Performance Analysis of N-Continuous OFDM Systems with Low-Interference Signal

Peng Wei,Yue Xiao,Shuxia Yan,Wei Xiang

1 School of Electronics and Information Engineering,Tiangong University,Tianjin 300387,China

2 Tianjin Key Laboratory of Photoelectric Detection Technology and System,Tiangong University,Tianjin 300387,China

3 National Key Laboratory of Science and Technology on Communications,University of Electronic Science and Technology of China,Chengdu 611731,China

4 School of Engineering and Mathematical Sciences,La Trobe University,Melbourne,VIC 3086,Australia

Abstract: N-continuous orthogonal frequency division multiplexing (NC-OFDM) is a promising multicarrier transmission waveform conceived for improving sidelobe suppression performance.To reduce the severe inband interference in traditional NC-OFDM,we have proposed low-interference signal modeling for NC-OFDM.However,spectral leakage and error may be undesirably increased owing to the limited continuous differentiability of the smooth signal.In this paper,the low-interference scheme is investigated in terms of power spectrum density (PSD) and error performance,under the parameters of the highest derivative order (HDO) and the length of the smooth signal,to prove and quantify its advantages over traditional NC-OFDM.In the context of PSD,sidelobe decay is evaluated upon considering two discontinuous points due to the finite continuity of the smooth signal and its higher-order derivatives.Furthermore,it was shown that the low-interference scheme incurs small signal-to-noise ratio (SNR) loss and bit error rate (BER) for a short length of the smooth signal or a small HDO compared to traditional NC-OFDM.Meanwhile,due to the cyclostationarity loss imposed by the smooth signal,an effective solution is suggested for the time synchronization in a practical system.Based on analyses and simulation results,the tradeoffs between sidelobe suppression and BER are studied.

Keywords: BER; low-interference; N-continuous OFDM;PSD;sidelobe suppression

I.INTRODUCTION

With the increased requirement of wide-band ubiquitous coverage in the future wireless communications,maritime communications have attracted much attention [1–6].To support a high data transmission rate and combat the inter-symbol interference (ISI)induced by frequency selective fading channels in a dynamic maritime environment,orthogonal frequency division multiplexing(OFDM)is a promising wireless transmission technology [7,8].However,the traditional rectangularly pulsed OFDM exhibits severe interference at adjacent channels[9],since its slow sidelobe decaying in the form of the sinc function leads to high out-of-band(OOB)power emission,which limits its application in higher spectral efficiency scenarios.

The issue of sidelobe suppression has attracted considerable research attention in recent years [10–17].Specifically,the classic windowing technique can smooth the edges of OFDM symbols to reduce OOB radiation [10].However,its extended guard interval will reduce the spectral efficiency.Then,aiming for sidelobes close to the band,a class of cancellation carrier techniques were investigated [11,12,16].Unfortunately,the additional power consumption of cancellation carriers causes a loss in the signal-to-noise ratio (SNR).Furthermore,the nonorthogonal carriers employed in [12]will result in severe ISI,when the value of the tradeoff parameter is large between sidelobe power in the targeted band and self-interference of the cancellation signal.In[16],the reserved carriers may impose an influence on pilot placement.Recently,precoding methods have received more attention based on matrix optimization [13–15,17].In these methods,the precoders in[13,15,17]in conjunction with frequency diversity may induce high complexity for improving the error performance compared to conventional OFDM,and Ref.[14]provided a lowcomplexity precoding matrix design,which,however,induced high interference for a steep OOB spectrum.

