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IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com

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- 1. Seshu Kumar .N, Bharat Kumar Saki, Manjula Bammidi, P Srihari / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.1904-1913 Optimized Joint Time Synchronization And Channel Estimation Scheme For Block Transmission UWB SystemsSESHU KUMAR .N1, BHARAT KUMAR SAKI2, MANJULA BAMMIDI3, P SRIHARI4, 1 PG Student, Department Of ECE, DIET,Anakapalli, Visakhapatnam, AP, INDIA. 2 PG Student, Department Of CSIT, JNTUK college of Engineering, Visakhapatnam, AP, INDIA 3 Associate Professor, Department Of ECE, DIET,Anakapalli, Visakhapatnam, AP, INDIA. 4 Associate Professor, HOD, Department Of ECE, DIET,Anakapalli, Visakhapatnam, AP, INDIA.Abstract In this paper, timing synchronization in synchronization accuracy is much more stringenthigh-rate ultra-wideband (UWB) block than conventional wideband systems. Moreover, thetransmission systems is investigated. A new joint highly dispersive nature of UWB channels presentstiming and channel estimation scheme is proposed additional challenges to timing acquisition. Thefor orthogonal frequency division multiplexing literature is abundant with research on preamble(OFDM) and single carrier block transmission assisted synchronization methods for OFDM inwith frequency domain equalization (SC-FDE) conventional wideband channels [4]-[7]. Since thisUWB systems. The scheme is based on a newly paper is concerned with timing synchronization,designed preamble for both coarse timing and thesubsequent channel estimation. Despite of the There are only a few publications in thepresence of coarse timing error, the estimated literature on synchronization for multi-band (MB)channel impulse response (CIR) is simply the OFDM UWB systems [9]—[13], all using thecyclic shifted version of the real CIR, thanks to preamble defined in the WiMedia standard [1] exceptthe unique structure of the preamble. The coarse [10]. Cross-correlation with the preamble template istiming error (CTE) is then determined from the used in [9]—[11] for synchronization, leading toCIR estimation and used to fine tune the timing high complexity. The methods in [12], [13] employposition and the frequency domain equalization autocorrelation of the received signal as the timingcoefficients. The proposed scheme saves preamble metric and take the maximum of the metric to be theoverhead by performing joint synchronization timing point, which tend to synchronize to theand channel estimation, and outperforms existing strongest path but not necessarily the first significanttiming acquisition methods in the literature in path. In this paper, we propose a generic joint timingdense multipath UWB channels. In addition, the and CE scheme for block transmission systems,impact of timing error on channel estimation and which begins with an auto-correlation based coarsethe performance of SC-FDE UWB systems are timing estimator using a newly designed preamble,analyzed, and the bit-error-rate (BER) followed by CE and CE assisted timing adjustment.degradation with respect to certain timing error is In the coarse timing stage, the coarse FFT windowderived. position is established and the initial CE is performed. The particular preamble structure makesIndex Terms—Synchronization, UWB, SC-FDE, the CE robust to the coarse timing error, and thetiming error analysis, joint timing and channel proposed technique to accurately estimate the coarseestimation. timing error allows both fine timing adjustment and CIR adjustment for later data demodulation. BothI. Introduction synchronization and CE are acquired from the For high data rate ultra-wideband (UWB) procedure. This design saves the preamble overheadcommunications, two block transmission systems and improves the synchronization performance inhave been proposed: orthogonal frequency division UWB channels.multiplexing (OFDM) [1] and single carrier with The fine timing estimation is related to thefrequency domain equalization (SC-FDE) [2], both time of arrival (ToA) estimation. For ToA estimationof which compare favorably to the impulse based in MB-OFDM UWB systems, energy detectors canUWB systems in terms of equalization complexity be found in [14], [15], and an energy jump detectorand energy collection in dense multipath UWB has been proposed in [16]. In this work, we propose achannels. SC-FDE has lower peak to average power newly designed energy jump detector and provide itsratio (PAPR) and lower sensitivity to carrier performance analysis. In addition, the impact offrequency offset (CFO) than OFDM, but is less timing error on SC-FDE systems has been analyzedrobust to timing error [3].Due to the very high data in this work, which is not available in the literature.rate of UWB systems, the requirement on the The remainder of the paper is organized as follows. 1904 | P a g e
