Broadband beam steering for misaligned multi-mode OAM communication systems

TIAN Zhengjuan ,CHEN Rui,* ,LONG Wenxuan, ,ZHOU Hong ,and MORETTI Marco

1.School of Communication Engineering,Xidian University,Xi’an 710071,China;2.Department of Information Engineering,University of Pisa,Pisa 56126,Italy

Abstract:Orbital angular momentum (OAM) at radio frequency(RF) has attracted more and more attention as a novel approach of multiplexing a set of orthogonal OAM modes on the same frequency channel to achieve high spectral efficiency (SE).However,the precondition for maintaining the orthogonality among different OAM modes is perfect alignment of the transmit and receive uniform circular arrays (UCAs),which is difficult to be satisfied in practical wireless communication scenarios.Therefore,to achieve available multi-mode OAM broadband wireless communication,we first investigate the effect of oblique angles on the transmission performance of the multi-mode OAM broadband system in the non-parallel misalignment case.Then,we compare the UCA-based RF analog and baseband digital transceiver structures and corresponding beam steering schemes.Mathematical analysis and numerical simulations validate that the SE of the misaligned multi-mode OAM broadband system is quite low,while analog and digital beam steering(DBS) both can significantly improve the SE of the system.However,DBS can obtain higher SE than analog beam steering especially when the bandwidth and the number of array elements are large,which validates that the baseband digital transceiver with DBS is more suitable for multi-mode OAM broadband wireless communication systems in practice.

Keywords:orbital angular momentum (OAM),uniform circular array (UCA),broadband wireless communication,misalignment,beam steering.

1.Introduction

Currently,explosive growth of emerging services,e.g.,high-definition (HD) video,virtual reality (VR),auto-pilot driving,and Internet of Things (IoT),calls for a large increase of the wireless data rate.However,it is inadequate to utilize the traditional techniques to catch up with the growing requirement of the data rate.Therefore,a number of effective techniques are developed.One available technique called spectrum extension,has attracted tremendous attention in recent years.For instance,millimeter wave and terahertz bands are being licensed [1].It is well-known that the initial commercial deployments of the fifth generation (5G) new radio (NR) networks,both at sub-6 GHz and at mmWave frequencies,were already under way during 2019 [2],which adopt larger bandwidth (e.g.,100 MHz for sub-6 GHz band,and 400 MHz for mmWave band) than the fourth generation (4G) networks to support three generic types of connectivity:extended mobile broadband (eMBB),ultra-reliable lowlatency communication (URLLC) and massive machinetype communication (mMTC) [3].Since the radio frequency spectrum resources are scarce,besides exploiting more frequency bandwidth,some innovative techniques,such as advanced modulation and coding schemes,intelligent cognitive radio (CR) and massive multiple-input multiple-output (MIMO),have been explored to enhance the system spectral efficiency (SE).

In essence,almost all existing wireless communication techniques are based on planar electromagnetic (EM)waves.Since the discovery in 1992,light beams with helical phase fronts can carry orbital angular momentum(OAM) [4],and a significant research effort has been focused on vortex EM waves as a novel approach for multiplexing a set of orthogonal OAM modes on the same frequency channel and achieving high SE [5−19].The OAM beams can be generated by spiral phase plate,spiral parabolic antenna,metasurface material,and uniform circular array (UCA) [15],where UCA is more common due to the flexibility of generating and receiving multi-mode OAM beams and realizing beam steering.Specifically,the single-mode OAM beam can be generated by a UCA through feeding its antenna elements with the same input signal,but with a successive phase shift from element to element [20].On this basis,after a full turn the phase has the increment of 2πℓ,where ℓ is an unbounded integer termed as OAM mode number [4].Thus,the multi-mode OAM beam can be achieved by synthesizing multiple single-mode beams through radio frequency (RF) analog or baseband digital approaches with corresponding transceiver structures [11].

In spite of vortex EM waves carrying OAM being deemed as a promising beyond 5G (B5G) technique [21,22],there are still some technical challenges for the practical application of OAM wireless communication.One challenge is that OAM wireless communication requires perfect alignment between the transmit and receive antenna arrays.It was analyzed in [12,23,24] that for OAM narrow-band communication systems,if the precondition is not accurately met,the system performance quickly deteriorates,and the beam steering method proposed in [12]can avoid the performance deterioration caused by misalignment.However,the effect of misalignment on the performance of OAM broadband systems and the corresponding beam steering method have not been studied.As a widely accepted application case of the OAM communication,future extremely high data-rate backhaul transmission will definitely adopt large bandwidth.