In recent years,due to its excellent sidelobe suppression performance,N-continuous OFDM(NC-OFDM)has been widely researched[18–28].NC-OFDM was first introduced by Beeket al.in [18,28]upon smoothing the OFDM signal and its firstNderivatives.Then,it was further refined in [19–27]to improve the error performance as opposed to conventional NC-OFDM [18].In [19],via zeroing the beginning and end of each OFDM symbol,including its firstNderivatives,a memoryless precoder was attained at the expense of an increased peak-to-averagepower ratio(PAPR),while its interference-constrained counterpart is with degraded sidelobe suppression performance.On the contrary,the optimized percoder in[20]can considerably reduce sidelobes on selected frequencies at the expense of self-interference.Then,to eliminate the interference,some improved precoders[21,22,27]were designed to facilitate signal recovery at the receiver at the expense of complexity.Furthermore,the precoders in[23,26]can effectively improve the error performance by just adding interference to the guard interval.Nevertheless,the noncyclic NC-OFDM symbol may be more sensitive to channelinduced ISI and inter-carrier interference(ICI).

Against the aforementionedN-continuous precoders designed in the frequency domain,a class of time-domain signal designs with low complexity were proposed for theN-continuity in[29–31].In[29,30],the concept of a smooth signal linearly combined with basis signals was presented forN-continuity.Furthermore,aiming at reducing the interference,a low-interference scheme was developed with smoothwindow truncated basis signals [31],while maintaining considerable OOB radiation suppression and low complexity comparable to the counterparts of[23,26].However,if the length of the smooth signal is too short,the sidelobe suppression performance of the low-interference scheme will be degraded,even if the highest derivative order (HDO)Nis large enough.In contrast,the excessive long-length smooth signal causes moderate error performance degradation as opposed to the conventional OFDM.Therefore,our objective in this paper is to quantify the performance of the low-interference scheme in terms of the power spectrum density(PSD),SNR,and bit error rate(BER),which are regarded as three of the most important performance measures for communication systems.

It is noted that although the average power spectrum expressions were formulated in [19,20,24,26],they cannot directly show the characteristics of the sidelobe suppression performance with the HDO ofN.According to the continuity-based principle of sidelobe decaying presented in [32],the authors of [30]derived a tight PSD expression,which shows that the sidelobe decays asf−N−2.Note that only one discontinuous point exists by differentiating the OFDM signal during an OFDM symbol period.Different from[30],due to the truncated design of basis signals,two discontinuous points associated with the length and continuity of the smooth signal are active.On the other hand,the short-duration smooth signal will also incur noncyclic signaling sensitive to synchronization for reception.Additionally,detailed BER analysis in multipath fading channels for the currentN-continuous methods is still desired.

Against the above backdrop,the novel contribution of this paper is that,considering the effects of HDO and the length of the smooth signal to the lowinterference NC-OFDM scheme,we derive an asymptotic PSD expression upon considering the finite continuities at two disjoint points,provide a procedure of the BER calculation in the multipath fading channel under perfect synchronization,discuss three exact average signal-to-interference-plus-noise ratio(SINR)expressions of the imperfectly synchronized signal received through the multipath fading channel,and give an effective solution for time synchronization based on the analysis of the error variance at the receiver,in the quest for more-compact spectrum and error-resilient transmission in the low-interference scheme.

The remainder of this paper is organized as follows.In Section II,the low-interference NC-OFDM scheme is introduced.In Section III,an analysis of the PSD is presented for three cases.In Section IV,the BER is first evaluated by deriving the closed-form expression of the SINR assuming perfect synchronization,followed by three analyzed SINR expressions associated with symbol time offset (STO) and carrier frequency offset(CFO),and the per-bit SNR at the transmitter and the error variance of time synchronization at the receiver are investigated.Our numerical results are outlined in Section V.Finally,our conclusions are offered in Section VI.

II.LOW-INTERFERENCE NC-OFDM

The block diagram of the low-interference NC-OFDM transceiver is portrayed in Figure 1.The binary data stream is first mapped to complex-valued dataxi,krdrawn from phase shift keying (PSK) or quadrature amplitude modulation(QAM).Then,theith data vector xi=[xi,k0,xi,k1,...,xi,kK−1]Tis modulated by OFDM on the subcarrierK={k0,k1,...,kK−1},attached by the cyclic prefix(CP)to generate theith CPOFDM symbolyi(m).Finally,the low-interference scheme employs the time-domain addition of the smooth signalwi(m) onto theith CP-OFDM symbolyi(m)as follows[30,31]

where the sample indexm ∈M={−Mcp,...,M−1}has the symbol lengthMand the CP lengthMcp,andyi(m)is expressed as

