Weighted Correlation Receivers for Ultra-wideband Transmitted Reference Systems

Transcription

1 Weighted Crrelatin Receivers fr Ultra-wideband Transmitted Reference Systems Yi-Ling Cha and Rbert A. Schltz Abstract This paper derives the average lielihd rati test ALRT receiver fr ultra-wideband transmitted reference systems in multipath envirnments with Rayleigh path strength mdels. A theretical weighting functin is btained by apprximating the ALRT receiver structure, and can be applied t crrelatin receivers. Cmpared t a simple crrelatin receiver [5], results shw that theretical weighted crrelatin receivers are mre rbust t different envirnments, and can reduce bit errr prbabilities even fr IEEE 8.5.3a ultra-wideband channel mdels with lgnrmal path strength mdels. I. INTRODUCTION Ultra-wideband UWB impulse radi systems, because f their fine time-reslutin capability, require Rae receivers with tens r even hundreds f crrelatin peratins t tae full advantage f the available signal energy [], []. Instead f using a Rae receptin, Hctr and Tmlinsn prpsed an UWB transmitted reference TR system with a simple receiver structure t capture all the energy available in a UWB multipath channel [3]. In this TR system, a reference pulse is transmitted befre each data-mdulated pulse fr the purpse f determining the current multipath channel respnse. The prpsed crrelatin receiver crrelates the data signal with the reference t use all the energy f the data signal withut requiring additinal channel estimatin and Rae receptin, and nly an analg delay line is needed t align the reference and signal pulses. The sampling frequency f the analg-t-digital cnverter is als reduced. Therefre, a TR system can simplify the implementatin nt nly by eliminating the Rae receptin and channel estimatin, but als by reducing the digital sampling frequency. This simple receiver structure has ne majr drawbac, namely the transmitted reference signal used as a crrelatr template is nisy. Averaging multiple reference pulses t prduce a cleaner template can imprve the receiver perfrmance [4], [5] but it needs t stre and prcess received wavefrms and might nt be pssible t implement using analg devices. In rder t imprve bit errr prbability BEP perfrmance and als maintain a simple receiver structure, we start by deriving the average lielihd rati test ALRT receiver in multipath envirnments with Rayleigh path strength mdels. A subptimal weighted crrelatin receiver, which multiplies the prduct f the reference and data wavefrms by a weighting functin befre the integratin, is develped frm the ALRT receiver. Yi-Ling Cha yilingch@usc.edu is a graduate student and Rbert Schltz schltz@usc.edu is a Prfessr with the Department f Electrical Engineering at the University f Suthern Califrnia. This wr was supprted by the Army Research Office under MURI Grant N. DAAD This weighted crrelatin receiver btained under the Rayleigh envirnment assumptin is evaluated in IEEE 8.5.3a UWB envirnments with lgnrmal path strength mdels [6]. II. UWB TR SYSTEM MODULATION The transmitted signal f a single user UWB TR system with antipdal mdulatin is s tr t = i= g tr t it f + b i/ns g tr t it f T d. Here g tr t is a transmitted mncycle wavefrm that is nnzer nly fr t, T w, and T f is the frame time. Each frame cntains tw mncycle wavefrms. The first is a reference and the secnd, T d secnds later, is a data-mdulated wavefrm. The data bits b i/ns {, } are equally liely. The index i/n s, i.e., the integer part f i/n s, represents the index f the data bit mdulating the data wavefrm in the i th