STAR RECEIVER IN MIMO-CDMAMULTIPATH CHANNELS

Xiaolei Wang and Athanassios ManikasCommunications and Signal Processing Research Group

Department of Electrical and Electronic EngineeringImperial College London, SW7 2AZ, UK

Abstract— This paper is concerned with the investigation ofa MIMO-CDMA communication system which is based on theconcept of SPATIOTEMPORAL ARRAY (STAR) manifold forhandling multipath fading channels. Firstly a MIMO frequencyselective channel is modelled as a function of both transmitter’sand receiver’s antenna-array manifold vectors. Then, assumingknowledge of the channel parameters of the desired user, twolinear receivers are proposed. The proposed receivers are ’near-far’ resistant and expressed in terms of the STAR manifold vectorof the desired user. The investigation is supported by computersimulation studies where the performance of the proposed re-ceivers is compared to the performance of conventional MMSEand RAKE receivers which have been also extended in order tobe expressed in terms of the STAR manifold vectors.

Index Terms— MIMO, CDMA, Array Processing, Subspace,STAR Manifold.

NOTATIONa,A Scalara,A Column VectorA MatrixIN Identity matrix of N ×N dimension0N Zero vector of N elementsdiag (A) Diagonal matrix formed from vector Ab·c Round down to integer⊗ Kronecker product

I. INTRODUCTIONWith multiple antennas at both receiver and transmitter,

Multiple Input Multiple Output (MIMO) wireless systems haveshown great potential to provide high bandwidth efficiencyfor the next generation broadband mobile communications [1].Recently, the study of multiuser MIMO systems has receivedmuch attention where the signals of different users overlap inboth time and frequency resulting in multiple-access interfer-ence (MAI), which eventually limits the promised performanceof MIMO systems [2].

A vast majority of the reported research on MIMO systemsadopts the assumption of multiple independent elements atboth sides of the transmission links [3]. However, fadingbetween different links are not independent in real propa-gation environment [4]. The fading correlation depends onthe physical parameters of multiple antennas and the scatterercharacteristics.

Array processing and communication techniques haveevolved into a well-established technology, from old conven-tional direction nulling and phase-arrays to advanced super-resolution arrays [5][6]. By exploiting array geometry and

spatial-temporal information of the channel, e.g. direction anddelay of multipath, an arrayed-MIMO system operating in amultiple-access environment provides enormous opportunitiesbeyond the just added diversity, or array gain, benefits.

The paper is organised as follows: In section II, a multiuserMIMO channel is modelled as a function of the antenna-array manifold vector of both transmitter and receiver. Thestudy concentrates upon the uplink of an asynchronous DS-CDMA channel. Both base station and mobiles are equippedwith an array of antennas. In Section III, two spatial-temporalreceivers are developed by applying array processing to projectthe received signal onto the null-space of the unwanted-signal subspace. In Section IV, simulation studies comparethe performance of the proposed receivers to that of theconventional MMSE and RAKE receivers which have beenalso extended in order to be expressed in terms of the STARmanifold vectors. The paper is concluded in Section V.

II. SYSTEM MODEL

Consider the uplink of an M -user asynchronous MIMO-CDMA wireless communication system operating in a multi-path propagation environment. The receiver has an array of Nantennas whereas each transmitter is equipped with an arrayof N antennas1. It is assumed that the channel is frequency-selective with a delay spread less than one data symbol period.

The ith user’s baseband transmitted signal vector mi (t) ∈CN×1 can be written as

mi (t) =∞X

n=−∞bi [n] cPN,i (t− nTcs) (1)

nTcs ≤ t ≤ (n+ 1)Tcswhere Tcs is the data symbol period, bi [n] is transmitted dataduring the nth symbol period, and cPN,i (t) denotes one periodof the PN-code sequence waveform of the ith user

cPN,i (t) =

Nc−1Xk=0

αi [k] p (t− kTc) (2)

with Tc representing the chip period, αi [k] = ±1 ( ith

user’s PN sequence of period Nc = Tcs/Tc) and p (t) is therectangular chip waveform.

1Note that a symbol with a ’bar’ at the top will denote a transmitter’sparameter, (•)H represents the Hermitian, (•)T the transpose, (•)∗ thecomplex conjugate, C the Set of complex numbers and R the Set of realnumbers.