- 2. Seshu Kumar .N, Bharat Kumar Saki, Manjula Bammidi, P Srihari / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.1904-1913Section II presents the system model and the structure for accurate CE whose result will be used insynchronization problem formulation. The proposed the subsequent timing adjustment and frequencytiming and CE scheme is introduced in Section HI. domain equalization. As shown in Fig. 1, in additionThe timing error impact on SC-FDE systems with to the CP, a cyclic suffix is padded after theminimum mean square error frequency domain preamble body. Note that the suffix is used only forequalization (MMSE-FDE) is analyzed in Section the preamble, and data blocks have CP only as inIV. Section V gives the performance evaluation and conventional OFDM. In the proposed preamblediscussion of the proposed method in UWB shown in Fig. 1, A and B represent the first half andchannels. Finally Section VI concludes the paper. the second half of one block generated byII. System Model N 1 1 Although the system model adopts cyclic Pt ( n ) N k 0 Pt (k )e j 2 nk / N o k N 1prefixed single carrier block transmission withfrequency domain equalization over UWB channels, (1)the synchronization technique developed directly where Pt(k) is a random ±1 sequence inapplies to an OFDM system. The pth transmitted frequency domain (FD), referred to as FD pilotblock is composed of digitally modulated symbols block, which results in flat amplitude in FD for goodxPk (0<k<N-1), where N is the block length. A cyclic CE performance. The [A B] structure can be repeatedprefix of length Ng is used to avoid IBI as in an M times and finally a prefix and a suffix are added toOFDM system. Denote the symbol-spaced equivalent have [B A B ... B A], where A and B are the first NxCIR by h= [h o, hl,..., hL], where L is the maximum samples of A and last Nx samples of B respectively.channel delay. The UWB channel is assumed to be The length of the prefix and suffix denoted by N xconstant over the data blocks. As long as the needs to be larger than the maximum coarse timingmaximum channel delay is shorter than the CP length error, and Nx should be larger than the channel length(L < Ng), there is no FBI in data demodulation with L to avoid IBI. By repeating [A B] we can achieveperfect timing. The task of synchronization is to find noise averaging for more accurate CE, and thethe beginning of the data block, also referred to as adjacent [A B] blocks serve as cyclic extension. Forfinding the FFT window position. Perfect the case of M—1, the whole preamble becomes [B Asynchronization in cyclic prefixed block transmission B A], thus the synchronization and CE overhead issystems can be achieved in the LBI-free region of the 2Nx+N, smaller than the preamble pattern proposedCP ([-Ng+L,0]) [5]. As long as the FFT window in [1] where the overhead is 2(N+Ng). Following thestarts within the IBI-free region, data demodulation conventional auto-correlation based metric function,can be carried out perfectly. Otherwise timing error we define our timing metric function aswill cause performance degradation to a system. In M(d) =Section IV, the effect of timing offset on the 2 N x ( M 1) N 1performance of SC-FDE UWB will be analyzed 2 C (d ) 2 rdx x rd N 1 . 2 N x ( M 1) N 1 k 0III. The Proposed Joint Timing and Channel R(d ) rd N 1 2 2Estimation Scheme ( rd 1 ) Timing within the IBI-free region does not k 0cause extra interference to the detection. However, in (2)UWB systems, the IBI-free interval given a short CP and coarse timing is established by finding theis often very small or even zero, depending on maximum of (2) as given by 6C = maxM(d). Notechannel realizations. Thus, it is necessary to locate the range of d to look for the maximal can be definedthe first significant arrival path of the transmitted as [9S, 9S + Q], where 6S is the point where thesignal block, which is defined as the exact timing metric function first exceeds the preset threshold andpoint in the following. With the definition of the Q can be set as N. Simply speaking, the metricexact timing point, we define the timing error as the calculator activates the maximization procedure onceoffset between the