Non-uniform sampling point selection in orthogonal frequency division multiplexing receiver with fractional sampling

Transcription

1 Published in IE Communications Received on 6th January 2010 doi: /iet-com ISSN Non-uniform sampling point selection in orthogonal frequency division multiplexing receiver with fractional sampling H Nishimura 1 M Inamori 1 Y Sanada 1 M havami 2 1 Department of Electronics and Electrical Engineering, Keio University, Hiyoshi, Kohoku, Yokohama, Kanagawa , Japan 2 Division of Engineering, UWB Communications roup, King s College London, Strand, London WC2R 2LS, UK harunishimuraki@sndeleckeioacjp Abstract: o achieve diversity with a single antenna, a fractional sampling (FS) scheme in OFDM has been proposed FS achieves path diversity with a higher sampling rate and multiple demodulation branches In the previous literatures for FS, the sampling interval is fixed even though the initial phase selection of the sampling has been investigated he fixed sampling interval limits the diversity gain through FS hus, in this study, a non-uniform sampling point selection scheme according to the frequency response of a channel is proposed Numerical results through computer simulation show that the proposed scheme improves the BER performance by about 2 db when the number of the demodulation branches is 2 While the proposed scheme further improves bit error rate (BER) performance it requires a larger rate and more computational complexity for the selection of the sampling points herefore the complexity reduction for the sampling point selection is also investigated he complexity reduction scheme eliminates the specific sets of the sampling points and reduces the computational complexity of the sampling point selection for the proposed scheme by a factor of 2 Computer simulation also shows that the low complexity version of the proposed scheme even achieves the equivalent BER compared to the original proposed scheme 1 Introduction Orthogonal frequency division multiplexing (OFDM) is a very attractive technique for reliable broadband wireless communications such as WLAN IEEE80211a/g, WiMAX etc [1 3] his is because of the efficient use of frequency spectrum and robustness to multipath channels with the use of multiple subcarriers o overcome the effect of signal fading in a multipath environment, multiple antennas are used at a receiver to achieve antenna diversity [4, 5] However, it may be difficult to implement multiple antenna elements in a small terminal [4] herefore to achieve diversity with a single antenna, a fractional sampling (FS) scheme in OFDM has been proposed [6] In the FS scheme, a received baseband signal is sampled with a rate higher than the Nyquist rate and they are combined to achieve path diversity on each subcarrier [6] Further, the effect of the pulse shaping filter in FS has been investigated in [6 8] It has been shown that the excessive bandwidth of the filter realises path diversity in FS In [9], a sampling rate selection algorithm for FS is proposed In this scheme, the sampling rate is selected according to the frequency response of a channel In [10], an initial phase of the sampling is selected according to the frequency response of the channel In [11], the initial phase selection of the sampling has been evaluated through experiments In these previous literatures for FS, the sampling interval is fixed [6 11] he selection of the sampling points makes the difference in the frequency response It is then possible to achieve lower BER performance as compared to the conventional schemes in [9] or [10] In other words, the number of demodulation branches in FS can be reduced for the same BER In this paper, a novel non-uniform sampling point selection scheme for FS based on the frequency response of the channel is proposed he proposed scheme fixes neither the initial sampling point nor the sampling interval Most of the bit errors occur on the subcarrier experiencing the deepest fade he proposed scheme selects the sampling points so that the smallest frequency responses of the subcarriers is maximised to improve the average BER his paper is organised as follows Section 2 explains the diversity scheme through FS he conventional and proposed sampling point selection schemes are also described he simulation results are shown in Section 3 Finally, conclusions are presented in Section 4 2 System model 21 Fractional sampling he block diagram of an OFDM system with FS is shown in Fig 1 [6] Suppose the information symbol on the kth subcarrier is s[k](k ¼ 0,, N 2 1), the OFDM symbol is 554 IE Commun, 2011, Vol 5, Iss 4, pp & he Institution of Engineering and echnology 2011 doi: /iet-com

