A Frequency Doain Approach to Design Constrained Aplitude Spreading Sequences for DS-CDMA Systes for Frequency Selective Fading Channels B.J.Peiris, K.R.Narayanan and S.L. Miller Dept. of Electrical Engineering Texas A&M University, College Station, TX 77843, USA. Eail: janath, krn, siller @ee.tau.edu Abstract Although there are attractive sequence design algoriths for frequency selective channels, unfortunately in ost of these algoriths, the designing has been done ith the assuption that the chips of the sequences have unconstrained aplitudes and phases. This increases the feedback bandidth and peak to average poer ratio (PAPR) at the transitter output. In this paper, e observe that the spectru of the spreading sequences designed according to existing algoriths are narroband and are located at the strongest spectral coponents of the users channels. The narrobandness of spreading sequences allos to represent each user s sequence ith feer nuber of paraeters and hence reduces the required nuber of feedback bits to represent those sequences. Further, the narrobandness of the spectra allos to break the global optiization of the users cross correlation atrix into siple sub optial local (grouped) optiizations. In this paper, e capitalize the aforeentioned observations to design constrained aplitude sequences, for frequency selective channels. I. INTRODUCTION FOr years, the research counity has been interested in the design of optiu spreading sequences for DS- CDMA systes. For non-frequency selective channels, it is knon that the optiu set of sequences, in the sense of iniizing the su of squared cross correlation of the users spreading sequences achieves the Welch loer bound. It is also shon that these Welch bound sequences achieve the su capacity (axiu su of rates) of the syste[1]. There are any different algoriths to design those spreading sequences either centrally (at the base station) or in a distributive fashion (at the obile station) [2]-[3]. In a frequency selective environent, each user s spreading sequence is transitted through a different channel and the correlation properties of the filtered spreading sequences ay be different fro that of the transitted spreading sequences. Further, although the input poer of the chip sequences is fixed, the poer of the received sequences can vary according to the channel characteristics and the chosen spreading sequences. Hence, it is not sufficient just to iniize the su of squared cross correlation of the users transitted spreading sequences. A better objective is to axiize the ratio beteen each user s received poer and the su of squared cross correlation plus noise. In literature, there are any efforts to find a good set of spreading sequences. Those algoriths are based on iterative algoriths and the chips of the sequences are assued to be unconstrained [4]-[7]. The sequences ith chips having unconstrained aplitudes and phases lead to the folloing difficulties in practical transission schees. First, once the sequences are designed either at the base station (central) or at the obile station (distributed), those sequences have to be transitted to the other end prior to the actual data transission. When the chips are unconstrained, transission overhead in the feedback channel can be sufficiently high. Secondly, unconstrained chip sequences deand ider dynaic range at the transitter aplifier output hich can be quantified by the PAPR of the unconstrained spreading sequences. To avoid the PAPR issue and to reduce the feedback bandidth considerably, e suggest using constrained chip sequences. This paper focuses only on the design of constrained aplitude sequences. In this paper e sho that the design of spreading sequences subjected to the axiization of signal to interference ratio (SINR) is siilar to the design of spreading sequences by concentrating their spectra around the spectral peaks of the channels frequency responses hile avoiding the overlapping of spectra of the sequences if possible. We further sho that the localization of spreading sequences at the spectral peaks allos to represent each user s spreading sequence ith feer paraeters in the frequency doain, than that of in the tie doain. Based on these observation of the spectral