established timing point and the the arrival of the signal is sensed. To reduce theexact timing point. computation complexity at the implementation stage, the metric can be implemented in an iterative mannerA. Preamble Structure and Coarse Timing similar to [4] as In dense multipath channels, coarse timingobtained by evaluating a metric function can not C(d + 1) = C(d) - rdrd+N+ rd+2Nx+(M-1)N rd+2Nx+MNprovide sufficient accuracy, since coarse timing often (3) andfalls far away from the exact timing due to the large R(d+l) = R(d)-rd2-rd+N2 + rd+2Nx+(M-1)N2 + channel dispersion. We propose to use the estimated rd+2Nx+MN2 (4)channel information to fine tune the timing position.A new preamble structure is designed to have the which implies a sliding window implementation. Forperiodical property for acquisition and cyclic every received sample, only 3 new multiplication and 1905 | P a g e
- 3. Seshu Kumar .N, Bharat Kumar Saki, Manjula Bammidi, P Srihari / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.1904-19136 add operations are needed to compute the iterative 1timing metric. This sliding window correlator saves and er i | g i k mod N | 2substantial complexity compared to the metrics k 0proposed in [7] and [9] which need to perform the 1whole correlation for every received sample. el i | g i k 1 mod N | 2 (9) k 0B. Channel Estimation and Timing Adjustment Where (gmax) is the maximum absolute In this step, CE is carried out using the same value of the elements in g and T) is a preset thresholdpreamble after the initial acquisition and the whose determination will be given in Section V.Fig.estimation accuracy is not affected by the timing 2 is an example of g obtained by simulation using theerror. Denote the coarse timing error (CTE) by ec CM2 channel model. In this example, the blocknumber of symbols. Consider the low complexity length is N = 128. Due to the coarse timing error, theleast square (LS) CE. If M = 1, the frequency domain actual CIR is cyclic shifted, and thus some strongCE is given by H(k) = 75$^, where PT(k) and Pt(k) taps at the end of the vector g are observed whichdenote the received and transmitted frequency correspond to the channel taps shifted. The firstdomain training symbol at the A:th subchannel significant tap is near the end of the vector g,respectively. The estimation can be improved by meaning that the current timing is several samples toemploying the repetitive [A B] structure to reduce the right of the exact timing point. The first tap of gthe noise effect. In addition, the cyclic structure can be determined by (6)-(9) and explained asgiven by B and A provides IBI-free CE as long as the follows. Once an estimated channel tap g(i) is abovecoarse timing error |ec| < Nx. In this case, the a normalized threshold ?75max» the energy of festimated channel is given by channel taps on the right side of i represented by Hk = Hke2ck/N + Wk er{i) is compared to the energy of £ channel taps on (5) the left side of i represented by e/(i), to evaluate thewhere Wk is the Gaussian noise in frequency domain. energy jump on the point i. The first channel tap to is Since the proposed preamble has a cyclic detected at the position where the energy jump is thestructure, the estimated frequency domain channel largest. Having a window of £ channel taps is tousing this preamble does not suffer from the coarse reduce the occurrence of mistakenly choosing a noisetiming error induced IBI that is usually present in tap as the first tap. Since before the first tap there areconventional schemes. The impact of coarse timing only noise, whereas er(i) contains energy from theerror on the CE with conventional preambles will be channel taps and noise, er(i) — ei(i) cancels the noiseshown in Section IV-C. Performing inverse FFT and can well detect the jump point from the noise(TFFT) on Hk, the estimated CIR g is the CIR h only region to the starting of the CIR g. Thecyclicly shifted by the timing error of c samples, determination of parameters i] and 4 will beplus the Gaussian noise. The CTE c can be discussed in Section V. Upon obtaining the coarseestimated from g and then used to adjust timing for timing error, the synchronizer can adjust timingthe subsequent data blocks, which has no cyclic position by ec samples, and at the same time, the CEproperty to use as the preamble does. The coarse result should also be modified corresponding to thetiming error estimation is obtained by an energy timing adjustment by Hk = Hker^^klN. Note that injump detector written as (6), we determine the timing error