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A simple blind CFO estimate technique for a QO-STBC based MC-CDMA uplink system
Accepted Manuscript A simple blind CFO estimate technique for a QO-STBC based MC-CDMA uplink system Tsui-Tsai Lin, Fuh-Hsin Hwang
PII: DOI: Reference:
S1051-2004(13)00223-6 10.1016/j.dsp.2013.10.004 YDSPR 1499
To appear in:
Digital Signal Processing
Please cite this article in press as: T.-T. Lin, F.-H. Hwang, A simple blind CFO estimate technique for a QO-STBC based MC-CDMA uplink system, Digital Signal Process. (2013), http://dx.doi.org/10.1016/j.dsp.2013.10.004
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A simple blind CFO estimate technique for a QO-STBC based MC-CDMA uplink system Tsui-Tsai Lin and Fuh-Hsin Hwang
Abstract We propose a simple technique for estimating the carrier frequency offset (CFO) of a quasi-orthogonal spacetime block coding (QO-STBC) multicarrier code-division multiple-access (MC-CDMA) uplink system without the knowledge of channel state information in a multipath fading channel. Based on the intrinsic properties of the considered system, a useful multiply constrained minimum output energy (MCMOE) receiver is developed, and the CFO estimate is achieved by using a one-dimensional linear search, whose objective is to maximize the output power of the developed receiver. In addition, an alternative based on the Taylor series expansion technique is designed to avoid the exhaustive search. By feeding the CFO-compensated data back to the input of the MCMOE receiver, we use an iterative approach to further enhance the estimator’s performance. Numerical results have been conducted to demonstrate that the proposed CFO estimator provides an accurate estimate within a few times of iteration.
I. I NTRODUCTION The multicarrier code-division multiple-access (MC-CDMA) scheme [1]-[2] is known to achieve frequency diversity as well as combat against the frequency-selective channel fading, and the space-time block coding (STBC) technique [3]-[4] is widely used to provide reliable data-transfer for multiple-input Tsui-Tsai Lin is with Department of Electronics Engineering, National United University, No. 1 Lien-Da, Kung-Ching Li, Miaoli 360, Taiwan; Email: [email protected]. Fuh-Hsin Hwang is with Department of Optoelectronics and Communication Engineering, National Kaohsiung Normal University, No.62, Shenjhong Rd., Yanchao, Kaohsiung 824, Taiwan; Email: [email protected].
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multiple-output (MIMO) communication systems [5]-[6]. Thereupon, the joint application of STBC and MC-CDMA, termed STBC MC-CDMA [7]-[8], achieves benefits from both the spatial and frequency diversity, and can be regarded as a practical solution of the physical layer for the air interface of the mobile communication systems. In spite of providing the inherent advantages, the STBC MC-CDMA scheme suffers from certain unfavorable effects which possibly exist in an MC-CDMA system, such as the carrier frequency offset (CFO) [9]-[10]. In a mobile environment, CFO may result from the mismatched frequency of local oscillators at transceivers or Doppler shift owing to user’s mobility. It prevents the subcarriers of an MC-CDMA system from being orthogonal, and thus degrades the system performance in case no countermeasure is applied. In the past, a number of effective algorithms regarding CFO estimation for MC-CDMA systems have been proposed. Thiagarajan [11] proposed a subspacebased CFO estimator, which is derived from the minimization of the determinant with respect to the noise subspace matrix. [12] presented a pilot-aided CFO estimation at the expense of spectral efficiency loss. By exploiting the correlative coding (CC), Kim and Kim [13] devised an intercarrier interference (ICI) suppressor to alleviate the transmission impairment, e.g. the phase noise (PHN) [14] and CFO, for Alamouti space-frequency block code (SFBC) OFDM system. However, this approach relied on the assumption that autocorrelation functions of PHN and the channel frequency responses were precisely known. These constraints are impractical and limit its applications in blind signal processing. A two-level scheme [15] was presented for blind CFO estimation using the fact that all the users’ CFOs were aligned, but the assumption only holds in downlink transmission. Despite providing efficient CFO estimation for downlink systems with aligned users’ CFO, this approach cannot be applied directly to uplink transmission, in which the CFOs for all users are quite different. In [16], Deng proposed an iterative MC-CDMA receiver by using the generalized sidelobe cancellation technique where the CFO estimate was determined by way of an exhaustive search. Despite gaining remarkable estimation performance, these aforementioned schemes cannot be directly applied to STBC MC-CDMA uplink systems owing to limitations of requirement in pilot symbols and heavy computational complexity (quite time-consuming). To the best of the authors’