Therefore,in this paper,we first present the OAM broadband communication system models for the UCAbased RF analog and baseband digital transceiver structures in a more general non-parallel misalignment scenario.Then,we analyze the effect of the oblique angles on the performance of the multi-mode OAM broadband communication system from the perspective of channel gain and inter-mode interference (IMI).Thereafter,we compare two beam steering schemes,i.e.,analog beam steering (ABS) and digital beam steering (DBS),corresponding to the two transceiver structures,in mitigating the destructive effect of oblique angles.Finally,we validate that both ABS and DBS can circumvent large performance deterioration,but the baseband digital OAM transceiver with DBS has a better performance for the misaligned multi-mode OAM broadband communication system.

2.System model

We consider a multi-mode OAM broadband communication system,where the multi-mode OAM beam is generated by anN-elements UCA at the transmitter and received by anotherN-elements UCA at the receiver.

2.1 Multi-mode OAM broadband OFDM systems

It is well known that orthogonal frequency division multiplexing (OFDM) technology is widely used in 4G networks,and is also the core technology of 5G NR,which has been specified in the physical layer fundamentals of 5G NR non-standalone air interface.In order to address diverse scenarios and deployments,5G NR will adopt a scalable OFDM technology,where the OFDM waveform is scalable in the sense that the subcarrier spacing of OFDM can be chosen according to 15×2nkHz,wherenis a specified integer and 15 kHz is the subcarrier spacing used in long term evolution (LTE) [25].Therefore,we consider OFDM technology for our OAM broadband communication system to support the generic types of connectivity in 5G NR.

In practice,the perfect alignment between the transmit UCA and the receive UCA is difficult to realize.For an arbitrary misalignment case,the beam steering needs performing at both the transmitter and the receiver [12].Corresponding to the two transceiver structures shown inFig.1,there are two beam steering schemes,i.e.,ABS and DBS,which are implemented by the RF analog transceiver structure and the baseband digital transceiver structure,respectively.InFig.1,Urepresents the number of OAM modes.

Fig.1 Simplified block diagram of UCA-based multi-mode OAM transceiver

It is apparent that the phase shifts induced by the phase shifters of the RF analog transceiver are fixed for the broadband signals to be transmitted at different frequencies,while for the baseband digital transceiver,phase shifts are easily applied to the broadband OFDM signals at different subcarriers. Hence,the beam steering matrices of ABS at the transmitter and the receiver are denoted asWAandPA,while the beam steering matrices of DBS at the transmitter and the receiver denoted asWD(kp) andPD(kp) are functions of the subcarrier frequency,wherekp=2π/λpis the wave number at thepth subcarier,and λpis the wavelength at thepth subcarrier.

Thus,in the transmission of theU-mode OAM beam that multiplexesUOAM modes in the free space channel at thepth subcarrier,the despiralized OFDM symbol vectors at the RF analog and baseband digital receivers,denoted asyA(kp) andyD(kp) respectively,take the form

For easier analysis,we only consider beam steering at the receiver as shown inFig.2,and the arbitrary misalignment case could be considered and verified similar to that in [12].

Fig.2 Geometrical model of the transmit and the receive UCAs in a general non-parallel misalignment case

Hence,according to (1) and (2),the despiralized OFDM symbol vectors at the RF analog and baseband digital receivers can be written as

2.2 Channel model

In free space communications,propagation through the RF channel leads to attenuation and phase rotation of the transmitted signal.This effect is modelled through multiplying by a complex constanth,whose value depends on the distancedbetween the transmit and the receive antennas and the wave numberkp[20]:

where 1/(2kpd) denotes the degradation of the amplitude,β models all constants relative to the antenna elements and their patterns,and the complex exponential term is the phase difference due to the propagation distance.

According toFig.2,the coordinate of thenth antenna element on the transmit UCA is (Rtcosφn,Rtsinφn,0) in theZ−XOYcoordinate system,and the coordinate of themth antenna element on the receive UCA in theZ−XOYcoordinate system,denoted by (Bx,By,Bz) can be expressed as

which is proved as follows.

whereRx(α) andRy(γ) represent the rotation matrices corresponding to thex-axis and they-axis,respectively,which are given by

After matrix multiplication and addition,the coordinate of themth antenna element on the receive UCA in theZ−XOYcoordinate system is shown in (6). □

Thus,the transmission distancedm,nfrom thenth(1 ≤n≤N) transmit antenna element to themth(1 ≤m≤N) receive antenna element can be calculated as

whereDis the distance between the transmit and the receive UCA centers,RtandRrare respectively the radii of the transmit and the receive UCAs,φn=[2π(n−1)/N+φt]and φm=[2π(m−1)/N+φr] are respectively the azimuthal angles of the transmit and the receive UCAs,φtand φrare the corresponding initial angles of the first reference antenna element in both UCAs,and α and γ are the oblique angles of the receive UCA as shown inFig.2.For easier analysis,we assume φt=0 and φr=0.

