Low Complexity Discrete Hartley Transform Precoded OFDM System over Frequency-Selective Fading Channel

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1 Low Complexity Dicrete artley Tranform Precoded OFDM Sytem over Frequency-Selective Fading Channel Xing Ouyang, Jiyu Jin, Guiyue Jin, and Peng Li Orthogonal frequency-diviion multiplexing (OFDM) uffer from pectral null of frequency-elective fading channel. Linear precoded (LP-) OFDM i an effective method that guarantee ymbol detectability by preading the frequency-domain ymbol over the whole pectrum. Thi paper propoe a computationally efficient and lowcot implementation for dicrete artley tranform (DT) precoded OFDM ytem. Compared to conventional DT-OFDM ytem, at the tranmitter, both the DT and the invere dicrete Fourier tranform are replaced by a one-level butterfly tructure that involve only one addition per ymbol to generate the time-domain DT- OFDM ignal. At the receiver, only the DT i required to recover the ditorted ignal with a ingle-tap equalizer in contrat to both the DT and the DFT in the conventional DT-OFDM. Theoretical analyi of DT-OFDM with linear equalizer i preented and confirmed by numerical imulation. It i hown that the propoed DT-OFDM ytem achieve imilar performance when compared to other LP-OFDM but exhibit a lower implementation complexity and peak-to-average power ratio. Keyword: OFDM, linear precoding, multipath diverity, dicrete artley tranform, DT, peak-toaverage power ratio, PAPR. Manucript received Apr ; revied Aug. 7, 04; accepted Aug., 04. Xing Ouyang (chritoyx@gmail.com), Jiyu Jin (correponding author, jiyu.jin@ dlpu.edu.cn), Guiyue Jin (guiyue.jin@dlpu.edu.cn), and Peng Li (lipeng@dlpu.edu.cn) are with the School of Information Science & Engineering, Dalian Polytechnic Univerity, Liaoning, China. I. Introduction Orthogonal frequency-diviion multiplexing (OFDM) i an attractive multicarrier tranmiion technique that ha been implemented in wirele and wired broadband communication ytem uch a LTE, WiMAX, and DVB [] [4]. It i pectrally efficient due to the minimized orthogonal ubcarrier pacing. By dividing the wideband ytem pectrum into narrowband ubchannel whoe repone i frequency flat, a ingle-tap frequency-domain equalizer i more efficient than a time-domain equalizer to compenate frequency-elective fading. owever, conventional OFDM ytem are enitive to the pectral null over frequency-elective fading channel becaue the information ymbol tranmitted over the ubchannel in the vicinity of deep fading may be overwhelmed by noie and can hardly be recovered at the receiver [5] [6]. Linearly precoded OFDM (LP-OFDM) ytem are tudied in the literature a attractive alternative to mitigate the pectral null problem [5] [7]. The baic idea of LP-OFDM i to pread information ymbol into either part of or the whole of the ytem band by unitary or trigonometric tranform before the ymbol are multiplexed by the invere dicrete Fourier tranform (IDFT). Therefore, in pite of the pectral null, ymbol can till be extracted at the receiver. In [6], it wa proved that the maximum achievable multipath diverity order of LP-OFDM equal the number of available channel path, even with a linear equalizer. In [7] [8], Debbah and other propoed an iometric random precoded OFDM cheme in which ymbol are pread by iometric matrice. The complex field coding wa preented to precode the 3 Xing Ouyang et al. 05 ETRI Journal, Volume 37, umber, February 05

2 ymbol by truncated matrice [9]. In [7] [9], the precoding matrice are retricted to being of full column rank; conequently, data rate lo occur. Linear contellation precoding (LCP) wa propoed for OFDM in [0] [] without the lo of data rate. To relieve decoding complexity, an optimal grouped LCP in term of the maximum available multipath diverity wa alo preented in [0]. owever, the complexity of thee cheme i till unatifactory due to the precoding and decoding procee and ymbol detection algorithm. The LP-OFDM ytem baed on linear tranform, for which fat calculation algorithm are devied, are attractive for their implicity and atifactory performance. For example, the dicrete coine tranform (DCT) and Walh adamard tranform (WT) precoded OFDM were invetigated in [] [5]. It wa hown that the LP-OFDM ytem can achieve attractive performance improvement in the frequency-elective