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knowledge, no work has been devoted to the development of a low complexity blind CFO estimator for an MC-CDMA STBC uplink system. In this paper, we revisit the STBC MC-CDMA scheme while considering an outdoor realistic uplink scenario. Recall that all users are supposed to have a common CFO in the related downlink system such that the estimate algorithm can be simplified efficaciously, but the assumption is no longer practical for the uplink systems. Consequently, the proposed CFO estimate approach in [15] is not suitable for the considered uplink system here. Specifically, the intent of the present paper is to solve the CFO problem for the uplink system of the hybrid quasi-orthogonal STBC (QO-STBC) MC-CDMA scheme [17]-[18], in which we assume each user suffers from an individual CFO [12]. Besides, we also present an improved iterative blind CFO estimator for the considered system. The proposed CFO estimation algorithm is primarily originated from the multiply constrained minimum output energy (MCMOE) receiver [15], which employs a set of correlators to collect the multipath signals emitted from the user-of-interest as well as suppress the multiple access interference (MAI). The principle of the MCMOE CFO estimate is based on the maximization of the output power of the MCMOE receiver. In the proposed technique, the peak location of the output power spectrum should be found. A possible way is to apply exhaustive searching, but it is very time-consuming. To conveniently achieve the maximization, we approximate the output power to a quadratic function in terms of CFO by using the first-order Taylor series expansion technique [19]. In this paper, a keypoint is demonstrated that the CFO estimate can be determined as the solution to the quadratic function instead of the time-consuming peak search. Finally, we introduce an iterative scheme to further enhance the estimate performance by feeding back the CFOcompensated data to the input of the MCMOE receiver during the next iteration period. Numerical results have been conducted to verify that the proposed blind estimator reaches a precision CFO estimation only within a few iterations. The remainder of this paper is organized as follows. Signal model and the MOE receiver are described in Section II. Section III develops our proposed blind iterative CFO estimation. Simulation results are given in Section IV, followed by conclusions in Section V.
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The notations used in this paper are defined as follows. Vectors and matrices are typed with boldface lower and capital letters. In and 0n represent an n × n identity matrix and an n × 1 all zero vector, respectively. diag{a1 , · · · , an } is a block-diagonal matrix with diagonal blocks taking from a1 , · · · , an . A(:, k) and [A]ij denote the kth column and (i, j)th entry of matrix A. The Kronecker product, complex conjugate, transpose and Hermitian transpose are denoted by ⊗, (·)∗ , (·)T and (·)H , respectively. Besides, {·}, tr{·}, E{·}, | · | and · denote the operations of taking the real part, trace of a matrix, ensemble average, absolute value, and the matrix/vector Frobenius norm, respectively.
II. S IGNAL M ODEL AND MOE R ECEIVER A. Signal Model The framework of the considered scheme is a quasi-synchronous1 K-user MIMO MC-CDMA uplink system with four transmit antennas and Nr receive antennas over a multipath fading channel of the maximum excess delay spread L. The Jafarkhani’s QO-STBC coding scheme [17] is employed to achieve diversity gain and coding advantages. For each user, the modulated signal at the pth transmit antenna is (p)
spread with an exclusive spreading code ck of length N where CFO exists at the receiver end. The cyclic prefix (CP) of length Ng is inserted into the data stream so as to reduce the intersymbol interference (ISI) where Ng ≥ L. In the uplink transmission, the transmitted data from each user with independent RF oscillator propagates through an independent multipath fading channel to the base station. Therefore, each user’s signal has different CFO and channel impulse response, which require to estimate before decoding. At the receiver end, after CP removal and FFT operation, the N × 1 data vector collected at the qth receive antenna corresponding to the ith time slot is given by y(q) (i) =
4 K
(p)
(p,q) (p) dk (i)
σk φi (k )S(k )diag{ck }F(:, 1 : L)hk
+ v(q) (i)
k=1 p=1 1
We assume proper time-synchronization is achieved for the desired user and all active users are asynchronous. In addition, multiple antennas
at the transmitter front end share a single radio frequency oscillator, achieving frequency-synchronization among collocated antennas at the transmitter.
