Optimal Bandwidth Allocation for Source Coding, Channel Coding and Spreading in a CDMA System

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1 ptima Bandwidth Aocation or Source oding, hanne oding and Spreading in a DA System Qinghua Zhao, Pamea. osman, and aurence B. istein Department o Eectrica and omputer Engineering, University o aiornia, San Diego, 5 Giman Drive, a Joa, A 3-47, U.S.A. {qizhao, pcosman, mistein}@ucsd.edu ABSTRAT This paper investigates the tradeos between source coding, channe coding and spreading in a DA system, operating under a ixed tota bandwidth constraint. We consider a system consisting o a uniorm source with a uniorm quantizer, a channe coder, an intereaver, and a direct sequence spreading modue. The system uses binary phase-shit keyed (BPSK) moduation. Rate-compatibe punctured convoutiona (RP) codes and sot decision Viterbi decoding are used or channe coding. The system is anayzed over both an additive white Gaussian noise (AWGN) channe and a at Rayeigh ading channe. A tight upper bound or rame error rate is derived or non-terminated convoutiona codes. The perormance o the system is evauated using the end-to-end mean squared error (SE). We show that, or a given bandwidth, an optima aocation o that bandwidth can be ound using the proposed method. KEY WRDS Bandwidth aocation, DS-DA, Rayeigh ading, rame error rate, convoutiona codes, mutiuser system. 1 Introduction Source coding, channe coding and spread spectrum are three o the main components in a DA communication system. They compete or the major shared resource bandwidth. Source coding rees up bandwidth or both orward error correction (FE) and spreading. Aocating more bandwidth to source coding aows more inormation rom the source to be transmitted, but reduces the bandwidth avaiabe or both FE and spreading. For dierent compression methods and rates, the bit stream coming out o the source encoder wi be more or ess sensitive to dierent types o error patterns. FE and spreading protect the transmitted bits rom noise and intererence. Depending on the channe conditions and the characteristics o the source coded bit stream, the system wi perorm better with either more FE or more spreading. Reated studies in the iterature are imited to the tradeo Acknowedgment: This research was partiay sponsored by the enter or Wireess ommunications o USD, the ore program o the State o aiornia, and Ericsson Wireess ommunications Incorporated. between either source coding and channe coding, e.g., [1, ], or channe coding and spreading, e.g., [3, 4]. In each case, research topics incuded anayzing a given system to ind the optima bandwidth aocation to each modue as in [1, 3], and joint design o coding agorithms or transmitter/receiver schemes or each category [, 4]. In [5], we studied the bandwidth aocation tradeo or a direct sequence DA system that incorporated an image coder, a RP [6] channe coder, and a RAKE receiver. Due to the compexity o the system, we obtained most o the resuts through simuations. In this paper, we wi investigate the tradeos anayticay. et and denote the source code rate (in bits-persource symbo), the channe code rate, and the processing gain, respectivey. I the source produces symbos per second, or a given bandwidth o chips per second, we have the oowing constraint: (1) where! " is a constant. We wi ind the bandwidth aocation #$ "$ % $ ' that optimizes system perormance. We wi aso address the question o how sensitive is the optima aocation to changes in the channe conditions, transmission rate, and bandwidth constraint. The paper is organized as oows. Section gives an overview o the system and our approach to the bandwidth aocation probem. Sections 3 and 4 anayze the system or both AWGN and at Rayeigh ading channes. Some representative resuts o tradeos among the three components are presented in Section 5, and the concusions are given in Section 6. System overview We consider a mutiuser scenario. The system or each user is simiar and is shown in Figure 1. The source input vector (*),+.- has cosed bounded support. Each component o the vector is considered to be one source symbo. The output o the source encoder is an / -bit binary index, and the source code rate /13. The source encoder is a

