Mistiming Performance Analysis of the Energy Detection Based ToA Estimator for MB-OFDM

Transcription

1 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS Mistiing Perforance Analysis of the Energy Detection Based ToA Estiator for MB-OFDM Huilin Xu, Liuqing Yang contact author, Y T Jade Morton and Mikel M Miller Abstract In this letter, we aly energy detection based tieof-arrival ToA estiation to ulti-band orthogonal frequencydivision ultilexing signals We analyze the istiing erforance of the ToA estiator in the Nakagai- channel Analysis shows that the sloe of the robability of istiing curve increases with the nuber of subbands and the Nakagai araeter This is known in counications as diversity Siulations are carried out in various channels to corroborate our theoretical analysis I INTRODUCTION In recent years, ulti-band orthogonal frequency-division ultilexing MB-OFDM has attracted wide interest since it was roosed as one of the candidate technologies for the ultra-wideband UWB syste [], [] With the large bandwidth, MB-OFDM can achieve a high data rate In addition, due to the huge bandwidth, MB-OFDM signals can rovide orders of agnitude iroveent on the tiing recision coared to conventional narrowband signals In this letter, we will investigate the tie-of-arrival ToA estiation with MB-OFDM signals Usually, ToA estiation is carried out by a channel estiator This eans that the channel iulse resonse CIR is first recovered and then the delay of the first ath autoatically becoes the ToA estiate As the MB-OFDM signal consists of several subbands each only containing art of the channel inforation, these subband signals need to be cobined at the receiver to obtain full knowledge of the channel Generally, two strategies can be taken to cobine the ulti-band channel inforation Under the first strategy, the channel inforation fro all subbands is jointly utilized for channel and ToA estiation, which is often tered as the coherent cobining see, eg, [3], [4] With the second strategy, the CIR is first estiated for each subband and then channel estiates fro all subbands are averaged for a better final ToA estiate [5], which is called the noncoherent cobining The coherent cobining can usually rovide a better tiing resolution than the noncoherent cobining at the rice of a higher coutational colexity see, eg, [3], [4] However, there are scenarios where one has to adot the noncoherent cobining For exale, the channel inforation can not Huilin Xu and Liuqing Yang are with the Deartent of Electrical and Couter Engineering, University of Florida, PO Box 63, Gainesville, FL 36, USA Eails: xuhl@ufledu, lqyang@eceufledu, Tel: , Fax: Y T Jade Morton is with the Deartent of Electrical and Couter Engineering, Miai University, 6P Engineering Building, Oxford, OH 4556, USA Eail: ortonyt@uohioedu Mikel M Miller is with the Air Force Research Laboratory, Eail: MikelMiller@eglinafil be coherently cobined when rando hase rotations are resent after carrier deodulation Even worse is that the channel itself ay be indeendent across subbands due to the frequency deendent fading as the channel fading and disersion statistics vary with the frequency [6] For these reasons, we aly the noncoherent cobining or the energy detection ethod for ToA estiation with MB-OFDM signals As the analysis of such a roble is very colicated for general channel environents, we will confine our discussion to the Nakagai- fading channel A series of channel caaigns have shown that the Nakagai- distribution can fit the channel easureent results very well [7] In addition, the Nakagai- distribution is a general fading odel, araeters of which can be adjusted to incororate several fading environents, such as the Rayleigh distribution = and the one-sided Gaussian distribution = 5 Therefore, the analysis and results of this letter are alicable to a wide range of channel fading environents Based on the analysis of the airwise istiing robability, we will rove that the tiing erforance can be iroved with MB signals by exloiting the diversity across subbands Although [5] touches uon a siilar issue, it assues a Rayleigh fading channel with indeendent ultiath coonents and a unifor ower delay rofile PDP Instead of these liiting constraints, we consider in this letter the Nakagai- fading channel with arbitrary ultiath correlation and arbitrary PDP Following the ethod in [8, Ch ], we calculate the airwise istiing robability based on the oent generating functions MGF of the robability density function PDF of the instantaneous signal to noise ratio SNR With the union bound analysis, we can derive the diversity order of the istiing robability curve II ENERGY DETECTION BASED TOA ESTIMATOR In this section, we will first introduce the MB-OFDM syste that is investigated in this letter Architectures of the MB-OFDM transitter and receiver are illustrated in Fig a and b as secified in [] Different fro the basic OFDM syste, for MB-OFDM, the baseband signal is carrier odulated on one of the B frequency bands at the transitter according to the frequency hoing attern, and the received wavefor is carrier deodulated accordingly at the receiver Note that the indeendent channel across subbands can also be induced by the dense ultiath However, this tye of indeendency will not revent one fro coherently cobining subband channel inforation

