Open-Loop Beamforming Technique for MIMO System and Its Practical Realization

Abstract

The concept of close-loop beamforming for MIMO system was well known proposed the singular value decomposition on channel matrix. This technique can improve the capacity performance, but the cost of feedback channel and the complexity processing discard the interest of implementation. Therefore, this paper aims to investigate the benefit of using an open-loop beamforming for MIMO system in practical approaches. The low-profile concept of open-loop beamforming which is convenient for implementation is proposed by just inserting Butler matrices at both transmitter and receiver. The simulation and measurement results indicate that the open-loop beamforming with Butler matrix outperforms the conventional MIMO system. Although, the close-loop beamforming offers a better performance than open-loop beamforming technique, the proposed system is attractive because it is low cost, uncomplicated, and easy to implement.

1. Introduction

The MIMO (Multiple-input-multiple-output) system is a good quality of service such as channel capacity. In general, MIMO systems consideration of channel capacity is based on the array antennas at both transmitter and receiver.Many works have proposed the method of eigen-beamforming technique [1–5]. This technique utilizes the properties of estimated channels by performing singular value decomposition on channel matrix. Then eigenvectors compositing of channel matrix are considered as pre- and postcoding schemes for MIMO systems. This technique can improve the capacity performance, but both transmitter and receiver have to perfectly know the channel information, named as close loop MIMO system. However, there are many unattractive issues of using eigen-beamforming in practice such as a requirement of high system complexity and many procedures for channel feedback transmission. In turn, for open loop beamforming, the transmitter sends independent information symbols from multiple transmit antenna elements to the receiver. The received channels are not sent back to the transmitter. So, the transmitter does not know the channel information. The pre- and postcoding schemes do not require any additional complexity like close loop system. Therefore, the study of using open loop is focused on in this paper.

In the research areas of MIMO system, many works such as [6–9] have been proposed to enhance the channel capacity in order to satisfy the user demand for high data rate applications. Some studies have been focused on theoretical works and some have been performed by measurements. Nevertheless, most of them develop the technique to enhance the channel capacity through channel behavior [10–12] such as adjusting transmitted powers according to eigenvalue of channels as so-called water filling method. In general, it can be noticed that the theoretical consideration of channel capacity is based on the array antennas at both transmitter and receiver, but the channel characteristic is composed of many angle parameters such as angle of arrival, angle of departure, and angle spread. Therefore, it is interesting to investigate the performance of MIMO system using the open loop beamforming instead of the conventional MIMO system. Recently, the authors in [13, 14] developed the channel estimation of MIMO-OFDM system based on open loop beamforming consideration. The applicability of open loop beamforming technique is dependent on the channel stochastic information available to the receiver. The design of suitable pilots is proposed by facilitating the direct implementation of open loop beamforming and analyzing the performances of different channel estimation techniques. Although the significant improvement on MIMO capacity can be expected by using open loop beamforming, so far in the literature, there is no work to illustrate the capacity benefit of using open loop beamforming. The reason is that the pre- and postcoding schemes of angle transformations increase the complexity on both transmitter and receiver. Hence, it is challenging to find the technique with low cost and complexity matching with the concept of open loop beamforming. In [15], the proposed scheme uses a discrete fourier transformation (DFT) to receive a signal vector in RF domain. This can be realized by placing a Butler matrix between the antenna elements and the receiver switch. This paper presents only the simulations results. There is no measurement result presented in this paper. Clearly the difference between [15] and our paper is on the measurements. In [16], the authors investigate the correlation coefficients (line of sight and nonline of sight) via both simulation and measurement results. However, the analysis of correlation coefficients has not been analytically discussed yet. This is the claim of our novelty, that our paper presents the analytical analysis of correlation coefficients. Moreover, the additional measurements have been done by rotating the array direction for many degrees. These measurements are different from [16] and give more insight of using Butler matrix.We used the method of open loop beamforming in [17] investigated the capacity performance of compared with conventional MIMO system by computer simulations.The results in [17] confirm the advantages of using open loop beamforming. In this paper, the analytical analysis of how open loop beamforming impacts on the channel matrix is explained. Then, it gives a reason why using open loop beamforming for MIMO system provides a better performance over a conventional MIMO system. Also, further work from [17] is carried out by manufacturing a Butler matrix and performing the experimental results due to the fact that only simulation results cannot claim the use of the proposed system in practice. The reason why Butler matrix is chosen is because Butler matrix is just a low-complexity hardware that can offer the open loop beamforming. In general, there are infinite choices to choose the set of orthogonal steering vectors to form an open loop beamforming. Therefore, it is hard to justify whether Butler matrix provides the best performance among others. To focus on hardware complexity, the other methods to form open loop beamforming might need 16 phase shifters to simultaneously form 4 beams while Butler matrix approach uses only one low-cost printed circuit board. Thus, the authors construct the 4×4 MIMO system employing an open loop beamforming by Butler matrix which is a low-profile concept and convenient for implementation. This matrix simultaneously forms multiple departure or arrival angles into four directions. By only inserting Butler matrix next to antenna arrays, the conventional MIMO systems can be transformed into the MIMO systems with open loop beamforming without the need of additional burden on processing units at both transmitter and receiver.

