A) Proposed LCX-MIMO system

Figure 3 shows the LCX-MIMO system proposed in this paper. Different from placing the LCXs in parallel in previous research, two LCXs with maximum radiation angles of $\theta _{1}$, $\theta _{2}$ are set perpendicular to each other. The main idea of the method is based on the fact that the location positioning accuracy of the direction perpendicular to the LCX cable is worse than that of the direction parallel to the LCX cable [Reference Oki, Hou, Higashino and Okada27]. The proposal adds an additional LCX to improve detection accuracy of single LCX in the vertical direction.

Fig. 3. Proposed LCX-MIMO system.

We set the X-axis as the direction along the LCX1 and the Y -axis as the direction along the LCX2. The mobile user is arranged in the overlapping wireless coverage area of the two LCXs. It is assumed that the propagation path of the signal radio wave in the space is a straight line when the signals transmitted from the user terminal reach both ends of each LCX. We use the LCX1 as the example to show the relationship between signal arrival time, radiation angle, and location coordinate of the user terminal. The path of the signal from the user to one end of the LCX1 include two parts: propagation in the air and propagation in the cable. Therefore, the arrival times can be calculated as

(2)\begin{equation} t_{1}= \frac{y_{1}}{c\cos\theta_{1}}+\frac{x_{1}+y_{1}\tan\theta_{1}}{v_{1}c}, \end{equation}

(3)\begin{equation} t_{2}= \frac{y_{1}}{c\cos\theta_{1}}+\frac{L_{1}-x_{1}+y_{1}\tan\theta_{1}}{v_{1}c}, \end{equation}

where $v_{1}$ is the shortening coefficient of signal wavelength in LCX1 and $c$ is the speed of light. $L_{1}$ is the length of the LCX1. Then the location of the user over X-axis can be calculated as

(4)\begin{equation} x_{1}= \frac{\left | t_{1}-t_{2} \right |v_{1}c+L_{1}}{2}. \end{equation}

Then, we can calculate the location over Y -axis using $x_{1}$, arrival time $t_{2}$, and radiation angle $\theta _{1}$ as

(5)\begin{equation} y_{1}= \frac{(ct_{2}-(({L_{1}-x_{1})}/{v_{1})})v_{1}\cos\theta_{1}}{v_{1}+\sin\theta_{1}}. \end{equation}

$(x_{1},y_{1})$ is the location detected by LCX1 and with the same process we can estimate the location detected by LCX2 as

(6)\begin{equation} y_{2}= \frac{\left | t_{3}-t_{4} \right |v_{2}c+L_{2}}{2}, \end{equation}

(7)\begin{equation} x_{2}= \frac{(ct_{4}-(({L_{2}-y_{2})}/{v_{2})})v_{2}\cos\theta_{2}}{v_{2}+\sin\theta_{2}}, \end{equation}

where $v_{2}$ is the shortening coefficient of signal wavelength in LCX2. $L_{2}$ is the cable length. It should be noted that in the positioning process of the LCX2 placed vertically, the Y-axis coordinate $y_{2}$ is first calculated from the time difference of arrival, and then the X-axis coordinate $x_{2}$ is calculated. Then we can get the $y$ coordinate as

(8)\begin{equation} y= \frac{y_{1}+y_{2}}{2}, \end{equation}

and the $x$ can be calculated by

(9)\begin{equation} x= \frac{L_{1}-\left (ct_{2}- \frac{y(v_{1}+\sin \theta_{1})}{v_{1}\cos \theta_{1}}v_{1} \right )+x_{1}}{2}. \end{equation}

B) Arrival time estimation by MUSIC method

The MUtiple SIgnal Classification (MUSIC) method is a relatively simple and efficient eigenstructure method of DOA estimation. The method estimates the noise subspace from available samples and searches for steering vectors that are as orthogonal to the noise subspace as possible. This is normally accomplished by searching for peaks in the MUSIC spectrum.

In this paper, we use the MUSIC method to estimate the arrival time of the signal from user terminal to cable end. Here, we assume the receiving signal, $x(n)$, consisting of $s$ complex exponentials in the presence of Gaussian white noise to be

(10)\begin{equation} x(n)=\sum_{i=1}^{s}A_{i}e^{-j2\pi f_{i}\tau }+w(n), \end{equation}

where $A_{i}$ is the amplitude of the signal and $f_{i}$ is the frequency. $w(n)$ is the Gaussian white noise. Given an $M \times M$ autocorrelation matrix as

(11)\begin{equation} \mathbf{Rx} =\sum_{i=1}^{M}d_{i}\mathbf{u}_{i}\mathbf{u}^{H}_{i} , \end{equation}

the $d_{i}$ is the eigenvalue, $\mathbf {u}_{i}$ is the eigenvector corresponding to the eigenvalue. We use the superscript $H$ to represent the Hermitian transpose. If the eigenvalues are sorted in decreasing order, the eigenvectors corresponding to the $s$ largest eigenvalues span the signal subspace. The remaining $M-s$ eigenvectors span the orthogonal space, where there is only noise. Therefore, the matrix $\mathbf {R}_{x}$ can be composed of two parts as

