On the Restriction of Utilizing Orbital Angular Momentum in Radio Communications

Transcription

1 2013 8th International Conference on Communications and Networking in China (CHINACOM) On the Restriction of Utilizing Orbital Angular Momentum in Radio Communications Yingjie Zhang, Student Member, IEEE, Wei Feng, Member, IEEE, and Ning Ge Member, IEEE, State Key Laboratory on Microwave and Digital Communications Tsinghua National Laboratory for Information Science and Technology Department of Electronic Engineering, Tsinghua University, Beijing , China Abstract The orbital angular momentum (OAM) is considered as one of the most fundamental physical quantities in optics. It was recently discovered that the optical OAM can be also applied to radio communications. Nevertheless, OAMbased radio communications are not applicable for long-distance scenarios. In the existing studies, both the transmitting and receiving antenna arrays are facing each other perfectly on the same axis in free space, which however is not the general case in practical wireless communication systems. In this paper, we analyze the performance of OAM-based radio communications with disalignment antenna arrays, indicating the existence of an oblique angle. Both numerical and simulation results demonstrate that the capacity of OAM-based radio communication decrease significantly with disalignment antenna arrays, and the rotation phase is seriously distorted. Moreover, the performance is poorer with larger oblique angles. Index Terms Orbital angular momentum (OAM); radio communication; antenna array; wireless communication; oblique angle I. INTRODUCTION Orbital angular momentum (OAM) of light beams is found to be one of the most fundamental physical quantities. If the phase structure of the light beam rotates, the light has OAM and helical phase fronts. It was shown by Allen et al. [1] that light beams with an azimuthal phase of exp(ilθ) (OAM state l) haveanoamoflħ per photon (where l is an integer value of topological charge, θ is the azimuthal angle and ħ is Plank s constant divided by 2π). This indicates that OAM is a natural property of helically phased beams and can be easily generated in optics labs. This discovery has been used in optical manipulation and quantum information processing [2]. Additionally, attention has been paid to the technique utilizing OAM beams in optical communications because of the orthogonality of various OAM states [3]: OAM can significantly increase the capacity and spectral efficiency of optical communication systems, by encoding information as orthogonal OAM states, or using different OAM states for multiplexing. Laguerre-Gaussian (LG) modes, which carry OAM, are often used to form a complete basis set for light beams [4] (as is shown in Fig.1). Unfortunately, the research of OAM-based radio communications was only restricted to the optical frequency range for more than 10 years. This changed when Thidé et al. show that antenna arrays can generate low-frequency radio beams with helical phase fronts and OAM modes similar to Fig. 1. [4]). Phase fronts of LG beams with different OAM states (taken from LG laser beams in optical frequency range [5]. The OAM radio beams can be both generated and detected by circular vector antenna arrays. The follow-up papers [6][7] show that OAM in radio can be unambiguously estimated by measuring electric field at one single point or two points with local measurements. OAM-based radio communication provides a possible scheme to transmit at the same carrier frequency within the same bandwidth. And a real-world experiment was done by Tamburini et al. [8] in Venice, The experiment demonstrated that it is possible to simultaneously transmit and receive two incoherent radio waves encoded in two different OAM states (l =0, 1) at the same frequency. Therefore, the OAM-based radio communication seems to provide a new opportunity for improving the link-level spectrum efficiency, which is already quite difficult to be dramatically enhanced utilizing conventional wireless communication techniques [9]. However, Edfors and Johansson [10] argue that OAM states in radio lead identical to the eigen-modes of traditional MIMO theory in free space and the OAM-based radio communication does not offer any additional capacity gain. Moreover, OAMbased radio communication can only achieve high performance in short-range distances because OAM states, especially high order states, become very weak beyond Rayleigh distance. In this paper, we focus on the work of [10]. In the existing work, the transmitting and receiving antenna arrays were facing each other on the same beam axis. In this paper, on the contrary, we analyze the capacity and rotation phase of OAM-based radio communication using antenna arrays when the transmitting and receiving antennas are not parallel, which means that the beam is not pointed in the direction orthogonal to the IEEE

