In this paper we demonstrate that in doubly-dispersive environments, a multicarrier (MC) system based on a fractional Fourier transform (FrFT) can achieve a better concentration of power near the main diagonal of the equivalent channel matrix compared to standard orthogonal frequency division multiplexing (OFDM). The resulting inter-symbol and inter-carrier interference in such a chirp- based MC system can therefore be suppressed with a reduced complexity equaliser. Simulations show that compared to an equalised OFDM system, the equalised FrFT-MC approach can either significantly reduce complexity or enhance performance in a time-varying environment

Solyman, Ahmed

Soraghan, John

Weiss, Stephan

English

University of Strathclyde Institutional Repository

Strathprints Institutional RepositorySolyman, Ahmed and Weiss, Stephan and Soraghan, John (2012) Chirp-based multicarriermodulation. In: 9th IMA International Conference on Mathematics in Signal Processing, 2012-12-17- 2012-12-20, Birmingham.Strathprints is designed to allow users to access the research output of the University of Strathclyde.Copyright c© and Moral Rights for the papers on this site are retained by the individual authorsand/or other copyright owners. You may not engage in further distribution of the material for anyprofitmaking activities or any commercial gain. You may freely distribute both the url (http://strathprints.strath.ac.uk/) and the content of this paper for research or study, educational, ornot-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to Strathprints administrator:mailto:strathprints@strath.ac.ukhttp://strathprints.strath.ac.uk/CHIRP-BASED MULTICARRIER MODULATIONAhmed A. A. Solyman, Stephan Weiss and John J. SoraghanDepartment of Electronic & Electrical Engineering, University of Strathclyde, Glasgow, Scotland, UK{ahmed amin,stephan,j.soraghan}@eee.strath.ac.ukABSTRACTIn this paper we demonstrate that in doubly-dispersive envi-ronments, a multicarrier (MC) system based on a fractionalFourier transform (FrFT) can achieve a better concentrationof power near the main diagonal of the equivalent channelmatrix compared to standard orthogonal frequency divisionmultiplexing (OFDM). The resulting inter-symbol and inter-carrier interference in such a chirp- based MC system cantherefore be suppressed with a reduced complexity equaliser.Simulations show that compared to an equalised OFDM sys-tem, the equalised FrFT-MC approach can either significantlyreduce complexity or enhance performance in a time-varyingenvironment.Index Terms— Multicarrier transmission; fractionalFourier Transformation; time varying channel; equalisation.1. INTRODUCTIONThe popularity of orthogonal frequency division multiplexing(OFDM) systems is based on the diagonalising property tothe discrete Fourier transform (DFT) for circulant matrices.This can be fully exploited under stationary conditions, butin the presence of e.g. carrier frequency offset [1] or Dopplerspread [2, 3, 4, 5, 6] both the circulant property of the effectivechannel matrix and therefore the optimality of OFDM throughthe introduction of inter-channel interference (ICI) are lost.OFDM is often required for transmission over doubly-dispersive channels carrying Doppler spread, such as e.g. fordigital video broadcast (DVB) to handheld devices (DVB-H).To mitigate ICI, MMSE equalisation schemes have been in-troduced [7, 8], whereby the inversion of matrices whose di-mension matches the number of subcarriers [7] — up to 213for DVB-H — has led to low-complexity schemes [8], whereonly a band-limited matrix structure is inverted.As an alternative to DFT-based OFDM, a multicarrier sys-tem using the fractional Fourier transform (FrFT) has been in-troduced in [9]. In stationary conditions DFT-OFDM since itdoes not