FREQUENCY modulated differential chaos shift key (FM-

Transcription

1 Accepted in IEEE 83rd Vehicuar Technoogy Conference VTC, 16 1 SNR Estimation for FM-DCS System over Mutipath Rayeigh Fading Channes Guofa Cai, in Wang, ong ong, Georges addoum Dept. of Communication Engineering, Xiamen University, Fujian 3615, China University of Québec, Écoe de Technoogie Supérieure, Montréa, Canada Emai: caiguofa6@16.com, wangin@xmu.edu.cn, ong.kong.1@ens.etsmt.ca, georges.kaddoum@etsmt.ca Abstract In this paper, we dea with the probem of maximum ikeihood M estimation of the signa-to-noise ratio SNR parameter for frequency moduated differentia chaos shift key FM-DCS system over mutipath Rayeigh fading channes. The M estimators are derived for various scenarios incuding data-aided, non-data aided and joint - estimation by using both the data and piot symbos. For comparison purposes, the Cramér-Rao ower bounds CRBs for the SNR estimators are derived. The performance of the estimators is evauated by simuations and comparing with CRBs in terms of the mean-square-error. Simuated resuts show that for a arge spreading factor the proposed scheme performs we over a wide SNR range in comparisons with CRBs. Index Terms FM-DCS; SNR estimation; Cramér-Rao ower bound CRB; mutipath Rayeigh fading channes. I. INTRODUCTION FREQUENCY moduated differentia chaos shift key FM- DCS system offers exceent performance under mutipath fading or time-varying channes without requiring channe estimation 1-, which has been used in utra-wideband UWB systems for short-distance communications 3-. So far, many researchers have studied the performance of the combination of advanced technoogy, e.g. cooperative communications 5-6, channe coding 7-8, and muti-input mutioutput MIMO9-1, etc., with FM-DCS moduation. To optimize these schemes, the receiver needs to rey on the estimates of signa-to-noise ratio SNR. For instance, in a coded FM-DCS system, the SNR estimation is used to make soft-decision decoding and adjust adaptive strategies; in a DCS cooperative system, the seection of reay nodes and adopting ampify and-forward AF or decode-and-forward DF mechanisms are based on the estimates of SNR; in a MIMO-DCS system, the power contro agorithms are aso dependent on the estimates of SNR. For traditiona moduations, incuding phase shift keying PS11, quadrature ampitude moduation QAM1-13, and frequency shift keying FS1-15, much work has been done to show the SNR estimation. The maximum ikeihood M based estimator, which has asymptotic properties of being unbiased, achieving to Cramér-Rao ower bound CRB, is overwhemingy the most popuar approach to obtaining practica estimators see Chapter 7 of 16. The CRB see Chapter 3 of 16 gives the minimum variance of unbiased estimators, which is a very usefu too for evauating the performance of an estimator. In this work, we derive M SNR estimators, which incude data-aided, non-data aided and joint - estimators, for FM-DCS system in mutipath Rayeigh fading channes. The CRBs for M SNR estimators are derived as a fundamenta benchmark that refects the best achievabe performance. II. SYSTEM MODE Consider a FM-DCS communication system, shown in Fig.1. In the FM-DCS moduator, every bit to be transmitted is represented by two sampe functions. For bit 1, the same FM chaotic signa is transmitted twice in succession whie bit is sent by transmitting the reference chaotic signa foowed by an inverted copy of the same signa. The ogistic map is empoyed as chaotic generator: x k1 = 1 x k. The chaotic signa is moduated by using FM moduator to generate FM chaotic signa. The FM chaotic signa z, after samping, obtains discrete points, where is defined as a spreading factor. The k-th transmitted signa is represented by a sequence of sampes of the chaotic signa, in which the j-th sampe is given by s j,k = { zj, j = k 1 1,..., k 1 c mk z j, j = k 1 1,..., k, 1 where c mk {1, 1}. The genera channe mode of spread spectrum wireess communication systems is used. The discrete time impuse response of the channe is given by hn = α δn τ, where is the number of paths, α and τ are the channe coefficient and the deay of the -th path respectivey. Here, we assume that α are the circuary symmetric compex independent random variabes with a Rayeigh distribution. In most appications 1718, the arge spreading factors are generay chosen, thus, the argest mutipath time deay is shorter than the bit duration, i.e. < τ max. In this case, the inter-symbo interference ISI is negigibe. Hence, simiary to 1718, at the receiver, the decision variabe is approximated by r k α z jτ n j α z jτ n j, j=1 c mk 3 16 IEEE. Persona use of this materia is permitted. Permission from IEEE must be obtained for a other uses, in any current or future media, incuding reprinting/repubishing this materia for advertising or promotiona purposes, creating new coective works, for resae or redistribution to servers or ists, or reuse of any copyrighted component of this work in other works.

