MITSUBISHI ELECTRIC RESEARCH LABORATORIES http://www.mer.com Optma and Suboptma Lnear Recevers for Tme-Hoppng Impuse Rado Systems Snan Gezc Hsash Kbayash H. Vncent Poor Andreas F. Mosch TR2004-052 May 2004 Abstract In tme-hoppng mpuse rado a number of frames are aocated for each nformaton symbo. In each of these frames one utra-wdeband puse s transmtted. Durng demoduaton of the receved sgna these puses need to be optmay combned n order to acheve the owest bt error probabty. for a snge-user system over an addtve whte Gaussan nose channe an optma near scheme s the one n whch sampes from the receved puses n dfferent frames are added wth equa weght. However n mutuser and/or frequency-seectve envronments the contrbutons from dfferent frames shoud be combned consderng the mnmum mean square error (MMSE) crteron n order to obtan ow bt error rates. Moreover n frequency-seectve envronments where the recever obtans sampes from dfferent mutpaths those mutpath components shoud aso be combned optmay. In ths paper we consder optma and suboptma near recevers for a gven user n a frequency-seectve mutuser envronment. The optma near recever combnng a the sampes from the frames and the mutpath components accordng to the MMSE crteron s desgned. Due to the compexty of ths recever two suboptma recevers are consdered: ) An optma frame combnng recever whch optmay combnes the sampes from the frames whe combng dfferent mutpath components suboptmay. ) An optma mutpath combnng recever whch combnes the sampes from dfferent mutpath components optmay whe combnng the sampes from the frames suboptmay. In ths paper these optma and suboptma near recevers are desgned and ther performance s evauated va smuatons. UWBST 2004 Ths wor may not be coped or reproduced n whoe or n part for any commerca purpose. Permsson to copy n whoe or n part wthout payment of fee s granted for nonproft educatona and research purposes provded that a such whoe or parta copes ncude the foowng: a notce that such copyng s by permsson of Mtsubsh Eectrc Research Laboratores Inc.; an acnowedgment of the authors and ndvdua contrbutons to the wor; and a appcabe portons of the copyrght notce. Copyng reproducton or repubshng for any other purpose sha requre a cense wth payment of fee to Mtsubsh Eectrc Research Laboratores Inc. A rghts reserved. Copyrght c Mtsubsh Eectrc Research Laboratores Inc. 2004 201 Broadway Cambrdge Massachusetts 02139
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Optma and Suboptma Lnear Recevers for Tme-Hoppng Impuse Rado Systems 1 Snan Gezc Hsash Kobayash and H. Vncent Poor 1 Department of Eectrca Engneerng Prnceton Unversty Prnceton NJ 08544 {sgezchsashpoor}@prnceton.edu Andreas F. Mosch 2 Mtsubsh Eectrc Research Laboratores 201 Broadway Cambrdge MA 02139 Andreas.Mosch@eee.org Abstract In tme-hoppng mpuse rado a number of frames are aocated for each nformaton symbo. In each of these frames one utra-wdeband puse s transmtted. Durng demoduaton of the receved sgna these puses need to be optmay combned n order to acheve the owest bt error probabty. For a snge-user system over an addtve whte Gaussan nose channe an optma near scheme s the one n whch sampes from the receved puses n dfferent frames are added wth equa weght. However n mutuser and/or frequency-seectve envronments the contrbutons from dfferent frames shoud be combned consderng the mnmum mean square error (MMSE) crteron n order to obtan ow bt error rates. Moreover n frequency-seectve envronments