Performance Analysis of Ultra-Wide Band Impulse Radio (UWB-IR) with Super-Orthogonal Turbo Codes (SOTC)

Transcription

1 rformance Analyi of Ultra-Wide Band Impule Radio UWB-IR with Super-Orthogonal Turbo Code SOTC Uman Riaz, Man-On Pun and C.-C. Jay Kuo Abtract Low-complexity low-rate uper-orthogonal turbo code SOTC are propoed in thi work to replace the implicit repetition code RC in an ultra-wide band impule radio UWB-IR ytem to improve BER, tranmiion range and ytem throughput. Variou receiver including matched-filter and RAKE receiver are adopted for data detection in the additive white Gauian noie AWGN channel and the indoor IEEE a channel. The performance of SOTC-encoded UWB i analyzed and validated by computer imulation. It i hown that the SOTC-baed UWB-IR ytem can achieve ignificant performance improvement over the conventional DS- UWB ytem encoded by RC. I. INTRODUCTION Ultra-wide-band impule-radio UWB-IR i a technique that ue very narrow low duty cycle radar-like pule of the order of nano econd for information tranmiion [1]. The employment of thee narrow pule reult in occupancy of everal Giga-Hertz of bandwidth. UWB-IR can be viewed a a bandwidth uncontrained but power pectral denity PSD contrained ytem. To increae the effective energy per ymbol, multiple pule namely, pule carrying the ame information ymbol are tranmitted. From the coding perpective, thi i equivalent to the repetition code RC. However, a large value of reduce ytem throughput and tranmiion energy efficiency per information ymbol for a given pule duration. Thi preent a evere problem to highpeed hort-range indoor application or low-peed long-range low-power outdoor application uch a enor network. In thi work, we focu on the latter application that have a trict contraint on tranmiion energy per ymbol. Furthermore, it i well known that increaing doe not provide any coding gain for RC if the SNR value meaured in term of E b /N 0 remain contant. Hence, more ophiticated coding cheme are deired to replace the exiting RC of UWB-IR to improve ytem throughput, the tranmiion range and energy efficiency. Some effort have been made in the pat. For example, low-rate uperorthogonal convolutional code SOCC [2] and an adaptive combination of turbo code and RC [3] were tudied in the literature. In our previou work [4], we propoed to replace the RC of UWB-IR with the uper-orthogonal turbo code SOTC. Being different from the conventional turbo code that have to correlate the received equence with all poible trelli branche, SOTC employ the invere Walh Hadamard Tranform IWHT in calculating the trelli branch metric, which reult in low decoding The author are with Department of Electrical Engineering and Signal and Image Proceing Intitute, Univerity of Southern California, Lo Angele, CA , USA. uriaz@uc.edu, mpun@ipi.uc.edu and cckuo@ipi.uc.edu complexity [5]. Moreover, we compared the performance of the SOTC-encoded UWB ytem uing two type of receiver, namely, the RAKE receiver and a imple matched filter MF receiver, in the AWGN and the IEEE SG3a UWB channel. The uperior performance of the SOTC encoded UWB ytem wa confirmed by computer imulation in [4]. However, theoretical analyi of the ytem performance wa not covered in [4]. In thi work, we firt derive the analytical performance bound for the SOTC-encoded UWB ytem. For the mediumto-high SNR range, the aymptotic performance bound can be obtained by reorting to the effective free ditance of contituent code of SOTC. Then, ince the SOTC-encoded UWB ytem operate primarily at the low SNR range, we ue a emi-analytical method to analyze the ytem in IEEE SG3a UWB channel without inter-ymbol interference ISI at low SNR. Furthermore, we invetigate the effect of on ytem throughput, bit error probability BEP and the tranmiion range for both RC and SOTC encoded UWB-IR ytem. Finally, analytical reult are verified by imulation. II. SYSTEM MODEL A. Signal and Channel Similar to [4], we conider the SOTC encoded UWB-IR uing DS-BPSK modulation in a ingle uer cenario. The tranmitted ignal i given by N 1 t = Eg b j a i gt jt it c, 1 j= where E g = E b i the energy of the tranmitted waveform with E b being the energy per bit and a j { 1, 1} i the j-th data ymbol. a i { 1, 1}, 0 i, i the peudorandom direct equence code while gt i a Gauian pule hape with g2 tdt =1. T c and T are the chip and the ymbol duration, repectively where T c in the equel i alo referred to a frame duration. Since each information bit i tranmitted with pule, the effective bit rate or throughput i R =1/T = T c 1. Two channel model are conidered in thi work. They are the traditional AWGN and the UWB multi-path channel pecified by IEEE a [6]. The IEEE SG3a channel i a modified S-V model that can be written a [7] h multi-path t =X N Kn n=1 k=1 i=0 α n,k δt T n τ n,k, 2 where X i the total multi-path gain and T n i the exponentially-ditributed arrival time intance of the n-th cluter. Parameter τ n,k and α n,k are the exponentially-ditributed

