BLIND CFO ESTIMATION USING APPROXIMATE CONJUGATE- SYMMETRY PROPERTY IN OFDM/OQAM

Transcription

1 BLIND CFO ESTIMATION USING APPROXIMATE CONJUGATE- SYMMETRY PROPERTY IN OFDM/OQAM S.MD.IMRAN (PG Scholar) 1 B.SIVA REDDY (Ph.D) 2 1 Dept of ECE, G Pulla Reddy College of Engineering & Technology, Kurnool, JNTUA 2 Assistant Professor, Dept of ECE, G Pulla Reddy College of Engineering & Technology, Kurnool, JNTUA ABSTRACT Orthogonal frequency-division multiplexing (OFDM) systems are highly sensitive to synchronization errors. We introduce an algorithm for the blind estimation of symbol timing and carrier frequency offset in wireless OFDM systems. We consider the problem of blind symbol timing (ST) estimation for pulse-shaping orthogonal frequency division multiplexing (OFDM) systems based on offset quadrature amplitude modulation (OQAM).This paper deals with the matter of blind synchronization for OFDM/OQAM systems. Specifically, by exploiting the approximate conjugate-symmetry property of the start of a burst of OFDM/OQAM symbols, attributable to the presence of the time offset, a replacement procedure for blind symbol timing and CFO estimation is proposed. By using computer simulations, the performances of Blind symbol estimators are analyzed. This result gives better Signal-to-Noise Ratio for proposed method. INTRODUCTION: The principle of multicarrier modulation (MCM) consists of rending up a broadband signal at a high data rate into many lower rate signals, every one occupying a narrower band. A usually recognized advantage of MCM is its lustiness against differing kinds of channel distortions, like multipath propagation and narrowband interference. Orthogonal frequency division multiplex (OFDM) is definitely, until now, the foremost necessary category of MCM. The OFDM word form usually recovers two differing kinds of modulation. Within the 1st one, as planned, for example, in [1], every carrier is modulated victimization construction modulation (QAM). During this theme, which is additionally known as OFDM/QAM [2], QAM symbols are formed with an oblong window. In a very second class of OFDM systems, that is additionally known as orthogonally multiplexed QAM in [3] (O-QAM) or OFDM with offset QAM in [2] (OFDM/OQAM), the modulation used for every may be a staggered offset QAM (OQAM). Each the OFDM/QAM and OFDM/OQAM modulation schemes on paper guarantee orthogonality and a most and identical spectral potency. Moreover, in follow, they can be implemented thanks to the Discrete Fourier Transform (DFT). A crucial distinction comes from the very fact that OFDM/OQAM, in contrast to OFDM/QAM, permits the introduction of AN economical pulse shaping that makes it less sensitive to the frequency offset thanks to the transmission and to the receiver. Therefore, if OFDM/QAM constitutes the modulation kernel of the celebrated questionable coded OFDM (COFDM) system, OFDM/OQAM is currently bestowed as being an honest candidate to induce still higher bit rates over wireless channels [2]. OFDM/OQAM systems, as all multicarrier systems, area unit additional sensitive to synchronization errors than single-carrier systems. For this reason, it's vital to derive economical synchronization schemes. Within the last years many studies are centered on blind and data-aided carrier frequency offset (CFO) and image temporal order (ST) synchronization for OFDM/OQAM systems. New proposals aim at simplifying the structure of the preamble so as to be ready to use it for synchronization and deed functions. In [3] a synchronization theme for preamble-based ST and chief financial officer estimation with strong acquisition properties in dispersive channels has been developed. In [4] a replacement preamble structure has been projected with helpful properties that change the utilization of a one-tap equalizer. The characteristics of the preamble derive from the requirement to change the procedures for channel estimation. The ensuing synchronization algorithms become obsessed with the actual preamble, whose utilization is clearly conditioned by the supply of a correct synchronization technique. Therefore, a general contribution to the event of synchronization algorithms needs the aptitude to control

