LOCAL DECODING OF WALSH CODES TO REDUCE CDMA DESPREADING COMPUTATION

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1 LOCAL DECODING OF WALSH CODES TO REDUCE CDMA DESPREADING COMPUTATION Albert M. Chan, Jon Feldman, and Raghu Madyastha (Vanu, Inc., Cambrdge, MA, USA, Potr Indyk and Davd Karger (Computer Scence & Artfcal Intellgence Laboratory, MIT, Cambrdge, MA, USA, ABSTRACT In tradtonal hardware mplementatons of the CDMA standard IS-9 [], sgnals receved at the base staton are despread at a rate of.2288 Megachps/sec pror to Walsh decodng. However, n a software mplementaton where low computatonal complexty s crtcal, despreadng at such a hgh rate mposes a stran on computatonal resources. In ths paper, we take advantage of the flexblty afforded by software mplementatons and develop three classes of Walsh decodng algorthms that do not requre full despreadng of ncomng sgnals at the base staton. Two proposed classes of algorthms explot the fact that Walsh codes are locally decodable codes, whch have the surprsng property that any bt of the message can be recovered (wth some probablty) by examnng only a small number of symbols of the codeword. We also descrbe a thrd class of algorthms based on code puncturng. All these algorthms enable tradng off computaton for performance or, from another perspectve, they enable the system to dynamcally adapt the computatonal requrements of the despreader and subsequent Walsh decoder to changng channel condtons such that a target bt-error rate (BER) s mantaned. These algorthms are applcable to other CDMA-based systems that use Walsh codes for orthogonal modulaton, and the thrd class s also applcable to CDMA-based systems such as UMTS (3G WCDMA) that use codes for channelzaton.. INTRODUCTION The desgn of recever algorthms for code dvson multple access (CDMA) cellular phone sgnals s challengng because CDMA sgnals must be processed at hgh rates compared to other cellular phone standards. In the IS-9 reverse lnk (moble to base staton communcaton n the second-generaton CDMA standard []), chp pulses are transmtted at a rate of.2288 Megachps/sec. In the recever despreadng process, each chp s multpled by a bt of the user s long code, where the long code s computed by applyng a mask to a lnear feedback shft regster (LFSR). After the despreadng process, the resultng sequence of bts must be further processed to determne the Walsh codewords transmtted by the moble. The computaton rate for long code computaton, recever despreadng and Walsh decodng s thus roughly some multple of.2288 MHz. Snce a CDMA sgnal may propagate va many paths from moble to base staton, the base staton recever typcally repeats the despreadng and Walsh decodng processes for a number of receved copes of the sgnal (usually four), thus scalng the computaton rate by another factor. Furthermore, for a base staton, the despreadng and Walsh decodng processes for multple paths must be performed for each moble n the system, scalng the computaton rate by yet another one to two orders of magntude. Takng all ths nto consderaton, these processes utlze a sgnfcant proporton of the avalable computatonal resources on a mult-ghz processor. In ths paper, we ntroduce classes of technques to reduce the computatonal requrements of processng IS-9 reverse lnk sgnals at the base staton. These technques are also applcable to other advanced CDMA-based recevers that use Walsh codes for orthogonal modulaton, and the code puncturng technque can also be modfed to a partal correlaton technque for other CDMA-based recevers that use codes for channelzaton. In partcular, we develop methods that process only a fracton of the despread sgnal to decode Walsh codewords. In turn, only a fracton of the receved sgnal needs to be despread, and only a fracton of the long code needs to be computed. The remander of ths paper s organzed as follows. We revew the reverse lnk of IS-9 n Secton 2 and Walsh codes, used for orthogonal modulaton n IS-9, n Secton 3. Three classes of Walsh decodng algorthms that use only a fracton of the despread sgnal are ntroduced n Secton 4. The frst s local decodng a well known concept n the computatonal complexty and cryptography communtes [2], but here we present the novel applcaton of local decodng to a noncoherent recever. The second algorthm class s generalzed local decodng, whch s a novel and elegant generalzaton of local decodng. The thrd s punctured decodng, whch follows from the observaton that the generator matrx of a Walsh code contans all dstnct columns. We analyze the performance of these algorthms Proceedng of the SDR Techncal Conference and Product Exposton. Copyrght 2 SDR Forum. All Rghts Reserved