where the recever obtans sampes from dfferent mutpaths those mutpath components shoud aso be combned optmay. In ths paper we consder optma and suboptma near recevers for a gven user n a frequency-seectve mutuser envronment. The optma near recever combnng a the sampes from the frames and the mutpath components accordng to the MMSE crteron s desgned. Due to the compexty of ths recever two suboptma recevers are consdered: ) An optma frame combnng recever whch optmay combnes the sampes from the frames whe combnng dfferent mutpath components suboptmay. ) An optma mutpath combnng recever whch combnes the sampes from dfferent mutpath components optmay whe combnng the sampes from the frames suboptmay. In ths paper these optma and suboptma near recevers are desgned and ther performance s evauated va smuatons. 1. INTRODUCTION Recenty communcaton systems that empoy utrawdeband (UWB) sgnas have drawn consderabe attenton. UWB systems occupy a bandwdth arger than 500 MHz; due to the arge spreadng factors and ow power spectra densty they can coexst wth ncumbent systems n the same frequency range. The recent Federa Communcatons Commsson (FCC) rungs ([1] [2]) specfy the reguatons for UWB communcaton systems n the US. Smar rungs are expected n the near future n Europe and Japan as we. 1 Ths research s supported n part by the Natona Scence Foundaton under grant CCR-99-79361 and n part by the New Jersey Center for Wreess Teecommuncatons. 2 Aso at the Department of Eectroscence Lund Unversty Lund Sweden. Impuse rado (IR) systems whch transmt very short puses wth a ow duty cyce are typcay empoyed to mpement UWB systems ([3]-[5]). In an IR system a number of frames are aocated to each nformaton symbo. In each frame a UWB puse s transmtted and ts poston n the frame s determned by a tme-hoppng (TH) sequence [3]. The number of frames/puses that are sent per nformaton symbo s denoted by. In a snge user system over an addtve whte Gaussan nose (AWGN) channe the receved sgna conssts of puses n frames. After matched-fterng and sampng the contrbutons from the frames are added wth equa weght to form the decson varabe [3]. In consderng a mutuser envronment the contrbutons from dfferent frames can have dfferent sgna-to-nterference-pus-nose ratos (SINR) dependng on the TH sequences of the users. Therefore equay-weghted contrbutons from the dfferent frames no onger form an optma decson varabe. Aso n a frequency-seectve envronment there can be sefnterference aso caed nter-frame nterference (IFI) due to mutpath whch affects the optma combnng of the frame components at the recever. Apart from dfferent contrbutons from frames there s aso dversty due to the frequency-seectve envronment. Optma combnaton of dfferent mutpath components s affected by mutpe access nterference (MAI) and IFI. In other words we need to consder the optma combnaton of contrbutons from both dfferent frames and the dfferent mutpath components. In ths paper we frst consder the optma near recever for a gven user n the frequency-seectve mutuser envronment whch combnes a the sampes from the receved sgna accordng to the mnmum mean square error (MMSE) crteron. Due to the compexty of ths optma recever we aso consder two suboptma recevers wth ower compexty. The frst recever s caed an optma frame combnng (OFC) recever whch combnes the sampes from dfferent frames accordng to the MMSE crteron and combnes the sampes from dfferent mutpath components accordng to the maxma rato combnng (MRC) scheme. The other recever s caed an optma mutpath combnng (OMC) recever whch combnes the contrbu-