In summary, the contribution of this paper falls into three main issues. At first, the analytical analysis of how open loop beamforming impacts on the channel matrix is originally provided. This helps the reader to understand in a true benefit of open loop beamforming. Secondly, the practical realization of open loop beamforming for 4×4 MIMO systems has been demonstrated. The third contribution is on the experimental comparisons in terms of channel capacity. All contributions confirm either new concept or actual benefit of employing MIMO with open loop beamforming. The paper is organized as follows. In Section 2, the details of SISO system, conventional MIMO system, close loop beamforming, and open loop beamforming technique are described. Then in Section 3, the analytical analysis of using open loop beamforming is explained. The feature of Butler matrix to apply for open loop beamforming is given in Section 4. Section 5 describes the details of channel measurements. Section 6 provides the simulation and measurement results of open loop beamforming realized by Butler matrix in comparing with conventional MIMO system. Finally in Section 7, the conclusion of this paper is given.

2. System Model

2.1. SISO System

For a memoryless SISO (single-input-single-output) system, the Shannon capacity is given by [8] 𝐶=log₂1+𝑃^𝑡ℎ𝑃_𝑁,(1) where 𝑃_𝑡 is the transmitted power, ℎ is the wireless channel coefficient, and 𝑃_𝑁 is the noise power in each branch of antennas at the receiver. Note that the signal-to-noise-power ratio (SNR) is defined as 𝑃_𝑡/𝑃_𝑁.

2.2. Conventional MIMO System

We consider the narrowband MIMO channel. Let 𝐱 be a vector of the transmitted signals with 𝑁_𝑡 transmitted antennas and let 𝐲 be a vector of the received signals with 𝑁_𝑟 received antennas. Then the relation between transmitted and received signals is given by 𝐲=𝐇𝐱+𝐧,(2) where 𝐧 is an 𝑁_𝑟×1 noise vector and 𝐇 is an 𝑁_𝑟×𝑁_𝑡 channel matrix. In this paper, the channel matrix is modelled by using the concept of geometrical two-ring model. This model is based on the extension of single-bounce two-ring scattering model for flat fading channels [19]. With this notation, channel output sequence can be written in matrix form as the following expressions:⎡⎢⎢⎢⎢⎢⎢⎣𝑦₁𝑦₂𝑦⋮_𝑁𝑟⎤⎥⎥⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎢⎢⎣ℎ₁₁ℎ₁₂⋯ℎ_1𝑁𝑡ℎ₂₁ℎ₂₂⋯ℎ_2𝑁𝑡ℎ⋮⋮⋱⋮_𝑁𝑟1ℎ_𝑁𝑟2⋯ℎ_{𝑁𝑟𝑁𝑡}⎤⎥⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎢⎣𝑥₁𝑥₂𝑥⋮_𝑁𝑡⎤⎥⎥⎥⎥⎥⎥⎦+⎡⎢⎢⎢⎢⎢⎢⎣𝑛₁𝑛₂𝑛⋮_𝑁𝑟⎤⎥⎥⎥⎥⎥⎥⎦.(3)