fading channel with a linear minimum mean quare error () equalizer. To further implify the complexity of LP- OFDM, everal work have focued on how to unite the precoding and the multiplexing; that i, making the IDFT procee into a impler algorithm. For example, the fat algorithm for calculating cacaded WT and DFT are tudied in [6] [9], in which WT and DFT are calculated jointly at a lower complexity than when calculating WT and DFT eparately. In [0] [], the fat algorithm wa applied to the WT-OFDM ytem, termed a T-tranform, to implement the ytem more efficiently. In addition, it wa hown that the peak-to-average power ratio (PAPR) of the WT-OFDM ignal i improved. In addition, the DT precoded OFDM (DT-OFD) wa propoed for it uperior PAPR performance [] [4]. In [5], we preent a imple algorithm to calculate the DT and IDFT jointly with only one addition operation per ymbol. In thi paper, we introduce a more computationally efficient and lower-cot DT-OFDM cheme than other LP-OFDM to counteract the pectral null problem baed on our previou work. In the propoed cheme, the tranmitter generate the time-domain DT-OFDM ignal by a imple one-level butterfly tructure that involve only one addition operation per ymbol. In the receiver, the DFT that i required in other LP- OFDM and in the conventional DT-OFDM i avoided, and the frequency-elective fading channel can be compenated by a ingle-tap equalizer with only a DT module. Thu, both the computational complexity and ytem deign of the propoed tranceiver are impler and more efficient than that of other LP-OFDM. Moreover, in thi paper, the theoretical BER performance of the propoed DT-OFDM with both zeroforcing (ZF) and equalizer i analyzed and confirmed by imulation. It i hown that the propoed cheme ha the ame ability to mitigate pectral null a other LP-OFDM cheme but feature much lower PAPR and complexity. The ret of thi paper i organized a follow. The ytem model of the conventional OFDM and DT-OFDM are illutrated in Section II. The low-complexity tranceiver deign for a DT-OFDM i introduced in Section III. The theoretical performance of the DT-OFDM i analyzed in Section IV, and a dicuion i provided in Section V. In Section VI, imulation i performed to confirm our analyi and the ability of the propoed cheme to counteract the pectral null problem. Finally, Section VII conclude thi paper. II. Sytem Model of DT-OFDM. Conventional OFDM In Fig., the ytem diagram of a conventional OFDM ytem (excluding the dahed frame) i illutrated. For any given number of ymbol, we can group the ymbol in the frequency domain a x = [x(0), x(),, x( )] T, where the upercript ( ) T denote the tranpoe operator and i the number of ubcarrier. The ymbol are multiplexed by an IDFT, which can be implemented by taking an invere fat Fourier tranform (IFFT), a in = W x, () where i the time-domain ignal vector, W i the DFT matrix with it (m, n)th element being (/ ) e jπmn, and ( ) denote the ermitian tranpoe operator. The time-domain ignal i then parallel-to-erial (P/S) converted. To keep the ignal from the interference of previouly delayed ignal and to convert the channel to be circulant, a cyclic prefix (CP), which i the lat c replica of, i attached to the beginning of. The length of CP, c, i choen to be larger than the maximum delay pread. The dicrete ignal i digital-to-analog (D/A) converted for tranmiion. Received ymbol Tranmit ymbol P/S & deciion S/P x x p y y p y r DT DT Fig.. Conventional OFDM ytem (excluding the dahed frame) and conventional DT-OFDM ytem (including the dahed frame). tap Equ. IDFT DFT P/S & CP Removing CP & S/P D/A Channel A/D ETRI Journal, Volume 37, umber, February 05 Xing Ouyang et al. 33