In order to effectively reduce the interference distributed in the whole CP-OFDM symbol in conventional NC-OFDM [18,30],the limited time-domain distribution ofwi(m) is designed,where the beginning ofwi(m) aligns with the beginning of theith CP-OFDM symbol and the tail ofwi(m) ends in the front of theith CP-OFDM symbol.Thus,to satisfy theN-continuity for(m),wi(m)and its derivatives(m)should follow

wheren ∈UN≜{0,1,...,N}andy(n)i(m) is thenth derivative ofyi(m).

Based on the linear combination of basis signals(m),the design ofwi(m)is formulated as

whereL≜{−Mcp,−Mcp+1,...,−Mcp+L−1}indicates the support lengthLand location ofwi(m).Based on the prototype basis signals[31]and a truncation windowg(m),the basis signals(m)are designed by

which belong to the basis setQdefined as

whereu(m)is the unit-step function,g(m)is a smooth window function with zero edge for truncation,such as the Hanning,triangular,or Blackman windows,and(m)is expressed by[31]

To calculate coefficientsbi,nsatisfying theNcontinuous property in(3),upon solvingN+1 equations by substituting(4)-(7)into(3),we have

where bi=[bi,0,bi,1,...,bi,N]T,is an(N+1)×(N+1)symmetric matrix,given by

P2=P1Φ,and Φ ≜diag(ejφk0,ejφk1,...,ejφkK−1)withφ=−2πMcp/M.

Finally,following the same way of addingwi(m)in(1)and the design in(4)with the preset(m)and the calculatedbi,n,as shown in Figure 1,theN-continuous symbol vectoris generated as

As shown in Figure 2,in the following Sections III and IV,the PSD and SNR of the low-interference NC-OFDM are analyzed to evaluate the effect of the smooth signal to the sidelobe suppression and error performance.Table 1 shows the main notation settings in the paper.

Table 1.Notation setting.

III.POWER SPECTRUM DENSITY ANALYSIS

In continuous time,the transmitN-continuous OFDM signal is expressed as

whereT=Ts+Tcpwith the symbol periodTsand the CP durationTcp,is theN-continuous data on subcarrierkr,andTLis the duration of the continuoustime smooth signalwi(t).

Assuming thats(t) and its firstNderivatives are continuous as well as its(N+1)th derivative has finite amplitude discontinuity [33],the PSD ofs(t) can be expressed by[30]

whereF{·}represents the Fourier transform in the interval of[−Tcp,Ts].

It is noted that in[30],Eq.(12)is conditioned suchthat the discontinuity of the (N+1)th derivative only appears between(t) and(t).However,by observing the design of basis signals(m) in (5),one can obtain that since the zero-edge truncation windowg(t) cannot possess infinite continuous derivatives at its end and its derivatives are unrelated toN,differentiating(t) will yield another discontinuity at the end of the smooth signalwi(t),where this discontinuity may not have the same derivative order as the one between(t) and(t).In this case,although(12)is still available for theN-continuous signal,since it is independent of the length of the smooth signal,it may not completely describe the PSD of the lowinterference scheme,depending on theN-continuity and the length of the smooth signal associated with discontinuities at two different times.In the following,without loss of generality,the symmetric Blackman window is taken as an example ofg(t),whereg(t)=0.42−0.5 cos(πt/TL) + 0.08 cos(2πt/TL)fort ∈[0,2TL]withTL=(L−1)Tsampand the sampling intervalTsamp=Ts/M.