Figure 1(a) shows the conventional 4×4 MIMO systems. There is an arbitrary number of physical paths between the transmitter and receiver [20]; the ith path having attenuation of 𝑎_𝑖, makes an angle of 𝜙_𝑡𝑖(Ω_𝑡𝑖∶=cos𝜙_𝑡𝑖) with the transmit antenna array and angle of 𝜙_𝑟𝑖(Ω_𝑟𝑖∶=cos𝜙_𝑟𝑖) with the receive antenna array. The channel matrix 𝐇 can be written as the following expressions: 𝐇=_𝑖𝑎^𝑏𝑖𝐞_𝑟Ω_𝑟𝑖𝐞_𝑡Ω_𝑡𝑖^∗,(4) where 𝑎^𝑏𝑖∶=𝑎_𝑖√𝑁_𝑡𝑁_𝑟exp−𝑗2𝜋𝑑^𝑖𝐞𝜆_𝑐,(5)_𝑡(Ω)∶=1√𝑁_𝑡⎡⎢⎢⎢⎢⎢⎢⎣exp−𝑗12𝜋Δ_𝑡exp−𝑗𝑁_𝑡⋮Ω𝐞−12𝜋Δ_𝑡Ω⎤⎥⎥⎥⎥⎥⎥⎦,(6)_𝑟(Ω)∶=1√𝑁_𝑟⎡⎢⎢⎢⎢⎢⎢⎣exp−𝑗12𝜋Δ_𝑟exp−𝑗𝑁_𝑟⋮Ω−12𝜋Δ_𝑟Ω⎤⎥⎥⎥⎥⎥⎥⎦.(7) Also,𝑑_𝑖is the distance between transmit and receive antennas along path ith. Note that (⋅)^∗ is the conjugate and transpose operation. The vector 𝐞_𝑡(Ω) and 𝐞_𝑟(Ω) are, respectively, transmitted and received unit spatial signatures along the direction Ω, 𝜆_𝑐 is the wavelength of the center frequency in a whole signal bandwidth. Assuming uniform linear array, the normalized separation between the transmit antennas is Δ_𝑡 (antenna separation/𝜆_𝑐) and the normalized separation between receive antennas is Δ_𝑟 (antenna separation/𝜆_𝑐). Note that the reason of normalization is because this proposed system can work in any frequency band. Hence, the normalization is made to neglect the unused parameter. When channel state information (CSI) is not available at the transmitter, the capacity of MIMO systems [21] expressed in bits per second per hertz (bps/Hz) can be written as 𝐶=log₂det𝐈_𝑁𝑟+𝑃^𝑡𝑃_𝑁𝑁_𝑡𝐇𝐇^∗,(8) where 𝐈_𝑁𝑟 is the identity matrix having 𝑁_𝑟×𝑁_𝑟 dimension, 𝐇 is the channel matrix having 𝑁_𝑟×𝑁_𝑡 dimension with 𝐇^∗ being its transpose conjugate. In this paper, the channel matrix 𝐇 is normalized by ‖𝐇‖^2𝐹=𝑁_𝑟𝑁_𝑡.

(a) Conventional MIMO system

(b) Close loop beamforming

(c) Open loop beamforming

Figure 1 

System overviews.

2.3. MIMO System with Close Loop Beamforming Technique