3 CFR (db) path uniform power delay profile multipath channel; 30 the normalized maximum delay i ,000 Subchannel index Fig.. 0-path uniform power channel in OFDM ytem with,04 ubchannel. At the receiver, the ignal-experiencing multipath channel i received, and CP i dicarded directly. The received ignal i r = + n, () where n i the additive white Gauian noie (AWG) and i the quai-tatic channel impule repone (CIR) matrix; that i, the channel i invariant during one OFDM block. 0 0 h h h h 0 h, 0 0 h0 with h(l), l = 0,,, L, to be the CIR of the lth channel path. The received ignal i tranformed by a fat Fourier tranform (FFT) to the frequency domain a (3) y = Wr = WW x + Wn = Λx + v, (4) where v i the AWG in the frequency domain and Λ i the channel frequency repone (CFR) matrix, which i diagonal with it (k, k)th element, a (k), repreenting the channel repone on the kth ubchannel. The frequency-domain ignal i equalized for deciion to recover the tranmitted information. In broadband OFDM ytem, although each ubchannel experience flat fading, the whole pectrum i till frequency elective. Therefore, ome of the ubchannel uffer from deep fading. In Fig., the CFR of a 0-path channel with uniform power delay profile (PDP) i ketched. There are deep notche over the pectrum. Thi how that ymbol on the ubchannel in the vicinity of null cannot be recovered correctly.. Conventional DT-OFDM The artley tranform i a Fourier-related tranform [6], and it dicrete form wa firt introduced by Bracewell in 983 [7]. The obviou difference compared with a DFT are that the forward DT i identical to the invere DT and i realvalued. Variou fat algorithm, termed a Fat artley Tranform, have been propoed to implify the implementation complexity, which i aid to be imilar to that of an FFT. The block diagram of the conventional DT-OFDM i illutrated in Fig. (including the dahed frame). In contrat to conventional OFDM, before the multiplexing of frequencydomain ymbol by an IDFT, the ymbol are tranformed by a DT into the artley domain at the tranmitter, and the precoded ignal i x p = Fx, (5) where F denote the DT matrix with it (m, n)th element to be ca (πmn/) / = [co (πmn ) + in (πmn / )] /. The precoded ignal then undergoe the ame ignal proceing procedure a the conventional OFDM. At the receiver, the received ignal i tranformed to the frequency domain a y = Wr = WW Fx + Wn = ΛFx + v, (6) where v i the frequency-domain noie, and the equalized ignal i y p = Gy = GΛFx + Gv, (7) where G i a diagonal weighting matrix baed on equalization criteria. The equalized ignal i tranformed by a DT for deciion. In the DT-OFDM, a well a the other LP-OFDM ytem, ZF i unfavorable ince the noie within the pectral null will be amplified and impoed on all ymbol after a DT, reulting in eriou performance degradation. The i a linear equalizer that ha better performance due to it capability to uppre noie effectively. The detailed analye on their performance will be provided in Section IV and V. III. Low-Complexity DT-OFDM In thi ection, we propoe a computationally efficient and low-cot implementation for the DT-OFDM illutrated in Fig. 3. At the tranmitter, the DT-OFDM ignal i generated by a one-level butterfly tructure that realize a DT and an IDFT jointly, and the receiver cheme require only a DT without a DFT to recover the ditorted ignal with implified ytem deign. In a DT-OFDM ytem, the baeband time-domain ignal,, i virtually obtained by ubtituting (5) into (), which give = W x p = W Fx. (8) The DT and DFT matrice can be expreed a F = C + S and 34 Xing Ouyang et al. ETRI Journal, Volume 37, umber, February 05

4 Received ymbol Tranmit ymbol P/S & deciion S/P DT x x(0) x() x() x( ) x( ) Phae rotation /4 and caled / y y p y r p r r(0) r() j r() j Equalization j j j j DT Phae rotation π /4 and caled / Fig. 3. Propoed low-complexity DT-OFDM ytem. (0) () () ( ) ( ) j j P/S & adding CP D/A r( ) r( ) W = C j S, repectively, where the (m, n)th element of C and S are, repectively, π π co mn and in mn. It can be readily deduced that the DT matrix can be expreed in term of a DFT matrix a F W W W W j j W. W j P/S & removing CP Channel A/D (9) (0) By exploiting the propertie of DFT; that i, W W = I with I being the unitary matrix, and WW = J, where J () i a flip matrix, and ubtituting (0) into (8), it can be derived that W Fx π j e 4 I jjx () Px. Obviouly, intead of the cacaded DT and DFT operation, the time-domain ignal can be generated by combining x and x flipped by J with a phae difference of π. Subtantially, the nth time ample can be obtained by adding the nth frequency ymbol with the ymbol on the ( n)th, n =,,,, ubchannel, a in π j n e 4 xn j x n, (3) where (n) and x(n) are the nth entry of and x, repectively. The propoed tranmitter of the DT-OFDM baed on the one-level butterfly tructure i illutrated in Fig. 3. The ymbol are firtly caled and phae hifted, and then x i obtained uing a one-level butterfly-like tructure. Thereby, the tranmitter i implified