2 subcarrier with none specific data regarding the structure of the preamble. Obviously, this not solely represents a preambleindependent contribution to the synchronization task, that permits a typical definition of the preamble structure free by the necessities of the synchronization algorithms, however conjointly paves the thanks to a rise of the spectral potency to be achieved by avoiding the preamble. The blind estimation rule projected in [5] relies on the exploitation of the second-order cyclostationarity of the transmitted OFDM/OQAM signal; the convergence of such a technique is especially slow (too several image periods have to be compelled to be processed) in order that it's not helpful in apply, unless severe signal/noise ratio ratios area unit thought of. Moreover, it's restricted to the case wherever chief financial officer is gift however it's not dedicated to the joint chief financial officer and temporal order offset estimation. However, [6] considers the case wherever each the offsets area unit collectively calculable by exploiting the cyclostationarity properties. In [7] associate rule for blind chief financial officer estimation is additionally projected in keeping with associate approximate (for an oversized variety of subcarriers) maximum-likelihood approach and it's shown its superior performance as compared with the cyclostationarity-based ways. Moreover, in [8] a most chance technique for blind chief financial officer estimation suited to eventualities of low Signal to- Noise quantitative relation is projected. However, the liability of each projected ways lies in their process complexness. During this paper, we have a tendency to analyze the conjugate-symmetry property that about holds within the starting of a burst of OFDM/OQAM symbols. Mistreatment such associate approximate property, a blind technique for joint ST and chief financial officer estimation is projected. Though the projected technique springs with relevancy an AWGN channel, it's analyzed by technique with relevancy commonplace multipath channels; the numerical results show that the projected technique will represent a helpful contribution to the blind temporal order synchronization once the OFDM/OQAM system operates over a multipath channel. Moreover, an equivalent analysis shows that the projected technique provides a helpful contribution to the coarse chief financial officer compensation only for adequate signal/noise ratio ratios. Preliminary results about the analysis of the approximate conjugate-symmetry property within the starting of a burst of OFDM/OQAM symbols and its exploitation for ST and chief financial officer estimation area unit reported in [9]. SYSTEM MODEL Let us take into account associate OFDM/OQAM system with a good variety M of subcarriers. The received signal once the Information-bearing signal s(t) presents a temporal order (timing) offset τ, a CFO normalized to r(t) = e e s(t τ) + n(t) (1) Where n(t) is a zero-mean complex-valued white Gaussian noise process with independent real and imaginary part, each with two-sided power spectral density N0. The signal s(t) is equal to s(t) = s (t) + j s (t T/2) (2) (3) s (t) = a, e ( ) g(t nt) (4) s (t) = a, e ( ) g(t nt) where T is that the OFDM/OQAM image interval, A {0,..., M 1} is that the set of size, M of active subcarriers, the sequences a, and a, indicate the important and imaginary part of a {complex range/ pure unreal number} of a of the complex information pictures transmitted on the m th subcarrier throughout the n th OFDM/OQAM symbol, authority is that the range of coaching job symbols, N is that the range of payload symbols, whereas g(t) is that the paradigm filter. Note that we have a tendency to tend to assume that every the coaching job symbols and thus the payload symbols unit of measurement unknown. The discrete-time low-pass version of the transmitted signal is given by s[i] s(t) = s (it ) + j s (i T ) (5) Where T T/M is that the sampling interval. Allow us to 1st contemplate the derivation of associate degree economical generation procedure for the signal s (t). The same derivations are often straight forwardly obtained for the signal s (t) since the continuous-time signal is generated by D/A conversion; we ve a tendency to contemplate the generation of its discrete-time samples s (t) s (it ) s (t) = a,. e ( ) g(it nmt ) s (t) = a,.*e ( ) g((i nm) T ) (6) The generation of the sequence s [i] are similar to the generation of the sequence of vectors d () outlined because the Mx1 vector whose m th part, say d (),, is up to s (nm + m) for m {0,1,.,M-1} this can be written as, d (), s [nm + m] ; m {0,1,2,.., M 1} = a, e ()