2 Chaotic Generator y k, j z j Deay s k, j cmkzj c r ak, =yk, jyk, j j= 1 Channe = c, c,..., c m m1 m m time of a FM-DCS sequence ength. Consider g piot symbos and data symbos, so that the tota packet is of the ength = g. The decision vector is defined as r a = r a1, r a,..., r ak,..., r a T, where. T denotes the transpose. The information symbo vector is defined by c m = c m1, c m,..., c mk,..., c m T. Deay Correator A. Data Aided Estimation Using Piot Symbos Fig. 1. System mode of the FM-DCS communication system. where n is a wideband AWGN, characterized by a circuary symmetric compex Gaussian density with zero mean and variance N variance N / in the rea and imaginary components, and denotes conjugation operator. In free ISI case, the accurate distribution of Eq. 3 can be cacuated by using the method of, but the computationa compexity is very high owing to using the residue cacuus. In , it was shown that for arge spreading factors, Eq. 3 coud be considered as the Gaussian distribution. For a arge spreading factor, the foowing approximated expression is used 17-18: j=1 z jτ z jτi, i. Hence, Eq. 3 is further approximated by r k c mk α z jτ α z jτ n j j=1 c mk α zjτ n j n j n j. 5 Actuay, the effect of mutipath fading eads to the oss of partia energy, however, by means of the above approximation we may consider as coecting a the energy. Hence, it is not difficut to derive the mean and variance of the rea part r ak, respectivey, which are given by E r ak = c mk α E b, 6 V arr ak = α E bn N, 7 where E b = j=1 z j is energy per bit, E. denotes the expectation operator and Var. is the variance operator. For convenience, we define h = α as the power factor and the transmitted power is supposed to be normaized to one, i.e. E b = 1. Using the mutipe observations {r ak },...,, the true SNR, γ, that we wish to estimate, is define as γ = h/n. It is mathematicay more convenient to use the parameter vector, θ = h N, and the function, gθ = h/n. III. SNR ESTIMATION Since the channe is sowy fading, we assume that the channe coefficients are constant during the transmission For this case, the probabiity density function PDF of r ak is written as exp r akc mk h/ N pr ak c mk, θ = hn. 8 πn hn With independent received symbos, the joint PDF of the g received symbos is given by g exp r akc mk h/ N pr a c m, θ = hn g. 9 πn hn Thus, the og-ikeihood function F for the g received symbos is given as, g Λr a ; θ = g n π g r ak c mkh nn hn N hn. 1 In order to maximize 1, by taking the first-order partia derivative of 1 with respect to h and N, the resuts are given by g N g c mk r ak c mkh h = N hn = N g N h N hn N g r ak c mkh N hn, 11 N h g r ak c mkh N hn. 1 Setting the above equations to zero, one obtains the foowing resuts g c mkr ak c mk h/ =, 13 g g N hn = r ak c mk h/. 1 Using 13, the estimate of h is given as ĥ = g c mkr ak g. 15 Using 1 and 15, the estimate of N is given by N = ĥ g g r ak c mkĥ. 16 Thus, using 15 and 16, the M estimation of the SNR is found, which is given by γ = ĥ/ N.