2 where y g [n], h g [n] and v g [n] (g ¼ 0,, 2 1) are polynomials of sampled y(t), h(t) and v(t), respectively, and are expressed as y g [n] := y(n s + D g s /) (4) h g [n] := h(n s + D g s /) (5) v g [n] := v(n s + D g s /) (6) where s / is the step size of the initial timing and D g is the normalised gth sampling point D g satisfies the following inequation 0 D 0, D 1,, D 1 1 (7) Fig 1 then given as OFDM receiver using FS u[n] = 1 N 1 s[k]e j(2pnk/n) (1) N k=0 where n(n ¼ 0, 1,, N 2 1) is the time index A guard interval (I) is appended before transmission N I is the length of the I he baseband signal at the output of the filter is given by x(t) = N+N I 1 u[n]p(t n s ) where p(t) is the impulse response of the baseband filter and s is the symbol duration his signal is upconverted and transmitted through a multipath channel with an impulse response c(t) he received signal after the down conversion is given as y(t) = N+N I 1 u[n]h(t n s ) + v(t) (2) where h(t) is the impulse response of the composite channel and is given by h(t) := p(t) w c(t) w p( t) and v(t) is the additive aussian noise filtered at the receiver For the multipath channel, h(t) can be expressed in a baseband form as h(t) = N m 1 a i p 2 (t t i ) (3) i=0 where p 2 (t) := p(t) w p( t) is the deterministic correlation of p(t) (w indicates convolution operator) and satisfies Nyquist s property It is assumed that the channel in (3) has N m path components, a i is the amplitude that is timeinvariant during one OFDM symbol (quasi-static channel model) and t i is the path delay If y(t) is sampled at the rate of s /, where is the oversampling rate, its polyphase components can be expressed as y g [n] = N+N I 1 l=0 u[l]h g [n l] + v g [n] After removing the I and taking DF at each branch, the received symbol is given by z[k] = H[k]s[k] + w[k] (8) where z[k] = [z 0 z 1 ], H[k] = [H 0 H 1 ] and w[k] = [w 0 w 1 ] are 1 column vectors, each gth component representing [z[k]] g := z g [k] = N 1 y g [n]e j(2pkn/n) (9) [H[k]] g := H g [k] = N 1 h g [n]e j(2pkn/n) (10) [w[k]] g := w g [k] = N 1 v g [n]e j(2pkn/n) (11) respectively As already stated, v(t) in (2) is filtered by the pulse shaping filter in the receiver, p(t) he power spectrum of the filter, P 2 (f ), has the following relation p 2 (t) = 1 1 P 2 (f )e j2pft df (12) Since the white noise is assumed to be input to the filter, the power spectrum of the filtered noise, V( f ), is expressed as and its autocorrelation is given as V( f ) = s 2 vp 2 ( f ) (13) R(t) = E[v(t)v (t + t)] = 1 = s 2 v 1 1 V (f )e j2pf t df 1 P 2 (f )e j2pf t df = s 2 vp 2 (t) (14) Suppose p 2 (t) satisfies the following condition p(n s ) = { 1 (n = 0) 0 (n = 0) (15) IE Commun, 2011, Vol 5, Iss 4, pp doi: /iet-com & he Institution of Engineering and echnology 2011

3 and the band limited noise, v(t), is sampled at the baud rate of 1/ s, the samples of v(t) are independent of one another However, when the sampling rate increases, the noise samples are correlated and noise-whitening is required to maximise the SNR [12] he whitening filter equalises the spectrum of the correlated noise samples [13] In order to perform noise-whitening, it is necessary to calculate a noise covariance matrix on the kth subcarrier, R w [k] he (g 1, g 2 )th element of the noise covariance matrix on the kth subcarrier is given as [R w [k]] g1 g 2 = E[w g1 w g 2 ] [( = 1 N 1 N E ( ) ) v n 1 s + D s g1 e j(2pkn 1/N) n 1 =0 ( ( ) )] N 1 v n 2 s + D s g2 e +j(2pkn 2 /N) n 2 =0 = 1 [ ( ) N 1 N 1 E v n N 1 s + D s g1 v n 1 =0 n 2 =0 ( )] n 2 s + D s g2 e j(2pk(n 2 n 1 )/N) (16) Using (14) [ ( ) ( )] E v n 1 s + D s g1 v n 2 s + D s g2 (( ) ( )) = s 2 vp 2 n 2 s + D s g2 n 1 s + D s g1 = s 2 vp 2 ((n 2 n 1 + (D g2 D g1 )/) s ) (17) hus, (16) is expressed as [R w [k]] g1 g 2 = s 2 v 1 N 1 N N 1 n 1 =0 n 2 =0 e j(2pk(n 2 n 1 )/N) p 2 ((n 2 n 1 + (D g2 D g1 )/) s ) and multiplying both sides of (8) by Rw (1/2) [k] (18) R (1/2) w [k]z[k] = R (1/2) w [k](h[k]s[k] + w[k]) (19) z [k] = H [k]s[k] + w [k] (20) where z [k] = Rw 1/2 [k]z[k], H [k] = Rw (1/2) [k]h[k] and w [k] = R (1/2) w [k]w[k] Since R w [k] is the Hermitian matrix, the covariance of the noise vector after noisewhitening, w [k], is E[w [k]w H [k]] = R (1/2) w [k]e[w[k]w[k]]r (H/2) = R (1/2) w R w [k]rw (H/2) = I (21) each subcarrier are obtained through FS Based on the frequency responses of the subcarriers estimated with the preamble symbols, those samples are combined to maximise the SNR expressed as the following equation [6] ŝ[k] = H H [k]z [k] H H [k]h [k] = (R (1/2) w [k]h[k]) H (R (1/2) w [k]z[k]) (Rw (1/2) [k]h[k]) H (Rw (1/2) [k]h[k]) where {} H is the Hermitian operator 22 Sampling point selection (22) In the preamble period, the received signal is sampled at the rate of times higher than the Nyquist rate ( ) sampling points out of are selected according to the frequency response of a channel, which is estimated with the preamble symbols of a packet In the data period, the received signal is sampled at the timing selected by the sampling point selection schemes In this paper, two sampling point selection schemes are evaluated 221 Channel estimation: In the preamble period, the received preamble on the kth subcarrier is expressed as z p [k] = H p [k]s p [k] + w p [k] (23) where z p [k]is 1 vector of the received preamble, H p [k]is the 1 vector of the frequency response of the channel, s p [k] is the preamble symbol that is known at the receiver side and w p [k] is the 1 noise vector In this paper, the frequency response of the channel, Ĥ[k], is estimated as follows Ĥ p [k] = Ĥ p 0[k] Ĥ p 1[k] Ĥ p 1[k] = z p [k] s p [k] = H p [k] + wp [k] s p [k] (24) In WLAN, for example, the estimation is carried out in two long preamble symbols [1, 2] and they are averaged together After the sampling point selection, the frequency response of the channel with the selected sampling points during the data period, H[k], is expressed as H 0 [k] Ĥ p D 0 [k] H 1 [k] Ĥ p H[k] = = D 1 [k] H 1 [k] Ĥ p D 1 [k] (25) where D g (g = 0, 1,, 1) indicates the gth sampling point defined in (7) 222 Conventional scheme: In the conventional scheme, uniform sampling interval is assumed and the initial phase of the sampling is selected he sampling point in (5) is given as where I is a identity matrix From (21), the noise samples are whitened After whitening, multiple outputs on D g (g u ) = D 0 (g u ) + g (26) 556 IE Commun, 2011, Vol 5, Iss 4, pp & he Institution of Engineering and echnology 2011 doi: /iet-com

4 where D 0 (g u ) = g u (/) indicates the initial phase of the sampling and g u (0 g u (/) 1) is the corresponding index Since the received signal is sampled at the oversampling rate of in the data period, there are / candidates of the initial phase Fig 2a shows the candidates of the initial phase when ¼ 2, as an example In the figure, and represent the samples corresponding to g ¼ 0 and 1, respectively As shown in Fig 2a, the number of possible initial phase of the sampling is / ¼ 4 he most of the bit errors occur on the subcarrier experiencing the deepest fade In the conventional scheme, the index, g u, for the initial phase is selected based on the frequency responses of the subcarriers according to the following expression argmax{min P [k, g u ]} (27) g u k where P [k, g u ] = H H [k, g u ]H [k, g u ] and H [k, g u ] is the frequency response on the kth subcarrier when the index of the initial phase is g u [10] 223 Proposed scheme: In the conventional schemes, the interval between the sampling points is fixed he proposed scheme employs non-uniform sampling point interval In the data period, the number of the possible combinations of the sampling points is C =!/!