to the left or right of the exact timing by the search result being in the first half or the second half of the CIR. Therefore, the N -T0 if 0 T0 N C N P T0 2 fine timing can handle at most coarse timing N 2 if T0 N error, which is usually more than sufficient. The 2 block diagram of the entire timing synchronizer for (6) SC-FDE UWB systems is shown in Fig. 3. When thewhere timing metric module detects the coarse timing point, it passes the received signal to the subsequent max E (i) 0 i N - 1, modules. The LS estimator estimates FD channel T0 response Hk based on the coarse FFT window, and (7) then the coarse timing error is estimated by the CTE estimator based on the coarse CE result. Finally the CTE estimation is used to adjust the FFT window if | g i | n | g max | position and the FDE channel coefficients.E i e r i el i 0 if | g i | n | g max | C. Performance of Fine Timing Adjustment (8) The performance of the fine timing adjustment is solely dependent on the accuracy of the 1906 | P a g e
- 4. Seshu Kumar .N, Bharat Kumar Saki, Manjula Bammidi, P Srihari / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.1904-1913first tap search, given a set of estimated channel taps. 2The conventional energy detector [6], [14], [15] variance . Similarly, the t adjacent taps on theusually measures the energy in a window of channel 1taps and takes the starting point of the maximal left side of (11) have approximately equal variancemeasurement window as the first tap position. 2However, this method requires the knowledge of the denoted by . Then both sides of (11) are scaled 2exact channel length, which is usually difficult to Chi-square distributed random variables (r.v.s) withestimate accurately in UWB channels. Setting the 2t degrees of freedom and the component Gaussianswindow too long may lead to an early timing while 2 2 2 2having the window too short may lead to a late variance + and + respectively,timing. The proposed energy jump detector 1 w 2 wovercomes this drawback of the conventional energy 2detector. Its window length does not need to equal where is the variance of the noise componentthe exact channel length and usually a window length wless than the channel length is sufficient. With a in the coarse CE result. Then the probability in (11)moderate window length, the energy jump detector can be written ascan achieve accurate timing. Note that another energy jump detector has been proposed in [16], PED (t ) f t ( x) f r ( y )dydxwhere the metric is in an energy ratio form while that 0 zof the proposed detector is in a subtraction form. The (12)detector in [16] is able to deal with the more realistic where fi(x) and fr(x) represent thechannel with continuously varying delays at the cost probability density function (PDF) of the left sideof higher complexity. That is, CIR is estimated more and right side of (11) respectively. The PDF of afrequently in [16]. In the following, we study the scaled Chi-square r.v. of 2t degrees of freedom istiming error probability of the proposed energy jump f x 1 x / 2 2detector in comparison with that of the conventional x t 1e , where 2 is the 2energy detector. The energy jump factor at the exact 2timing point d can be written as component Gaussian’s variance. Substituting the PDF of the Chi-square r,v. into (12), the error 1 -1 probability can be written asE d | g d k mod N | 2 | gd - k - 1 mod N | 2 k 0 k 0 1 PED (t ) 1 = r t | h k d k mod N | 2 | d k 1 mod N | 2 12 2 2 k 0 t 1 2 1 1 t k (10) where Wk is the noise component of the kth k! t k = k 0 12 2element of the estimated CIR vector. We study the 1 2 2 case where the window length is less than the 1 channel length (£ < L), usually resulting in late 1 t 1 1 0k t k (13)timing with the conventional energy detector. For thecase of f > L, the analysis is similar. r t k 0 k! 1 0 t k Consider a late timing event of t sampleswhere t > 0. For the conventional energy detector, where . is the Gamma function withthe error probability that a late timing of t samples 12 2occurs is given by k k - 1! and 0 Indeed 70 2 1 2 t -1PED (t ) Pr | hk d k mod N | 2 < represents the energy ratio of the estimated channel tap (including noise) at the actual first channel tap k 0 position over that at the + 1 actual tap position. This | h t 1 d k mod N | 2 . …(11) definition takes into account the power decaying k over time.k 0 For the proposed energy jump detector, the Following [15], we consider the complex event of timing error of t samples happens when E(d)Gaussian channel taps with variances decaying along < E(d + t). Similar to (10),the index. Assume the small number of t adjacentchannel taps on the right side of (11) have the same 1907 | P a g e
