Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Simulation of Free-Space Communication using the Orbital Angular Momentum of Radio Waves Áron Demeter, Csaba-Zoltán Kertész Department of Electronics and Comuters, Transilvania University, Brașov, Romania [email protected], [email protected] Abstract—The Orbital Angular Momentum (OAM) property of radio waves can be efficiently used to combine multiple free-space radio-communication channels into a single carrier frequency. This paper presents a study of such a multichannel system using Simulink model of radio-waves with different OAM value combined together. The study focuses on proving that OAM channels remain orthogonal at radio frequencies, and the OAM modulation of FSK, PSK and QAM modulated signals results in bit-error rates close to standard communication channel behavior in noisy environment. (a) I. I In 1902 Poynting discovered, that the circularly polarized light carries Spin Angular Momentum (SAM), which is associated with photon helicity and depends on the polarization, therefore it can be present in left- or right-handed circularly polarized light. The SAM value for one photon is 𝑆 = ±ℏ, where ℏ is the Planck constant. The other part of angular momentum was identified by C. Darwin in 1932 as the product of linear momentum of photon and radial distance, which is independent of spin [1]. So the total angular momentum 𝐴 is given by the sum: 𝐴=𝑆 +𝐽 (1) where 𝐽 represents the Orbital Angular Momentum (OAM) of the photon, which is associated with the orbital phase profile of the beam, measured in the direction orthogonal to the propagation axis [2]. In 1992, Allen et al. discovered that using cylindrical lenses to generate Laguerre-Gaussian (LG) modes from Hermite-Gaussian (HG) modes, a helically phased light beam can be obtained which has got the OAM property. The OAM value for one photon is defined in (2), where 𝑙 is an integer number. 𝐽 = ±𝑙ℏ (2) In a plane perpendicular to the propagation axis, the phase of the electric and magnetic vector fields have an 𝑙⋅𝜙 dependence (𝜙 - angular coordinate), which means that for 𝑙 ≠ 0 the phase fronts of beams are not planar but helical [3]. For a given 𝑙 value, the beam will contain 𝑙 intertwined helical phase fronts. A forked diffraction grating can also produce helical beams with optical vortex. Allen et al. showed that a 𝜓𝑙 beam with transversal complex radial amplitude 𝐴(𝑟) depending on the radial (𝑟) and angular (𝜙) coordinates, presented in (3), will contain 𝑙ℏ OAM state for each photon. [4]. 𝜓𝑙 = 𝐴(𝑟) ⋅ 𝑒−𝑖𝑙𝜙 978-1-4799-5183-3/14/$31.00 ' 2014 IEEE (3) (b) (c) Fig. 1. OAM=3 state. (a) wave structure of 40 waves, where the colors from blue to red are indicating the increasing amplitude and the continuous red line interconnects the same phase points of all waves - in this case at the signal maximum, (b) helical phase structure, (c) phase front, where the colors from red to blue correspond to the phase between 0 and 2𝜋. Based on (3), the algorithm for generating twisted waves can be implemented, resulting the waveforms on Fig.1. Because the LASER beams are composed of a narrow band of wavelength (monochromatic) and they are collimated, the light energy is restricted in the direction transverse to the propagation direction to form a narrow beam, which is ideal for OAM eigenstates [5]. The first known successful experiment in communications, with OAM-carrying waves, was done in 2004 by Gibson et al., the information encoded in OAM states was transmitted in air over 15m using computer generated holograms on Spatial Light Modulators (SLM) [6]. In 2012, a 2.56 Tbit/s data transfer rate was successfully made over 1m using OAM Multiplexing (OAMM). The spectral efficiency can be increased combining OAMM with polarization multiplexing [7]. In radio domain only a few experiments has been made, but without much success. In 2011, Bo Thidé and his team demonstrated that it is possible to simultaneously transmit more radio channels on the same frequency band over 442m, by encoding them in different OAM states [2]. However, there were lots of reports of disagreement about this experiment 846 2 declaring, that it was just an implementation of the Multiple In Multiple Out (MIMO) technology. In the above experiment, a modified dish has been used to transmit the helical waves, and a two-antenna interferometer to receive them. In most of the articles, the circular antenna arrays are considered as the best solution for generating OAM-carrying radio beams [8], which can lead to the same results as in optics [9]. To evaluate the performance of the circular antenna arrays we created a Simulink model. It simulates the propagation of the helical waves using different types of modulations. 