A Novel Prefiltering Technique for Downlink Transmissions in TDD MC ...

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 5, SEPTEMBER 2005

A Novel Prefiltering Technique for Downlink Transmissions in TDD MC-CDMA Systems Michele Morelli, Member, IEEE, and L. Sanguinetti

Abstract—We discuss a prefiltering technique for interference mitigation in the downlink of a time division duplex (TDD) multicarrier code-division multiple access (MC-CDMA) system. The base station (BS) is equipped with multiple transmit antennas, and channel state information (CSI) is obtained at the transmitter side by exploiting the channel reciprocity between uplink and downlink transmissions. The prefiltering coefficients are designed so as to minimize a proper cost function that depends on the signal– to-interference-plus-noise ratios (SINRs) at the mobile terminals (MTs). The resulting scheme allows using a simple despreading receiver, thereby eliminating the need for channel estimation and equalization. Numerical results show the advantages of the proposed scheme over some existing solutions. Index Terms—Multicarrier code-division multiple access (MC-CDMA), prefiltering, time division duplex (TDD) systems.

I. I NTRODUCTION

M

ULTICARRIER code-division multiple access (MCCDMA) is a transmission technique that combines orthogonal frequency-division multiplexing (OFDM) with direct sequence CDMA [1], [2]. Due to its favorable features such as robustness to frequency-selective fading, high spectral efficiency, and flexibility to support integrated applications, MC-CDMA is considered as a promising candidate for the physical layer of future high-speed wireless communications. In an MC-CDMA system, the data of different users are spread in the frequency domain using orthogonal signature sequences. In the presence of multipath propagation, however, the spreading codes lose their orthogonality. This gives rise to multiple-access interference (MAI) at the receiver, which strongly limits the system performance. Several advanced multiuser detection techniques are available for interference mitigation [3]. In spite of their effectiveness, however, these methods are not suitable for downlink applications due to their relatively high complexity. As an alternative to multiuser detection, prefiltering techniques can be employed in the downlink to mitigate MAI and channel distortions. The idea behind prefiltering is to vary the complex gain assigned to each subcarrier so that interference is reduced and the signal at the receiver appears undistorted. In this way, channel estimation is not necessary at the mobile terminal (MT), and detection schemes with low complexity can be employed. In order to work properly, however, prefiltering techniques require channel state information (CSI) at Manuscript received February 1, 2004; revised July 22, 2004; accepted August 5, 2004. The editor coordinating the review of this paper and approving it for publication is J. K. Tugnait. This work was supported by the Italian Ministry of Education under the FIRB project PRIMO. The authors are with the Department of Information Engineering, University of Pisa, 56100 Pisa, Italy (e-mail: [email protected]). Digital Object Identifier 10.1109/TWC.2005.853894

the transmitter. This can be achieved in time division duplex (TDD) systems by exploiting the channel reciprocity between alternative uplink and downlink transmissions [4]. If channel variations are sufficiently slow, the channel estimate derived at the base station (BS) during an uplink time slot can be reused for prefiltering in the subsequent downlink time slot. Prefiltering schemes for downlink MC-CDMA transmissions have recently been proposed in conjunction with multiple transmit antennas [5]–[7]. In particular, the method discussed in [5] is based on the zero-forcing criterion and selects the prefiltering coefficients to eliminate the MAI at each MT. The resulting scheme dispenses with channel knowledge at the receiver side and has good performance when multiple antennas are employed at the BS. Zero-forcing prefiltering is also proposed in [6] in conjunction with equal-gain-combining (EGC) reception. Compared with [5], this scheme has better performance but requires channel estimation at the MT. The method discussed in [7] aims at maximizing the signal-to-interference-plus-noise ratio (SINR) at the MTs. Unfortunately, this leads to a complicated joint optimization problem for the transmit filters of all the active users. To overcome this obstacle, the authors propose a suboptimum approach based on a modified SINR. In this letter, we present a novel prefiltering scheme for the downlink of TDD MC-CDMA systems equipped with multiple transmit antennas. In designing the prefiltering coefficients, we aim at minimizing a proper cost function that depends on the SINRs of all the active users. As we shall see, this problem has no closed-form solutions. As an alternative, we propose a suboptimal procedure in which the zero-forcing approach of [5] is first employed to completely eliminate the MAI, and the result is then exploited to minimize the cost function in closed form. In doing so, we impose a constraint on the overall transmit power allocated to the active users. This is in contrast to the methods in [5]–[7], where separate power constraints are imposed for each user. Simulations indicate that significant performance improvements can be achieved in this way with respect to existing prefiltering schemes. The rest of the paper is organized as follows. Section II describes the signal model and introduces basic notation. In Section III, we discuss the power constraint condition and we compute the prefiltering coefficients. In Section IV, simulation results are presented, while in Section V, conclusions are drawn. II. S IGNAL M ODEL A. MC-CDMA Transmitter We consider the downlink of an MC-CDMA network in which the total number of subcarriers N is divided into smaller

