1 Introduction

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1 Published in IET Communications Received on 25th November 2008 Revised on 15th December 2009 Correlated noise cancellation in fractional sampling orthogonal frequency and code division multiplexing with alternative spreading code M. Inamori 1 H. Nishimura 1 Y. Sanada 1 M. Ghavami 2 1 Department of Electronics and Electrical Engineering, Keio University, Hiyoshi, Kohoku, Yokohama, Kanagawa , Japan 2 King s College London, Division of Engineering, UWB Communications Group, Strand, London WC2R 2LS, UK mamiko@snd.elec.keio.ac.jp ISSN Abstract: Orthogonal frequency and code division multiplexing (OFCDM) has received large attention as a modulation scheme to realise high data rate transmission. The OFCDM system with fractional sampling (FS) is investigated. FS is a diversity scheme with a single antenna, which achieves path diversity through oversampling and parallel signal reception. However, the correlation among noise components may deteriorate the bit error rate performance at the receiver as the number of subcarriers and oversampling ratio increases. To overcome this problem, the alternative spreading code is used for the FS OFCDM system. Numerical results through computer simulation show that the proposed scheme can improve the performance of the receiver with the large oversampling ratio and the number of subcarriers. 1 Introduction Orthogonal frequency and code division multiplexing (OFCDM) has received large attention as a modulation scheme to realise high data rate transmission based on code division multiple access (CDMA) technique [1 3]. The OFCDM system transmits signals using more than 1000 subcarriers that are orthogonally overlapped in the frequency domain. On the other hand, various diversity schemes have been actively investigated for the OFCDM-based system [4, 5]. One of the typical diversity schemes is the antenna diversity in which multiple antenna elements are implemented at the receiver [4]. However, it may be difficult to implement multiple antenna elements in small devices. Therefore a new diversity scheme called fractional sampling (FS) has been proposed in [6]. This scheme tries to acquire diversity gain through the signal sampled faster than the Nyquist rate at the receiver and achieves path diversity. In [6], subcarrier-based noise whitening and maximal ratio combining (MRC) have been investigated because of their low complexity. However, as the number of subcarriers and the oversampling ratio increase, the correlation among the noise components over different subcarriers deteriorates the bit error rate (BER) performance. To solve this problem, an OFCDM system with an alternative spreading code is investigated in this paper. This spreading code has positive and negative components, alternatively. Therefore the OFCDM system with the alternative spreading code can cancel the correlated noise components. The performance of the FS OFCDM system with alternative spreading code will also be evaluated through computer simulation. We start with the brief description of the system model in Section 2. Section 3 explains the effects of correlated noise components in the FS OFCDM system. The proposed scheme is then discussed in Section 4. Numerical results IET Commun., 2010, Vol. 4, Iss. 10, pp & The Institution of Engineering and Technology 2010

2 are demonstrated in Section 5. Finally, conclusions are presented in Section 6. 2 System model 2.1 Transmitter model Fig. 1a shows the block diagram of an OFCDM transmitter [1, 2]. The input data are modulated with quadrature phase shift keying (QPSK) and is serial-to-parallel (S/P) converted to N /S f parallel sequences, where N denotes the number of subcarriers and S f denotes the spreading factor in the frequency domain. Each modulated symbol is duplicated into S f parallel copies. Each branch of the symbol stream is then multiplied by a chip from the spreading code with the repetition period of S f, which is represented as s[(x 1)S f + i] = d[x]q i 1 x N /S f, 0 i S f 1 where s[(x 1)S f + i]istheith spread data component of the xth data symbol transmitted over the [(x 1)S f ]th subcarrier, d[x] isthexth data symbol and q i is the ith spreading code. The spread data sequence is modulated to the multi-carrier signal by inverse discrete fourier transform (IDFT), and the guard-interval (GI) is inserted to the modulated signal. 