2 V ) R # W ) # W X R Quantizer Encoder i Index Assignment π( i) onvoutiona Encoder n bits Intereaver Spreading 54 n Ns chips uti-access hanne Y j R Quantizer Decoder j Inverse Index Assignment π( j) Viterbi Decoder n bits Deinter De- -eaver spreading n 67 Ns chips Figure 1. System overview. quantizer with distortion ;=<?>A@ : B EGF3HJIKI NP IKI Q"R TS () XW D ;=<?>A@ D + - is a partition o into disjoint regions, )Y+Z-, represents the []\_^ power o the usua Eucidean ` ; ( is the probabiity density unction o. where UV each IKI IKIQ o which is represented by code vector norm, and ( In our system, we take to be a one-dimensiona (N ) uniorm source over a bc ed. We aso take [1g, so that the end-to-end distortion is the mean-squared error. The quantizer we use here is designed to be optima or a noiseess channe, and it has partitions and code points ha ij% > #kim n > e > #o3ij e (3) respectivey, where ipbq %m r, and is the centroid o V. In [7], we prove that even though the anaysis is done or a uniorm source, the resuts can be appied to a wide variety o source distributions. Since /s "t/, we wi use / and interchangeaby. The / -bit binary representation, i U bc %uuu' v w #kit U bq %uu%u' r, o a source symbo is mapped to an / -bit index by the index assignment bock 1. Its purpose is to rearrange the indices so that those with sma Hamming distances between them represent quantization eves which are cose. This way, the distortion caused by the most probabe errors is sma, and thus the tota distortion is sma. There are many dierent index assignment schemes possibe or a scaar quantizer []. We pick a random index assignment [1], where the mapping w #Tx is a one-to-one mapping rom indices b through b through to a random permutation o the indices. Since the permutation is random, the index assignment can be good or bad. To measure its distortion, we must average over a possibe permutations, i.e., we use the expectation o the distortion to evauate 1 The index assignment bock is a part o the source coding. We separated it out or ease o anaysis. its perormance. The use o random indexing simpiies anaysis, athough the method proposed in this paper wi work or any speciic index assignment. A channe encoder with rate?/1y encodes the indices and passes them to the intereaver. The intereaver output is mutipied by the spreading sequence assigned to the given user, with spreading actor. The output o the spreading is moduated and passed to the channe. Here we consider DS-DA systems, with channe symbo rate z, chip rate z, and processing gain Yz z. The system has { active users, with the b \_^ user taken as the reerence user. At the receiver, the received signa is demoduated, despread, and decoded by the channe decoder to / -bit indices. These indices are mapped by the inverse index assignment bock and decoded by the quantizer decoder. By comparing the reconstructed source with the origina source symbo, we can cacuate the end-to-end distortion. From [], the expected mean-squared error o a system with a uniorm source, a uniorm scaar quantizer, and a random index assignment, is #k/} =~ e > ; ~ ƒ # P > > ; ~ ƒ (4) where ~ is the probabiity o index error, i.e., at east one bit o the / -bit index is in error, so the index is incorrect. In earier work, without proo, [1] gives a simiar resut or an uncoded memoryess binary symmetric channe. Equation (4) works or a channe codes and channes. The vaue o ~ depends on the channe code, moduation scheme, channe conditions and receiver structure. Generay, inding the expression or ~ is no trivia task. In this paper, we wi ind a cose upper bound or ~, and thus an upper bound or the distortion #k/} =~ e, as a unction o the three parameters, and. We denote this upper bound by #k % ', and ind the optima bandwidth aocation #$ $ % $ ' or. By operating the system with this bandwidth aocation, we can guarantee a system perormance better than #$ $ $.

