpful and harmful for distance estimation. While multipath increases spatial diversity of the signals, it also leads to additional peaks in the correlation function that reduce the robustness of travel time estimation. The main feature in this work is the proper treatment and utilization of the beneficial properties of rich multipath while controlling its negative effects. To achieve this goal, we propose statistical peakselection algorithms that significantly increase the localization accuracy.
We demonstrate, therefore, that passive, correlationbased radio localization is feasible in wireless indoor environments.
Organization
The paper is organized as follows. Section 2 provides a brief motivation for using correlationbased methods for distance estimation. In Section 3, we consider the problem of travel time estimation using cross correlations. Section 4 presents different approaches for improving the pairwise traveltime estimation based on correlationbased methods. Section 5 briefly presents how we use MDS to find position estimates, discusses the results from applying our algorithms and MDS to the simulated and measured data, and demonstrates the effect of transmitter positions using the simulated data. With Section 7, we conclude the paper. Appendices A.1 and A.2 provide brief descriptions of the raytracing simulations and the measurement data we use in this paper.
2 Motivation for the Use of Cross Correlations in Distance Estimation
we see that the resulting cross correlation is a pulse at the delay difference ∆τ = τ_{1}  τ_{2}. This also holds for arbitrary source signals, as long as they have certain autocorrelation properties, as shown in the next section.
By finding the peak in the received signals cross correlation, we can estimate the distance between the receivers as $\widehat{d}=\Delta \tau \phantom{\rule{0.3em}{0ex}}{c}_{0}$, with c_{0} indicating the speed of light. When the transmitter is on a straight line going through the two receivers, this estimated distance is the exact distance between the nodes [10]. However, when there is an angle α between the direction of the plane wave front and the straight line between the receivers, the distance estimate will give $\widehat{d}=d\mid cos\left(\alpha \right)\mid $, which carries a systematic error.
While basing the distance estimation on a single plane wave is questionable because of the rather large systematic error, real radio propagation environments provide directional diversity by multiple sources and by multipath.
Multipath is both advantageous and challenging: (i) The receiver cross correlation gets multiple peaks providing more information about the propagation environment, which improves distance estimation, (ii) By reflections, the length of some paths can actually exceed the distance between the nodes.
Note that in this scheme, the existence of a direct line of sight (LOS) or nonLOS between the nodes is of reduced interest. More important is whether a wave can travel unobstructed over a pair of nodes. While we may observe an obstructed direct LOS between the nodes, we may still get a good distance estimate from another wave front connecting the node pair from a different propagation angle.
The way to exploit this signal diversity and how to obtain a robust distance estimate is the topic of the rest of this paper.
3 Computation of Cross Correlations
where h_{ kl }(t, τ) denotes the time and frequency selective radio channel from the l th source to the k th receiver.
when the source signals fulfill the condition in (4). This CCF provides information about the delay lag between the two receivers R_{ k }and R_{k'}as discussed in the previous section.
For the actual implementation, all convolutions and correlations in delay domain are implemented as multiplications in frequency domain.
It is well known [10] that for an infinite number of (uncorrelated) orthogonal sources, isotropically distributed in space, the resulting CCF has a rectangular shape, centered at zero and having a width of 2d/c_{0}. The range resolution is limited by the bandwidth of the source signal and is given by c_{0}/B [15] due to using peaksearch in a signal of limited bandwidth. In our setup, c_{0}/B = 1.25 m. Since in our simulations and measurements (cf. Appendices A.1 and A.2) only a finite number of transmitting antennas contribute to the signal recorded at each receiving antenna, we rely on sufficient scattering in the environment for enhancement of directional diversity. This leads to a tradeoff between two effects: (i) Multipath increases the signal diversity and thus creates peaks in the CCF that better represent the true distance, but (ii) multipath also generates "wrong" (additional) peaks from propagation paths that do not directly travel through the receivers, which in turn reduce the accuracy of distance estimation.
4 Improved Distance Estimation Method
The distance estimation can be improved by combining four ideas: (i) using shorttime estimates of the CCF, (ii) using multiple peaks from the CCF for distance estimation, (iii) using relative weighting on the peaks from the CCF to distinguish between peaks of comparable height (power), and (iv) using multidimensional scaling (MDS) to jointly improve the distance estimation and produce a location estimate.