- 5. Seshu Kumar .N, Bharat Kumar Saki, Manjula Bammidi, P Srihari / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.1904-1913 -1 -1 1,t t 1 t l 2!E d t | g d t k mod N |2 | g d t k 1 mod N | 2 t 1 k 0 k 0 t 1! 2 1 t l 1 = (20) 1 d k t mod N | 2 2,t t 1 t l 2! | h k 0 k t t 1 t 1!1 2 t l 1 - 1t t 1 | d k 1 mod N |2 | hk d k mod N |2 . (21) 1 0 k 0 k 0 Defining , and substituting (17)(14) 4 22 1Then the probability of the event for the proposed and the PDF of the Chi-square r,v. into (16), we haveenergy jump detector can be written as 0t 1t 2t 2 t t 1 PEJ (t ) Pr E d E d t = PEJ t t k 1 l 1 i 0 t 1 Pr 2 | hk d k mod N | 2 k ,l k i l t 1 k 0 k i t -1 e 0 k x dx i!l 1! 0 t -1 (| h d k | 2 | d - k mod N | 2 )). 0t 1t 2t 2 t t 1 k 0 = t k 1 l 1 i 0 (15)The left side of (15) is a scaled Chi-Square r.v. with k ,l k ki l t 1i t 2t degrees of freedom and the component Gaussians = i!l 1! 0 k i t 2 2variance 2( + 1l 1 t l 2!i t ), while the right side of (15) t t 1 1 1 wis a sum of two scaled Chi-Square random variables t l 1 i 0 t 1!i!l 1!with It degrees of freedom and the component 2 2 2 1i l 1 2 t and Gaussians variances . Then t l 1 1 i t + 3 w w 2 1 1the timing error probability can be written as: 2l 1 1t i PEJ (t ) f l ( x) f r ( y )dydx f L ( x) Ϋ i t 0 x 0 1 2 1 2 t l 1x dx (16) (22)Where Ϋ x T y dy. According to [17], The energy ratios defined by are = X related to the signal-to-noise-ratio (SNR) and theΫ (x) can be written as UWB channel statistics, and they determine the Ϋ (x) = performance of the estimator. Fig. 4 shows the k ,t k k x e 2 t t -1 i k z timing error probability comparison of the proposed1t 2t t - 1! t 11 energy jump detector and the conventional energy k 1 t 1 k i 0 i! detector with specific energy ratios. In the proposed (17) energy jump detector, the first channel tap searchingWhere algorithm has slightly higher computation com- plexity than the conventional energy detector. It 1 1 requires calculating the energy of two windows and 2 22 2 performing a subtraction while the conventional detector only computes one. However, the window in (18) the energy jump detector is usually shorter than the channel-long window of the conventional energy 1 2 detector. 2 2 IV. The Impact of Timing Error on the SC- (19) FDE Performance The timing error impact on a general OFDM system and on an MB-OFDM system was 1908 | P a g e
- 6. Seshu Kumar .N, Bharat Kumar Saki, Manjula Bammidi, P Srihari / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.1904-1913analyzed in [18] and [19] respectively. However, the and BFFT, the demodulated block can be written astiming error impact on SC-FDE UWB systems hasnot been well studied. The sub-symbol level timing x p = F H CAFxm p F H CF h p CFnoffset analysis and timing jitter analysis on SC-FDE mUWB systems were studied in [3], where it was 1shown that the sub-symbol level timing offset (26)determines the sampling position of the equivalent Where F and FH are the N x N FFT andchannel and even the worst sub-symbol level timing IFFT operation matrices respectively, and A isoffset only results in small performance degradation. diagonal representing the estimated frequencyTiming jitter was found to be more detrimental to domain channel response. In the presence of timingsystem performance. Here, we study the symbol offset, the quality of CE is degraded as will be shownlevel timing offset which will be shown to have more in Section IV-C. However, employing our proposedadverse effect than sub-symbol level timing offset. preamble and the joint timing and CE algorithm in Section m, the effect of timing offset on CE is only aA. Small Delay Spread Channels phase shift in the frequency domain. Therefore, Hk = Small delay spread channels refer to Hke:>2*krn/N ignoring the noise effect, where thechannels with the maximum delay shorter than CP, phase shift is directly included in the CE. Let *1>mi.e. Ng > L. Let m denote the timing error, and = FHCFhy and the (fe,n)th (the matrix index startsassume m < Ng — L. When m. < 0, the starting from 0) element of *1m is given bypoint of the established FFT window falls in the IBI-free part of CP, leading to the same performance as that of the exact timing point. However, when m > 0, the established FFT window is on the right of the 0 n N-mcorrect window, meaning that the current block m k1,, n window contains m samples of the next block. N 1 N 1 2i m - t h N - m n N -1Assuming carrier frequency is perfectly 1 C h j Nsynchronized, the pth received block as shown inAppendix I can be written as N i 0 k 0 i t n Y P h1 x1 p n m m (27) Then the demodulated block can be rewritten as (23)where h is the Toeplitz channel matrix with N 1 x p x p I 1,m x 1 p n 1 m h0 0 ... 