characteristics of the spreading sequences, e propose to ethods to design constrained aplitude spreading sequences. In the first ethod, e focus on the sequence designed algorith in [5] and at each iteration, e project optiized unconstrained spreading sequences onto a unit circle. In the second ethod, e select a faily of narroband unit circle sequences proposed in [9] and assign the carefully so that each user can gain a significant SINR iproveent. Note that, since the spectru of each user s sequence is narroband, it is required only to optiize the users ith overlapping spectra. The rest of the paper is organized as follos. The next section discusses the CDMA syste odel and an existing good sequence design technique for ultipath channels. Section III investigate the frequency doain characteristics of those ell designed unconstrained spreading sequences for frequency selective channels. In Section IV, e utilize the frequency doain characteristics of the unconstrained spread-
! i ing sequences and propose different approaches to design constrained aplitude spreading sequences. Section V presents the results that verify the perforance of the proposed schees by coputer siulations. Section VI concludes the paper. II. SYSTEM MODEL AND BACKGROUND A. Syste odel We consider a user synchronous uplink DS-CDMA syste here is the th user s length spreading sequence. th user s channel is a frequency selective slo fading one ith taps given by the vector "$# # %# &'. Assuing ()(", that is ignoring the inter-sybol interference (inter-chip interference still exists), the output of the chip level atched filter at the base station for the * th chip interval is given by It can be clearly seen fro (2) that the th user s spreading sequence is an eigen vector of the atrix G a T 0 G. By initializing ith a properly noralized rando vector of length, the final value for the th user s spreading sequence can be found by calculating in (2) iteratively until it converges [5]. To reduce the overhead in the feedback path, Rajappan and Honig [6] have also introduced a reduced-rank transitter adaptation schee here the signature sequences are constrained to loer diensional subspaces spanned by soe orthogonal basis vectors. For exaple, if g is the hoji atrix hose i coluns are the basis vectors of the th user, the optiu spreading sequence is a linear cobination of the i coluns of gk given by H"gkel. In [6], it is assued that the basis vectors for gk are randoly selected / 0 &00 orthonoral vectors. The corresponding l value, ith an +-,. 4 21' 3., 3 # 36587, MMSE receiver is given by 9;:<*):<>=@? (1) 1' here 4 g a gon 0 g a G a T G g l dcpl Df (3). is the data bit of user during the A th epoch, 7, Siilar is the AWGN ith variance BDC /. Equation (1) can be ritten in vector for as E.F 0 21'HG 4. 5JI. to that of in (2), l is an eigen vector of the atrices gk a gon gk a, here G a Tq 0 G Ygk. The appropriate eigen I. 7 7 K% 7 0, E. L+ M. + 2. + vector is selected by finding the eigen vector that axiizes 0N. the SINR at the receiver output. For the and G is an OP th user, l has to atrix hose first colun is be iteratively calculated and once it is found, the optiu $# # %# &' Q9R%Q9 and the rest of the coluns spreading sequence can be calculated by @g l. are obtained by cyclically donshifting the first colun. B. Joint transitter receiver adaptation [5] To understand the frequency doain characteristics of the spreading sequences hich are optiized for a given set of frequency selective channels, e briefly look into one of the existing good sequence design techniques: the joint transitter receiver adaptation schee, proposed by Rajappan and Honig [5]. In this ethod, a centralized adaptation technique in hich the users jointly adapt to optiize the total syste perforance and a distributed adaptation technique in hich each user adapts to optiize his/her on perforance are considered. The distributed adaptation is suitable for peer to peer ireless netorks and the centralized adaptation is suitable for ultiple access systes. In our analysis e restrict ourselves to the distributed adaptation technique hich can be used both for uplink and don link scenarios. Further, e assue that perfect channel inforation is available both at the transitter