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Fig. 1.

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Block diagram of the transmitter and receiver.

groups of M elements. The BS is equipped with P antennas and employs the subcarriers of a given group to communicate with K active users (K ≤ M ). The latter are separated by their specific spreading codes, usually chosen from an orthogonal set. Without loss of generality, we concentrate on a single group and assume that the M subcarriers are uniformly spread over the signal bandwidth to better exploit the channel frequency diversity. We denote {in ; 1 ≤ n ≤ M } the subcarrier indexes in the considered group, with in = 1 + (n − 1)N/M . Fig. 1 shows the block diagram of the system under investigation. The symbol ak of the kth user is spread over M chips using the code sequence ck = [ck (1), ck (2), . . . , ck (M )]T , where √ ck (n) ∈ {±1/ M } and the notation (·)T means transpose operation. The resulting vector ak ck is fed to P parallel prefiltering units, one for each antenna branch. The output from the pth unit is ak uk,p , where uk,p = [uk,p (1), uk,p (2), . . . , uk,p (M )]T is a vector with entries uk,p (n) = ck (n)wk,p (n),

n = 1, 2, . . . , M

2 (i) = ables with zero mean (Rayleigh fading) and power σk,p 2 E{|hk,p (i)| }. The transmit antennas are sufficiently separated in space to consider vectors hk,p statistically independent for different p’s. The CIRs are practically constant over the downlink time slot (slow fading) and perfect channel knowledge is assumed at the BS. The impact of channel variations on the system performance is addressed later by simulation.

C. MC-CDMA Receiver As shown in Fig. 1, the P waveforms arriving at the receiver side are implicitly recombined by the receive antenna and passed to an OFDM demodulator. We concentrate on the mth MT and denote X m = [Xm (1), Xm (2), . . . , Xm (M )]T the demodulator outputs corresponding to the M subcarriers of the considered group. Assuming ideal frequency and timing synchronization, we have

(1) Xm =

T

and wk,p = [wk,p (1), wk,p (2), . . . , wk,p (M )] are the prefiltering coefficients of the kth user at the pth antenna. As shown later, the elements of wk,p are computed on the basis of CSI generated from the uplink transmission. The contributions of all users are summed chip-by-chip to form the following multiuser signal at the pth antenna sp =

K

ak uk,p ,

p = 1, 2, . . . , P.

(2)

k=1

Finally, each sp (p = 1, 2, . . . , P ) is mapped on M subcarriers using an OFDM modulator for each antenna branch. An NG -point cyclic prefix [longer than the channel impulse response (CIR)] is inserted in the transmitted signals to avoid interference between adjacent OFDM blocks. B. Channel Model The signals transmitted by the BS array propagate through multipath channels and undergo frequency selective fading. We assume that the MTs are equipped with singleantenna receivers and denote hk,p = [hk,p (0), hk,p (1), . . . , hk,p (Lk,p − 1)]T the discrete-time CIR between the pth transmit antenna and the kth MT. The entries of hk,p are modeled as complex-valued independent Gaussian random vari-

P K

ak H m,p uk,p + nm

(3)

p=1 k=1

where H m,p is a diagonal matrix H m,p = diag {Hm,p (i1 ), Hm,p (i2 ), . . . , Hm,p (iM )}

(4)

and nm = [nm (1), nm (2), . . . , nm (M )]T is thermal noise, which is modeled as a Gaussian vector with zero-mean and covariance matrix σn2 I M (I M is the identity matrix of order M ). In (4), Hm,p (in ) is the channel frequency response over the in th subcarrier and is computed by taking the discrete Fourier transform (DFT) of hm,p , i.e., Lk,p −1

Hm,p (in ) =

hm,p ()e

−j2πin N

,

n = 1, 2, . . . , M.