frame. Hence each bit is transmitted in N s successive frames t achieve an adequate bit energy in the receiver, and the channel is assumed invariant ver ne bit time. In this TR system, T d is greater than the multipath delay spread T mds t assure that there is n interference between reference signal and data signal. The frame time is designed t be T f T d > T mds s that n interframe interference exists. Because the single user case is cnsidered here, the time-hpping r direct sequence mdulatin which is used t reduce multiuser interference is eliminated fr simplicity, but withut lss f generality. III. ALRT OPTIMAL RECEIVER AND WEIGHTED CORRELATION RECEIVERS A. ALRT Optimal Receiver Assuming the multipath channel is invariant ver ne bit time, the received TR signal f bit b is mdelled as rt = r s t + nu, t, r s t = N s i= [p α g rx t it f 3 +p α b i/ns g rx t it f T d ] where nu, t represents Gaussian receiver nise with twsided pwer spectral density N. This received signal adpts the tapped delay line channel mdel and assumes the existence f K specular prpagatin paths. The th multipath cmpnent s prpagatin delay is dented by, and its amplitude is

2 dented by p α in which p {+, } with equal prbability, and α has Rayleigh distributin f α = α { } exp α, 4 where is the average pwer. It als assumes p and α j are independent fr any and j, and any tw paths are spatial uncrrelated. The received mncycle wavefrm g rx t arriving ver a single path can differ in shape frm the transmitted wavefrm [8]. In the design and analysis f the ALRT ptimal receiver, we assume that g rx t is nwn and can be used as a template in a crrelatr. We nw detect the bit b based n the bservatin r f rt, t, N s T f. Minimizing the bit errr prbability using ALRT, the decisin rule is f the frm p r b = p r b =. 5 Define α = [α, α,..., α ], p = [p, p,..., p ], then { p r b =, p, α exp } Ns T f [rt r s t] dt 6 N = exp{p α C α E}, 7 where is an equivalent symbl which ignres irrelevant cnstants in the right hand side f 6. Equatin 7 is simplified frm 6 by using the reslvable multipath assumptin which is made fr analytical reasns, and parameters in it are defined as E = C R = C D = C = N s N i= NsT f [g rx t it f ±g rx t it f T d ] dt, N s i= N s i= Ns T f Ns T f N [C R + C D ]. rtg rx t it f dt, rtg rx t it f T d dt, Using the assumptin that α and p are independent, we can separate the averaging prcess f nuisance parameters in 7 int tw steps. Because f the spatial uncrrelated paths assumptin which is ften adpted in multipath mdelling, the prbability density functin pdf f α and p can be decmpsed as fα = fp = fα = fp = α exp{ α }, 8 ] [ δ Dp + δ Dp +, 9 where δ D is a dirac delta functin. By ding sme straight frward integratin, the nuisance parameters α can be averaged ut frm 7. The result is shwn in withut cmputatin details, p r b =, p = Q [ w σ p r b =, p, αfαdα + p C πw 3 wc exp ], p C w and wt is defined as w = σ + σ E. It can be seen immediately that E is the average signalenergy-t-nise-pwer-spectral-density rati SNR in the th path, and w is always psitive. Then p can be eliminated using 9 and, and the result is wc w p r b = exp σ exp wc + C πw [ Q C w Q C ]} w. The lielihd functin p r b = can be cmputed using the same manipulatin, and the result is wd w p r b = exp σ 3 exp wd + D πw [ Q D w Q D ]} w, where D = N [C R C D ]. Substituting and 3 int 5, taing natural lgarithm ln n bth sides, and eliminating cmmn terms, the decisin rule can be simplified t where q r b = = + q r b = πwc q r b = = + πwd [ wc q r b =, + ln exp wc Q wc }], 4 [ wd + ln exp wd Q wd }].