3 Œ B z Since we use non-trivia channe codes, ~, and thus, #_/P r~, are decreasing unctions o, i.e., i both and are given, the arger (or a given eve o compexity) is, the better the perormance o the system. Thereore, we can repace the inequaity constraint, Equation (1), by an equaity constraint:? ˆu (5) Hence, the probem we need to sove is to minimize #k under constraint (5). Frame Error rate, inormation bits per rame Rayeigh Upper Bound Rayeigh Simuation AWGN Upper Bound AWGN Simuation 3 System The bit stream out o the index assignment bock is encoded by a non-terminated convoutiona encoder with code rate. At the receiver, a sot decision Viterbi decoder decodes the noise-contaminated bit stream to indices. The output o the intereaver is mutipied by a ong pseudo-random sequence assigned to the given user and transmitted using BPSK moduation. Since we transmit the indices by sequentiay passing them through a non-terminated convoutiona code, the / -bit index error rate is aso the rame error rate o this convoutiona encoder. A rame o size / consists o / consecutive inormation bits. The error rate o an inormation rame o size / is the probabiity that at east one o the / bits in the rame is decoded incorrecty. In [11], an upper bound or the rame error probabiity was given heuristicay, but a requirement o very arge /s was imposed. In [7], we derive a tight upper bound or rame error rate or any coded rame engths which are arger than the constraint ength: where / ~ ~ #k/s? B #=#_` Œ is the inormation rame size, ` #_SŽˆ (6) is the number o branches o the treis that are in a rame, #os % is the pairwise probabiity o two sequences that have Hamming distance S, is deined, as in [6], by z, and % % z I DA u (7) % In Equation (7), z % % are the coeicients o the compete path eumerator [1]: zš# z where S is the Hamming weight o the encoder output o a path, is the ength o a path, and Sq both go rom 1 to œ. Vaues o Œ or the RP codes in [6] are isted in [7]. Figure compares the bound in (6) with simuation resuts or the rate 1/ code in [6] with memory 4. From the pot, we can see that the theoretica upper bound is quite tight. For both AWGN and Rayeigh ading channes, we cacuate #os and then optimize the end-to-end distortion o the system E b in db Figure. Upper bound on rame error rate or rate 1/, convoutiona code with memory AWGN channe For a direct sequence DA system with a arge number o users, the pairwise error probabiity or the AWGN channe is approximatey given by #osž Ÿn S #_{ t 3m ž Ÿn %S #_{ 3 " 1S where ' () u () #_{ j«ª 3 #_{ A«ª 3 Aso is the energy per channe bit; ; is the power spectra density o the white Gaussian noise, is a constant which depends on the puse shape, and equas when we use square-shaped chips, { is the tota number o users; ˆ m± is the processing gain, and is the energy per chip (which is kept constant). Substituting () into (6) and then into (4), we have #k/}? >J; ² " 1S ƒ v B # #o` 3. Fat Rayeigh ading channe a ³ ; d T #k/} Œ '?u (1) Assume, where ³ is the ade ampitude and has a Rayeigh density, and assume the ading seen by the channe decoder is uncorreated rom bit to bit. For a direct sequence DA system, the conditiona pairwise error probabiity, conditioned on the ading parameters, is given by

4 @ È Å I À ; d É ² S B ; ³ Å S S ƒ i Å Ë S u ; À S / / > å / É > i u u [13] I W >m@ ; ~ ; #os U ³ µ ³ #_{ «ª 3J >m@ ¹ ³ #_{ n«ª3 3 >m@ µ»º¼ ½ ¼ W Averaging ~ ; #_S U ³ over the distribution o a ³, i! bc %uu%ue =S ] PÀ >m@ B #_SŽ Ãi  D  where Æ ;, #_SŽ can be ur- and Å ther simpiied to #_SŽ¹ Y where É #_SŽ¹ Æ 3 yieds the pairwise error probabiity [13, #_{ a ³ j "S Ä. When Å 3S A«ª333 Æ Æ Æ!Ç µ > (11) 3S 3 É #_SŽ > Substituting (1) into (6) and then into (4), we have #k/}? 4 ptimization > ; #os > B" # #_` "S T #_/P = ÊŒ (1) u (13)?u (14) In the equations or the upper bound o the distortion, (1) and (14), #_/P = is not a simpe unction o, i.e., or a given set o RP codes, the spec-, cannot be written as a unction o. trum, and Œ Thus, we cannot ind the optima bandwidth aocation by taking derivatives o with respect to. To ind the optima tripet #J$ /} "$ $ ', we irst ix, and ind the optima aocation #k/ìë3 and the minimum distortion Ë #_/Í or this. Then by comparing the minimum Ë #k/s s or dierent, we ind the best tripet. For a given channe code rate, we can use the bandwidth constraint and substitute Î j / into #k/} ', so that the upper bound becomes a unction o a singe variabe /. We denote this new unction as #k/s. 4.1 AWGN channe Substituting ž / into (1), we have >J; B3 #k/s¹ % ƒ #r#_` T ÊŒ ² ÌS / u (15) Dierentiating #k/s with respect to /, and setting it equa to zero, resuts in bœ #k/s > ; B! "# ÏÑÐ!!! B3 > ; B3 #=#_` "# #=#_` T ÊŒ ÏÑÐ T ÊŒ B ÒÓŽ Ÿ Ä 1SŽ% / NÔ >?