( )! Fig 2b shows the candidates of the sampling points when ¼ 2, as an example In this figure, and represent the samples corresponding to g ¼ 0 and g ¼ 1, respectively As shown in Fig 2b, the number of the possible combinations of sampling points is C = 28 In this figure, g n is the index for the combination of the sampling points, S (g n ) = {D g }(g = 0, 1,, 1) and 0 g n C 1 (28) Most of the bit errors occur on the subcarrier experiencing the deepest fade In the proposed scheme, the index, g n, for the combination of the sampling points is selected based on the frequency responses according to the following evaluation function argmax{min P [k, g n ]} (29) g n k where P [k, g n ] = H H [k, g n ]H [k, g n ] and H [k, g n ] is the frequency response on the kth subcarrier for g n 23 Complexity reduction in non-uniform sampling point selection he complexity of the non-uniform sampling point selection scheme is relatively large o select the sampling point, it needs to calculate 8 C 2 = 28 or 8 C 4 = 70 frequency responses for ¼ 2 or 4, respectively o reduce the amount of computational complexity, it is desirable to eliminate the combinations of the sampling points which cannot achieve path diversity Correlation between subcarrier responses is shown in Fig 3, which is obtained by E[H g1 [k]hg [C[k]] Dg1,D g2 = 2 [k]] E[H g1 [k]hg 1 [k]] E[H g2 [k]hg 2 [k]] (30) From (30), when two given sampling points are next to each other, which means D g1 D g2 =1 or 7, the correlation is quite high In this case, diversity gain is relatively small he sampling point indices that include the highly correlated samples are eliminated from the candidates in the proposed sampling point selection scheme he number of the candidates reduces from 8 C 2 = 28 to 20 for ¼ 2 and from 8 C 4 = 70 to 2(¼conventional scheme) for ¼ 4 Fig 2 Candidates of sampling point ( ¼ 8, ¼ 2) a Uniform sampling b Non-uniform sampling Fig 3 Correlation between samples (32 path Rayleigh) IE Commun, 2011, Vol 5, Iss 4, pp doi: /iet-com & he Institution of Engineering and echnology 2011

5 3 Numerical results 31 Simulation conditions Simulation conditions are presented in able 1 he parameters in the table are based on the IEEE80211a/g standard [1, 2] he frequency response of the channel is assumed to be constant during one OFDM packet he structure of the OFDM packet is shown in Fig 4 Relevant to the IEEE80211a/g standard, the number of subcarriers, N, is 64 (48 subcarriers are for data symbols, four subcarriers are for pilot symbols and the rests are null subcarriers) [1, 2] For channel estimation (CE), two symbols, 1 and 2 are used as shown in Fig 4 and averaged together In the simulation, perfect and imperfect CEs for equalisation are investigated, respectively For the imperfect CE, the frequency responses are estimated with the preamble symbols, 1 and 2 Ideal carrier recovery is assumed here he total response of the transmitter and receiver filters is assumed to be truncated sinc pulse with the duration of 2 s [6] he oversampling rate in the preamble period is fixed to ¼ 8 In the data period, g u and g n are selected according to the evaluation functions in (27) and (28) In this simulation, the oversampling ratio at the data period,, is selected among {1, 2, 4} As channel models, 32 path Rayleigh fading with uniform delay profile and Indoor Residential A are assumed [14] Figs 5, 9 and 11 show the BER performance for the above channel models with perfect CE he BER curves with imperfect CE are shown in Fig 10 In the figures, Fixed means the BER curves with the fixed initial sampling point (n 0 = 0, g u = 0) Conv denotes the BER curves for the conventional scheme in (27) Pro indicates the BER curves with the proposed scheme in (29) he low complexity version of the proposed scheme is indicated as Low-Comp Fig 5 32 path Rayleigh fading model 32 Uniform sampling point selection and non-uniform sampling point selection Fig 5 shows the BER performance on the 32 path Rayleigh fading channel with perfect CE he conventional uniform sampling scheme improves the BER when ¼ 1 whereas the improvement is less significant when ¼ 2 he BER curve of the proposed scheme with ¼ 2 is slightly better as compared to that of the fixed sampling point scheme with ¼ 4 On the other hand, the BER curve of the able 1 Simulation conditions modulation scheme first: QPSK second: OFDM FF size 64 number of subcarriers 64 number of data subcarriers 48 number of OFDM packets per trial number of OFDM symbols per packet 10 bandwidth of subcarriers 3125 khz preamble length (I + preamble) ms OFDM symbol length (I + data) ms CE perfect Imperfect (estimated by t 1 and t 2 ) Fig 6 Frobenius norm of R w 2(1/2) [k] Fig 4 OFDM packet structure for simulation a g n ¼ g ns b g n ¼ g nl 558 IE Commun, 2011, Vol 5, Iss 4, pp & he Institution of Engineering and echnology 2011 doi: /iet-com

