Detection Techniques for Direct Sequence & Multicarrier Variable ...

Detection Techniques for Direct Sequence & Multicarrier Variable Rate Broadband CDMA Lars K. Rasmussen & Teng J. Lim

Centre for Wireless Communications National University of Singapore Kent Ridge Crescent Singapore 119260 [email protected] - [email protected] Abstract| Multicarrier CDMA and direct sequence CDMA on these principles have been reported by Adachi et al. in have been suggested for both narrowband and broadband [4]. A way to achieve variable rate transmission without multiple access systems. A series of concepts have further aecting the bandwidth of the system is the use of mulbeen proposed for variable rate broadband multimedia transmission within a CDMA system. These principally dierent tiple modulation formats. In order to provide a constant techniques exhibit certain common characteristics that can SNR it is however necessary to signi cantly increase be exploited. In this paper we propose a generalised unify- bit the power for modulation formats with constellations larger ing discrete-time model for describing both MC-CDMA, DSCDMA and hybrid DS-MC-CDMA narrowband and variable than QPSK [6]. rate broadband schemes. The model leads to a system deAn alternative option related to the concept of multiple scription which is similar to the matrix algebraic formulation channels is the use of multicarrier (MC) CDMA techniques. of a traditional DS-CDMA system. Performance enhancing multiuser detection techniques based on matrix algebra are Several approaches have been proposed for both narrowsuggested. band and broadband applications (see [7] for references). These approaches rely on the spreading operation being I. Introduction performed in the frequency domain. Jung et al. reports For second generation digital cellular mobile communi- a detailed study of such a pure frequency spread system cations systems both TDMA and CDMA technologies have in [7]. Another option is to combine direct sequence (DS) been accepted as multiple accessing techniques. These sys- CDMA with a multitone technique. Such a system was rst tems have primarily been narrowband systems intended for suggested by Vandendorpe in [8]. In eect this system is a voice communications. The demand for toll quality speech DS-CDMA overlay on an OFDM scheme. In a CDMA based mobile radio system, the signal proand variable rate broadband services in the wireless mobile cessing for the downlink and for the uplink dier considerenvironment is expected to increase rapidly in the near fuably. In this paper we focus on the uplink and develop ture. Variable data rates of up to 2 Mbps are suggested for a uniform approach to variable rate, broadband CDMA third generation systems, depending mainly on the proptransmission. A generalised discrete-time matrix algebraic agation environment. Common suggestions for maximum model similar to the model developed in [7], but which endata rates are 2 Mbps for an oce environment, 144 Kbps compasses all the principal variable rate broadband schemes for slow moving users, and 64 Kbps for fast moving users. suggested in the literature is derived. Narrowband MC and CDMA techniques oer some attractive features for proCDMA systems are also described by the model which esviding variable data rates in a multiple access environment sentially is a DS-CDMA overlay on a MC-CDMA system. [1]. A high level of exibility can be obtained to accommoIt is shown that all schemes share certain characteristics date variable data rates and bit rate on demand. Features and are described by a matrix algebraic equation similar to such as high user capacity, simpli ed frequency planning, the traditional DS-CDMA case. and soft capacity are among the possibilitites. Furthermore in a broadband scenario, toll quality speech codecs at 16 or II. DS Overlaid Multicarrier CDMA 32 Kbps are feasible. Several strategies have been proposed for variable rate broadband CDMA. One or more of the deMulticarrier transmission techniques are derived from orscribing system parameters can be varied to accommodate thogonal frequency division multiplexing (OFDM) [9]. In varying rates. Two dierent approaches can be followed an OFDM system the overall bandwidth Bu is divided into for variable rate transmission depending on whether the QT subbands each with equal bandwidth Bs and individual data symbol interval Tsk and/or the chip interval Tck are subcarriers. The relation between the subcarrier bandwidth kept constant or variable. In case Tsk is varying and Tck is Bs and the overall bandwidth Bu is given by constant we experience a varying processing gain. The use Bu = QT
Bs : of multiple processing gains was considered for a practical test system in [1] and is suggested for the IS-665 broadband Since orthogonal subcarriers allow for spectral overlap, Bs CDMA standard [2]. When Tck is varying according to Tsk is chosen equal to 1=Ts where Ts is the transmission symwe have a xed processing gain for varying data rates. We bol interval. In an OFDM system independent data symthen experience varying chip rates. Variable chip rates have bols are transmitted over each subcarrier. If the same data been suggested for variable rate transmission in [3], [4]. symbols are transmitted over Q subcarriers, 1 < Q < QT , In case Tsk remains constant variable rate transmission frequency diversity is obtained. This more general concept can be achieved in a CDMA system by assigning multiple is known as multicarrier techniques [7]. In order to facilspreading codes to the same user. A series of narrowband itate simple receiver structures, most MC-CDMA systems channels is assigned on demand to a speci c user according suggested in the literature are designed to avoid ISI. This is to the desired data rate [5]. A practical test system based only possible by introducing guard intervals between suc-