3 a () subcarrier spacing = Δf * T and a carrier part offset φ, is written as.* g((nm + m n M)T = (j a, ) e T ). g((m + (n n )M) (7) = b, g((m + (n n )M) T ) Where b, (8) is just quantity i.e. (j a, ) e ; ]{0,1,2,.., M 1} (9) Which is the IDFT of the sequence j a, with respect to the index m. If we define the vector b as the M 1 vector whose m th component (for m { 0, 1,...,M 1} ) is b, in the above equation, now we can synthetically write as, b () = IDFT [w X a () ] (10) Here IDFT operator applied on the input vector and, m A, the m th component w of the M-vector w is w =j and the m th component of the vector a () is the () symbol a, in equation (3) and the components of a () are not relevant. Equation (10) is only defined for n {0,1,, N + N 1} but we can directly extend it to any n provided that we assume that a () b () 0 when n not belongs to {0,1,, N + N 1 }. Now we can write equation (8) as d (), = b g (11) Where g is defined so that so that its m th component g, is g, g(m + nm)t (12) Since the prototype filter g(t) can be not zero only in the interval [0,KT), it follows that the vector g can be not zero only for n {0, 1,...,K 1}, where K is the overlap parameter, that is, the ratio between the length of the prototype filter g(t) and the multicarrier symbol interval T. So that, (11) can be rewrite as d () = g b () + g b () () g b () (13) w I F F T g b ( ( b () b g g g. () b ( PPN A diagram of the structure for the economical generation of the term s [i] in (6) is according in Fig. 1. Analogously, the generation of the sequence s [i] in (5) is () comparable to the generation of the sequence of vectors d outlined because the output of the PPN: d () = g b () () + g b () + g b () (14) Here, b () is given by, b () IDFT [a () w] (15) Where the m th component (m A) of the vector a () is the symbol a, in (4) THE EXACT CONJUGATE-SYMMETRY PROPERTY IN OFDM In OFDM systems the vector w in (15) has unit components; the PPN and the offset of M/2 samples are not present, i.e., the vector d () () + jd is defined as IDFT [ a () + j a () ] and it is transmitted after cyclic-prefix extension. Therefore, if a () = 0, the vector d () () + jd = () d possesses the well-known CSP, as synthetically depicted in Fig. 2. M X a c b X b # c a # M/4-1 M/4-1 M/4-1 M/4-1 Fig : 2. Structure of OFDM symbol for a () = 0 where a and b denote the positions where the CSP holds with respect to the positions a # and b #, respectively, c denotes a single position where the CSP holds with respect to the position c but it is not used, and, finally, denotes positions where no CSP holds. In the transmitted multicarrier symbol, the cyclic prefix is followed by M samples that possess the CSP: if such M samples are collected in the vector [u0 u1 um/2 u2] (where the length of both vectors u1 and u2 is M/2 1), then u1 = u# 2 where u# 2 denotes the flipped and conjugate version of u2 (i.e., the m th entry of u1, denoted as u,, is equal to u,, for m=0,1,.m/2-2). In the signal received on a flat channel in the absence of noise and frequency offset, each cyclic prefix is followed by M samples that possess the CSP: if such M samples are collected in the vector [u e u e u / e u e ] [v v v / v ], where P

4 denotes phase offset, and then v e = [v e ] # = e # # v or it can be written a v s, = v e (16) This relation defines the CSP that is exactly possessed by the OFDM transmitted signal. Such a property are often used for synchronization functions in OFDM systems with real information symbols as planned in [17]: the top of the cyclic prefix and therefore the part offset are often determined by scanning the received signal and looking wherever such CSP is best approximated consistent with a least-squares approach, that is, {θ, } = arg min, v (θ) # v e (17) where θ denotes an estimate of the normalized delay θ = τ/ts assumed to be an integer and, moreover, the M vector [ v (θ) v (θ) v / (θ) v (θ) ] has been extracted at the candidate delay θ. Note that the maximization procedure in (17) provides a closed-form solution for the phase estimate θ and, then, the ST estimation requires only a one-dimensional search: θ # = arg max { 2 v (θ). v (θ) v (θ) v (θ) } (18) A slightly different approach, which is aimed at simplifying the threshold setting, is often suggested (see [18] and references there in ): θ argmax (). # () () () (19) Note that, though not explicitly represented in the notation, the statistics in (19) depends on the length of the interval [v0 v1 vm/2 v2] from which the vectors v1 and v2 of length M /2 1 are extracted. In the present section a length M is considered while in the next