Adaptive MC-CDMA Multiple Channel Estimation and ... - IEEE Xplore

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 6, JUNE 2007

Adaptive MC-CDMA Multiple Channel Estimation and Tracking Over Time-Varying Multipath Fading Channels Bor-Sen Chen, Fellow, IEEE, and Jung-Feng Liao, Student Member, IEEE

Abstract— Multicarrier code-division multiple access (MC– CDMA) combines multicarrier transmission with direct-sequence (DS) spread spectrum techniques. In this study, several possible channel models of MC-CDMA systems are considered as a multi-state switching Markov process to match the time-varying multipath fading channel. A novel multiple channel estimator, which consists of several possible channel models and can be self-adjusted to fit different channel uncertain conditions, is proposed for MC-CDMA systems to effectively estimate channel coefficients in the frequency domain over time-varying multipath fading channels. Based on the proposed channel estimation, an enhanced minimum mean-square error (MMSE) equalizer is employed for symbol detection of MC-CDMA systems. Simulation results demonstrate that the proposed multiple channel estimator can track the channel more accurately than the conventional channel estimators. With the consideration of channel estimation error as a channel noise to improve symbol detection, the enhanced equalizer based on the proposed multiple channel estimator performs much better than those based on the conventional channel estimators in bit error rate (BER). Index Terms— Channel estimation, MC-CDMA, MMSE equalizer, multicarrier-CDMA, multiple channel estimator.

I. I NTRODUCTION RTHOGONAL frequency division multiplexing (OFDM) is a modulation scheme in which multiple symbols are transmitted in parallel using different subcarriers [1,2]. Since inverse discrete Fourier transform (IDFT) and discrete Fourier transform (DFT) have been adopted for OFDM systems, the spectrum of each symbol on each subcarrier is overlapping but mutually orthogonal. Therefore, the spectrum utilization of OFDM systems is highly efficient. It appears that the OFDM system is the best solution for delay spread and spectrum efficiency, but it cannot provide multiple access ability on the same channel bandwidths. To achieve multiple access and spectrum efficiency, MC-CDMA has been proposed as a suitable modulation technique for the next generation mobile communication system [3]. MCCDMA, which is a hybrid multiple access system combining DS-CDMA and OFDM techniques, divides the available bandwidth into a large number of narrow subcarriers [4]

O

Manuscript received November 3, 2005; revised December 4, 2006; accepted December 6, 2006. The associate editor coordinating the review of this paper and approving it for publication was P. Jung. The authors are with the Department of Electrical Engineering, National Tsing-Hua University, 101 Section 2, Kuang Fu Road, Room 813, EE Building, Hsin-chu 30013 Taiwan, R. O. C. (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TWC.2007.05860.

and spreads each data symbol in the frequency domain by transmitting all the chips of a spread symbol at the same time, but in different orthogonal subcarriers. Since all the chips of these symbols, which form a multicarrier block symbol, overlap in time, high data rate can be achieved even when the system uses a large multicarrier symbol duration, which can drastically reduce the serious inter-symbol interference (ISI). When MC-CDMA spreads the transmitted symbols in a non-flat fading frequency channel, the inner product of different spreading codes will no longer be zero. This situation leads to the destruction of orthogonality between different users and introduces multiple access interference (MAI) to degrade the performance. In order to preserve the orthogonality between different users, the channel impairment should be estimated accurately and equalized efficiently. Conventional investigations [5-7] have always been conducted under the assumption of perfect channel estimation in the frequency or time domain. Recently, the impact of channel estimation errors on the performance of MC-CDMA detectors has attracted significant research interest, and different approaches have been adopted for channel estimation and tracking. Pilot symbolaided channel estimation in both time and frequency domains has been proposed [8,9]. The estimations of the channel coefficients are then obtained through two-dimensional (2-D) linear filtering. Another approach [10,11] considers an explicit channel estimation based on channel-sounding. In addition, a decision-directed channel estimation in the frequency domain using Kalman-based channel estimator has also been proposed [12,13]. Furthermore, in real cellular networks, variation of the fading channel is dependent on the different Doppler spreads according to the velocity change of the mobile station [14]. Under this situation, we cannot just use a simple channel model to exactly describe the time-varying multipath fading channel for MC-CDMA systems. In this study, based on the channel uncertain conditions of MC-CDMA systems over the time-varying multipath fading channels, a novel multiple channel model, which includes several possible channel models of Autoregressive (AR) process [15] according to different ranges of Doppler frequencies (or mobile velocities), is constructed to describe the timevarying fading channel. Several possible channel models are considered to switch as a multi-state Markov process to match the time-varying fading channel. Then, we can propose an adaptive multiple channel estimator to accurately estimate

c 2007 IEEE 1536-1276/07$25.00

CHEN AND LIAO et al.: ADAPTIVE MC-CDMA MULTIPLE CHANNEL ESTIMATION AND TRACKING

c1 (0) c (0) 2

d1 (n)

d 2 ( n)

⊗ ⊗

d N u ( n)

codes with length N . Then the signal is translated by the multicarrier modulation (i.e., IDFT), and produces the OFDM symbols in time domain as (see Fig. 1):

sng (0)

∑

sng (1)

S n ( 0)

⊗

d1 (n)

Symbol Replication N

d N u ( n)

P/S

c2 ( N − 1)

d 2 ( n)

⊗ ⊗

d N u ( n)

N −1 j2πmk 1 Sn (m)e N sn (k) = N m=0

DAC

IDFT N

c1 ( N − 1)

d1 (n)

∇

sng (k )

c N u ( 0)

∑

Sn ( N − 1)

⊗

2329

sng ( N + G − 1)

Cyclic prefix

c Nu ( N − 1)

Fig. 1. Block diagram of MC-CDMA Transmitter for base station (downlink system).

the channel coefficients in the frequency domain. Based on the estimated channel, an enhanced MMSE equalizer with the consideration of channel estimation error is employed for symbol detection. Simulation results illustrate that the proposed adaptive multiple channel estimator has a significant improvement in mean-square error (MSE) performance than the conventional channel estimators. Besides, the estimated channel by the proposed multiple channel estimator is superior to those of the conventional channel estimators in BER performance. This paper is organized as follows. Section II describes the MC-CDMA system model. Section III proposes an adaptive multiple channel estimator over the time-varying fading channels. Section IV describes an enhanced MMSE equalizer. Section V presents the performance measures and computer simulation results. Finally, we conclude this paper in Section VI.