cessive data symbols. When DS-CDMA is superimposed on a MC system it is not possible to alleviate ISI between chips by use of guard intervals since the chip period is usually considerably smaller than the guard interval [8]. We do therefore not consider the use of guard intervals here. Both MC and DS allow for multiple access through user-speci c spreading codes. The DS-CDMA concept is well-known [10]. In a MC system the spreading is eectively performed in the frequency domain. In a MC-CDMA scheme with overlaid DS-CDMA (DS-MC-CDMA), joint design of the two principally dierent techniques for spreading can be incorporated to allow for better user separation in a broadband system. In a MC-CDMA scheme QT = Q
P where we denote the number of independent data symbols per symbol interval by P . If we allow each user in the system to use only a subset of the available subcarriers we obtain more design exibility since now QT  Qk
Pk ; (1) where k is the user index. This exibility facilitates the variable rate concept of multiple channel use. For a DSMC-CDMA system the overall maximum bandwidth is determined by (2) B = QT + N ? 1 ; u

Ts

where N is the length of the DS spreading codes within one symbol duration Ts . The basic chip interval for the DS spreading is determined by N and Ts as

Tc = TNs :

(3)

To allow for multiple chip rates we introduce ~ Tck = t~k Tc = tkNTs (4) where t~k is an integer determining the actual user-speci c chip interval t~k Tc. In this case t~k Ts represents the interval during which a length N spreading code is used for modulation. In a system where only the concept of multiple chip rates are used this will also be the data symbol interval. To allow for multiple processing gains as well we introduce t^k which is an integer determining the actual user-speci c processing gain Nk = t^k N . The corresponding Tck is determined by Eqn. (4) so the user-speci c data symbol interval is described by Tsk = t^k t~k Ts ; (5) Eqn. (3){(5) are visualised in Fig. 1. For an mk -ary trans~ TTsk = t^k t~k Ts ck = tk Tc t~k Ts -

 -T Ts-

-

from the set of design parameters to accommodate system requirements. III. Uplink System Model

In this section the baseband equivalent uplink model for the generalised DS-MC-CDMA communications system considered throughout this paper is described. The uplink model is based on a discrete-time, chip synchronous, symbol asynchronous system, assuming single path channels1 and the presence of stationary additive white Gaussian noise with zero mean and variance 2 = N0 =2. With reference to the concepts presented in the previous section we introduce a series of indices to be used throughout the section. A speci c user is identi ed by k, a transmission symbol interval by l, a user-speci c data symbol interval by lk , an interval spanned by a length N spreading code within a data symbol by ~lk , a transmission symbol interval within a length N spreading code interval within a data symbol by l^k , a chip within a length N spreading code by i, a chip within a length N~ spreading code by iv , a chip within the received signal vector by mv , an arbitrary subcarrier by %, a group of subcarriers transmitting identical data symbols by p, the individual subcarriers within such a group by q and each individual subcarrier used by user k by k . The K users in the system each transmit nite data symbol sequences selected from an mk -ary symbol set V k . Each user transmits Pk data symbols per data symbol interval in a total of Mk data symbol intervals of duration t^k t~k Ts . Therefore, a total of M symbol intervals of duration Ts , where M = t^k t~k Mk , are used by each user. The data rate rd;k is determined by Eqn. (6) with Qk consecutive subcarriers transmitting the same data symbol2 . Let P~k = Pk =t^k t~k be the fractional number of independent data symbols transmitted in each transmission symbol interval. PIn a K user system we then appear to transmit Pt = Kk=1 P~k independent data symbols per transmission symbol interval, where low rate transmission is accomplished by transmitting identical data symbols over consecutive transmission symbol intervals. We de ne the data symbol vector for user k at data symbol interval lk as dk;lk = (dk;1;lk ; dk;2;lk ; :::; dk;Pk ;lk )> dk;p;lk 2 V k : As mentioned in Section II the available bandwidth is divided into QT subcarriers each with bandwidth 1=Ts. The subcarrier baseband frequencies are chosen as f = % ? 1 % = 1; 2;