3 3 y 6 x wc x 5 Fig.. wc versus y fr =,,..., K. The decisin statistic, q r b =, is a functin f wc. Withut receiver nise, C is equal t zer when b =. Therefre, with mderate r high SNR, C and als wc shuld be clse t zer if b =, and shuld have significant psitive values if b =. Define y as the term in the bracets in 4. Figure shws that y is clse t wc /. This indicates the effect f the lgarithm term in q r b = is small. The same explanatin can apply t q r b =. Therefre, abandning the lgarithm term is a reasnable way t simplify the receiver structure, and the subptimal receiver is wc wd. 5 Expanding C and D, and eliminating cmmn terms n bth sides, the decisin rule in 5 is equivalent t wc R C D. 6 The meaning f 6 can be explained as fllws. The transmitted bit is embedded in the phase difference f the reference wavefrm and data wavefrm. Therefre, withut receiver nise, the plarity f C R C D fr all shuld be the same and is the transmitted bit. It is pssible that we have detectin errrs by using C R C D as a decisin statistic because f the receiver nise, but the errr prbability decreases as the SNR increases. Therefre, we shuld give C R C D different weights fr different accrding t bth the a priri infrmatin and received infrmatin, i.e., the average SNR and sample SNR in the th path. Fr the subptimal detectin in 6, we discard the infrmatin f sample SNR, i.e., lgarithm terms in q r b = and q r b =, in rder t reduce the receiver cmplexity. We nly generate weights fr {C R C D } accrding t the a priri nwledge, and the weights are {w}. Fr a specified value f E, w is an increasing functin f, and C RC D gets a large weight if the average pwer f the th multipath cmpnent is large. B. Weighted Crrelatin Receivers If we ignre the weights in 6, the decisin rule becmes C R C D, 7 which is recgnized as the generalized lielihd rati test fr UWB TR systems [5]. Cmparing 7 and the decisin rule f a simple crrelatin receiver [5] which is rewritten in 8 N s j= jtf+t d+t mds jt f+t d rt T d rtdt 8 in which rt T d and rt represent the received reference and data wavefrms, sme similarities can be seen between them. The integratin, rt T d, and rt in 8 crrespnd t the summatin, C R, and C D in 7, and the summatin in 8 is imbedded in C R and C D. Therefre we can als multiply a weighting functin wt, which is a cntinuus functin interplated frm {w}, t rt T drt in 8 befre the integratin t prvide the a priri infrmatin in the decisin. Then the decisin statistic f the weighted crrelatin receiver becmes D s u = N s j= jtf +T d +T mds jt f +T d rt T d rtwt jt f T d dt, 9 and we say the transmitted bit is if D s u >, therwise. This is called a theretical weighted crrelatin receiver, and wt is a theretical weighting functin WF. The derivatin f the BEP is eliminated because f the space limitatin. Given a channel impulse respnse, the BEP is Tmds Ns P bit = Q wtg tdt Tmds N w tg tdt + BN Tmds w tdt where gt is the cnvlutin f the transmitted pulse and the channel impulse respnse, and B is the ne-sided receiver bandwidth. In channel mdelling, multipath channels are ften assumed having an expnential average pwer prfile, i.e., = Ω exp Γ where Ω is the average pwer in the first path and Γ is the time cnstant. This assumptin is reasnable even fr channel mdels with clusters lie IEEE 8.5.3a which can be bserved frm the figures in [6]. Use this expnential average pwer assumptin, w = σ + σ E Ω = + A-SNR exp Γ P P σ exp Γ exp Γ, where A-SNR dentes the average signal-energy-t-nisepwer-spectral-density rati f the received reference/data