=0

(5) To keep the complexity of the MT as low as possible, the decision statistic for am is obtained by feeding X m to a simple despreading unit. This produces y m = cH mX m

(6)

where (·)H means Hermitian transposition. In this way, both channel estimation and equalization are avoided at the MT, leading to a very simple and low-power-consuming receiver.

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An estimate of am is finally obtained feeding ym to a threshold device. Substituting (3) into (6) yields ym =

P K

am q H m,p uk,p

+

cH m nm

Furthermore, to limit the overall transmit power, we impose the following constraint: K

(7)

p=1 k=1

HH m,p cm .

where q m,p = At this stage, it is useful to separate the contribution of the mth user from that of the others. This produces K ym = am q H u + ak q H c H nm m m m uk + m k=1, k=m desired signal thermal noise MAI

(8) where we have introduced the P M -dimensional vectors q m = H H H H H H H [q H m,1 q m,2 · · · q m,P ] and uk = [uk,1 uk,2 · · · uk,P ] . The abovementioned expression indicates that ym is the sum of three terms: 1) the desired signal; 2) the MAI caused by the loss of orthogonality among users; and 3) the noise contribution after despreading. Note that both the desired signal and the MAI depend on the prefiltering coefficients through uk , k = 1, 2, . . . , K. In the next section, we address the design of uk rather than wk,p (p = 1, 2, . . . , P ). The design criterion aims at mitigating MAI while enhancing the desired signal component.

uk 2 = K

(12)

k=1

where the notation · denotes the Euclidean norm. Note that this is in contrast with the methods discussed in [5]–[7], where the same power is allocated to all the active users, i.e., uk 2 = 1 for k = 1, 2, . . . , K. Intuitively, the constraint (12) ensures more degrees of freedom and allows the BS to give more power to the weaker users. Collecting (9) and (11), we see that the direct minimization of J2 is analytically unfeasible as it leads to a joint optimization problem for the prefiltered sequences of all active users. For this reason, we follow a suboptimal approach in the sequel in which MAI is eliminated first, and the result is then exploited to minimize J2 in closed form. B. MAI Elimination Inspection of (8) reveals that ak (q H m uk ) is the interference that the signal of user k produces at the mth MT. Therefore, the elimination of the MAI induced by the kth user at the other MTs requires that

qH m uk = 0,

1≤m≤K

and

m = k

qH k uk = λ k III. P REFILTERING A LGORITHM A. Optimality Criterion The optimality criterion for the design of the prefiltered spreading sequences uk is based on the SINR at the MTs. Assuming statistically independent data symbols with zero mean and unit variance, from (8) we see that the SINR at the mth MT is given by

K

σn2 +

|q H m uk |

SINRk .

(15)

where uk satisfies the following system of K linear equations and P × M unknowns

where we have used the identity cH m cm = 1 that holds with unitenergy spreading codes. Intuitively, we would like to maximize SINRm for m = 1, 2, . . . , K. Thus, it makes sense to choose the uk to maximize the sum K

uk = λk uk

(9) 2

k=1, k=m

J1 =

where λk is a positive real parameter that is chosen to meet the constraint (12). To proceed, we define the matrix Q = [q 1 q 2 · · · q K ] and denote bk the K-dimensional vector with entries

1, if n = k bk (n) = (14) 0, otherwise. Then, from (13), we see that uk can also be written as

H q m um 2

SINRm =

(13)

(10)

QH uk = bk .