to generate the baeband time-domain ignal without multiplication except for a calar and a phae rotation of π /4. In thi paper, a receiver cheme with only a DT i propoed, a hown in Fig. 3. Intead of a DFT, the received ignal after A/D and S/P converion i tranformed by P = (W F) a π j 4 rp P r e I jjr W F r FWW FxFWn FΛFxFv. (4) The third equation in (4) can be readily deduced from (). The received ignal, r p, i obtained by the one-level butterfly tructure by weighting the received ignal with P. Therefore, the frequency-domain ignal can be obtained by a DT, ince a DT i unitary; that i, F F = I, a y = Fr p = ΛFx + v. (5) At the receiver, the frequency ignal can be compenated by a ingle-tap equalizer baed on ZF or. For the ZF equalizer, the weighting matrix G i Λ. The correponding equalized ignal in (7) can be given by y ZF = x p + Λ v, (6) and the ignal tranformed to the artley domain by a DT i F FΛ yzf yzf x v xz, ZF (7) where z ZF i the noie vector after ZF equalization. It can be een that the channel can be completely compenated, but x will be everely corrupted by the amplified noie. The equalizer i an attractive equalizer that can uppre the noie at pectrum null, and the equalized ignal in the artley domain i y FG y F Λ Λ I Λ ΛF xv FD I DFxFD I Λ v, (8) where G i the coefficient matrix baed on, λ i the ignal-to-noie ratio (SR), and D = Λ Λ i an by real diagonal matrix with it (k, k)th element repreented by (k). ETRI Journal, Volume 37, umber, February 05 Xing Ouyang et al. 35

5 IV. Performance Analyi In thi ection, the BER performance of the conventional OFDM and DT-OFDM baed on both ZF and equalizer i analytically tudied to provide inight into the performance improvement of the DT-OFDM. Quadrature phae-hift keying (QPSK) i conidered in thi ection for analyzing the performance, but the principle can be extended to other modulation format. The BER in term of the SR in the AWG channel for QPSK i P where Q(x) i defined a Q (9) QPSK, t Qx e d. t (0) x π In both the AWG and the frequency-flat fading channel, the DT-OFDM obtain the ame performance a the conventional OFDM ytem ince there i no notch over the pectrum; thu, it provide no frequency diverity. One of the advantage of DT-OFDM i that it i more robut to frequency-elective fading channel by averaging the deep fading over the whole pectrum, and it performance largely depend on the criteria of equalization.. BER Performance of Conventional OFDM Sytem For the conventional OFDM ytem, if the ZF equalizer i ued, then the equalized ignal on the kth ubchannel i given by y k x k k v k () ZF, and, if the equalizer i ued, then the ignal i * k k y k x k v k, () k k where the upercript * denote the complex conjugate operator. It can be een that the conventional OFDM ytem under the frequency-elective fading channel get the ame received SR, (k) λ, for both ZF and. Thu, the error probability of the OFDM can be given by POFDM E PQPSK k k 0 E PQPSK k (3) k 0 E Q k, k 0 where E[ ] i the expectation operation.. BER Performance of DT-OFDM Sytem The analyi on BER performance of DT-OFDM ytem i omewhat complicated ince the precoded ignal i mutually correlated, and the precoding matrix will affect the ditribution of noie on each ubchannel after equalization. A. ZF Equalizer for DT-OFDM For the ZF equalizer, a ignal can be compenated completely; thu, the equivalent channel gain (ECG) i equal to one, a hown in (7). The noie on the kth ubchannel, from (7), i π zzf k kn n v n n0 ca. (4) The noie variance after equalization can be deduced a E, * n z kz k n (5) ZF ZF ZF k0 n0 where σ n i the power of noie v. Since the noie z ZF (k) i a function of the independent random variable v(n) and (n), n = 0,,, ((n) are mutually correlated), the noie power impoed on the kth ubchannel can be deduced a k E z kz k * ZF ZF ZF π n0 n ca kn n. ence, the BER of the DT-OFDM with ZF equalizer i ZF E PQPSK k 0 ZF k P E E Q π, k 0 ca kn n n0 where E i the ymbol energy, which i a contant for PSK. B. Equalizer for DT-OFDM (6) (7) For the, the ymbol in the artley domain are hown in (8), and the deired kth ymbol at the receiver i given by π m x k ca km x k m 0 m k x k, (8) where '(k) i defined a the ECG of the kth frequencydomain ymbol for. The reidual inter-carrier interference (ICI) and the noie on the kth ymbol are 36 Xing Ouyang et al. ETRI Journal, Volume 37, umber, February 05