4 4 weighting functin weighting functin time nsec gamma=8ns, A SNR=6dB gamma=6ns, A SNR=6dB gamma=4ns, A SNR=6dB gamma=3ns, A SNR=6dB A SNR=6dB, gamma=4ns A SNR=8dB, gamma=4ns A SNR=4dB, gamma=4ns A SNR=3dB, gamma=4ns time nsec Fig.. Weighting functins with different values f A-SNR and Γ. wavefrm. The theretical WF wt fr the crrelatin receiver can be btained frm by interplatin and is expressed as exp t Γ wt = + A-SNR P exp t Γ exp Γ. It is wrth nting that wt can be multiplied by any cnstant withut affecting the BEP. Figure shws the shape f wt with different values f A-SNR and Γ. We can see that increasing Γ lengthens the tail f wt, while increasing A-SNR increases the height. The theretical weighting functin wt is nt the nly chice fr a weighting functin. If we inspect 8 in details, the length f the integratin interval can als affect the BEP perfrmance. The integratin time, which is defined as T crr here, shuld fllw the cnstraint that T crr T mds. Under the cnstraint, increasing T crr can increase the incming signal energy, but we als get mre nise pwer thrugh the nise crss nise term. Since the tail respnse f the channel is usually small, it might nt be gd t use T crr = T mds lie in 8. We shuld adjust T crr accrding t the channel respnse t minimize the BEP, and this adjustment is equivalent t applying a square weighting functin {, t Tcrr w s t =, therwise. The decisin rule in 8 is mdified t N s j= jtf +T d +T mds jt f +T d and the BEP given a channel respnse is [ P bit = Q N N s ηe f rt T d rtw s t jt f T d dt, 3 + BT crr N N s ηe f ] 4 where E f = T mds g tdt is the received signal energy in ne frame, and η = T crr g tdt/ T mds g tdt is the efficiency factr. IV. NUMERICAL RESULTS Tw parameters decide the shape f a theretical weighting functin wt. One is A-SNR, and the ther ne is Γ which is the time cnstant f the expnential average pwer prfile f channels. Theretical weighted crrelatin receiver is tested n the IEEE 8.5.3a UWB channel mdel cm, cm, cm3, and cm4 [6], which have lgnrmal path strength mdels and pissn arrival clusters and rays, representing fur different envirnments. The single received pulse g rx t used in simulatins is a secnd derivative Gaussian pulse with duratin.7 nsec, the ne-sided receiver bandwidth B = 4GHz, =.7 nsec, and N s =. In this sectin,, 4, and equal pwer channel realizatins fr each channel mdel in IEEE 8.5.3a are used t btain the average BEPs. Althugh paths arrive in clusters in IEEE 8.5.3a, equivalent channels mdels withut clusters and with expnential average pwer prfiles exist fr analytical reasns [7]. The parameter A-SNR can be predicted frm the lin budget cmputatin. Given A-SNRs, equivalent Γs fr fur envirnments are acquired by minimized average BEPs which are e-4 at thse A-SNRs, and listed in Table I. Fr w s t, we can btain the ptimal T crr s which mae the crrelatin receiver achieves the average BEP=e-4 with minimum /N, and list thse values in Table I. Using thse ptimal parameter values in Table I fr wt and w s t, BEP curves in fur envirnments are pltted in Figure 3. Curves shw BEPs depend n applicatin envirnments, and using theretical weighting functin utperfrms using square weighting functin. This als means applying a theretical weighting functin t a simple crrelatin receiver can imprve the BEP perfrmance. CM CM CM3 CM4 theretical WF, wt A-SNR db Γ nsec square WF, w s t Tcrr nsec TABLE I OPTIMAL PARAMETER VALUES OF THEORETICAL AND SQUARE WEIGHTING FUNCTIONS AT BEP=E-4. Figure 4 shws the perfrmance degradatin f a theretical weighted crrelatin receiver in fur envirnments if the value f A-SNR r Γ is nt ptimized. The x-axis is A-SNR upper figure r Γ lwer figure, and the y-axis is the required /N t achieve average BEP=e-4. The circle n each line mars the psitin f the ptimal value f A-SNR r Γ. Results shw the receiver is nt sensitive t the A-SNR deviatin. As fr Γ, underestimatin degrades the perfrmance mre than verestimatin. Since Γ is related t applicatin envirnments, we shuld cnsider large values f Γ if the parameter is nt adjustable in the receiver. Figure 5 cmpares the perfrmance degradatin in cm upper figure and cm4 lwer figure envirnments using theretical and square weighting functins while parameters are nt ptimized. Because wt and w s t have different parameters,