Assuming that the transmit and the receive UCAs are placed in the far-field distance of each other,i.e.,D≫RtandD≫Rr,we can approximatedm,nin (11) as

where (12) uses the method of completing the square and the conditionD≫Rt,Rras same as the simple casea−b/a,a≫b,(13) is directly obtained from the condition ofD≫Rt,Rr.Then,substituting (13) into(5) and abbreviatingh(dm,n,kp) tohm,n(kp),we can obtain the channel coefficient from thenth transmit antenna element to themth receive antenna element as

where (14) neglects a few minor terms in the denominator of the amplitude term and thus only 2kpDis left.In the end,the channel matrix of the UCA-based free space OAM communication system can be expressed asH(kp)=[hm,n(kp)]N×N.

3.Effect of oblique angles on the performance of multi-mode OAM broadband communication systems

In this section,we analyze the effect of oblique angles on the performance of an ideal multi-mode OAM broadband system from the perspective of channel gain and IMI.

3.1 Effect of α and γ on channel gain and IMI

In [10],the OAM channel is defined asFUH(kp)which will become a diagonal matrix ΛU(kp) due to the circularity ofH(kp) when α=0 and γ=0.However,in the case of misalignment,H(kp) is not a circulant matrix any more due to the effect of α and γ.

In the misalignment case,theuth-row andvth-column element ofcan be calculated as

where η(kp)=β/2kpDN· exp(−jkpD),s=n−mZ,t=ℓu−ℓvZ refers to the difference between the two OAM modes,Z represents the integer field,(15) applies the approximation cosα ≈ 1−α2/2 for cosα and cosγ ≈1−γ2/2 for cosγ in the case that α and γ are relatively small and neglects a few quadratic terms,and

For easier expression,we refer to(u,u)j2as the channel gain,andas the IM Is. j·j denotes modulus.From (15) and (16),we can observe that,when α ≠0 or γ ≠0 andt≠0,then ξ ≠0,which indicates that α and γ result in IM Is and thuscannot be transformed into a diagonal matrix in the misalignment case.

For further study,we let sinγ=ν,sinαcosγ=µ,then(16) can be w ritten as

where η′(kp)=η(kp)N2·exp(jπℓv/2−jπt−jϕt)=Nβ/2kp·D· exp(−jkp D+jπℓv/2−jπt−jϕt),andJt(·) represents thet-order Bessel function of the first kind.

According to the characteristic of the Bessel function,it is easy to obtain that,whent≠0,if ρ ≠0 (α ≠0 or γ ≠0),Jt(kp Rrρ)≠0,which w ill result in large IM Is even if α and γ have small values due to largekp.Moreover,whent=0,we can find that if ρ=0,Jt(kp Rrρ)=1,if cates that α and γ w ill result in much loss of the channel ρ≠0,Jt(kp Rrρ) w ill be much less than 1,which indigain.

3.2 Analysis of signal-to-interference-plus-noise ratio (SINR)

The received modulation symbols at theuth mode and thepth subcarrier at the receiver in the m isalignment case,denoted asy(kp,ℓu),can be w ritten as

Therefore,the SINR at theuth mode and thepth subcarrier can be formulated as

Additionally,from (18) and the expression of η′(kp),we can observe that,when other parameters are fixed,the value oflinearly increases withN,which means both channel gain and IM Is linearly increase withN.In order to analyze the effect of largeNon the SINR,we define(u,v)=(u,v)/N.Then,SINRu(kp)can be calculated as

where (21) holds whenNis large enough,and SIRu(kp)is defined as the signal-to-interference ratio (SIR) at theuth mode and thepth subcarrier.Hence,using a large number of antennas can elim inate the uncorrelated noise but has no effect on the IM Is of the m isaligned OAM communication system,which is different from the massive M IMO technology being able to alleviate the interference between multiple data streams [27].It follows that interference cancellation technology,such as beam steering,is necessary for a practical OAM communication system.