Fg. 2. sgna. Fg. 1. An exampe tme-hoppng mpuse rado sgna wth pusebased poarty randomzaton where Nf 6 Nc 4 the tme hoppng sequence s {212310} and the poarty codes are {+1+11+1-1+1}. tons from dfferent mutpath components optmay whe combnng the contrbutons from dfferent frames suboptmay. The remander of the paper s organzed as foows. Secton 2 descrbes the sgna mode for an IR system and presents a dscrete-tme representaton of the receved sgna. Secton 3 nvestgates the optma near recever that combnes a the components of the receved sgna accordng to the MMSE crteron. The OFC and the OMC recevers are presented n Secton 4 and Secton 5 respectvey. After smuaton resuts are presented n Secton 6 some concusons are drawn n Secton 7. 2. S IGNAL M ODEL We consder a synchronous bnary phase shft eyed TH-IR system wth K users n whch the transmtted sgna from user s represented by: s E X stx (t) d b ptx (t jtf cj Tc ) Nf j j bj/nf c where ptx (t) s the transmtted UWB puse E s the bt energy of user Tf s the average puse repetton tme (aso caed the frame tme) Nf s the number of puses representng one nformaton symbo and bbj/nf c {+1 1} s the bnary nformaton symbo transmtted by user. In order to aow the channe to be shared by many users and avod catastrophc cosons a tme-hoppng (TH) sequence {cj } where cj {0 1... Nc 1} s assgned to each user. Ths TH sequence provdes an addtona tme shft of cj Tc seconds to the jth puse of the th user where Tc s the chp nterva and s chosen to satsfy Tc Tf /Nc n order to prevent the puses from overappng. The random poarty codes dj are bnary random varabes tang vaues ±1 wth equa probabty ([6]-[8]). Assumng a tapped-deay-ne channe mode wth mutpath resouton Tc the dscrete channe mode α [α1 αl ] s adopted for user where L s assumed wthout oss of generaty to be the number of mutpath components for each user. Then the receved Match-fterng sampng and despreadng of the receved sgna can be expressed as s K L X E X X r(t) α dj bbj/nf c Nf j 1 1 prx (t jtf cj Tc ( 1)Tc ) + σn n(t) (2) where prx (t) s the receved unt-energy UWB puse and n(t) s a zero mean whte Gaussan nose wth unt spectra densty. Consder a fter matched to the UWB puse prx (t) as shown n Fgure 2. The output of ths fter s samped at nstants when the paths L arrve n each frame where L {1... M } wth M L. Due to possbe cosons the actua number N of tota sampes per nformaton symbo can be smaer than Nf M. The sampes at the output of the matched fter are despread3 by the poarty code of a user of nterest say user 1 (Fgure 2). The dscrete sgna at the th path of the jth frame can then be expressed for the th nformaton bt as4 rj stj Ab + nj (3) for 1... M and.. ( + 1)Nf 1 p j Nf.p where A dag{ E1 /Nf... EK /Nf } b (K) [b b ]T and nj N (0 σn2 ). sj s a K 1 vector whch can be expressed as a sum of the desred sgna part (SP) nter-frame nterference (IFI) and mutpe-access nterference (MAI) terms: (SP ) sj sj (IF I) + sj (M AI) + sj (4) where the th eements can be expressed as ( h α 1 (SP ) sj (5) 0 2... K ( P h dj (IF I) (nm) Aj dm αn 1 sj 0 2... K (6) ( h 0 1 (M AI) sj P dj dm αn 2... K (nm) B j (7) 3 In the context of IR systems spreadng and despreadng by random poarty codes are not ntended for expandng the bandwdth of the sgna. It many heps reduce the effect of MAI [6] and emnate the spectra nes [8]. 4 Note that the dependence of r j on the ndex of the nformaton bt s not shown expcty.