2 ÔÁ Ò Ó Û ¼Ø 9.6 kbps 28.8 kbps 28.8 kbps Rate /3 Block 64 ary Convolutonal Interleaver Orthogonal Encoder Modulator 92 bts 76 bts 76 bts 4.8 kcwps.2288 Mcps 37.2 ksps 96 Walsh codewords 644 Walsh symbols ÔÄ Ò Long PN Code Spreadng.2288 Mcps ½ ¾ 2476 PN chps 2476 PN chps ÔÉ Ò Quadrature Short PN Code Spreadng Fgure : IS-9 reverse lnk physcal layer at moble. Baseband Flter Baseband Flter Ò Û ¼Ø usng the smulaton of addtve whte Gaussan nose (AWGN) channels. The dfferent classes of technques are characterzed by dfferent tradeoffs between computatonal complexty and bt-error rate (BER). Secton explores a realstc wreless scenaro n whch base statons encounter reverse lnk channels wth fluctuatng nstantaneous SNR. We present Walsh decoders based on local decodng, generalzed local decodng, and punctured decodng that adapt the amount of computaton accordng to the nstantaneous SNR, so that a target BER s acheved wth mnmal computaton. Conclusons are presented n Secton IS-9 REVERSE LINK Ths secton descrbes physcal layer processng for the reverse lnk of IS-9 []. For smplcty, we focus on the 9.6 kbps data rate of Rate Set I on the reverse traffc channel, but the concepts are smlar for other vocoder rates and for the reverse access channel. Fgure shows the physcal layer processng of the reverse traffc channel at the moble. The traffc channel bts for Rate Set I are encoded usng a rate-/3 convolutonal encoder followed by block nterleavng. After nterleavng, the coded bts, at 28.8 kbps, are mapped sx bts at a tme nto one of 64 possble Walsh codewords of length 64. Ths modulaton provdes addtonal codng gan and smplfes noncoherent detecton of the data at the base staton. The resultng ± symbol stream, now at a rate of 28.8 x 64/6 = 37.2 ksps, s spread by a long code generated based on a mask assgned to the specfc user. Specfcally, each symbol of every Walsh codeword s spread onto four chps of the long pseudo-nose (PN) code, resultng n a spread spectrum sgnal wth rate.2288 Megachps/sec. After long PN code spreadng, the chps are fed nto the quadrature spreadng and modulaton structure, whch ncludes multplcaton of the long code spreadng output by short n-phase and quadrature-phase PN codes, a delay of the resultng quadrature-phase sgnal by half a PN chp, baseband flterng, and quadrature modulaton. Unlke the forward lnk of IS-9, the reverse lnk does not utlze a plot channel and thus sgnal processng at the base staton s noncoherent. The term noncoherent means that the random phase assocated wth the receved carrer s unknown, so detecton s done by observng the envelope of the receved sgnal. Under the assumpton that proper tmng synchronzaton has been acheved, the optmal noncoherent recever structure for one path of the IS-9 reverse lnk [3] s depcted n Fg. 2. It s suffcent for our purposes to note that correlatons of each of the 64 possble Walsh codewords wth two sgnals derved from the receved sgnal are combned nto a set of nonnegatve squared correlaton values. The 64 squared correlaton values ndcate the lkelhood of each of the 64 possble Walsh codewords. Takng nto account the nterleaver and the sx bts that each Walsh codeword represents, the set of 64 correlaton values can be converted nto soft nformaton to be fed nto a convolutonal decoder [4]. 3. WALSH CODES The Walsh code of length n = 2 s a set of perfectly orthogonal codewords that can be defned and generated by k k the rows of a 2 2 Hadamard matrx []. Startng wth a matrx H = [ ], hgher-order Hadamard matrces can be generated by the followng recurson: k H k H k 2 2 H k =. 2 k k H H 2 2 For example, H 4 can be recursvely generated as H = [ ], H 2 =, H 4 = so that the Walsh codewords of length 4 n bpolar ( ± ) form are w = [ +, +, +, ], w = [ +,, +, ], + 2 = [ +, +,, ] w 3 = [ +,,, + w, and ]. Note that the codewords are mutually orthogonal; that s, the dot product of any par of codewords s zero. Proceedng of the SDR Techncal Conference and Product Exposton. Copyrght 2 SDR Forum. All Rghts Reserved