2 arrival time intance and the amplitude of the k-th ray within the n-th cluter, repectively. α n,k i defined a α n,k = p n,k β n,k, 3 where p n,k i a dicrete binary random variable with equally probable ±1 and β n,k i a log-normal random variable. To maintain the overall channel gain unity, we aume X = 1 and n,k α2 nk =1in the equel to take into account the multipath energy capture without the pathlo factor. For the IEEE a channel, the received ignal can be written a rt = N Kn n=1 k=1 α n,k t T n τ n,k +vt, 4 where vt i zero-mean complex AWGN of variance σ 2. While the performance of RC i determined by it free ditance, d free-rc with BEP of Q 2N E g N 0, the aymptotic performance of SOTC with uniform interleaver analyi depend on the effective free ditance of it contituent SOCC, d free-eff [8]. In the following, we firt derive the effective free ditance of the contituent SOCC baed on minimum weight w min =2. Then, the reulting d free-eff i exploited to obtain the aymptotic performance of SOTC encoded UWB-IR in the AWGN and the ISI-free UWB channel. Since the aymptotic analyi i only applicable to the medium-to-high SNR range, a emi-analytical bound i propoed to predict ytem performance at low SNR. Finally, we analyze the tradeoff among ytem throughput, BEP and tranmiion range for both RC and SOTC encoded UWB-IR ytem. B. Receiver Structure Two type of receiver are conidered in thi work, namely, the MF-baed receiver and the RAKE receiver. Among the RAKE receiver, three type of receiver are examined: All RAKE ARAKE that combine all the multi-path alo referred to a ideal RAKE, Selective RAKE SRAKE that combine L path with the tronget ignal power and Partial RAKE PRAKE that elect the firt L p path. The correlation mak ued in the RAKE receiver can be expreed a m corr-rakej = L ω l m corrj t τ l, 5 l=1 where {τ l ; l =1, 2,,L} i a ubet of path delay choen from τ n,k and ω j i the weight for the j-th multipath. The correlation mak, m corr t, i given a m corrj t = N 1 E g a i gt jt it c. 6 i=0 In thi tudy, we conider an ISI-free IEEE a CM1 channel [6]. Let j t i the ignal correponding to the jth information bit in 1 and r j t be the correponding received ignal in 4. Then, for a given multi-path delay τ L,M, the output of the RAKE receiver correponding to the j-th information bit can be approximated a N 1 i+1tc ξ j = E g r j tm corrj t τ L,M dt 7 i=0 it c E g b j αlm. 2 8 By capturing more multi-path, the RAKE receiver can exploit the implicit time diverity exiting in a multi-path channel. III. PERFORMANCE ANALYSIS We conider an SOTC cheme coniting of two recurive ytematic SOCC with primitive feedback polynomial [1, 5] oct that can be expreed a GD =1+D 2, where D if the one-element delay of the hift regiter. The polynomial repreentation for a rate- 1 4 SOCC can be determined a [ 1+D 3 1+D + D 3, 1+D2 + D 3 1+D + D 3, 1, 1+D + D2 + D 3 ] 1+D + D 3. 9 A. Effective Free Ditance S initial m 2 m-2 WD 2 m-1 D S 2 m-1 +1 S 1 D 0 WD 2 m-1 S final Fig. 1. Simplified diagram for all enumeration of input weight 2 We employ the input-output weight enumerating function IOWEF to obtain an cloed-form expreion of d free-eff. Thank to the pecial tructure of the SOCC with orthogonal output, the effective augmented tate diagram can be reduced to a tate tranition diagram, baed on which all poible path of input weight 2 can be calculated for given m. Fig. 1 how the implified diagram for SOCC uing the orthogonal branch metric, where W and D are polynomial whoe degree repreent the weight of the input and the output equence, repectively. To calculate the enumerator function for the minimum weight codeword correponding to input weight 2 and it compound error path, we conider a hift regiter with memory m. Letd m1 be the minimum weight error equence and d mi with i 1be the compound error equence. We have d mi = 2 m + iɛ m 2 m 2, 10 where i {1, 2, }. For SOCC with the primitive feedback polynomial, d free-eff i given a d free-eff m = min[d mi ]=2 m 2 [4 + ɛ m ], 11 where ɛ m capture the effect of the output weight increae during the intermediate tage a a function of m. ɛ m can be iteratively computed by the following equation with ɛ 1 =0. ɛ m = ɛ m 1 +2 m B. AWGN Channel The aymptotic average upper bound on BEP of a Turbo Code in AWGN for medium to high SNR i given a [8] N P b e = W w A PCC w, Z Z=exp 1 E b, 13 N N 0 w=1