and at the receiver. Let the th user s receiver S be a iniu ean squared error (MMSE) detector given by S UT G, and let the SINR output of the user s MMSE detector be given by V / WYXZ[6XZ0[ Z W Z, here T ] W XZ [ XZ0\ Z [ Z W Z 1'Y]_^ 1`G ] ] a] G a] 5 BDCYb is the interference plus noise covariance atrix. As shon in [5], the th user s spreading sequence that axiizes his/her on SINR, ith an MMSE receiver, under distributed adaptation is given by G a T G dce f (2) III. FREQUENCY DOMAIN CHARACTERISTICS OF GOOD SPREADING SEQUENCES If e consider the distribution optiization schee in [5] for a single user scenario (assuing rs? ), the optiu spreading sequence is the solution to (refer equation (2)) G a G tce f (4) It can be clearly seen that the optiu spreading sequence is the eigen vector corresponding to the axiu eigen value of G a G. Further, for large the eigen vectors of G a G ill be approxiately sinusoids ( toneburst [8]). Hence, the optiu spreading sequence for a single user syste is a sequence hose spectru is an ipulse at the frequency here the channel frequency response is the strongest. As e have entioned earlier, for a ultiuser CDMA syste, it is not just enough to axiize the individual user s perforance and the goal is the axiization of each user s perforance hile keeping the cross correlation aong the users spreading sequences at a sufficiently lo value. According to Parseval s theore, the cross correlation beteen to sequences in the frequency doain is the sae as that of in the tie doain. That is, e can design the sequences in the frequency doain instead of designing the in the tie doain considering the criteria for iniizing the interference aong the users as the iniization of the frequency doain cross correlation aong the users. To observe ho the optiized spreading sequences are positioned in the frequency doain under a
v v ž ž 14 12 10 user A user B user C spectru, then the corresponding optiu spreading sequence is approxiated by channel C Š l v f (6) Aplitude 8 channel B channel C channel A 6 4 2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Noralized Frequency Fig. 1. Poer spectra of the users spreading sequences designed according to [5]. This allos us to select the best sallest set of vectors as each user s basis. Once the best i basis vectors for each user s channel is selected, the optiu l for the th user can be calculated according to (3). Ability to siplify the existing sequence design algoriths: Since the bandidths of the spreading sequences designed for frequency selective channels are narro, the user s spreading sequences can be grouped into different frequency regions. Hence, the optiization (to axiize SINR) can be done separately for separate groups ith less coplexity. ultiuser environent, e have coputed the optiu set of spreading sequences, ith spreading length 16, using the iterative algorith in (2) for a 12 user syste. Each user s channel is assued to be a randoly generated to path channel and as kept fixed throughout the siulation. As seen in Fig. 1, the frequency spectru of each user s spreading sequence is narroband and lies around the frequencies of that user s strongest channel frequency. Further, the figure illustrates that for users hose strongest channel frequencies occur at different frequency points, the spectra of the optiu spreading sequences of those users have little or no overlap. It as also verified that hen to or ore channels have the sae strongest channel frequency, although their spectra overlap, the correlations aong those sequences are sall. With these observations e dra soe iportant conclusions. The ability to represent each user s spreading sequence ith lesser nuber of paraeters: Since the length of each user s spreading sequence is, it is possible to represent each user s spreading sequence ith linearly independent basis vectors. For exaple, e can select gk as gku here vx is given by v _v Y,?