wth and A j {(n m) : n {1... L} m F m j mt f + c m T c + nt c jt f + c j T c + T c } (8) B j {(n m) : n {1... L} m F mt f + c m T c + nt c jt f + c j T c + T c } (9) where F {... ( + 1) 1}. Note that A j s the set of frame and mutpath ndces of puses from user 1 that orgnate from a frame dfferent from the jth one and code wth the th path of the jth puse of user 1. Smary B j s the set of frame and path ndces of puses from user that code wth the th path of the jth puse of user 1. For smpcty of the anayss we assume a guard nterva between nformaton symbos that s equa to the ength of the channe mpuse response (e.g. [9]) whch avods nter-symbo nterference (ISI). Therefore for bt we ony consder the nterference from the puses n the frames of the current symbo namey from the puses n frames... ( + 1) 1. 3. LINEAR MMSE RECEIVER In ths secton we consder a near recever for user 1 that combnes a the sampes from the receved sgna optmay accordng to the MMSE crteron. Let r be an N 1 vector denotng the dstnct sampes r j for ( j) L {1... }: [ ] T r r 1 r j 1 1 r j m M r 1 j (M) 1 M j m (M) M (10) where M 1 m N denotes the tota number of sampes wth N M. Usng (3) r can be expressed as r SAb + n (11) where A and b are as n (3) and n N (0 σni). 2 S s a sgnature matrx whch has s T j (see (4) through (7)) for ( j) C as ts rows where C {( 1 j 1 )... ( 1 j m 1 )... ( M j (M) 1 )... ( M j m (M) M )}. From (4)-(7) S can be expressed as S S (SP ) + S (IF I) + S (MAI). Then after some manpuatons r becomes r b E 1 (α + e) + S (MAI) Ab + n (12) where α [ 1 1 T m 1 M 1 T m M ] T wth 1 m denotng an m 1 vector of a ones and e s an N 1 vector whose eements are e j d j (nm) A j d m α n for ( j) C. A near recever combnes the eements of r and obtans a decson varabe as foows: y 1 θ T r (13) where θ s the weghtng vector. The MMSE weghts that maxmze the SINR of the receved sgna n (12) can be obtaned [10] as θ MMSE R 1 w 1 (α + e) (14) where w 1 S (MAI) Ab + n and R w1 E{w 1 w T 1 }. Assumng equprobabe nformaton symbos the correaton matrx can be expressed as R w1 S (MAI) A 2 ( S (MAI)) T + σ 2 n I. (15) Then the near MMSE recever becomes ˆb sgn { r T R 1 w 1 (α + e) }. (16) Note that ths recever requres the nverson of an N N matrx (N M ). Hence t can be very compex n some stuatons. Therefore we nvestgate some suboptma near recevers n the foowng subsectons. 4. OPTIMAL FRAME COMBINING (OFC) In ths case the mutpath components n each frame are added accordng to the MRC crteron. Then those combned components n the frames are combned accordng to the MMSE crteron. That s the decson varabe s gven by y 2 γ j L r j (17) where γ Nf... γ (+1)Nf 1 are the weghtng factors for the th bt. From (3) y 2 can be expressed as ( y 2 γ T Ŝ Ab + ) ˆn (18) L L where γ [γ Nf γ (+1)Nf 1] T s the vector of weghtng coeffcents ˆn [n Nf n (+1)Nf 1] T s the nose vector whch s dstrbuted as N (0 σni) 2 and Ŝ s an K matrx whose jth row s s T +j 1. Usng (4)-(7) Ŝ can be expressed as Ŝ Ŝ (SP ) + Ŝ(IF I) + Ŝ(MAI). Then we get ( [ y 2 γ T b E 1 ( L ) 2 1 Nf + L ê ] + w 2 ) (19) where ê s an 1 vector whose jth eement s e Nf +j 1 d +j 1 (nm) A d Nf m α n +j 1 and w 2 L α Ab + L α ˆn. From (19) the MMSE weghts can be obtaned as ) Ŝ (MAI) ( γ MMSE R 1 w 2 ( ) 2 1 Nf + L L ê (20)