(2)

for k = 0, 1, . . . , N −1. A guard interval is inserted between successive OFDM symbols in order to avoid ISI effects by using a cyclic prefix technique [1]. After the parallel-to-serial conversion, the combination of the cyclic prefix with the IDFT output sequence is given by: sgn (k) = sn (k + N − G)N

(3)

for k = 0, 1, . . . , N +G−1, where (k)N denotes the modular of k by N , and the subscript n and superscript g denote the nth transmitted signal block with a guard interval. Thus, the duration of the MC-CDMA block symbol is T = (N + G)Tc , where GTc is the guard interval, and Tc is the sampling time. The block diagram of the MC-CDMA transmitter is illustrated in Fig. 1. B. Receiver Model of Downlink In the downlink case of MC-CDMA systems, all usertransmitted signals from base station are synchronous and experience the same multipath fading channel. The sampling data yng (k) of the matched filter output at the kth instant of the nth MC-CDMA block symbol can be expressed as yng (k) =

L−1

sgn (k − l) hn (k, l) + vng (k),

(4)

l=0

II. MC-CDMA S YSTEM M ODEL In this section, we describe the transmitter and receiver models of MC-CDMA systems with Ns subcarriers. In practice, the nth multicarrier block symbol (duration T ) for user j consists of μ symbols d1j (n), . . . , dμj (n), which are spread by the user’s spreading code cj with length N . Thus, these μ spread symbols of the jth user are placed into Ns = μN available subcarriers and are transmitted in parallel. The block diagram of the transmitter for MC-CDMA systems is illustrated in Fig. 1. In the rest of this paper, for simplicity of notation, we focus only on the case that each user transmits one symbol (μ = 1) in each MC-CDMA block symbol. Therefore, the total number of subcarriers is Ns = N . Below, AT denotes the transpose of matrix A, and AH denotes the Hermitian of matrix A. A. Transmitter Model We consider a multiple access system in which the data from Nu users are transmitted from base station (downlink) in a synchronous manner. The nth symbols with spread sequence of all Nu users transmitted on the mth subcarrier can be written as Nu cj (m)dj (n), (1) Sn (m) = j=1

for m = 0, 1, . . . , N − 1, where dj (n) is the nth symbol of the jth user, and cj (m) is the jth user’s normalized orthogonal

for k = 0, . . . , N + G − 1, where hn (k, l) and vng (k) denote the lth path sampling of the complex time-varying fading channel with length L and the additive white Gaussian noise (AWGN) at the kth instant of the nth MC-CDMA block symbol, respectively. For a higher data rate transmission, we can reasonably assume that the channel is constant during one MC-CDMA block symbol interval T [12], thus hn (0, l) = hn (1, l) = · · · = hn (N + G, l) for l = 0, · · · , L − 1, where the index k of hn (k, l) would be ignored, and can be simply rewritten as hn (l). However, the channel variation during the successive symbol intervals is allowed. Since the guard interval of the MC-CDMA symbol is longer than the channel’s maximum delay spread, the ISI effects can be completely eliminated by the cycle prefix technique [1]. After removing the guard interval from (4), the received signal can be described as yn (k) =

yng (k

+ G) =

L−1

sn (k − l)N h n (l) + v n (k), (5)

l=0

for k = 0, . . . , N − 1. The received samples are demodulated by taking the N -point DFT, and can be represented as Yn (m) = DF T {yn (k)} = Sn (m)Hn (m) + Vn (m) Nu = cj (m)dj (n)Hn (m) + Vn (m) j=1

(6)

2330

∇


y (t ) ADC

yng (k )

yng (0)

Yn (0)

yng (1)

Yn (1)

The autocorrelation function of the Jake’s model approximates a Rayleigh fading channel, which can be represented as [14] MMSE Equalizer and Despreading

S/P DFT N

yng ( N + G + 1)

¯ T ), Φ(¯ n) = E{h n (l)hHn−¯n (l)} = σh2 n,l J0 (2π fd n

dˆ1 (n)

Yn ( N − 1)

β n|n −1

Hˆ n|n - 1

remove CP

Yn

Multiple Channel Estimator

Fig. 2. Block diagram of MC-CDMA Receiver with the proposed channel estimator.

for l = 0, 1, . . . L − 1, in which J0 is the zero order of first kind Bessel function, σh2 n,l is the fading channel variance of the lth path at the nth MC-CDMA symbol, fd is the maximum Doppler frequency shift, and T is the MC-CDMA symbol period with the guard interval. A higher order AR model can closely match the spectral characteristics of the Rayleigh fading channel. Here, we focus on the second order AR model because of its simplicity and accuracy. The lth path of channel impulse response at the nth MC-CDMA symbol can be modeled by an AR(2) process h n (l) = −a1 h n−1 (l) − a2 h n−2 (l) + wn (l),

for m = 0, 1, . . . , N − 1. We can re-express (6) in a matrix form by stacking the N consecutive subcarriers as follows Yn = Hn Sn + Vn = Hn Cdn + Vn

(7)

where Yn = [Yn (0) Yn (1) · · · Yn (N − 1)]T , C = [C1 C2 · · · CNu ], Cj = [cj (0) cj (1) dn = [d1 (n) d2 (n)

··· ···

Hn = diag{Hn (0) Hn (1)

cj (N − 1)]T , dNu (n)]T , ···

Hn (N − 1)},

Sn = [Sn (0) Sn (1)