3 Ó Û ¼Ø LPF Sampler Quadrature Short PN Code Despreadng Long PN Code Despreadng Walsh Correlatons ½ ¾ Ô Á Ò» Û Ò Ó Û Ò µ ¾» Û Ò Û Ò Ó µ ¾ Ô Ä Ò Ö Øµ Ô É Ò Þ» Û Ò Û Ò ¾ ÓÖ ½ ¾ ½ ¾ Ô É Ò» Û Ò Ò Û Ò µ ¾» Û Ò Û Ò Ò µ ¾ LPF Sampler Ô Ä Ò Ò Û ¼Ø Ô Á Ò Fgure 2: Optmal noncoherent recever structure for one IS-9 sgnal path. Because of the specal structure nherent n Walsh codes, t s well known that the correlatons of a sequence k wth the 2 Walsh codewords can be effcently computed n O( n log n) operatons usng the fast Hadamard transform (FHT) []. The FHT has the same butterfly structure as the fast Fourer transform (FFT) [6], but the coeffcents for the FHT are ± rather than complex exponentals. Thus, the correlatons n the optmal noncoherent recever descrbed n Secton 2 can be performed by two FHTs of length SUBCODE-BASED WALSH DECODING Tradtonally, Walsh codes are decoded by optmal Walsh decodng wth fxed computatonal complexty. In ths secton we ntroduce algorthms that effcently trade off computatonal complexty and BER based on the mathematcal structures underlyng Walsh codes. These algorthms are suboptmal but effectve and form the bass of the dynamc adaptaton algorthms of Secton. A generator matrx for a bnary lnear code C s a matrx G whose rows form a bass for C. Thus, the codewords n bnary form can be generated by computng c=xg for all k bnary row vectors x. The generator matrx for the (64,6) Walsh code used n IS-9 s a 6 64 matrx whose columns are the 64 dstnct 6 bnary vectors;.e., K G =. The specal structure of ths generator matrx makes t possble to effcently decode the Walsh codeword wthout examnng the entre codeword. In ths secton, we descrbe three such technques. 4. Local Decodng The specal structure of Walsh codes permts an estmate of one of the sx nput bts by observng only two of the 64 symbols of the receved Walsh codeword, va a process known as local decodng, whch s well known n the computatonal complexty and cryptography communtes [2]. To understand ths technque, we label the symbols of the codeword as c = c c, c, K, ], where the subscrpts [, 2 c63 are the decmal values of the correspondng bnary column vectors n G. Suppose we wsh to decode the zeroth bt of x = x, x, x, x, x, ] n the absence of recever nose. [ x We can obtan the value of x by modulo-2 addng any par of codeword components c and c j for whch the bnary representatons of and j (and the correspondng columns n G) dffer only n the zeroth bt poston: c c j = = s= = x = x s= ( G G ) ( G G ) xs ( Gs Gs ). x G x s s s s t= sj s= x G For example, c 2 and c 34 form one of the 32 possble pars that can be used to decode x. In a practcal stuaton n whch Walsh codewords are represented n bpolar ( ± ) format as opposed to bnary format, the modulo-2 addton s replaced by normal multplcaton of c and c j. t tj Proceedng of the SDR Techncal Conference and Product Exposton. Copyrght 2 SDR Forum. All Rghts Reserved

4 The sgnal-to-nose ratos (SNRs) encountered n practce are usually too low to relably decode a bt by examnng only one par of codeword symbols. Instead, the sum of products of up to 32 bpolar pars can be quantzed to ± to decode one bt n a soft votng procedure. Our use of local decodng n a noncoherent system s novel. The resultng recever structure s the same as n Fg. 2 except that the two Walsh correlators are replaced by two local decoders that sum the products of the same pars, and the subsequent squarers are elmnated snce the multplcaton of symbol pars already squares cos θ and sn θ. The soft votes from the two local decoders are then added together to elmnate any dependency on θ, and the sum s quantzed to ± to decode one bt. The smulated AWGN BER for local decodng of the IS-9 reverse lnk as a functon of both the number of pars used to decode each bt and the SNR per nformaton bt s depcted n Fg. 3. (For the purposes of computng SNR per nformaton bt n ths paper, nformaton bt shall refer to bts at the nput of the Walsh encoder and not the bts at the nput of the convolutonal encoder.) On the leftmost part of the graph s the baselne curve of optmal noncoherent decodng. The successve curves to the rght are generated by locally decodng each nput bt usng 32, 6, 8, 4, 2 and par(s) of codeword symbols. Although local decodng usng 32 pars of codeword symbols makes use