Effects of High-Power Amplification on Linear Multiuser Detectors Performance in DS-CDMA Frequency-Selective Fading Channels

Transcription

1 Effects of igh-power Amplification on Linear Multiuser Detectors Performance in DS-CDMA Freuency-Selective Fading Channels L. Rugini P. Banelli S. Cacopardi D.I.E.I. University of Perugia Via G.Duranti 93/a 65 PERUGIA Italy Abstract-The aim of this paper is to examine the effects of the nonlinear distortions introduced by high-power amplifiers (PAs in the downlin of direct-seuence code-division multiple-access (DS-CDMA systems. By modelling the nonlinear distortion noise as coloured Gaussian noise a semi-analytical expression for the symbol-error rate ( of linear multiuser detectors (LMDs in freuency-selective fading channels has been derived. Simulation results confirm the effectiveness of the approach. I. INTRODUCTION DS-CDMA is a widely employed techniue in satellite and cellular mobile communication systems. This techniue is characterised by an almost constant envelope for uplin transmissions thereby reducing with respect to other techniues lie multicarrier CDMA the system sensitivity to nonlinear amplifiers. Anyway such a characteristic is lost in the downlin scenario where the DS-CDMA signal transmitted by the base station is the superposition of many independent signals that belong to the different users. Thus the resulting signal has a non constant envelope as a conseuence of the constructive and destructive superposition of each user contribution. Conseuently for downlin transmissions the performance degradation introduced by the nonlinear PA has to be carefully taen into account. Indeed even if a predistortion techniue [] is applied to counteract the PA nonlinearity a residual clipping cannot be avoided because of the maximum amplifier output power. The nonlinear distortion effects and the conseuent performance degradation induced in the system lin budget have been analysed in [] for the matched filter (MF detection of DS-CDMA signals in channels and in [3] for the linear decorrelating detector (LDD in and flat Rayleigh fading channels. Although the freuency-flat condition can be appropriate under some circumstances in many cases DS-CDMA systems are subject to freuency-selective fading due to the multipath propagation. This effect is especially highlighted in CDMA cellular systems (e.g. IMT- characterised by a wide freuency band. Since freuencyselective channels destroy the user orthogonality high performance improvements can be obtained maing use of LMD techniues [4] because of their capability in reducing the multiuser interference at the receiver side. In this paper we consider the evaluation of LMDs in presence of a PA at the transmitter. Firstly we extend to the minimum mean-suared error (MMSE detector some of the results obtained in [][3] for channels. Moreover the degradation induced by the nonlinear amplifier is analysed in freuency-selective channels obtaining the expression for uaternary phase-shift eying (PS modulations. Simulation results which validate the analytical approach are presented for the MMSE the LDD and the RAE receivers. II. SYSTEM MODEL The baseband signal transmitted by the base station to the th user in the downlin of a DS-CDMA system is expressed by x( t A b[] i s( t it ( i where T is the symbol duration A and s ( t are the amplitude and the spreading waveform respectively and b [ i ] is the ith symbol of the user. The spreading waveform is expressed by s( t c[ j] p( t jtc ( N j where N is the processing gain Tc T / N is the chip duration p(t is the chip pulse shaping waveform and c[ j ] is the jth value of the th user spreading code with c[ j ]. The symbols { b [ i ]} belong to a set of independent and euiprobable random variables with E{ b [ i ] }. The base station transmits synchronously the sum of the signals belonging to each user by a PA that if supposed to be instantaneous can be modelled by its AM/AM and AM/PM distortion curves G( and Φ ( respectively [] or euivalently by a complex nonlinear distortion F( G(exp[ jφ(]. ence the sum x(t of the users signals j arg( xt ( xt ( x( t xt ( e (3 is transformed by F( into arg( ( wt ( F( xt ( e j xt (4 which represents the baseband input-output relationship for the nonlinear amplifier. The output signal w(t can be expressed as the input signal multiplied by a complex coefficient α which represents the average linear amplification gain plus a nonlinear distortion complex noise nd ( t as expressed by Notations: We use lower (upper bold face letters to denote vectors (matrices superscripts T and to represent complex conjugate transpose ermitian and Moore-Penrose pseudo-inverse operators respectively E{} to represent the statistical expectation x and csgn( x to denote the smallest integer greater than x and the complex signum of x respectively. / The -function is defined as ( x ( π x exp( ν / dν. The symbols and denote the convolution operator and the ronecer matrix product respectively M N is the M N all-zero matrix and I N is the N N identity matrix. We define ( A mn as the (mnth entry of the matrix A ( A : n (( A m as the nth column (mth row vector of the matrix A : ( a m ( ( a n as the mth (nth entry of the column (row vector a.

