Separate Least Mean Square Based Equalizer with Joint Optimization for Multi-CAP Visible Light Communication

Jianli Jin,Jianping Wang,Huimin Lu,Danyang Chen

School of Computer and Communication Engineering,University of Science and Technology Beijing,Beijing 100083,China

Abstract: Visible light communication(VLC)is expected to be a potential candidate of the key technologies in the sixth generation (6G) wireless communication system to support Internet of Things (IoT) applications.In this work, a separate least mean square(S-LMS) equalizer is proposed to compensate lowpass frequency response in VLC system.Joint optimization is employed to realize the proposed S-LMS equalizer with pre-part and post-part by introducing Lagrangian.For verification,the performance of VLC system based on multi-band carrier-less amplitude and phase (m-CAP) modulation with S-LMS equalizer is investigated and compared with that without equalizer,with LMS equalizer and with recursive least squares(RLS)-Volterra equalizer.Results indicate the proposed equalizer shows significant improved bit error ratio (BER) performance under the same conditions.Compared to the RLS-Volterra equalizer, SLMS equalizer achieves better performance under low data rate or high signal noise ratio (SNR) conditions with obviously lower computational complexity.

Keywords: visible light communication; Internet of Things;equalization;least mean square

I.INTRODUCTION

As the fifth generation (5G) wireless communication system is maturing towards commercial applications,the prospective studies about the sixth generation(6G)system are increasing recently.Although it is not clear what changes will 6G bring to the society, researchers still have concluded that 6G system should support massive Internet of Things (IoT) devices [1].Visible light communication (VLC) is listed as a potential technology in 6G system and expected to provide ubiquitous access, high data rate transmission,energy harvesting and indoor positioning[2-4],which could benefit the IoT applications.Some key technologies, such as Light Fidelity (LiFi), visible light positioning (VLP) and self-powering receiver have been investigated [5, 6].In the future, VLC could become one of the important roles in IoT applications due to the superiorities of license-free bandwidth,no electromagnetic interference and widely employed light emitting diodes (LEDs) [7, 8].However, compared with the bandwidth for laser diode (LD) light sources, the narrower bandwidth for the LED light sources with visible wavelengths, such as only several MHz bandwidth for commercial phosphor-coated white LEDs, significantly limits the achievable data rate of VLC system [9].To expand the modulation bandwidth of LEDs and improve the achievable data rate,many equalization methods including analog/digital and pre/post-equalizers were investigated.

For the analog equalizers, the resistor-capacitor(RC) based pre-equalizers are widely used to overcome the bandwidth limitations in VLC system.A cascaded pre-equalization circuit was proposed for bandwidth expanding of VLC system using white LED light source [10].Four independent RC preequalizers were introduced to high speed imaging multiple-input multiple-output (MIMO) VLC system for low-pass frequency response compensation [11].Based on the active-passive hybrid network, a novel RC equalizer was applied in baseband transmission VLC system [12].However, compared to the digital equalizers, analog equalizers rely on extra hardware implementation, leading to the lack of flexibility and the increasing of complexity.Therefore,some researches have focused on the digital equalizers for expanding bandwidth of VLC system,which could be realized using digital signal process and corresponding numerical algorithms.An LMS-Volterra based joint MIMO equalizer was proposed to compensate the super-Nyquist carrier-less amplitude and phase of CAP signals[13].The improved bit error rate(BER)performance had been realized using an improved frequency-domain decision feedback equalizer [14].Furthermore, machine learning based algorithm was also introduced in VLC system, for example, a Kmeans clustering based post-equalizer was proposed for VLC system achieving 400 Mb/s data rate with complex network [15].In the researches mentioned above, the digital equalizers were generally designed as post-equalizers which could enhance the power of not only high-frequency signal but also channel noise [16].Some solutions combining pre-equalizer and post-equalizer to compensate the low-pass channel showed better performance [17, 18].A joint processing algorithm combining pre-equalizer and postequalizer was employed in optical system to mitigate the nonlinearity and imbalance between in-phase and quadrature signals [19].It can be attributed that the pre-equalizers could enhance the high frequency part of signal and the enhancement requirement for the post-equalizers can be reduced.In this way, the channel noise enhancement in the post-equalizer could be alleviated.Some researches showed that the joint equalization could improve data rate of VLC system but there was lack of joint optimization between preequalizer and post-equalizer [20, 21].The equalizers based on nonlinear structure are more complex than the linear equalizer because the linear equalizers can be seen as a special case with only first-order of nonlinear equalizer.Thus the nonlinear equalizer is not suitable for the devices with limited computing resource such as cameras,monitors and vehicles.

