Fast Hybrid DF/D Architecture for OFDM in ognitive Radio System Zhu hen, Moon Ho Lee, Senior Member, EEE, hang Joo Kim 3 nstitute of nformation&ommunication, honbuk ationa University, Jeonju, 56-756,Korea 3 Radio technoogy Research Group of ER E-mai: {chenzhu, moonho}@chonbukackr 3 cjkim@etrirekr Abstract n this paper, the sparse matrix decompositions for DF matrix and D matrix are proposed Based on these propositions, we deveop a fast hybrid DF and D architecture for OFDM n addition, we address the OFDM based on DF or D in ognitive Radio system An adaptive OFDM based on DF or D in ognitive Radio system has the capacity to nuify individua carriers to avoid interference to the icensed users herefore, there coud be a consideraby arge number of zero-vaued inputs/outputs for the DF/DF or D/D on the OFDM transceiver Hence, the standard methods of DF and D are no onger efficient due to the wasted operations on zero Based on this observation, we present a transform decomposition on two dimensiona (-D) systoic array for DF/DF and D/D, this agorithm can achieve an efficient computation for OFDM in ognitive Radio system ndex erms D, DF, OFDM, ognitive Radio ntroduction Orthogona frequency division mutipexing (OFDM) has recenty received considerabe attention for its efficient usage of avaiabe frequency bandwidth and robustness to frequency seective fading environments OFDM systems normay use the discrete Fourier transform (DF) for muticarrier moduation of the data to be transmitted However, under certain channe conditions, throughput is enhanced when using the discrete cosine transform (D) rather than the DF[] herefore, it has high practica vaue to design a hybrid DF and D architecture for OFDM he basic idea of ognitive Radio (R) is to provide a system with the abiity to sense avaiabe spectrum sots not occupied by existing users and detect whether any primary user is demanding the bands that the R system currenty uses Since R ony used non-contiguous band in the spectrum, OFDM is considered a suitabe transmission technique [][3] n spectrum pooing, OFDM is proposed as the baseband transmission scheme hose subcarries which cause the interference to the icensed user shoud be nuified herefore, there are zero-vaued inputs for the DF or D of the transmitter and zero-vaued outputs for the DF or D of the receiver When zero vaued inputs/outputs outnumber non-zero inputs/outputs, the standard DF/DF or D/D used for OFDM is no onger efficient f there is a arge number of zero inputs/outputs, we propose to use computationay efficient DF/DF and D/D based on the transform decomposition [-6] he rest paper is organized as foows n Section, the OFDM-ognitive Radio framework wi be introduced n Section 3, we wi present the fast hybrid DF and D architecture for OFDM he efficient computation for DF and D in OFDM- ognitive Radio system is discussed in Section Finay, Section 5 concuded this paper OFDM-ognitive Radio Framework A genera schematic of an OFDM-ognitive Radio (OFDM-R) transceiver is shown in Fig Without oss of generaity, a high speed data stream, x( n ), is moduated hen, the moduated data stream is spit into sower data streams using a seria-to-parae (S/P) converter ote that the subcarriers in the OFDM-R transceiver do not need to be a active as in conventiona OFDM Moreover, active subcarriers are ocated in the unoccupied spectrum bands, which are determined by dynamic spectrum sensing and channe estimation
techniques he DF or D is then appied to these moduated subcarrier signas Prior to transmission, a 3 Fast Hybrid D and DF Architecture for OFDM n this section, we derive the recursive formuas for D and DF matrices he resuts show that the D and DF matrices can be unified by using the same sparse matrix decomposition agorithm and recursive architecture within some characters changed (a) ransmitter (b) Receiver Fig Schematic of an OFDM transceiver guard interva with a ength greater than the channe deay spread is added to each OFDM-R symbo using the cycic prefix (P) bock in order to mitigate the effects of intersymbo interference Foowing the parae-to-seria (P/S) conversion, the baseband OFDM-R signa, sn ( ), is then passed through the transmitter radio frequency (RF) chain, which ampifies the signa and up converts it to the desired center frequency he receiver performs the reverse operation of the transmitter, mixing the RF signa to baseband for processing, yieding the signa rn ( ) hen, the signa is converted into parae streams using S/P converter, the P is discarded, and DF or D is then appied After compensating distortion introduced by the channe using per-tone equaization, the data in the active subcarriers is mutipexed using a P/S converter, and demoduated into a reconstructed version of the origina high-speed input, x '( n ) From this system overview, we observe that the DF (D) and DF (D) bocks are critica components of the transceiver n the next section, we wi investigate the hybrid architecture of DF and D (we confine a the foowing discussions on DF and D due to the same computation structure of DF and DF as we as D and D) Based on this architecture we can provide two schemes to achieve OFDM: OFDM based on DF and OFDM based on D, the adopted scheme wi be determined by channe conditions Whie in Section, we wi describe how it is possibe to impement efficient versions of these bocks 3 Sparse matrix decomposition for D matrix A typica D matrix is the D- case (we consider D- in this paper), which is defined by () k, / Bk cos, kn, 0,,,, () kn /, k 0 Bk From the definition,, k,,, the -by- D matrix is given as / / / / 3 5 7, () 6 6 3 7 5 i cos( i / ) is the cosine unit for D computations Before we introduce a simpe matrix factorization agorithm, we define some matrices he row permutation matrix Pr is defined by Pr and Pr pri, j,, (3) pri, j, if i j,0 j /, and pri, j, if i ( j ) mod,0 j / pri, j 0, others i, j {0,,, } Further, we define a reversibe permutation matrix Pc as foows, / 0 0 0 0 / 0 0 Pc Pc, () 0 0 0 / 0 0 / 0 hus we can write 0 Pr Pc, (5) 0 B
denotes the transpose of a matrix and 3 B 3 Generay, the permuted D can be recursivey formed by using matrix / / / 0 Pr Pc / / 0 B /, (6) B can be cacuated by: / f( mn, ) B, f ( m,) m, (7) / mn, / f ( m, ) f ( m, n) f ( m,), m, n {,,, / } Furthermore, the submatrix B can be represented by B K D, () 0 0 0 K 0 D diag,, n n and i i, i { 0,,, } From (6) and (), the D matrix can be written by 0 0 0 (9) Pr Pc 0 K 0 0 D Finay, the genera recursive form is given by 0 0 Pr Pr 0 K 0 K 0 Pc 0 D 0 (0) Pc 0 D o simpify (0), we can rewrite it by using 0 0 Pr 0 K 0 K 0 0 D 0 Pc () 0 D he butterfy data fow diagram corresponding to () is shown as in Fig 3 Sparse matrix decomposition for DF matrix he DF is a Fourier representation of a given sequence x (m), 0 m and it is defined by nk X ( k) x( n) W, 0 k, () n 0 j/ e W, j he -point DF matrix can be denoted by nk F W hus we can write a permuted -by- DF matrix by using F 0 F F Pr, (3) E 0 j E Generay, the -by- j permuted DF matrix is F 0 F F, () Pr 0 E F F, and the submatrix E can be written by E Pr F W, (5) 0 W diag W, W,, W, and W is the compex unit for -point DF matrix Simiar to (9), we can rewrite the permuted DF matrix by using F 0 F 0 Pr F W 0 F 0 0 (6) 0 Pr 0 F 0 W As a resut, the genera recursive form for DF matrix can be represented by F Pr F 0 0 Pr F 0 Pr 0 Pr 0 0 (7) 0 W 0 W
Fig Butterfy data fow diagram of the proposed computation of the -by- D matrix Fig3 Butterfy data fow diagram of the proposed computation of the -by- DF matrix eary, the form of (7) is the same as that of (), we ony need to change K to Pr and D to W, with the parameters {,,,, / } he butterfy data fow diagram corresponding to (7) is shown as in Fig3 As iustrated in Fig, and Fig3, we find that the DF computation can be from the computation of the D matrix by repacing the submatrix D to W, and the permutation matrix K Pr to As a resut, a simpe generaized bock diagram for D and DF hybrid architecture and its fast agorithm can be shown as in Fig n this figure, we joint D and DF computations into one chip or one kind of processing architecture, we use one switching box to contro the output data fow his resut is usefu to deveop the united chip for OFDM K / / / Pr / D / W / / / Efficient omputation for DF and D in OFDM-R System n this section we wi introduce the agorithms on -D systoic array for DF and D, which can resut in a faster execution time in OFDM-R system ransform decomposition on -D systoic array for DF Based on the definition of DF in (), we assume that ony L outputs are nonzero and that there exists a P such that P divides and define Q / P, using the variabe substitution nq n, () 0,,, P, n 0,,, Q, we can rewrite the DF as foows: Q P ( nq n) k n0 0 Q P k p nk xnq ( n) W W, (9) n0 0 X( k) x( nqn ) W Fig A simpe D and DF hybrid architecture
p denotes reduction moduo P, and k takes on any L vaues between 0 and Breaking this up into two equations X n and Q X( k) X ( k ) W, (0) P 0 P n 0 n nj P p nk ( j) x( nq n ) W () x ( nw ) j 0,,, P, () 0 n nj P x ( n ) x( nq n ) (3) n nspecting (), it can be seen that sequence over which the DF has to be computed is two dimensiona and hence depends on n hus a DF has to be computed for each different vaue of n, and there are Q such ength P DF s We consider the case L outputs are nonzero Because the index K ony consists of L nonzero vaues, ony L twidde factors are mutipied with each X ( k p) for n 0,,, Q n his mutipication part resuts in a reduction of the computation he mathematica derivation for the transform