2.2 Receiver structure with fractional sampling At the receiver side, FS and MRC are used to achieve diversity over a multipath channel [6]. The block diagram of an OFCDM receiver with FS is shown in Fig. 1b. (1) The transmitted signal with the GI, u[l ], is given as u[l] = 1 N 1 s[k]e j2pkl/n, l = 0,..., P 1 (2) N k=0 where N is the IDFT length, s[k] is the symbol transmitted on the kth subcarrier, P is the sum of the IDFT length and the length of the GI. The received signal, y(t), is expressed as follows y(t) = P 1 u[l]h(t lt s ) + v(t) (3) l=0 where 1/T s is the baud rate, h(t) is the impulse response of the composite channel and is given by h(t) = (p w c w p )(t), w denotes convolution, p(t) is the impulse response of the pulse-shaping filter, p (t) = p( t), c(t) is the impulse response of the physical channel and v(t) is the additive white Gaussian noise [6]. The received signal which is sampled at a rate of G/T s is expressed as follows y g [n] = P 1 u[l]h g [n l] + v g [n], g = 0,..., G 1 l=0 where n is the time index, y g [n] = y(nt s + gt s /G), h g [n] = h(nt s + gt s /G) and v g [n] = v(nt s + gt s /G). The demodulated signal received on the kth subcarrier, z[k], is derived after the removal of the GI and demodulation by the DFT at the receiver for each g. z[k] is expressed as where (4) z[k] = H[k]s[k] + w[k], k = 0,..., N 1 (5) z[k] = [z 0 [k],..., z G 1 [k]] T (6) z g [k] = 1 N 1 y g [n]e j2pkn/n (7) N n=0 H[k] = [H 0 [k],..., H G 1 [k]] T (8) H g [k] = L 1 h g [n]e j2pkn/n (9) n=0 w[k] = [w 0 [k],..., w G 1 [k]] T (10) w g [k] = N 1 n=0 and L is the number of multipath. v g [n]e j2pkn/n (11) Figure 1 Transmitter and receiver models a OFCDM transmitter block diagram b OFCDM receiver block diagram When sampling at the receiver is carried out at the baud-rate of 1/T s, we have a usual OFDM input/output relationship with white noise. However, when sampling is performed at the multiple of the baud rate, the noise is coloured. Noise whitening is necessary because MRC 1218 IET Commun., 2010, Vol. 4, Iss. 10, pp & The Institution of Engineering and Technology 2010

3 maximises the signal-to-noise ratio (SNR) when the noise is white. In order to take subcarrier-by-subcarrier MRC combining approach, subcarrier-based noise whitening is carried out. The covariance matrix of the noise on the kth subcarrier is given as R w [k] = E[w[k]w H [k]] (12) where E[ ] denotes the expectation and H represents Hermitian transpose. After noise whitening, (5) is converted as R 1/2 w [k]z[k] = R 1/2 w [k]h[k]s[k] + R 1/2 w [k]w[k] (13) This equation turns to the following expression z [k] = H [k]s[k] + w [k] (14) where R 1/2 w [k]z[k] = z [k], R 1/2 w [k]h[k] = H [k], and [k]w[k] = w [k]. The estimate of s[k], ŝ[k], through R 1/2 w MRC is then given as ŝ[k] = H H [k]z [k] H H [k]h [k] w [k]h[k]) H R 1/2 w [k]z[k] w [k]h[k]) H Rw 1/2 [k]h[k] = (R 1/2 (R 1/2 (15) The combined signal is despread and demodulated, which is represented as ˆd[x] = S f 1 ŝ[(x 1)S f + i]q i (16) i=0 3 Effects of correlated noise In order to investigate the effect of the noise whitening, the received signal is expressed in the vector form in this section. From (5), the received signal for all N subcarriers is expressed as where z = Hs + w (17) z = [z T [0],..., z T [N 1]] T (18) H = diag[h[0],..., H[N 1]] (19) s = [s[0],..., s[n 1]] T (20) w = [w T [0],..., w T [N 1]] T (21) The coloured noise vector, w, can be expressed as w = R 1/2 w v (22) where v is the white noise in the vector form given as v = [v T [0],..., v T [N 1]] T (23) v[k] = [v 0 [k],..., v G 1 [k]] T (24) and v g [k] is the white noise of the gth sample component on the kth subcarrier. The noise covariance matrix is R w := E[ww H ]whose(k 1 G + g 1, k 2 G + g 2 )th element is given by E[w g1 [k 1 ]w g2 [k 2 ]] = s 2 1 v N N 1 N 1 n 1 =0 n 2 =0 p 2 ((n 2 n 1 + (g 2 g 1 )/G)T s ) e j(2p/n )(k 2 n 2 k 1 n 1 ) (25) where p 2 (t) is the composite response of the filters given as p 2 (t) = (p w p)(t), s 2 v is the variance of v(t), {k 1, k 2 } = 0,..., N 1 and {g 1, g 2 } = 0,..., G 1. After subcarrierbased noise whitening, (17) is converted as R ww z = R ww Hs + R ww w (26) where R ww = diag[rw 1/2 [0],..., Rw 1/2 [N 1]]. Equation (26) results in the following equation where and z = R ww z z = H s + w (27) = [z T [0],..., z T [N 1]] T (28) H = R ww H = diag[h [0],..., H [N 1]] (29) w = [w T [0],..., w T [N 1]] T = R ww w = R ww R 1/2 w v I G R n [0, 1]... R n [0, N 1]. R n [1, 0] I.. G. = R n [N 1, 0] I G v[0] v[1]. (30) v[n 1] where R n [k 1, k 2 ]istheg G matrix, which corresponds to the (k 1, k 2 )th subblock of the NG NG matrix, R ww R 1/2 w.the IET Commun., 2010, Vol. 4, Iss. 10, pp & The Institution of Engineering and Technology 2010