of the entre Walsh codeword, ts performance s nearly 3 db worse than optmal decodng because the relatonshps between symbols n dfferent pars are not fully exploted. Each reducton of the number of local decodng pars by a factor of two causes an addtonal loss of 2 db Generalzed Local Decodng To decode two nput bts usng local decodng, one par of symbols n the Walsh codeword can decode one bt and another par can decode the other bt. Whle local decodng can decode bts usng few computatonal operatons and examnng potentally few codeword symbols, one mght expect to get a lower BER by jontly decodng both nput bts when constraned to examne only 4 of the 64 symbols of a receved Walsh codeword. We have developed an elegant generalzaton of local decodng to perform such jont decodng, whch uses certan groups of four symbols n the Walsh codeword to jontly decode two nput bts at low BERs. In local decodng, the two symbols c and c used to determne the bt x are j chosen such that the bnary representatons of and j dffer only n the t th bt poston. In what we call generalzed local decodng, the four symbols c, c, c and c j used to determne the par of bts g x s and t h x t are Probablty of Walsh Decoder Bt Error chosen such that the bnary representatons of g, h, and j all dffer now n both the s th and t th bt postons. These four symbols are processed n a way that generalzes local decodng. In local decodng, x t s determned by takng the product of codeword components c and c n bpolar format and quantzng to ±. Ths Optmal Decodng Local Decodng, 32 Pars Local Decodng, 6 Pars Local Decodng, 8 Pars Local Decodng, 4 Pars Local Decodng, 2 Pars Local Decodng, Par 2 Fgure 3: BER of 2-pont local Walsh decodng for AWGN channels as a functon of SNR per nformaton bt. j process s equvalent to performng a 2-pont FHT on the two-component vector c, ] where < j, and [ c j determnng x t to be the bnary ndex of the maxmum component of the FHT. That s, f the zeroth FHT component s largest then decode as, otherwse the frst FHT component s largest and decode as. In generalzed local decodng, a 4-pont FHT s performed on the fourcomponent vector c, c, c, c ] where g < h < < j, and [ g h j x s and x t are decoded as the bnary ndex of the maxmum component of the FHT where s < t. Generalzed local decodng works also for groups of 8, 6, and 32 codeword symbols, for whch 3, 4, and bts are decoded respectvely. Note that at one extreme, generalzed local decodng for groups of 2 codeword symbols s smply local decodng, whle at the other extreme generalzed local decodng for the entre 64 symbols of the Walsh codeword s optmal Walsh decodng mplemented by a 64-pont FHT. All sx nput bts can be decoded usng combnatons of varous-szed FHTs. For example, one could use three 4- pont FHTs or two 8-pont FHTs. As n the prevous secton, multple groups of codeword symbols can be processed and combned to decode nput bts. In local decodng, up to 32 pars of codeword symbols can be used to decode one nput bt because there are 32 choces for the fve fxed bts n the bnary representaton of 6 p the codeword symbol ndces. In general, up to 2 groups p of 2 codeword symbols can be used to decode p nput bts Proceedng of the SDR Techncal Conference and Product Exposton. Copyrght 2 SDR Forum. All Rghts Reserved

5 Probablty of Walsh Decoder Bt Error Optmal Decodng Generalzed 8 Pont Local Decodng, 8 Groups Generalzed 8 Pont Local Decodng, 4 Groups Generalzed 8 Pont Local Decodng, 2 Groups Generalzed 8 Pont Local Decodng, Group 2 Fgure 4: BER of generalzed 8-pont local Walsh decodng for AWGN channels as a functon of SNR per nformaton bt. 6 p because there are 2 choces for the fxed 6 p bts n the bnary representaton of the codeword symbol ndces. Because the reverse lnk of IS-9 s noncoherent, the 6 p components of the 2 FHTs are squared before beng added component-wse. Fgure 4 shows the smulated AWGN BER for generalzed local decodng of the IS-9 reverse lnk as a functon of both the number of 8-pont FHTs used to decode three bts and the SNR. From Fgs. 3 and 4, t s clear that f the number of codeword symbols examned to decode each nput bt s fxed, better performance s obtaned wth hgher-order FHTs Punctured Decodng Thus far, we have been explotng ( p) p, 2 codes that are found wthn the (64,6) Walsh code that s, we have been usng a subset of the