2 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS Pilot bits ex jπf ct Fig Constellation Maing ADC IFFT Coarse Sync Tie-Frequency Code GI Insertion Tie-Frequency Code a b GI Reoval DAC Decision FFT exjπf ct ToA Estiation Equalization MB-OFDM transceiver block diagras a Transitter; b Receiver Let s b t denote the baseband signal transitted on the bth subband The noise-free baseband received signal r b t can be exressed as r b t = s b t g b t c b t, b [, B] where denotes the convolution oeration, g b t is a linear tie invariant filter including ossible transit and receive filters as well as the bandliiting effect of transitting a finite bandwith signal, and c b t is the baseband channel for the bth subband L c c b t = A b,l δt τ b,l, b [, B] l= where {A b,l } Lc l= and {τ b,l} Lc l= are alitudes and delays of the L c channel aths Note that due to the frequency deendent fading [6], c b t can be indeendent across subbands The baseband received signal r b t is saled at the rate of /T s which is essentially the bandwidth of the transitted signal s b t The noise-free discrete tie sales r = r b t t=nts can be exressed as r = s h, b [, B] 3 where s is the nth sale transitted on the bth subband and h is the equivalent discrete tie channel h = [c b τ g b τ] τ=nts, b [, B], n [, L ] 4 with L being the nuber of channel tas Reresenting the delay sread of the channel, L is assued to be known at the receiver In order to facilitate the ToA estiation, the discrete tie channel 4 is first estiated for all B subbands The nth ta of the estiated channel for the bth subband can be exressed as follows: h = h L + η, n [L, L + L ] η, n [, L ] and n [L + L, L + L + L ], b [, B] 5 with L < L and L < L Due to the lack of tiing inforation, the channel is estiated with a larger nuber of tas than L Aong the L + L + L tas in 5, L consecutive tas contain the noise-containated channel inforation and the rest L + L tas only contain noise Following [5], we also assue that the noise η is indeendent zero ean circularly syetric colex Gaussian with unit variance In [5], the following assutions are ade: channel tas are indeendent within each subband; the channel is uncorrelated over all B subbands; 3 channel tas are Gaussian rando variables, and the channel has a unifor PDP In this letter, we will consider the correlation aong channel tas both inside each subband and over all subbands This is ore realistic since subbands always share soe coon channel inforation In addition, the analysis is erfored when the channel ta h has a Nakagai- distribution with arbitrary PDP After the channel is estiated, a sile energy detection based synchronizer can be adoted for ToA estiation In articular, for a single band, this estiator detects the start of the channel by seeking the axiu total energy of a length L segent in the channel estiate sequence given by 5 With the availability of ultile subbands, this ToA estiator sily cobines the energy fro all subbands Then, the index of the first channel ta can be estiated by k = arg ax B +L n= h 6 According to the synchronization criterion 6, istiing occurs when the following inequality holds for any l [, L ] or l [ L, ]: B L +L+l n=l +l h > B L +L n=l h 7 After the coon ters are canceled in 7, the inequality becoes and B B L +L+l n=l +L L n=l +l η > η > B B L +l n=l h L + η, L +L n=l +L+l for l [, L ] h L + η, for l [ L, ] III ANALYSIS OF MISTIMING PROBABILITY In this section, we will analyze the istiing robability of the energy detection based ToA estiator We will first derive the airwise robability of istiing by an arbitrary nuber of l tas Then we use the union bound to obtain an uer bound of the istiing robability of the ToA estiator The high SNR aroxiation of the uer bound will clearly reveal the relationshi between the channel araeters and the tiing erforance In the following, we will focus on the l > case The analysis can be directly alied to the l < case 8 9