; Q : (7) %

Ts

T

Based on (7) the centre frequency of the system carrier with bandwidth Bu is given by c fc = QT2T? 1 6= 0: s Fig. 1. Illustration of Eqn. (3){(5) for N = 5, t~k = 3 and t^k = 2. The Obviously, the centre frequency of the system carrier is not processing gain is Nk = 10 and the data symbol interval is 6Ts . equal to zero in the equivalent baseband domain. Howmission format for user k the user-speci c data rate is de- ever, the choice of subcarrier frequencies facilitates a simtermined by ple mathematical modelling of the discrete-time DS-MCSingle path channels are assumed for simplicity. More realistic multi(6) rd;k = ~m^k Pk ;

tk tk Ts

where mk allows for multiple modulation formats. For a given overall bandwidth and data rates we can now choose

1

path radio channel models can be included as described in [11]. 2 To achieve further frequency diversity, the Qk subcarriers could be evenly distributed over the set of available subcarriers. Here we use consecutive subcarriers for notational simplicity.

CDMA system structure. The centre frequency f% given by (7) of subcarrier % results in a phase deviation identip ed by (1= QT ) exp(j 2f% t) [7]. This form lends itself to a simpli ed mathematical notation. Each subcarrier is modulated by a user-speci c length N DS-CDMA spreading sequence with chips selected from an mck -ary chip symbol set V ck . Multiple chip rates are provided by using an actual chip rate which is a multiple of the basic chip rate. In this case a spreading code of length N extends over t~k consecutive transmission symbol intervals of duration Ts . To ease the matrix algebraic modelling we introduce the t~k -oversampled spreading code vector ~sk;%;lk for spreading code interval ~lk . This spreading vector is of length t~k N where blocks of t~k samples constitute a chip from the length N spreading code. This vector is then divided into t~k spreading vectors sk;%;l of length N , one for each transmission symbol interval Ts . ? ~sk;%;lk = sk;%;lk ;0 ; sk;%;lk ;1 ; :::; sk;%;lk ;t~k ?1 ; :::; >

sk;%;lk ;t~k (N ?1) ; :::; sk;%;lk ;t~k N ?1 ?
= sk;%;t^k t~k lk ; sk;%;t^k t~k lk +1 ; :::; sk;%;t~k t^k lk +t~k ?1 > ; where sk;%;lk ;i 2 V ck .

In order to facilitate the use of only a subset of subcarriers for each user and the transmission of identical data symbols on several subcarriers we distinguish between subcarriers with identical data symbols q and groups of subcarriers with dierent data symbols p. The received signal is then described by

e(t) =

where

Qk MX Pk X K X ?1 k ?1 t^X k ?1 t~X k ?1 NX X

wk;lk dk;p;lk k=1 p=1 q=1 lk =0 ~lk =0 ^lk =0 i=0   j 2  (  ? 1) t k + n(t);
sk;k ;l;i u(tu ) exp Ts

tual chip rate 1=Tv = Bu = (QT + N ? 1)=Ts = N=Ts for the DS-MC-CDMA system. The discrete-time equivalent signal for the mth v chip is then Qk MX Pk X K X ?1 k ?1 t^X k ?1 NX k ?1 t~X X

wk;lk dk;p;lk k=1 p=1 q=1 lk =0 ~lk =0 ^lk =0 i=0   j 2  (  ? 1) m k v + nmv ;
sk;k ;l;i  (bm~ v c) exp ~

emv =

N

where m~ v = 0; 1; :::; N~ (M + 1) ? 1, (
) is the Dirac delta v function, m ~ v = Nm N~ ? i ? Nl ? k , bxc is the largest integer less than x, and nmv is an additive white Gaussian noise sample with zero mean and variance 2 = N0 =2. We can de ne a set of equivalent spreading codes corresponding to the virtual chip rate of the system. ^sk;p;l =

s^k;p;l;iv j



>

s^k;p;l;0 ; s^k;p;l;1


; s^k;p;l;N~ ?1 ;