To reduce the complexity and improve the performance of VLC system, a novel separate least means square (S-LMS) equalizer for VLC channel compensation is proposed and verified in this paper.Compared to the conventional LMS equalizer, the S-LMS equalizer has two separate parts,that is pre-and postequalizer, in which tap coefficient can be jointly optimized using Lagrangian algorithm under minimum mean square error (MMSE) criterion.Using proposed S-LMS equalizer, the VLC simulation system based on multi-CAP (m-CAP) modulation is implemented.As one of the multi-carriers modulations,m-CAP could achieve frequency division multiplexing with low complexity and is suitable for the resource limited devices.On this basis, the system performance with S-LMS equalizer is investigated and compared with that without equalizer with LMS equalizer and nonlinear recursive least squares (RLS)-Volterra equalizer.Results indicate that the proposed S-LMS equalizer could support IoT transmission based on VLC with low computational complexity and improve the performance especially in short reach,high signal noise ratio(SNR)or low data rate conditions.As one of the advanced digital equalizer,Volterra has been investigated in VLC system for high data rate transmission.

Rest of the paper is organized as follow: Section II gives the structure, the derivation of the proposed algorithm and the simulation system model.Section III shows the simulation results and discussion.At last,Section IV concludes this paper.

II.SYSTEM MODEL AND S-LMS EQUALIZER DESIGNS

Firstly, a 5-CAP based VLC system is implemented and shown in Figure.1.At the transmitter, a random binary sequence is generated as original data.After serial-to-parallel (S/P) conversion, quadrature amplitude modulation (QAM) mapping and shaping filter pairs,the 5-CAP signalsd(t)could be expressed as:

wherenis the subcarriers index,the QAM order is 16,In(t)andQn(t)are the real and imaginary parts of the QAM signals,respectively.The impulse responses of the shaping filter pairs in then-th subcarrier including both square root raised cosine(SRRC)and sine/cosine functions can be given by[22]:

whereTcis the symbol duration,αis the roll-off factor.With SRRC filters, the spectra of 5-CAP signals can be adjusted flexibly thus higher spectral efficiency compared to baseband signals can be obtained.

The VLC channel is modeled according to the experimental data which is measured in our previous work [23] and shown in the Section III.In the simulation, a line-of-sight link is considered with generalized Lambertian pattern [24] which is widely used in VLC channel.The received optical power could be expressed as:

wherePtis the emitted optical power,H(0) is the channel direct current,mis the order of Lambertian emission,APDis the area of photo diode,dis the distance between LED and PD,φis the angle of irradiance,ψis the angle of incidence,Ts(ψ) andg(ψ) is the gain of the optical filter and concentrator respectively.As a general nonlinear model, Rapp model is introduced to represent the nonlinearity of LED channel and is given in [25]:

whereVois the output voltage of an LED,Vi,Vt,Vmaxandqis the input voltage,threshold voltage,maximumsaturation voltage and knee factor respectively.At the receiver,the photoelectric conversion is completed by the PD and the overall channel noise varianceNccontaining shot and thermal noise is introduced from [26]:

whereandare the shot noise and thermal noise respectively.With Lambertian pattern,Rapp model and channel noise model, the received signal after VLC channel could be obtained.Table 1 lists the parameters used in simulation.

Table 1. Parameters in simulation.