According to the first and second derivatives of the Blackman window function,0.5(π/TL)sin(πt/TL)−0.16(π/TL)sin(2πt/TL)and 0.5(π/TL)2cos(πt/TL)−0.32(π/TL)2cos(2πt/TL),we obtain that at the edges of 0 and 2TL,the former is continuous with zero while the latter is discontinuous with zero but has finite amplitude.Furthermore,according to (5),the end ofwi(t) corresponds to the edge ofg(t)at time 2TL.Thus,as shown in Figure 3,whenN ≥1,discontinuities of(t)may appear at times−Tcpor−Tcp+TL,which correspond to the beginning and end ofwi(t),respectively.When(t) is differentiated byN+1 times,discontinuities at these two disjoint times just simultaneously occur forN=1.Thus,as shown in Figure 2,three cases includingN=0 (e.g.N <1),N=1,andN ≥2(e.g.N >1)arise as discussed below.

Additionally,to facilitate the analysis,according to(4)-(8),wi(t)is first written as

whereanrandbnrare the(n+1)th row and(r+1)th column elements ofand,respectively,andf(n)(t)andg(t)consisting of(t)can be expressed as

3.1 N=0

where sinc(x)≜sin(πx)/(πx),fr=kr−Tsf,

whereβ1=Tcp/Tsandβ2=TL/Ts.

By substituting(17)-(19)into(16),we have

which shows that whens(t)is continuous,the PSD ofs(t)decays asf−4for a largef.

3.2 N=1

Since the discontinuities of(t) appear at both−Tcpand−Tcp+TL,according to (12),the PSD ofs(t)can be expressed as

where

Substituting(11)and(13)-(15)into(22)yields

where

with the binomial coefficient

It follows from(22)-(24)that(21)can be expressed as

It is shown in(25)that whens(t)ands(1)(t)are both continuous,the PSD ofs(t)decays asf−6for largef.

3.3 N≥2

Different from the above two cases,upon differentiatings(t) forN ≥2,discontinuity first occurs at time−Tcp+TLand then at timeTs.It can be inferred in (25) that the sidelobe ofs(t) decays as at leastf−3.Furthermore,sinceg(−Tcp+TL)=g(1)(−Tcp+TL)=0 and whenTLis largeg(2)(−Tcp+TL)=0.18(π/TL)2is small,the discontinuity of(−Tcp+TL) will be small.Consequently,the sidelobe decaying will approachf−N−2,which is mainly determined by the discontinuity at time−Tcp.

To estimate the effect of the HDO and support length ofwi(t) on the PSD,we introduce another smooth signal(t) to eliminate the discontinuity of(t) at time−Tcp+TL.Thus,according to (12),we can derive that

According to the derivation in Appendix A,Eq.(26)can be expressed as

whereF2(n,f)is given by

It can be inferred from(27)that whenfis large,the decaying of the PSD is affected by two parts,which have sidelobe decays off−N−2andf−3.WhenTLincreases,sinceF2(n,f)in(28)is in inverse proportion toTL,F2(n,f)will have a small value.Furthermore,due to the zero ends ofg(t)andg(1)(t)and the small end ofg(2)(t)as analyzed above,(−Tcp+TL)also becomes small.The above two small values make the last term in(27)smaller.Thus,whenfis large,asTLincreases,the decaying of the PSD ofs(t)approachesf−2N−4.On the contrary,with a reducedTL,the PSD decaying asymptotically arrives atf−6.

Figure 4 further validates our PSD analysis in the above three cases compared to simulation results with differentLandN,where the parameter configuration will be shown in Section V.It is shown that the curves of (20),(25),and (27) match well with their simulation results.As concluded in Figure 2,whenNandLincrease,the sidelobe power significantly decays at the speed off−2N−4.On the contrary,decreasingNorLdegrades the sidelobe suppression performance,such as fromL=144 to 36 forN=3 in Figure 4.Due to the design of (62) with finite basis signals instead of the ideal design,forN ≥2,there will be a gap between the analytical and simulated results whenfis large.