codeword symbols to decode a subset of the nput bts. In ths secton, we use a subset of the codeword symbols to decode all 6 nput bts. Snce the columns of the generator matrx G for the (64,6) Walsh code nclude all possble 6-bt bnary vectors, any (n,6) bnary lnear code wth n < 64 and dstnct generator matrx columns can be obtaned by puncturng those symbols n the (64,6) Walsh code that correspond to unwanted columns n G. For a gven n < 64, the set of all possble (n,6) lnear codes created by dstnct puncturng patterns has a wde range of BERs. For example, some (n,6) codes have a row of zeros n ther generator matrces and thus have a mnmum Hammng dstance of zero, whle others maxmze the mnmum dstance. In noncoherent systems such as IS- 9, the (n,6) code that maxmzes the mnmum dstance does not necessarly lead to the best BER, and can even lead to performance much worse than randomly choosng a (n,6) code. Take for nstance the (32,6) borthogonal code, whch s the (32,6) lnear code wth the maxmum mnmum dstance. However, the complement of each codeword s also a codeword, so each codeword can be mstaken for ts complement and vce-versa n a noncoherent system wth unknown carrer phase. A more approprate crteron for selectng a (n,6) code for a noncoherent system s to maxmze the mnmum dstance and smultaneously mnmze the maxmum dstance. Whle tables of codes that maxmze mnmum dstance are readly avalable and are constantly mproved and expanded [7], we are not aware of smlar tables that smultaneously consder both metrcs. Although the development of such tables would be an nterestng area for future research, t s well known n codng theory that randomly generated codes are good codes wth hgh probablty, especally for long codeword lengths (large n ) [8]. In lght of ths, we can randomly select a subset of n codeword symbols from the (64,6) Walsh code, thereby effectvely obtanng an (n,6) code that, wth hgh probablty, has good BER performance. The maxmum-lkelhood decoder for the resultng (n,6) code s the mnmum-dstance decoder whch, n the case of bpolar codewords, s equvalent to takng the correlatons of the 64 possble codewords of length n wth the punctured receved codeword, and selectng the codeword correspondng to the maxmum. Ths set of 64 correlatons can stll be effcently computed usng an FHT on the receved codeword wth the 64 n unwanted codeword symbols set to zero. In a noncoherent system such as IS-9, the recever structure s smlar to Fg. 2, except that puncturng s pror to the Walsh correlatons. Fgure shows the smulated AWGN BER curves for randomly punctured codes and for codes wth only mnmum dstance optmzed, wth parameters (6,6), (6,6), (48,6), (32,6), and (6,6). The fact that the borthogonal (32,6) code has ndstngushable codewords n noncoherent systems s reflected by the poor BER when compared to the ensemble of (32,6) codes. For the other code lengths smulated, the performance of the code wth maxmum mnmum dstance s smlar to the average performance of codes n the ensemble, and even slghtly better n the hgh SNR regme where errors are perhaps mostly caused by codewords at the mnmum dstance.. ADAPTIVE SUBCODE-BASED WALSH DECODING Lke typcal wreless channels, the IS-9 reverse lnk encounters large fluctuatons n nstantaneous SNR. The fluctuatons due to path loss and shadowng are partally mtgated by open-loop and closed-loop power control, but the rapd SNR fluctuatons due to multpath fadng reman, especally at hgh moble speeds. The technques presented n the prevous secton provde a varety of operatng ponts n the tradeoff plane between BER and computatonal Proceedng of the SDR Techncal Conference and Product Exposton. Copyrght 2 SDR Forum. All Rghts Reserved