After transmission, the received electrical signalr(t) is synchronized based on autocorrelation detection.The signaly(t) is sent to phase recovery model and demodulated after being compensated by postequalizer.The BER of the system is calculated.As a comparison,a conventional LMS and an RLS-Volterra based post-equalizers are employed at the receiver.To further improve the performance of VLC system by overcoming the limitation of conventional LMS postequalizer,we propose an S-LMS equalizer containing both pre- and post-parts mentioned above.The diagram of the proposed S-LMS equalizer is shown in Figure.2.Note that the uppercase letters denote the signals expressed in frequency domain and the lowercase letters denote the signals in time domain.D(k)andY(k) are the transmitted training signal and the signal after post-equalization for thek-th iteration,respectively, between which the error based on MMSE criterion is expressed ase(k).Wc(k) is the amplitude frequency response of VLC channel andNc(k)is the additive white Gaussian noise in the transmitting process.For the designed S-LMS equalizer,Wpre(k)andWpost(k) are the amplitude-frequency responses for the pre-equalizer and post-equalizer based on the separate factorp(k+1),respectively.

Figure 1. The diagram of 5-CAP VLC system with proposed S-LMS equalizer.

Figure 2. The diagram of proposed S-LMS equalizer.

As shown in Figure.2,the signalY(k)compensated by proposed S-LMS equalizer can be expressed as:

The compensated signalY(k) can be realized by minimizing the error function based on MMSE criterion,which is expressed as:

where the signal hasN-points sampling andiis the index of sampling point.Compared to the conventional MMSE criterion,the error function is expanded from single sampling point toN-points statistical signal,thus the performance of the error estimation could be enhanced.Therefore,the corresponding cost function of the overall equalizer can be defined as:

where E()is the statistical expectation.As the equalizer is separated according to factorp, the cost function subjects to(1)and(3)0≤p ≤1.As the channel state information(CSI)is unknown in practical communication systems,a conventional LMS equalizer is employed to obtain the estimated channel response at the beginning of the process.Then the S-LMS equalizer allocates the estimated to the pre- and post-equalizer according to the factorp.To optimize both the pre-and post-equalizer,Lagrangian is introduced to solve the joint optimization problem based on the criterion mentioned above:

whereλ1,λ2,λ3,andλ4are the Lagrange multipliers and set to 0, 1, 0.3, 0.3 respectively.The gradient of the Lagrangian respect top(k)is given as:

whereAis 10 logis the additive white Gaussian noise and can be estimated by

whereR(k) is the received signal.Using stochastic gradient algorithm [27], the update function of separate factor with iteration step sizeµcould be written as:

In each iteration, the optimizedp(k) is obtained at the receiver and feedback to the transmitter then the corresponding amplitude frequency responses for preand post-equalizer can be determined.It is worth noting that the feedback information only containsp(k)(about thousands of bits) and can be transmitted by most of the uplinks in VLC.The tap coefficients in time domain could be generated from respective amplitude frequency responses.The iteration continues untile(k) converges.Thus the design process of the proposed S-LMS equalizer is completed.It is worth noting that the fluctuation of MSEs is stable once the error function converges.It can be attributed to the design of the proposed S-LMS.In Eq.(13),to keep positive relation with the update value resulting in small fluctuation around the optimal solution.Moreover,the S-LMS equalizer could still work utilizing a single conventional LMS post-equalizer when the feedback link is unavailable.Letξ1andξ2denote given error limits,the proposed S-LMS is shown in Algorithm 1.

Algorithm 1. Separate least mean square equalizer.Input: p(0),D(k),µ,λ(1,2,3,4),N,ξ1,ξ2.Step1: Employ LMS.1: Set p(0)=0 2: Equalize with conventional LMS 3: Estimate channel response ˆWc Step2:Employ S-LMS.1: Set Wc = ˆWc 2: Equalize using Eq.(7)3: while e(k)>ξ2 do 4: Estimate ˆNc using Eq.(12)5: Update p(k)using Eq.(13)6: end while 7: Find the optimal p(k)and continue equalization

Computational complexity denoting the occupied memory resource is introduced to measure the complexity of equalizers.The computational complexity of the proposed S-LMS is analyzed and compared with conventional LMS and second-order RLSVolterra [28] in Figure.3.All the equalizers have the same memory lengthnfor a fair comparison.The memory length of the second-order RLS-Volterra equalizer for linear terms and nonlinear terms is 0.5nfor simplicity.nis even and the expansion of secondorder Volterra series could be expressed as:

Figure 3. The computational complexity of different equalizers.

wherex() is the input series,a1standa2ndare the weights of the linear and nonlinear terms.Hence the computational complexity of the proposed Volterra equalizer could be expressed asO(0.75n+0.125n2).The computational complexity of the proposed S-LMS and conventional LMS could be expressed asO(2n)andO(n)respectively.