(12)\begin{equation} \mathbf{Rx} =\sum_{i=1}^{s}(\lambda _{i}+\sigma ^{2}_{w})\mathbf{u}_{i}\mathbf{u}^{H}_{i}+\sum_{i=s+1}^{M}\sigma ^{2}_{w}\mathbf{u}_{i}\mathbf{u}^{H}_{i}. \end{equation}

Here, $\lambda _{i}$ is the eigenvalue of the correlation matrix consisting of signal only and $\sigma _{w}$ is the variance of white noise. The estimation function for MUSIC can be defined as

(13)\begin{equation} g(\tau)=\frac{1}{\sum_{i=s+1}^{M}\left | \mathbf{u}^{H}_{i}\mathbf{e}(\tau) \right |^{2}}, \end{equation}

where $\mathbf {e}(\tau )$ is known as the steering vector and can be expressed as

(14)\begin{equation} \mathbf{e}(\tau )=\left [ 1,e^{-j2\pi f_{1}\tau},e^{-j2\pi f_{2}\tau},\ldots,e^{-j2\pi f_{M}\tau} \right ]^{T}, \end{equation}

(15)\begin{equation} \mathbf \tau = nT_{r},(n=0,1,2,\ldots,N_{s}-1) . \end{equation}

$T_{r}$ is the time resolution of MUSIC method. $N_{s}$ is the number of elements in pseudo spectrum, and $N_{S}T_{r}$ is the maximum observation time for signal arrival. The orthogonality between the noise subspace and the steering vectors will minimize the denominator equation (13) to 0 value. However, it is a small value in practice because of the noise. As a result, it will give rise to peaks in $g(\tau )$ in equation (13). The largest peak corresponds to the signal arrival and then we can estimate the ToA.

A) LCX-MIMO and channel capacity with power allocation method

As introduced in the previous section, when signals are fed to both sides of the LCX cable, the intersection angle of the two radiation directions will be 2$\theta _{-1}$ degree. Their radiation characteristics have low correlation with each other and we can obtain a good 2-by-2 LCX-MIMO channel. In addition, we can realize a 4-by-4 LCX-MIMO channel by using two LCX cables with different radiation property.

MIMO channel capacity named as Shannon ergodic capacity shows a fundamental property and limitation of MIMO system [Reference Goldsmith, Jafar, Jindal and Vishwanath33]. The channel capacity can be described under two conditions. One is that the channel state information (CSI) is unknown for the transmitter side. It usually uses the equal power (EP) allocation method by allocating power to each stream equally to obtain promising channel capacity. The EP allocation method can be expressed as

(16)\begin{equation} C_{EP} =\mathbb{E}_{\mathbf{H}}\bigg[\log_2 \bigg(\det[\mathbf{I}+\frac{\zeta}{N}\mathbf{H}\mathbf{H}^{H}]\bigg)\bigg]. \end{equation}

Here, we assume that the CSI $\mathbf {H}$ is only known at receiver side and $\zeta$ is the average signal-to-noise ratio (SNR). $N$ and $\mathbf {I}$ are the antenna number at transmitter and the identity matrix. The superscript $H$ represents the Hermitian transpose. In the other condition, if the CSI is known, we can get the optimal capacity by water-filling (WF) algorithm method [Reference Khalighi, Brossier, Jourdain and Raoof34]. The channel capacity with WF method can be expressed as

(17)\begin{equation} C_{WF} =\mathbb{E}_{\mathbf{H}} \bigg[\sum_{i=1}^{N}\log_2 (1+\lambda_iP_i)\bigg]. \end{equation}

Here $\lambda _i$ is the corresponding eigenvalue of CSI matrix. $P_i$ is the power allocated for the $i$th MIMO eigenchannel as

(18)\begin{equation} P_i = \left(\mu-\frac{1}{\lambda_i}\right)^+, \end{equation}

where $x^+$ means the result of max($x$, 0) and $\mu$ is water-fill level. The WF algorithm fill each eigenchannel with power to the level $\mu$ iteratively and the MIMO channel capacity is represented as

(19)\begin{equation} C_{WF} =\mathbb{E}_{\mathbf{H}} \bigg[\sum_{i=1}^{N}(\log_2 (\lambda_i\mu))^+\bigg]. \end{equation}

Equations (16) and (19) above are normalized by bandwidth.

B) LCX-MIMO channel capacity with user location

In addition to EP allocation method and WF allocation method, the research in [Reference Hou, Zhu, Denno and Okada32] has proposed and evaluated a low-complexity transmission power allocation method for LCX-MIMO channel using the position information (PI) of the user. The results show that the proposed power allocation method using user PI can improve the channel capacity with almost the same low computational complexity as the EP method. The main idea of the method is adjusting the power over receiver side in consideration of the known constant power loss in the LCX cable. We use Fig. 4 to show the method, user $U$ with two antennas is set in the wireless coverage of the LCX power is allocated to each port according to the ratio of the distance between the user and both ends of the cable. The equation of the method in 2-by-2 LCX-MIMO can be given by

(20)\begin{equation} P=P_{1}+P_{2}, \end{equation}

(21)\begin{equation} \frac{P_{1}}{L/2+x}= \frac{P_{2}}{L/2-x}, \end{equation}

where $P$ is the total power, $L$ is the length of the cable. $x$ is the X-axis coordinate of the user.