1.5 1 Ay 0.5 −0.5 −1 −1.5 II. T S M AWGN AWGN Channel HelixFDemodulator StoringFaFperiod 1 input_signal In1 Buffer sim_time Tx 1 output_signal Out1 Clock MATLABFFunction StoringFaFperiod 2 In2 input_signal Reshape Buffer1 sim_time Clock1 Rx output_signal 2 Out2 MATLABFFunction1 Fig. 2. The Helix Modulator and the Helix Demodulator connected to the communication channel −2 −2 −1.5 −1 −0.5 0 Ax 0.5 1 1.5 2 (a) 3 2 1 Ay Because the system is very sensible to the phase changes, we assume, that the transmitter and receiver antenna arrays are perfectly facing each other in the line-of-sight, like a fixed wireless system. At lower frequencies and lower OAM states greater mismatch is allowed. For signal modulation and demodulation, existing Simulink digital baseband blocks has been used, containing the complete implementation of the selected modulation, including the coherent or non-coherent energy detection at the receiver part. Therefore, the main parts of the model are the Helix Modulator and the Helix Demodulator blocks, presented in Fig.2. They contain a Matlab function block which implements the wave-twisting and -untwisting algorithms. The buffers are storing 𝑁 samples from the incoming complex signal incorporating at least one full period. The clocks are required to control the output frame indexes per simulation step. They are assumed to be perfectly synchronized, considering that the phase shifting can be done passively or using phased-array controllers, which are not part of the simulation. The Helix Modulator’s Matlab function clones the incoming signal to the number of waves and shift each of them with 𝑙⋅𝜙, where 𝑙 is the OAM value and 𝜙 is the phase, uniformly distributed between the antennas. In other words, the transmitter is a serial to parallel converter and a phase shifter at the same time. Every shifted signal feeds one antenna from the array. While the optical beams contains a lots of waves, in radio domain the number of antennas is limited. To ensure a similar HelixFModulator 0 0 −1 −2 −3 2 1 40 30 0 20 −1 Ax 10 −2 0 Ns (b) Fig. 3. Helix modulator output frame (a) in the case of 4 circles and 40 antennas. (b) is the 3D representation of the same wave for one period and one antenna circle ( Ns is the samples used in one wave period ). phase profile to the LG beams, at least 2 ⋅ 𝑙 + 1 antennas are required [9]. The Helix Demodulator’s Matlab function shifts the incoming waves to the same phase and makes the sum of them, acting like a parallel to serial converter and phase shifter at the same time. In every simulation step, the transmitter outputs a frame which contains 𝑎𝑛𝑡𝑒𝑛𝑛𝑎𝑛𝑢𝑚𝑏𝑒𝑟×𝑐𝑖𝑟𝑐𝑙𝑒𝑛𝑢𝑚𝑏𝑒𝑟 samples. For example if there are four concentric circles, each of them with 40 waves (antennas), the outgoing frame will look like the one on Fig.3. These frames are connected to an AWGN channel simulating noisy environment. Also a multipath propagation channel was tested using Rician fading blocks to simulate reflected signals. Because the OAM works only in line-of-sight transmission and reflected signals also change OAM states, multi-path effects were studied more in terms of interference to the generic OAM transmission channel. The optical beam width scales with √𝑙 which means, that the receiver aperture will have a physical limitation on higher OAM values [6]. In a similar manner the antenna array dimensions (circle radius) in radio domain will also result in the same OAM limitation [9]. There are several methods for detecting the OAM states of radio waves [3], which are not part of our aim, so the simulation uses well defined states in the Helix Modulator 847 III. T S A. FSK modulation Fig. 4. Results of the twisted (center) and untwisted (bottom) waves of an incoming sine wave (top) in case of matched OAM state (a) and unmatched OAM state (b) and Demodulator blocks. When the OAM states are set to the same value at the transmitter and the receiver, the sum will give a strong signal, but when they are set different, the result will be unrecognizable, or close to zero. Even if the OAM states are set to adjacent, the result will be wrong, so it is easy to distinguish them. Fig.4 shows what happens with a simple