Qibiao Zhu , Tao Jiang＊

1 School of Electronic Information and Communications, Huazhong University of Science and Technology, Wuhan 430074, China

2 School of Information Engineering, Nanchang University, Nanchang 330031, China

I. INTRODUCTION

Recently, orbital angular momentum (OAM)has emerged as an additional spatial degree of freedom to provide high capacity for future mobile networks by OAM multiplexing in the radio fi eld [1]. The corresponding OAM-based radio system was referred to as a radio vortex(RV) system [2]. When the RV system was combined with multiple-input multiple-output (MIMO) system in practice, the vortex channel matrix model based on the free-space channel transfer function was derived theoretically [3], since OAM multiplexing combined with conventional spatial multiplexing could further enhance the communication capacity in Line-of-Sight (LOS) millimeter-wave communications [4]. Furthermore, an OAM-mode division multiplexing (MDM) system [5], an inband full-duplex communication system [6]and a trellis-coded OAM-quadrature amplitude modulation (QAM) union modulation method [7] were presented to improve the performance of the RV system, respectively.In the RV system, negative and non-integer topological charges can be used to obtain orthogonal vortex channels if they meet the selection criteria [8]. Theoretically, the partial arc sampling receiving scheme was able to demultiplex arbitrary number of topological charges as long as sufficient power and phase change of each vortex signal were sampled[9]. Although the vortex signals can transmit over vortex channels with different frequency bands, such as the 60 GHz radio channel [10],the vortex channel model was not involved.In the near- field zone, the vortex channel exhibited a good isolation of topological charges[11]. However, in the far-field zone, the vortex channel was considered as the free-space propagation channel. Therefore, the expression of the vortex channel model was replaced by the channel transfer function of the freespace propagation model [12]. Note that, the replacement is not available since there exists no theoretical model for the vortex channel in free space, and the existing free-space propagation channel model does not include the topological charge which is the eigenvalue of the vortex signal. Although the vortex channel matrix model of the RV-MIMO system was derived in [3], the model was based on the free-space propagation model and only available for the RV-MIMO system rather than the ordinary RV system. Thus, it is essential to study the vortex channel model for the RV system.

In this paper, a freespace vortex channel model of the radio vortex system was proposed.

In this paper, a free-space vortex channel model is proposed for the RV system. The proposed free-space vortex channel model is derived from the conventional free-space channel transfer function of the free-space propagation model in the radio fi eld and the electric fi eld expression of the Laguerre-Gaussian (LG)modes in the optical fi eld, which are suitable for the free-space propagation scenario due to the OAM transmission characteristics. The aim of this paper is to derive the reasonable vortex channel model of the RV system theoretically. The key contribution is to derive the free-space vortex channel model. Finally,simulation results validate the characteristics achieved by the proposed free-space vortex channel model.

Fig. 1. The RV system model.

Fig. 2. Different phase fronts for different signals.

The rest of this paper is organized as follows. The RV system model is presented in Section II. Then, the free-space vortex channel model of the RV system is analyzed and derived in Section III. Moreover, simulations are conducted to demonstrate the characteristics of the proposed model in Section IV. Finally,conclusions are drawn in Section V.

II. SYSTEM DESCRIPTION

According to previous studies [2], [3], the received vortex signal should be related to time,distance and the topological charge while the vortex signal is transmitted.

As shown in figure 1, the vortex signal of the RV system generated by the vortex antenna transmits over the vortex channel. Then the received vortex signal could be modelled as

wherey(d,t,?) is the received vortex signal,v(t,?) denotes the transmitted vortex signal with?being the azimuthal angle,hv(d) is the free-space vortex channel transfer function, andn(t) represents the independent and identically distributed (i. i. d.) zero mean complex Gaussian random variable with unit variance. Therefore, the transmitting vortex signal can be written as

As illustrated in figure 2, the radio signal in the conventional radio system has planar phase fronts. However, the mentioned vortex signal in the RV system has a helical phase front. The distortion of the helical phase front is related to the topological charge. If the topological charge is zero, the phase front will be planar.In this case, the vortex signal is changed into the radio signal. Namely, the radio signal is a special case without the topological charge.

