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1 A Class of Low-Interference N-Continuous OFDM Schemes Peng Wei, Lilin Dan, Yue Xiao, Wei Xiang, and Shaoqian Li Abstract—N-continuous orthogonal frequency division multi- padding OFDM (NCSP-OFDM) was recently developed in plexing (NC-OFDM) was demonstrated to provide significant [10] at the cost of increased complexity in the transmitter. sidelobesuppressionforbasebandOFDMsignals.However,itwill Meanwhile, to enable low-complexity signal recovery in NC- 6 introducesevereinterferencetothetransmitsignals.Henceinthis 1 OFDM, several techniques [8]-[9] have been proposed. How- letter,wespecificallydesignaclassoflow-interferenceNC-OFDM 0 schemes for alleviating the introduced interference. Meanwhile, ever, those techniques will also result in complexity load or 2 we also obtain an asymptotic spectrum analysis by a closed- efficiency reduction. n formexpression.Itisshownthattheproposedschemeiscapable For reducing the interference of NC-OFDM and avoiding a of reducing the interference to a negligible level, and hence to complex signal recovery at the receiver, this letter proposes a J save thehighcomplexityof signal recovery at thereceiver, while low-interference NC-OFDM scheme by adding an improved maintainingsimilarsidelobesuppressionperformancecompared 8 to traditional NC-OFDM. smooth signal in the time domain. Firstly, in the proposed 1 scheme, the smooth signal is generated in a novel way as Index Terms—N-continuous orthogonal frequency division ] multiplexing(NC-OFDM);sidelobesuppression;time-domainN- the linear combination of designed basis signals related to T continuous OFDM (TD-NC-OFDM). rectangular pulse. Secondly, the basis signals are smoothly I truncated by a preset window function to obtain the short- . s duration smooth signal with shorter length than a cyclic- c I. INTRODUCTION [ prefixed OFDM symbol and low interference. Thirdly, the ORTHOGONAL frequency division multiplexing short-duration smooth signal is overlapped onto only part of 1 (OFDM) has been widely adopted in wireless theOFDMsymboltoreducetheinterference.Furthermore,we v communications due to its high-speed data transmission 4 give an asymptotic expression of the spectrum feature of the and inherent robustness against the inter-symbol interference 8 low-interferenceNC-OFDM signal, toshowthatthe proposed 6 (ISI). However, traditional rectangularly pulsed OFDM schemecanmaintainthesimilarsidelobesuppressiontotradi- 4 exhibits large spectral sidelobes, resulting in severe out-of- tionalNC-OFDM.Lastly,thecomplexityanalysisdenotesthat 0 band power leakage. In this case, traditional OFDM will the proposed scheme can significantly reduce the complexity . 1 introduce high interference to the adjacent channels. of NC-OFDM. 0 For alleviating this problem, various methods have been 6 proposed for sidelobe suppression in OFDM [1]-[10]. More 1 II. N-CONTINUOUS OFDM specifically, the windowing technique [1] extends the guard : v interval at the cost of spectral efficiency reduction. Cancella- In conventional NC-OFDM [5], the ith transmit symbol i tion carriers [2] will consume extra power and incur a signal- y¯(t) follows X i to-noise ratio (SNR) loss. Precoding methods [3]-[4] require r K−1 a complicate decoding processing to eliminate the interference. y¯(t)= x¯ ej2πkr∆ft,−T ≤t<T (1) Against this background, NC-OFDM [5] is a class of effi- i i,kr cp s r=0 cient