2 plane of the antennal arrays. Numerical and simulation results indicate that capacity decrease significantly and the rotation phase is seriously distorted with disalignment antenna arrays. And the performance is worse with larger oblique angles. Our work shows another restriction of utilizing OAM in radio communications and can be considered to be an effective supplement of [10]. This paper is organized as follows. Section II gives a review of OAM-based radio communications. Section III investigates the capacity and the rotation phase of OAM-based radio communication with disalignment antenna arrays. Finally, further discussions and conclusion are drawn in Section IV. II. REVIEW OF OAM-BASED RADIO COMMUNICATION In order to generate radio waves with OAM properties, the authors of [5] and [6] use circular antenna arrays (shown as Fig.2). Each antenna element of the array is located equidistantly around the perimeter of the circle. The elements are fed with the same input signal, but with a successive phase delay of 2πl/N (where N is the element number and l is the desired OAM state) such that phase has increased by 2πl after a full turn. There s a upper limit of OAM states that can be generated: l <N/2. And vector sensing antennas such as tripoles are advised because they can rotate into any given direction under software control. Additionally, conventional antenna pattern optimization can be applied to the circular antenna arrays for a better directivity. Fig. 2. Eight-element circular antenna arrays to generate OAM state k (taken from [10]). On the other hand, the authors of [10] conclude that OAMbased radio communication does not provide any capacity gain when compared to traditional MIMO communication. Fig.3 shows the model of the antenna arrays in [10]. The arrays are placed on a common beam axis at a distance D with radius R TX and R RX, respectively. Let the angle between the first element of the transmitting array and the first element of the receiving array be φ. Given the rotation angle θ nrx,n TX between a transmitting antenna n TX and a receiving antenna n RX, the distance between is denoted as d(θ nrx,n TX ). And the free space loss and additional phase rotation transfer function between a pair of antenna elements is described as h(d) =β λ 4πd exp( j2π d λ ) (1) where d denotes the distance of the two elements, λ denotes wavelength of the radio, and β denotes the attenuation constant. For the same number of elements N TX = N RX = N on both sides, the channel matrix of the system becomes h 1,1 h 1,2... h 1,N h 2,1 h 2,2... h 2,N H = (2) h N,1 h N,2 h N,N where h nrx,n TX = h(d nrx,n TX ) is given as (1) and only depends on (n RX n TX ) and φ. Fig. 3. Model of transmitting and receiving antenna arrays (taken from [10]). After a series of derivation, the expression of channel matrix identifies with the singular value decomposition (SVD) of the eigen-modes of traditional MIMO communication system. This implies that OAM-based radio communication does not offer any capacity gain because OAM-based radio communication is a subset of the schemes of traditional MIMO. Further simulations indicate that high-order OAM states become quite weak beyond Rayleigh distance. The conclusion is that OAMbased radio communication is an optimal scheme at short distance under certain conditions. To increase Rayleigh distance for a better performance, larger array radii and higher carrier frequencies are advised. As a matter of fact, the research about generating OAM in higher 60-GHz millimeter-wave frequency domain has been studied [11], but these work will not be discussed in this paper. III. RESTRICTION OF DISALIGNMENT ANTENNA ARRAYS The authors of [10] have mentioned some of the restrictions of utilizing OAM in radio communications. However, the research was only studied when the antenna arrays were parallel and facing each other on the same axis in free space. However, this is often not the case in practical communication systems. We start by theoretically analyzing the capacity of OAM-based radio communication system when a pair of arrays are facing obliquely. Then we use Matlab simulation platform to calculate the condition number of the channel matrix and the capacity of the system. Moreover, we use Ansoft HFSS simulation platform to analyze the rotation phase front of the beams generated by the antenna arrays. Simulation results show that OAM-based radio communication system has a poor performance and the uniform rotation phase is distorted. 272