3 where N i the interleaver ize, w i the weight of the input equence and A PCC i the conditional IOWEF of the equivalent turbo code in a PCCC configuration. For a uniform interleaver analyi, A PCC w, Z igivenby[8] A PCC w, Z w! N 2ηmax w [Aw, Z, η max ] 2, 14 η max where Aw, Z, n i the parity enumerating function with n-concatenated error event caued by input weight w. η max i the maximum number of error event in the trelli caued by input weight w. For SOTC with recurive SOCC, the aymptotic performance i determined by the parity enumerating function A2,Z,1 that give the parity weight of all poible path generated by input weight w =2. Calculation of A2,Z,1 for a given SOCC involve all enumeration of input weight w = 2. Baed on Fig. 1 and 11, all poible enumeration for input weight w =2in the indeterminate Z can be obtained a A2,Z,1 = Z z min+pɛ m2 m 2 Z z min =, 15 ɛm2m 2 1 Z p=0 where z min = d free m 2. Uing 13-15, we have the aymptotic BEP for SOTC with a uniform interleaver a N 2 P b e k=1 2k N 1 Z 2+2zmin k Z=exp k 1 Z ɛm2m 2 2k 1 E b. N N 0 16 Thu, a large increae the length of the SOTC orthogonal output of the trelli branche, which i effectively tranlated into a SOTC proceing gain with BEP monotonic in. C. UWB Channel To derive the aymptotic bound in UWB channel, we firt compute the pairwie error probability PEP of the coded equence. We conider turbo code a equivalent block code of length M = N with N being the interleaver length and ignoring the trailing bit to terminate the contituent code to zero tate due to large interleaver length aumption. The probability of error in a coded equence i computed with reference to the all zero coded equence. Let c i be the M 1 coded equence with c 0 a the all zero coded equence. The correponding BPSK modulated codedword equence i x i where x i { 1, 1} and x i =2c i 1 with c i {0, 1}. Given the channel model in 2 and the RAKE correlation mak in 5, the firt bit output from RAKE i given a L 1 r 0 = x 0 E g αl 2 + L 1 E g α l n 0 l, 17 where n 0 l = nt,gt τ i i the inner product between nt and gt τ i.letñ 0 = L 1 E g α ln 0 l, where ñ 0 ha zero mean and variance given a σ 2 ñ 0 L 1 = E g σ 2 αl where {α l } i the ubet of {α n,k }. Collecting the RAKE output of all bit of the coded equence in an ISI-Free channel the PEP can be hown a P e c 0 c i α exp E L 1 b d αl 2, 19 N 0 where d i the hamming ditance between the pair of codeword and α = [α 0,,α L 1 ] T i the collected vector of channel impule repone and L i the number of RAKE finger. The unconditional pairwie error probability i P e c 0 c i = E α {P e c 0 c i α}, the cloed form expreion of which i non-trivial to evaluate due to log-normal ditribution of channel amplitude and hence we proceed with the conditional pairwie error probability. Let c 0 c i α be the pairwie error probability in 19 without the hamming weight d. Finally, replacing Z in 16 with the conditional PEP P e c 0 c i α, we have the average analytical bound on the conditional error probability N 2 2k P b e α N 1 Z 2+2z min k. k 1 Z ɛm2m 2 2k Z= P ec 0 c i α k=1 20 To conclude, the aymptotic performance bound in AWGN i given by E α {P b e α}, where E α { } i the expectation operator with repect to α. Since the cloed-form expreion in 20 i non-trivial, we evaluate the aymptotic upper bound by reorting to numerical method. D. Semi-Analytical Bound SAB The aymptotic analyi hown in Sec. III-B and Sec. III-C i only applicable to the medium-to-high SNR value E b /N 0 [5]. For low SNR analyi, one can either reort to the iterative convergence analyi uing EXIT chart or the imple bound baed on IOWEF of the code. Inpired by [9], we propoe a imple but effective emi-analytic method to predict the ytem performance for low SNR. Thank to the ISI-free aumption, the ignal at each RAKE finger undergoe an independent flat fading. From 17, we can eaily how that the effective SNR for a particular realization of α i given by SNR eff 1 E b L αl N 0 Similar to [9], we approximate the BEP value conditioned on given α, P e,eff SNR eff α, with the BEP at SNR eff obtained in the AWGN channel. Next, the emi-analytical BEP i computed by averaging P e,eff SNR eff α over a ufficient number of channel realization a P e,sab = E α {P e,eff SNR eff α}. 22 By exploiting the BEP curve in AWGN together with 21 and 22, we can eaily predict the BEP performance for any given multipath channel. However, being different from [9], our analyi i applied to coded ytem. Since coded ytem have the optimal performance in AWGN, the emi-analytical BEP obtained in 22 correpond to the performance upper bound with perfect interleaving in fading channel. In the equel, we refer thi emi-analytical bound a SAB.