sinusoidal carrier wave if the OAM states are matched or nonmatched. This proves that our simulation model is consistent with the theoretical orthogonality of the OAM states. Bernoulli Binary 2-FSK Bernoulli Binary Generator2 M-FSK Modulator Baseband1 The Bit-Error Rate (BER) curves of the baseband modulation are very close to the theoretical non-coherent BER curve. Fig.5 shows the scheme of the simulation model and the BER of the 2-FSK signal for 𝑙 = 3, 𝑙 = 6 and 𝑙 = 9. The simulations were made with the minimum number of waves for every OAM state. For the antenna array, the 2-FSK passband modulation uses the same configuration as the baseband model, but the baseband signal is upconverted to 1GHz carrier frequency. The BER curves and the simulation model are shown on Fig.6 in a similar manner to the baseband simulation. In the OAM-based communication the focus is on the orthogonality of the OAM-states which makes possible to transmit more information on the same carrier frequency. The simulation model on Fig.7 is demonstrating, that different communication channels can be used on the same frequency band without any interference. This model is using a single Helix Demodulator whose OAM parameter is the one of the parameters set in the Helix Modulators. In this case, the demodulator in the figure was set to decode the OAM = 5 modulated channel. Every channel is containing different information generated by the Bernoully Binary blocks with different probability of zero. The system is using the minimum Bernoulli Binary Bernoulli Binary Generator1 Helix Modulator 0 Tx ErrordRate Calculation Rx 0 101 BER Wrong Total Data AWGN Channel3 Modulator5 Tx AWGN Rx Error Rate Calculation5 Display5 Helix Modulator2 0 Error1Rate Calculation Error Rate Calculation6 Rate Transition2 0 101 BER AWGN Wrong Channel2 Total AWGN Display6 ZOH 2-FSK Unbuffer FSK Data M-FSK Demodulator Baseband Buffer1 FSK Helix Demodulator Demodulator1 0 10 −1 10 Helix Demodulator2 0 10 Theoretical=2−FSK l===0 l===3 l===6 l===9 Theoretical92−FSK l9=90 l9=93 l9=96 l9=99 −1 10 −2 BER BER 10 −2 10 −3 10 −3 10 −4 10 −5 10 0 −4 2 4 6 8 10 10 12 Eb/N0 (dB) Fig. 5. Baseband FSK simulation model and the BER diagram of the baseband 2-FSK signals 0 2 4 6 Eb/N0 (dB) 8 10 12 Fig. 6. Passband FSK simulation model and the BER diagram of the passband 2-FSK signals 848 Error Rate Calculation4 905347 BER Wrong Total Bernoulli Binary Bernoulli Binary Generator4 Bernoulli Binary Bernoulli Binary Generator5 Bernoulli Binary Bernoulli Binary Generator6 Bernoulli Binary Bernoulli Binary Generator7 Data FSK Modulator6 Data 9 BER Wrong Total Helix Modulator OAM = 3 ErrorORateTx Calculation Rx 54 191 Error Rate Calculation1 Display4 9 ErrorORateTx Calculation Rx 191 Display1 FSK ZOH Modulator1 Data FSK FSK AWGN Channel4 Add1 Modulator2 Data AWGN Helix Modulator OAM = 5 Helix Demodulator3 Demodulator2 43 191 Display3 FSK Display2 Modulator4 Rate Transition3 ErrorORateTx Calculation Rx 904257 BER Wrong Total Helix Modulator OAM = 7 Data BER Wrong Total Helix Modulator OAM = 8 905149 52 191 Error Rate Calculation2 ErrorORateTx Calculation Rx Error Rate Calculation3 Fig. 7. Passband FSK simulation model for a 4 combined communication channels with single channel demodulation ZOH Bernoulli Binary Bernoulli Binary Generator8 Data FSK Modulator9 FSK Helix Modulator1 OAM = 3 Data Demodulator5 Helix Demodulator1 OAM = 3 Rate Transition6 ZOH Bernoulli Binary Bernoulli Binary Generator9 Bernoulli Binary Bernoulli Binary Generator10 Data FSK Modulator3 Data FSK FSK Modulator7 AWGN Helix Modulator1 OAM = 5 Add2 Data Demodulator4 Helix Demodulator1 OAM = 5 Rate Transition5 ZOH AWGN Channel5 FSK Helix Modulator1 OAM = 7 Data Demodulator3 Helix Demodulator1 OAM = 7 Rate Transition4 ZOH Bernoulli Binary Bernoulli Binary Generator11 Data FSK Modulator8 FSK Helix Modulator1 OAM = 8 Fig. 8. Helix Demodulator1 OAM = 8 Data Demodulator6 Rate Transition7 Tx ErrorHRate Calculation Rx Error Rate Calculation10 Tx ErrorHRate Calculation Rx Error Rate Calculation7 Tx ErrorHRate Calculation Rx Error Rate Calculation8 Tx ErrorHRate Calculation Rx Error Rate Calculation9 0 0 101 BER Wrong Total Display11 0 0 101 BER Wrong Total Display8 0 0 101 BER Wrong Total Display9 0 0 101 BER Wrong Total Display10 Passband FSK simulation model for a 4-channel simultaneous communication number of antennas for the maximum OAM state, in this case 19 antennas. The BER calculator shows zero for the selected