3 These results can be considered to be an effective supplement of [10]. In most of practical communication systems, transmitting and receiving antenna arrays cannot face each other perfectly on the same axis. It s more likely that they face obliquely, instead of parallelly, with an oblique angle α, as shown in Fig.4. Assuming both centers of the two antenna arrays and the rotation angle φ are fixed, the oblique angle α is the only variable that determines the relative location of the two antenna arrays. R TX D Fig. 4. Model of disalignment transmitting and receiving antenna arrays, with an oblique angle α. The parameters are the same as mentioned above in [10]. In this case, the distance between a pair of antennas n TX and n RX, with angles θ and α, is given as d(θ nrx,n TX,α)=[D 2 + RTX 2 + RRX(cos 2 2 θ nrx,n TX + sin 2 θ nrx,n TX cos 2 α)+2dr RX sin 2 θ nrx,n TX sin 2 α (3) 2R TX R RX cos θ nrx,n TX ] 1/2 And the rotation angle between n TX and n RX is given as θ nrx,n TX =2π n RX n TX + φ (4) N Therefore, we obtain that the distance between a pair of antennas only depends on φ, (n RX n TX ) and α. According to (1), the elements of channel matrix is given as λ h nrx,n TX = β 4πd(n RX n TX,α) exp( j2π d(n (5) RX n TX,α) ) λ And the normalized capacity of the channel becomes r C = log 2 (1 + P iλ 2 i ) (6) N 0 i=1 where λ 1,λ 2...λ r are the non-zero eigenvalues of the channel matrix H, r is the rank of H and considered to be the spacial degrees of freedom, P 1,P 2 P r are the water-filling power allocations: P i =(μ N 0 λ 2 ) + (7) i r with μ satisfying the total power constraint P i = P. α θ R RX i=1 ϕ If α =0, which is the case in [10], H becomes circulant, and H is diagonalized by the N N unitary DFT matrix, which implies that this is identical with eigen-modes of traditional MIMO theory. Using FFT algorithm and water-filling principle, the capacity of the OAM-based communication systems can be maximized with low computational complexity, even though it does not provide any additional capacity gain. However, if an oblique angle α > 0 exist, the channel matrix H cannot be diagonalized by DFT matrix, because the element h nrx,n TX depends not only on (n RX n TX ), but also on α. So that the SVD of the channel matrix, as well as transmitter and receiver beam-forming, will be done with high computational complexity, and capacity maximization with water-filling will become difficult. There will be a trade-off between computational complexity and capacity maximization. Therefore, the capacity will decrease in most cases. On the other hand, the condition number of the channel matrix is considered as an important index in numerical analysis of capacity. The condition number of the channel matrix H is defined as max i λ i /min i λ i (λ i are non-zero eigenvalues of the channel matrix H). If the condition number is close to 1, the channel is well-conditioned and will facilitate communication in high SNR regime [12]. However, this is not the case if α > 0. We calculate the condition number for 8 8 antenna arrays, at a SNR of 30 db when the oblique angles α = π/9 and α =2π/9 exist. The simulation results in Fig.5 show that the condition number increase significantly in most cases when an oblique angle exists. This implies that the channel matrix becomes not well-conditioned and the capacity will decreases if an oblique angle exits. Condition Number α=0 α=π/9 α=2*π/ Array Distance (λ) Fig. 5. The condition number of the channel matrix with circular antenna arrays of size 8 8 at a SNR of 30 db. The radii of both antenna arrays are 10λ, and the condition number is calculated for distances of from 100 times to 450 times the wavelength. Oblique angles of α =0, α = π/9 and α = 2π/9 are considered. Furthermore, we calculate the normalized capacity of the OAM-based communication system, as defined in (6), with circular antenna array of size 4 4, 8 8 and 16 16, at a SNR of 30 db when the oblique angles α = π/9 and α = 273

4 2π/9 exist. The results are shown as Fig.6. We can see that the normalized capacity decreases significantly with both two oblique angles in all three cases, and the capacity decreases as well at larger distances, which corresponds to the results in [10]. Moreover, we calculate the normalized the normalized capacity at a distance of 150λ with oblique angles of the range [0,π] under the same conditions. The results in Fig.7 indicates that the performance is worse with larger oblique angles, and the plot is symmetrical because of the symmetry of the antenna arrays. and l =3and oblique angles of α =0, α = π/9, α = π/6 and α =2π/9 are considered. The results of three different OAM states are shown in Fig.8, Fig.9 and Fig.10, respectively. From the results we can see that the phase (Poynting vector) is distorted and no more uniformly rotary any more in all the three cases and the rotation phase is more distorted with larger oblique angles. (a) l=1, α=0 (b) l=1, α=π/ α=0 α=π/9 α=2π/9 Normalized Capacity (bps/hz) (c) l=1, α=π/6 (d) l=1, α=2π/ Array Distance (λ) Fig. 6. Normalized capacity of the OAM-based radio communication system with circular antenna