Figure 1(b) showed the close loop beamforming representation of MIMO systems. We can refer to eigen-beamforming technique. Considering an MIMO channel with 𝑁_𝑟×𝑁_𝑡 channel matrix H that is known to both the transmitter and the receiver, the eigenvectors can be found by applying SVD technique to channel matrix as shown below: 𝐇=𝐖𝐒𝐕^∗,(9) where 𝑁_𝑟×𝑁_𝑟 matrix 𝐖 and the 𝑁_𝑡×𝑁_𝑡 matrix 𝐕 are unitary, and 𝐒 is an 𝑁_𝑟×𝑁_𝑡 diagonal matrix. These two matrices are used as pre- and postcoding matrices at transmitter and receiver, respectively. So the channel matrix when applying pre and post matrices can be written as follows: 𝐇^𝑒=𝐖^∗𝐇𝐕.(10) Thus, the capacity of MIMO system is given by𝐶=log₂det𝐈_𝑁𝑟+𝑃^𝑡𝑃_𝑁𝑁_𝑡𝐇^𝑒𝐇^𝑒∗.(11)

2.4. MIMO System with Open Loop Beamforming Technique

Figure 1(c) showed the open loop beamforming representation of MIMO systems. The open loop beamforming can be represented by the transmitted and received signals. The signal arriving at a directional cosine Ω into the receive antenna array is along the unit spatial signature𝐞_𝑟(Ω) given by (7). Hence, the particularly fixed sequence of Ω can form a special set of 𝑁_𝑟×1 vectors given as𝜉_𝑟∶=𝐞_𝑟(0),𝐞_𝑟𝐿1_𝑟,…,𝐞^𝑟𝑁^𝑟−1𝐿_𝑟.(12)

In (12), it can be noticed that there is a set of orthogonal basis for the received signal space. This basis provides the representation of received signals in the open loop beamforming.

It is similarly defined for the open loop beamforming representation of the transmitted signal. The signal transmitted at direction Ω is along the unit vector 𝐞_𝑡(Ω), defined in (6). The set of 𝑁_𝑡×1 sequential vectors is given as𝜉_𝑡∶=𝐞_𝑡(0),𝐞_𝑡𝐿1_𝑡,…,𝐞^𝑡𝑁^𝑡−1𝐿_𝑡,(13) where 𝐿_𝑡=𝑁_𝑡Δ_𝑡 and 𝐿_𝑟=𝑁_𝑟Δ_𝑟 are the normalized antenna array lengths of the transmitter and receiver, respectively. Let 𝐔_𝑡 and 𝐔_𝑟 be the unitary matrices whose columns are the basis vector in (13) and (12), respectively, which can be written as𝐔_𝑡=1√𝑁_𝑡⎡⎢⎢⎢⎢⎢⎢⎣11𝑒1_{(−𝑗2𝜋/𝑁𝑡)}⋯1⋯𝑒^{(−𝑗2𝜋(𝑁𝑡−1)/𝑁𝑡)}⋮1𝑒⋮_{(−𝑗2𝜋(𝑁𝑡−1)/𝑁𝑡)}⋱⋮⋯𝑒^{(−𝑗2𝜋(𝑁𝑡−1)(𝑁𝑡−1)/𝑁𝑡)}𝐔⎤⎥⎥⎥⎥⎥⎥⎦,(14)_𝑟=1√𝑁_𝑡⎡⎢⎢⎢⎢⎢⎢⎣11𝑒1_{(−𝑗2𝜋/𝑁𝑟)}⋯1⋯𝑒^{(−𝑗2𝜋(𝑁𝑟−1)/𝑁𝑟)}⋮1𝑒⋮_{(−𝑗2𝜋(𝑁𝑟−1)/𝑁𝑟)}⋱⋮⋯𝑒^{(−𝑗2𝜋(𝑁𝑟−1)(𝑁𝑟−1)/𝑁𝑟)}⎤⎥⎥⎥⎥⎥⎥⎦.(15)