From a practical perspective, the helical phase front can be generated by the spiral phase plate (SPP) which introduces a termejk?in theory [2]. Therefore, the vortex signal can be achieved by multiplying a factorejk?to a radio signal theoretically. As a result, the vortex signal can be denoted as

wheres(t) is the radio signal andkdenotes the topological charge. Thus, the transmitting vortex signal can be rewritten as

III. THE VORTEX CHANNEL MODEL

The optical OAM signal carries the vortex phase and the radial amplitude distribution in the LG modes, which can be easily generated and stably propagated in free space [13]. The electric field of the LG modes traveling along the propagation direction can be expressed in cylindrical coordinates as

whereris the radial distance from the propagation axis,dthe propagation distance,?the azimuthal angle,pthe radial mode number,kthe topological charge,μ= 2π/λthe propagation constant withλbeing the wavelength.In (5), the beam radius

wherew0is the radius of the zero-order Gaussian beam at the waist, the Rayleigh distanceTherefore,w0can be denoted as

By substituting (7) into (6), the equation (6)can be transformed into

Note that,is an associated Laguerre polynomial

Thek-th LG mode has the azimuthal angular dependence of the formejk?. According tomode corresponds to a zero-order Gaussian beam whenk= 0. In addition, whenfor anyk. In this case, the amplitude of the LG mode is a ring with a radius proportional toThus, the associated Laguerre polynomial is reduced as

In figure 3, different radial mode numbers(p= 0, 1) for the amplitude and the phase patterns are demonstrated when the radial distancertakes 1. For the amplitude pattern, themode shows thep+ 1 concentric rings. For example, there are two concentric rings for thep= 1 LG mode. For the phase pattern, thep= 0 LG mode has one phase pattern, while thep= 1 LG mode has two phase patterns.

In practice, the phase of thep= 0 LG mode is the ideal phase for its simple phase pattern.Furthermore, it is difficult and unnecessary to fabricate the vortex antenna with multiple radial mode numbers. Hence, the radial mode numberpcould be considered as zero. Based on this fact, the LG modes could be denoted as

Fig. 3. Three-dimensional amplitude distribution (a, e), amplitude pattern (b, f)and phase pattern (c, g) of the Laguerre-Gaussian (LG) modes: (Top)and (bottom)

Since the radial distance can be considered as the aperture of the vortex antenna in the radio fi eld [15], [16], the radial distancercould be defined as a parameterRwhich can be derived by the initial conditions of the radio vortex system in practice. Therefore, thecould be expressed as

whereis the transfer function, which contains variations of the amplitude and the phase for the transmission of the vortex signal. Compared (13) with (4), assume thats(t)takes 1 for a certain timet,could be considered as vortex channel transfer functionof the RV system. Based on this idea,the free-space vortex channel transfer function is denoted as

As seen in figure 2, the radio signal is a special case for the vortex signal when the topological chargek=0. Consequently, the vortex channel in free space becomes the freespace channel ifk=0.

In Ref. [12], the free-space channel transfer functionh(d) was given by

whereβis the channel gain coefficient, which contains amplitude attenuation and phase rotation, λ the wavelength, and d the distance between the transmitting antenna and the receiving antenna. For the time domain, the expression can be transformed as

Assume that the initial transmitting powerPtis a constant P0dB, the received powerPr=P0when the propagation distance d tends to be 0 m. Therefore,

Note that, the Rayleigh distance is defined as

whereRAis the vortex antenna radius [12].Substituting (23) into (22), the parameterRcan be denoted as

By substituting (7), (8), (23), and (24) into(18), the free-space vortex channel transfer function can be modelled as

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In addition, electromagnetic waves travel at the speed of light through air: c = 3.0 × 108m/s, and the propagation distancedand timethave the relation ofd= c ×t, hence, the vortex channel response of the time domain can be calculated as

To predict the received signal strength in typical wireless environments, the free-space channel transfer function is employed to achieve the path loss. The empirical free-space path loss,PLFS(d), can be directly derived from the free-space channel transfer function(15) under the assumption of the transmitting powerPt= 1 (Watts) as

wherePrrepresents the received signal power.For the vortex channel, the free-space vortex channel path loss is similar to the empirical free-space path loss which is obtained by

In conclusion, compared with the free space channel transfer function of the freespace path-loss model, the expression of the proposed free-space vortex channel model contains the topological charge k which describes the variations of the amplitude and vortex phase during the vortex signal transmission.