4 g 1 th element of w [k] is expressed as w g 1 [k 1 ] = N 1 G 1 k 2 =0 g 2 =0 = v g1 [k 1 ] + [R n [k 1, k 2 ]] g1,g 2 [k 2 ] N 1 G 1 k 2 =0 g 2 =0 [R n [k 1, k 2 ]] g1,g 2 [k 2 ] (31) where [R n [k 1, k 2 ]] g1,g 2 is the (g 1, g 2 )th element of R n [k 1, k 2 ]. The second term of the right side of this equation gives the correlation between the noise components after subcarrierbased noise whitening. 4 Proposed scheme 4.1 Despreading with non-alternative spreading code Suppose that the spreading code with the following condition is employed q i = q i+1 = =q i+sf 1 (32) From (31), the correlated noise after despreading from the kth to the (k + S f 1)th subcarriers, g non [k, k + S f 1], is expressed as g non [k, k + S f 1] S f = +k 1 G 1 k 1 =k = G 1 g 1 =0 N 1 k 2 =0 N 1 G 1 g 1 =0 k 2 =0 g 2 =0 G 1 g 2 =0 H g 1 [k 1 ][R n [k 1, k 2 ]] g1,g 2 [k 2 ] (H g 1 [k][r n [k, k 2 ]] g1,g 2 + H g 1 [k + 1][R n [k + 1, k 2 ]] g1,g H g 1 [k + S f 1][R n [k + S f 1, k 2 ]] g1,g 2 ) [k 2 ] (33) The correlation among the noise components, R 1/2 w,over the subcarriers and oversampling indexes (k 1 G + g 1 100, k 2 G + g 2 100) is shown in Fig. 2, where the nondiagonal elements of this matrix are almost periodic over the neighbouring indexes. Moreover, from (25), if the number of subcarriers increases and the spreading factor is small enough, the following approximation on the noise covariance matrices can be assumed. R 1/2 w [k] R 1/2 w [k + 1]... R 1/2 w [k + S f 1] (34) Figure 2 Correlation of the noise components (logarithmic representation of absolute value) Thus, from (34) [R n [k, k 2 ]] g1,g 2 [R n [k + 1, k 2 ]] g1,g 2 [R n [k + S f 1, k 2 ]] g1,g 2 (35) If the correlation of the channel responses of the subcarriers is high, the following approximation is also derived H g 1 [k] H g 1 [k + 1] H g 1 [k + S f + 1] (36) From (35) and (36), the correlated noise after despreading can be approximated as follows g non [k, k + S f 1] G 1 N 1 G 1 g 1 =0 k 2 =0 g 2 =0 S f (H g 1 [k][r n [k, k 2 ]] g1,g 2 ) [k 2 ] (37) If the size of the matrix and the spreading factor increases, the total amount of the correlated noise in (31) grows. Therefore the BER performance is deteriorated with the correlated noise component. 4.2 Despreading with alternative spreading code If the number of the subcarriers and the oversampling ratio increases, the total amount of the correlated noise in (31) grows. To solve this problem, a spreading code that has the following property is used q 2i = q 2i+1, i = 0, 1,..., S f /2 1 (38) This code is referred to as an alternative spreading code in this paper. The one way to generate the alternative spreading code is that half length of the sequences are used and then doubled in length using this relation, for example, { } is converted to { } IET Commun., 2010, Vol. 4, Iss. 10, pp & The Institution of Engineering and Technology 2010

5 In general, if c is the original code with length S f /2, then the alternative code can be constructed as where T is the S f S f /2 matrix q = Tc (39) T = (40) From (15) and (35), the correlated noise is approximated as g alt [k, k + S f 1] = S f +k G 1 N 1 G 1 k 1 =k g 1 =0 k 2 =0 g 2 =0 q (k1 k) H g 1 [k 1 ] [R n [k 1, k 2 ]] g1,g 2 [k 2 ] (41) From (36), (38) and (41), the correlated noise after despreading from the kth to (k + S f 1)th subcarriers, g alt, is given as g alt [k, k + S f 1] S f +k G 1 k 1 =k g 1 =0 N 1 H g 1 [k] ( 1) k 2 =0 (k k 1 ) G 1 g 2 =0 [R n [k 1, k 2 ]] g1,g 2 [k 2 ] (42) The inside of the braces in (42) cancels between the k 1 th and (k 1 + 1)th subcarriers because of the element ( 1) (k k 1 ). This element is based on the property of the alternative spreading code given in (38). Thus, despreading with the alternative spreading code cancels most of the correlated noise components. However, the drawback of this scheme is that the number of available spreading codes reduces to half. This is also due to the constraint shown in (38). 