where R w2 L + 1 L 2 L Ŝ (MAI) A 2 L (Ŝ(MAI) ) T 1 2 E{ˆn 1 ˆn T 2 }. (21) It s straghtforward to show that E{ˆn 1 ˆn T 2 } σ 2 ni for 1 2. When 1 2 the eement at row j 1 and coumn j 2 [ E{ˆn 1 ˆn T 2 } ] j 1 j 2 s equa to σ 2 n f j 1 N c + c j 1 + 1 j 2 N c + c j 2 + 2 and zero otherwse (j 1... and j 2... ). We note from (19) and (20) that the OFC recever ˆb sgn{y 2 } requres the nverson of an matrx. The reducton n compexty compared to the optma near MMSE recever of the prevous secton s due to the suboptma combnaton of the mutpath components. The SINR of the system can be expressed as SINR OF C E 1 x T 2 R 1 w 2 x 2. (22) where x 2 L (α ) 2 1 Nf + L α ê. 5. OPTIMAL MULTIPATH COMBINING (OMC) Now consder a recever that combnes dfferent mutpath components optmay accordng to the MMSE crteron whe empoyng equa gan combnng (EGC) for contrbutons from dfferent frames. In ths case the decson varabe s gven by y 3 L β r j (23) where β [β 1 β M ] T s the weghtng vector. Usng (3) y 3 can be expressed as y 3 β T [ S j Ab + ñ j ] (24) where ñ j [n 1 j n M j] T s the nose vector whch s dstrbuted as N (0 σni) 2 and S j s an M K sgnature matrx whose mth row s s T m j. Usng (4)-(7) S can be expressed as S S (SP ) + S (IF I) + S (MAI). Then we get y 3 β T b E 1 α + ẽ j + w 3 where α [ 1 mth eement s e mj d j w 3 S(MAI) j (25) M ] T ẽ j s an M 1 vector whose (nm) A mj d m α n and Ab + ñ j. From (25) the MMSE weghts are chosen as where β MMSE R 1 w 3 R w3 + j 1 α + S(MAI) j A 2 ẽ j (26) ( S (MAI) j ) T (27) j 2 E{ñ j1 ñ T j 2 }. (28) It can be observed that E{ñ j1 ñ T j 2 } σni 2 for j 1 j 2. When [ j 1 j 2 the eement at row 1 and coumn 2 E{ˆnj1 ˆn T j 2 } ] s equa to σ 2 1 2 n f 1 N c + c 1 + j 1 2 N c +c 2 +j 2 and zero otherwse ( 1 L and 2 L). We note from (25) and (26) that the OMC recever ˆb sgn{y 3 } needs to nvert the M M matrx R w3. The reducton n the compexty compared to the optma near recever n Secton 3 s the resut of suboptma combnaton of the contrbutons from dfferent frames. The SINR of the system can be expressed as SINR OMC E 1 x T 3 R 1 w 3 x 3. (29) where x 3 α + ẽ j. From the prevous equatons the foowng property foows: Property: Consder a snge user system where the puses n a frame never code wth any puse n another frame. In other words there s no IFI and MAI. In ths case the expressons for SINR can be shown to reduce to SINR OF C SINR OMC E 1 σ 2 n α 2. (30) L 6. SIMULATION RESULTS In ths secton we consder the downn of a TH- IR system wth 5 users (K 5) where E 1. The number of chps per frame N c s equa to 10 and the dscrete channe s gven by α [ 0.4019 0.5403 0.1069 0.0479 0.0608 0.0005] [11]. The TH sequences and poarty codes of the users are chosen from approprate unform dstrbutons and the resuts are averaged over dfferent reazatons. In the frst scenaro the number of frames per symbo s equa to 10 and the frst three mutpaths are samped at the recever; that s L {1 2 3}. Fgure 3 shows the bt error probabty (BEP) for dfferent SNR vaues. From the fgure t s observed that the optma near recever performs the best as expected. The OFC recever performs better than the OMC recever n ths case whch mpes that there s a greater dversty gan n combnng 10 dfferent frames than combnng 3
Bt Error Probabty 10 0 10 1 10 2 10 3 Optma Combnng Optma Mutpath Combnng Optma Frame Combnng Conventona RAKE 10 4 0 5 10 15 SNR (db) Fg. 3. BEP versus SNR for a 5-user TH-IR over the channe [ 0.4019 0.5403 0.1069 0.0479 0.0608 0.0005] where N c 10 and E 1. The number of frames per nformaton bt s 10 and the frst three paths are combned at the recever. Bt Error Probabty 10 0 10 1 10 2 Optma Combnng Optma Mutpath Combnng Optma Frame Combnng Conventona RAKE 10 3 0 2 4 6 8 10 12 14 16 18 20 SNR (db) Fg. 4. BEP versus SNR for a 5-user TH-IR over the channe [ 0.4019 0.5403 0.1069 0.0479 0.0608 0.0005] where N c 10 and E 1. The number of frames per nformaton bt s 