(16)

Since K ≤ M , the abovementioned system may be underdetermined (more unknowns than equations). In such a case, there exists an infinite number of sequences uk satisfying (16). The problem of finding the best uk is addressed later.

k=1

As is shown later, however, the maximization of J1 does not lead to any useful prefiltering algorithm. As an alternative, we propose the minimization of J2 =

K

(SINRk )−1 .

k=1

(11)

C. Minimization of J2 From (15), we see that the power constraint (12) may be rewritten as K m=1

λ2m um = K. 2

(17)


Furthermore, collecting (9), (11), and (13) produces J2 = σn2

K 1 . 2 λ m=1 m

(18)

Thus, we must compute the coefficients {λk ; k = 1, 2, . . . , K} that minimize J2 under the constraint (17). The solution to this problem is found looking for the minimum of the augmented cost function K K 1 2 2 2 +µ K − λm um (19) J = σn λ2 m=1 m m=1 where µ is a Lagrange multiplier. Taking the derivative of J with respect to λk and setting it to zero produces λk =

γ uk

(20)

where γ is found substituting (20) into (17) and reads K γ= . K um

J2 =

K

K

(21)

.

(22)

m=1

uk

Clearly, the sequence that minimizes J2 is the minimumnorm solution of (16) and reads [8, p. 412] uk = Q(QH Q)−1 bk .

1 × Q(QH Q)−1 bk , ρk,k

k = 1, 2, . . . , K. (25)

The only difference between (24) and (25) is in the multiplicative scalar parameter, which is chosen in (25) to meet the set of power constraints uk = 1 for k = 1, 2, . . . , K. For this reason, the proposed algorithm is called the modified TIR (m-TIR) in the sequel. 3) Equations (20) and (21) provide the coefficients λk that minimize J2 under the power constraint (17). It is interesting to compute the coefficients λk maximizing the cost function J1 in (10) instead. After some manipulations (not shown for space limitations), we obtain



0,

K

u

2

, if k =

(26)

otherwise

where = arg min1≤k≤K {uk }. Clearly, this solution has no practical interest as it allocates the overall transmit power to a single user. IV. S IMULATION R ESULTS

2 um

uk = √

λk =

At this stage, we are left with the problem of finding the best sequence uk satisfying the linear system (16), where the optimality criterion is still the minimization of J2 . To this purpose, we substitute (20) and (21) into (18) to obtain

2) The prefiltering algorithm (24) is reminiscent of the total interference removal (TIR) scheme discussed in [5], which reads

 

m=1

σn2

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(23)

From the abovementioned expression, it follows that uk = √ ρk,k , where we have denoted ρi,j the (i, j)th entry of (QH Q)−1 . In summary, collecting (15) and (20)–(23), we see that the set of prefiltered spreading sequences leading to the complete elimination of MAI and minimizing J2 is given by K uk = × Q(QH Q)−1 bk , K √ ρm,m · ρk,k m=1

k = 1, 2, . . . , K. (24) The following remarks are of interest. 1) Inspection of (24) reveals that the crux in the calculations is the inversion of QH Q. On the other hand, QH Q is a K × K matrix, irrespective of the spreading factor and the number of transmit antennas. In addition, a single matrix inversion is needed for the computation of all the sequences {uk ; k = 1, 2, . . . , K}. This makes the proposed algorithm very attractive for practical implementation.

A. System Parameters We consider an indoor scenario inspired by the TDD Hiperlan/2 standard. The transmitted symbols belong to a quaternary phase-shift keying (QPSK) constellation and are associated to the information bits through a Gray map. The total number of subcarriers is N = 64, and Walsh–Hadamard codes of length M = 8 are used for spreading purposes. The signal bandwidth is B = 20 MHz, so that the useful part of each MC-CDMA block has length T = N/B = 3.2 µs. A cyclic prefix of TG = 0.8 µs is adopted to eliminate interblock interference. This produces an extended block (including the cyclic prefix) of 4 µs. The downlink time slot has duration 1.0 ms and consists of 250 blocks. All the CIRs hk,p have the same length Lk,p = 8. The 2 (i) = exp(−i) (i = 0, 1, . . . , 7) channel taps have power σk,p and are generated by filtering statistically independent white Gaussian processes in a third-order low-pass Butterworth filter. The 3-dB bandwidth of the filter is taken as a measure of the Doppler shift fD = f0 v/c, where f0 = 5 GHz is the carrier frequency, v denotes the MT velocity, and c = 3 × 108 is the speed of light. B. System Performance The performance of the proposed prefiltering algorithm is computed in terms of averaged bit error rate (BER) computed over all active users. Comparisons are made with the TIR scheme discussed in [5] and a third method proposed by Sälzer

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Fig. 2. BER performance over a static channel with four users (half load).