···

Sn (N − 1)]T ,

Vn = [Vn (0) Vn (1)

···

Vn (N − 1)]T ,

where Yn is the output vector of the DFT, which is illustrated in Fig. 2; C is the normalized orthogonal code matrix of all users with the dimension N × Nu ; dn is the symbol-mapped data of all users; Hn denotes a block diagonal matrix of the subcarrier channel coefficients which are the DFT conversion of the channel in time domain; and Vn is the frequency domain noise vector, which is the DFT output of the channel noise in time domain. The architecture of the receiver for MCCDMA systems is illustrated in Fig. 2, in which the received vector Yn is fed into the proposed channel estimator to ˆ n and the predicted channel generate the estimated channel H ˆ Hn|n−1 , and then an MMSE equalizer employs the received ˆ n|n−1 and spreading code vector Yn , predicted channel H ˆn of matrix C to accurately detect the transmitted symbols d all users. III. A DAPTIVE M ULTIPLE C HANNEL E STIMATOR OVER T IME -VARYING M ULTIPATH FADING C HANNELS A. Subcarrier State Space Model Although Jake’s model is always employed for the fading channel simulation, it is difficult to use the model to estimate channel variation. An AR process can match the spectral characteristics of the Rayleigh fading channel [15].

(8)

(9)

for l = 0, 1, . . . L − 1, where wn (l) is a complex zero mean white Gaussian process. The parameters a1 and a2 observed in [15] are closely related to the physical parameters of the underlying fading process √ a1 = −2rd cos(2π 2fd T ), a2 = rd2 (10) in which the parameter rd is the pole radius that corresponds to the steepness of the peaks of the power spectrum and should be chosen very close to one to model the spectral peaks at the maximum Doppler frequency of the fading channel. The autocorrelation function of the AR(2) process justified in [15] is close to that of a Rayleigh fading process. In this study, the channel estimation is performed in the frequency domain. Hence, the time domain AR(2) model of Rayleigh fading channel can be transformed to a frequency domain AR(2) model. Taking the N -point DFT of hn (l), we have Hn (m) = =

N −1 l=0 L−1

hn (l) e

−j2πlm N

=

L−1

hn (l) e

−j2πlm N

l=0

[ −a1 hn−1 (l) − a2 h n−2 (l) + wn (l)) ] e

−j2πlm N

l=0

= −a1 Hn−1 (m) − a2 Hn−2 (m) + Wn (m), where Wn (m) =

L−1

wn (l)e

(11)

−j2πlm N

l=0

for m = 0, . . . , N −1, and Hn (m) is the channel coefficients of the mth subcarrier at the nth MC-CDMA block symbol in the frequency domain. From the above equations, we obtain the frequency domain relationship of the AR(2) model for the multipath fading channel. The frequency domain AR(2) process in (11) is similar to the time domain AR(2) process in (9) for the multipath fading channel. Consequently, the frequency domain AR(2) model in (11) and the observation equation in (6) of the MC-CDMA system can be described by the dynamics of state equations for the mth subcarrier as follows (12) Xn+1 (m) = FXn (m) + Wn (m), Yn (m) = Gn (m) Xn (m) + Vn (m),

(13)


2331

p1M

where T

Xn (m) = [Hn (m) Hn−1 (m)] , −a1 − a2 , F= 1 0

p11

Ch Model 1

Gn (m) = [Sn (m) 0], Sn (m) =

Nu

p12

Ch Model 2

p21

(a)

k=1

(14)

for m = 0, 1, . . . , N − 1. The state vector Xn (m) contains the mth subcarrier channel coefficients, Wn (m) is the driving noise vector in the frequency domain, and Vn (m) is the frequency domain channel noise. From (6), the covariance of the frequency domain channel noise Vn (m) can be expressed as QVn (m) = E{Vn (m)VnH (m)} N −1 N −1 −j2π m k j2π m r = E{ v n (k) e N × v Hn (r) e N } =

r=0

k=0 N σv2

Yn (m)

Xˆ 1, ( n|n ) (m)

0]T ,

(15)

where σv2 is the variance of the time domain channel noise which is white Gaussian with zero mean. From (11) and (14), the covariance matrix of the driving noise Wn (m) can be derived as QWn (m) = E{Wn (m)WnH (m)} L−1 1 0 2 σw = , l 0 0

pMM

pM 1

ck (m)dk (n),

Wn (m) = [Wn (m)

Ch Model M

(16)

Yn (m) Rx signal m subcarrier

From the dynamics of state equations in (12) and (13), a Kalman filter can be employed to efficiently estimate the channel coefficients of the desired user. Unfortunately, in a real cellular network, the time-varying multipath fading channel is dependent on the different Doppler spreads according to the change in velocity of the mobile station, i.e., a1 , a2 or F in (14) will change according to different Doppler frequencies due to the velocity change of the mobile station. Since the velocity of the mobile station is unavailable, we cannot just use a simple channel model (i.e. with a time-varying F in (14)) to exactly describe the time-varying multipath fading channel. Therefore, we propose an adaptive multiple channel model to accurately describe the time-varying fading channel. To simplify the notations, we neglect the argument m in (12) and (13). From the dynamics of the state equations in (12) and (13), the proposed multiple channel model with the channel transition described by a Markov process, which switches among different channel models according to different ranges

Combination of Ch Estimates and Ch Prediction

Ch Model 2 Estimator

β n|n −1

MMSE Symbol Detector

dˆ1 (n)

μ j , n ( m) Xˆ M , ( n| n ) (m)

Ch Model M Estimator

decision feedback

Likelihood Function Markov Switching Probability

Mode Probability

Λ j , n ( m)

pij

(b) Fig. 3. (a) Markov switching multiple channel models; (b) Architecture of the proposed multiple channel estimator and MMSE symbol detector for MC-CDMA systems

of Doppler frequencies, can be represented by the following switching system Xj,n+1 = Fj Xj,n + Wj,n ,