sidelobe suppression techniques by making the OFDM X and signal and its first N derivatives continuous. However, a y¯(n)(t) = y¯(n)(t) , (2) disadvantage of NC-OFDM is the high implementation com- i t=−Tcp i−1 t=Ts plexity in the transmitter. For reducing the complexity, some (cid:12) (cid:12) wherex¯ is the prec(cid:12)odedsymbolon th(cid:12)e rth subcarrierwith simplified schemes have been proposed in [6] and [7]. In i,kr (cid:12) (cid:12) [11],time-domainNC-OFDM(TD-NC-OFDM)wasproposed thesubcarrierindexsetK={k0,k1,...,kK−1},yi(n)(t)isthe nth-orderderivativeofy (t)withn∈U ,{0,1,...,N},the for reducing the complexity of NC-OFDM, by transforming i N traditionalfrequency-domainprocessingintothetimedomain. frequency spacing is ∆f = 1/Ts, Ts is the symbol duration, and T is the cyclic prefix (CP) duration. Ontheotherhand,traditionalNC-OFDMwillintroducesevere cp For satisfying Eq. (2), the N-continuous processing can be interference so as to degrade the bit error rate (BER) per- summarized as [5] formance. For alleviating this problem, N-continuous symbol x¯ =x , i=0 TheauthorsarewiththeschoolofNationalKeyLaboratoryofScienceand i 0 (3) Technology onCommunications, UniversityofElectronic ScienceandTech- (x¯i =(IK −P)xi+PΦHx¯i−1, i>0, nologyofChina,Chengdu,China(e-mail:[email protected];{lilindan, xiaoyue}@uestc.edu.cn). Wei Xiang is with the school of Mechanical and where x =[x ,...,x ]T, x¯ =[x¯ ,...,x¯ ]T, Electrical Engineering Faculty ofHealth, Engineering andSciences, Univer- i i,k0 i,kK−1 i i,k0 i,kK−1 sityofSouthernQueensland, Austrialia (e-mail:[email protected]). IK is the identity matrix, P = ΦHAH(AAH)−1AΦ, 2 Φ , diag(ejϕk0,ejϕk1,...,ejϕkK−1), ϕ = −2πβ with On the otherhand,by substitutingEqs. (6)-(9) into Eq.(5), β =T /T , and the coefficients b can be calculated as cp s i,n 1 1 ··· 1 bi =Pf−˜1(P1xi−1−P2xi), (10) k0 k1 ··· kK−1 A= ... ... ... . where Pf˜is a (N+1)×(N+1)symmetric matrix, given as   k0N k1N ··· kKN−1 f˜(0)(−Mcp) ··· f˜(N)(−Mcp)   f˜(1)(−M ) ··· f˜(N+1)(−M ) cp cp III. LOW-INTERFERENCENC-OFDM SCHEME Pf˜= .. .. , . . TD-NC-OFDM is recently proposed in [11] for transform-   f˜(N)(−M ) ··· f˜(2N)(−M )  ingtheprocessingofNC-OFDMintothetimedomain.InTD-  cp cp    NC-OFDM,thesmoothsignalisaddedwiththetimesampling 1 ··· 1 interval T =T /M where M is the length of the symbol. FollowingsathmepbasicsideaofTD-NC-OFDM,theproposedlow- P = 1  j2πk.0/M ··· j2πkK.−1/M , interferenceschemealsoaddsasmoothsignalwi(m)ontothe 1 M .. ..   OFDM signal yi(m) in the time domain, given as (j2πk0/M)N ··· (j2πkK−1/M)N   y¯i(m)=yi(m)+wi(m), (4) and P2 =P1Φ. Finally, w (m) is added onto only part of the CP-inserted where m ∈ M = {−M ,−M +1,...,M −1} and M i cp cp cp OFDM symbol to achieve the N-continuous symbol y¯ , as is the length of a CP. Thus, to make the OFDM signal N- i continuous, wi(m) should satisfy y + Qf˜bi 0≤i≤M −1, w(n)(m) = y(n)(m) − y(n)(m) . (5) y¯i = i (cid:20) 0(M−L)×1 (cid:21) s (11) i m=−Mcp i−1 m=M i m=−Mcp  Qf˜bi i=Ms, (cid:12) (cid:12) (cid:12) To cons(cid:12)(cid:12)truct wi(m) with lo(cid:12)(cid:12)w interference, (cid:12)(cid:12)wi(m) is just where Qf˜= {q0,q1,...,qN}, Ms is the number of the added in the front part of each CP-prefixed OFDM symbol. OFDM symbols, and x−1 is initialized as x−1 =0K×1 since Thus, the proposed linear-combination design of wi(m) is the back edge of y−1 equals to zero. described as In general, the proposed low-interference NC-OFDM