0 h L h L-1... h1 1xN as its first row. xm (p) k isi N k 0is the N x 1 desired pth data block left cyclic shiftedby m (> 0), n is the Gaussian noise, (28)x m p = 2 | Hk | Where k , I isi is the | H k | N 0 / 2 Eh 20,... 0, x p 1, - ... N g Tp 0, x p 1, - N G 1 - Tp,1, ..., inter-symbol interference from the current block due N m to the fact that an MMSE receiver is not inter-symbol interference (ISI) free [2], and n is the Gaussianx p 1, N g m1 - x p,m-1 T N x1 noise N term N-1 with N-1 variance i2lk (24) 02 2N 2 n | C e I 0 K 0 k N | 2 . The ith elementwhere xpn denotes the nth (—Ng < n < N — 1) N 1symbol in the pth block, and 0 N -m xm of I isi is S i x n 0 n p ,n Where h0 0 ... 0 n i 0 N - m x N m 1 N -1 S n i k e j 2 n-i k / N . The third term in mh1 h 1 h 0 ... 0 0mx n m N k 0 (28) is the timing error induced interference. Define hm 1 ... h0 the index set [0, N—1] as U and the subset [0, m— 1] as U. Then for i e U, the demodulated symbol The received block in (23) can be seen as can be written asthe combination of two convolutions plus theAWGN: one is the circulant convolution between xm 1 N 1 (p) and the channel h, and the other one is the linear x pmi k - i, N mi x p ,i 1, mconvolution between h and xy(p). After FFT, FDE, N k 0 1909 | P a g e
- 7. Seshu Kumar .N, Bharat Kumar Saki, Manjula Bammidi, P Srihari / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.1904-1913 + in block ISI k -1 k N Pe x p ,i Q N -1 m 1 S n i x p,n S n i iI, , N mn x p,n + , i U / U1 m 0 n2 N s i En m n 0 2 n i m 1 b iI, , N mn x p1, N 9 n nij m (33)n 0 where 5R{-} is taking the real part of a complex number. Then the overall average BER can be IBI N 1 P x . 1 (29) obtained by Pe and for i U/U 1, the demodulated symbol can be e p ,i N i 0written as B. Large Delay Spread Channels In the large delay spread channels, there is N -1 m 1 often no IBI-free region within the CP, and hence 1 x p ,i i1,, N m n x p 1, N g n mx p ,i k timing error on both sides of the exact timing point N k 0 n o may cause performance degradation. When m > 0, IBI the established EFT window contains multipath + components from both the previous and the next in block. Considering the case when timing error is block ISI small, i.e., m < L-Ng, and letting v = L — Ng — m, N -1 m 1 Sn i x p,n S n i iI, , N mn x p,n ni m the demodulated block can be written as n m n 0 x p F H CAFX m p F H CF h1 x p + m (30) The interference includes the in-block ISI H F CFn, (34)which comes from the current block and the IBI x 2 p and h 2 are given by coming from the succeeding block. There are a large Wherenumber of independent interference symbols in (29)and (30), therefore Gaussian approximation can beapplied to the sum of ISI according to the x 2 p x p-1, N - x p , N-L, x p -1, N - 1 x p , N L 1, Lyapunovs central limit theorem as shown in the TAppendix B. The variance of the interference for thezth desired symbol is denoted by a2 (i) which can be …., x p 1, N-1 x p , N- N g 1, 0, ...0 (35) Nx1written as N - h h L-2 hL N-1 m1 L-1 s2 i | S n i | 2 | S n i - i, N mn | 2 m 1, 0 h L-1 h L- v 1 0x N - n m n 0 h2 0 n i 0 0 m 1 0 + | n 0 1, m i, N mn |2 . (31) 0 (N- )x h L-1 0 N - x N (36)Without loss of generality, considering the BPSK The IBI has contribution from the preceding blocksymbols, the bit error rate (BER) is given by Pe(xPii) and the succeeding block. Note that= P(5R(£Pii) > 0xp_i = — 1). Therefore, the error h1 x 2 p h 2 x1 p 0, and then (34) can be m mprobability is given by rewritten as Pe x p ,i Q N 0 m k -1 p k i1,, N m i , i U1 x p F H CAFxm p FH CF h1 h 2 x1 p x 2 p m m 2 N s2 i n H + F CFn. (37) CF h h Eb 2, m, m Let F H 1 2 and define the (32) index subsets U2 = [0, m, -1) and U3 = [N – L + v, N – Ng]. Following the same procedure in the previous subsection, the variance of the interference can be obtained as 1910 | P a g e