6 complexty at a fxed SNR, so n theory f we knew the nstantaneous SNR, we could choose the specfc technque (and thus, operatng pont) at that tme to acheve the target BER wth as lttle computaton as possble. However, ths knd of nformaton s not necessarly avalable or accurate at the base staton. To address ths ssue, we can test a seres of suboptmal Walsh decodng algorthms wth operatng ponts that smultaneously move towards hgher computatonal complexty and lower BERs untl the target BER s acheved. Although we cannot know the true BER, we can get an ndcaton by montorng a relablty metrc generated by the Walsh decodng algorthm untl t exceeds a threshold. The suboptmal Walsh decodng technques can be chosen such that only an ncremental amount of computaton s requred for the next, slghtly more complex technque. Ths dea of ncrementally ncreasng computaton untl suffcent relablty s acheved s smlar n sprt to decodng rateless codes [9,]. We test our adaptve algorthms by smulatng the IS-9 reverse lnk, assumng that the fluctuatng SNR per nformaton bt at the base staton has a lognormal dstrbuton wth mean 4.7 db and varance. db. Each adaptve Walsh decodng technque at a partcular stoppng threshold corresponds to a pont n the tradeoff plane between BER and computatonal complexty, averaged over many IS-9 frames experencng ndependent fades. Due to mplementaton-specfc software optmzatons, t s not very meanngful or realstc to count the total number of algorthmc operatons for long code generaton, despreadng, and Walsh decodng as a measure of computatonal complexty. Rather, we take as a measure the average number of Walsh codeword symbols used to decode all sx nput bts, because the computaton for long code generaton, despreadng, and Walsh decodng s roughly lnear n ths quantty. The tradeoff between BER and average computatonal complexty for adaptve local decodng s shown n Fg. 6. In ths algorthm, the number of pars used to locally decode one of the sx nput bts s ncreased by ncrements of one untl the magntude of the sum of the par products exceeds a threshold. Snce there are 64 possble termnatons to the algorthm, we call ths a 64-stage algorthm. Each pont was generated by assumng a specfc stoppng threshold, and assumng that the order n whch the pars are examned s chosen at random. The curve s traced out from left to rght as the threshold ncreases from to nfnty; at low thresholds the algorthm typcally stops after lookng at few pars, whle at hgh thresholds the algorthm typcally stops after lookng at many pars. The reason the tradeoff curve here s not a very good one s that even f few pars are used to decode each of the sx bts, most of the pars for each of the sx bts do not correspond to the same Walsh codeword symbols, and computatonal complexty becomes very hgh. Probablty of Walsh Decoder Bt Error Optmal Decodng Optmzed (6,6) Decodng Random (6,6) Decodng Optmzed (6,6) Decodng Random (6,6) Decodng Optmzed (48,6) Decodng Random (48,6) Decodng Optmzed (32,6) Decodng Random (32,6) Decodng Optmzed (6,6) Decodng Random (6,6) Decodng 2 Fgure : BER of punctured Walsh decodng for AWGN channels as a functon of SNR per nformaton bt. Fgure 6 also shows the tradeoff curve for an adaptve generalzed local decodng algorthm usng 8-pont FHTs. Ths 8-stage algorthm ncrements the number of 8-pont FHTs used to decode three of the sx nput bts untl the metrcs for the three bts all cross a threshold. When each addtonal 8-pont FHT s used, the squared correlatons are added to the sum of squared correlatons of the 8-pont FHTs already used. The metrc for each bt, for the purposes algorthm termnaton, s the absolute dfference between the maxmum sum of squared correlatons over all three-bt nput patterns wth a zero n that bt poston and the maxmum sum of squared correlatons over all those patterns wth a one n that bt poston. Ths adaptve generalzed local decodng algorthm creates a better tradeoff than adaptve local decodng. The tradeoff curve for adaptve punctured decodng s also shown n Fg. 6, correspondng to a 64-stage algorthm n whch up to all 64 Walsh codeword symbols are used to decode all sx nput bts. When an addtonal codeword symbol s used, the length of the 64 correlatons s extended by that addtonal symbol. The metrc for each bt, for the purposes of stoppng the algorthm, s the absolute dfference between the maxmum squared correlaton over all sx-bt nput patterns wth a zero n that bt poston and the maxmum squared correlaton over all those patterns wth a one n that bt poston. Ths 64-stage algorthm approaches the BER of optmal noncoherent decodng (the BER of the rghtmost pont on the curve) wth lower computatonal complexty. Note that adaptve random punctured decodng has the better tradeoff at low BERs, whle adaptve generalzed 8-pont local decodng has the better tradeoff at hgher BERs. In Fg. 7, we compare the BER and computatonal complexty of optmal noncoherent decodng specfcally to adaptve random punctured decodng. Whle optmal decodng requres a fxed amount of computaton regardless Proceedng of the SDR Techncal Conference and Product Exposton. Copyrght 2 SDR Forum. All Rghts Reserved