6 and x ICI k x n n0, nk π π m ca kmca mn m 0 m (9) * n ca, (30) n 0 π z k kn v n n repectively. Thu, the noie power on the kth ymbol i k E z kz k * n π m ca km. m0 m (3) Compared with ZF, even if there i a deep fading; that i, a mall (n), the noie power can be effectively uppreed. Moreover, the ymbol are no longer ICI free. The kth deired ymbol will be interfered by ICI a indicated in (9), and the ICI power i k E x k. (3) ICI Since x(k) are mutually independent, (3) can be given a m E π m ca km. m 0 m E π m ICI k ca km m0 ICI (33) The firt term in (33) i the power of ignal y' (k) and the econd term i the deired ignal power '(k) E, which can be readily deduced from (8) and (8). The ICI can be modeled a Gauian if i large, and the interference plu noie can be deduced a k k k ICI E π m ca km m0 m n π m ca m0 k E km m E π m ca km k E m 0 m E k k, (34) where '(k) i defined in (8) a the equivalent channel gain of the kth ymbol. Therefore, the ignal-to-interference-plunoie ratio (SIR) can be deduced from (8) and (34) a Γ k k k π m ca km m 0 m. π m ca km m 0 m By exploiting the equation π (35) ca km, m0 (36) the SIR in (35) can be further implified a Γ k m0 m0 π m ca km m. π ca km m (37) Therefore, the error probability of the DT-OFDM with i P Q Γ k E. (38) k 0 To how the reitance of DT-OFDM againt pectral null, Fig. 4(a) provide a naphot obervation of CFR of a 0-tap multipath channel at SR = 0 db. It i obviou that in the conventional OFDM the ymbol in the deep notche can hardly be detected correctly, while the ECG of a DT-OFDM in the artley domain '(k) i flat over the whole pectrum without notche. Although the ECG in the ZF cae i alway equal to one, the noie will be amplified ignificantly by G(k), a hown in Fig. 4(b) and 4(c), and pread over all ymbol after the DT, a hown in Fig. 4(d), with pronounced SR degradation. The achieve better performance ince the coefficient of the tap at null can be efficiently uppreed, a hown in Fig. 4(b). Thu, noie power i effectively limited over all the ymbol, a hown in Fig. 4(d). The cumulative ditribution function (CDF) of received SR after the DT i illutrated in term of ECG '(k) to noie power z(k) in Fig. 5. At an SR of 7 db, the ditribution of the conventional OFDM and DT-OFDM of uperpoed together. For the DT-OFDM with ZF equalizer, about 5 db SR lo i incurred if Prob(SR > SR 0 ) = 0.9. ence, it can be expected that the DT-OFDM with achieve imilar performance a the conventional ETRI Journal, Volume 37, umber, February 05 Xing Ouyang et al. 37

7 0 0 DT-OFDM, ZF 30 0 ZF (k) (db) DT-OFDM, OFDM Subchannel index (k) (a) G(k) (db) Subchannel index (k) (b) 0 50 G(k)v(k) (db) ZF Subchannel index (k) (c) z(k) (db) DT-OFDM, ZF DT-OFDM, 70 DT-OFDM, reidual interference Subchannel index (k) (d) Fig. 4. Snaphot of the conventional OFDM and DT-OFDM under a frequency-elective fading channel: (a) CFR of the conventional OFDM and equivalent channel gain of the DT-OFDM after ZF and equalization in the artley domain, (b) equalizer weight for both ZF and in the DT-OFDM, (c) noie v(k) after equalization G(k), and (d) noie being tranformed into the artley domain z(k) by a DT. Prob( (k) / z(k) < SR0) db 3 db 5 db OFDM DT-OFDM, ZF DT-OFDM, to the conventional OFDM; that i, at Prob(SR > SR 0 ) = 0, the DT-OFDM with ee around a 3 db to 6 db improvement for SR value of 3 db to 5 db. Moreover, a 0 db SR gain occur at Prob(SR > SR 0 ) = 0 3 at an SR of 5 db. Thu, it can be concluded that the performance improvement of the DT-OFDM with i more evident at high SR for it diverity gain. V. Dicuion In thi ection, the PAPR of the DT-OFDM i evaluated and the ytem complexity i compared with other LP- OFDM SR 0 (db) Fig. 5. CDF of received SR for OFDM and equivalent SR for DT-OFDM with ZF and equalizer with different tranmitted SR. OFDM at low SR and that the DT-OFDM with ZF equalizer exhibit certain performance degradation. A the SR increae, the DT-OFDM with equalizer experience a ignificant performance improvement compared. PAPR of DT-OFDM A dicued in () and (3) in Section III, the time-domain ignal of the DT-OFDM i the um of two ubcarrier modulated by the mirror ymmetrical ymbol, rather than the um of the overall orthogonal ubcarrier modulated by different ymbol in conventional OFDM ytem. Thu, it can be inferred that the peak power of the DT-OFDM ignal can be reduced ignificantly, a dicued in []. The PAPR of the OFDM ignal i defined a 38 Xing Ouyang et al. ETRI Journal, Volume 37, umber, February 05