N scheme is illustrated as follows. b f˜ (m), m∈L w (m)= i,n n (6) i n=0 P 0, m∈M\L, Low-Interference NC-OFDM Algorithm where L , {−Mcp,−Mcp+1,...,−Mcp+L−1} indi- Input: xi, yi with CP Output: y¯ catesthelocationofw (m)withlengthL,andthebasissignals i i fñ(m) belong to the basis set Q, defined as 1 Initialization : x−1 =0K×1, i=0, Qf˜, Pf−˜1, P1, P2; 2 while i≤Ms−1 do Q, qn˜ qn˜ = fñ˜(−Mcp),fñ˜(−Mcp+1),..., 3 Calculate the coefficients bi,n by Eq. (10); (cid:26) (cid:12)(cid:12)(cid:12) fhñ˜(−Mcp+L−1) T,n˜ ∈U2N , (7) 54 AGdendewratieotnhteostmheooOthFDsiMgnasilgwnail=byQEf˜qb.i(;11); i (cid:27) 6 i=i+1; where U2N = {0,1,...,2N}. For achieving the low- 7 end interferencesmoothsignalwi(m)inEq.(6),thedesignofthe 8 if i=Ms then basis signals fñ˜(m) and the linear combination coefficients 9 wi is appended to the transmit signal as Eq. (11); bi,n ∈bi =[bi,0,bi,1,...,bi,N]T will be specified as follows. 10 else On the one hand, the time duration of fñ˜(m) is truncated 11 Terminate. by a preset window function g(m), which is considered as a 12 end smooth and zero-edged window function, such as triangular, Hanning, or Blackman window function. Then, the truncated According to the above processing, the main advantage basis signals can be given by of the improved smooth signal w (m) is that its length has i f˜ (m)= f(n˜)(m)g(m)u(m), m∈L (8) been effectively truncated to L, by a careful design in Eq. n˜ 0, m∈M\L, (8). Furthermore, for L < M + M , only parts of the (cid:26) cp OFDM signal are overlapped with the interference w (m). where u(m) denotes the unit-step function, and f(n˜)(m) is i More especially, since the front part of the OFDM symbol calculated by the rectangularly OFDM pulse [11], given as is CP, the interference to the real data is further mitigated. In f(n˜)(m)=1/M(j2π/M)n˜ kn˜ejϕkrej2πkMrm. (9) Section V, we will showthat the influenceof wi(m) hasbeen r effectively reduced. kXr∈K 3 IV. THEORETICAL ANALYSIS OF SPECTRUMAND thesimulationresultsmatchwellwith thetheoreticalanalyses COMPLEXITY with varying highest derivative order N. In Section V, with the above parameters,we will show the proposedscheme can A. Spectral Analysis maintainthesimilarspectralroll-offtotraditionalNC-OFDM. Assume that the first N-1 derivatives of the smoothed OFDM signal are continuous, and the Nth-order derivative hasfinite amplitudediscontinuity[14]. Thus,accordingto the 0 Theory definition of the power spectrum density (PSD) [4] and the Simulation relationship between spectral roll-off and continuity in [12], -20 thePSDofthelow-interferenceNC-OFDMsignalisexpressed by -40 N=0 Ψ(f)=Ul→im∞2U1TE(cid:12)(cid:12)(cid:12)iU=X−−1UF(njy2¯i(πNf))(Nt)oe−j2πfiT(cid:12)(cid:12)(cid:12)2, (12) PSD (dB) --8600 NN==12 (cid:12) (cid:12) N=3 where T =Ts+Tcp.(cid:12)(cid:12) (cid:12)(cid:12) -100 Eq. (12) indicates that the spectrum of the low-interference NC-OFDM signal is related to the expectation of y¯(N)(t) i multiplied by f−N. In this letter, the conventional Black- -120-10 -5 0 5 10 man window function is used as an example, given as Frequency (MHz) g(t) = 0.42 − 0.5cos(2πρt) + 0.08cos(4πρt) where ρ = 1/((2L−2)T ). By substituting Eqs. (5) (6) (8) and (9) samp Fig. 1. PSD comparison between the analytical and simulation results in into Eq. (12), the PSD of the smoothed OFDM signal is low-interference NC-OFDM. expressed by U−1 Ψ(f)= lim 1 E e−j2πfiT (fT )−N B. Complexity Comparison s U→∞2UT ((cid:12)i=−U