III.RESULT AND DISCUSSION

In this section,the S-LMS equalizer based on the Lagrangian model is obtained for above the 5-CAP VLC system.Furthermore, the performance of the VLC system using S-LMS equalizer is investigated and compared with that using conventional LMS equalizer and RLS-Volterra equalizer.Monte Carlo simulations based on over a million source data for a run are used in the analysis process of system performance.Unless otherwise noted, the memory length of all the equalizers is set to 32 which means the computational complexity of S-LMS, conventional LMS and RLSVolterra is 64, 32 and 152 respectively.16QAM is applied in this VLC system model.Considering thatVinis an instantaneous value and can hardly represent the nonlinearity,Vppis deployed to evaluate the nonlinearity of the proposed system [28].The low level voltage of signals is set to 0 V,Vtis 1 V, peak-topeak voltageVppis set to 1.2 V and 1.4 V to represent slight and serious nonlinear impairment respectively.Only 200 sampling points of the training signal are utilized for each iteration that makes it possible to transmit user data and training signal in a frame.For the VLC system with 400 Mb/s data rate,Figure.4 is the convergence of the error function values defined in Eq.(8) with respect to number of iterations of the SLMS equalizer where the MSEs with differentµare measured and compared.It can be observed that all the error functions could keep converging and the step size has obvious influence to the MSE.Generally, a big step size could make the error function converge faster but with fluctuations in tracking process.In theory,a small step size converges slower but tracks stably.On one hand, as shown in Figure.4, the MSE could not converge in 300 iterations whenµis 0.003.The instability of step size of 0.003 can be attributed to the failure of convergence and the sub-optimal solution.On the other hand,although the convergence is fast, the MSE shows unstable fluctuations in tracking phase whenµis 0.01.As a result,µis set to 0.005 in this paper and the error decreases from 0.23 to 0.02 within 50 iterations.

Figure 4. The error function values varying with number of iterations in the design process of S-LMS equalizer of the VLC system.

With the convergence of the error function value,the optimized separate factorpvarying with frequency is obtained and shown as the red dashed line in Figure.5.However, the equalizers are hardly realized according to the converged factor due to the limited memory length of filters.Thus the converged factor has to be smoothed and the smoothed actual factor is shown as the blue solid line in Figure.5(a).It is observed that the post-equalizer part compensates more than the prepart especially in high frequency domain.This indicates the post-part of the S-LMS equalizer works better in the low frequency domain while the pre-part is suitable in the high frequency domain.This is because the S-LMS allocates more gains to the post-part than the pre-part.The amplitude-frequency responses of the pre-and post-equalizer are shown in Figure.5(b).The joint response denotes the response estimated by conventional LMS equalizer and is the allocation basis of S-LMS.The measured channel response is normalized into the same range of equalizer response for a better comparison with joint response.As the bandwidth of the transmitted signal is 137.9 MHz, it can be observed that the response of the post-part has an obvious drop at this frequency point.This is because the channel response using in S-LMS is estimated by LMS equalizer in which an approximate response of matched low-pass filter is contained.As a result, the joint response of the S-LMS equalizer is inverse to the channel response within the signal spectrum.

Figure 5. (a) Converged separate value p and (b)amplitude-frequency response of pre- and post-equalizers for the designed S-LMS equalizer.