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=
K
(1) (1,q) (2) (2,q) (3) (3,q) (4) (4,q) σk φi (k )S(k ) Ck hk , Ck hk , Ck hk , Ck hk dk (i) + v(q) (i),
(1)
k=1 (p)
where σk2 is the signal power of user k out of the total K users, and dk (i) is the symbol transmitted from (p)
the pth transmit antenna. Suppose that dk (i)’s are independent and identically distributed (i.i.d.) random variables with zero mean and unit variance. φ(k ) = ej2πk (N +Ng )/N is defined as the phase difference between two consecutive symbols where k is the corresponding frequency offset normalized with respect to the subcarrier spacing. The N × N Toeplitz unitary matrix S(k ) = FE(k )FH presents the impact of frequency offset in (1) among the subcarriers, where E(k ) = diag{1, ej2πk /N , · · · , ej2(N −1)πk /N }. The √ N × N matrix F is the fast Fourier transform (FFT) matrix with the (m, n)th entry ej2π(m−1)(n−1)/N / N . (p,q)
The L×1 vector hk
denotes the channel impulse response from the pth transmit antenna to the qth receive
antenna. Suppose the channel parameters keep constant during the time interval for calculating the time(p)
(p)
averaged correlation matrix, but vary from one interval to another. Furthermore, Ck = diag{ck }F(:, 1 : (1)
(2)
(3)
(4)
L) and dk (i) = [dk (i), dk (i), dk (i), dk (i)]T denote the post-FFT spreading matrix and the transmitted symbol vector associated to the kth user at the ith time slot, respectively. Finally, v(q) (i) is the complex white Gaussian noise with zero mean and variance σn2 IN , which is assumed to be independent of user symbols. Stacking Nr received data vectors, we have the N Nr × 1 received array data vector y(i) of the ith time slot T y(1)T (i), y(2)T (i), · · · , y(Nr )T (i) K (1) (1) (4) (4) = σk φi (k ) [INr ⊗ S(k )] (INr ⊗ Ck )hk , · · · , (INr ⊗ Ck )hk dk (i) + v(i)
y(i) =
= =
k=1 K k=1 K
i
σk φ (k )Φ(k )
(1) (2) (3) (4) Γk , Γ k , Γ k , Γ k
diag
(1) (2) (3) (4) hk , hk , hk , hk
σk φi (k )Φ(k )Γk Hk dk (i) + v(i),
where (p)
=
(p,1)T
hk
(p,2)T
, hk
dk (i) + v(i) (2)
k=1
hk
(p,Nr )T
, · · · , hk
T
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v(i) =
T
v(1)T (i), v(2)T (i), · · · , v(Nr )T (i)
Φ(k ) = INr ⊗ S(k ) (p)
(p)
= I Nr ⊗ C k (1) (2) (3) (4) = Γk , Γk , Γk , Γk
Γk
Γk
(1) (2) (3) (4) Hk = diag hk , hk , hk , hk .