- 8. Seshu Kumar .N, Bharat Kumar Saki, Manjula Bammidi, P Srihari / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.1904-1913 s2 i | S i 2 , m , 3, v where F CF h 2 and U4 is defined as [N - H n i , N mi |2 + nU z Ng -v,N - Ng]. Thus, the ith symbol error probability n i can be derived as | S i n 2 , m , i , N Lm 2 | +nU z k 0 k i3,, i N L m n i N 1 m, v P x p ,i Q ,iU nU | S n i | 2 | i2nm, | 2 . (38) , 4 , 2 N s i nU 2 U nU U 3 2n i Eb For large timing error m > L - Ng > 0, the problem (44)becomes the same as Subsection IV-A.Then the error probability of the ith symbol of the pth block is given by k 01 k N N -1 P x p ,i Q , i U and i U 4 . 2, k 2 N s2 i n P x p ,i Q k 0 i , N m 1 Eb , i U2 2 N s2 i (45) Ek C. The Impact of Timing Error on Channel (39) Estimation In the literature, CE and synchronization often employ separate preamble N 1 k 0 k i2, i L N m, v blocks instead of the single preamble proposed in , P x p ,i Q ,iU Section IE, for example in the Multi-band OFDM 3 standard proposal [1]. Conventional preambles 2 N s i 2 usually begin with synchronization blocks, followed Ek by CE pilot blocks and then data symbols. This subsection considers timing error induced (40) degradation in CE with conventional preamble blocks. Note that channel estimation in SC-FDE is identical to OFDM. Consider the least square CE with one pilot block p*. Under a timing error of m (> k 01 k N P x p ,i Q 0) samples, the frequency domain received pilot , i U U 2 U 3 . block can be written as 2 N s2 i P r m mPt m m Ek F h1 P t n. (41) (46)When the timing point is to the left of the exact where and Pt are diagonal matrix with elementstiming point, i.e., m < 0, the interference comes onlyfrom the previous block, and the demodulated block e j 2im / N and the frequency domain pilot block respectively, isis given by defined asx p F CAFx p F CF x p F CFn H m H 2 H 0....,0, , x ,N 0 g pt , 0, ; x 0, N g 1 Pt , ...x0 , N g m Pt ,m1 T Nx1 h where xo is the data block following the pilot block. (42)where L N g, resulting in more multipath Define Nxm as the last m columns of the matrix mcomponents from the previous block. The variance of Fh 1 , and its (k, n)th element is given bythe interference can be written as nU 1 i | S n i 3, | | S n i | | i3 n |2 m-n ,n m hl e j 2k (n m) / N , 2 2 2 i n N Lm k m 0. nU nU 4 n 0 n i n i l0 (43) (47) Then the fcth sub-carrier of Pr can be written as 1911 | P a g e
- 9. Seshu Kumar .N, Bharat Kumar Saki, Manjula Bammidi, P Srihari / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.1904-1913 Pr k / m H k e j 2km / N Pt k + 3T5,3T;] is used. Since the focus of this paper is timing synchronization, perfect carrier frequencym 1 x pt ,n n k , synchronization is assumed in the simulation. The k ,n 0,n design parameters Nx and £ are determinedn 0 according to the channel statistics. Fig. 5(a) gives the (48) coarse timing error statistics, which shows theThe estimated channel at the kth sub-carrier is empirical cumulative distribution function (CDF) of Pr k / m the coarse timing error at ^ = 10 dB in CM1-CM4Hk . Pt k UWB channels. It can be seen that the coarse timing error is most severe in CM4 which is the mostThe signal-to-interference-ratio (SIR) on the kth sub- dispersive channel model, indicating that thecarrier can be written as performance of the auto-correlation based coarse timing estimator is highly dependent on the channel | H k Pt k | 2 condition. The more dispersive the channel is, the SIRk = larger the coarse timing error. According to Fig. 5(a) m 1 | ,n x0,n Pt ,n | 2 k we set Nx = 30. The choice of design parameter £ in n 0 the CTE estimator depends on the channel length | H k | | Pt k | 2 2 statistics. If £ is too small, the estimation error (49) probability is higher; Otherwise, the computationm 1 m 1 complexity is higher. We choose £ to be the average| n 0 k ,n 2 | s | n 0 k ,n pt , n | 2 number of channel taps carrying 50% of the channel energy for different channel models. Fig. 5(b) showswhere Es is the average energy of the data symbols. the CDF of number of taps carrying 50% of theFor m < 0, since the length of the channel delay channel energy, using the channel sampling rate of 2spread spans L sampling periods, only q = max(L — ns, and 1000 channel realizations for each channel(Ng + m),0) block symbols are corrupted by the model. It indicates that in CM4 £ can be chosen asmultipath components from the previous block. 