7 Probablty of Walsh Decoder Bt Error Fgure 6: Tradeoff between BER and average computatonal complexty for the three classes of adaptve subcode-based Walsh decodng. The SNR per nformaton bt s lognormally dstrbuted wth mean 4.7 db and varance. db. Probablty of Walsh Decoder Bt Error 2 32 Stage Local Decodng 8 Stage Generalzed 8 Pont Local Decodng 64 Stage Random Punctured Decodng Average Computatonal Complexty... BER of Optmal Decodng. BER of Adaptve Random Punctured Decodng BER of Adaptve Random Punctured Decodng, Target BER = 2 Complexty of Optmal Decodng Complexty of Adaptve Random Punctured Decodng Complexty of Adaptve Random Punctured Decodng, Target BER = 2 e Fgure 7: BER and average computatonal complexty of adaptve random punctured decodng as a functon of average SNR per nformaton bt. The varance of the SNR s. db. of SNR, smlar BERs can be acheved by adaptve random punctured decodng wth an amount of computaton that decreases as SNR ncreases. If the goal s to acheve a target 2 BER of, the computaton of adaptve random punctured decodng s even less. Although we have used the same threshold at each stage of these mult-stage algorthms for convenence, there s no reason that dfferent thresholds cannot be used at dfferent stages. For a gven algorthm, varyng the thresholds at each stage generates dfferent tradeoff curves. Thus, unlke Fg. 6, the set of ponts n the tradeoff plane achevable by a partcular adaptve algorthm s a regon rather than a curve. Whle t s dffcult to analytcally or numercally determne the optmal thresholds that acheve ponts that le on the Average Computatonal Complexty lower left border of the achevable regon, t s possble to fnd dstnct thresholds for each stage that generate mproved tradeoff curves. Ths s an nterestng area for future research. Furthermore, adaptve algorthms wth fewer stages may be more practcal. For example, a 2-stage adaptve local decodng algorthm looks at a certan number of pars, and then decdes whether to examne the remanng pars. If desgned correctly, the reducton n the number of stages can smplfy the algorthm and yet retan a smlar tradeoff. 6. CONCLUSIONS In ths paper, we have developed several classes of algorthms that reduce the amount of computaton requred for Walsh decodng and, n turn, for despreadng and long code generaton. These algorthms are made possble by software rado s nherent flexblty, whch allows us to montor relablty metrcs as successvely larger portons of a Walsh codeword are examned and to detect the codeword when relablty s suffcent. Software rado compels a paradgm shft n the desgn of new sgnal processng algorthms. Ths paper provdes a flavor of how algorthms can be desgned for software rather than tradtonal hardware. 7. REFERENCES [] EIA/TIA/IS-9 Moble Staton-Base Staton Compatblty Standard for Dual-Mode Wdeband Spread Spectrum Cellular System, Telecommuncatons Industry Assocaton, July 993. [2] L. Trevsan, Some Applcatons of Codng Theory n Computatonal Complexty, Quadern d Matematca, Vol. 3, pp , 24. [3] J.S. Lee and L.E. Mller, CDMA Systems Engneerng Handbook, Artech House, Boston, 998. [4] A.J. Vterb, CDMA: Prncples of Spread Spectrum Communcaton, Addson-Wesley, Readng, MA, 99. [] R.K. Yarlagadda and J.E. Hershey, Hadamard Matrx Analyss and Synthess, wth Applcatons to Communcatons and Sgnal/Image Processng, Kluwer Academc, Boston, 997. [6] A.V. Oppenhem and R.W. Schafer, wth J.R. Buck, Dscrete- Tme Sgnal Processng, 2 nd ed., Prentce Hall, Upper Saddle Rver, NJ, 999. [7] V.S. Pless and W.C. Huffman, Eds., Handbook of Codng Theory, Vol., Elsever Scence, Amsterdam, 998. [8] S.J. MacMullan and O.M. Collns, A Comparson of Known Codes, Random Codes, and the Best Codes, IEEE Transactons on Informaton Theory, Vol. 44, No. 7, pp , Nov [9] M. Luby, LT-codes, n Proceedngs of the 43 rd Annual IEEE Symposum on the Foundatons of Computer Scence (FOCS), pp.27-28, 22. [] A. Shokrollah, Raptor codes, n Proceedngs of the IEEE Internatonal Symposum on Informaton Theory, p. 36, 24 Proceedng of the SDR Techncal Conference and Product Exposton. Copyrght 2 SDR Forum. All Rghts Reserved

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