(3)
Without loss of generality, throughout the paper we assume that the desired user is the first one, and the other users’ signals are referred to as the MAI. Then the problem of interest here is to design a simple blind scheme to estimate the CFO 1 from y(i). Using the intrinsic structure of Jafarkhani’s QO-STBC, the coding (transmitted) data vector dk (i) and the information data vector bk (i) is characterized by the following relations: dk (4i) = bk (i); dk (4i + 1) = Pb∗k (i); dk (4i + 2) = JPb∗k (i); dk (4i + 3) = −Jbk (i),
(4)
where bk (i) = [bk (4i), bk (4i + 1), bk (4i + 2), bk (4i + 3)]T , with bk (i) being the ith information symbol for user k, and
⎡
⎤
⎡
⎤
⎡
⎤
⎢ ˜I O ⎥ ⎢ O ˜I ⎥ ⎢ 0 −1 ⎥ ⎥, J = ⎢ ⎥ and ˜I = ⎢ ⎥. P=⎢ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ O ˜I −˜I O 1 0
(5)
According to (4), the four consecutively received blocks are given by
y(4i) =
K
σk φ(4i) (k )Φ(k )Γk Hk bk (4i) + v(4i),
k=1
y(4i + 1) =
K
σk φ(4i+1) (k )Φ(k )Γk Hk Pb∗k (4i) + v(4i + 1),
k=1
y(4i + 2) =
K
σk φ(4i+2) (k )Φ(k )Γk Hk JPb∗k (4i) + v(4i + 2), and
k=1 K
y(4i + 3) = −
k=1
σk φ(4i+3) (k )Φ(k )Γk Hk Jbk (4i) + v(4i + 3).
(6)
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The ensemble correlation matrix of each block of the received signal data can be derived and written by R(m) = E{y(4i + m)yH (4i + m)} =
K
H H 2 σk2 Φ(k )Γk Hk HH k Γk Φ (k ) + σn IN Nr ,
m=1, 2, 3 and 4.
(7)
k=1 (m) Note that we have used the fact that E{bk (4i)bH k (4i)} = I4 . Since the ensemble correlation matrix R
cannot be obtained practically, we alternatively calculate the time-average to approximate R(m) . Given ˆ (m) constructed by Ns data vectors, the sample correlation matrix R Ns −1 1 (m) ˆ R = y(4i + m)yH (4i + m) Ns i=0
(8)
is a consistent estimate of R(m) under the assumption of ergodicity.
B. MOE Receiver (p)
The objective of the MOE receiver [20] is to extract d1 (i) for p = 1, 2, 3 and 4 while MAI is suppressed from the received signal y(i). When the channel state information matrix H1 is available at the receiver, the design of the MOE receiver for the above scenario is based on the criterion of minimizing the receiver (p)
output power, which is subject to the constraint that the desired signal d1 (i) receives a unit gain in the presumed vector H1 (:, p). Specifically, the weight vector wp can be determined by solving the following constrained optimization problem: min wp
subject to:
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E{|wpH y(4i + m)|2 } = wpH Rwp
m=0
wpH Γ1 H1 (:, p) = 1,
p = 1, 2, 3 and 4,
(9)
where R=
3
R(m) = 4R(m) .
(10)
m=0
Applying the method of Lagrange multipliers [21], the solution to (9) is given by wp =
R−1 Γ1 H1 (:, p) , H −1 HH 1 (:, p)Γ1 R Γ1 H1 (:, p)
p = 1, 2, 3 and 4.
(11)
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III. P ROPOSED B LIND I TERATIVE CFO E STIMATOR It has been shown that the MOE receiver can achieve high output signal-to-interference-plus-noise ratio (SINR) as long as the carrier frequency is correctly synchronized, i.e., 1 = 0 [22]. However, in presence of CFO, the MOE receiver exhibits severe degradation in performance because of the desired signal cancellation effect. To provide reliable performance, CFO estimation is an imperious demand before decoding. In this section, we propose a blind CFO estimator depicted in Fig. 1 for a QO-STBC based MIMO system.
A. Nonblind MOE CFO Estimation In the past, the min-max based CFO estimation has been shown to provide reliable performance for MC-CDMA systems while considering an AWGN channel only [23]. This method with modification of the constraint equation can be applied to the considered hybrid MC-CDMA uplink system, in which the channel matrix H1 is pre-known. In particular, by using the frequency offset estimate of the desired user, which is to be determined next, the constraint equation in (9) can be modified to wpH ()Φ()Γ1 H1 (:, p) = 1,
p = 1, 2, 3 and 4.
(12)
The weight vector can be completely determined by R−1 Φ()Γ1 H1 (:, p) H H −1 HH 1 (:, p)Γ1 Φ ()R Φ()Γ1 H1 (:, p) −1 −1 R Φ()Γ1 H1 (:, p), = HH 1 (:, p)Ψ()H1 (:, p)
wp () =
p = 1,2, 3 and 4.