channel, but the other channels BER values are high. With 19 antennas the system is capable to distinguish 9 channels; therefore every introduced antenna is increasing the total capacity. Another simulation model is shown on Fig.8, where are four Helix Demodulators, one for each channel. Every antenna array is receiving all of the vortex channels, but each decodes only one of them. In this case, the BER results are zero in every calculator block. The OAM state orthogonality was also tested on single channel transmission with multi-path effect simulated with Rician fading blocks. These enable simulating various reflection path between the transmitter and the receiver. The ratio between the power in the line-of-sight component and the power in the diffuse component was set to 𝑘 = 6, and also considering some moving objects near the system, the maximum diffuse Doppler shift was set to 40Hz. Theoretically signals reflected loose their OAM state, so only the line-ofsight channel will be picked up by the receiver, however in practice the reflected signals also present a large incoming interference which increases BER levels. This increase is on the other hand less than the effect of multi-path on the plain FSK signals. The model and result of multi-path analysis is presented on Fig.11. B. PSK modulation The PSK passband simulation model and the BER diagram for the passband simulation is shown in Fig.9. It can be seen, that the results are close to the FSK modulation’s curves, but compared to the theoretical 2-PSK curve, they are worse at higher 𝐸𝑏 /𝑁0 ratio. This shows the sensibility of the system to the phase shifting operation. It is expected that the imprecision introduced by the the floating point calculations within all the phase shifting operations to influence also the PSK demodulation. This also implies that on physical implementation a greater care is needed on the phase shifting lines leading to the antennas. The figures are terminating at the 𝐸𝑏 /𝑁0 point where no more errors are detected at 100 kbits. Because of the one period process at each Helix block, the Error Rate Calculation block is set to delay the detection by two bits. As the number of period processing increase, the delay also increase, but this does not affect the BER result. 849 Bernoulli Binary Data Bernoulli1Binary Generator1 Modulator5 Tx Rx Error1Rate Calculation 0 261 Data BernoulliABinary Generator1 Helix1Modulator2 0 Error1Rate Calculation6 Rate1Transition2 Bernoulli Binary PSK BER AWGN Wrong Total Channel2 Modulator5 Tx AWGN QAM Rx 0 ErrorARate Calculation ErrorARate Calculation6 Display6 ZOH HelixAModulator2 0 165 BER Wrong AWGN Total Channel2 AWGN Display6 ZOH Data PSK Demodulator1 Data RateATransition2 Helix1Demodulator2 −1 QAM Demodulator1 HelixADemodulator2 0 10 10 TheoreticalS2−PSK lS=S0 lS=S3 lS=S6 lS=S9 −2 10 Theoretical=8−QAM l===0 l===3 l===6 l===9 −1 10 −2 BER BER 10 −3 10 −3 10 −4 10 −4 10 −5 10 0 −5 1 2 3 4 5 Eb/N0 (dB) 6 7 8 10 9 0 2 4 6 Eb/N0 (dB) 8 10 12 Fig. 9. Passband PSK simulation model and the BER diagram of the passband 2-PSK signals Fig. 10. Passband QAM simulation model and the BER diagram of the passband 8-QAM signals The Rate Transition block is not necessary, but useful when the difference between the carrier frequency and bit rate is high. To get closer to the theoretical 2-PSK curve, the number of samples and the amplitude of the waves can be increased. When simulating the multiple channels of OAM states combined with a similar model as presented for the FSK modulation, the results were nearly the same meaning that the OAM multiplexing has no further effect on the signals. Multi-path effect on PSK signals is also similar to the FSK case (Fig.12), but with BER values higher comparatively to theoretical values than were with FSK due to the sensibility to phase changing of the PSK modulation. phase+amplitude modulation is used. To get closer to the theoretical 8-QAM curve, also the amplitude and the number of samples can be increased. C. QAM modulation In radio frequency free-space communications, the FSK and PSK modulations are not so frequently used as QAM modulation. A QAM modulated channel has much higher bit rate capability than the PSK modulated channel, and the error probability is much lower than at the FSK. The passband QAM simulation model together with the BER diagram of the passband model is shown in Fig.10. As at the other simulations, the simplest constellation, 8-QAM