We can transform the conventional MIMO system into the open loop beamforming by the following expression: 𝐇^𝑎∶=𝐔^∗𝑟𝐇𝐔_𝑡.(16) Thus, the capacity of MIMO systems can be given by 𝐶=log₂det𝐈_𝑁𝑟+𝑃^𝑡𝑃_𝑁𝑁_𝑡𝐇^𝑎𝐇^𝑎∗,(17) where 𝐈_𝑁𝑟 is the identity matrix having 𝑁_𝑟×𝑁_𝑟 dimension, 𝐇^𝑎 is the channel matrix having 𝑁_𝑟×𝑁_𝑡 dimension.

Figure 2 shows the simulated channel matrices from statistical modeling adopted by [20]. The basis for the statistical modeling of MIMO fading channels is approximated by the physical paths partitioning into angularly resolvable bins and aggregated to form resolvable paths, whose channel gains are 𝐇^𝑎𝑘𝑙. Assuming that 𝑎_𝑖 of the physical paths is independent. We use (4)–(7) to find channel matrix for conventional MIMO and (14)–(16) to find channel matrix for open loop beamforming.

(a)

(b)

(c)

(d)

Figure 2 

Examples of 𝐇a with different angle spreads at transmitter and receiver.

3. Analytical Analysis of Open Loop Beamforming

It has been demonstrated in the literatures [22–27] that the channel capacity depends on the channel correlation. The large amplitude of correlation coefficient degrades the capacity performance. In this section, the analytical analysis of how open loop beamforming improves the channel capacity is described by showing the impact of using open loop beamforming on the channel correlation.

Let the channel matrix be modeled as𝐇=𝚿^𝑟1/2𝐇_𝑖𝑖𝑑𝚿^𝑡1/2,(18) where 𝐇_𝑖𝑖𝑑 is the complex random matrix in which its entries are identically independent distribution, Ψ_𝑟 and Ψ_𝑡 represent the matrices of channel correlation at receiver and transmitter, respectively.

Using Conventional MIMO System
According to (4), the (k,l)th entry of correlation matrix at receiver is able to be given as 𝚿_𝑟||_𝑘,𝑙=𝐸_𝑖𝑎^𝑏𝑖𝑒^{−𝑗2𝜋𝑘Δ𝑟Ω𝑟𝑖}𝐞_𝑡Ω_𝑡𝑖^∗×_𝑖𝑎^𝑏𝑖𝑒^{−𝑗2𝜋𝑙Δ𝑟Ω𝑟𝑖′}𝐞_𝑡Ω_𝑡𝑖^∗^∗,(19) where E{⋅} is the expectation operation. Assuming all the propagation paths are independent from each other, the correlation element in (19) can be reduced to 𝚿_𝑟||_𝑘,𝑙=_𝑖||𝑎^𝑏𝑖||²𝑒^{−𝑗2𝜋(𝑘−𝑙)Δ𝑟Ω𝑟𝑖}.(20) Same as the receiver, the (k,l)th entry of correlation matrix at transmitter is given by 𝚿_𝑡||_𝑘,𝑙=_𝑖||𝑎^𝑏𝑖||²𝑒^{𝑗2𝜋(𝑘−𝑙)Δ𝑡Ω𝑡𝑖}.(21)
In literatures, the degradation of channel capacity depends on the magnitude of correlation coefficient. As seen in (20) and (21), the magnitude is varied by multipath attenuation factors and angle of arrival or departure. However, in order to provide more insight on correlation coefficient, this paper assumes the Gaussian distribution for the attenuation 𝑎_𝑖 and the uniform distribution for the directional cosine within the range of angle spread 𝜎. As a result, the mean of correlation coefficient at receiver is given by 𝚿_𝑟|||_𝑘,𝑙=_𝑖||𝑎^𝑏𝑖||²𝑒^{−𝑗2𝜋(𝑘−𝑙)Δ𝑟Ω𝑟𝑖}prob𝑎^𝑏𝑖,Ω_𝑟𝑖𝑑𝑎^𝑏𝑖𝑑Ω_𝑟𝑖.(22)
Because the property of each path is independent from each other and also the attenuation is independent from the directional cosine, then 𝚿_𝑟|||_𝑘,𝑙=_𝑖||𝑎^𝑏𝑖||²𝑒^{−𝑗2𝜋(𝑘−𝑙)Δ𝑟Ω𝑟𝑖}sin𝜋(𝑘−𝑙)Δ_𝑟𝜎_𝑟𝑖𝜋(𝑘−𝑙)Δ_𝑟𝜎_𝑟𝑖,(23) where 𝑎^𝑏𝑖 is the mean value of 𝑎^𝑏𝑖 and Ω_𝑟𝑖is the mean value of Ω_𝑟𝑖.
In (23), it can be noticed that the mean of correlation coefficient depends on the angle spread. This mean correlation is close to 0 when the angle spread is large. If the angle spread is very small and assuming the same mean of directional cosine for all paths, then the magnitude of mean correlation can be given by |||𝚿_𝑟|||=_𝑖||𝑎^𝑏𝑖||².(24)
With the same derivation as receiver, the mean of correlation coefficient at transmitter is given by |Ψ_𝑡|=∑_𝑖|𝑎^𝑏𝑖|².