IV.ERROR PERFORMANCE ANALYSIS

In this section,as shown in Figure 2,considering perfect synchronization,we first analyze the BER of the low-interference scheme in the multipath fading channel.Then,with the imperfect frequency and time synchronization at the receiver,the error performance is evaluated in terms of the average SINR.Third,based on the correlation function,the solution for time synchronization at the low-interference NC-OFDM receiver is analyzed.Finally,the power consumption of the smooth signal at the transmitter is investigated in terms of the per-bit SNR,e.g.,Eb/N0.

4.1 Average SNR-Based Error Rate Analysis with Perfect Synchronization

Without loss of generality,let us assume that the channel delays follow 0=τ1< τ2< ··· <=τmax.When both STOδ1and CFOδ2are considered at the receiver.Then,based on [34],through a multipath channel,theith received signal(t)can be expressed as

for−Tcp≤t < Ts+,where the complex-valueddenotes the time-domain channel coefficient on thepath,following an independent and identical distribution with mean zero and variance=2},indicates the time delay along thepath,and the additive white Gaussian noise (AWGN) noiseni(t)has zero mean and a variance ofσ2n,

Under the assumption of perfect synchronization,as shown in Figure 1,after CP removal followed by anM-point discrete Fourier transform(DFT)for the signal in (29),the received signal on subcarrierkrand pathbecomes

where,,andare the frequencydomain expressions of(t),(t),and(t),respectively,is the frequency-domain channel response,andWi,kris the interference caused by the delayedwi(t).Meanwhile,we also have

which denotes that the amplitude of the frequencydomain channel response in each path is a constant.

According to [35],the total average SNRcan be expressed as the sum of average SNRin each channel path.We first assume that the energy of the channel response is normalized,that is=1,and the dataxi,krare i.i.d.random variables with zero mean and unit variance.We also assume that the AWGN noise(t)in thepath has zero mean and a variance ofsatisfying=σ2n.Thus,based on the Paswar theorem,the average SNRγin(32)can be expressed as

where based on (13),the interference power of the smooth signalis expressed by

Figure 5 compares the analytical average SINR in(34) and the simulation results in the 3GPP LTE Extended Vehicular A(EVA)channel model[36],whose parameters will be given in Section V.We observe that the analyticalγprovides a good evaluation of the simulation results.It also reveals that the SNR loss is negligible in the low-interference NC-OFDM.Even if the length of(l) becomes longer and the HDO increases,such asN=4 andL=1000,the lowinterference scheme still performs closely to conventional OFDM in the sense of SINR,and is much better than conventional NC-OFDM [18]and time-domain NC-OFDM(TD-NC-OFDM)[30].

In the following,an approximated BER expression is derived with the aid of the average SNR in (34).We first approximate the distribution of the smoothed signal to a complex Gaussian distribution with zero mean and variance of.According to[35],the BER expression of an uncoded QAM-based OFDM over the Rayleigh fading channel is approximated by

whereJis the size of the square QAM constellation,au1=2u1−1,and⎿x」indicates the largest integer tox.

wheregu2=3(2u2+1)2/(2(J−1)).

Finally,the average BER can be expressed as

wherepb,kr(E)ispb(E)in(38)with the subcarrier indexkr.

4.2 Average SNR Analysis with Imperfect Synchronization

Based on the aforementioned analysis,the key to analyzing the error performance is calculating the average SINR in(34).When imperfect synchronization is considered,channel-induced interference will be complicated.As shown in Figure 6,depending on the location of the estimated starting time of the OFDM symbol,three cases of imperfect synchronization will be discussed below.Meanwhile,we assume that the end of the channel-delayed smooth signal is after the estimated starting time,and before the exact end of the current OFDM symbol.

4.2.1 Case I

Since the estimated starting time of the OFDM symbol is after the exact time,as shown in Figure 6,after removing the CP,the received signal(t)consists of a part of the current OFDM symbolzi(t)followed by a part of the next onezi+1(t),which is expressed as

To show the interference attributed to STO,CFO,and the smooth signal,Eq.(40)can be rewritten as

where

(·)Tsdenotes the modulo ofTs,for 0 ≤t

and forTs−δ1≤t

wherezi(t+δ1)−(t+δ1) andzi+1(t+δ1−T)represent the intra-and inter-symbol interferences,respectively.