(18)

Yn = Gn Xj,n + Vn ,

(19)

where

l=0

2 where σw = E{wn (l)wnH (l)} is the covariance of the driving l noise at the lth path, which has been derived in (1.88) of [17] to satisfy the relation of an AR(2) process by solving the YuleWalker equations 2 σw 1 + a2 l (17) E{h n (l)hHn (l)} = 1 − a2 ((1 + a2 )2 − a21 )

Hˆ n|n - 1

Ch Model 1 Estimator

a1,j

−a1,j −a2 Fj = , 1 0 √ = −2rd cos(2π 2fd,j T), a2 = rd2

for j = 0, 1, . . . , M and the M different channel models are with different parameters a1,j according to different ranges of Doppler frequencies fd,j due to the different velocity ranges of the mobile station, for example, slow, medium and high velocities. Based on the multiple channel hybrid system, we propose an adaptive multiple channel estimator for MCCDMA switching system (18) and (19) in the next subsection. B. Adaptive Multiple Channel Estimator Based on the multiple channel model (18) and (19) of the MC-CDMA systems over time-varying multipath fading channels, an adaptive multiple channel estimator is proposed for channel estimation in the frequency domain. The multiple channel models in time-varying multipath fading channels are considered to switch as a multi-state Markov process, which is shown in Fig. 3(a). Then, we develop an adaptive multiple channel estimator to accurately estimate the subcarrier channel coefficients over time-varying multipath fading channels. The architecture of the proposed adaptive multiple channel estimator is illustrated in Fig. 3(b), which consists of two major parts: Kalman-based switching channel estimator and combination. The proposed adaptive multiple channel estimator is described in the following.

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1) Kalman-Based Switching Channel Estimator: From the dynamics of the state equations in (18) and (19) and Kalman filter algorithm [17], we can obtain ˆ j,(n|n−1) = Fj X ˆ j,(n−1|n−1) , X

(20)

ˆ j,(n|n−1) + Kj,n ej,n ˆ j,(n|n) = X X

(21)

for j = 1, . . . , M , where ˆ j,(n|n−1) ej,n = Yn − Gn X

(22)

denotes the prediction error according to the jth Kalman-based switching channel estimator of the mth subcarrier. Kj,n is the corresponding Kalman gain, which is employed to minimize the covariance matrix E{ej,n eH j,n } of the state prediction error according to the jth Kalman-based switching channel estimator. The Kalman gain and the error covariance matrices of the state prediction and estimation are obtained from the following equations [17]

Kj,n

Pj,(n|n−1) = Fj Pj,(n−1|n−1) FH j + Qj,Wn −1 H = Pj,(n|n−1) GH n Gn Pj,(n|n−1) Gn + QVn Pj,(n|n) = [I2 − Kj,n Gn ] Pj,(n|n−1)

(24) (25)

(26) } can be respectively, and the error covariance E{ej,n eH j,n expressed as (27)

2) Combination of Multiple Channel Estimations: The final channel estimation is obtained by weighting the combination of multiple channel estimators in (21) according to their channel mode probabilities, which are determined by the likelihood function of each channel estimation. The likelihood function Λj,n and channel mode probability μj,n for the jth Kalman-based switching channel estimator can be updated, respectively, by Λj,n = N (ej,n ; 0, Bj,n ) 1 −1 = exp(− (1/2) eH j,n Bj,n ej,n ) (2π Bj,n )1/2 μj,n =

M 1 1 pij μ i,n−1 = Λj,n c¯j Λj,n c c i=1

c=

j=1

Λj,n c¯j

M

pij μi,n−1

(31)

.

i=1

The physical meaning of the likelihood function in (28) is that if the absolute state estimation error ej,n is large, then the likelihood function is small with a less weighting in the combination of multiple channel estimations, and vice versa. In (29), the channel mode probabilities are updated according to the Markov transition probabilities and the likelihood function in every iteration. The final channel estimation and error covariance matrices can be obtained by combining the results of the M Kalmanbased switching channel estimators according to their respective channel mode probabilities as follows ˆ (n|n) = X

M

ˆ j,(n|n) μj,n X

(32)

j=1

P(n|n) =

M

ˆ j,(n|n) − X ˆ (n|n) Pj,(n|n) + X j=1

ˆ j,(n|n) − X ˆ (n|n) X

H

(33)

μj,n .

In order to evaluate the performance of the proposed adaptive multiple channel estimator, the normalized mean-square error (NMSE) of the channel estimation for the mth subcarrier is defined as ˆ n (m)|2 } E{|Hn (m) − H (34) N M SEm = E{|Hn (m)|2 } This adaptive multiple channel estimation algorithm of MCCDMA systems requires the knowledge of the transmitted symbols {Sn (m)}, for m = 0, 1, . . . , N −1. The transmitted symbols can be obtained from the training sequence in the training mode or decision feedback in the tracking mode, respectively. In the tracking mode, the output symbols from the decision device are fed back to the proposed adaptive multiple channel estimator to update the subcarrier channel coefficients. Then, from (18) and (32), one-step prediction of the MC-CDMA channel coefficients for the mth subcarrier is performed by ˆ (n+1|n) = X

M

ˆ j,(n|n) μj,n Fj X

.