ÕJÖT ØrÙÛÚ Ü < 1SŽ w /ÍÞ Ô > ÕJÖT Ü < Øvá ÙÛÚ Ò%Ó wß à Ø=â Ú Ô > ÕJÖT Ü < Ø Ù Ú ÌS w /ÍÞ (16) The approximation in the ast step o (16) is vaid when is arge. It is easy to show that 1SŽ /ãä1s #k/s is a convex cup, so, soving (16) numericay with any good root inding agorithm gives the optima /PË or an AWGN channe. 4. Fat Rayeigh channe Substituting?ž '/ into (14) resuts in > ; B #k/s ƒ #_SŽe#o v ² #=#_` ÊŒ T/ Setting the derivative o #k/s equa to zero, we obtain É b #k/s > ; B3 ž "# ÏKÐ"AÊ #_SŽ ² #o v ÒÓ#=#_` ÊŒ S%/ u (17) >A@ (1) As was the case with (16), (1) needs to be soved numericay. Note that or arge signa-to-noise ratios, n ±, we can ignore non-minimum distance error events and thus use simper orms o å æ ç or both cases above.

5 Ë ƒ Ó Ó Ë 5 Resuts Figure 3 shows the upper bound or the end-to-end distortion,, versus the source code rate and channe code rate or an uncorreated Rayeigh â ading channe, under the bandwidth constraint â ± Ú è é"b b. Here Ú is Sê, and the active number o users in the system b. The RP codes used are rom Tabe 1 o is {ë [6]; their spectra are isted in [7]. From Figure 3, we can see that, or each given channe code rate, there is an optima source code rate /. that achieves the smaest or this. The goba optimum aways as at the smaest, i.e., the strongest channe coding. This is true or both AWGN and at Rayeigh ading channes when no intererence suppression is impemented. reduces the raw error rate into the channe decoder. As Ú increases, the channe gets better, and we do not need as arge an, so we can decrease and aocate more o the avaiabe bandwidth to source coding, i.e., increase /. Aternativey, or each set o curves which have the same bandwidth constraint, we see that as the number o users increases, /1Ë decreases. This is because we need to aocate more bandwidth to spreading to suppress the muti-user intererence. We can aso see that the increase (decrease) o /1Ë"# is sower or a arger number o users. This is because with more users, the muti-user intererence dominates the therma noise, whie the eects are comparaby ess signiicant. o the change o Ú 14 1 = 3 = 64 og 1 D u optima source code rate m From top to bottom K=1, 5, 1, m r c chip energy to noise ratio E /N in db c o =3 = 64 Figure 3. Distortion vs. source code rate / and channe code rate. For any ixed, by soving (16) and (1), we aso show in the oowing igures how /ÌË and Ë vary when the channe conditions change. Figure 4(a) shows the variation o the optima / with the chip energy-to-noise ratio,, and Figure 4(b) shows Ú anaogous resuts or the optima n vaue o. There are two sets o curves on each igure, one or bandwidth constraint "b, and the other or ƒ È b. The curves are parameterized by the number o users {Y vìg bc v3b. Aso, È a curves correspond to the use o the memory, rate " convoutiona code in [6], and an uncorreated at Rayeigh ading channe. For each curve in Figure 4, where the number o users, {, is ixed, we see that as Ú increases, /1Ë increases, and decreases. This is because the processing gain,, has two eects on the perormance o the system: 1) A arger suppresses more intererence rom other users; ) a arger eads to a arger * ±rí â Ú Ú, and thus optima processing gain N s From top to bottom K=1, 5, 1, chip energy to noise ratio E /N in db c o Figure 4. /1Ë and Ë vs. chip energy-to-noise ratio Fat Rayeigh ading channe. Figure 5 iustrates how /ÌË and Ë change as the bandwidth constraint? changes. The system used in this igure is the same as in Figure 4. From this igure, we see that as increases, /ÌË increases, and /1Ë increases aster when there are ewer users in the system. This is because when there is ess intererence, as Ú increases, the channe con- dition improves aster than when there is more intererence..