3 3 B. Non-Data Aided Estimation Using Data Symbos For this case, the PDF of r ak is written as r akh/ exp N hn r akh/ N hn exp pr ak θ = πn hn exp r ak h N = hn πn hn cosh N hn, 17 where expx expx = coshx is used. With independent received symbos, the joint PDF of the received symbos is given by exp r ak h N hn cosh hrak N hn pr a θ = 18 πn hn Thus, the F for the received symbos is given as, h Λr a ; θ = n π nn hn N hn r ak N hn n cosh N hn. 19 Simiary, taking the first-order partia derivative of 19 with respect to h and N resuts in N h = N hn N N hn h N hn h r ak tanh N rak N hn N hn, = N N h h h rak N N hn N hn N h N hn r ak tanh N hn. 1 Putting the above equations equa to zero and soving them simutaneousy gives us foowing resuts: h hn N = h = r ak tanh N hn. 3 r ak, where expxexpx expxexpx = tanhx is used. It is noticed that for high SNR, where r ak N, the term exp r ak /N is approximated as zero. Thus, the summation term is approximated as r ak tanh N hn r ak. 5 Thus, the estimate of h is given as ĥ = r ak. 6 Using and 6, the estimate of N is given by N ĥ ĥ r ak =. 7 C. Estimation Using Piot and Data Symbos Assuming independent received symbos, the joint PDF is the product of PDFs resuting from the piot and data symbos, thus the F is given as Λr a ; θ = n π nn hn g r ak c mkh N hn k=g1 n cosh h rak k=g1 N hn N hn. 8 Using simiar approximations as the Sec. III-B and taking the first-order partia derivative of 8 and setting them to zero resut in the estimates of h and N as ĥ = g c mkr ak r ak, 9 k=g1 N ĥ ĥ r ak =. 3 IV. CRAMÉR-RAO OWER BOUND It is noticed that we can have different CRBs for the different estimators in FM-DCS system. The competey or fuy data-aided estimator serves as a bench mark on the variance for a estimators, which is the same as but uses a information in the entire packet as training sequence 131. Hence, we derive the CRB for the estimator. Furthermore, the CRB for the estimator is aso derived. Since the unknown parameter is a vector, the CRB for SNR is given as 16 CRB = gθ The above noninear equations seem to prohibit the cosed θ I1 θ gθt θ, 31 form soutions for estimates of h and N. Using the fact, where the derivative of the function, gθ θ, is given by tanhx = tanhx, the summation term is given by gθ h rak θ = 1 h N N, 3 r ak tanh N hn = r ak tanh N hn and I 1 θ is the inverse matrix of the Fisher information = r ak 1 exp r matrix FIM Iθ, which is given by 16 ak /N γ γ, 1 exp r ak /N γ γ Iθ = E Λr a ;θ r a h E Λr a ;θ ra h N, 33 E ra Λr a ;θ N h E ra Λr a ;θ N where E ra fr a = fr apr a θdr a.