Fig. 4. The power allocation method with user location.

On the other hand, it is impossible to obtain the perfect location of the user and usually localization error will occur due to many factors such as localization algorithm error and noise in the process, etc. As the Fig. 4 shows, ${U}'$ is the mis-detected user position due to localization error and channel capacity can be worse with the power allocation method using the wrong PI $({x}',{y}')$. This paper will also investigate the capacity performance of LCX-MIMO with our proposed positioning method.

A) Experiment configuration

In the experiment, we measured the channel matrix between the monopole antennas (receiver) and the LCXs (transmitter) with a multi-port vector network analyzer (VNA) in an anechoic chamber. The measurement frequency was set at 2.4 GHz band. As Fig. 5 shows, we place two LCXs perpendicular to each other in the experiment environment and define the intersection as the origin point of the coordinate. LCXs are set with over 1 m height from the ground of the anechoic chamber. We set 25 measurement points in the wireless coverage area with the X-axis range from 1 to 3 m and the Y -axis range from 1 to 3 m. The equal interval between the measurement points is set as 0.5 m. All the measurement points are set at the same height as the LCX. Receiving antennas will be placed at these points and LCX system will detect the location of the receiver terminal with the estimation of signal arrival time using MUSIC method. The distance between the two adjacent monopole antennas was set as 6 cm, which is half wavelength of the signal for 2.4 GHz band. Antenna gain of conventional monopole antenna is 1 dBi. Voltage Standing Wave Ratio (VSWR) is 1.1 for both LCX.

Fig. 5. Experiment configuration.

As the time resolution of the estimation in equation (15), $T_{r}$ can be set to infinitesimal. However, the value of $T_{r}$ has a limitation in real-time estimation, and we should set the value considering the actually measured MUSIC spectrum. For comparison, we set three values for $T_{r}$ as 5, 0.5, 0.1 ns. $N_{s}$ in equation (15) will be set as a suitable value to ensure that the signals arrive within the maximum observation time. We evaluated the localization accuracy over three frequency bands, and Table 1 shows the specifications of the experiment. $N$ is the number of noise space elements which depends on the number of measured samples in frequency domain.

Table 1. Experimental specifications

For channel capacity performance evaluation, we calculate the capacity of our 2-by-2 LCX-MIMO using user's location under three conditions that the user location is perfect, user location is detected by single LCX, and user location is detected by our proposed method. For comparison, the capacity results with EP method and WF method will be also provided.

B) Experimental results

We calculate the direct distance from the user correct position to the detected user position as the localization error. Figure 6 shows the comparison result of the localization error with conventional method and proposed method. Frequency range is from 2403.9 to 2420.1 MHz with bandwidth as 16.25 MHz, Time resolution $T_{r}$ is set as 0.5 ns. From the result, our proposal performs better than the conventional method, the maximum error of proposed method is with 1.5  m, the minimum error is within 0.5 m. In addition, we take the average value of all errors as the localization accuracy. Our proposed method has the accuracy as about 0.8 m and the conventional method is 1.15 m.

Fig. 6. Results of LCX localization accuracy ($B=16.25$ MHz, $N$=52, $T_{r}=0.5$ ns).

Figure 7 is the results of Localization accuracy with different $N$ values. When $N$ is 52, the minimum error is small, but the maximum error is large. As $N$ increases, the range of error distribution shrinks. When $N$ is 244, all measured localization errors with proposed method are within 1 m and the minimum error is about 0.6 m. We also provide the result of the conventional method for comparison, and our proposed method is better than conventional method in general.

Fig. 7. Results of localization accuracy with different $N$.

Figure 8 shows the results of the localization error with different time resolution $T_{r}$. It can be found that large value of $T_{r}$ will reduce the localization accuracy as the spatial resolution becomes worse. On the other hand, when $T_{r}$ is set as small as 0.1 ns, the localization accuracy is not much improved than the result with $T_{r}$ as 0.5 ns.

Fig. 8. Results of localization accuracy with different time resolution $T_{r}$.

Figures 9 and 10 are the channel capacity results of 2-by-2 LCX-MIMO over 2.4 GHz frequency band. The SNR is set as 6 and 10 dB. The PI method has promising capacity when using proposed positioning method and is better than that using conventional method. In addition, LCX system with proposed PI method can have a good capcaity even with localization errors.

Fig. 9. CDF of the capacity results (2.4 GHz band, $SNR=6$ dB).

Fig. 10. CDF of the capacity results (2.4 GHz band, $SNR=10$ dB).

We did these measurements in the anechoic chamber as the experimental environment, and this will make the accuracy of the ToA estimation higher. In real environment, interference from noise and other radio wave may deteriorate the ToA estimation. For higher localization accuracy in real environment, we can measure multiple times and take the average of the distribution of the measurement results to reduce the estimation error. In addition, we can use more than two LCX cables with different length and radiation angles, it can also improve the localization accuracy.