2 and a the mutpath components are combned at the recever. mutpath components. Aso note that the OFC recever needs to nvert a 10 10 matrx whe the OMC recever needs the nverson of a 3 3 matrx. Fnay the conventona RAKE recever whch performs EGC across the frames and MRC across the mutpath components has the hghest BEP vaues due to ts suboptma combnng schemes n both dversty domans. In Fgure 4 2 and a the paths of the receved sgna are samped; that s L {1 2 3 4 5 6}. From the pot t s observed that the optma near recever s the best and the conventona RAKE s the worst as expected. However n ths case the OMC recever performs better than the OFC recever because s sma n ths case and optmay combnng these two components s ess mportant than the optma combnaton of the sx mutpath components. Aso note that the SNR s ncreased by decreasng the nose power. Hence after some pont the BEP does not decrease much snce the errors are many due to MAI and IFI. 7. CONCLUDING REMARKS We have consdered optma and suboptma near recevers for TH-IR systems. The optma near recever performs MMSE combnng of a the receved sampes. It gves the best BEP performance but ts compexty s usuay very hgh. Therefore we have proposed the OFC recever whch combnes the contrbutons from the frames optmay whe performng MRC for the receved mutpath components. Fnay we have consdered the OMC recever whch combnes the components from dfferent frames wth equa weght whe usng the MMSE crteron for the mutpath components. Dependng on the system parameters the OMC recever coud beat the OFC recever and vce versa. These recevers may not be very practca n rea envronments. However they provde mportant theoretca references for more practca recevers. Furthermore these recevers may be feasbe n downns of some TH-IR systems where the base staton (or the pconet coordnator) transmts nformaton about the TH sequences and poarty codes of a the users. REFERENCES [1] FCC 00-163: Notce of Proposed Rue Mang. [2] FCC 02-48: Frst Report and Order. [3] M. Z. Wn and R. A. Schotz Impuse rado: How t wors IEEE Communcatons Letters 2(2): pp. 36-38 Feb. 1998. [4] R. A. Schotz Mutpe access wth tme-hoppng mpuse moduaton Schotz Proc. IEEE Mtary Communcatons Conference 1993 (MILCOM 93) vo. 2 pp. 447-450 Bedford MA Oct. 1993. [5] M. Z. Wn and R. A. Schotz Utra-wde bandwdth tmehoppng spread-spectrum mpuse rado for wreess mutpeaccess communcatons IEEE Trans. on Communcatons vo. 48 ssue 4 pp. 679-691 Apr 2000. [6] E. Fsher and H. V. Poor On the tradeoff between two types of processng gan 40th Annua Aerton Conference on Communcaton Contro and Computng Montceo IL Oct. 2-4 2002. [7] S. Gezc H. Kobayash H. V. Poor and A. F. Mosch Performance Evauaton of Impuse Rado UWB Systems wth Puse-Based Poarty Randomzaton n Asynchronous Mutuser Envronments IEEE Wreess Communcatons and Networng Conference (WCNC 04) Atanta GA March 2004. [8] Y.-P. Naache and A. F. Mosch Spectra shape of UWB sgnas nfuence of moduaton format mutpe access scheme and puse shape Proceedngs of the IEEE Vehcuar Technoogy Conference (VTC 2003-Sprng) vo. 4 pp. 2510-2514 Jeju Korea Apr 2003. [9] J. Evans and D. N. C. Tse Large System Performance of Lnear Mutuser Recevers n Mutpath Fadng Channes IEEE Trans. Inform. Theory IT-46(6):2059-2078 September 2000. [10] S. Verdú. Mutuser Detecton Cambrdge Unversty Press Cambrdge UK 1998. [11] C. J. Le-Martret and G. B. Gannas A-dgta PAM mpuse rado for mutpe-access through frequency-seectve mutpath Proceedngs of the IEEE Goba Teecommuncatons Conference (GLOBECOM 2000) vo. 1 pp. 77-81 San Franssco CA Nov. 2000.