As explained in detail above, given sufficient source diversity and a weakly scattering environment, the peak of the crosscorrelation of signals recorded by two sensors in the environment corresponds to the travel time between them. However, littletono theory exists for the case of limited source diversity and a strongly scattering environment. In this situation, we have multiple strong peaks where possibly none correspond to the correct travel time. As a result, we developed an empirical approach to peak selection that tries to utilize the information we have from both multiple peaks in the correlation functions and multiple realizations of the multipath in the environment. Others have studied how to address multiple peaks in crosscorrelation in reverberant environments and developed strategies using secondary peaks, weighting, and a type of fourth order correlation function [16, 17]. Peak selection in an optimal way is a challenging problem that will be the subject of future work.
4.1 Shorttime Estimates of the CCF
The longtime averaging applied in the original approach in (8) may reduce information about the propagation environment. By using the shorttime estimates of the CCF from (6), individual differences in the propagation environment, caused by fading, can be utilized to improve the distance estimation as follows.
4.2 Multiple Peaks for Distance Estimation
where ${\widehat{d}}_{k,{k}^{\prime}}$ and ${\widehat{\tau}}_{k,{k}^{\prime}}$ denote the distance estimate and travel time estimate, respectively.
A direct way to estimate the delay between two receivers is to identify the largest peak in their CCF. This approach does not perform well in multipath environments. Instead, we consider a more robust statistical approach based on multiple peaks in the CCF. The problem is how to choose and how to use the peaks in the CCF.
which corresponds to taking the top ⌊pM⌋ = 4 peaks in our data set. The value of p should balance the tradeoff between choosing enough peaks to average both the under and overestimation of the travel time and choosing few enough to exclude peaks that do not add useful information to traveltime estimation. Our choice for p is based on our empirical observations of the data.
where the choice of the weights w_{k,k'}(t, n) is described in the next section.
4.3 Crosscorrelations with Weights
To improve the distance estimation further, we propose to distinguish between dominant peaks and peaks of similar amplitude. For this reason, we weigh the peaks based on their relative amplitude.
The estimates computed by this statistical procedure can subsequently be improved by taking geometrical considerations into account as shown in the next section.
4.4 Multidimensional Scaling
Multidimensional scaling (MDS) algorithms are statistical techniques dating back 50 years, that take as its input a set of pairwise similarities and assign them locations in space [12, 13]. Recently, it was applied to a different, but related problem, of node localization in sensor networks [3].
In our problem, the input is the distance estimates between all receiver pairs. Multidimensional scaling, after introducing a few more assumptions as stated below, improves these individual distance estimates by jointly estimating receiver positions. The estimated receiver positions are also of much interest in this problem and are not simply a byproduct in improving pairwise distance estimates.
 1.
To fix our initial receiver location, we first choose the receiving antenna R_{ k }_{(1)} that has the smallest average estimated distance from the other receiving antennas and place it at the origin, i.e., R_{ k }_{(1)} = (0,0).
 2.
The second receiver R_{ k }_{(2)} is then chosen to be the one with the smallest estimated distance from the first receiver and is placed at ${R}_{k\left(2\right)}=\left({\widehat{d}}_{k\left(1\right)k\left(2\right)},0\right)$.
 3.
The third receiver R_{ k }_{(3)} is then chosen to be the one with the smallest estimated distance from receivers R_{ k }_{(1)} and R_{ k }_{(2)} and placed at the point in the first quadrant ${\widehat{d}}_{k1k3}$ from R_{ k }_{(1)} and ${\widehat{d}}_{k\left(2\right)k\left(3\right)}$ from R_{ k }_{(2)}. Should the third receiver fall on a line with the first two anchors, the triangle inequality is not valid and the space not properly spanned. In this case, another third receiver is chosen.
 4.
The rest of the receiving antennas are placed using the iterative leastsquares lateration procedure in [19].
With position estimates computed using MDS, which jointly uses the pairwise distance estimates, we can compute new pairwise distance estimates. These distance estimates should be an improvement as they are "jointly computed" and explicitly use the geometry of the setup, i.e., the receivers lie in a 2D plane.