IV. NUMERICAL SIMULATION AND ANALYSIS

In this section, the free-space vortex channel model is simulated to validate the transmission characteristics of the vortex signals. Note that,the proposed free-space vortex channel model is the same as the free-space propagation model when the topological chargek= 0. The main parameters of the two channel models are listed in table 1. These parameter values are some demonstrated values which can be changed according to the actual propagationscenario.

Table I. Simulation parameters.

Fig. 4. Path loss comparison.

Fig. 5. The relationship between the relative magnitude and distance for the phase effect.

Fig. 6. Phase variations for different distances.

Figure 4. depicts the path loss of the two channel models ascending with the increasing propagation distance. The larger the topological chargek, the faster the path loss increases in free space since the magnitude decreases with the increase of the topological charge according to (18). Note that, there is an inflection point when the distance is 10.0 m due to the effect of the topological charges. Therefore, it is necessary to study the influence of the phase related to the topological charges on the signal magnitude in short distances.

As shown infigure 5, the magnitude value of the frees-pace path-loss model at 5.0 m is employed as the reference value, other values divided by this reference value can yield a relative magnitude. The values of the relative magnitude can be positive or negative since the phase effect is related to the topological charges. Therefore, the absolute value of the relative magnitude does not decrease as the topological charge increases.

The variations of the path loss and the relative magnitude can be seen in the long and short distances, respectively. Moreover,the shorter distance can be used to study the smaller phase variation.

As illustrated in figure 6, the phase of the different channel models varies with the distance for different topological charges. The phase of the two models varies periodically in one wavelength range. However, the number of phase periods in one wavelength is different. The larger the topological chargek, the more the number of phase periods.

For the time domain, the magnitude values are rapidly attenuated over time. There are periodic reductions for the free-space propagation model and the topological chargek= 1,2 of the proposed model. However, there is no periodic variation for the topological chargek= 3 of the proposed channel model since the magnitude attenuation is accelerated with the increase of the topological charge, as seen infigure 7.

Figure 8. shows the magnitude-frequency responses of the two channel models. For the free-space propagation model, the magnitude-frequency response curve is approximately fl at. However, the magnitude-frequency re-sponse curves of the proposed free-space vortex channel model demonstrate the frequency selective fading.

Figure 9. describes the phase-frequency responses of the two channel models. For the frequency of 30.0 GHz, the phase-frequency response curves are flat except for the topological chargek= 2 of the proposed model.Therefore, thek= 2 vortex channel model shows the frequency selectivity infigure 8 at 30.0 GHz.

The magnitude-frequency responses and the phase-frequency responses of the channel models are the Fourier transform of the time-domain impulse responses of the channel models. They are the reflection of the channel characteristics in the frequency domain. They can well reflect the influence of the channel on the frequency components of the signal.However, for the transceiver, the bit error rate(BER) is an important performance parameter to evaluate the communication quality of the system which contains the channel. Therefore,the BER of the two channel models are simulated based on the binary phase shift keying(BPSK) modulation.

As shown infigure 10, compared with the free-space propagation model, the proposed channel model does not change the BER performance although the topological charges are different. Therefore, the proposed free-space vortex channel model has the same bit error rate performance as the free-space propagation model.

In summary, the proposed free-space vortex channel model indicates the impact of the topological charge on the vortex channel for the radio vortex system in free space. Compared with the free-space propagation model often used in the radio vortex system, the proposed free-space vortex channel model can more accurately reflect the vortex channel characteristics.

V. CONCLUSION

In this paper, a free-space vortex channel model of the radio vortex system was proposed.

Fig. 7. Magnitudes of time domain.

Fig. 8. Magnitude-frequency responses.

Fig. 9. Phase-frequency responses.

Fig. 10. BER comparison.

The proposed free-space vortex channel model was derived from the Laguerre-Gaussian modes and the free-space propagation model according to the practice in free space. Simulation results showed that the proposed model could significantly reflect the vortex channel characteristics compared with the existing free-space propagation model for the radio vortex system.

ACKNOWLEDGEMENT

This work was supported in part by National Science Foundation for Distinguished Young Scholars of China with Grant number 61325004, Major Program of National Natural Science Foundation of Hubei in China with Grant number 2016CFA009, the Fundamental Research Funds for the Central Universities with Grant number 2015ZDTD012, the National Natural Science Foundation of China under Grant No. 61463035, and the Research Foundation of the Education Department of Jiangxi Province under Grant No. GJJ150198.

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