It is also assumed thatxi,krare independent identically distributed (i.i.d.) random variables with zero mean and unit variance.Then,we haveIKand=0K×K.Thus,according to(13),(34),and(42)-(44),the average SINR of ¯ri(t)in(41)can be expressed as

where Re{·}represents the real part of a complex value and Λ1=1 for≤δ1and Λ1=0 for1.

4.2.2 Case II

As depicted in Figure 6,since the estimated starting time of the OFDM symbol is before the exact time and after the end of the channel-delayed previous OFDM symbol(t),there is no ISI in the current OFDM symbol(t).Then,for 0 ≤t < Ts,theith received signal(t)can be expressed as

Thus,according to (11),(13),(34),and (47),the average SINR is expressed as

where the interference energydue to the STO and the smooth signal is given by

4.2.3 Case III

In this case,as shown in Figure 6,since the estimated starting time of theith OFDM symbol is prior to the end of the channel-delayed(i−1)th OFDM symbol,theith received signal(t) is composed of a part of the current symbolzi(t) and a part of the previous symbolzi−1(t).Thus,for 0 ≤t

Similarly,to indicate the interference,Eq.(50) is rewritten as

where for 0≤t<δ1+cp,

and forδ1+cp≤t

wherezi(t−δ1)−(t−δ1)andzi−1(t+T−δ1)are the intra-and inter-symbol interferences,respectively.

As mentioned above,we assume that the channel responses satisfy=in theith CP overlapped by the (i−1)th channel-delayed symbol.Then,according to (11),(13),(30),(31),(34),and (51)-(53),by calculating the energy of the interference(t) in[0,δ1+cp) and [δ1+cp,Ts],the average SINR is expressed as

where the interference energyis expressed as

where Λ2=1 for≥Tcp−δ1and Λ2=0 forcp−δ1.

Figure 7 plots the analyzed average SINR versus per-bit SNR of the above three cases.From the curves,we observe that for a fixed value of per-bit SNR and the given values ofδ1andδ2,the increasedNwill lead to a small decrease in SINR.Furthermore,the high intra-and inter-symbol interferences associated with STO and CFO in Cases I and III cause a much lowerγIandγIIIcompared toγII,due to no intra-and inter-symbol interferences in Case II of Section 4.2.2.

4.3 Solution for Time Synchronization

In this subsection,we will focus our attention on the error performance of the STO estimation to obtain the effective solution of time synchronization for the low-interference scheme.According to[34]and[38–40],due to the cyclostationarity of the CP-OFDM signal,the correlation function of the received OFDM signal is important for time synchronization.However,since the smooth signal is just added into the front part[−Tcp,−Tcp+TL]of each CP-OFDM symbol,it destroys the cyclostationarity between the CP in [−Tcp,−Tcp+TL]and its corresponding copy in[−Tcp+TL,Ts].Then,we first define the correlation function of the received OFDM signal(t) asThus,fort ∈[−Tcp,−Tcp+TL]andt+ ∆∈[−Tcp+TL,Ts],R(t,∆)is expressed as

According to [38],for the CP-OFDM signal,if no subcarrier weighting is employed,R(t,∆)in(56)includes the information of the time synchronization parameters for ∆=Tsonly.In this case,under the assumption ofTL≤Ts,Eq.(56)can be rewritten as

Figure 8 verifies that the maximum value of|R(t,∆)|is associated with ∆=Tsfort ∈[−Tcp,0].However,due to the smooth signal,R(t,Ts) in (57)and Figure 8 imply that(t)does not completely follow the cyclostationarity distribution.Furthermore,astdecreases from 0 to−Tcp,corresponding to the valley of|R(t,Ts)|in Figure 8,the weakened correlation will cause an accuracy reduction for time synchronization.In contrast,when increasingtto 0,the reduced instantaneous power of the smooth signal increases the value of|R(t,Ts)|.Consequently,decreasing the value ofNorLis capable of improving the performance of time synchronization.As observed in Figure 9,upon using the CP-based STO estimation technique[34,39]over a Rayleigh fading channel (EVA channel in[36]),when the value ofNorLdecreases,the mean-squared error variances of the low-interference scheme are effectively reduced.However,since most power of the smooth signal is concentrated in the front part of each OFDM symbol,if the value ofNorLis inappropriately reduced,the sidelobe suppression performance will be significantly degraded.