( )im t

1iS1N ×

VectorAdder

[ ]ix n

1iτ

1iβ

2iβ

iiKβ

2iS

iiKS

( 1)i iiK i Kτ τ −−

1N ×

2 1i iτ τ−

1N ×

1N ×

1N ×

1iS

2iS

iiKS

D

I

1N ×

Tapped Delay Linesof 2Nc length

2 1cNN ×

J

sTcsT

( )ix t

( )n t1N ×

Fig. 1. Arrayed MIMO Channel of the ith user.

Let us assume plane-wave propagation with transmittedsignals of the ith user arriving at the receiver via Ki multipaths.Furthermore, consider that the jth path of the ith user’s di-rection of departure (DOD) (azimuth, elevation) =

¡θij , ϕij

¢and direction of arrival (DOA)

¡θij , ϕij

¢. The array manifold

vector of the receiver Sij of the jth path of the ith user canbe formulated as

Sij = exp(−j · [rx, ry, rz] · kij) (3)

where [rx, ry, rz] ∈ RN×3 defined as the Cartesiancoordinates of the receiver antenna array element andkij = (2πFc/c) · [cos θij cosϕij , sin θij cosϕij , sinϕij]T isthe wavenumber vector pointing towards the DOA of the jth

path of the ith user. The transmitter array manifold vector asSij for the jth path of the ith user can be defined in a similarfashion to Eqn. (3) but with the parameters associated withthe transmitters. Without loss of generality, all the signals areassumed to propagate on x-y plane, i.e., ϕij = ϕij = 0.

Based on the channel model depicted in Fig. 1, the receivedcomplex baseband signal vector at the input of the receiveantenna array (point I) can be expressed as

x (t) =MXi=1

KiXj=1

βijSijSH

ijmi (t− τ ij) + n (t) (4)

where βij and τ ij represent the fading coefficient and thedelay of the jth path of the ith user, and n (t) denotes thecomplex white Gaussian noise vector with zero mean andcovariance matrix σ2IN . Note that transmit power is assumedidentically distributed among all transmit antennas, and isabsorbed by the fading complex coefficient βij .

The baseband received signal vector x (t) is initially sam-

pled with a period Ts = Tc, Thus, τ ij mod Tcs =¡lij + ρij

¢Ts, where lij is the integer part, lij =j

τ ij mod TcsTs

kand ρij ∈ [0, 1) is the factional part. The

phase shift exp¡−j2πFcρijTs¢ is absorbed by the complex

coefficient βij .At point J, the contents of a bank of N tapped delay lines

(TDL), each of length 2Nc, are concatenated to obtain

x [n] =hx1 [n]

T, · · · , xN [n]

TiT

∈ C2NNc×1 (5)

where xk [n] ∈ C2Nc×1, k = 1, · · ·N , represents the contentsof the TDL at kth receiver antenna associated with the nth

data-symbol period. Note that because of the lack of symbolsynchronization, xk [n] contains contributions from not onlythe current (nth) data-symbol, but also the (n− 1)th (previous)and (n+ 1)th (next) data-symbol.

To model the above effects, the array manifold vector Sijgiven in (3) is expanded to STAR manifold vector for the jth

path of ith user,

Sij ⊗¡Jlij ci

¢ ∈ C2NNc×1 (6)

where J =∙0T2Nc−1 0I2Nc−1 02Nc−1

¸and

ci =£α [0] , α [1] , · · · , α [Nc − 1] , 0TNc

¤T ∈ C2Nc×1

Using the STAR manifold vector defined by (6), the sampledbaseband signal vector x [n] can be rewritten as

x [n] =MXi=1

⎧⎪⎪⎪⎨⎪⎪⎪⎩Hi,prev diag

³βi

´SHi bi [n− 1]+

Hi diag³βi

´SHi bi [n] +

Hi,next diag³βi

´SHi bi [n+ 1]

⎫⎪⎪⎪⎬⎪⎪⎪⎭+ n [n]where n [n] ∈ C2NNc×1 denotes the sampled noise vector withidentical independent Gaussian distribution, and

Hi =£Si1 ⊗

¡Jli1ci

¢, · · · , SiKi

⊗ ¡JliKi ci¢ ¤ ∈ C2NNc×Ki

Si =£Si1, Si2, · · · , SiKi

¤ ∈ CN×Ki

βi=£βi1, βi2, · · · , βiKi

¤T ∈ CKi×1

Hi,prev =³IN ⊗

¡JT¢Nc

´Hi ∈ C2NNc×Ki

Hi,next =¡IN ⊗ JNc

¢Hi ∈ C2NNc×Ki .