6 optima source code rate m optima processing gain N s = db = 4dB From top to bottom K=1, 5, 1, = db = 4dB From top to bottom K=1, 5, 1, bandwidth constraint bandwidth constraint Figure 5. a) /1Ë and b) Ë vs. bandwidth constraint. Fat Rayeigh ading channe. Thus we do not need as arge a processing gain, and we can aord to aocate more bandwidth to the source coding. Simiar resuts or the AWGN channe are presented in [7]. 6 oncusions In this paper, we studied the bandwidth aocation probem or a DA system which empoyed RP channe coding and sot decision Viterbi decoding. Under a bandwidth constraint, we optimized the system perormance by combining anaytica and numerica techniques. For the system we considered, our resuts show that or both AWGN and at Rayeigh ading channes, or a given, it is aways beneicia to use the strongest channe code possibe when the compexity o the system is not a concern. We aso showed, or a given, how the optima aocation between Ë and Ë changes when the channe conditions number o interering users, channe noise, or bandwidth constraints change. Reerences [1] B. Hochwad and K. Zeger, Tradeo between source and channe coding, IEEE Trans. Ino. Theory, vo. 43, pp , Sept. 17. []. Zhao, A. A. Aatan, and A. N. Akansu, A new method or optima rate aocation or progressive image transmission over noisy channes, IEEE Proc. D, pp. 13, ar.. [3] K. H. i and. B. istein, n the optimum processing gain o a bock-coded direct-sequence spreadspectrum system, IEEE Journa on Seected Areas in omm., vo. 7, pp , ay 1. [4] D. J. V. Wyk, I. J. ppermann, and. P. inde, Perormance tradeo among spreading, coding and mutipe-antenna transmit diversity or high capacity space-time coded DS/DA, Proc. o I 1, vo. 1, pp. 33 7, Sept. 1. [5] Q. Zhao, P.. osman, and. B. istein, Tradeos o source coding, channe coding and spreading in requency seective Rayeigh ading channes, The J. o VSI Signa Processing - Systems or Signa, Image, and Video Tech., vo. 3, pp. 7, Feb.. [6] J. Hagenauer, Rate-compatibe punctured convoutiona codes (RP codes) and their appications, IEEE Trans. omm., vo. 36, pp. 3 4, Apr. 1. [7] Q. Zhao, Bandwidth Aocation and Tradeos o Source oding, hanne oding, and Spreading or DA Systems. Ph.D Thesis, USD, /3. [] A. ehes and K. Zeger, Binary attice vector quantization with inear bock codes and aine index assignments, IEEE Trans. Ino. Theory, vo. 44, pp. 7 4, Jan. 1. [] Q. Zhao, P.. osman, and. B. istein, n the optima aocation o bandwidth or source coding, channe coding and spreading in a coherent DS- DA system empoying an SE receiver, Euro. Wireess on. Proc., vo., pp , Feb.. [1] Y. Yamaguchi and T. S. Huang, ptimum binary ixed-ength bock codes, Quartery Progress Report 7,.I.T. Research ab. o Eectron., ambridge, ass, Juy 165. [11] G. aire and E. Viterbo, Upper bound on the rame error probabiity o terminated treis codes, IEEE omm. etters, vo., pp. 4, Jan. 1. [1] R. J. ceiece, The Theory o Inormation and oding. Addison-Wesey Pubishing ompany, Inc., 177. [13] J. G. Proakis, Digita ommunications. cgraw- Hi, 3rd ed., 15.