8 Prob (PAPR > PAPR0) OFDM WT-OFDM DCT-OFDM DT-OFDM Table. Comparion of real arithmetic operation between conventional and propoed DT-OFDM. Abbr. Conventional DT-OFDM Propoed DT-OFDM Operation reduction (%) Add. Multi. Add. Multi. Add. Multi. 64 5,504 3,38,688,79 5.% 46.% 56 9,84 7,408 3,84 9,6 5.6% 47.%,04 45,408 86,06 67,584 45, % 47.6% 4, ,30 409,600 39,488,99 54.% 48.0% 0 3 addition are PAPR 0 Fig. 6. CCDF of PAPR for OFDM, DCT-OFDM [0], WT- OFDM [], and propoed DT-OFDM. max n PAPR, E n (39) and the complementary cumulative ditribution function (CCDF) of the PAPR i defined a Prob PAPR PAPR, (40) which repreent the probability of the PAPR of an OFDM ignal exceeding a threhold PAPR 0. In Fig. 6, the CCDF of the PAPR i preented for different 6 QAM LP-OFDM ytem, and the number of ubchannel i =,04. It can be een that, both the DCT and the WT precoded OFDM propoed in [] [9] how lower PAPR than the conventional OFDM, with around.5 db and 0.5 db PAPR improvement at Prob(PAPR > PAPR 0 ) = 0 4. For the DT-OFDM, the PAPR i reduced further, and the PAPR i reduced by another 3.5 db and 5.5 db when compared to the DCT and the WT precoded OFDM, repectively.. Signal Proceing and Sytem Deign Complexitie In the propoed DT-OFDM a illutrated in Fig. 3, the DFT i aved by utilizing the propertie of both the DT and the DFT, and two DT and two one-level butterfly tructure are required. If the input i complex, then the DT take log real multiplication and 3log real addition, and the onelevel butterfly algorithm take real addition. In addition, the ingle-tap equalizer at the receiver take 4 real multiplication and real addition. Therefore, if the input i complex, then the total number of real multiplication and 0 and add. = 6(log + ) (4) multi. = 4(log + ), (4) repectively. In the conventional DT-OFDM a hown in Fig., an FFT, an IFFT, and two DT are required. If the input i complex, then the FFT (IFFT) take log real multiplication and 4log real addition, and the total number of real arithmetic operation i and add. = 4log + (43) multi. = 8log + 4, (44) repectively. In Table, their complexitie are compared, and it can be een that the complexity of the propoed tranceiver i around half of that of the conventional DT- OFDM. In the DCT-OFDM, beide the DFT and the IDFT, both the DCT and the IDCT, whoe complexity i imilar to a DT, are required. Thu, it can be expected that the complexity of the propoed cheme i about half of that of the DCT-OFDM. In WT-OFDM, although two T-tranform are applied to perform the WT and IDFT operation joint at the tranmitter [5] [9], two additional WT are required at the receiver, reulting in higher implementation complexity than the propoed DT-OFDM. VI. Simulation Reult In thi ection, imulation are performed uing MATLAB to evaluate and confirm the advantage of the propoed DT- OFDM cheme to provide frequency diverity to improve the performance in frequency-elective channel. The ytem bandwidth i 0 Mz with,04 ubchannel, and QPSK and 6 QAM are employed. Both the multipath Rayleigh fading ETRI Journal, Volume 37, umber, February 05 Xing Ouyang et al. 39

9 0 0 QPSK 6 QAM Conv. OFDM DT-OFDM 0 0 OFDM, QPSK Theoretical OFDM DT-OFDM, QPSK, Theoretical DT-OFDM, DT-OFDM, QPSK, ZF Theoretical DT-OFDM, ZF Rayleigh flat fading BER 0 BER 0 AWG E b / 0 (db) Fig. 7. BER of conventional OFDM and DT-OFDM under flatfading channel with QPSK and 6 QAM. channel and the more realitic ITU channel model B are invetigated. The channel repone i aumed to be invariant within one OFDM ymbol block, and the channel tate information i perfectly known at the receiver. The path power i normalized a Σ l h(l) =. In Fig. 7, imulation i performed under both the AWG and the flat-fading Rayleigh channel. The conventional OFDM and DT-OFDM achieve the ame BER performance ince the DT-OFDM provide no frequency diverity in the frequency flat-fading channel. In addition, the performance of the DT-OFDM baed on ZF and are identical ince there i no pectral null in a flat-fading channel and the noie will not be amplified. A frequency-elective channel i conidered in Fig. 8. In Fig. 8 and 9, a 0-path equal-gain