5 Numerical results 5.1 Simulation conditions The FS OFCDM system with the alternative spreading code is evaluated through computer simulation. Simulation conditions are shown in Table 1. The data are modulated with QPSK. The received signal is sampled at the rates of 1/T s,2/t s and 4/T s. The spreading factor, S f, is set from 2 to 16 in this simulation. Channel estimation is assumed to be ideal. Here, two channel models are considered. One is a 16-path Rayleigh fading Table 1 Simulation conditions bandwidth 80 MHz number of subcarriers 256/512/1024 guard interval 12.8/25.6/51.2 (ms) subcarrier spacing Df 78.1/39.1/19.5 (khz) number of IDFT points 256/512/1024 DFT sampling speed T s 12.5 (ns) data modulation QPSK channel estimation ideal fractional sampling ratio G 1,2,4 spreading factor S f 2/4/8/16 channel model Rayleigh fading 16-path uniform/24-path exponential model with a uniformed delay profile as shown in Fig. 3a [6]. The interval between the path delays in this model is T s /4. The other one is a 24-path Rayleigh fading model with an exponential delay profile as shown in Fig. 3b [2]. Theinterval between the path delays is 5T s. The composite impulse response of the transmit and receive pulse shaping filters is assumed to be a sinc pulse with a duration of 2T s [6]. Fig. 4 shows the frequency response of the pulse shaping filters with the impulse response of the rectangular pulse and the truncated sinc pulse. If the rectangular pulse is applied, equals 0, which means no correlated noise components. Therefore the BER performance with the rectangular pulse is equivalent to that of asimomodel[7]. the non-diagonal elements of R 1/2 w 5.2 BER improvement with alternative spreading code Figs. 5 and 6 show the BER performance on the 16- and 24-path Rayleigh fading channel models, respectively. The number of subcarriers is 1024 and the spreading factor is 2. Numerical results of the 1 4 SIMO model are shown in the same figure as a reference. This SIMO model is assumed to receive uncorrelated signals at each antenna and combine those signals with MRC. From these figures, when the oversampling ratio is 1 or 2, the BERs with both alternative and non-alternative spreading codes are almost the same. On the other hand, if the oversampling ratio is 4, the BER with the non-alternative spreading code is larger than that with the oversampling ratio of 2. This is due to the correlated noise components in (31). On the contrary, the BER with the alternative spreading code reduces as the oversampling ratio increases. This is because the alternative spreading code cancels the correlated noise components among the adjacent subcarriers as shown in (42). In Fig. 6, the improvement on the BER curve is limited in IET Commun., 2010, Vol. 4, Iss. 10, pp & The Institution of Engineering and Technology 2010

6 Figure 3 Multipath channel models a 16-path Rayleigh fading model with uniform delay spread b 24-path Rayleigh fading model with exponential delay spread comparison with Fig. 5. The reason is that the delay spread assumed in the exponentially decay model is larger. Therefore the fluctuation on the channel responses among the adjacent subcarriers is larger and the approximation in (36) becomes inaccurate. Thus, the residual component of the correlated noise after despreading limits the improvement of the BER performance. 5.3 Number of subcarriers In Figs. 7 and 8, the relationship between the BER and the number of subcarriers for both alternative and non-alternative spreading codes is presented. Here, E b /N 0 is set to 15[dB], the spreading factor is set to 2, and the oversampling ratio is set to G = {1, 2, 4}. In Fig. 8, the BERs are deteriorated when the number of subcarriers is 256. This is because the delay spread is large on the 24-path Rayleigh fading channel and some of the paths have larger delays than the GI. When the number of subcarriers is more than 512, the largest delay of the paths is accommodated within the GI. Figure 5 BER performance against E b /N 0 on the 16-path Rayleigh fading channel with the uniform delay profile (number of subcarriers: 1024, S f ¼ 2) Figure 4 PSD against normalised frequency with different pulse shapes Figure 6 BER performance against E b /N 0 on the 24-path Rayleigh fading channel with the exponential delay profile (number of subcarriers: 1024, S f ¼ 2) 1222 IET Commun., 2010, Vol. 4, Iss. 10, pp & The Institution of Engineering and Technology 2010