(35)

j=1

(28)

(29)

in which pij is the Markov transition probability from model i to model j, N (ej,n ; 0, Bj,n ) denotes the Gaussian density function of ej,n with zero mean and covariance Bj,n , and c is a normalized factor given by M

c¯j =

(23)

where IP denotes the identity matrix of dimension P ×P , and covariance matrices QVn and Qj,Wn are defined in (15) and (16), respectively. The error covariance matrices of the state prediction and estimation according to the jth Kalman-based switching channel estimator are given by

H ˆ ˆ Pj,(n|n−1) = E Xn − Xj,(n|n−1) Xn − Xj,(n|n−1)

H ˆ ˆ Pj,(n|n) = E Xn − Xj,(n|n) Xn − Xj,(n|n) ,

H E{ej,n eH j,n } = Bj,n = Gn Pj,(n|n−1) Gn + QVn .

where

(30)

The predicted MC-CDMA channel coefficients are employed to detect the next symbol in the equalizer of the MC-CDMA system. After channel estimation, an enhanced MMSE equalization algorithm of MC-CDMA systems will be introduced in the next section. IV. E NHANCED MMSE E QUALIZER FOR S YMBOL D ETECTION A. Equalization using MMSE for Symbol Detection In this section, we employ an enhanced MMSE equalizer for the symbol detection of MC-CDMA systems. In Section III, we propose an adaptive multiple channel estimator to provide


an accurate estimation and prediction of the time-varying fading channel. The channel prediction is necessary for the decision-directed scheme to overcome the delay problem. A conventional decision-directed scheme [12] adopted the previous estimates to perform the current MMSE equalization. However, when the channel variation is fast, the previous estimates will not be suitable for the current equalization due to the large channel variation. Therefore, with the prediction ˆ n|n−1 obtained from the proposed multiple channel channel H estimator in (35), we can employ an enhanced MMSE equalizer to detect the transmitted symbols over the time-varying fading channel. The MMSE equalization for MC-CDMA systems under the assumption of a perfectly known channel has been examined in [18]. Here an enhanced MMSE equalizer for each subcarrier is employed with the consideration of channel estimation error as a channel noise. In this situation, observation equation in (7) can be re-expressed as ˆ n|n−1 Sn + H ˜ S + Vn (36) Yn = Hn Sn + Vn = H n|n−1 n where ˜ n|n−1 = Hn − H ˆ n|n−1 H

.

The design objective

of MMSE equalizer is to find an ˆ optimal Φn = diag φˆn (0), φˆn (1), . . . , φˆn (N − 1) based on the channel prediction such that φôn (m) = argφˆn (m) min E

2 ˆ (m) − φ (m)Y (m) Sn , n n

(37) for m = 0, . . . , N − 1 and the optimal MMSE equalization by solving (37) with the channel prediction as follows ˆH H n|n−1 (m) , φôn (m) = 2 ˆ Hn|n−1 (m) + βn|n−1 (m) + QVn (m)/Es (38) for m = 0, . . . , N − 1, where Es = E{|Sn (m)|2 }, the ˆ n|n−1 (m) can be obtained from (35), and channel prediction H the covariance of the channel estimation error βn|n−1 (m) = 2 ˜ (1,1) = Pn|n−1 (m) is derived in (26) and (33), E H n|n−1 (m)

the mth subcarrier is given by Zn (m) = φô (m)Yn (m) = φô (m)(Sn (m)Hn (m) + Vn (m)) n

B. Performance Analysis for the Proposed MC-CDMA System In this subsection, we evaluate the performance of the symbol detection algorithm for the proposed MC-CDMA system. From (6) and (38), the MMSE equalizer output of

n

ˆH H n|n−1 (m)Hn (m) = Sn (m) 2 ˆ Hn|n−1 (m) + βn|n−1 (m) + QVn (m)/Es ˆH H n|n−1 (m) Vn (m). + 2 ˆ Hn|n−1 (m) + βn|n−1 (m) + QVn (m)/Es (39) Therefore, the average signal to noise ratio of the MMSE equalizer output for the mth subcarrier is defined as SN R(m) H H ˆH ˆ E{H n|n−1 (m)Hn (m)Sn (m)Sn (m)Hn (m)Hn|n−1 (m)} = ˆH ˆ E{H (m)Vn (m)VnH (m)H (m)} n|n−1

n|n−1

H ˆH ˆ Es × E{H n|n−1 (m)Hn (m)Hn (m)Hn|n−1 (m)} = . (40) ˆ QVn (m) × E{|H (m)|2 } n|n−1

where Es and QVn (m) are defined in (38) and (15), respectively. If we assume that the proposed multiple channel estimator can accurately estimate and predict the channel ˆ n|n−1 (m) ≈ Hn (m), we can further coefficients, i.e., H simplify (40) to SN R(m) ≈

2 ˆ Es × E{|H n|n−1 (m)| }

QVn (m)

=

Es × E{|Hn (m)|2 } N σv2

(41)

Besides, the dispreading data output of the first user can be expressed as D1 (n) = =

(1,1)

in which Pn|n−1 (m) is the (1,1) element of Pn|n−1 (m). Therefore, the optimal MMSE detector is enhanced by considering the channel estimation error βn|n−1 (m), which can be obtained from the proposed adaptive multiple channel estimator. If the channel estimation error βn|n−1 (m) is not considered in (38), it would become a conventional MMSE equalizer. The block diagram of signal flow is illustrated in Fig. 3(b).

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= ≈

N −1 m=0 N −1 m=0 N −1 m=0 N −1

Zn (m)c1 (m) φôn (m)(Sn (m)Hn (m) + Vn (m)) c1 (m) Nu φôn (m)( ck (m)dk (n)Hn (m) + Vn (m)) c1 (m) k=1

[φôn (m)d1 (n)Hn (m) + φôn (m)Vn (m) c1 (m)]

m=0

(42) where the last approximation in (42) is based on the orthogonality of the normalized Walsh-Hadamard codes, and the decision output of the nth symbol for user 1 is dˆ1 (n) = dec(D1 (n)). For the simplicity of analysis, only the case of BPSK will be considered. Assuming that the total interference in (42) can be approximated by a joint Gaussian distribution with zero mean, the symbol detection error probability of the