-y{z} Q~ y{z} Q~Q pƒ y z} Q~K 9;: + :d>=@? f v It is clear that, corresponds to a sinusoid hose frequency response is an ipulse at the + th frequency index and the th user s spreading sequence can be represented by 1' l v. According to Fig. 1, the optiized spreading sequences are narro band. Hence, it is possible to represent each user s spreading sequence ith a feer nuber of sinusoids (basis vectors) that are located at the strongest frequencies of the corresponding user s channel, ith significant accuracy. For exaple, if a user s channel s strongest M i (i>()(l ) frequency coponents are located at $ˆ, in decreasing order of agnitude of the C K% (5) IV. CONSTRAINED AMPLITUDE SEQUENCE DESIGN A. Constrained aplitude sequence design by apping unconstrained sequences onto a unit circle In this subsection e focus on the sequence designed algorith in [5], and at each iteration, e siply ap the chips of the optiized unconstrained spreading sequences onto a unit circle. That is, in each user iteration, th user s optiized sequence is replaced by a unit circle sequence -Œ k hich is given by -Œ 6 *YnŽ< e } % *YnQn n 98: *: =?. The unit circle apping allos to reduce the PAPR. But, since the phases of the chips of a unit circle sequences can take any value beteen 9 and Mš, to transit the phase inforation it needs considerable aount of feedback bits. For exaple, if e quantize each chip s phase ith bits, the required nuber of feedback bits is LOœ. Instead of directly quantizing phases of the chips, e propose the folloing algorith. First, e observe the narrobandness of each spreading sequence and represent each ith reduced nuber of basis vectors (say i ) according to the equations (5), (6). Secondly, e quantize the coefficients l of the basis vectors ith bits before apping the reconstructed onto a unit circle sequence. That is, if the quantized coefficients of the basis vectors are l, then the corresponding unit circle sequnce is given by -Œ 6 e Ÿ Y R Š l ž v n n. Note that i and are selected such that the required nuber of ovearhead bits i as sall as possible hile the perforance of the reduced rank-quantized sequences is very close to that of full rank unquantized sequences. Nuerical values for i and are given in section V. It should be noted that to reconstruct the unit circle sequences both at the receiver and the transitter, it needs only the inforation of the quantized coefficients l% ž s of the basis vectors. Hence, it is sufficient to transit only l% ž s fro the designed end to the other end and no additional inforation has to be transitted regarding the phases of the chips. is
š n! B. The Design of constrained aplitude sequences using already existing faily of sequences ([9]) As discussed in section III, it is possible to design nearly optiu spreading sequences for ultipath channels by assigning spreading sequences ith narroband spectra centered around the strongest channel frequencies, hile aintaining a lo cross correlation aong the sequences that have overlapping frequency spectra. If it is possible to coe up ith a set of spreading sequences ith a fixed aplitude alphabet and that have the above given characteristics, then that set ill provide good perforance hile aintaining a lo PAPR at the transitter output. To tackle this proble, e first select a subset of spreading sequences fro the set of Opperann sequences. An Opperann sequence [9] is a unit aplitude sequence here the ebers of the spreading sequence set are given by, { } ǹ =?n e. ] 5 9q: here A«7 are any real nubers. The triple AH 7 :<>=ª?M specifies the sequence set and the refers to the th sequence in the faily. It is shon in [9] that, if is set to 1, all the sequences ill have the sae autocorrelation agnitude. Hence, the corresponding spectra ill have the sae shape hile localized at different frequencies. Further, it is shon that if is prie, correlation beteen any to sequences in the sae faily is zero. We select the Opperann sequences ith? and A ±? and vary 7 until e get a set of spreading sequences hose spectra are narro band and centered around different discrete frequency points. If e assign these spreading sequences for users such that each user s strongest channel frequency atches ith the center frequency of the spreading sequence, e can guarantee that the received filtered sequences ill also have lo cross correlation properties. The reason is that, since the spreading sequences are narro band sequences, after filtering through the corresponding channels, the received filtered sequences ill also have spectra close to those of the transitted sequences. But