4 A. Estimation Using the fact see 18, pp. 86, i.e. for a Gaussian random variabe X, X Nm, σ, one has { k!σ k EX m n n = k = k k! 3 n = k 1, and taking the second-order partia derivatives of 11 and 1, where is instead of g, with respect to h and N, we can obtain the eements of Iθ. Hence, the FIM of estimator is derived as Iθ = N N hn N N hn hn N hn N hn N h N hn N hn. 35 Furthermore, using 3, 11 and 1, one has E ra h = E ra N =. It shows that Eq. 1 satisfies the reguarity conditions. Thus, according to the Theorem 7.1 in Chapter 7 of 16, this estimator is unbiased. Substituting 3 and the inverse matrix of 35 into 31, the CRB for estimator is derived as = B. Estimation γ γ γ γ. 36 We define the foowing equations as: E ra rak = N hn h, 37 E ra r ak tanh N hn = h, 38 E ra rak cosh N hn = N f γ, 39 where f γ = exp γ γ πγ t exp γ t cosh tγ γ dt. Simiary, taking the second-order partia derivatives of and 1 with respect to h and N, using 37, 38, and 39, we can obtain the eements of Iθ. Hence, the FIM of estimator is derived as γ 1 fγ γ γ 1 γfγ γ Iθ = C γ 1 γfγ, γ γ 1 γ fγ γ where C =. Simiary, using 37, 38, 39, N γ and 1, one has E ra h = Era N =. Hence, the estimator is aso unbiased. Substituting 3 and the inverse matrix of into 31, the CRB for estimator is derived as CRB N = γ 3 γ γ γ γ fγ γ. γ γ γ fγ CRB N SNRdB Fig.. vs. true SNR for different proposed estimators, where spreading factor is set to CRB N SNRdB Fig. 3. vs. true SNR for different proposed estimators, where spreading factor is set to 3. For arge SNR, fγ is approximated as zero. Eq. 1 is further approximated as γ γ γ γ, which is the same as Eq. 36 if equas to. Note that, the proposed expressions can be appied into DCS systems, e.g. 5, 6, 9, 18. At the time, because the term N / of Eq. 7 is repaced by N / in DCS system, / is instead of for Eqs. 16, 7, 3, 36 and 1. Hence, the estimation performance for DCS system is the same as that of FM-DCS system. V. RESUTS AND DISCUSSIONS For mutipath Rayeigh fading channes, three paths = 3 are considered having equa average power gain, i.e. E α 1 = E α = E α 3 = 1/3, with the time deays τ 1 =, τ = and τ 3 = 5, respectivey. The estimators performance is evauated in terms of the normaized mean squared error using simuations and compared to the normaized CRB defined as: γ = E{γ γ } γ, NCRBγ = CRBγ γ,

5 CRB N Simuated resuts show that for a arge spreading factor the proposed scheme performs we over a wide SNR range in comparison with CRBs. The proposed expressions are vaid for any binary differentia spread spectrum BDSSsystem. ACNOWEDGEMENTS This work was supported in part by the Nationa Natura Science Foundation of China under Grant No and the NSERC discovery grant SNRdB Fig.. vs. true SNR for different proposed estimators without ISI, where spreading factor is set to 8. where γ is the true vaue of SNR and γ is the estimated vaue. In simuation, a short packet is considered, comprising 8 piot symbos and 8 data symbos. The performance of the proposed estimators is evauated. Fig. shows the vs. true SNR for different proposed estimators, where the spreading factor is set to 8. It is observed that estimator outperforms and joint cases in the ow SNR region, where it is attributed to the approximations of the noninear equations in the section III-B causing the performance oss, whie and joint cases outperform case in the medium SNR region, because and joint cases use a data of the packet to estimate SNR but case ony adopts a sma amount of piot data. However, in the high SNR region a estimators perform not we and their s are the same, because Eq. is non-zero and the system has serious ISI for a sma spreading factor, where these two factors aso cause the error of channe information estimation resuting in error propagation in noise variance estimation. Here, these factors are referred as to interference term. In a word, the variance of interference term, which is arger than the very sma true noise variance at high SNR region, causes