2 wt ( αxt ( + nd ( t. (5 This paper assumes that the signal w(t which is transmitted by the base station passes through a slowly varying multipath channel characterised by an impulse response j g( τ βe θ δτ ( τ (6 where is the number of paths and β θ and τ are the gain the phase-shift and the propagation delay of the th path of the channel respectively and δτ ( is the delta Dirac function. By exploiting (5 the signal rt ( at the input of the receiver can be rt ( g( τ wt ( τ dτ + nt ( rsig ( t + r ( t + nt ( (7 jθ + jarg( α where rsig ( t β α e x( t τ is the useful jθ received signal r ( t βe nd( t τ is the nonlinear distortion noise and nt ( is the thermal noise. At the receiver side assuming perfect synchronisation and channel state information rt ( is filtered by a chip-matched filter and successively sampled at the chip rate obtaining rn[] l r( t p ( t lt ntc dt (8 rnsig [] l + rn [] l + rn [] l. The received chip rn [] l (8 is characterised by three additive parts: rn SIG [] l r SIG ( t p ( t lt ntc dt is the useful component related to xt ( r [] n l ( r ( t p t lt ntc dt is the in-band nonlinear distortion noise introduced by the uantity nd ( t and rn [] l n( t p ( t lt ntc dt is the in-band thermal noise with power σ E{ rn [ l] }. Assuming that the maximum delay spread of the channel is smaller than the symbol interval (i.e. max{ τ } < T the channel spreads the information related to a particular symbol over two symbol intervals and conseuently N consecutive chips contain all the energy related to the symbol of interest. ence we will consider a receiving window of two symbol intervals. Defining the channel order L max{ τ} Tc and r [ l ] [ r[ l] r [ ]] T l r [] l [ r[] l r []] T l r [] l [ r[ l] r [ ]] T l A diag( A A b [ l ] [ [ ] [ ]] T b l b l gl [] g( τ Rpp ( t τ dτ where t ltc R ( pp τ is the autocorrelation function of the pulse waveform p( t r[] l [[] r l T r [ l+ ] T ] T r [] l [ r[] l T r [ l+ ] T ] T r [] l [ r[] l T r [ l+ ] T ] T A I3 A b [] l [[ b l ] T b[] l T b [ l+ ]] T T gl [ ] g[] G (9 gl [ ] g [] N N+ L c[] c [] C L L C N C N C N ( c [ ] c [ ] N N C where C ( C we obtain N L+ : N : j arg( α r[] l α e GCAb[] l + r [] l + r [] l. ( Focusing on the user the estimate b ˆ [] l of the PS symbol b [ l ] is obtained by applying the N-row vector d to the received signal r [] l in ( as ˆ [] / b l csgn( dr [] l. ( The extension to other constellation mapping or longer delay spreads is straightforward. III. PERFORMANCE IN NOINEAR CANNELS First of all we survey some nown results for channels which turn out to be useful for freuency-selective channels also. The bacground hypotheses are herein summarised as: i igh number of active users or euivalently high number of physical channels which belong to users with < (multicode transmission. ence also the processing gain N must be sufficiently high. ii The amplitudes { A } are almost eual. Under these assumptions the PA input xt ( in (3 can be approximated by a cyclostationary Gaussian random process maing possible the application of the Bussgang theorem as in []. As a conseuence of this theorem the linear component α xt ( and the nonlinear one nd ( t in (5 are mutually uncorrelated and the PA output autocorrelation function can be evaluated [5] as Rww ( τ α Rxx ( τ + Rn ( dn τ where all the uantities are averaged over the cyclostationarity period T [6] with d i+ R ( τ γ R ( τ. (3 ndnd i xx i The coefficients { γ i } and F( and on the PA input power or as in [7]. In channels since α which depend on the PA function σ can be calculated as in [5] L the channel matrix G in (9 becomes G g[] I N and a receiving window of one symbol interval (N samples is wide enough to recover the symbol of interest. In this situation it holds true jϕ r[] l α g[] e CAb[ l] + r[ l] + r [ l] (4 where ϕ arg( α + arg( g[] is the phase-shift due to the channel and the PA. Therefore the receiver can be expressed by a N-row vector j d e ϕ ( D : (5 where D is a N matrix that depends on the used detec- x