this kind of assigning of spreading sequences is not alays possible since there is a possibility that to or ore users ay have the sae strongest channel frequency. We have developed the folloing strategy for assigning sequences for such users. (For this technique, e select to sets of Opperann sequences ith narro band and C, 9;: :d>=@?.) Let the! th user s channel be and the corresponding spectra given by frequency indices of the user s + strongest channel coponents be C. 1) For the! th user s channel, select the corresponding Opperann sequence 3, such that ² 3 n; (e (7) define ² as the operator that gives frequency index of the strongest frequency of a sequence). If the selected sequence 3 has not been assigned for the users?-k%)=ª?, assign that sequence for the th user. 2) If the selected sequence 3 has previously been assigned to a user ³!' ith channel then, e search the second set of Opperann sequences C and select the Opperann sequence C. such that ² C. n µ² 3 n M. This guarantees that the received poer at the th user s receiver is high. But, since the sequences of the users and ³ are not fro the sae Opperann sequence set, the cross correlation beteen the to users is not necessarily sall. To reduce the cross correlation, 3 e ultiply the th user s Opperann sequence by y so that the spreading sequence for user is given by 3 C. y. The is selected such that the real part of the cross correlation beteen the filtered sequences of the users and is zero. That is ¹D a3 C. e Ÿ n»¼½9. Where 3! ] ¾ 3 and C. ¾ C.. The value of is given by, œáà Y à 3 a C. näå 3 a C. n f Note that this is possible only if C. has not been assigned previously. The algorith does not optiize ore than to users at a tie since optiization ith ore users leads to an increase in the coplexity of the algorith. 3) If both 3 and C. have been assigned previously, then select an unassigned Opperann sequence, fro the set such that ², nh for Æ: : "=ª?. The selection should be done such that the algorith searches for the next strongest channel frequency in the descending order. 4) Carry out this for all the users in the syste. Fro the above given algorith, it is possible to obtain a set of spreading sequences hose filtered outputs have saller cross correlation hile producing sufficiently high SNR output at the receiver. Further, the optiization here reduces to optiizing the perforances of to users ( and ³ ) at a tie rather than optiizing all users jointly. The latter is prohibitively coplex for a set of spreading sequences ith a fixed aplitude alphabet. Another advantage of using Opperann sequences is that only Y A«7 need to be sent fro the transitter to the receiver if the optiization is centralized. V. SIMULATION RESULTS To verify the perforance of the proposed unit circle sequences, siulations ere carried out for a DS-CDMA syste. Each user s channel is assued to be a 2-tap frequency selective channel here taps are dran fro a Gaussian distribution and the poer is noralized to one. Very slo fading siilar to that of in [5], [6] is assued and, therefore, in the siulations the channels do not change over the length of the siulations. Spreading sequence length is 16 and the nuber of users is 12 corresponding to a 75% loading. Fig. 2 shos the perforance of an uplink DS-CDMA syste ith chips of the spreading sequences are apped
10 0 10 0 10 1 10 1 10 2 10 2 BER 10 3 BER 10 3 10 4 10 4 RH un quantized MMSE 10 5 RH un quantized PIC single user AWGN Kuar 4phase RH qauntized unit circ D=5 MMSE RH quantized unit circ D=5 PIC 10 6 0 1 2 3 4 5 6 7 8 Eb/No RH un quantized MMSE 10 5 RH un quantized PIC Single user AWGN Kuar 4phase Opperann sequenecs PIC Opperann sequencs MMSE 10 6 0 1 2 3 4 5 6 7 8 Eb/No Fig. 2. The perforance of the unit circle sequences ith frequency doain quantized coefficients. K=12, N=16. RH-quantized = Sequences designed according to (6) ith quantized Ç'ÈÉ Ê. RH-unquantized = Sequences designed according to (6) ith unquantized Ç'ÈÉ Ê and Ë<Ì Í. onto a unit circle according to the subsection A in section IV. For coparison, e have also included the single user perforance in an AWGN channel, perforance of the unquantized sequences ([5], iîï ) and the perforance of coplex sequences proposed in [10]. Since it requires at