5 5 average bit errr prbability cm4, theretical WF cm3, theretical WF cm, theretical WF cm, theretical WF cm4, square WF cm3, square WF cm, square WF cm, square WF /N db /N db /N db cm 8.5 cm A SNR db cm cm cm3 cm gamma nsec cm4 cm3 Fig. 3. BEP perfrmance using theretical and square weighting functins fr cm, cm, cm3, and cm4. Fig. 4. and Γ. /N required t achieve BEP=e-4 fr different values f A-SNR we have t find a fair way t cmpare the degradatin. Table I shws the ptimal values f A-SNR and Γ fr wt, and the ptimal values f T crr fr w s t in cm, cm, cm3, and cm4 envirnments. Fr the theretical weighting functin,,, 3, and 4 at the x-axis in Figure 5 represent that A-SNR, Γ=7.8, 7.5, 8.45,., 9.35,.5, and.4, 39.5 which are the ptimal values f A-SNR and Γ in envirnments cm, cm, cm3, and cm4. Fr the square weighting functin,,, 3, and 4 at the x-axis in Figure 5 represent that T crr =, 5, 5, and 75 which are the ptimal values f T crr in envirnments cm, cm, cm3, and cm4. The y-axis in Figure 5 is the required /N t achieve average BEP=e-4. Under this definitin f the x-axis, the cmparisn is fair. In the upper figure, we can see the best perfrmance happens while x= because parameters in bth weighting functins are ptimized fr cm envirnment. Fr the same reasn, the best perfrmance happens while x=4 in the lwer figure. Bth the upper and lwer figures shw the perfrmance degradatin by using a theretical weighting functin is smaller than using a square weighting functin when the values f parameters are nt ptimized. Using theretical weighting functins maes the receiver mre flexible in different envirnments. V. CONCLUSION This paper derives ALRT and subptimal receivers fr single user UWB TR system in Rayleigh envirnments. A theretical weighting functin is btained frm the subptimal receiver, and applied t the simple crrelatin receiver. Besides this theretical weighting functin, applying a square weighting functin t a simple crrelatin receiver is als examined. Althugh the theretical weighting functin is derived with a Rayleigh path strength assumptin, results shw it can imprve the receiver perfrmance in lgnrmal envirnments as well. The mst imprtant thing is receivers using a theretical weighting functin is mre flexible in different envirnments cmpared t using a square weighting functin. /N db /N db cm envirnment 3 4 ptimal parameters in fur envirnments cm4 envirnment 3 4 ptimal parameters in fur envirnments theretical weighting functin square weighting functin theretical weighting functin square weighting functin Fig. 5. Perfrmance degradatin f theretical and square weighting functins by using nn-ptimal parameter values. REFERENCES [] M. Z. Win, R. A. Schltz, On the rbustness f ultra-wide bandwidth signals in dense multipath envirnments, IEEE Cmmun. Lett., vl., pp. 5-53, Feb [] M. Z. Win, R. A. Schltz, On the energy capture f ultra-wide bandwidth signals in dense multipath envirnments, IEEE Cmmun. Lett., vl., Sep. 998, pp [3] R. T. Hctr and H. W. Tmlinsn, An verview f delay-hpped transmitted-reference RF cmmunicatins, Technique Infrmatin Series: G.E. Research and Develpment Center, January. [4] J. D. Chi and W. E. Star, Perfrmance f ultra-wideband cmmunicatins with subptimal receivers in multipath channels, IEEE JSAC, vl., n. 9, December, [5] Y.-L. Cha and R. A. Schltz, Optimal and Subptimal Receivers fr Ultra-wideband Transmitted Reference Systems, Glbcm, December, 3. [6] IEEE P8.5-/368r5-SG3a, Channel Mdeling Sub-cmmittee Reprt Final, Nvember 8,. [7] R. D. Wilsn and R. A. Schltz, Cmparisn f CDMA and mdulatin schemes fr a UWB radi in a multipath envirnment, Glbcm, December, 3. [8] J. M. Cramer, R. A. Schltz, and M. Z. Win, Evaluatin f an ultrawideband prpagatin channel, IEEE Trans. n Antennas and Prpagatin, vl. 5, n. 5, May, pp