entirely decouple subcarriers. However, in doubly-dispersive channels reduced ICI has been reported [9] and fur-ther improved by MMSE equalisation in [10].In this paper we present a low complexity MMSE equaliserfor FrFTOFDM systems that depends on the LDLH factor-ization [11, 12]. The proposed system exploits the enhancedenergy concentration close to the main diagonal of the equiv-alent channel matrix compared to what can be obtained onDFT-based OFDM. The complexity of the FrFT-OFDM sys-tem is approximately the same as for classic DFT-OFDM andthe equaliser complexity [9] is linear with the number of sub-carriers which exhibits a lower complexity compared to theblock MMSE. It will be demonstrated that the performanceof the proposed low complexity system is approximately thesame as traditional OFDM systems employing a completeMMSE equaliser.2. FRFTMULTICARRIER SYSTEM MODEL2.1. Fractional Fourier TransformThe FrFT is a generalised form of the Fourier transform,which analyses signals w.r.t. chirps rather than complex ex-ponentials. A factor a ∈ [−1, 1] determines the chirp rate andtherefore selects a representation between the time (a = 0)and frequency domains (a = 1) [13]. Using the discretisationproposed in [13] leads to the transformation of a signal x[n]defined on the interval n ∈ [0, (N − 1)],Xa[k] =N−1∑0x[n]Ka[n, k] , k = 0 . . . (N − 1), , (1)whereby Ka[n, k] are mutually orthonormal chirp signalsspanning the basis of the discrete FrFT. The FrFT coefficients(c)(b)(a) time index n time index n time index nfrequencyindexkFig. 1. Time=frequency tilingsof the TFrFT bases functionsKa[n, k] for (a) a = 0 (time division duplex), (b) a = 1(frequency division duplex), and (c) 0 < a < 1 (FrFT).dn p/s dˆnCP Fas/ph[m, ν]F−aP CP−1PTWnNNa N N Na Nasn rnNr˜nN+LN+Lv[m]r[m]s[m]Fig. 2. FrFT-OFDM system using N subcarriers and a cyclic prefix of length L to transmit over a doubly-dispersive channelh[m, ν], whereby P maps symbols onto Na ≤ N active subcarriers and Fa is the FrFT matrix.Xa[k] can therefore be obtained by a matrix operationya =Xa[0]Xa[1]...Xa[N − 1] = Fax[0]x[1]...x[N − 1] = Fax ,(2)with F0 = I an identity and F±1 = T±1 based on thenormalised DFT matrix T. Since an N -point FrFT matrixFa is unitary, the inverse discrete FrFT (IFrFT) is definedas x = FHa ya = F−aya, where (·)H denotes Hermitiantranspose. These discrete forms will be referred to as FrFTand IFrFT in the following. The time-frequency tilings ofthe basis functions Ka[k, n] are sketched in Fig. 1, where theFrFT bases are shown for different values of a, creating time-division or frequency division duplex bases in the extremecases of Fig. 1(a) and (b) and a linear chirp basis in the caseof Fig. 1(c).s2.2. FrFT-Based Multicarrier SystemA conventional OFDM systems applies an inverse discreteFourier transform (IDFT) matrix to a data vector sn and in-troduces a cyclic prefix (CP) prior to multiplexing the OFDMsymbol across a doubly dispersive channel h[m, ν] corruptedby additive white Gaussian noise v[m]. After demultiplexingthe received signal r[m] =∑∞ν=0 h[m, ν]s[m − ν] + v[m]and removal of the cyclic prefix, a DFT matrix reconstructsthe transmitted data vector sˆn.q In a DFrFT-based multicar-rier system, the DFT matrix is replaced by a FrFT matrixFa [9, 10] as shown in Fig. 2.The signal rn = Hnsn + vn at the output of the demulti-plexer in Fig. 2 is characterised by th equivalent channel ma-trix Hn with elements[Hn]i,j ={h[n− L+ i, i− j] i ≥ j,h[n− L+ i, L+ i− j − 1] i < j.