In order to mitigate the effect of the smooth signal on time synchronization and maintain a compact spectrum,invoked by the small power ratio of the smooth signal in [0,Ts],STO estimation using the training(reference)symbol is employed[34,40].Figure 9 also presents the mean-squared error variance versus per-bit SNR of the original OFDM and the lowinterference scheme for AWGN and Rayleigh fading channels,when using the training symbol based STO estimation technique.The length and HDO of the smooth signal areL=1000 andN=4,respectively.For large values ofNandL,we notice that the error variances of the low-interference scheme are close to those of the original OFDM.

4.4 Analysis of SNR Loss at the Transmitter

According to [41],the linear relationship in dB between the per-bit SNR and the average SINRover the AWGN channel can be expressed as

Meanwhile,the energy ofwi(t) in (13) can be expressed as

Thus,it follows from(58)and(60)that the relationship between Eb/N0andwi(t)can be expressed as

It can be shown in Figure 10 that the analyzed Eb/N0of (61) is reduced by increasingNandL,which simultaneously increases the power ofwi(t).Furthermore,the Eb/N0of the low-interference scheme is significantly higher than that of the conventional NC-OFDM,and close to that of the original OFDM.Corresponding to the analysis in Section 4.1,it is also implied that the low-interference scheme is capable of achieving a small error performance degradation as opposed to the original OFDM.

V.NUMERICAL RESULTS

In this section,we present our performance results characterizing the low-interferenceN-continuous OFDM system employing 16-QAM onK=256 subcarriers,having a subcarrier spacing of ∆f=1/Ts=15KHz and transmitting the signal with a sampling intervalTsamp=Ts/2048 and a CP duration ofTcp=144Tsamp.To portray the PSD performance,Welch’s averaged periodogram method [42]is employed under the setting of a 2048-sample Hanning window and 512-sample overlap.To show the BER performance in the multipath fading environment,we adopt the EVA channel model provided in[36]having the excess tap delay[0,30,150,310,370,710,1090,1730,2510]ns and the relative power [0,-1.5,-1.4,-3.6,-0.6,-9.1,-7,-12,-16.9]dB.

Figure 11 shows the impact of the HDONand the length of the smooth signalLon the PSD of the low-interference scheme as opposed to conventional OFDM and NC-OFDM,whereN=0,2,4 andL=72,144,500.Meanwhile,three truncation windows,including Blackman,triangle,and Chebyshev windows,are adopted for the low-interference scheme.As can be observed from the results,the low-interference scheme can achieve sidelobe decaying close to conventional NC-OFDM.WhenNis large andLis small,e.g.,N=2 andL=72,the sidelobe decaying rate of the low-interference scheme reduces as opposed to NC-OFDM.As expected,with an increase inL,such as fromL=36 toL=144 in Figure 11b,a steeper OOB spectrum is attainable.More importantly,since the sidelobe decaying of the Chebyshev window is slightly lower than that of the Blackman window and much higher than that of the triangle window,the lowinterference scheme using the Blackman window has the lowest OOB power emission.

Figure 12 illustrates the simulated and analytical BER performance versus the per-bit SNR among the original uncoded OFDM,conventional NC-OFDM,and the low-interference NC-OFDM scheme over the multipath Rayleigh fading channel.Zero-forcing(ZF)channel equalization is adopted.It confirms,not surprisingly,that the smooth signal has a slightly negative effect on the BER performance of the low-interference scheme in the multipath fading channel,corresponding to the analyses of the SINR and BER in Section IV.Figure 12 further verifies that as the values ofLandNincreases,a small BER performance degradation is incurred by the low-interference scheme as opposed to conventional NC-OFDM.