inaccurate noise variance estimation. In order to expain this point, we show the performance of the proposed estimators for a arge spreading factor and under no ISI scenario. Fig. 3 pots the vs. true SNR for different proposed estimators, where the spreading factor is set to 3. Fig. depicts the vs. true SNR for different proposed estimators without ISI, where the spreading factor is set to 8. It can be seen that at a arge spreading factor the proposed scheme performs we over a wide SNR range, incuding high SNR region, from Fig. 3, where their performance can achieve the eve of no ISI case in Fig.. Hence, for a arge spreading factor, the variance of interference term can be negected. VI. CONCUSIONS In this paper, the M SNR estimators and CRBs for SNR estimators serving as a benchmark in FM-DCS system have been derived under mutipath Rayeigh fading channes. REFERENCES 1 M. P. ennedy, G. oumban, G. is, and Z. Jako, Performance evauation of FM-DCS moduation in mutipath environments, IEEE Trans. Circuits Syst.-I, vo. 7, no. 1, pp , Dec.. Y. Xia, C.. Tse, and F. C. M. au, Performance of differentia chaos shift-keying digita communication systems over a mutipath fading channe with deay spread, IEEE Trans. Circuits Syst.-II, vo. 51, no. 1, pp , Dec.. 3. Wang, X. Min, and G. Chen, Performance of SIMO FM-DCS UWB system based on chaotic puse custer signas, IEEE Trans. Circuits Syst.-I, vo. 58, no. 9, pp , Apr. 11. Y. Fang, P. Chen, and. Wang, Performance anaysis and optimization of a cooperative FM-DCS UWB system under indoor environments, IET Networks, vo. 1, no., pp , May 1. 5 W. Xu,. Wang, and G. Chen, Performance of DCS cooperative communication systems over mutipath fading channes, IEEE Trans. Circuits Syst.-I, vo. 5, no. 1, pp. 196-, Jan Y. Fang, J. Xu,. Wang, and G. Chen, Performance of MIMO reay DCS-CD systems over Nakagami fading channes, IEEE Trans. Circuits Syst.-I, vo. 6, no. 3, pp , Mar C. Zhang,. Wang, and G. Chen, Promising performance of PA coded SIMO FM-DCS communication systems, Circuits, Systems and Signa Processing, vo. 7, no. 6, pp , Nov Y. yu,. Wang, G. Cai, and G. Chen, Iterative receiver for M-ary DCS systems, Accepted by IEEE Trans. Commmun., Apr G. addoum, M. Vu, and F. Gagnon, Performance anaysis of differentia chaotic shift keying communications in MIMO systems, in Proc. 11 IEEE ISCAS, May 11, pp P. Chen,. Wang, and F. C. M. au, One anaog STBC-DCS transmission scheme not requiring channe state information, IEEE Trans. Circuits Syst.-I, vo. 6, no., pp. 1-11, Apr N. S. Aagha, Cramer-Rao bounds of SNR estimates for BPS and QPS moduated signas, IEEE Commun. ett., vo. 5, no. 1, pp. 1-1, Jan P. Gao and C. Tepedeeniogu, SNR estimation for nonconstant moduus consteations, IEEE Trans. Signa Process., vo. 53, no. 3, pp , Mar F. Beii, R. Meftehi, S. Affes, and A. Stephenne, Maximum ikeihood SNR estimation of ineary-moduated signas over time-varying fatfading SIMO channes, IEEE Trans. Signa Process., vo. 63, no., pp. 1-56, Jan S. A. Hassan and M. A. Ingram, SNR estimation for M-ARY noncoherent frequency shift keying systems, IEEE Trans. Commun., vo. 59, no. 1, pp , Oct S. A. Hassan and M. A. Ingram, SNR estimation in a non-coherent BFS receiver with a carrier frequency offset, IEEE Trans. Signa Process., vo. 59, no. 7, pp , Juy S. M. ay, Fundamentas of Statistica Signa Processing, Estimation Theory. Upper Sadde River, NJ: Prentice Ha, Wang, G. Cai, G. Chen, Design and performance anaysis of a new mutiresoution M-ary differentia chaos shift keying communication system, IEEE Trans. Wireess Commun., vo. 1, no. 9, pp , Sept G. addoum, F. Gagnon, and F.-D. Richardson, Design and anaysis of a muti-carrier differentia chaos shift keying communication system, IEEE Trans. Commun., vo. 61, no. 8, pp , Aug J. G. Proakis, and M. Saehi, Digita Communications, McGraw-Hi, 7. A. Abe, W. Schwarz, and M. Gotz, Noise performance of chaotic communication systems, IEEE Trans. Circuits Syst.-I, vo. 7, no. 1, pp , Dec..