5 Results
In the subsections that follow, we apply these distance estimation and localization methods to both a simulated dataset and data from an indoor radio channel measurement campaign.
5.1 Raytracing Simulations
With the simulated channel bandwidth of 240 MHz, our theoretical resolution is limited to an accuracy of c/B = 1.25 m. Our final results produced an average pairwise distance estimation error of 4.55 m.
Looking at this figure, we notice that the error is mostly in the xdirection. The reason for this is the strong directionality of the waves coming mostly from top/bottom, but not from left/right. This naturally leads to an underestimation of the distance between the horizontallyspaced node pairs. We also observe that the receiving antennas that are lying centrally have the smallest position estimation errors. This is due to the increased diversity of the source signals. We find an average position estimation error of 3.66 m, with a minimum error of 1.25 m, a maximum error of 5.87 m, and a standard deviation of 1.56 m.
CRLB versus localization errors
CRLB  $\mathbf{3}\sqrt{\mathbf{\text{CRLB}}}$  ${\mathit{R}}_{\mathit{k}}{\widehat{\mathit{R}}}_{\mathit{k}}$  $\frac{{\mathit{R}}_{\mathit{k}}{\widehat{\mathit{R}}}_{\mathit{k}}{}^{\mathbf{2}}}{\mathit{CRL}{\mathit{B}}_{\mathit{k}}}$  

Simulations  1.74 m^{2}  3.92 m  3.66 m  9.69 
Experiments  0.51 m^{2}  2.09 m  2.10 m  10.03 
If our estimator is optimal (fulfilling the CRLB), then the mean value of the last column should be 1. In our case, the mean value is ≈10. In other words, the variance of our distance estimator is about 10 times higher than the one of the CRLB; however, this estimate is based on just 14 samples.
Additionally, we can recompute pairwise distance estimates from the position estimates. The empirical cdfs of the distance estimation error are shown by the solid lines in Figure 6. While we expect these new distance estimates to be improved because they are computed jointly with the other receiver pairs constrained to lie on the plane via MDS, we see that this is not the case with the simulations. Since the distance estimates in our simulations are almost always underestimated, the MDS fails to improve over the initial distance estimates and rather makes the whole "environment" smaller.
5.2 Effect of Transmitter Locations
As expected, using the top and bottom transmitters results in good location estimation in the ydirection while using the left and right transmitters gives good location estimation in the xdirection. Comparing the location estimates of the top left and bottom right scatterers in Figure 9d to their position estimates using all of the transmitters (plot (a) in the same figure), one can observe that including the transmitter closest to the true receiver location results in that receiver's estimated position error being larger. This is also consistent with the intuition brought forward in Section 2. Sources close to the receiver nodes will most likely lead to an underestimation of the distance.
5.3 Performance in a Measured Environment
Looking at the quadrangle of the bottom four nodes, we observe that the estimates are placed in a rhomboid. The reason for this is the strong directionality of the waves coming mostly from left/right, but not from top/bottom. This naturally leads to an underestimation of the distance between the verticallyspaced node pairs. The result is that the nodes appear squeezed in the ydirection, but do have the correct distance in the xdirection. We also observe that the receiving antennas that are lying centrally have the smallest position estimation errors. This is due to the increased diversity of the source signals. We find an average position estimation error of 2.1 m, with a minimum error of 0.4 m, a maximum error of 3.36 m, and a standard deviation of 0.92 m.
Again, we compare these results to a benchmark using the CramerRao lower bound. The results are summarized in Table 1. The variance of our distance estimator is about 10times higher than the one of the CRLB; however, this estimate is based on just 8 samples. For our estimation scheme, this is quite a good result, leading to useful estimates in indoor environments. Note that even though the direct LOS between some nodes is sometimes obstructed by people, the distance estimation is still reasonable. This is due to wave fronts from other directions, which are not obstructed. Thus, our algorithm is inherently robust against NLOS problems, as long as wave fronts can propagate over both nodes in a nonobstructed way.