Moreover,it is obvious that SINRu(kp) is the function of the subcarrier frequency as shown in (20).In order to explain it intuitively,we plot the relationship between SINR at theuth mode and the subcarrier frequency inFig.3with the system parameters specified inTable 1,where ℓu=1,and α=γ=1◦,5◦,10◦,15◦,20◦.InFig.3,we can observe that the SINR values decrease with the increase of the oblique angle,and for a fixed oblique angle the SINRs vary with the subcarrier frequency,which indicates that the phase compensation through phase substraction [28] or beam steering [12] for the received signals at different subcarriers should be distinct.

Fig.3 Relationship between SINR and subcarrier frequency

Table 1 System parameters for the multi-mode OAM broadband system

4.Beam steering for the misaligned multimode OAM broadband system

To alleviate the effect of oblique angles on the channel gain and IMIs induced by the misalignment of transmit and receive UCAs,we consider applying beam steering to the misaligned multi-mode OAM broadband communication system.Beam steering can steer the beam pattern towards the direction of the incident OAM beam and thus compensate the changed phases caused by α and γ at the receive UCA.We consider two beam steering schemes,i.e.,ABS and DBS,and compare their differences in this section.

4.1 ABS

Through calculating the phase difference between the reference element and themth element of the receive UCA,the beam steering matrix of ABS at the receiverWAcan be written as

m=1,···,N,kA=2πfA/c,fA=fL+A·∆fis the fixed frequency of the RF analog receiver,fLis the lowest frequency of the broadband system,∆fis the subcarrier spacing,Af1,2,···,Pg,⊗ denotes the Kronecker product,1Udenotes aU×1 vector of ones.After involving these phases into the original phases inFUat the receive UCA,the effective multi-mode OAM channel after ABS at the receiver can be expressed as

Theorem 1For a multi-mode OAM broadband communication system composed of anN-element transmit UCA and anN-element receive UCA,after ABS,there still remain IMIs induced by α and γ in a non-parallel misalignment case.

Proof Theuth-row andvth-column element in matrixcan be obtained as

wherekg=kp−kA=2π(fp−fA)/c=2πg∆f/c,c is the speed of light,g=p−A,andg∆frepresents the frequency deviation between the channel at thepth subcarrier and the RF analog receiver.In order to guarantee the minimum whole frequency deviation of the OAM broadband system,we chooseA=P/2,namely,fA=fc,wherefcis the center frequency of system.From (25),we can see that there still remain IMIs after ABS and its value depends onkgRrρ.□

Based on the above analysis,after ABS,the SINR at theuth mode and thepth subcarrier can be formulated as

Remark 1It is revealed in (25) that the value of jJt(kgRrρ)j will increase withkg.Thus,once the bandwidth of the OAM system is substantially increased for future applications,the IMIs after ABS will become larger because of the increased frequency deviation.That is to say,the IMIs induced by ABS become more serious in the OAM system with a larger bandwidth.

4.2 DBS

As for DBS,the receiver beam steering matrix at thepth subcarrierWD(kp) can be calculated as

andm=1,···,N.After involving these phases into the original phases inFUat the receive UCA,the effective OAM channel matrix after DBS at the receiver becomes

Theorem 2For a multi-mode OAM broadband communication system composed of anN-elements transmit UCA and anN-elements receive UCA,the beam steering matrix of DBSWD(kp) can eliminate IMIs induced by α and γ in a non-parallel misalignment case.

where η′′(kp)=Nβ/2kpD·exp(−jkpD+jπℓv/2).It is obvious that the value ofis not sensitive to oblique angles.Specifically,whent≠0,even through α ≠0 and γ ≠0,(u,v)≈0.That is to say,the non-diagonal elements of the effective multi-mode OAM channel matrix after DBS becomes vanishingly small,which proves that DBS can eliminate the IMIs almost completely.□

Thus,after DBS,the SINR at theuth mode and thepth subcarrier can be formulated as

4.3 Applicability analysis of ABS and DBS

Comparing (18) and (25),we can find that the difference between the misaligned OAM channel element and the effective channel element after ABS is the second Bessel function term,and their values are respectively dependent on the size ofkpRrρ andkgRrρ.In order to explain it intuitively,we give the specific values ofkpRrρ andkgRrρ inTable 2with the system parameters specified inTable 1.It is clear thatkgRrρ ≪kpRrρ as shown inTable 2.According to the characteristics of the Bessel function,we can infer that whent=0,jJt(kgRrρ)j will be much larger than jJt(kpRrρ)j,and whent≠0,jJt(kgRrρ)j will be much lower than jJt(kpRrρ)j if oblique angles are relatively small. Therefore,compared with misalignment cases,ABS can obtain much larger channel gain and much lower IMIs,and thus effectively alleviate the destructive effect of oblique angles on the transmission performance of the misaligned multi-mode OAM broadband system.However,it is obvious thatkgRrρ increases with oblique angles and jJt(kgRrρ)j becomes large when oblique angles are relatively large (e.g.,α,γ>10◦),which indicates that there remain large IMIs even after ABS.