Using Open Loop Beamforming
According to (16) and (18), the correlation matrices of open loop beamforming can be formed as 𝐇^𝑎=𝚿^{𝑟𝑎(1/2)}𝐇_𝑖𝑖𝑑𝚿^{𝑡𝑎(1/2)}.(25) Then, the (k,l)th entry of correlation matrix at the receiver is expressed by 𝚿^𝑎𝑟||_𝑘,𝑙=𝐸𝐞_𝑟𝐿𝑘_𝑟^∗_𝑖𝑎^𝑏𝑖𝐞_𝑟Ω_𝑟𝑖𝐞_𝑡Ω_𝑡𝑖^∗×𝐞_𝑟𝐔^𝑡𝐿𝑙_𝑟^∗_𝑖𝑎^𝑏𝑖_′𝐞_𝑟Ω_𝑟𝑖^′𝐞_𝑡Ω_𝑡𝑖^′^∗𝐔^𝑡^∗.(26) Assuming all the propagation paths are independent, 𝚿^𝑎𝑟||_𝑘,𝑙=_𝑖||𝑎^𝑏𝑖||²𝐞^𝑟𝐿𝑘_𝑟^∗𝐞_𝑟Ω_𝑟𝑖𝐞_𝑟𝐿𝑙_𝑟^∗𝐞_𝑟Ω_𝑟𝑖^∗.(27)
For the mean value of open loop beamforming, the same assumption with conventional MIMO has been used. From (26), the mean of correlation coefficient at receiver is given by 𝚿^𝑎𝑟|||_𝑘,𝑙=_𝑖||𝑎^𝑏𝑖||²𝐞^𝑟𝐿𝑘_𝑟^∗×𝐞_𝑟𝐞_𝑟Ω_𝑟𝑖𝐿𝑙_𝑟^∗𝐞_𝑟Ω_𝑟𝑖^∗prob𝑎^𝑏𝑖,Ω_𝑟𝑖𝑑𝑎^𝑏𝑖𝑑Ω_𝑟𝑖.(28) Due to the independent property, then 𝚿^𝑎𝑟|||_𝑘,𝑙=_𝑖||𝑎^𝑏𝑖||²𝑁1_𝑟𝑁^𝑟−1_𝑛=0𝑒^{(𝑗2𝜋𝑛Δ𝑟((𝑘/𝐿𝑟)−Ω𝑟𝑖))}1𝑁_𝑟×^𝑁𝑟−1_𝑛_′=0𝑒^{(−𝑗2𝜋𝑛′Δ𝑟((𝑙/𝐿𝑟)−Ω𝑟𝑖))}probΩ_𝑟𝑖𝑑Ω_𝑟𝑖=_𝑖||𝑎^𝑏𝑖||²1𝑁_𝑟2𝑁^𝑟−1_𝑁𝑛=0^𝑟−1_𝑛=0𝑒^{(𝑗2𝜋Δ𝑟((𝑛𝑘−𝑛𝑙/𝐿𝑟)))}×𝑒^{−𝑗2𝜋(𝑛−𝑛′)Δ𝑟Ω𝑟𝑖}sin𝜋𝑛−𝑛^Δ_𝑟𝜎_𝑟𝑖𝜋(𝑛−𝑛^)Δ_𝑟𝜎_𝑟𝑖.(29) For a small angle spread, 𝚿^𝑎𝑟|||_𝑘,𝑙=_𝑖||𝑎^𝑏𝑖||²𝑁1_𝑟⋅1−𝑒^{(𝑗2𝜋𝑁𝑟Δ𝑟((𝑘/𝐿𝑟)−Ω𝑟𝑖))}×11−𝑒^{(𝑗2𝜋Δ𝑟((𝑘/𝐿𝑟)−Ω𝑟𝑖))}𝑁_𝑟⋅1−𝑒^{(𝑗2𝜋𝑁𝑟Δ𝑟((𝑙/𝐿𝑟)−Ω𝑟𝑖))}1−𝑒^{(𝑗2𝜋Δ𝑟((𝑙/𝐿𝑟)−Ω𝑟𝑖))}^∗=_𝑖||𝑎^𝑏𝑖||²𝑒^{(𝑗𝜋(𝑁𝑟−1)(𝑘−𝑙)/𝑁𝑟)}×⎛⎜⎜⎝sin𝜋𝑁^𝑟Δ_𝑟𝑘/𝐿^𝑟−Ω_𝑟𝑖×⎛⎜⎜⎝𝑁sin^𝑟sin𝜋𝑁𝜋_𝑟Δ_𝑟𝑘/𝐿^𝑟−Ω_𝑟𝑖⎞⎟⎟⎠Δ_𝑟𝑙/𝐿^𝑟−Ω_𝑟𝑖𝑁_𝑟sin𝜋Δ_𝑟𝑙/𝐿^𝑟−Ω_𝑟𝑖⎞⎟⎟⎠.(30) For the transmitter, the mean of correlation coefficient can be given as 𝚿^𝑎𝑡|||_𝑘,𝑙=_𝑖||𝑎^𝑏𝑖||²𝑒^{(−𝑗𝜋(𝑁𝑡−1)(𝑘−𝑙)/𝑁𝑡)}×⎛⎜⎜⎝sin𝜋𝑁^𝑡Δ_𝑡𝑘/𝐿^𝑡−Ω_𝑡𝑖×⎛⎜⎜⎝𝑁sin^𝑡sin𝜋𝑁𝜋_𝑡Δ_𝑡𝑘/𝐿^𝑡−Ω_𝑡𝑖⎞⎟⎟⎠Δ_𝑡𝑙/𝐿^𝑡−Ω_𝑡𝑖𝑁_𝑡sin𝜋Δ_𝑡𝑖/𝐿^𝑡−Ω_𝑡𝑖⎞⎟⎟⎠.(31) Due to the fact that sin=𝜋𝑁^𝑡Δ1^𝑡𝑘/𝐿𝑘_𝑡/𝐿^𝑡−Ω_𝑡𝑖/𝑁^𝑡sin𝜋Δ_𝑡𝑘/𝐿^𝑡−Ω_𝑡𝑖=Ω_𝑡𝑖<1𝑘/𝐿_𝑡≠Ω_𝑡𝑖.(32)
The magnitude of mean correlation, for 𝑘≠𝑙, at the receiver when using open loop beamforming is given by |||𝚿^𝑎𝑟|||=|||𝚿^𝑎𝑡|||<_𝑖||𝑎^𝑏𝑖||².(33) Comparing between (24) and (33), the magnitude of mean correlation achieved by using open loop beamforming is less than using conventional MIMO system. Therefore, according to the results in the literature [22–27], it can be implied that the capacity of MIMO system can be improved when applying open loop beamforming instead of conventional MIMO.