7 Figure 7 BER performance against number of subcarriers on the 16-path Rayleigh fading channel with the uniform delay profile (S f ¼ 2, E b /N 0 ¼ 15 (db)) Figure 9 BER performance against spreading factor S f on the 16-path Rayleigh fading channel with the uniform delay profile (number of subcarriers: 1024, E b /N 0 ¼ 15 (db)) Figure 8 BER performance against number of subcarriers on the 24-path Rayleigh fading channel with the exponential delay profile (S f ¼ 2, E b /N 0 ¼ 15 (db)) In both figures, the BERs of G = {1, 2} remain fairly constant. As the number of subcarriers increases, the BER with the non-alternative spreading code increases when G ¼ 4 and the number of subcarriers is This is because of the correlated noise components between the adjacent subcarriers. The reason is that the value of the element in R n [k, j] in (30) is of the order of and the total amount of the correlated noise components becomes close to that of the white noise. 5.4 Spreading factor S f In Figs. 9 and 10, the relationship between the BER and the spreading factor for both alternative and non-alternative Figure 10 BER performance against spreading factor S f on the 24-path Rayleigh fading channel with the exponential delay profile (number of subcarriers: 1024, E b /N 0 ¼ 15 (db)) spreading codes on the different channel models is presented. Here, E b /N 0 is set to 15[dB] and the oversampling ratio is set to G = {1, 2, 4}. The number of subcarriers is In Fig. 9, the BERs with G = {1, 2} remain fairly constant. As the spreading factor increases, the BER with the alternative spreading code decreases when G ¼ 4. This is due to the effect of frequency diversity. In contrast, when G ¼ 4, the BER with the non-alternative spreading code is deteriorated as the spreading factor increases. The reason behind is the correlated noise given in (37). If the spreading factor, S f, increases, the variance of the noise also grows with the factor of S 2 f. As for the results corresponding to the alternative spreading code, the BER is improved as the spreading factor IET Commun., 2010, Vol. 4, Iss. 10, pp & The Institution of Engineering and Technology 2010

8 increases. This is because the alternative spreading code can cancel the correlated noise. More diversity gain is obtained with the proposed spreading code in Fig. 10. This is due to the assumed channel model. Since the delay spread assumed in the exponential delay model is larger than that in the uniform decay model, the correlation among the channel responses of the subcarriers is smaller in Fig Spreading code In order to show the validity of the alternative spreading code, the relationship between the BER and G for various spreading codes on the two different channel models is presented in Figs. 11 and 12. Here, E b /N 0 is set to 15[dB] and the number of subcarriers is The first, second, third, fifth, and ninth rows of the Walsh-Hadamard matrix with the size of 16 are used as the spreading codes [2]. Table 2 shows the spreading code used in this simulation. The second row corresponds to the proposed alternative spreading code. Both Figs. 11 and 12 show that BERs with G = {1, 2} are almost the same for all spreading codes. Moreover, the figures also show that the alternative spreading code gives the best result when G ¼ 4 the worst result is given by the first row (non-alternative spreading code). The results with the other codes (third, fifth, ninth) are spread between them. The reason is that the pair of Figure 11 BER performance against G with different spreading codes on the 16-path Rayleigh fading channel with the uniform delay profile (number of subcarriers: 1024, E b /N 0 ¼ 15 (db)) Figure 12 BER performance against G with different spreading codes on the 24-path Rayleigh fading channel with the exponential delay profile (number of subcarriers: 1024, E b /N 0 ¼ 15 (db)) the blocks of 1 and 21 in the spreading code cancels the correlated noise components if the channel responses corresponding to those blocks are sufficiently close. As the size of the block reduces, the difference between the corresponding channel responses in those blocks decreases and the residual of the correlated noise components diminishes. 