30, in CM3 f = 25, while in CM1 and CM2 £ = 15Assuming that there is a synchronization block and £ = 20 respectively.preceding the CE block in a non-joint scheme andwith the same transmitting power, it can be derivedthat the SIR is given by | H k | 2 | Pt k | 2 q0SIRk q 1 , | | | P | q 1 q0 k ,n k ,n t , p n 2 2 s n 0 n 0 (50) nfor m < 0 where k , n h i 0 L n i e j 2ik / N and p= N — Ng — q. It can be seen that timing error alsoaffects the CE accuracy and will further degrade thesystem performance derived in Subsections FV-A Fig. 10. (a) Average SINR of the coarse CE in CM1;and IV-B when the conventional preamble pattern is (b) MSE performanceof the channel estimator, withused. proposed preamble pattern and with theconventional preamble pattern.V. Simulation Results and Discussions MB-OFDM and SC-FDE block The MAE performance shows that thetransmission schemes are for high-rate UWB proposed CE aided timing method outperformssystems. Thus, in the simulation, we set the block conventional OFDM timing method significantly inlength N = 128, CP length Ng = 32, following [1]. UWB channels. The timing performanceThis block length is determined based on practical improvement of the proposed method over theimplementation complexity. However, the resulting conventional method comes from the more accurateCP does not guarantee to be always longer than a coarse CE and the first tap searching algorithm. TheUWB channel. The UWB channels simulated follow proposed cyclic extended preamble structure canCM1-CM4 models proposed by the IEEE 802.15.3a provide an IBI-free coarse stage CE regardless. ForWorking Group [1]. A root raised cosine (RRC) example, when the coarse timing position is on thepulse with a roll-off factor of 0.5 and span of [— right side of the true position, the coarse CE in [6] 1912 | P a g e
- 10. Seshu Kumar .N, Bharat Kumar Saki, Manjula Bammidi, P Srihari / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.1904-1913will include samples of IBI. According to system has been analyzed. Simulation results have(49) in Section IV, the SIR of the coarse CE shown that the proposed scheme yields accurateis given by timing and superior CE performance in UWB | H k | 2 | Pt k | 2 channels.SIRk 1 | ,n |2 s | ,n Pt ,n |2 n 0 k k n 0 REFERENCES [1] "Multi-band OFDM Physical Layer Proposal for IEEE 802.15 Task Group 3a," Where n is defined as in (47). The k, Mar. 2004.SINR results of the coarse CE obtained from [2] Y. Wang and X. Dong, "Cyclic prefixedsimulation are plotted in Fig. 10(a) for the proposed single carrier transmission in ultra-scheme and that of [6] in CM1. In addition, the MSE wideband communications," IEEE Trans.performance of the channel estimator in CM3 and Wireless Commun., vol. 5, pp. 2017-2021,CM4 are compared in Fig. 10(b). Furthermore, the Aug. 2006.first tap searching algorithm in [6] is a conventional [3] "Comparison of frequency offset and timingenergy detector, whose performance is inferior to the offset effects on theperformance of SC-FDEproposed energy jump detector as described in and OFDM over UWB channels," IEEESection III-C. Trans. Veh. Technol, vol. 58, no. 1, pp. 242- 250, Jan. 2009. [4] T. M. Schmidl and D. C. Cox, "Robust frequency and timing synchronization for OFDM," IEEE Trans. Commun., vol. 45, pp. 1613-1621, Dec. 1997. [5] M. Speth, F. Classen, and H. Meyr, "Frame synchronization of OFDM systems in frequency selective fading channels," in Proc. IEEE Veh. TechnoLConf., vol. 3, May 1997, pp. 1807-1811. [6] H. Minn, V. K. Bhargava, and K. B. Letaief, "A robust timing and frequency synchronization for OFDM systems," IEEE Trans. Wireless Commun., vol. 2, pp. 822- 839, July 2003. [7] C. L. Wang and H. C. Wang, "An optimizedFig. 11. MAE performance of the proposed scheme joint time synchronization and channelwith different parameters. estimation scheme for OFDM systems," in Proc. IEEE Veh. Technol. Conf., May 2008, Fig. 11 shows the timing performance with pp. 908-912.different design parameters, such as the number of [8] E. G. Larsson, G. Liu, J. Li, and G. B.repeated [A,B] (M) and in the first tap search Giannakis, "Joint symbol timing andalgorithm. It can be seen from Fig. 11 that in short channel estimation for OFDM basedchannel conditions such as CMI or CM2, increasing WLANs," IEEE Commun. Lett., vol. 5, no.does not improve performance significantly, due to 8, pp. 325-327, Aug. 2001.noise averaging (though not shown here), thusleading to better timing performance. However,increasing M in long channel conditions such asCM3 and CM4 cannot improve the performance athigh SNRs whereas increasing is more effective.VI CONCLUSIOIN A joint timing and CE scheme that employsa unique preamble design and CIR assisted finetiming has been presented for UWB blocktransmission systems. The preamble proposed savesthe overhead and has the cycle property resulting inCE and fine timing robust to coarse timing error.Based on the CE result, a novel algorithm forestimating the coarse timing error has beendeveloped for UWB channels. Furthermore, theimpact of symbol level timing error on an SC-FDE 1913 | P a g e

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