4. Practical Realization Using Butler Matrix

Figure 3 shows a block diagram of Butler matrix [18] which is applied to the concept of open loop beamforming for 4×4 MIMO systems. It is constituted by four 90° hybrid couplers, 2 phase shifters 45°, and a crossover. The fixed beamforming matrix is a bidirectional transmission. Hence, it can be used for either receiver or transmitter.

Figure 3 

Block diagram of Butler matrix [18].

It is easily shown that the weight vectors corresponding to each port presented in Table 1 are mutually orthogonal. Therefore, instead of using (14) and (15), the unitary matrix of applying Butler matrix can be written by the following expressions:𝐁_𝑡=1√𝑁_𝑡⎡⎢⎢⎢⎢⎢⎢⎣𝑒^{−𝑗(−45∘)}𝑒^{−𝑗(−180∘)}𝑒^{−𝑗(45∘)}𝑒^{−𝑗(−90∘)}𝑒^{−𝑗(0∘)}𝑒^{−𝑗(−45∘)}𝑒^{−𝑗(−90∘)}𝑒^{−𝑗(−135∘)}𝑒^{−𝑗(−135∘)}𝑒^{−𝑗(−90∘)}𝑒^{−𝑗(−45∘)}𝑒^{−𝑗(0∘)}𝑒^{−𝑗(−90∘)}𝑒^{−𝑗(45∘)}𝑒^{−𝑗(−180∘)}𝑒^{−𝑗(−45∘)}𝐁_𝑟⎤⎥⎥⎥⎥⎥⎥⎦,=1√𝑁_𝑟⎡⎢⎢⎢⎢⎢⎢⎣𝑒^{−𝑗(−45∘)}𝑒^{−𝑗(−180∘)}𝑒^{−𝑗(45∘)}𝑒^{−𝑗(−90∘)}𝑒^{−𝑗(0∘)}𝑒^{−𝑗(−45∘)}𝑒^{−𝑗(−90∘)}𝑒^{−𝑗(−135∘)}𝑒^{−𝑗(−135∘)}𝑒^{−𝑗(−90∘)}𝑒^{−𝑗(−45∘)}𝑒^{−𝑗(0∘)}𝑒^{−𝑗(−90∘)}𝑒^{−𝑗(45∘)}𝑒^{−𝑗(−180∘)}𝑒^{−𝑗(−45∘)}⎤⎥⎥⎥⎥⎥⎥⎦.(34)

Table 1 

Element phasing, beam direction, and interelement phasing for the Butler matrix shown in Figure 3 (conceptual).

Figure 4 shows a configuration of manufactured Butler matrix. The dimensions of Butler matrix can be calculated from transmission line theory. The manufactured product is also confirmed by measuring inter-element phasing and beam direction which are shown in Table 2. In this table, the distributions of all inter-element phasing are similar to conceptual Butler matrix but they are slightly deviated by ±10 degree. However, the beam direction is deviated by only 0.6 degree. All parameters used for experiments are presented in Table 3.

Table 2 

Element phasing, beam direction, and interelement phasing for the Butler matrix shown in Figure 4 (manufactured).

Table 3 

The parameters used for measurements.

Figure 4 

Configuration of manufactured Butler matrix.

Figure 5 illustrates the beam direction of applying Butler matrix to both transmitter and receiver. It is interesting to see that the concept of open loop beamforming is successfully achieved by simply adding Butler matrices next to antenna elements. Then, the channel matrix realized by Butler matrix can be written as 𝐇^𝑏∶=𝐁^∗𝑟𝐇𝐁_𝑡,(35) where 𝐁_𝑡 and 𝐁_𝑟 are the unitary matrices whose columns are the basis vector in four directions for transmitter and receiver and 𝐇 is channel matrix of size 𝑁_𝑟×𝑁_𝑡 to get conventional MIMO. Thus, the capacity of MIMO systems when applying Butler matrix is given by 𝐶=log₂det𝐈_𝑁𝑟+𝑃^𝑡𝑃_𝑁𝑁_𝑡𝐇^𝑏𝐇^𝑏∗.(36)

Figure 5 

Illustration of applying Butler matrix for 4×4 MIMO systems.

5. Measurement

Figure 6 shows a block diagram of measurement setup for 4×4 MIMO system. The network analyzer is used for measurement channel coefficients in magnitude and phase. The power amplifier (PA) is used at transmitter to provide more transmitted power. Low-noise amplifier (LNA) is used at the receiver to increase the received signal level [28]. The channel measurements are undertaken five times at each location. In each location, two modes of MIMO operation (conventional MIMO and open loop beamforming) are measured. The Butler matrices are inserted at both transmitter and receiver when measuring MIMO channels with open loop beamforming.

Figure 6 

Block diagram of measurement setup.

Figure 7 shows measurement scenarios. We choose a large room to provide various test conditions. The location of the transmitter is fixed as shown in Figure 7 with rectangular symbol. There are five measured locations for the receiver represented by circular symbol in Figure 7. It is easy to measure both conventional MIMO and open loop beamforming by using switches presented in Figure 6. The measured results achieved by network analyzer are used as a channel response in MIMO systems. Also seen in Figure 6, apart from Butler matrix, all components of conventional MIMO and open loop beamforming are the same. Therefore, the measured channels can be directly compared to each other as presented in the next section.

Figure 7 

Measurement scenarios.

6. Results and Discussion

6.1. Simulation Results

The simulations are undertaken by MATLAB programming, and the capacity results are evaluated by using (8), (11), and (36). For conventional MIMO approach, the channel matrix H is found by assumptions in (5), (6), and (7). For simulations, the channel matrix is generated by randomizing the attenuation 𝑎_𝑖 (range from 0 to 1) for each path. The authors use 100 scattering paths for summation in (4). For angle of arrival/departure (𝜙_𝑟𝑖/𝜙_𝑡𝑖), firstly the authors fix the angle spreads at transmitter and receiver. Then the angles are randomly generated within these spreads. In this paper, four cases are considered as (i) 60° spread at transmitter and 360° spread at receiver, denoted as 60-360, (ii) 360° spread at transmitter and 60° spread at receiver, denoted as 360-60, (iii) 60° spread at transmitter and 60° spread at receiver, denoted as 60-60, and (iv) 360° spread at transmitter and 360° spread at receiver, denoted as 360-360. Note that case (iii) is equivalent to line-of-sight scenario while case (iv) is equivalent to Rayleigh fading channel.