(3)In stationary conditions, Hn is circulant, and can be decou-pled by Fa with a = ±1, whereby the case a = 1 representsthe conventional OFDM system.We additionally introduce a binary matrix P ∈ ZN×Na ,which assigns a data vector dn ∈ CNa to N subcarriers, ofwhich only Na are active according toP =[0Na×(N−Na)/2 INa 0Na×(N−Na)/2]T, (4)where 0L×M is an L×M matrix with zero entries, and IL anL × L identity matrix. The equaliser matrix Wn ∈ CNa×Nain the receiver operates on the inputr˜n = PHFaHnF−aPdn +PHFavn= Cndn + v˜n , (5)with a system matrix Cn ∈ CNa×Na . The purpose of thebinary matrix P is not only to reduce out-of-band emissions,but also to eliminate components in the upper right and lowerleft corners of Cn.3. LOW COST EQUALISATION3.1. MMSE EqualiserAssuming perfect knowledge of the channel matrix Hn, theapproach in [8] can be extended to the system in Fig. 2. In theideal case, a linear block MMSE equaliser is defined based onthe system matrix Cn. Below, we restrict the calculation ofWn to the first Q sub- and super-diagonals of Cn by meansof a binary masking matrix M with elements[M]ij ={1 0 ≤ |i− j| ≤ Q,0 Q < |i− j| < Na.(6)The shape of this matrix is imprinted on the masked matrixBn = M ⊙Cn, where ⊙ represents element-wise multipli-cation. Based on the masked matrix, analogously to [8] theMMSE equaliser can be defined asWn,MMSE = BHn (BnBHn + γ−1I)−1 , (7)where γ is the signal to noise ratio (SNR) at the input to theequaliser, assuming corruption by white Gaussian noise. Thematrix inversion in (7) requires O(N3) flops which is notpractical for high values of Na, such as found in DVB-T.The masking is justified since the equivalent system ma-trix in the fractional domain is approximately banded [9].This enables a low-cost approach to equalisation, which isintroduced next.3.2. Equalisation Using Band LDLH FactorizationThe band structure of Bn with Q off-diagonal terms belowand above the main diagonal leads to a band structure for(BnBHn ), where only the first 2Q off-diagonal terms aboveand below the diagonal contain finite elements. This can sim-plify the calculation of the MMSE equaliser in (7). However,since (7) is time-dependent, we will below calculate dˆn =Wn,MMSEr˜n without explicitly determining Wn,MMSE.The LDLH factorisation of the Hermitian band matrixBnBHn+γ−1I = LDLH is numerically straightforward [11],and leads todˆn = BHn (LDLH)−1r˜n = BHnxn . (8)Instead of calculating the inverse in (8), the systemLDx1,n︷ ︸︸ ︷LHxn︸ ︷︷ ︸x2,n= r˜n (9)is solved by forward substitution to obtain x2,n via the lowerleft triangular matrix L and a rescaling by the diagonal matrixD−1 to calculate x1,n. Finally backsubstitution with the up-per right triangular LH yields xn, which can be inserted into(8) in order to determine dˆn.The overall complexity for obtaining dˆn is (8Q2+22Q+4)Na multiply-accumulates (MACs) [8], such that the selec-tion of the parameterQ involves a trade-off between accuracyand cost.4. SIMULATIONS AND RESULTS4.1. Simulation SetupTo assess the performance of the proposed system, we assumean OFDM transmission with N = 128 subcarriers of whichNa = 96 are active, a cyclic prefix of length L = 8, andQPSK modulation. Simulations are performed over an en-semble of 104 Rayleigh fading channels defined by an expo-nential power delay profile with an RMS delay spread of threesampling periods. This channel model uses the same statisticsas in [14], including a maximum Doppler spread ΩD equal to15% of the carrier spacing.4.2. Power ConcentrationTo assess the impact of the masking level Q, we compare theratio between the power of components in the original systemmatrix Cn and in the reduced matrix Bn after masking byM. Averaged over the ensemble and using the trace operatortr{·}, this power ratio is defined asρ[Q] = E{tr{BnBHn}tr{CnCHn}}(10)4 6 8 10 12 14 16 18 200.880.90.920.940.960.981masking level Qρ[Q]FrFT MCOFDMFig. 