multipath Rayleigh channel with a normalized maximum delay of 0.5 i adopted, and QPSK and 6 QAM are employed, repectively. The equalizer obtain ignificant improvement compared with the conventional OFDM and ZF. Thi i becaue the frequency null are averaged and the noie i effectively uppreed, a analyzed in (3) and (37). Due to the noie enlargement, the DT-OFDM with ZF i even inferior to conventional OFDM, epecially at low SR. It performance curve i aymptotically approaching the conventional OFDM a the SR increae. In addition, the numerical reult follow the analytical reult a tudied in (7) and (38). In Fig. 0, imulation i performed under ITU channel model B with QPSK, 6 QAM, and 64 QAM format. It i oberved that for 6 QAM and 64 QAM, the conventional OFDM outperform DT-OFDM with at low SR, and croover occur at db and 0 db, repectively. A E b / 0 (db) Fig. 8. BER of conventional OFDM and DT-OFDM under 0- tap multipath fading channel with QPSK. BER OFDM, 6 QAM Theoretical OFDM DT-OFDM, 6 QAM, Theoretical DT-OFDM, DT-OFDM, 6 QAM, ZF Theoretical DT-OFDM, ZF E b / 0 (db) Fig. 9. BER of conventional OFDM and DT-OFDM under 0- tap multipath fading channel with 6 QAM. imple explanation i that, at low SR, the received SR of all the ymbol i averaged below a certain detectable SR in DT-OFDM, but for OFDM ome of the ymbol are till recoverable. In Fig., imulation are performed for the WT, DCT, and DT-OFDM under ITU channel model B. In both the ZF and cheme of different LP-OFDM, they achieve almot the ame BER performance. Thu, it can be inferred that the propoed DT-OFDM cheme inherit the ame ability to exploit the frequency diverity. 40 Xing Ouyang et al. ETRI Journal, Volume 37, umber, February 05

10 BER OFDM, QPSK DT-OFDM, ZF DT-OFDM, OFDM, 6 QAM DT-OFDM, ZF DT-OFDM, OFDM, 64 QAM DT-OFDM, ZF DT-OFDM, pectrum to exploit the frequency diverity. Compared with other LP-OFDM cheme; for example, WT and DCT precoded OFDM, the propoed cheme exhibit much lower PAPR and implified ignal proceing complexity. Thu, thee advantage make the propoed cheme a promiing low-cot olution for frequency-elective fading channel. Reference E b / 0 (db) Fig. 0. BER of conventional OFDM and DT-OFDM under ITU channel model B with QPSK, 6 QAM, and 64 QAM. BER OFDM WT-OFDM DCT-OFDM DT-OFDM E b / 0 (db) Fig.. BER performance of DCT, WT, and DT-OFDM under ITU channel model B. VII. Concluion Thi paper preent a low-complexity DT precoded OFDM ytem to mitigate the pectral null problem under frequency-elective fading channel. The precoded ignal i obtained uing a one-level butterfly tructure that require only addition rather than DT and IFFT, and only a DT i required at the receiver. Theoretical analyi and imulation confirm that the propoed cheme i robut in frequencyelective channel by preading ymbol into the whole ZF []. Cvijetic, D. Qian, and J. u, 00 Gb/ Optical Acce Baed on Optical Orthogonal Frequency-Diviion Multiplexing, IEEE. Commun. Mag., vol. 48, no. 7, July 00, pp [] G.L. Stuber et al., Broadband MIMO-OFDM Wirele Communication, Proc. IEEE, vol. 9, Feb. 004, pp [3] T. wang et al., OFDM and It Wirele Application: A Survey, IEEE Tran. Veh. Technol., vol. 58, no. 4, May 009, pp [4] J. Zhao and A.D. Elli, Advantage of Optical Fat OFDM over OFDM in Reidual Frequency Offet Compenation, IEEE Photon. Technol. Lett., vol. 4, 0, pp [5] Z. Wang and G.B. Giannaki, Linearly Precoded or Coded OFDM againt Wirele Channel Fade? Proc. IEEE Workhop Signal Proce. Adv. Wirele Commun., Taoyuan, Taiwan, Mar. 0 3, 00, pp [6] C. Tepedelenlioglu, Maximum Multipath Diverity with Linear Equalization in Precoded OFDM Sytem, IEEE Tran. Inf. Theory, vol. 50, no., Jan. 004, pp [7] M. Debbah et al., Analyi of Certain Large Iometric Random Precoded Sytem, IEEE Tran. Inf. Theory, vol. 49, no. 5, May 003, pp [8] M. Debbah, P. Loubaton, and M. de Courville, Aymptotic Performance of Succeive Interference Cancellation in the Context of Linear Precoded OFDM Sytem, IEEE Tran. Commun., vol. 5, no. 9, Sept. 004, pp [9] Z. Wang and G.B. Giannaki, Complex-Field Coding for OFDM over Fading Wirele Channel, IEEE Tran. Inf. Theory, vol. 49, no. 3, Mar. 003, pp [0] Z. Liu, Y. Xin, and G.B. Giannaki, Linear Contellation Precoding for OFDM with Maximum Multipath Diverity and Coding Gain, IEEE Tran. Commun., vol. 5, no. 3, Mar. 003, pp [] Z. Liu, Y. Xin, and G.B. Giannaki, Space-Time-Frequency Coded OFDM over Frequency-Selective Fading Channel, IEEE Tran. Signal Proce., vol. 50, no. 0, Oct. 00, pp [] Y.-P. Lin and S.-M. Phoong, BER Minimized OFDM Sytem with Channel Independent Precoder, IEEE Tran. Signal Proce., vol. 5, no. 9, Sept. 003, pp [3] X. Ouyang et al., Interleaved Multiplexing Optical Fat OFDM without the Interference between Subchannel, IEEE Photon. ETRI Journal, Volume 37, umber, February 05 Xing Ouyang et al. 4