4 E. Effect of Here, we invetigate the effect of on ytem throughput for a given ditance and a pecified BEP. The average tranmit power i contrained by the following inequality P tavg τ d P max, 23 where τ d = τg T c i the pule duty cycle with τ g being the pule duration and P max = f 2 f 1 P fdf i the maximum allowed tranmit power with P f being the power pectral denity of the frequency band over [f 1,f 2 ]. The propagation lo in the free pace i a function of ditance l and frequency f of the following form Ll, f =4πlf 2 /c The average received ignal power can be obtained from and read c 2 Pmax f 2 f 1 P ravg τ d 4π l 2 25 f 2 f 1 Equating the received energy to noie ratio with the target energy to noie ratio, we have P ravg T c N o f 2 f 1 = M L Eb N 0 target 1, 26 f 2 f 1 where M L i the link margin. Subtituting 25 into 26 yield [ Eb c ] 2 Pmax f 2 f 1 N T c M L τ d N 0 4π l 2. f target 2 f 1 N 0 27 Finally, we get [ 16π l 2 2 f2 f 1 N0 M L E b /N 0 target c 2 f 2 f 1 T c P max τ d ]. 28 Since the bit interval time i T, the correponding throughput i R =1/T =1/ T c. IV. SIMULATION RESULTS Computer imulation have been conducted to validate our analytical reult. The SOTC decoder employ the Max-Log- Map/Min-Sum algorithm with an interleaver of ize N = Unle otherwie pecified, four iteration are performed in SOTC decoding. The BEP performance i averaged over 1000 block. A. AWGN Channel Fig. 2 how the imulation reult for SOTC- and RCencoded ytem with =3and 7 in AWGN. Due to the orthogonality of the concatenated code, the ytem performance with SOTC improve monotonically with the increae of. Note that the performance of RC i independent of for afixede b /N 0. Fig. 2 how that SOTC with =7ha more than 8 db improvement over RC at BEP of.the aymptotic analytical upper bound computed from Eq. 16 i alo hown againt the imulation reult. Thee aymptotic bound are only ueful in the medium-to-high E b /N 0 range where imulation i not a practical mean to meaure ytem performance [8]. Fig RC Simulated =3 Simulated N =3 Aym. Bound N =7 Simulated N =7 Aym. Bound E b /N 0 db SOTC and RC receiver in the AWGN channel. B. IEEE a UWB Channel We compare the performance of SRAKE for = 3, 7 and L =10in Fig. 3. A hown in Fig. 3, the performance of SRAKE improve with. The aymptotic bound were obtained from 20. In contrat with the aymptotic bound that are only applicable in the high E b /N 0 region, the SAB derived from 22 can accurately predict ytem performance in the low E b /N 0 region =7 =3 =3 Simulated =3 SAB =3 Aym. Bound =7 Simulated =7 SAB =7 Aym. Bound E b /N 0 db Fig. 3. The SRAKE performance in the UWB Channel with L =10. Fig. 4 how the ytem performance a a function of the SRAKE finger number for different value of.iti apparent that uing more RAKE finger for a given can capture more path, which reult in better ytem performance. Furthermore, the ytem performance increae with. However, Fig. 4 indicate that the performance improvement from = 7 to = 15 i much le than that from = 3 to = 7. Thi i mainly due to the fact that the energy per pule for = 15 i roughly half of that for = 7. A a reult, increaing the ame number of RAKE finger for =15provide much le captured energy than that for =7. Hence, the marginal performance improvement diminihe a increae. C. Impact of In thi experiment, we invetigate the impact of on the tranmiion ditance and ytem throughput baed on 28.