was used. If the number of waves (antennas) increase, then the amplitude of the waves in the Helix Demodulator and the QAM Demodulator will also increase. The simulation needs the processing quantity to be one period of carrier frequency in order to change the phase correctly even if IV. C The simulation presented in this paper prove that OAMbased radio channels remain orthogonal with acceptable BER on well-known modulation types. The OAM multiplexing of radio waves does not bring anything basically new, as long as it can be implemented with the traditional beam forming techniques using MIMO antenna matrixes or phased arrays. The simulated BER curves shows that there are no big differences between the different OAM states, so theoretically an infinite number of channels can be used without any BER degradations. Compared to the modified parabolic dish, a circular antenna array can generate more than one OAM state without any mechanical modifications, given that at least 2 ⋅ 𝑙𝑚𝑎𝑥 + 1 antennas are used. The results shows that the OAMM is compatible with the digital multiplexing techniques even if they are phasechanging modulations. However the phase shifting circuits has to be implemented with high precision to avoid introducing error into the phase fronts needed by the latter. Also the phasechanging modulations are much more susceptible to multipath effects, which need to be addressed in the design of the communication system. Compared to the optical domain, only one circle of waves is enough to generate the OAM states. 850 Bernoulli Binary Data Bernoulli Binary FSK Reshape BernoulliBBinary Generator1 Modulator5 Tx Rx 0 412 ErrorBRate Calculation6 AWGN Channel2 AWGN Rician Fading Tx MultipathBRician FadingBChannel1 Rx HelixBModulator2 0 ErrorBRate Calculation 0 3606 ErrorBRate Calculation6 Display AWGN Channel2 Rician Fading AWGN MultipathBRician FadingBChannel1 Display ZOH Data RateBTransition2 FSK Data Demodulator1 RateBTransition2 HelixBDemodulator2 0 −1 10 Demodulator1 HelixBDemodulator2 0 Theoretical=2−PSK l===0 l===3 l===6 l===9 −1 10 −2 −2 10 BER 10 −3 −3 10 10 −4 −4 10 10 −5 −5 0 PSK 10 Theoretical=2−FSK l===0 l===3 l===6 l===9 10 BER Modulator5 0 ErrorBRate Calculation PSK Reshape BernoulliBBinary Generator1 HelixBModulator2 ZOH 10 Data 2 4 6 8 10 Eb/N0 (dB) 12 14 16 10 18 0 2 4 6 Eb/N0 (dB) 8 10 12 Fig. 11. Multi-path FSK simulation model and the BER diagram of the passband 2-FSK signals affected by multi-path effect Fig. 12. Multi-path PSK simulation model and the BER diagram of the passband 2-PSK signals affected by multi-path effect As with the optical domain, the line of sight it is also important in the radio domain, as the OAM of the reflected waves can be affected making impossible the correct decoding in the receiver. This can mean that the channel security is increased (eavesdropping out of the line-of-sight is impossible) on the other hand practical use is limited to fixed wireless communications. There is a possibility to use intelligent antennas for beam-forming purpose, however this needs further studies. [4] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman et al., “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Physical Review A, vol. 45, no. 11, pp. 8185–8189, 1992. [5] E. J. Galvez, “Gaussian beams,” 2009. [6] G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Optics Express, vol. 12, no. 22, pp. 5448–5456, 2004. [7] J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur et al., “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nature Photonics, vol. 6, no. 7, pp. 488–496, 2012. [8] C. Deng, W. Chen, Z. Zhang, Y. Li, and Z. Feng, “Generation of OAM radio waves using circular Vivaldi antenna array,” International Journal of Antennas and Propagation, vol. 2013, 2013. [9] B. Thidé, H. Then, J. Sjöholm, K. Palmer, J. Bergman, T. Carozzi, Y. N. Istomin, N. Ibragimov, and R. Khamitova, “Utilization of photon orbital angular momentum in the low-frequency radio domain,” Physical review letters, vol. 99, no. 8, p. 087701, 2007. R [1] C. G. Darwin, “Notes on the theory of radiation,” Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, pp. 36–52, 1932. [2] F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New Journal of Physics, vol. 14, no. 3, p. 033001, 2012. [3] H. Then, B. Thidé, J. Mendonça, T. Carozzi, J. Bergman, W. Baan, S. Mohammadi, and B. Eliasson, “Detecting orbital angular momentum in radio signals,” arXiv:0805.2735v1 [astro-ph], 2008. 851