6 Implementation and Complexity
A realistic implementation of these methods would of course require the consideration of several practical issues, including timing synchronization and information exchange between the receiver node and the central entity, and optimal selection of the radio band for providing enough ambient signal strength. Also note that the central entity can be connected to the nodes by any means of wired or wireless communication.
Since the central entity performs all the calculations, there is no communication or ranging overhead between the nodes. The central entity can also ensure the synchronization between the nodes [22].
As for all delaybased localization algorithms, the receiving nodes need to sample the ambient signals with a high sampling rate (and thus bandwidth) using a fast analogtodigital converter. The advantage of our approach is that the sampling can be done with a low bit resolution, against which our approach is robust. Of course, the recorded data can be further compressed before sending it on to the master.
As numerical example, we consider a bandwidth of B = 240 MHz, and a maximum resolvable distance of d_{max} = 50 m, a minimum number of 2Bd/c samples must be acquired. An ADC resolution of 8 bit is more than sufficient for the correlationbased ranging, thus the sampled data to be transmitted equals to 80 bytes per recorded block. When employing data compression algorithms, this number is reduced even further (which is most beneficent when recording multiple blocks). Thus, only a limited amount of data needs to be transmitted, leading to a much smaller necessary communication bandwidth than what was needed for sampling the ambient signals.
The complexity of the localization problem can be segmented into the complexity for ranging, and the complexity for estimating the node positions. Since the latter complexity is the same for both kinds of algorithms, we just compare the ranging complexity.
The complexity of our scheme per distance estimation sample for K nodes is comprised of K signal sampling, storing, and communication events (nodes transmitting the recorded signals to the central entity), and K(K 1)/2 computations of the signal cross correlations and peak searches. Note that the computation of cross correlations can be done computationally efficiently using the fast Fourier transform. In contrast, conventional TDOAbased schemes need K(K  1)/2 ranging actions (i.e., ranging between all pairs of nodes). Subsequently, the ranging information must be communicated by at least K  1 nodes. Thus, the main difference in complexity lies in the computation of the cross correlations.
The complexity increase of the proposed scheme needs to be seen together with the advantage of the algorithm's robustness against NLOS environments, strongly reduced communication and coordination overhead, and the much higher available bandwidth and thus ranging resolution.
7 Conclusions
In this paper, we consider the feasibility of radio localization in a richscattering indoor environment using correlationbased techniques, where nodes only use unknown ambient signals for localization and do not probe actively.
We presented a systematic way to use peaks in the cross correlations of the received signals for computing pairwise distance estimates and spatial location estimates for a passive network of wireless receiving nodes (sensors). The robustness of the estimation is enhanced by multipath due to scattering but its accuracy is diminished by it. The increased signal diversity improves the estimation robustness while generating many peaks in the cross correlations. To enhance inter receiver (sensor) distance estimation, we use statistical methods that exploit multipath effects by taking into account multiple fading realizations of the channel.
The advantages of this scheme are threefold: (i) there is no ranging or communication overhead between the nodes, all communications and calculations are done by a central entity; (ii) the algorithm is inherently robust against nonlineofsight between nodes; (iii) by using just unknown ambient signals for localization, there are no regulatory limitations of bandwidth (and thus resolution).
We demonstrated the feasibility of our approach using both simulated and real measurements in a cubiclestyle office environment. In our simulations, we use a 3D raytracing tool, operating at 2.45 GHz, to measure the radio channels between 14 transmitters and 14 receivers in a simulated cubical office environment with diffuse scattering. In the real measurements, the radio channels between eight transmitters and eight receivers were measured using the RUSK Stanford channel sounder, operating at 2.45 GHz with a bandwidth of 240 MHz. The experimental equipment is special and favors our localization approach. Realistic implementation would require several practical aspects to be considered. Most importantly, a master node would be necessary to centralize the computation and synchronize the receiver nodes. However, using our equipment, we have demonstrated the feasibility of correlationbased radio localization techniques.
Despite the lack of a large number of transmitting antennas, we were able to utilize the spatial diversity of the strongly scattering room by using our improved estimation methods. The main result is that with the real data we were able to estimate spatial antenna locations with less than 2 m error when the theoretical resolution limit is 1.25 m.
Endnotes
^{a}These weights are not to be confused with the weighting of the peaks.