3. Percentage of power of Cn contained in Bn, mea-sured by ρ[Q] in dependence of the number of off-diagonal el-ements Q considered by M, comparing FFT-OFDM (a = 1)and FrFT-OFDM (a = 0.2).with 0 ≤ ρ[Q] ≤ 1.Results in Fig. 3 clearly indicate that FFT-OFDM experi-ences a spread of energy away from the main diagonal due toDoppler fading which is not limited to nearby off-diagonals,hence requiring a high value for Q to capture most of thepower contained in Cn. FrFT-OFDM does not manage todiagonalise Cn, but in contrast to FFT-OFDM, the leakedpower is contained in neighbouring off-diagonal element, anda much lower value of Q suffices to capture most of the com-ponents of Cn in Bn.4.3. BER PerformanceMotivated by the above power concentration, we performMMSE equalisation for simulations based on channels witha maximum Doppler frequency equivalent to 15% of the sub-carrier spacing. The low complexity equaliser introduced inSec. 3 is operated with Q = {5, 96}, whereby the second set-ting is equivalent to a standard block MMSE equaliser. Thesystem uses chirp rates a = {0.2, 1}, which for the secondsetting is identical to standard FFT-OFDM.Performance results in terms of bit error ratio are shown inFig. 4. The FFT-OFDM curves match those reported in [8].The proposed FrFT system with Q = 5 shows a degrada-tion over the full scheme at high SNR, but only requires 3.4%of the computational cost in terms of MACs and still outper-forms FFT-OFDM even with a full block MMSE equaliserwith a computation cost of O(N3a ) MACs.5. CONCLUSIONSWe have proposed a low cost equaliser for an FrFT-OFDMsystem, which generalises an FFT-OFDM system when as-suming a chirp rate a = 1. Equalisation is based on a lim-itation of the considered system matrix to the first Q off-0 5 10 15 20 25 30 35 4010−410−310−210−1Eb/No / [dB]bit error ratioOFDM low complexity MMSEOFDM block MMSEFrFT MC low complexity MMSEFrFT MC block MMSEFig. 4. Bit error ratio for MMSE equalisation using block(Q = Na = 96) and low-cost (Q = 5) approaches for FrFT-(a=0.2) and FFT-OFDM (a=1).diagonal terms and uses a numerically efficient LDLH fac-torisation. The fact that in a doubly-dispersive scenario theFrFT-based multicarrier system concentrates the power in itssystem matrix closer to the main diagonal than a standardFFT-OFDM scheme leads to a significantly improved perfor-mance w.r.t. standard OFDM, and a considerable cost reduc-tion w.r.t. previous FrFT equalisation approaches. This per-formance mildly depends on a, whose optimisation based onchannel parameters is the subject of on-going research.6. REFERENCES[1] T. Pollet, M. Van Bladel, and M. 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A multicarrier system based on the fractional Fourier transform for time-frequency-selective channels,”

Analysis of the loss of orthogonality through doppler spread

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BER sensitivity of OFDM systems to carrier frequency offset andWienerphasenoise,” IEEETrans.Communications,

Bounding erformance and suppressing intercarrier interference in wireless mobile

Bounds on the interchannel interference of ofdm in time-varying impairments,”

Interchannel interference analysis of OFDM in a mobile environment,”

Low-complexity banded equalizers for OFDM systems in Doppler spread channels,”

Low-complexity bandedequalizersforOFDM systems in Dopplerspread channels,” EURASIPJournalofAppliedSignalProcessing,

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Low-complexity equalization of OFDM in doubly selective channels,”

Low-complexity LSMR equalisation of FrFTbased multicarrier systems in doubly dispersive channels,”

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Max-SINR ISI/ICI-shaping multicarrier communication over the doubly dispersive channel,”

Multi-tap equalization algorithm for the OFDM system based on the fractional Fourier transform,”

Chirp-based multicarrier modulation

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