3 tor. For the MF receiver the detector acts lie a simple despreader therefore DMF C (6 and the performance can be obtained in closed form by modelling the multiple-access interference (MAI and the nonlinear distortion noise as Gaussian random variables. These approximations are reasonably justified by the high number of active users and by the high processing gain N []. ence we obtain where g[] A ( ( P f σ σ + σ + σ (7 SIG MAI f ( x ( x ( x σ σ σ SIG α MAI jaj j j σ α g[] ( R and R C C is the matrix containing the crosscorrelation coefficients of the users' spreading codes. In [] the nonlinear distortion noise power σ has been evaluated supposing a rectangular chip pulse shaping waveform p(t leading to σ g[] R ww( αg[] R xx (. Anyway this assumption is not always realistic in bandlimited channels. By supposing a bandlimited pulse shaping waveform p(t the nonlinear distortion noise power can be evaluated as σ g[] ( C ψc where the N N matrix ψ is expressed by ( ψ mn [ Rnn ( τ R ( ] d d pp τ and Rnn ( τ is computed using (3. τ ( n m T d d c If we use the decorrelating detector [4] as D C (8 LDD and assuming that the code matrix C is full ran the MAI is completely eliminated and the performance can be expressed [3] as in (7 with σ g[] ( C ψc σ SIG α g[] A σ ( R σ where the uantity ( R is the thermal noise amplification factor due to the decorrelating operation and σ MAI. Also in this case the nonlinear noise ( Cr [ l] can be considered Gaussian because of the high processing gain N [3]. A scaled version of the MMSE receiver is [8] D MMSE M (9 where M ( I + αg[] AC W CA AC W and W is the inverse of the nonlinear-plus-thermal noise covariance matrix W g[] ψ + σ I N. As far as the of the MMSE detector is concerned we propose as in [3] to model the nonlinear distortion noise as Gaussian. Moreover even the residual MAI at the MMSE output is well approximated by a Gaussian random variable as shown in [9] leading to a that can be (7 as well where σ SIG α g[] A ( MC σ g[] ( MψM σ ( MM σ σ MAI α g[] ( MC jaj. j j IV. IN NOINEAR FREUENCY-SELECTIVE CANNELS For freuency-selective channels the detector j arg( α d e ( D : ( has to consider not only the MAI but it has also to combine the resolvable paths. The received vector in ( suggests to regard the multipath situation summarised by G as the scenario in (4 with a modified code matrix GC. As a conseuence liewise (6 (8 and (9 for freuency-selective channels we can define the RAE receiver the LDD and the MMSE detector as DRAE D LDD D MMSE M ( where GC M ( I3 + α A W A A W W GψG +σ I N and ψ is euivalent to the suared matrix ψ with higher dimension N+ L. By applying the detector D in ( to the received signal r [] l in ( we obtain v Dr s + m + n + a ( [] l j arg( α e [] l [] l [] l [] l [] l s l SIG [ l] m l MAI [ l] j arg( α j arg( α [] l e [] l where [] α D Ab is the useful signal of the user [] α D Ab is the intersymbol interference (ISI plus MAI term n[] l e Dr [] l is the nonlinear noise a Dr is the thermal noise part and SIG [ N ( : N ] and MAI [( :: N ( : + :3] are obtained by partitioning the channel-code matrix GC in ( as SIG + MAI. As in the scenario we suppose that both the MAI and the nonlinear distortion noise can be approximated as Gaussian. Since these approximations wor reasonably well in channels a similar behaviour is expected in multipath channels because the received signal can be thought as the superposition of many replicas of -lie contributions and the sum of Gaussian random variables is still Gaussian. By ( the decision variable dr [] l in ( is eual to the (th element of the vector v [] l in (. Therefore in order to evaluate the conditioned to a given channel realisation g( τ we need to calculate the power of the (th element of the vectors in the right hand side of (. As far as the thermal noise is