Appendix
A.1 Raytracing Simulations
RayTracing (RT) is a sitespecific geometrical technique that evaluates propagation paths generated by rays as they interact with the environment. A key feature of indoor propagation channels is diffuse scattering. For this reason, in this work, we model the channel with a classic 3D RT tool [23], improved with penetration and diffuse scattering [24]. The model of diffuse scattering is described in [25]. A geometrical description of the environment, frequency, number of interactions and dielectric properties of materials are some of the input parameters of a RT tool. In the following sections, the ones used in this work are discussed.
A.1.1 Setup Parameters
The simulation frequency has been set to 2.45 GHz. Antenna radiation patterns are the ones of vertically polarized dipoles both at receive (Rx) and source/transmit (Tx) side. Whenever Tx are placed at walls, they radiate only into the relative halfspace. A maximum of three reflections, single diffraction, and singlebounce scattering has been used in the simulations. A directive scattering pattern model with scattering coefficient S = 0.4 and beamwidth α_{ r }= 4 has been chosen. These paths were filtered using a rectangular filter in frequency domain with a bandwidth of 240 MHz to resemble the measurements.
A.1.2 Simulated Environment
A cubiclestyle office scenario has been used as input for the RT tool. The dimensions of the room have been set to 50 × 20 m with a height of 4 m. The walls, the ceiling, and the floor are supposed to be made of concrete. Cubicles are 4 × 3 × 1.8 m and are organized in two rows. Cubicles are represented by their metallic frames that have been treated as perfect electric conductors. A number of 14 receiving nodes were placed in these cubicles, while the ambient noise signals are generated by 14 sources placed at the outer wall of the room. At both sides, the antennas are placed at an height of 1 m. Figure 4 shows a 2D map of the simulated environment as well as the positions of the receivers and sources. A rough model for the human body as a rectangular parallelepiped has been used. For the human body, a classic twothirds muscle homogeneous model [26, 27] has been used to get realistic values. The time variance of the channel has been modeled by randomly placing ten persons in the scenario in 100 different realizations. The relative dielectric permittivity ε_{ r }was set to 9 for concrete walls, and 35.2 for the human body, while the conductivity σ was set to 0.06 and 1.16, respectively.
A.2 Radio Channel Measurements
In this paper, we use channel measurements obtained during the Stanford July 2008 Radio Channel Measurement Campaign. More details on the full campaign can be found in [14]. In this appendix, we briefly summarize the most important features of the measurement setup.
A.2.1 Environment
To provide good input data for our localization algorithms, we set up the test environment as shown in Figure 10. We took measurements in a cubiclestyle office environment with rich scattering due to the metallic frames of the cubicles and highly reflective walls. The room size was around 34 × 12 m. Eight receivers were placed in two squares, while the transmitters were positioned at the outer walls. To simulate real timevariant environments, people were moving in the room while the measurements were being recorded.
A.2.2 Measurement Equipment
The measurements were taken with the RUSK Stanford channel sounder at a center frequency of 2.45 GHz with a bandwidth of 240 MHz, and a test signal length of 3.2 μ s. The transmitter output power of the sounder was 0.5 W. A rubidium reference in the transmit (Tx) and receive (Rx) units ensured accurate timing and clock synchronization. The sounder used fast 1 × 8 switches at both transmitter and receiver, enabling switchedarray MIMO channel measurements of up to 8 × 8 antennas, i.e., 64 links. The Tx and Rx antennas were offtheshelf WiFi antennas, which were connected to the switches of the sounder units using long lowloss cables.
The full 8 × 8 channel was sounded every 100.76 ms. We recorded a total of T = 1, 200 samples, capturing the time variations of the channel. By proper calibration, we removed the RF effects of the equipment and of the cables so that the resulting data only contain the impulse responses of the channels, denoted as h_{ kl }(t, τ).
Declarations
Acknowledgements
Part of this work was supported by US Army grant W911NF07200271, by AFOSR grant FA95500810089, by the Austria Science Fund (FWF) through grant NFN SISE (S10607). FTW is supported by the Austrian Government and by the City of Vienna within the competence center program COMET.
Authors’ Affiliations
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