6 Fig 7 a g n ¼ g ns b g n ¼ g nl proposed scheme with ¼ 4 is worse than that with ¼ 2 his is because the noise generated from the other subcarriers through noise-whitening in (19) deteriorates the BER performance In Figs 7a and b, the power spectrum of the noise before and after whitening by (19) is shown when ¼ 4 and g n = g ns or g nl, respectively hose two sets of the sampling points, g ns and g nl, presented in Figs 6a and b have the smallest and largest average Frobenius norms of R (1/2) w Noise power spectrum [k] From Fig 7a, he whitening filter equalises the spectrum of the correlated noise samples when g n = g ns On the other hand, when g n = g nl, the noise power grows on the 18th and 48th subcarriers On these subcarriers, the Frobenius norm of Rw (1/2) [k] is also large as shown in Fig 6b he BER curves between g n = g ns and g nl when ¼ 4 are compared in Figs 8a and b In the Fig 8a, E b /N 0 is fixed to 20 db As exploited in the Fig 8a, if the Frobenius norm of w [k] is large, the BER also increases as shown in Fig 8b As given in Appendix, this is due to the correlated noise component after noisewhitening [15] R (1/2) 33 Complexity reduction in non-uniform sampling point selection path Rayleigh fading model: Fig 9 shows the numerical results on the 32 path Rayleigh fading channel with perfect CE he low-complexity scheme, ¼ 2 Low- Comp, achieves almost the same BER as that of ¼ 4 Fixed Channel correlation and noise correlation from the Fig 8 BER comparison of g n ¼ g ns and g nl a BER against data subcarrier E b /N 0 ¼ 20 [db] b BER against E b /N 0 Fig 9 32 path Rayleigh fading model (perfect CE) other subcarriers increase with In this channel model, delay paths have the same average power However, for a certain instance, it is possible that strong paths arrive nonuniformly in terms of their delays herefore ¼ 2 Low- Comp can extract the stronger paths and less correlated noise than ¼ 4 Fixed his is the reason why ¼ 2 Low-Comp achieves the same performance as ¼ 4 Fixed From Figs 5 and 9, this means that the low complexity version of the proposed scheme achieves almost the same performance as the original proposed scheme IE Commun, 2011, Vol 5, Iss 4, pp doi: /iet-com & he Institution of Engineering and echnology 2011

7 Fig path Rayleigh fading model (perfect and imperfect CE) Fig 10 shows the BER curves with perfect and imperfect CEs For the imperfect CE, the BER performance is deteriorated by about 2 db because of estimation error owing to the noise However, the same as Fig 9, the proposed low complexity scheme with ¼ 2 achieves almost the equivalent performance with the conventional scheme with ¼ 4 In the imperfect CE case, the BER performance of the proposed low complexity version of ¼ 2 is slightly better than that of ¼ 4 Fixed It is because ¼ 2 Low-Comp has less noise effect than ¼ 4 Fixed in both preamble and data period 332 Indoor Residential A: he BER performance on the Indoor Residential A is evaluated in Fig 11 On the Indoor Residential A, which has the smallest delay spread of the four, the improvement of the BER curves for ¼ 2 is relatively insignificant as compared to the 32 path Rayleigh fading model herefore if the delay spread is not large enough, path diversity through FS and sampling point selection are less effective with both perfect and imperfect CEs 333 BER against delay spread: In Fig 9, the 32 path Rayleigh fading model has enough delay spread to improve the BER performance through FS and sampling point selection he BER performance of ¼ 2 Low-Comp is slightly better than that of ¼ 4 Fixed On the other hand, on Indoor Residential A, the BER improvement is relatively insignificant as compared to the other channel model as shown in Fig 11 he relationship between the Fig 12 BER against delay spread BER and the delay spread is investigated as shown in Fig 12 he BER curves with perfect CE are investigated in Fig 12 When the delay spread, t RMS, is smaller than 60 ns, the BER of ¼ 4 Fixed is lower than that of ¼ 2 Low-Comp When t RMS is larger than 90 ns, the BER of ¼ 2 Low-Comp is slightly better than that of ¼ 4 Fixed It is the same for