concerned the elements of the vector a [] l are jointly complex Gaussian random variables because obtained as linear combination D of jointly complex Gaussian random variables contained in r [] l. The covariance matrix Φ of a [] l is Φ E{ a[] l a[] l } σ DD (3 and hence the (th element of the thermal noise vector a [] l in ( has power σ A σa ( DD σ. (4 For the nonlinear noise we observe that the element ([] n l + of the vector n [] l is the sum of the N elements of r [] l weighted by the N elements of j arg( α e ( D : and conseuently if the processing gain N is high enough the nonlinear distortion noise ([] n l + can be well approximated as a Gaussian random variable. The accuracy of

4 this approximation depends not only on the processing gain N but also on the channel realisation g( τ contained in GC which affects the values of the weighting vector j arg( α e ( D : and on the power input bac-off (IBO to the PA. Indeed if few elements of ( D : are characterised by a higher modulus with respect to the others the central limit theorem and conseuently the Gaussian approximation will tend to fail. On the contrary if many elements of ( D : have significant modulus the approximation accuracy is very good because many elements of the vector n [] l are weighted with coefficients having almost-eual values. Moreover if the IBO is too high most of the elements of r [] l are close to zero (at least for class A or ideally predistorted amplifiers and conseuently n [] l is practically obtained by the linear combination of few significant elements thus violating the central limit theorem hypothesis as well. The covariance matrix Φ of the vector n [] l is Φ E{ n[] l n[] l } DGψG D (5 and therefore the (th element of the nonlinear distortion noise vector n [] l in ( has power σ N σ N ( DGψG D. (6 As far as the ISI plus MAI term m [] l in ( is concerned the good accuracy of the Gaussian approximation has been already tested in [] for the RAE and MMSE receivers in linear scenarios. Moreover in many cases (e.g. when + max{ L } N [6] the LDD is able to eliminate both ISI and MAI i.e. ( m[ l ] :. The covariance matrix ΦMAI of the column vector m [] l is MAI { [] [] Φ E m l m l } α DMAI A MAI D (7 and therefore the (th element of the vector m [] l in ( is characterised by a power σ M σ M α ( DMAI A MAI D. (8 The power of the (th element of the signal vector s [] l in ( can be obtained as in the previous case leading to σ S α ( DSIG A SIG D (9 which is eual to α A for the LDD [6]. Being the symbol-error probability conditioned to a given channel realisation g( τ eual to ( S M N A P ( g f σ ( σ + σ + σ (3 where the uantities σ S σ M σ N and σ A depend on the detector that is chosen among the ones in ( the average can be obtained by averaging (3 over the joint probability density function p( g of the channel parameters β... β θ... θ τ... τ as PSEL P( g p( g dg. (3 We want to point out that our approach used to obtain (3 is uite similar to the one of [] for linear scenarios. The main difference is that since we assume the spreading codes as fixed the average over the spreading codes is not needed. V. SIMULATION RESULTS The situation in which the base station transmits data to 4 users is considered. The amplitudes { A } are eual for all users. Gold seuences of length N 63 have been chosen for the short spreading codes { c [ j ]}. For the chip pulse shaping waveform p(t a suare-root raised cosine with roll-off factor eual to ρ. has been chosen. The nonlinearities considered are the soft-limiter model (3 x x Asat G( x Φ ( x (3 Asat x > Asat which is the envelope input-output characteristic of an ideal predistorted PA and the Saleh model (33 x π x G( x Φ ( x. (33 + x 3 + x The apparent signal-to-noise ratio ( SNR app is defined at the input of the LMD as SNR app ( E{ rnsig []} l + E{ rn[]} l σ (34 while the IBO and the output bac-off (OBO are defined as IBO Pz max Pz OBO Pwmax