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first user for the nth symbol can be approximated as Pe (n) ⎛ ⎞ N −1 H o H o ⎜ E{φˆn (m)Hn (m)Hn (m)φˆn (m)} ⎟ ⎜ m=0 ⎟ ⎜ ⎟ (43) = Q ⎜ ⎟ N −1 H ⎝ ⎠ 2 o o ˆ ˆ N σv E{φn (m)φn (m)} m=0

where φôn (m), QVn are defined in (38) and (15), respectively, 2 ∞ and Q(x) = √12π x e−(t /2) dt, for x ≥ 0. In on-line applications, the complexity of computation is a very important consideration for the channel estimation of MC-CDMA systems. Here, we make a rough estimate of the number of multiplications for the conventional Kalman filter and the proposed algorithm. The number of multiplications for the conventional Kalman filter (20)-(25) is about 12 in every iteration. Next, the number of multiplications for the combination of the proposed algorithm (28)-(32) is about 6 in every iteration. The total number of multiplications for the proposed algorithm is about 12M + 6 in every iteration, which is about M -times more than that of the conventional Kalman filter. Consequently, the proposed multiple channel estimation algorithm has linear complexity with the multiple channel number M when it is compared with the conventional Kalman-based channel estimator. V. S IMULATION R ESULTS This section presents several simulation results to demonstrate the performance of the proposed multiple channel estimator for MC-CDMA systems. In the following simulations, the data modulation scheme is 16-QAM and the number of subcarriers is Ns = 64 for simplicity. A time-varying Rayleigh fading channel is produced by Jake’s model [14], where the channel response in each path is generated by Jake’s method. A three-path channel model of the Rayleigh fading channel with a multipath delay spread of 2μs is considered as the mobile radio channel with path powers, E(0) = 1, E(1) = 0.7, and E(2) = 0.85, respectively, to produce a frequency-selective fading channel. The MC-CDMA block is composed of 65 samples, one for cyclic prefix. Perfect carrier and symbol synchronization are assumed in these simulation cases. The carrier frequency is 1 GHz with a bandwidth of 500 kHz, and the symbol duration is T = 65 × 2 = 130μs. The length of the adopted Walsh-Hadamard spread code is N = 16 chips. Thus the MC-CDMA system can support the maximum number of 16 active users and the maximum number of symbols is μ = 4 for each user in each MC-CDMA block symbol. We assume that the number of active users is Nu = 5 in the following simulation cases, and the symbol rate of the 16-QAM for each user is 30.77 Ksymbol/sec (bit rate 123.08 Kbit/sec). The AR(2) model parameters, a1 and a2 , are defined in (10) according to the Doppler frequency. Here, the parameter of the pole radius is chosen by rd = 0.998 for these simulation cases [15]. Thus the AR(2) model parameters can be determined according to different Doppler frequencies due to different mobile velocities.

TABLE I T HE M C HANNEL M ODELS FOR THE S IMULATION E XAMPLES

Channel Model No.

Range of Doppler Frequency

According Mobile Velocity

1

0 Hz

0 km/hr

2

10 Hz

10.8 km/hr

3

50 Hz

54.0 km/hr

4

100 Hz

108 km/hr

A. Multiple Channel Estimator Performance In order to evaluate the performance of the proposed algorithm more realistically, two different time-varying channel conditions according to the mobile stationary case and random movement case are considered in the following simulation examples. In the first simulation case, the Doppler frequency fd of a fading channel varies from 0 Hz to 5Hz according to the mobile velocity change from 0 km/hour to 5.4 km/hour for the mobile stationary condition, respectively. In the second simulation case, the Doppler frequencies vary according to the mobile random movement condition, on which the velocities are changed from 0 km/hour to 20, 40, 60, 80, 100, and 120 km/hour, and then changed back from 120 km/hour to 100, 80, 60, 40, 20, and 0 km/hour. And the training symbols length is 200. In this simulation, we simply employ four channel models, which include different ranges of Doppler frequencies fd = 0, 10, 50 and 100Hz as channel conditions for the switching channel models of the proposed adaptive multiple channel estimator. The state number of the Markov process for the MC-CDMA system is equal to M = 4, and the M channel models are defined in Table 1. Then, the M -state Markov transition probability matrix is chosen by ⎤ ⎡ p11 p12 · · · p1M ⎢ p21 p22 · · · p2M ⎥ ⎥ ⎢ P=⎢ . .. ⎥ .. . . ⎣ .. . . ⎦ . pM1 pM2 ⎡ 0.998000 ⎢0.000667 =⎢ ⎣0.000667 0.000667

· · · pMM 0.000667 0.998000 0.000667 0.000667

0.000667 0.000667 0.998000 0.000667

⎤ 0.000667 0.000667⎥ ⎥ (44) 0.000667⎦ 0.998000

This choice has been taken by assuming that the Doppler frequency varies within a frame of MC-CDMA symbols. We will show a sensitivity test for the Markov transition probability pii for i = 1, . . . M in the sequel. The initial channel mode probabilities for the proposed adaptive multiple channel estimator are set to μ1 (0) = μ2 (0) · · · = μM (0) = 0.25 , because we assume that each channel mode has the same initial probability in the MC-CDMA systems. Since the mobile velocity v ≤ 108km/hour corresponds to Doppler frequency fd ≤ 100Hz, we have the coherence time of the channel about (Δt)c ≈ 1/fd from eq. (14.1-19) in [19], and can obtain the relation (Δt)c /T > 76. Therefore, it is reasonable to assume that the channel remains approximately constant during one MC-CDMA symbol period. Fig. 4 and Fig. 5 compare the estimated 10th subcarrier channel coefficient by the conventional Kalman-based channel estimator (using the fixed channel model, fd = 10Hz and


3

3

2.5

Known Channel Estimated channel

2.5

Hn(m)

n

H (m)


2

2

1.5

1.5

1

1

0.5

0.5

0 0

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500

1000 n (symbols)

1500

0 0

2000

Fig. 4. Comparison of the known channel with the estimated channel by the Kalman-based estimator with parameter fd = 10Hz for the mobile random movement case (Doppler frequency fd varies from 0Hz to 111Hz and then back to 0Hz), Nu = 5, SNR = 30dB and m = 10.