imperfect CE when t RMS is more than 40 ns as the inaccuracy of the CE diminishes the difference From Fig 12, the oversampling rate is required to be higher as the delay spread is small On the other hand, when t RMS is large, the oversampling rate can be kept small 334 Computational complexity: In this subsection, the computational complexity of the proposed low complexity scheme is evaluated as compared to that of ¼ 4 Fixed in terms of the number of multiplications he number of multiplications for each process such as DF, noisewhitening, MRC and sampling point selection in (27) or (29) are shown in able 2 he total amount of computational complexity for the fixed sampling point scheme with ¼ 4 ( ¼ 4 Fixed ) the proposed nonuniform sampling point selection scheme with ¼ 2 ( ¼ 2 Pro ), and the proposed low complexity scheme with ¼ 2 ( ¼ 2 Low-Comp ) per OFDM packet are shown in able 3 Here, it is assumed that the oversampling rate is 4 during both the preamble period and the data period in ¼ 4 Fixed while it is 8 during the preamble period and 2 during the data period in the proposed schemes In this table, 10 symbols and 100 symbols represent the number of OFDM symbols per packet able 2 Computational complexity Number of multiplications DF N log 2 N noise-whitening 2 MRC sampling point selection able 3 otal computational complexity Fig 11 Indoor Residential A (perfect CE) 10 symbols 100 symbols ¼ 4 Fixed ¼ 2 Pro ¼ 2 Low-Comp IE Commun, 2011, Vol 5, Iss 4, pp & he Institution of Engineering and echnology 2011 doi: /iet-com

8 transmitted As compared with ¼ 4 Fixed in able 3, ¼ 2 Low-Comp reduces the computational complexity by a factor of 2 when the number of symbols per packet is 100 his difference is because the fixed sampling point scheme requires four DF processes per symbol, whereas the proposed scheme requires only two in the data period hus, even though the selection of the sampling point requires additional 28 DF processes per preamble symbol, the total amount of complexity decreases 4 Conclusions In this paper, the sampling point selection schemes for the FS OFDM receiver have been proposed In the conventional scheme, the interval between the sampling points is fixed to S / and it cannot extract the multipaths which arrive nonuniformly in the delay domain In this paper, non-uniform sampling point selection has been proposed In this scheme, the interval between the sampling points is not fixed It then improves diversity gain by about 2 db when the oversampling rate is 2 However, it actually deteriorates the performance when the oversampling rate is 4 his is because of the correlation of the noise samples In addition as the oversampling rate increases, the complexity for the sampling point selection grows exponentially herefore this paper has also proposed the low-complexity sampling point selection scheme to eliminate the specific sets of the sampling points which lead to large noise correlation In the low-complexity non-uniform sampling point selection scheme, the complexity for the sampling point selection with ¼ 2 has been reduced by about 70% of the original one while it maintains the equivalent BER As compared to the fixed sampling point selection scheme, the total complexity including the demodulation of 100 symbols decreases by a factor of 2 he proposed scheme with ¼ 2 achieves slightly better BER than the fixed sampling point scheme with ¼ 4 when the RMS delay spread is larger than 40 ns for imperfect CE herefore both path diversity and complexity reduction have been achieved with the proposed scheme 5 Acknowledgments his work is supported by International Communications Foundation and in part by a rant-in-aid for the lobal Center of Excellence for High-Level lobal Cooperation for Leading-Edge Platform on Access Spaces from the Ministry of Education, Culture, Sport, Science and echnology in Japan and Japan Society for the Promotion of Science 6 References 1 IEEE 80211a-Part 11: Wireless LAN medium access control (MAC) and physical layer (PHY) specifications; highspeed physical layer in the 5 Hz Band, &arnumber= IEEE 80211g-Part 11: Wireless LAN medium access control (MAC) and physical layer (PHY) specifications; highspeed physical layer in the 24 HZ Band, &arnumber= IEEE 80216e-Part 16: Air