Q X





= sk;k ;l; exp j 2(k~? 1)iv N q=1 for iv = 0; :::; N~ ? 1; (8) k

where  = Niv =N~ . The eect of multicarriers is therefore equivalent to increasing the length of the complex DS spreading code by QT . It is clear from (8) that for a DSMC-CDMA system it is not possible to let consecutive virtual chips be identical without violating the multicarrier structure. It is therefore not possible to allow for multiple chip rates in a MC based system. Users at lower data rates will invariably experience a larger processing gain. It is however possible to use multiple chip rates at the DS level. To get a matrix algebraic description, the received chips for the entire transmission are collected into one column vector

 > l = ^lk + t~k (~lk + t^k lk ); l = 0; :::; t^k t~k Mk ? 1; e = e1 ; e2; :::; eN~ (M +1) k = q + Qk (p ? 1);  = 1; :::; Qk Pk ; tu = t ? iTc ? lTs ? k0 ; of length N~ (M + 1). Assuming, without loss of generalp and wk;lk = k;lk = QT includes the path gain and the nor- ity, that the users are ordered according to increasing data malisation constant, dk;p;lk denotes the data symbol trans- rates, i.e., user K has the highest data rate, then de ne mitted in data symbol interval lk by user k on subcarrier A = (A1;1 ; A2;1 ; :::; AK;1 ; A1;2 ; :::; AK;MK ); group p, sk;k ;l;i is the ith chip of the relevant spreading Ak;lk = (ak;1;lk ; ak;2;lk ; :::; ak;Pk ;lk ); code, u(t) is a rectangular pulse of duration Tc, k0 = k Tc ? with k 2 [0; 1;


; N ? 1] being the chip synchronous user ak;p;lk = 0lN~ +~k ; ~sk;p;t^k t~k lk ; ::: speci c delay due to the random access of users on the > ~sk;p;t^k t~k (lk +1)?1 ; 0; :::; 0 ; uplink, the phase deviation for subcarrier k is captured by the exponential function and n(t) is an additive white Gaussian noise process. where A is a matrix of dimensions N~ (M + 1)  MPt , Ak;l In traditional DS-CDMA a discrete-time model is ob- is of dimensions N~ (M + 1)  Pk , and ak;p;l is a columnk k tained either through matched ltering or through sam- vector of length N~ (M + 1) with t~ t^ N non-zero elements. k k pling. It is desirable to digitize the system as early as Here 0 ~ denotes a row of zeros of length l N + ~k where jlN~ +~kk possible due to the advantages of DSP over analogue pro~ N cessing. Chip matched ltering is therefore commonly sug- ~k = k N ,

gested for DS-CDMA. This approach is however not possiW = diag(W 1;1 ; W 1;2 ; :::; W K;MK ); ble for the generalised DS-MC-CDMA system since the chip pulse shape is modi ed by the subcarrier phase deviation W k;lk = diag(wk;lk ; :::; wk;lk ); for dierent subcarriers. An equivalent discrete-time model is therefore obtained through sampling. The sampling rate where W is a diagonal matrix of dimensions MPt  MPt is Bu as determined by Eqn. (2). This corresponds to a vir- and W k;lk is a diagonal matrix of dimensions Pk  Pk and

nally

d =

>



d>1;1 ; d>2;1 ; :::; d>K;1 ; d>1;2 ; :::; d>K;MK ; >



Hybrid solutions, combining constrained and unconstrained minimisation have been suggested as well [12], [13]. Here we solve for d^ = arg minC min je ? AWdj2 ; m MP ?C

n = n1 ; n2 ;


;


nN~ (M +1) ; dc 2V k du 2R t where d is a column vector of length MPt and n is a column vector of length N~ (M + 1). We can then express the entire where dc of dimension C represents the data detected in the constrained domain and du represents the data detected in transmission as the unconstrained domain. (9) e = AWd + n: Duel-Hallen has suggested the decorrelating decision feed(

)

This matrix equation is identical in form to the description back detector (DDFD) in [20]. This detector has been furof traditional DS-CDMA given in [11]. All the schemes ther developed in [21]. The DDFD structure relies on the encompassed by the model can thus be regarded as DS- partially decorrelating noise whitening matched lter,  ? CDMA schemes with approach-speci c spreading codes. M w = F> A>; 1