sections a length M/ 2 is used. THE APPROXIMATE CONJUGATE-SYMMETRY PROPERTY IN OFDM/OQAM In OFDM/OQAM systems, as noted in section II, the m th component of the vector w in (15) is not unit but it is equal to j = exp(j2π (1/4)m), while the vectors a () and a () are real-valued; consequently, from (10) and (15) it follows that each vector b () and b () possesses a particular structure that derives from the cyclic shift of M/4 samples on the left. As shown in Figure 3, this implies that, for each vector of length M, two adjacent intervals of length M/2 can be singled out; in both intervals, the classical CS property reported in Figure2 holds. Therefore we can use the same test in (19) with reference to intervals of length M/ 2 in order to estimate the position of such intervals if we had to operate on sequence of vectors {b () }. According to relations (2), (13), signal. However, at the start of the burst transmission, some properties are present that are independent of the transmitted symbols and can be therefore exploited for blind synchronization. For sake of clearness, we illustrate such properties with reference to the prototype filter derived in [9] by using the frequency sampling technique and with in the case wherever the overlap parameter is K=4. Allow us to initial note that from (11) d () (), = g, X b, d () (), = g, X b, d (), = g, X b () (), + g, X b, d (), = g, X b () (), + g, X b, d (), = g, X b (), + g, X b () (), + g, X b, where we denote with d (),, g,, and b (), (for k = 0, 1, 2) the vectors obtained from d () (),, g and b respectively, by extracting the first M/2 components; moreover, we denote with d (), g, and b (), the vectors obtained from d (), g and b (), respectively, by extracting the last M/2 components. Analogously, relation (20) can be re-written with reference to the PPN in (14) by replacing the superscript R with I. The impulse response of the considered prototype filter is reported in Figure 4 where the intervals used to extract from the prototype filter the vectors g,, g,, g,, g,, and g, introduced in (20), are specified. Let us remind that the vector b () exhibits the properties described by Figure 3 and, therefore, the vector b (), exhibits the classical property described in Figure 2 over an interval of length M/2. The multiplication by g, in (20) implies () that d, does not possess exactly the CS property. However, such a property is approximately present since the entries of g, are all negative real values. In fact, the original scalar product in (19) for two equal vectors # v and v resulting from the exact CS property in b (), = [v0 v1 vm/4 v2] v. v # = v, v, = v, (θ) (v, denotes the mth component of the vector vi for m {0, 1,..., M 4 2}) becomes, when referred to d (), = g, X b, () = v, g,, v, g,, noise but also the entries of d and d are weak since the two leading vectors are g, and

5 and (14), the conjugate symmetry (that is exhibited by b () ) is destroyed by the fact that there s not any specific property () capable to pass such properties from b to the transmitted = v, g,, g,, (22) where g0,i,m denotes the mth entry of g0,i for m {0, 1, 2,..., M2 1}. Since now g0,i,m+1 = g0,i,m 2 1 m there is not a conjugate symmetry but since g0,i,m+1 g0,i,m 2 1 m 0, the sum in the right-hand side of (22) is still positive and much larger than the value obtained for positions where such an approximate property does not hold. This permits the use of the test in (19) over intervals of length M 2 in order to detect the positions where the CS property holds if we had to operate over a sequence of vectors {d(r) n }. Let us note that this approximately holds also if the sign of g0,i,m+1g0,i,m 2 1 m is not constant; in fact, if such a sign assumes the same value for a large percentage of the M/4 1 values of the sum in (22), then the term in the right-hand side of (22) is much larger than that obtained for a generic position where such an approximate property does not hold. Therefore, the approximate CS property can be detected by the test in (19) provided that a large percentage of terms in the sum in (22) has the same sign. The delay of M/2 samples that the output of the PPN in (14) exhibits with reference to the output of the PPN in (13) implies that d () = g, X b, d = g, X b () (), + g, X b, d = g, X b (), + g, X b () (), + +g, X b, d = g, X b (), + g, X b (), + () () + g, X b, +g, X b, d = g, X b (), + g, X b (), + +g, X b (), + g, X b () (), + g, X b, (23) where the vector d contains the first block of