3 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 3 L+l Let v l = B n=l h L + η and y l = B L+L+l n=l η +L, l [, L ] The airwise robability of istiing by l tas is Pry l > v l, l [, L ] which can be further exressed as f V v l f Y y l dy l dv l v l where f V v l and f Y y l are PDFs of v l and y l, resectively Since variance of the noise is unity, ie, E{η } =, b and n, the rando variable y l has a chi-square distribution with Bl degrees of freedo f Y y l = Bl! y l ex y l Using the result in [9], we have the airwise robability of istiing by l tas n= f V v l v n l ex v l dv l 3 Before deriving P l, we first calculate the conditional airwise robability of istiing, given the channel fading coefficients For certain λ l = B l n= h, the rando variable v l obeys a noncentral chi-square distribution with PDF [8, Ch ] f V v l ; λ l = vl λ l ex v l + λ l I v l λ l 4 where I a x is the odified Bessel function of the first kind Using 3 and roerty of the Lalace transfor: L{vl nf V v l ; λ l } = n F n V s; λ l, the conditional airwise robability of istiing P l λ l can be exressed as P l λ l = n= = n= n f V v l ; λ l v n l ex v l dv l F n V ; λ l 5 where F V s; λ l is the Lalace transfor of f V v l ; λ l Fro the MGF, the Lalace transfor of f V v l ; λ l can be exressed as see [8, Ch 9] ex sλl F V s; λ l = L{f V v l ; λ l } = E{e sv +s l ; λ l } = + s Bl 6 The nth order derivative of F V s; λ l can be exressed as n F n n V s; λ l = f s; λ lf n s 7 = where f s; λ l = ex sλl +s and f s = + s Bl According to the Faà di Bruno s forula [], the th order derivative of f s; λ l can be exressed as f s; λ l =f s; λ l λ k l B,k g 8 s, g s,, g k+ s with B,k being the Bell olynoial and gs = s +s Alying 8 to 7 and cobining ters with the sae order of λ l, we have F n V s; λ l = ex sλl n λ k l f n,k s 9 + s with f n,k s being a function of s n n f n,k s = f n s =k B,k g s, g s,, g k+ s The conditional airwise robability of istiing can then be silified using 5 and 9 P l λ l = ex λ l c B,l,k λ k l where c B,l,k = n n=k f n,k The airwise istiing robability P l is the exectation of P l λ l f Λ λ l P l λ l dλ l where f Λ λ l is the PDF of the rando variable λ l Sae as Eq, the integral of can also be obtained with the aid of MGF The channel tas undergo Nakagai- fading with PDF [8] γ fγ ; = A γ Γ ex γ 3 A γ where γ = h, b [, B], n [, L ], A reresents the PDP effect of the nth channel ta in the bth frequency band, γ is the average SNR and Γx is the Gaa function For correlated Nakagai- distributed rando variables γ, a sile for PDF can not be found for λ l = B l n= γ [], but the Lalace transfor of λ l can be written as [] where ρ i,j is the ower correlation between γ and γ with i = b l+n and j = b l+n Using roerty of the Lalace transfor, we can obtain the airwise istiing robability P l [cf and ] k c B,l,k F k Λ 5 where c B,l,k is defined after Eq At high SNR as γ, the Lalace transfor F Λ s can be aroxiated by [] F Λ s D B,l B l n= sa γ 6 where D B,l denotes the deterinant ter in 4 when γ Then P l can be aroxiated at high SNR by γ Bl B DB,l l n= A c B,l,k k Bl! Bl k! 7

4 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 4 F Λ s = det B l n= + sa γ ρ, + sa, γ ρ, + sa, γ ρ, + sa B,l γ ρ, + sa B,l γ ρ, + sa ρ,, γ + sa, γ 4 Fro 7, the airwise robability of istiing by l tas can be aroxiated at high SNR by C B,,l γ Bl, l [, L ] 8 where C B,,l = Bl D B,l B l n= A c B,l,k k Bl! Bl k! is a constant indeendent of γ For l [ L, ], the airwise robability of istiing can be derived in the sae way which has the for of C B,,l γ Bl, l [ L, ] Fro the union bound, the robability of istiing P t is uer bounded by the suation of the airwise robability of istiing This eans that L P t < C B,,l γ Bl + l= l= L C B,,l γ Bl 9 When the SNR γ is high, the right hand side of 9 aroxiates C B,, + C B,, γ B As a result, the sloe of the robability of istiing curves is G d = B in log-log scale In the field of counications, this sloe is often defined as the diversity of the syste error erforance Eq 9 ilies that the diversity gain of the estiator is G d = B which increases with both the subband nuber and the Nakagai- araeter This shows that by use of the noncoherent cobining, the energy detection based ToA algorith can achieve a higher diversity gain, roortional to the nuber of subbands and the diversity achieved in each channel IV SIMULATIONS Siulations are first carried out for the discrete tie channel with L = indeendent Nakagai- distributed tas The channel has an exonentially decaying PDP with the last ta being db weaker than the first ta The average ower of the first channel ta is noralized Assue that due to the lack of tiing inforation, L = L = 5 ure noise ters are involved in the estiated channel at the front and rear arts The robability of istiing is siulated for the energy