IV. Multiuser Detection for DS-MC-CDMA

The model in Eqn. (9) is identical to the description of traditional DS-CDMA given in [11], [13]. It is therefore possible to directly apply receiver structures developed for DSCDMA in the general DS-MC-CDMA case. Thus far only a few receiver structures for either joint multiuser detection (JD) or interference cancellation (IC) have been suggested for MC-CDMA or DS-CDMA overlaid OFDM. In [14] and [15] multi-stage detectors based on the structure developed by Varanasi and Aazhang [16] are proposed for MC-CDMA. A simpler multi-stage IC scheme is suggested in [6] for variable rate DS-CDMA. Another joint IC and equalisation scheme based on the minimum mean square error (MMSE) criterion is suggested for a DS-CDMA-OFDM system in [17]. A more mathematically systematic approach is followed by Jung et al. in [7]. Based on a discrete-time model similar to (9) they develop the zero forcing block equalisation detector and the MMSE block equalisation detector with and without decision feedback. We take a similar approach here where we exploit the similarities to traditional DS-CDMA and list a series of receivers based on matrix algebra which are straightforwardly adapted to the DS-MC-CDMA case. In the following we assume that A and W are known at the receiver. The conventional CDMA detector is a bank of lters matched to the individual spreading codes. Based on Eqn. (9) then we get the matched lter output as y = A>e = A>AWd + A>n: The optimal constrained detector suggested by Verdu [18] performs a constrained exhaustive maximum likelihood hypotheses search to select an estimate of d. d^ = arg minMPt je ? AWdj2 :

d2V k

It is well-known that the optimal detector is prohibitively complex. An alternative is to conduct an unconstrained ML-search, i.e., d^ = arg min je ? AWdj2 ; mMP

d2R

t

where R denotes the set of real numbers and m is the maximum dimensionality of the input symbols. The solution to this equation is the well-known decorrelator [19], Wd = Me; where   ? M = A>A A>: 1

where F>F = A>A and F is lower left triangular. Minimum mean square error (MMSE) receivers can also be developed based on (9). Following standard derivations for the MMSE criterion the steady state solution for the MMSE lter becomes 

M m = WA>AW + N20 I

?1

WA>:

The well-known fact that for a negligible noise level the MMSE detector converges to the decorrelator is noted. The MMSE detector is mainly attractive through adaptive realisations as demonstrated in [22]. Successive interference cancellation (SIC) schemes [6] are usually considered a more practical approach to multiuser detection. For SIC there is a distinction between linear and non-linear techniques [23]. In the linear case the correlator output is used directly for cancellation where the nonlinear approach makes use of hard decisions to re-generate the corresponding signal contribution for cancellation. The linear SIC schemes can be appropriately described by linear matrix ltering [24] while non-linear SIC is based on linear matrix ltering in conjunction with a sub-optimal tree search. The linear SIC scheme is related to the decorrelator. In a multi-stage scenario, the linear SIC is based on alternate projections and will in fact converge to the decorrelator [23]. The non-linear scheme is closely related to the DDFD. It is shown in [25] that the DDFD performs complexityconstrained ML detection while the SIC scheme is a suboptimal approach. Again a multi-stage realisation provides better performance. All the detector structures presented here are based on a discrete-time matrix-algebraic representation. Since the generalised DS-MC-CDMA scheme is described by a matrixalgebraic relation (Eqn. (9)), the multitude of matrix-algebraic detector structures are automatically applicable. The concern is of course the involved complexity. As opposed to binary DS-CDMA, in this general case we deal with contineous-valued multi-dimensional spreading codes. For variable rate scenarios the detector structure must also deal with a signi cant amount of \book keeping" in order to provide the advantages of joint detection. The application of multiuser detection for variable rate broadband CDMA does not alleviate the inherent problems encountered. It rather translates the problems into a direct question of processing complexity versus required performance.

V. Concluding Remarks

In this paper we have developed a mathematically unifying approach to DS-CDMA, MC-CDMA and hybrid DSMC-CDMA systems. Four dierent approaches to variable rate transmission for broadband CDMA is also encompassed by the model. All the schemes are described by a matrix algebraic model similar to traditional DS-CDMA. MC-CDMA and hybrid DS-MC-CDMA can therefore be viewed as DS-CDMA systems with approach-speci c spreading codes. As a consequence of the matrix description, it was noted that the rich array of multiuser detection developed for DSCDMA are directly applicable to the generalised scenario of DS-MC variable rate broadband CDMA. Even with low complexity sub-optimal detection strategies however, the variable rate broadband CDMA application still remains a signi cant challenge for processing complexity.

[12]

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