M/2 samples of the transmitted signal while d contains the second block of M/2 samples of the same signal and so on. Let us take into account that the choice of the prototype filter derived in [9] implies g g ; g, g, and ; g, g,. Finally, g, g, and, g, = g,. Now equation (23) can be approximated as d () = g, X b, () d g, X b, () d g, X b, () d g, X b, d g, X b () (), + g, X b, (24) g,, which include a large fraction of the prototype-filter energy. The vector d is the first vector which is contributed by one of the two leading vectors so that d is the first vector whose power is not negligible at usual signal-to-noise ratios; moreover, g, and g, possess a constant sign of the folded vector while g, possesses a non-marginal percentage of its M/4 entries of the folded vector with a different sign. Therefore, we can deduce that the test for approximate CS property achieves its maximum for d while the values relative to d and d are much smaller. Finally, note that the vectors d for k 4 are contributed both by g, and g, and, consequently, they do not exhibit significant values to the test (19). In fact, by using the approximation (24) for d in the scalar product of the test (19), we do not obtain any guarantee that the M/4 1 values summed up in the scalar product maintain themselves coherent. Thus, when we look for a position of conjugate symmetry in the received signal r(i) (seen as a sequence of vectors d ) observed at the channel output by using the test (19) over intervals of length M /2 φ(θ) () () () () (25) We obtain the argument of the top value (θ =θ0) of the statistics Ψ(θ) in correspondence of the interval where the vector d is present. The beginning of such an interval is shifted by 3M/2 samples at the right of the beginning of the burst and θ1, which denotes the central value of such an interval according to (25), is shifted of 7M/4 samples from the first sample of the burst. Moreover, M samples on the left of the top value, we find another maximum (θ = θ3) of the statistics due to the properties of d and, in the middle point between them, we find a second maximum (θ = θ2) due to the properties of d. Note that, though g,, g, the value of the maximum in θ2 is only slightly larger than that in θ3 since the signs of the components of g1,s, unlike those of g, are not constant. There is also a fourth maximum (θ = θ4) due to the properties ofd, buried in noise at usual signal-tonoise ratios. Therefore, the argument of the top value (θ = θ1) of the statistic in (25) can be used for blindly estimating the beginning of a burst of OFDM/OQAM symbols as θ1 7M 4. In Fig. 5 we report a typical realization of the statistics in (25) in AWGN for Eb/No = 20 db and M = Although derived for the sake of clearness with reference to the case K = 4, often considered in OFDM/OQAM literature, the method can be straightforwardly extended to other values of K. Let us finally note that the statistics Ψ (θ) in (25) The terms d for k 5 share the structure of d since for k 5 the two vectors with stronger entries are both present.

6 From the previous discussion, it follows that, for k {0, 1, 2, 3}, d possesses an approximate conjugate symmetry. With reference to the choice here considered, the entries of the vector g, are very weak so that d is usually buried in the is independent of the frequency offset ε; the derivation of the method B in Section V implicitly sketches the prove of this for the scalar product in the numerator of (25). BLIND CFO ESTIMATION In this section we introduce two different methods for blind CFO estimation. The first method, dubbed method A, can be derived from the fact that from (24), D () () (),, D () (26) Where the division between the two vectors is defined component-wise. Consequently, in the presence of a normalized frequency- offset ε and neglecting the presence of noise and the effects of the channel, the entries of the () vector D / D () are all equal to exp j2π M = exp(j2πε) since D () is extracted M samples on the left of D ().Therefore, we evaluate the angle of the average of the entries of D () / D () and, then, we estimate the unknown CFO as /2π. In calculating the average, we do not use the entries of D () / D () with amplitude larger than 2 as they are assumed to be outliers. Note that the extraction of the vectors D () requires a previous ST estimation and that the errors in timing compensation worsen the performance of the CFO estimator. The method B uses the property that, under ideal condition, the angles in