Pw (35 where P zmax ( P wmax and P z ( P w represent the maximum and the average PA input (output power respectively. Fig. shows the performance of the MMSE receiver in channels when the soft-limiter model (3 is used for the PA. It is evident that there is a very good agreement between the analytical model and the simulation results for high OBO and for very low OBO values while there is a little mismatch at high SNR for low OBO values (OBO.3 db. As explained in [3] this mismatch is due to the Gaussian approximation of the nonlinear distortion noise. Fig. shows the performance of the three linear receivers (RAE LDD MMSE in freuency-selective fading channels. The soft-limiter model (3 with OBO.99 db has been chosen for the PA. The amplitudes { βexp( jθ } of the 5 channel paths are modelled as independent zero-mean complex Gaussian random variables with variance E{ β } while the path delays { τ } are multiple of the chip duration T c. The has been evaluated by averaging (3 over N ch 4 independent channel realisations. As expected the MMSE receiver outperforms the other two detectors. Figs. 3-4 show the performance of the MMSE detector with the freuency-selective channel model described above using the PA model (3 and (33 respectively. As in channels there is a good agreement between analytical model and simulation results especially if the OBO is very low or uite high. Moreover little mismatch is present when the OBO is roughly db (Fig. 4. VI. CONCLUSIONS An analytical framewor to evaluate the performance of LMDs for DS-CDMA systems subject to amplifier nonlinear distortions in freuency-selective fading channels has been introduced. Results for PS mapping with suare-root

5 raised cosine pulse shaping have been presented. Simulation results have shown how the analytical model is appropriate under the hypotheses of high number of users and high spreading gain. REFERENCES [] A.R. aye D.A. George and M.J. Eric Analysis and compensation of bandpass nonlinearities for communications IEEE Trans. on Comm. vol. October 97 pp [] A. Conti D. Dardari and V. Tralli On the performance of CDMA systems with nonlinear amplifier and IEEE Proc. ISSSTA vol. pp [3] L. Rugini P. Banelli and S. Cacopardi Performance analysis of the decorrelating multiuser detector for nonlinear amplified DS-CDMA signals IEEE Proc. ICC vol. 3 pp [4] S.Verdú Multiuser Detection Cambridge Univ. Press 998. [5] W.B. Davenport and W.L. Root An Introduction to the Theory of Random Signals and Noise McGraw-ill 958. [6] L. Rugini P. Banelli and S. Cacopardi Theoretical analysis and performance of the decorrelating detector for DS-CDMA signals in nonlinear fading channels submitted to IEEE Trans. on Wireless Comm.. [7] P. Banelli and S. Cacopardi Theoretical analysis and performance of OFDM signals in nonlinear channels IEEE Trans. on Comm. vol. 48 March pp [8] S.M. ay Fundamentals of Statistical Signal Processing: Estimation Theory vol. Prentice all 993. [9].V. Poor and S. Verdú Probability of error in MMSE multiuser detection IEEE Trans. on Inf. Theory vol. 43 May 997 pp [] M. Juntti and M. Latva-aho Bit error probability analysis of linear receivers for CDMA systems IEEE Proc. ICC 999 vol. pp Ideal predistortion 4 eual power users OBO.53 db (sim. OBO.84 db (sim. OBO.3 db (sim. OBO.99 db (sim. OBO 3. db (sim. OBO 4.37 db (sim Ideal predistortion 4 eual power users 5 taps OBO.84 db (sim. OBO.3 db (sim. OBO.99 db (sim. OBO 3. db (sim. OBO 4.37 db (sim (db Fig.. of the MMSE detector in channels (db Fig. 3. of the MMSE in freuency-selective channels. - - Ideal predistortion (OBO.99 db 4 eual power users 5 taps - - Saleh model 4 eual power users 5 taps OBO.3 db (sim. OBO.46 db (sim. OBO.6 db (sim. OBO.93 db (sim. OBO 4.6 db (sim RAE (th. RAE (sim. -5 LDD (th. LDD (sim. MMSE (th. MMSE (sim (db Fig.. comparison in freuency-selective channels (db Fig. 4. of the MMSE in freuency-selective channels.