Fig. 3.

BER performance over a static channel with eight users (full load).

and Mottier (S&M) in [7]. The latter employs the following sequences −1 bk , uk = µk × Q QH Q + σn2 I K

k = 1, 2, . . . , K (27)

where the scalar coefficients µk are designed to meet the constraints uk = 1 for k = 1, 2, . . . , K. Note that S&M reduces to the TIR as σn2 becomes vanishingly small. Fig. 2 illustrates the BER with prefiltering techniques vs. Eb /N0 , where Eb is the transmitted energy per bit and N0 /2 is the two-sided noise power spectral density at the receiver side. There are four active users (half-load) and the number of transmit antennas is P = 1, 2, or 4. The CIRs are kept fixed over the downlink time slot (static channel) but vary from slot to slot. We see that the m-TIR gives the best results irrespective of the number of antennas. TIR and S&M have virtually the same performance and, at an error probability of 10−3 , they lose 2 dB with respect to m-TIR. Note that doubling the number of antennas entails a gain of approximately 4 dB for all schemes. Fig. 3 shows results in the same operating conditions of Fig. 2, except that the system is now fully loaded (K = 8). The m-TIR outperforms the other schemes for P = 2 and 4, whereas S&M is better for P = 1. The loss with respect to the corresponding curves of Fig. 2 is approximately 1.5 dB when P = 2 and reduces to 1 dB with P = 4. Much larger degradations occur in the case of a single transmit antenna (P = 1). The impact of channel variations on the system performance is addressed in Fig. 4, where the BER is shown as a function of the mobile speed v for Eb /N0 = 10 dB. The system is fully loaded and the number of antennas is either P = 2 or P = 4. The sequences uk (k = 1, 2, . . . , K) are computed at the beginning of each downlink slot and are kept fixed over the slot while the CIRs vary from block to block due to Doppler effects. As expected, the BER degrades as v increases, but the performance loss is negligible for mobile speeds up to 3 m/s.

Fig. 4. BER performance versus the mobile speed v with eight users (full load).

This means that the considered schemes are well suited for indoor applications. V. C ONCLUSION We have discussed a novel prefiltering algorithm for downlink TDD MC-CDMA systems endowed with multiple transmit antennas. Channel state information is obtained at the BS by exploiting the channel reciprocity between uplink and downlink transmissions. The proposed scheme completely eliminates MAI at the MTs while constraining the overall transmit power instead of the powers of the single users. In this way, prefiltering and power control are performed jointly at the BS. A simple despreading operation is required at the receiver to detect the transmitted symbols, thereby keeping the power consumption at the MT at a very low level. Simulations indicate that the proposed scheme is well suited for indoor applications and outperforms existing prefiltering techniques.


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[5] A. Silva and A. Gameiro, “Pre-filtering antenna array for downlink TDD MC-CDMA systems,” in Proc. Vehicular Technology Conf. (VTC-Spring), Jeju, Korea, Apr. 2003, pp. 641–645. [6] ——, “Pre-filtering techniques using antenna arrays for downlink TDD MC-CDMA systems,” in Proc. Multi-Carrier Spread Spectrum (MCSS), Oberpfaffenhofen, Germany, Sep. 2003, pp. 307–314. [7] T. Sälzer and D. Mottier, “Downlink strategies using antenna arrays for interference mitigation in multi-carrier CDMA,” in Proc. MultiCarrier Spread Spectrum (MCSS), Oberpfaffenhofen, Germany, Sep. 2003, pp. 315–325. [8] S. Haykin, Adaptive Filter Theory. Englewood Cliffs, NJ: PrenticeHall, 1991.