the scalar products corresponding to the top value (θ = θ1) of the statistics in (25) and the value in θ = θ2, which is M/2 samples at its left, differ only due to the presence of the CFO. More specifically, such a difference is equal to 2πε+π and the condition is ideal since it neglects the mismatch in the timing synchronization (obtained at the previous step), the effects of the channel and the presence of noise. Under such ideal conditions, the scalar product in (25) at θ = θ2 can be written as follows: v (,) (θ ). v (,)# (θ ) = v (,), (θ ) v(,), (θ ) (27) with the m th component v (,), (θ ) of the vector v (,) (θ ) (m {0,1,., 2} and l ε {1, 2}) defined as v (,), (θ ) v, (θ ) g,, e e v (,), (θ ) v, (θ ) g,,. e () e e (28) For m {0, 1 2}, where v, (θ ) is the m th component v (θ ) is the m th component of the vector g, (for m 0,1,. 2) and is a constant phase offset. The CS property implies that v1( θ ) = v # ( θ ) (i.e., v, (θ ) = v #, (θ )) m 0,1,. 2) and consequently, (27) becomes v (,) (θ ). v (,)# (θ ) = v, (θ ) g,, e e.. * ((v, (θ ) g,,.. e e e ) v, (θ ) g,, g,, =. e e e = e e e (29). v, (θ ) g,, g,, Analogously, the scalar product in (25) at θ = θ1 can be therefore written as follows: v (,) (θ ). v (,)# (θ ) = = v, (θ ) v, (θ ) (30) with the m th component v (,), (θ ) of the vector v (,) (θ ) (m {0,1,., 2} and l ε

7 {1, 2}) defined as v (,), (θ ) v, (θ ) g,, e e e v (,), (θ ) v, (θ ) g,, e () e e e For m {0, 1 2}, where v, (θ ) is the m th component v (θ ) is the m th component of the vector g, (for m 0,1,. 2) and is a constant phase offset. The CS property implies that v1( θ ) = v # ( θ ) (i.e., v, (θ ) = v #, (θ ) ) m 0,1,. 2) and consequently, (30) becomes v (,) (θ ). v (,)# (θ ) = v, (θ ) g,, e e.. * ((v, (θ ) g,,. e e e ) v, (θ ) g,, g,, =. e e e = e e e (32). v, (θ ) g,, g,, In order to evaluate the phase difference between the quantities in (29) and (32), we have to note that (see also Figure 4) g,, g,, < 0 (33) g,, g,, > 0 (34) While (34) is obvious, (33) is verified with reference to the considered prototype. The method is straightforwardly adapted to a different prototype provided that the quantities in the left hand side of (34) and (33) are non null. The limited variations with m of v, (θ ) and v, (θ ) imply that v, (θ ) g,, g,, = π (35) v, (θ ) g,, g,, = 0 (36) From (29), (32), (35), and (36) it follows that (v (,) (θ ). (v(,) (θ )) (v(,) (θ ). (v (,)# (θ )) = 2πε + π (37) Finally, we introduce the method C that defines its output as the average of the results obtained by the methods A and B. For all three methods, to avoid ambiguities in the estimate, the condition ε < 0.5 must be satisfied since they estimate ε through the complex quantity exp (j2πε). SIMULATION RESULTS: g[n] n Fig: The prototype filter and the. The time axis refers to the choice M= 2048 but the same partition holds also for other values of M

8 sigh(theta) RMSE ITUA-s=0.1 ITUB-s=0.1 ITUA-s=0.18 ITUB-s=0.18 ITUA-s=0.25 ITUB-s= No of samples Fig: A typical realization of the statistics Ψ(θ) Ebn0 Fig: RMSE of the proposed ST estimators 10 0 AWGN ITU-A-2048 ITU-B-2048 ITU-A-4096 ITU-B ITUA-MET-A ITUA-MET-B ITUA-MET-C ITUB-MET-A ITUB-MET-B ITUA-MET-C ---Biterror 10-1 RMSE Ebn0 Fig: Bit error rate over AWGN, ITU-A and ITU-B channels Fig: RMSE of the proposed ST estimators over AWGN, ITU-A and ITU-B channels Ebn0 Fig: RMSE of the proposed CFO estimators CONCLUSION: In this paper, the problem of blind symbol timing (ST) estimation for pulse-shaping orthogonal frequency division multiplexing (OFDM) systems based on offset Quadrature amplitude modulation (OQAM).This paper deals with the matter of blind synchronization for OFDM/OQAM systems. Specifically, by exploiting the approximate conjugate-symmetry property of the start of a burst of OFDM/OQAM symbols, attributable to the presence of the time offset, a replacement procedure for blind symbol timing and CFO estimation is proposed. By using computer simulations, the performances of Blind symbol estimators are analyzed. Those results are shown in simulation results. These results show better Bit Error Rate and Signal-to-Noise Ratio for proposed method. REFERENCES: [1] S. B. Weinstein and P. M. Ebert, Data transmission by frequency-division multiplexing using the discrete Fourier transform, IEEE Trans.Commun. Technol., vol. COM-19, pp , Oct [2] B. Le Floch, M. Alard, and C. Berrou, Coded orthogonal frequency division multiplex, Proc. IEEE, vol.

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