500

1000 n (symbols)

1500

2000

Fig. 6. Comparison of the known channel with the estimated channel by the proposed Multiple Channel Estimator for the mobile random movement case (Doppler frequency fd varies from 0Hz to 111Hz and then back to 0Hz), Nu = 5, SNR = 30dB and m = 10.

3 −2

2

n

H (m)

Kalman Estimator[12] fd=10Hz Kalman Estimator[12] fd=50Hz Multiple Channel Estimator

10


Normalized Mean Square Error

2.5

1.5

1

−3

10

−4

10

0.5

0 0

−5

500

1000 n (symbols)

1500

2000

Fig. 5. Comparison of the known channel with the estimated channel by the Kalman-based estimator with parameter fd = 50Hz for the mobile random movement case (Doppler frequency fd varies from 0Hz to 111Hz and then back to 0Hz), Nu = 5, SNR = 30dB and m = 10.

50Hz, respectively) with the known channel coefficient in the frequency domain for the mobile random movement case. In Fig. 4, the estimated subcarrier channel coefficients can track accurately in the small Doppler frequency condition, but a large distortion is presented in the large Doppler frequency condition. Fig. 5 presents the converse situations by the conventional Kalman-based channel estimator with fd = 50Hz, and a large distortion of the channel tracking is presented in the small Doppler frequency condition. When the proposed multiple channel estimation algorithms (20)-(33) are employed for the MC-CDMA system, Fig. 6 compares the estimated MC-CDMA channel with the known channel in the mobile random movement case. Simulation results confirm that the estimated channel coefficients by the proposed algorithm can accurately track the channel variation for different ranges of Doppler frequencies in the mobile random movement case. For NMSE defined in (34), Fig. 7 and Fig. 8 illustrate the comparison of MSE performance between the conventional

10

10

15

20

25 SNR (dB)

30

35

40

Fig. 7. Comparison of MSE performance between the proposed multiple channel estimator and the Kalman-based channel estimators in the mobile stationary case for user number Nu = 5.

Kalman-based channel estimator and the proposed adaptive multiple channel estimator for the mobile stationary and random movement cases, respectively. These results demonstrate that the proposed adaptive multiple channel estimator performs much better than the conventional Kalman-based channel estimators in MSE for MC-CDMA systems. Besides, the sensitivity test of Markov transition probability is also included in Fig. 8 by changing the Markov transition probability pii for i = 1, . . . , M from 0.998 to 0.950. Simulation results demonstrate that the proposed adaptive multiple channel estimation algorithm is less sensitive to these parameters. B. MC-CDMA Symbol Detection Performance Simulation results are presented to compare the BER performance of the proposed adaptive multiple channel estimator with the conventional Kalman-based channel estimators in symbol detection. Fig. 9 shows the BER performance in the

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−1

10

−1

10

−2

10 −2

10

10

15

20

25 SNR (dB)

30

35

−3

10

40

Fig. 8. Comparison of MSE performance between the proposed multiple channel estimator and the Kalman-based channel estimators in the mobile random movement case (Doppler frequency fd varies from 0Hz to 111Hz and then back to 0Hz) for user number Nu = 5.

Kalman Estimator[12], fd=10Hz Kalman Estimator[12], fd=50Hz Multiple Channel Estimator Known Channel

−2

10

BER

Kalman Estimator[12], fd=10Hz Kalman Estimator[12], fd=50Hz Multiple Channel Estimator Known Channel

BER

Normalized Mean Square Error

Kalman Estimator[12] fd=10Hz Kalman Estimator[12] fd=50Hz Multiple Channel Estimator P=0.950 Multiple Channel Estimator P=0.998

−3

10

−4

10

−5

10

10

15

20

25 SNR (dB)

30

35

40

Fig. 9. Comparison of BER performance among the known channel, the proposed multiple channel estimator, and the conventional Kalman-based channel estimators in the mobile stationary case for user number Nu = 5.

fading channel of the mobile stationary case with the timevarying Doppler frequency fd from 0Hz to 5Hz. The BER bound of the known channel is also shown for comparison. As anticipated, simulation results illustrate that the enhanced MMSE equalizer based on the proposed multiple channel algorithm performs symbol detection much better than those based on the conventional Kalman-based channel estimators. Fig. 10 shows the same simulation results in the fading channel of the mobile random movement case with the time-varying Doppler frequency fd from 0Hz to 111Hz and then back to 0Hz. From these simulation results, the proposed adaptive multiple channel estimator is found to have a significant improvement compared to the conventional Kalman-based channel estimators in BER performance. VI. C ONCLUSIONS The proposed adaptive multiple channel estimator combines the multiple channel estimation results with the corresponding

10

15

20

25 SNR (dB)

30

35

40

Fig. 10. Comparison of BER performance among the known channel, the proposed multiple channel estimator, and the conventional Kalman-based channel estimators in the mobile random movement case for user number Nu = 5.

channel mode probabilities to accurately estimate the subcarrier channel coefficients over the time-varying multipath fading channels. The probabilities of multiple channels are adaptively updated to track the time-varying fading channel via the likelihood functions according to their estimation errors. Simulation results demonstrate that the proposed adaptive multiple channel estimator has a significant improvement in the MSE performance than the conventional channel estimators over the multipath fading channel with a time-varying Doppler frequency. With the consideration of channel estimation error as a channel noise to improve symbol detection, the channel equalization based on the proposed multiple channel estimator can perform much better than those based on the conventional channel estimators in BER.