Furthermore,the complexity of the low-interference NC-OFDM scheme is shown in Figure 13.According to [30]and [31],we plot the number of real multiplications of the low-interference NCOFDM scheme,conventional NC-OFDM,and TDNC-OFDM at the transmitter using the parameters ofL=36,72,144,1000,a constant number of subcarriersK=256 for Figure 13a,and a fixed value of HDON=2 for Figure 13b.As observed in Figure 13,compared to conventional NC-OFDM and TDNC-OFDM,the low-interference NC-OFDM scheme has a much higher computational efficiency,while it can maintain a compact spectrum close to conventional NC-OFDM and TD-NC-OFDM,as shown in Figure 11,and a small BER performance degradation as opposed to the original OFDM,as shown in Figure 12.Furthermore,contrary to the conventional NCOFDM receiver,the low-interference scheme is capable of avoiding extra signal recovery at the receiver,due to its slight BER performance degradation.Meanwhile,the low-interference scheme is capable of attaining the same PAPR as the original OFDM,and so are NC-OFDM and TD-NC-OFDM systems.

Finally,we summarize the performance comparison of OFDM,NC-OFDM,TD-NC-OFDM,and lowinterference NC-OFDM in Figure 14.As observed in Figure 11,whenN≥2,N-continuous schemes satisfy the requirement of the spectrum emission mask of the user equipment (UE) [36].Thus,the values of HDO and length of the smooth signal areN=2,L=72 andN=4,L=144 in Figs.14a and 14b,respectively.It can be seen that forN≥2 and a value ofLsmaller than or equal to the length of the CP,the low-interference NC-OFDM is capable of achieving an elegant balance among out-of-band power emission,BER,complexity,and PAPR,compared to its counterparts.

VI.CONCLUSION

In this contribution,the PSD and error performance of the low-interference NC-OFDM with various HDOs and lengths of the smooth signal were thoroughly investigated and compared.The PSD was first evaluated in the form of an asymptotic expression with the aid of the newly introduced smooth signal.It was concluded from our analytical and simulated results that when the HDO and the length of the smooth signal increase,significant improvement of sidelobe suppression will be observed.However,as the length of the smooth signal decreases,the sidelobe suppression performance of the low-interference scheme might be significantly degraded,even if using a large HDO.Then,assuming perfect synchronization,we provided an asymptotic expression for quantifying the BER of the lowinterference scheme in the multipath fading channel by deriving a closed-form expression of the instantaneous SNR.Furthermore,the average SINR performance of the low-interference scheme was evaluated over the multipath fading channel,when both STO and CFO were considered at the receiver.We also have studied the effects of the length and the HDO of the smooth signal on the per-bit SNR performance of the low-interference scheme.It revealed that the increased length of the smooth signal and HDO causes a higher BER,and on the contrary,a smaller BER close to the original OFDM is achieved.Finally,the analyzed error variance indicated that the training(reference) symbol is an effective alternative of time synchronization in the low-interference scheme.From the results,we conclude that if the above two parameters are set reasonably,the low-interference scheme is capable of achieving considerable sidelobe suppression with small BER performance degradation,while having a relatively low complexity without PAPR regrowth as opposed to the original OFDM.

ACKNOWLEDGEMENT

This work was supported in part by the National Natural Science Foundation of China under Grant 61901298; in part by the Natural Science Foundation of Tianjin under Grant 20JCQNJC00300; and in part by the Science & Technology Development Fund of Tianjin Education Commission for Higher Education under Grant 2018KJ211.

NOTE

This is the 7th paper accepted for the Feature Topic of Maritime Communications in 5G and Beyond Networks published in the September issue ofChina Communicationsthis year.

APPENDIX A:DERIVATION OF(26)

By substituting(62)into(63),the coefficientsare calculated by

Furthermore,we can derive that

Finally,substituting(65)-(68)into(26),we can obtain(27).