decomposition agorithm with zero inputs is rather simiar, detais can be found in [] ransform decomposition on -D systoic array for D Reposing on a cose study of the D [7], we can aso achieve the transform decomposition for -D D he D of a date sequence f ( n ), n0,,, is defined by () k Fk ( ) / Bk f( n)cos, () n0 /, k 0 Bk, k,,, Supposing is even, we define a new point sequence x( n ) by: x( n) f( n), x ( n) f(), n0,,, /, (5) substituting (5) into (), it can be rewrite as () k Fk ( ) / k ( ) xn ( )cos (6) n0 hus the D Fk ( ) can be evauated as jk Fk ( ) / k ( )Re exp( ) Zk ( ), (7) the date sequence x(n) s DF is nk Zk ( ) xnw ( ) () n0 jk Using symbo A to denote / ( k)exp( ), we can obtain a concise equation Fk ( ) Re AZk ( ) (9) From the anaysis of transform decomposition for -D DF previousy, we can easiy obtain the transform decomposition for -D Z( k ) as foows: Q Zk ( ) Z ( k ) W, (30) P n 0 n Z ( j) x( nqn ) W n 0 P nj xn( ) WP 0 p nj P nk j 0,,, P, (3) and PQ, n Qn n, n 0,,, P, n 0,,, Q Substituting (30) into (9), we obtain: Q nk Fk ( ) Re Z ( k ) W A n p (3) n 0 Obviousy, we can aso impement the efficient computation for D Fig5 Bock diagram over transform decomposition for D We show the computationa structure for D in Fig5 Basicay the computation can be divided into two stages: DF and mutipications with recombination
3 Simuation Resuts he number of mutipications for -point radix - FF is M DF ( / )og (33) Under the definition of D-, the precise count of rea mutipications is shown in the beow [], M og (3) D Based on the transform decomposition for -D DF and D, we make quantitative anaysis on the computationa compexity by counting the number of compex mutipications n this case, the number of mutipications for DF and D are presented as foowing: M D DF og P ( Q ) L (35) MDD og P Q( Q) L (36) system when ony a sma number of subcarriers are avaiabe for ognitive Radio 5 oncusion n this paper, we present a fast hybrid DF and D architecture for OFDM based on the decomposition for DF and D matrix n terms of the decomposition for -D DF and D, we proposed an efficient computation for DF and D which can resut in a faster execution time in the case that a arge number of subcarriers are nuified in the OFDM-R system 6 Acknowedgement his work was partia supported by the M, under R supervised by A, ER SO and the Korea Foundation for nternationa ooperation of Science & echnoogy (KOS) through a grant provide by the Korean Ministry of Science & echnoogy (MOS) in 006-0003 7 References Fig6 omparison of computation between the traditiona way and the transform decomposition for -D DF/D n Fig6 we compare the computation compexity between the conventiona method and the transform decomposition for -D DF and D During the simuation, we assume the tota number of subcarriers for OFDM is 0, when the number of nonzero outputs L is reativey few, we can get significanty reduced computationa compexity for both DF and D based on the transform decomposition For exampe, when ony 6 out of 0 subcarriers are avaiabe for R, transform decomposition offers % and 37% saving of computations for the OFDM-DF system and the OFDM-D system respectivey Because the DF/DF or D/D are the most computationa intensive parts in an OFDM transceiver, the savings can significanty reduce the computationa compexity of the overa system herefore, transform decomposition for -D DF or -D D can be an efficient option for an OFDM in ognitive Radio [] Giridhar D Mandyam, On the Discrete osine ransform and OFDM Systems, EEE nternationa onference on Acoustics, Speech, and Signa Processing, vo, pp5-57 6-0 Apri 003 [] J D Poston and W D Horne, Discontiguous OFDM considerations for dynamic spectrum access in ide V channes, in Proc EEE nt Symp ew Frontiers Dynamic Spectr Access etworks, vo, (Batimore, MD, USA), pp 607 60, ov 005 [3] M P Wyie-Green, Dynamic spectrum sensing by mutiband OFDM radio for interference mitigation, in Proc EEE nt Symp ew Frontiers Dynamic Spectr Access etworks, vo, (Batimore, MD, USA), pp 69 65, ov 005 [] Henrik V Sorensen and Sidney Burrus, Efficient omputation of the DF with Ony a Subset of nput or Output Points, EEE rans on Signa Processing, Mar 993 [5] Qiwei Zhang, Andre BJ Kokkeer, Gerard JM Smit, An Efficient FF for OFDM Based ognitive Radio On A Reconfigurabe Architecture, EEE nternationa onference on ommunication, Jun 007, Gasgow, UK [6] Moon Ho Lee, High speed mutidimensiona systoic arrays for discrete Fourier transform, EEE rans ircuits Syst, vo 39, pp 76 79, Dec 99 [7] Moon Ho Lee, On computing -D Systoic Agorithm for Discrete osine ransform, EEE rans ircuits Syst, vo37, no0 990 [] H V Sorensen, D L Jones, M Heideman, and S Burrus, "Rea-vaued fast Fourier transform agorithms," EEE rans Acoust Speech Sig Processing ASSP-35, pp9 63, 97