By defining the composite channel transfer matrix of the ith

userGi = Hi diag

³βi

´SHi ∈ C2NNc×N

the MIMO-CDMA system model can be simplified by

x [n] =MXi=1

[Gi,prev,Gi,Gi,next] bi [n] + n [n] (7)

Gi,prev = Hi,prev diag³βi

´SHi , Gi,next = Hi,next diag

³βi

´SHi

and

bi [n] =hbi [n− 1]T , bi [n]

T , bi [n+ 1]TiT

∈ C3N×1

III. MIMO-CDMA RECEIVERS

Assumed that the 1st user is the desired one, the decisionvariable vector during the nth symbol period (after the receiverprocessor W1) can be decoupled into four terms representingthe desired, ISI, MAI and noise terms:

d1 [n] = d1,desired [n] + d1,isi [n] + d1,mai [n] + d1,noise [n]

= WH1 G1b1 [n] +WH

1 [G1,prev,G1,next]

∙b1 [n− 1]b1 [n+ 1]

¸+

WH1

MXi=2

[Gi,prev,Gi,Gi,next] bi [n] +WH1 n [n] (8)

Then the nth data-symbol is detected by (for BPSK mod-ulation) b̂1 [n] = sign (Re {d1 [n]}). Clearly, the optimummatrix W1 should be capable of coherently combining themultipath components of the desired signals as well removingthe unwanted MAI and ISI. Note that the weight matrix isnormalized to have unity norm.

A. Single-user Spatial-Temporal (ST) ReceiversUsing the signal model proposed in (7), the two well-known

single-user receivers (MMSE and Rake) can be extended tospatial-temporal receivers and redefined as

Rake Receiver: W1 = G1MMSE Receiver: W1 = R−1xxG1

where Rxx = E ©x [n]xH [n]ª. These two receivers will beused for comparison with the two receivers proposed in thefollowing subsection2.

B. Subspace-based ST ReceiversWith the aid of the array manifold, the directional parame-

ters of the channel can be employed by the receiver to enhancethe desired signals while alleviating the influence of unwantedsignals. A subspace-based weight matrix is proposed as

W1 = κ1PnH1¡HH1 PnH1

¢−1 SH1 ³S1 diag³β∗1´SH1 ´−1(9)

where κ1 is the normalization factor and Pn represents the con-structed projection-operator matrix onto the subspace spannedby the noise eigenvectors of the noise-plus-MAI/ISI space.

Substituting (9) into (8), it can be derived that

d1,desired [n] =WH1 G1b1 [n] = κ1b1 [n]

In the rest of this section, based on different channel knowl-edge, two approaches of constructing the projection operatorPn are proposed.

If the receiver is capable of obtaining the channel parameters(including the spreading codes) of all the users, the projectorcan be constructed by

Pn = projection operator onto subspace L [G1]⊥

= I2NNc − bH1 ³bHH1bH1´−1 bHH

1

2Note that, all these linear receivers (including the following proposedreceivers), only aim at handling the effect ISI/MAI. To cope with theeffect of Inter-Channel-Interference, system designers will need to implementSuccessive Cancellation (SUC) algorithms [1].

Fig. 2. Decision variables for 400 data symbols for a single run ofthe simulation. The subspace-based ST receivers show significant near-farresistant ability.

where bH1 = £H1,prev, H1,next, H2,total · · · , HM ,total

¤and Hi,total =

£Hi,prev, Hi, Hi,next

¤. Eqn. (9) in connec-

tion with the above defined projection operator Pn will becalled ST-1 receiver.

When the channel parameters of only the desired user areknown, a interference–plus-noise-cancelling (INC) projectionoperator Pn can be obtained from the matrix Runwanted associ-ated with the last three terms of expression (8)

Pn = projection operator onto the noise subspace of Runwanted

where Runwanted = Rxx−G1GH1 . This projection operator with

(9) will be called ST-2 receiver.With these two proposed approaches, the weight matrix W1

is made orthogonal to the estimated unwanted-signal subspacecontaining the contributions of ISI and MAI. The power of theresidual interference is determined by the angle between theestimated unwanted-signal subspace and the actual MAI+ISIsubspaces.