5 RC SOTC =3 Simulated =7 Simulated =15 Simulated Ditancem Number of RAKE finger Fig. 4. S-RAKE ytem performance a a function of the finger number for different at E b /N 0 =1dB ThroughputMbp Fig. 6. Throughput a a function of the tranmiion ditance in the ISI-free UWB channel for BEP=. We primarily invetigate the effect in AWGN with T c =3n and τ g =1n for a link margin M L db =7. Fig.5 how the tranmiion ditance a a function of for a given target BEP of in the AWGN channel. The correponding E b /N 0 target for RC are calculated from cloed form expreion of BEP for RC and the E b /N 0 target for SOTC ha been calculated from it performance curve for BEP V E b N 0 uing emi-analytical bound. It reveal that SOTC can achieve a ignificantly longer tranmiion ditance than RC for the ame throughput i.e.,. In particular, the improvement increae exponentially with. However, a large increae the number of trelli tate, and ubequently the memory required in decoding. V. CONCLUSION The performance of a DS-UWB ytem coded with SOTC in AWGN and ISI-free UWB channel wa tudied in thi work. The SOTC cheme i attractive due to it low rate and lowdecoding-complexity feature. The aymptotic performance bound for large SNR value were derived. Then, a emianalytic method wa propoed to predict the ytem performance at the low-to-medium SNR range. Through analyi and imulation, it wa hown that the SOTC-encoded UWB ytem require fewer pule per information bit to achieve a given BEP than the conventional RC-encoded ytem, which lead to higher ytem throughput and/or a longer tranmiion range depending upon the application contraint.. Ditance m RC SOTC Fig. 5. Tranmiion ditance a a function of for a given BEP=. We ee from Fig. 5 that the SOTC-encoded ytem require a much maller than the RC-encoded ytem to achieve the ame tranmiion ditance, which lead to higher throughput. Thi effect i further magnified in Fig. 6. In the ISI-free UWB channel, we have to increae the frame duration, T c,to avoid interference from the adjacent ymbol. A a reult, the throughput in the ISI-free UWB channel depend on both and T c. Fig. 6 indicate that the RC receiver uffer a ignificant amount of throughput lo due to the increae of T c = 40 n. For the SOTC-encoded ytem, the advere impact of a larger on throughput i alleviated by the fact that SOTC with a larger ha better BEP performance. A a reult, the SOTC-encoded ytem tranmiion range i much higher than the RC-encoded ytem in the ISI-free UWB channel for the ame throughput and BEP. REFERENCES [1] M. Z. Win and R. A. Scholtz, Impule radio: how it work, IEEE Commun. Lett., vol. 2, no. 2, pp , Feb [2] A.R. Forouzan, M. Nairi-Kenari, and J.A. Salehi, rformance analyi of time-hopping pread-pectrum multiple-acce ytem: uncoded and coded ytem, IEEE Tran. on Comm., vol. 1, no. 4, pp , Oct [3] N. Yamamoto and T. Ohtuki, rformance evaluation of adaptive internally turbo coded ultra wideband-impule radio AITC-UWB-IR in multipath channel, VTC2004 Fall, Lo Angele, CA, vol. 2, pp , Sep [4] U. Riaz, M.O. Pun, and C.-C. Jay Kuo, Ultra-wide band impule radio UWB-IR with uper-orthogonal turbo code otc, 39th Ailomar, vol. 49, no. 1, pp , Nov [5] P. Komulainen and K. hkonen, A low complexity uperorthogonal turbo-code for CDMA application, Proc, IEEE PIMRC 96, Taipei, vol. 2, pp , Feb [6] IEEE WPAN high rate alternative PHY tak group 3a TG3a, [7] A. Saleh and R. Valenzuela, A tatitical model for indoor multipath propagation, JASC, vol. 5, no. 2, pp , Feb [8] S. Benedetto and G. Montori, Deign of parallel concatenated convolutional code, IEEE Tran. Com., vol. 44, no. 5, pp , May [9] A. Rajewaran, V.S. Somayazulu, and J.R. Foerter, RAKE performance for a pule baed UWB ytem in a realitic UWB indoor channel, Proc. ICC, vol. 5, pp , 2003.