detection based ToA estiator when the channel is indeendent for different subbands Analysis has shown that by using B subbands, the energy detection based ToA estiation can obtain a diversity gain of B in the Nakagai- channel This is verified by siulations, as the robability of istiing curves with the sae B are roughly arallel to each other see Fig In addition, Probability of istiing 3 4 B=6, = B=3, = B=, =3 B=, =6 Union bounds SNR db Fig The robability of istiing with energy detection for B = 6 The araeter B is the nuber of subbands and is the Nakagai- fading araeter Probability of istiing B=4 B= B= SNR db Fig 3 The robability of istiing with energy detection for Saleh- Valenzuela channels The araeter B is the nuber of subbands

5 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 5 given B, the arallel curves have different horizontal shifts In the literature of counications, this shift is defined as the coding gain of the robability of istiing curves The siulation result is reasonable since a higher coding gain can be obtained when ore signal energy is collected fro ore subbands Union bounds are also lotted in Fig by using Eq 9, which confirs that these bounds are tight at high SNR We then siulate the ToA estiator in the ore realistic Saleh-Valenzuela S-V channel odel [3] The channel has a cluster structure with the cluster arrival rate of Λ = / ns and the ray arrival rate of λ = / ns The clusters and rays inside each cluster decay exonentially with tie constants Γ = 6 ns and γ = ns, resectively The channel rays are generated as rando variables with indeendent Rayleigh distributed alitudes and indeendent uniforly distributed hases Note that in addition to the rando and irregular PDP, closely saced ultiath arrivals will also give rise to correlations aong the discrete tie equivalent channel tas MB-OFDM signals with the bandwidth of MHz are odulated to different subbands around the 5 GHz center frequency and assed through the channel For each subband, the discrete tie channel is generated by saling the subband channel at the rate of /T s = MHz Since each subband only contains art of the entire channel inforation, the equivalent discrete tie channels differ across subbands but are correlated Nuerical analysis shows that given these channel araeters, the correlation coefficient between subbands can be u to about 5 As shown by Fig 3, given the Nakagai araeter =, the diversity gain increases linearly with the nuber of subbands available B This again verifies the diversity gain analysis of the energy detection based ToA estiator [7] H Suzuki, A statistical odel for urban radio roagation, IEEE Trans on Counications, vol 5, no 7, , July 977 [8] M K Sion and M-S Alouini, Digital Counication over Fading Channels John Wiley & Sons, [9] J N Pierce, Theoretical diversity iroveent in frequency-shift keying, Proc IRE, vol 46, no 5, 93 9, May 958 [] W P Johnson, The curious history of Faà di Bruno s Forula, Aerican Matheatical Monthly, 7 34, March [] N C Beaulieu and A A Abu-Dayya, Analysis of equal gain diversity on Nakagai fading channels, IEEE Trans on Counications, vol 39, 5 34, February 99 [] Y-C Ko, M-S Alouini, and M K Sion, Outage robability of diversity systes over generalized fading channels, IEEE Trans on Counications, vol 48, no, , Noveber [3] A Saleh and R Valenzuela, A statistical odel for indoor ultiath roagation, IEEE J Select Areas Coun, vol, no 5, 8 37, February 987 V CONCLUSIONS In this letter, we analyzed the robability of istiing erforance for an energy detection based ToA estiator for MB-OFDM systes in Nakagai- channels Theoretical analysis shows that a higher diversity gain can be achieved by using ore subbands in channels with a lower aount of fading larger Siulations in both the Nakagai discrete tie channel and the S-V channel odel have corroborated our theoretical analysis REFERENCES [] IEEE 85 WPAN High Rate Alternative PHY Task Grou 3a TG3a [] ECMA-368: High rate ultra wideband PHY and MAC standard, st edition, Deceber 5 [3] E Saberinia and A H Tewfik, Ranging in ultiband ultrawideband counication systes, in IEEE Trans Vehicular Technology, vol 57, no 4, 53 53, July 8 [4] H Xu, C-C Chong, I Guvenc, and L Yang, High-resolution TOA estiation with ulti-band OFDM UWB signals, in Proc of International Conf on Counications, Beijing, China, May 9-3, 8 [5] C R Berger, S Zhou, Z Tian, and P Willett, Perforance analysis on an MAP fine tiing algorith in UWB ultiband OFDM, IEEE Trans on Counications, vol 56, no, 66 6, October 8 [6] W Q Malik, D J Edwards, and C J Stevens, Frequency deendence of fading statistics for ultrawideband systes, IEEE Trans on Wireless Counications, vol 6, no 3, 8 84, March 7