11 Technol. Lett., vol. 5, no. 4, Feb. 03, pp [4] B. Gaffney and A.D. Fagan, Walh-adamard Tranform Precoded MB-OFDM: An Improved igh Data Rate Ultra Wideband Sytem, Proc. IEEE Int. Symp. PIMRC, elinki, Finland, Sept. 4, 006, pp. 5. [5] X. Ouyang, Single-Tap Equalization of Fat OFDM Signal under a Generic Linear Channel, IEEE Commun. Lett., vol. 8, no. 8, Aug. 04, pp [6] M.T. amood and S. Bouakta, Fat Walh-adamard-Fourier Tranform Algorithm, IEEE Tran. Signal Proce., vol. 59, no., ov. 0, pp [7] T. Su and F. Yu, A Family of Fat adamard-fourier Tranform Algorithm, IEEE Signal Proce. Lett., vol. 9, no. 9, Sept. 0, pp [8] X. Ouyang et al., Walh-adamard Fourier Tranform Baed OFDM with Space-Multipath Diverity, Proc. IEEE Conf. TECO, Xi an, China, Oct. 5, 03, pp. 5. [9] J. Zhao, DFT-Baed Offet-QAM OFDM for Optical Communication, Opt. Exp., vol., no., 04, pp [0] M.S. Ahmed et al., OFDM Baed on ew Tranform with BER Performance Improvement acro Multipath Tranmiion, Proc. IEEE Int. Conf. Commun., Cape Town, South Africa, May 3 7, 00, pp. 5. [] M.S. Ahmed, S. Bouakta, and B. Sharif, OFDM Baed on Low Complexity Tranform to Increae Multipath Reilience and Reduce PAPR, IEEE Tran. Signal Proce., vol. 59, no., Dec. 0, pp [] B. Imran and J. Varun, PAPR Analyi of DT-Precoded OFDM Sytem for M-QAM, Int. Conf. Intell. Adv. Syt., Kuala Lumpur, Malayia, June 5 7, 00, pp. 4. [3] B. Imran and J. Varun, A ew Dicrete artley Tranform Precoding Baed Interleaved-OFDMA Uplink Sytem with Reduced PAPR for 4G Cellular etwork, J. Eng. Sci. Technol., vol. 6, no. 6, 0, pp [4] X. Ouyang et al., Low Complexity Dicrete artley Tranform Precoded OFDM for Peak Power Reduction, Electron. Lett., vol. 48, no., Jan. 0, pp [5] I. Ali et al., A DT Precoded OFDM Sytem with Full Diverity and Low PAPR, IEEE Int. Symp. Per. Indoor Mobile Radio Commun., Sydney, Autralia, Sept. 9, 0, pp [6] R.V.L. artley, A More Symmetrical Fourier Analyi Applied to Tranmiion Problem, Proc. IRE, vol. 30, no. 3, Mar. 94, pp [7] R.. Bracewell, Dicrete artley Tranform, J. Opt. Soc. America, vol. 73, no., 983, pp Xing Ouyang received hi MS degree in information cience and engineering from Dalian Polytechnic Univerity, China, in 03 and i now tudying at Tyndall ational Intitute, Univerity College Cork, Ireland. i main reearch interet include multicarrier tranmiion technique and digital ignal proceing in both wirele and optical communication ytem. Jiyu Jin received hi PhD degree in information and communication engineering from Yeungnam Univerity, Gyeongan, Rep. of Korea, in 007. From 007 to 008, he wa a pot-doctoral reearcher at the School of Electrical Engineering and Computer Science, Seoul ational Univerity, Rep. of Korea. From 008 to 009, he wa an aitant profeor of information and communication engineering at Yeungnam Univerity. e joined the ilandwe Communication Technology Co., Ltd., Dalian, Liaoning Province, China, a a technical director in 00. e i currently an aociate profeor at the School of Information Science and Engineering, Dalian Polytechnic Univerity, China. i reearch interet include wirele/mobile communication ytem and Internet of thing. Guiyue Jin received her PhD degree in computer engineering from Yeungnam Univerity, Gyeongan, Rep. of Korea, in 009. She i currently an aitant profeor at the School of Information Science and Engineering, Dalian Polytechnic Univerity, China. er reearch interet include pervaive computing and wirele communication. Peng Li received hi PhD degree in information and communication engineering from arbin Intitute of Technology, China, in 009. e i currently an aociate profeor at the School of Information Science and Engineering, Dalian Polytechnic Univerity, China. i reearch interet include ad hoc network and optical communication ytem. 4 Xing Ouyang et al. ETRI Journal, Volume 37, umber, February 05