In Figure 8, the average capacity comparison between 4×4 MIMO systems with conventional MIMO, open loop beamforming, close loop beamforming, and SISO system is presented. It is clearly seen that all MIMO systems offer better performance than SISO system. The results also indicate that using the open loop beamforming realized by Butler matrix offers better performance than conventional MIMO system for all cases of angle spreads. The range of capacity enhancement is from 2 to 5 dB depending on characteristic of fadings. Although the best performance is achieved by the optimum close loop beamforming technique but it requires the full knowledge of wireless channel at the receiver. Instead of additional complexity to implement close loop beamforming, the open loop beamforming can provide the significant improvement of MIMO capacity by just inserting Butler matrices and no extra complexity is required.

Figure 8 

Average capacity (bits/s/Hz) versus SNR (dB) for 4 conditions of angle spread, Δ𝑡=Δ𝑟=0.5.

6.2. Measurement Results

The channel matrices 𝐇 and 𝐇^𝑏 are found by measured data from network analyzer. The channel fading environments are measured by changing the locations of the receiver. Five locations are considered in Figure 7. We also believe that the mismatches among RF circuits in transmitting/receiving components and mutual coupling effects are included in the measured channel. The simulations are undertaken by utilizing measured data into MATLAB programming and the capacity results are evaluated by using (8) and (36).

Figure 9 shows comparison of conventional and open loop beamforming channels of 4×4 MIMO systems at location 5, where 𝐇_𝑖𝑗 is referred to the channel coefficient at ith receive antenna and jth transmit antenna. It can be observed that channels of conventional MIMO and open loop beamforming are relatively different. The amplitude deviation is about ±5 dB and the phase deviation is about ±100°. These deviations are dominant to the capacity performance of MIMO systems. For other locations, the deviations of amplitude and phase are similar to location 5.

Figure 9 

Measured 4×4 MIMO channels of conventional MIMO system and open loop beamforming (Butler matrix), at location 5.

In Figure 10, the average capacity versus signal-to-noise ratio (SNR) at each location is presented. The results indicate that using the open loop beamforming realized by Butler matrix offers better performance than using conventional MIMO system. To compare between the simulation results presented in Figure 8 and the measurement results presented in Figure 10, the gaps between conventional MIMO and open loop beamforming in Figure 10 are different from the results in Figure 8. This is because the real propagation channel does not behave exactly as the channel modeled in simulations. However, both results provide the same agreement on improving MIMO capacity when using open loop beamforming technique.

Figure 10 

Average capacity (bits/s/Hz) versus SNR (dB) at each location.

For the measurements resulting in Figure 10, the set of transmitting and receiving antenna direction is based on face-to-face installation or 0° direction depicted in Figure 11. However, the proposed open loop beamforming may be very sensitive to the antenna directions because both the transmitter and receiver have the fixed directivities. Accordingly, it is necessary to investigate on other array directions. The additional measurements have been done by rotating the array direction for 0°, 45°, and 90° degrees as depicted in Figure 11. The additional experiments are undertaken in all locations. Figure 12 shows the average capacity (bits/s/Hz) versus signal-to-noise ratio (SNR) in various directions at location 5. The results still indicate the significant improvement of using open loop beamforming in comparson with conventional MIMO system at each array direction. In order to justify the results of other locations, the numeric values of average capacity at SNR = 10 dB are given in Table 4. It is obviously noticed that the benefit of open loop beamforming is pronounced for all locations and all directions. Please remember that the improvement of MIMO capacity costs the little expense of inserting Butler matrices at both transmitter and receiver without any extra complexity.

Table 4 

Average capacity (bps/Hz) of all locations when rotating the direction of array antennas for SNR = 10 dB.

Figure 11 

The rotating directions of array antennas for measurement scenarios.

Figure 12 

Average capacity (bits/s/Hz) versus SNR (dB) for various array directions at location 5.

7. Conclusion

This paper presented the performance of MIMO systems using open loop beamforming realized by Butler matrix. The simulation results reveal that the proposed system outperforms the conventional MIMO system for every fading case. Then, this paper verified the benefit of using open loop beamforming technique for 4×4 MIMO systems by measured results. The open loop beamforming realized by Butler matrix has been implemented and compared with conventional MIMO system. The results revealed that the open loop beamforming outperforms the conventional MIMO system for all fading locations. Hence, the proposed systems are very attractive to practically implement on MIMO systems due to low cost and complexity.

Acknowledgments

This paper is financially supported by Suranaree University of Technology, Thailand and the Royal Golden Jubilee Program of Thailand Research Fund, Thailand.