interface for fixed and mobile broadband wireless access systems-amendment for physical and medium access control layers for combined fixed and mobile operation in licensed bands (Amendment and Corrigendum to IEEE Std ), 4 Rangaraj, V, Jalihal, D, iridhar, K: Exploiting multipath diversity in multiple antenna OFDM systems with spatially correlated channels, IEEE rans Veh echnol, 2005, 54, (4), pp hoen, S, der Perre, LV, yselinckx, B, Engels, M: Performance analysis of combined transmit-sc/receive-mrc, IEEE rans Commun, 2001, 49, (1), pp epedelenlioĝlu, C, Challagulla, R: Low-complexity multipath diversity through fractional sampling in OFDM, IEEE rans Signal Process, 2004, 52, (11), pp Shinkai,, Nishimura, H, Sanada, Y: Improvement on diversity gain with filter bandwidth enlargement in fractional sampling OFDM receiver, IEICE rans Commun, 2010, E93-B, (6), pp Shinkai,, Nishimura, H, Inamori, M, Sanada, Y: Effect of baseband filter bandwidth in fractional sampling OFDM on indoor channel model with measured impulse responses, IE Commun, o be published 9 Nishimura, H, Inamori, M, Sanada, Y: Sampling rate selection for fractional sampling in OFDM, IEICE rans Commun, 2008, E91-B, (9), pp Nishimura, H, Inamori, M, Sanada, Y: Initial sampling point selection in OFDM receiver with fractional sampling Proc Int Workshop on Vision, Communications and Circuits, Xi an, China, November Saito, K, Shinkai,, Nishimira, H, Sanada, Y: Experiment investigation of sampling point selection in fractional sampling OFDM receiver Proc IEEE Pacific Rim Conf on Communications, Computers and Signal Processing, August 2009, pp Haykin, S: An introduction to analog and digital communications (John Wiley & Sons, Inc, oronto, 1989), p Papoulis, A, Pillai, SU: Probability, random variables and stochastic processes (Mcraw-Hill, New York, 2000, 4th edn), pp Joint echnical Committee of Committee 1 R1P14 and IA R4633/ R4544 on Wireless Access: Draft Final Report on RF Channel Characterization, Paper no JC(AIR)/ R4, 17 January, Inamori, M, Nishimura, H, Sanada, Y, havami, M: Fractional sampling OFCDM with alternative spreading code Proc Eleventh IEEE Int Conf on Communications Systems, uangzhou, China, November Appendix: Correlation of noise samples In this subsection, the effect of noise-whitening on the received signal is investigated From (8), all the received signals on N subcarriers are expressed as where z = Hs + w (31) z = [z [0],, z [N 1]] (32) H = diag[h[0],, H[N 1]] (33) s = [s[0],, s[n 1]] (34) w = [w [0],, w [N 1]] (35) Noise vector w is the coloured noise and expressed as w = R 1/2 w v (36) where R w is the noise correlation matrix and v is the white noise in vector form given as follows v = [v [0],, v [N 1]] (37) v[k] = [v 0 [k],, v 1 [k]] (38) where v g [k] is the gth white noise on the kth subcarrier hrough noise-whitening in (19), (31) is converted to the following equation R ww z = R ww Hs + R ww w (39) IE Commun, 2011, Vol 5, Iss 4, pp doi: /iet-com & he Institution of Engineering and echnology 2011

9 where R ww = diag[r (1/2) w [0],, R (1/2) w [N 1]] As a result, (39) is expressed as where z = H s + w (40) z = R ww z = [z [0],, z [N 1]] (41) H = R ww H = [H [0],, H [N 1]] (42) w = [w [0],, w [N 1]] = R ww w = R ww R 1/2 w v I R n [0, 1] R n [0, N 1] R n [1, 0] I = R n [N 1, 0] I v[0] v[1] v[n 1] (43) where R n [k 1, k 2 ]isa matrix It corresponds to the (k 1, k 2 )th subblock of the N N matrix and is expressed as R ww R 1/2 w he g 1 th element of w [k 1 ] is expressed as follows w g 1 [k 1 ] = N 1 1 k 2 =0 g 2 =0 = v g1 [k 1 ] + k2 =k1 k 2 =0 R n [k 1, k 2 ] g1,g 2 v g2 [k 2 ] 1 g 2 =0 R n [k 1, k 2 ] g1,g 2 v g2 [k 2 ] (44) where [R n [k 1, k 2 ]] g1,g2 is the (g 1, g 2 )th component of R n [k 1, k 2 ] he second term of this equation represents the noise intruded from the others subcarriers after noisewhitening his component deteriorates the BER performance in the receiver o improve the BER performance, the Frobenius norm of R n [k 1, k 2 ] must be kept small From (43), R n [k 1, k 2 ] is given as R n [k 1, k 2 ] = R (1/2) w [k 1 ]R 1/2 w [k 1, k 2 ] (45) where R 1/2 w [k 1, k 2 ] is the (k 1, k 2 )th subblock of R 1/2 w 562 IE Commun, 2011, Vol 5, Iss 4, pp & he Institution of Engineering and echnology 2011 doi: /iet-com