5.6 Multiuser environment In Fig. 13, the BER with both the alternative and nonalternative spreading codes on the 24-path Rayleigh fading channel model is evaluated in multiuser environment, where the number of users is varied. Here, E b /N 0 is 15[dB] and the number of subcarriers is It is also assumed that the Walsh-Hadamard codes with the length of 16 is employed. The first and second row of the matrix is used as the non-alternative and alternative spreading codes, respectively. From Fig. 13, it is observed that the BERs with both the alternative and non-alternative spreading codes are deteriorated in accordance with the increase in the number of users. This is because the delay spread assumed in this channel model is relatively large and the orthogonality among code-multiplexed channels is degraded. As a result, the effect of the inter-code Table 2 Spreading code first row second row third row fifth row ninth row IET Commun., 2010, Vol. 4, Iss. 10, pp & The Institution of Engineering and Technology 2010

9 and Technology in Japan. This work is also supported in part by the KDDI Foundation. M. Ghavami gratefully acknowledges the financial support provided by the Japan Society for the Promotion of Science (JSPS) in the preparation of this work. 8 References [1] ATARASHI H., SAWAHASHI M.: Investigation of inter-carrier interference due to Doppler spread in OFCDM broadband packet wireless access, IEICE Trans. Commun., 2002, E85- B, (12), pp Figure 13 BER performance agaisnt number of users on the 24-path Rayleigh fading channel with the exponential delay profile (number of subcarriers: 1024, E b /N 0 ¼ 15 (db)) interference is more significant for a large number of users. It is clear from the figure that the proposed scheme reduces the BER performance in the multiuser environment when the number of users is less than 8. 6 Conclusions The FS OFCDM system with the alternative spreading code has been investigated in this paper. In the FS OFCDM system, the correlation between the noise components may deteriorate the BER performance at the receiver with the increase of the number of subcarriers and oversampling ratio. The proposed spreading code mitigates the effect of the correlated noise components and improves the BER performance, especially when the oversampling rate is 4. It has also been shown that the FS OFCDM system with the alternative spreading code can obtain frequency diversity effect. 7 Acknowledgments This work is supported in part by a Grant-in-Aid for the Global Center of Excellence for high-level Global Cooperation for Leading-Edge Platform on Access Spaces from the Ministry of Education, Culture, Sport, Science [2] ATARASHI H., ABETA S., SAWAHASHI M.: Variable spreading factor orthogonal frequency and code division multiplexing (VSF-OFCDM) for broadband packet wireless access, IEICE Trans. Commun., 2003, E86-B, (1), pp [3] KISHIYAMA Y., MAEDA N., HIGUCHI K., ATARASHI H., SAWAHASHI M.: Field experiments on throughput performance above 100 Mbps in forward link for VSF-OFCDM broadband wireless access, IEICE Trans. Commun., 2005, E88-B, (2), pp [4] MAEDA N., ATARASHI H., ABETA S., SAWAHASHI M.: Antenna diversity reception appropriate for mmse combining in frequency domain for forward link OFCDM packet wireless access, IEICE Trans. Commun., 2002, E85-B, (10), pp [5] MIKI N., ATARASHI H., SAWAHASHI M.: Effect of time diversity in hybrid ARQ considering space and path diversity for VSF-OFCDM downlink broadband wireless access. Proc. PIMRC 04, Barcelona, Spain, September 2004, vol. 1, pp [6] TEPEDELENLIOGLU C., CHALLAGULLA R.: Low-complexity multipath diversity through fractional sampling in OFDM, IEEE Trans. Signal Process., 2004, 52, (11), pp [7] KATO Y., INAMORI M., SANADA Y.: Multipath diversity through fractional sampling in MB-OFDM. Proc. APWCS 08, Sendai, Japan, August 2008 IET Commun., 2010, Vol. 4, Iss. 10, pp & The Institution of Engineering and Technology 2010