The complexity comparison, quantified by the numbers of (cid:12) X (cid:12) N real additionsand multiplications, among NC-OFDM, NCSP- · kNx sinc(cid:12)(f (1+β))ejπfr(1−β)+1 b OFDM, and the low-interference scheme is shown in Table r i,kr r T i,n kXr∈K nX=0 I. Compared to other methods, the proposed low-interference N 2 scheme has notable complexity reduction. When L is equal · Nn¯ (j2π/Ts)n−n¯ krN−n¯+nGn¯(f) , (13) to or smaller than the length of CP, the complexity is nX¯=0(cid:18) (cid:19) kXr∈K (cid:12)(cid:12)) significantly reduced in the transmitter. For example, when (cid:12) L=M =144, K=256, N=2, and M=2048, at the transmitter (cid:12) N cp where sinc(x) , sin(πx)/(πx), fr = kr −Tsf, n¯ is the side, the complexity of the proposed low-interferencescheme binomial coefficient, (cid:18) (cid:19) is, respectively,47.4%and 20.3%of those of NC-OFDM and NCSP-OFDM. sin(πµf ) 0.5(2πρ)n¯cos(πµf ) G (f)=ejf˜r 0.42(n¯) r − r n¯ πfr/Ts 1−(ρTs/fr)2 COMPLEXITYCOMPARISONAMOTNAGBNLEC-IOFDM,NCSP-OFDMANDTHE cos(πn¯/2) sin(πn¯/2) LOW-INTERFERENCESCHEMEINTRANSMITTER · −πρ jπfr/Ts (πfr/Ts)2! Scheme Numberofrealmulti- Number ofreal additions plications 0.08(4πρ)n¯sin(πµf ) cos(πn¯/2) sin(πn¯/2) NC-OFDM O(4(N+1)K) O(2(N+1)(2K−1)) r + 1−(2ρTs/fr)2 πfr/Ts −j2πρ(πfr/Ts)2!!, NOCFDSPM- O(8(N+1)K) O(8(N+1)K−4(N+1)) Low- f˜r=πfr(Tp−2Tcp)/Ts with Tp=1/(2ρ), and µ=Tp/Ts. Interference O(2NK+(N+1)L) O((N+1)(2K+L+N−2)) Eq. (13) shows that the power spectral roll-off of the Scheme smoothed signal, whose first N-1 derivatives are continuous, decays with f−2N−2. Moreover, the expression of G (f) n¯ reveals that the sidelobe is affected by the length of w (t), V. NUMERICAL RESULTS i so that the selection of L is important to balance BER and Simulationsareperformedina baseband-equivalentOFDM sidelobe suppression performance. Fig. 1 compares the theo- system with K=256, M =144, and 16-QAM digital modu- cp retical and simulation results of the low-interference scheme lation. The carrier frequency is 2GHz, the subcarrier spacing withL=M =144,M=2048,wherethelengthofw (t)isequal is ∆f = 15KHz, and the time-domain oversampling factor cp i to that of CP. Considering the ratio of CP is only 7% of is 8. The PSD is evaluated by Welch’s averaged periodogram the whole symbol, the assumption is reasonable for practical method [15] with a 2048-sample Hanning window and 512- wireless systems such as LTE [13]. It is shown in Fig. 1 that sample overlap. In order to show the system performance in 4 themulti-pathfadingenvironment,LTEExtendedVehicularA (EVA) channel with 9 paths of Rayleigh fading channels [13] is considered. 10-1 Fig. 2 compares the PSDs between NC-OFDM and the proposed low-interference scheme with different N. The low- interference scheme can obtain the sidelobe suppression per- 10-2 formancesimilartotraditionalNC-OFDM.Moreover,AsLin- R creases,thesidelobesuppressionperformanceoftheproposed E B scheme is improved. For example, L is increased from 36 to 10-3 72 and 144. In general, we show that L=M =144 is a good cp choice for maintaining the sidelobe suppression performance Original OFDM as traditional NC-OFDM. 10-4 NC-OFDM NCSP-OFDM Low-interference scheme 0 5 10 15 20 25 30 35 40 0 E /N(dB) Original OFDM b 0 NC-OFDM -20 Low-interference scheme Fig.3. BERsofNC-OFDM,NCSP-OFDM,andthelow-interferencescheme -40 withN=3andL=144intheRayleigh fading channel. 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