arrays of size 4 4, 8 8 and at a SNR of 30 db. The radii of both antenna arrays are 10λ, and the capacity is calculated for distances of from 100 times to 500 times the wavelength. Oblique angles of α =0, α = π/9 and α =2π/9 are considered. Fig. 8. Poynting vectors generated by a 8-element circular antenna array in state l =1is considered. (a) l=2, α=0 (b) l=2, α=π/ Normalized Capacity (bps/hz) (c) l=2, α=π/6 (d) l=2, α=2π/ pi/8 pi/4 3*pi/8 pi/2 5*pi/8 3*pi/4 7*pi/8 pi Oblique Angle α Fig. 7. Normalized capacity of the OAM-based radio communication system with circular antenna arrays of size at a SNR of 30 db. The radii of both antenna arrays are 10λ, and the capacity is calculated for distances of 150 times the wavelength. Oblique angles of the range [0,π] are considered. Additionally, we use Ansoft HFSS 14.0 simulation platform for further analysis. We analyze the Poynting vectors for an 8-element transmitting antenna array with radius R TX =10λ at a distance of D = 150λ in 2.4 GHz WiFi frequency band of a wavelength of λ = 125mm. OAM states of l =1, l =2 Fig. 9. Poynting vectors generated by a 8-element circular antenna array in state l =2is considered. From the numerical and simulation results given above, we demonstrate that the disalignment antenna arrays (oblique angles) have a significant negative effect on the performance 274

5 (a) l=3, α=0 (c) l=3, α=π/6 (b) l=3, α=π/9 (d) l=3, α=2π/9 reflection or refraction [13]. So that OAM-based radio communications cannot be used in NLOS multipath environments, because this is the case in most practical communication systems. However, the restrictions do not exist in the optical regime. In optical communications, the diversity provided by different OAM states can be exploited to tremendously increase the capacity and spectral efficiency of the system. The restrictions of utilizing OAM in radio communication cannot be neglected, and we believe that OAM-based optical communication will have a brighter future than OAM-based radio communication on this account. Fig. 10. Poynting vectors generated by a 8-element circular antenna array in state l =3is considered. of the OAM-based radio communication systems. This should be taken as another serious restriction of utilizing OAM in radio communication. IV. DISCUSSION AND CONCLUSION In this paper, we investigate the performance of the OAMbased radio communications with disalignment transmitting and receiving antenna arrays. Firstly, we obtain the expression of the channel matrix of the system with an oblique angle, according to the model given in the literature. Then we analyze the normalized capacity and the rotation phase of the system, based on both the Matlab and the Ansoft HFSS simulation platforms. The results show that the normalized capacity decreases significantly and the rotation phase is seriously distorted. This implies that the OAM-based radio communication is unfeasible in practical radio wireless communication scenarios, where the relative locations and angles of both transmitter and receiver are varying all the time in general and it would cause the system s severe performance drop. Disaligment antenna arrays can be considered to be a major restriction of utilizing OAM in radio communications, and this conclusion is an effective supplement of the existing work. We agree with the conclusion in [10] that OAM-based radio communication does not bring anything conceptually new and does not provide any capacity gain in the area of wireless communications. And OAM-based radio communication is an optimal choice only in short-range free-space environments. Besides the restrictions mentioned in the existing work and the restriction of disalignment antenna arrays provided in this paper, some other conditions also impose restrictions on OAMbased radio communications. For example, OAM states cannot be maintained in NLOS multipath channels, and the information carried in rotation phase can be completely lost after ACKNOWLEDGMENT This work was partially supported by the National Natural Science Foundation of China ( , , , ), the National S&T Major Project(2011ZX ), the China Postdoctoral Science Foundation funded project (2012M510026, 2013T60114), the Tsinghua University Initiative Scientific Research Program(2011Z05117), and the Tsinghua National Laboratory for Information Science and Technology (TNList). REFERENCES [1] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J.P. Woerdman, Optical angular momentum of light and the transformation of Laguerre- Gauss laser modes, Phys. Rev. A, vol. 45, no. 11, pp , [2] G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas ko, S. M. Barnett, and S. Franke-Arnold, Free-space information transfer using light beams carrying orbital angular momentum, Opt. Express, vol.12, no.25, pp , Nov. 21, [3] J. 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