IV. SIMULATIONS

To evaluate the performance of the proposed receivers,firstly consider the uplink of a 3-user MIMO-CDMA systemwith the transmitters and receiver employing uniform lineararray (ULA) of half-wavelength intersensor. The receiversoperate in the presence of multipath effects from other users(MAI), as well as the ISI introduced in a frequency-selectivefading channel. Using gold sequences of length 31, we arbi-trarily assume 10 multipaths for 1st (desired) user, 8 multipathsfor 2nd user and 6 multipaths for 3rd user. Furthermore, itis assumed that all required channel parameters have beensuccessfully estimated, the input SNR associated with thedesired user is 20 dB and the interfering signals are muchstronger than the desired signal such that

0 2 4 6 8 10 12 14 16 18 20

10-4

10-3

10-2

10-1

100

SNR (dB)

BE

R

RakeMMSEST-2ST-1

R A K E

Fig. 3. BER performance of different receivers in a MIMO-CDMA systemwith 10 users, each with 3 multipaths. N = 2 and N = 2, with half-wavelength spacing.

20 log10

³°°°βi

°°° /°°°β1

°°°´ = 20 dB, for i = 2, 3

The decision variables at the output of MMSE, ST-1 and ST-2receivers are plotted in Fig. 2. It is shown that the proposedsubspace-based ST receivers are near-far resistant.

In Fig. 3 and 4, the Bit-Error-Rate (BER) performance ofvarious receivers is evaluated for N = 2 and different value ofN . There are 10 users, each with 3 multipaths. The simulationis executed for 1000 bursts of 400 symbols each. The resultsdemonstrate that the two proposed ST receivers outperformRAKE and MMSE receivers. The BER performance of all STreceivers improves as the number of receive antennas increasesfrom 2 to 4. However, the performance of the MMSE receiverdecreases because with more receive antennas it needs moresnapshots to properly estimated the receive correlation matrix.It is important to point out that because of near-far effect,the scenario considered here is MAI-dominant, which explainswhy the performance of MMSE or RAKE receivers do notimprove by increasing the SNR.

When the transmitters and the receiver have roughly thesame number of antennas, the ST-2 receiver performs betterthan ST-1 receiver in low SNR situations. This is because theST-1 receiver only projects the received signal onto the nullspace of the interference, while the ST-2 receiver projects thereceived signal onto the null space of interference-noise space.When SNR is high, MAI becomes the dominant source of theerror, so the ST-1 performs better.

The performance of the receivers is expected to decreasewith the increase of the number of users in the CDMA systemand simulation results presented in Fig. 5 show such effect.For a (2,3) MIMO channel with 2 multipaths per user. Theperformance improvement of the proposed ST receivers isclearly shown in Fig. 5.

V. CONCLUSIONS

In this paper, the uplink of a multiuser asynchronousMIMO-CDMA wireless communication system operating infrequency selective fading channel has been studied usingthe concept of STAR manifold vector. Two spatial-temporal

0 2 4 6 8 10 12 14 16 18 20

10-4

10-3

10-2

10-1

100

SNR (dB)

BE

R

MMSERakeST-2ST-1

RAKE

Fig. 4. BER performance of different receivers in a MIMO-CDMA systemwith 10 users, each with 3 multipaths. N = 2 and N = 4. All arrays areULA with half-wavelength spacing.

0 2 4 6 8 10 12 14 16 18 2010-4

10-3

10-2

10-1

100

SNR (dB)

BE

R

15 users10 users

ST-decorrelating

MMSE

ST-INC

Rake

ST-1

ST-2

RAKE

Fig. 5. BER performance of MIMO-CDMA system of 10 and 15 users,using 2-element ULA (half-wavelength spacing) at transmitters, 3-elementULA (half-wavelength spacing) at receivers, and assuming perfect channelestimation.

receivers have been proposed based on projecting the receivedsignals onto the null-space of the interference or interference–plus-noise subspace. Under the assumption that only the chan-nel parameters of the desired user are available, simulationresults show that, the subspace-based ST-2 receiver outperformRAKE and MMSE receivers.

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