Using the designed S-LMS equalizer, the performance of the 5-CAP VLC system with different data rates is simulated and compared with that using LMS, RLS-Volterra equalizers and without equalizer as shown in Figure.6.The constellation diagrams of 16QAM signals with 500 Mb/s data rate are also shown.Figure.6(a)is the BER performance with 1.2 V Vpp which means slight nonlinear impairment is introduced.The data rate with BER less than 3.8×10-3is 400 Mb/s,465 Mb/s and 500 Mb/s for conventional LMS, S-LMS and RLS-Volterra respectively.It is worth noting that the proposed S-LMS equalizer has better BER performance compared with RLS-Volterra equalizer with low data rate.To expand the data rate,S-LMS equalizer increases about 32 memory length for 65 Mb/s improvement while RLS-Volterra equalizer increases about 60 for 25 Mb/s improvement.With the increasing of memory length, the computational complexity of the RLS-Volterra grows rapidly.The same trend could be observed in Figure.6(b)where theVppis set to 1.4 V and higher nonlinear impairment is introduced causing significant decreasing of system performance.The data rate under limit is 350 Mb/s, 375 Mb/s and 500 Mb/s for the conventional LMS, S-LMS and RLS-Volterra equalizers respectively.It is revealed that the S-LMS equalizer has better performance than the conventional LMS equalizer and RLS-Volterra equalizer in relatively low data rate scenario.For high data rate transmission,the BER performance of S-LMS equalizer deteriorates rapidly with the deterioration of LMS.It can be attributed to the imperfect estimated channel response introduced by LMS equalizer.As a result,S-LMS could achieve both low BER and computational complexity for the communications in low data rate area.

Based on the Lambertian pattern LED source mentioned above,the SNR can be determined by the transmission distance.Figure.7 gives the BER performance versus distance for the VLC system with 500 Mb/s data rate and different equalizer.Vppis set to 1.2 V.The constellation diagrams of QAM signal after 1.2 m transmission and equalization are also shown,from which the difference of signal quality can be revealed.It is demonstrated that the S-LMS could outperform conventional LMS in the given distance.The BER performance of S-LMS is better than RLS-Volterra when the distance is less than 1 m which means the SNR is high.With the increasing of the transmission distance,the BER performance of the S-LMS decreases rapidly with the deterioration of LMS.Other things being equal,the maximum transmission distances of the 500 Mb/s VLC system are about 1.2 m, 1.4 m and 1.6 m with conventional LMS, S-LMS and RLS-Volterra equalizers respectively.The S-LMS equalizer extends the transmission distance about 0.2 m compared with the conventional LMS equalizer.As a comparison,the transmission distance of RLS-Volterra equalizer is extended about 0.4 m but about 5 times computational complexity is paid compared with the S-LMS equalizer.The difference can be attributed to the designs of equalizers.The proposed S-LMS has better lowpass response compensation due to the hybrid preand post-equalizer and that makes it outperform the Volterra equalizer in high SNR condition.But for the low SNR condition,the received signal can be hardly recovered due to the low SNR,even the low-pass frequency response has been compensated.As a comparison, the Volterra equalizer can compensate both the low-pass response and the nonlinear response, which makes it outperforms the proposed S-LMS equalizer in the low SNR condition.As a result, the S-LMS equalizer has the best BER performance and acceptable computational complexity increasing for the short distance VLC system.

Figure 6. The BER performance versus data rate and constellation diagram for the VLC system with S-LMS equalizer, with conventional LMS equalizer, with RLS-Volterra equalizer and without equalizer(a)Vpp = 1.2V (b)Vpp =1.4V.

Figure 7. The BER performance versus distance and constellation diagram for the VLC system with S-LMS equalizer, with conventional LMS equalizer, with RLS-Volterra equalizer and without equalizer.

IV.CONCLUSION

In this paper, an S-LMS equalizer including pre- and post-parts is implemented by joint optimizing based on Lagrangian model and verified for a 5-CAP VLC system.The design of the proposed S-LMS equalizer is completed for an optimized separate pre- and post-parts when the error function value is converged within 50 iterations.The results show that, by using the proposed S-LMS equalizer,the BER performance is significantly improved compared with that using the conventional LMS and RLS-Volterra equalizers for the VLC system with low data rate and short transmission distance.Compared with the nonlinear RLS-Volterra equalizer,the S-LMS equalizer could extend the transmission distance with obviously lower computational complexity in slight nonlinear channel.As a result,for this VLC system with less than 1.4 m transmission distance,500 Mb/s data rate can be realized when the proposed S-LMS equalizer is used.Therefore,the proposed S-LMS equalizer could enhance transmission performance with limited computational complexity which is suitable for IoT devices with limited computing resource in the 6G era.

ACKNOWLEDGEMENT

This work was supported by National Natural Science Foundation of China (No.61671055) and Scientific and Technological Innovation Foundation of Shunde Graduate School,USTB(BK19BF008).