least { bits to represent any coplex spreading sequences (e.g. QPSK), to ake a fair coparison beteen the unit circle sequences and any other type of coplex sequence, e have selected i and such that iðòñ and l is uniforly quantized ith 3 bits per diension. Hence, the required nuber of feedback bits is alost {. It can be seen that ith the use of a parallel interference canceller (PIC) and an MMSE detector, the perforance of the unit circle sequences is 0.5-1.4 db aay fro that of the un-quantized sequences. But, even ith the MMSE receiver, unit circle sequences outperfor single user AWGN perforance. This is because, each user s unit circle sequences is designed to exploit the channel gain. Perforance of the coplex sequences proposed in [10] is 0.8 db aay fro that of the proposed unit circle sequences. Fig. 3 shos the perforance of the syste hen the chips of the spreading sequences are derived fro a subset of opperann sequences. These spreading sequences are fro the selected subset of Opperann sequences ith N=15. Here, 7 has been varied to obtain to failies of orthogonal narro band sets of sequences, as discussed in section IV. It can be seen fro Fig. 3, the proposed schee to design spreading sequences ith a fixed aplitude alphabet, provides a 1.6 db gain over the single user AWGN perforance ith the use of a PIC, and it provides 2 db gain over coplex sequences proposed in [10]. Further, the perforance of the syste ith Opperann sequences is only a fe tenths of a db aay fro that of the unconstrained sequences. For an MMSE receiver, the unit circle sequence outperfors AWGN perforance and the coplex sequences proposed in [10] by about 0.4dB and 0.7dB respectively. Fig. 3. Peforance of the Opperann sequences. K=12. VI. CONCLUSION In this paper, e have proposed a novel ethod of designing spreading sequences ith constrained aplitude chips for frequency selective channels by analyzing the characteristic of the spreading sequences in frequency doain rather than analyzing the in the tie doain. siulation results are given to copare the perforance of the proposed unit circle sequences ith existing unconstrained sequences and coplex sequences [10]. REFERENCES [1] M.Rupf, J. L.Massey, Optiu sequence ultisets for synchronous code-division ultiple-access channels, IEEE Transactions on Inforation Theory, vol. 40, No.4,pp.1261 1266, Jul. 1994. [2] S. Ulukus, R. D. Yates, Iterative construction of optiu signature sequence sets in synchronous CDMA systes, IEEE Transactions on Inforation Theory, vol. 47, No.5, pp.1989 1998, Jul. 2001. [3] P. Visanath,V. Ananthara, and D.N.C. V.; Tse, Optial sequences, poer control, and user capacity of synchronous CDMA systes ith linear MMSE ultiuser receivers, IEEE Transactions on Inforation Theory, vol. 45, No.6, pp.1968 1983, Sept. 1999. [4] W. M. Jang, B. R. Vojcic, and R. L. Pickholtz, Joint transitter-receiver optiization in synchronous ultiuser counications over ultipath channels, IEEE Transactions on Counications, vol. 46(2), pp. 269 278, Feb. 1998. [5] G. S. Rajappan, M. L. Honig, Spreading code adaptation for DS-CDMA ith ultipath, IEEE Milco,vol. 2, pp. 1164 1168, 2000. [6] G. S. Rajappan, M. L. Honig, Signature sequence adaptation for DS- CDMA ith ultipath, IEEE Journal on Selected Areas in Counications, vol. 20, No.2, pp. 384 395, Feb. 2002. [7] J. I.Concha, S. Ulukus, Optiization of CDMA signature sequences in ultipath channels, Proc. of IEEE Vehicular Technology Conference, pp. 1978-1982 spring 2001. [8] D. C.Popescu, C.Rose, Interference avoidance applied to ultiaccess dispersive channels, Thirty-Fifth Asiloar Conference on Signals, Systes and Coputers, vol. 2, pp. 1200-1204,Nov. 2001. [9] J. Opperann, B. S. Vucetic, Coplex spreading sequences ith a ide range of correlation properties, IEEE Transactions on Counications, vol.45, No. 3, March 1997. [10] S. Bostaz, R. Haons and P. Y. Kuar 4-phase sequences ith near optiu correlation properties, IEEE Transactions on Inforation Theory, vol. 38, No.3, pp.1101 1113, May. 1992. [11] B. J. Peiris, K. R. Narayanan and S. L. Miller, A Technique to design spreading sequences for the uplink of a DS-CDMA syste in Frequency selective fading channels, Global Telecounications Conference, vol. 4, pp. 1867 1871, Dec. 2003.