Table 2 Values of kpRrρ and kgRrρ

In order to explain it more directly,we give a toy example inTable 3with the system parameters specified inTable 1,where MA represents the misalignment case,ℓu=1,α=γ=5◦,15◦,andfp=fL.We can observe that the results inTable 3are coincident with above analyses.Especially,when α=γ=15◦,the IMI of DBS is so small that it can be neglected,while the IMI of ABS is comparable to the noise power,and thus decreases the system performance to a certain extent compared with DBS.Therefore,although ABS can improve the system performance,DBS is the better choice for the misaligned multi-mode OAM broadband system.

Table 3 Comparison of channel gain and IMI between misalignment cases,ABS and DBS

5.Numerical simulations and results

In this section,the practicability of DBS for the misaligned multi-mode OAM broadband system is verified through being compared with perfect alignment,misalignment case and ABS.Except for special explanation,the simulation parameters are listed inTable 1,and the SNR is defined as the ratio of the transmit power versus the noise power.

Then,we compare the performance of ABS and DBS in the misaligned multi-mode OAM broadband system.As shown inFig.4,of DBS can remain basically unchanged and is approximately consistent with that of perfect alignment,whileof ABS has a visible decrease,especially in large oblique angles.Correspondingly,of ABS increases as oblique angles increase butof DBS always remains in a quite low level.Therefore,the transmission performance of the misaligned multi-mode OAM broadband system with ABS will decrease in a certain degree compared with that of DBS.

Fig.4 Effect of beam steering on the misaligned multi-mode OAM broadband system

Thereafter,we calculate SE to measure the overall transmission performance of the misaligned multi-mode OAM broadband system,where α=γ=10◦,15◦,20◦.InFig.5,we can observe that the SE in the misalignment case is so small that the misaligned OAM system without beam steering can not work normally.Both ABS and DBS can greatly improve the SE of the misaligned multimode OAM broadband system.However,the SE of ABS continues to decrease as oblique angles increase,and is always lower than that of DBS.Specifically,the maximum SE of ABS is about 3 bit/s/Hz lower than that of DBS,and about 5.5 bit/s/Hz lower than that of perfect alignment when α=γ=20◦.This decline cannot be ignored under the serious situation of scarce communication resources. Therefore,benefiting from higher SE,DBS is more practical for the misaligned multi-mode OAM broadband system.

Fig.5 Comparison of SE between perfect alignment,misalignment,ABS and DBS in the misaligned multi-mode OAM broadband system

Finally,we analyze the effect ofNon the SE of the misaligned multi-mode OAM broadband system,where α=γ=20◦,andN=15,32.Fig.6illustrates that the SE of misalignment case cannot get noticeable improvement even if we increase the number of antennas.Fortunately,after applying ABS and DBS,the SEs get an apparent increment.Moreover,we can find that the difference of SE between ABS and DBS increases with the number of antennasN.For example,the difference of SE between ABS and DBS reaches 6 bit/s/Hz whenN=32,which verifies DBS is more suitable for the misaligned multimode OAM broadband system again,especially in the system with large antennas.

Fig.6 Effect of N on the SE of the misaligned multi-mode OAM broadband system

6.Conclusions

In this paper,aiming at the application challenge of UCAbased OAM broadband communication,we first quantitatively investigate the effect of oblique angles on the transmission performance of the multi-mode OAM broadband communication system in a non-parallel misalignment case,which shows the performance of the OAM broadband system decreases significantly even under small oblique angles,and has little improvement asNincreases.Therefore,corresponding to the RF analog and baseband digital transceiver structures,we adopt two beam steering schemes,i.e.,ABS and DBS,to mitigate the destructive effect of the misalignment.Mathematical analysis and numerical simulations validate that ABS and DBS both can greatly improve the SE of the misaligned multi-mode OAM broadband system,but DBS can obtain a higher SE than ABS,and the larger the number of antennas,the better the performance.This fact indicates that the baseband digital transceiver with DBS is a better option for the multi-mode OAM broadband communication system in practice.