R EFERENCES [1] J. E. Kim, S. H. Yoon, S. J. Kang, and C. E. Kang, “Multi-carrier CDMA system using a code orthogonalising filter,” IEEE Electron. Lett., vol. 34, pp. 1557-1558, Aug. 1998. [2] V. N. Richard and P. Ramjee, OFDM for Wireless Multimedia Communications. Boston, MA: Artech House, 2000. [3] S. Nahm and W. Sung, “A synchronization scheme for multi-carrier CDMA systems,” in Proc. IEEE ICC, June 1998, vol. 3, pp. 1330-1334. [4] M. J. M. Peacock, I. B. Collings, and M. L. Honig, “Analysis of multiuser peer-to-peer MC-CDMA with limited feedback,” in Proc. IEEE ICC, June 2004, vol. 2, pp. 968-972. [5] Y.-L. Li and Y. Lee, “A novel low-complexity near-ML multiuser detector for DS-CDMA and MC-CDMA systems,” in Proc. IEEE GLOBECOM, Nov. 2002, vol. 1, pp. 493-498. [6] H. K. Park, E. Kim, Y. Lee, and K. H. Tchah, “Multi-carrier CDMA system with parallel interference cancellation for multipath fading channels,” IEEE Pers., Indoor Mobile Radio Commun., vol. 2, pp. 513517, Sep. 1998. [7] D. N. Kalofonos and J. G. Proakis, “Performance of the multistage detector for a MC-CDMA system in a Rayleigh fading channel,” in Proc. IEEE GLOBECOM, Nov. 1996, vol. 3, pp. 1784-1788. [8] S. Kaiser and P. Hoeher, “Performance of multi-carrier CDMA with channel estimation in two dimensions,” in Proc. IEEE Symp. Pers. Indoor Mobile Radio Commun., Sep. 1997, pp. 115-119. [9] J. Choi, “Channel estimation for coherent multi-carrier CDMA systems over fast fading channels,” in Proc. IEEE Veh. Technol. Conf., May 2000, vol. 1, pp. 400-404.


[10] S. Cacopardi, F. Frescura, F. Gatti, and G. Reali, “Channel estimation and tracking of an indoor orthogonal multicarrier DS-CDMA system using measured channel delay profiles,” in Proc. IEEE Veh. Technol. Conf., Apr.-May 1996, pp. 1559-1563. [11] S. Cacopardi, F. Frescura, and G. Reali, “Performance comparison of multicarrier DS-SS radio access schemes for WLAN using measured channel delay profiles,” in Proc. IEEE Veh. Technol. Conf., May 1997, pp. 1877-1881. [12] D. N. Kalofonos, M. Stojanovic, and J. G. Proakis, “Performance of adaptive MC-CDMA detectors in rapidly fading Rayleigh channels,” IEEE Trans. Wireless Commun., vol. 2, no. 2, pp. 229-239, Mar. 2003. [13] T. J. Lim and Y. Ma, “The Kalman filter as the optimal linear minimum mean-square error multiuser CDMA detector,” IEEE Trans. Inform. Theory, vol. 46, no. 7, pp. 2561-2566, Nov. 2000. [14] W. C. Jakes, Microwave Mobile Communications. New York: Wiley, 1974. [15] P. H.-Y. Wu and A. Duel-Hallen, “Multiuser detectors with disjoint Kalman channel estimators for synchronous CDMA mobile radio channels,” IEEE Trans. Commun., vol. 48, no. 5, pp. 752-756, May 2000. [16] G. L. Stüber, Principles of Mobile Communication, 2nd ed. Boston, MA: Kluwer Academic Publishers, 2001. [17] S. Haykin, Adaptive Filter Theory, 4th ed. Upper Saddle River, NJ: Prentice Hall, 2002. [18] A. Chouly, A. Brajal, and S. Jourdan, “Orthogonal multicarrier technique applied to direct sequence spread spectrum CDMA systems,” in Proc. IEEE GLOBECOM, Nov. 1993, pp. 1723-1728. [19] J. G. Proakis, Digital Communications, 4th ed. New York: McGraw-Hill, 2001. [20] J.-F. Liao and B.-S. Chen, “Robust mobile location estimator with NLOS mitigation using interacting multiple model algorithm,” IEEE Trans. Wireless Commun., vol. 5, no. 11, pp. 3002-3006, Nov. 2006. [21] J.-F. Liao, C.-L. Tsai, and B.-S. Chen, “Robust adaptive channel estimation and multiuser detection for ultra wideband in a realistic indoor channel,” in Proc. IEEE ICC, May 2005, vol. 4, pp. 2845-2851.

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[22] J. M. Mendel, Lessons in Estimation Theory for Signal Processing, Communications, and Control. Upper Saddle River, NJ: Prentice Hall, 1995. Bor-Sen Chen (M’82-SM’89-F’01) received the B.S. degree from Tatung Institute of Technology, Taiwan, the M.S. degree from National Central University, Taiwan, and the Ph.D degree from the University of Southern California, Los Angeles, in 1970, 1973, and 1982, respectively. From 1973 to 1987, he was a Lecturer, Associate Professor, and Professor at Tatung Institute of Technology. Currently, he is a Chair Professor with the Department of Electrical Engineering at National Tsing Hua University, HsinChu, Taiwan, R.O.C. His current research interests include control, signal processing, and systems biology. Dr. Chen has received the Distinguished Research Award from National Science Council of Taiwan four times. He is a Research Fellow of the National Science Council and the Chair of the Outstanding Scholarship Foundation. He was the Chairman of the Taipei Chapter of IEEE Control Society in 1991. He is now an Associate Editor of the IEEE T RANSACTIONS ON F UZZY S YSTEMS and the Editor of the Journal of Asian Control. He is a member of the Editorial Advisory Board of the International Journal of Fuzzy Systems and the International Journal of Control, Automation, and Systems. Jung-Feng Liao received the M.S. degree in electrical engineering from National Tsing Hua University, Hsinchu, Taiwan, R.O.C., in 1987. He is currently working for the Ph.D. degree in electrical engineering at National Tsing Hua University. His current research interests are channel estimation and equalization for multiuser communication systems.