(13)
H −1 where Ψ() = ΓH 1 Φ ()R Φ()Γ1 . Since the MOE receiver can effectively suppress the MAI, and
block the noise power as much as it can, the receiver output, which is chiefly composed of the desired signal and noise [20], is approximately expressed by (p)
wpH ()y(i) ≈ σ1 φi (1 )wpH ()Φ(1 )Γ1 H1 (:, p)d1 (i) + wpH ()v(i).
(14)
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According to (14), the total output power of the MOE receiver is given by PM OE () =
4 3 p=1 m=0
≈
4 3
2 E wpH ()y(4i + m) 2 H (p) 4i+m E wp ()[σ1 φ (1 )Φ(1 )Γ1 H1 (:, p)d1 (4i + m) + v(4i + m)]
p=1 m=0
=
4σn2
4 σ2 1
p=1
σn2
H H H × wpH ()Φ(1 )Γ1 H1 (:, p)HH 1 (:, p)Γ1 Φ (1 )wp () + wp ()wp ().
(15)
For high input signal-to-noise ratio (SNR), i.e., σ12 /σn2 1, (15) reduces to PM OE () ≈
4
H H 4σ12 wpH ()Φ(1 )Γ1 H1 (:, p)HH 1 (:, p)Γ1 Φ (1 )wp (),
(16)
p=1
which achieves a maximum value as = 1 [24]. It, therefore, provides a convenient solution for the CFO estimation problem, and the CFO can be effectively estimated by finding the frequency at which the maximum output power of the MOE receiver is reached. This principle is analogous to that of the direction-of-arrival (DOA) estimation for the antenna array system [25]. Following this principle, the CFO estimate is efficiently determined by: ˆ1 = arg max PM OE () =
= arg max
= arg max
= arg max
4 3
2 E wpH ()y(4i + m)
p=1 m=0 4
wpH ()Rwp ()
p=1 4
H H −1 HH 1 (:, p)Γ1 Φ ()R Φ()Γ1 H1 (:, p)
−1
p=1 4
HH 1 (:, p)Ψ()H1 (:, p)
−1
,
(17)
p=1
which can be regarded as a one-dimensional search problem. We refer this approach to as the MOE CFO estimation. Because the value of the normalized CFO is taken out of the interval (−0.5, 0.5), we divide the interval into nu sections with a uniform spacing μ = n−1 u , termed ”granularity”. Given nu , the output power expression (17) should be computed for nu + 1 times, and thus the computational complexity of this approach is inversely proportional to the value of μ [26]. Since the precision of estimation depends on the amount of μ, there is a trade-off between the estimate accuracy and computational complexity for the above-mentioned CFO estimation.
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B. Blind MCMOE CFO Estimation The MOE CFO estimation is nonblind and explicitly requires the channel state information matrix H1 , which is typically obtained by the training sequence or pilot symbols. To reach correct estimate of H1 , the pilot symbol should be sufficiently long in the presence of strong MAI and CFO, especially, but this may not be practical owing to the limitation of channel resources. In addition, the MOE receiver is sensitive to the mismatch between the true channel matrix H1 of the received signal in (6) and the hard constraints in (12). In the presence of slight errors in the channel estimation, the MOE receiver exhibits severe degradation in performance. This makes the MOE CFO estimation fail to provide a reliable performance. To avoid estimating H1 in the constraint equation, the proposed CFO estimate can be reached while we introduce multiple constraints with the to-be-determined CFO into the optimization problem of the MOE receiver. In MOE processing, the adverse phenomenon of mutual cancellation usually occurs because of the coherence between the multipath signals of the desired user [27]. It is necessary to decouple the multipath signals by forcing a unit gain on one of the multipaths while putting hard null constraints on the others. Applying these linear constraints to the minimization of the MOE receiver, the optimization problem is equivalent to find the weight vectors w1l ’s for l = 1, 2, · · · , 4LNr :
min
w1l ()
Subject to:
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H H E |w1l ()y(4i + m)|2 ≡ w1l ()Rw1l (),
m=0 H w1l ()Φ()Γ1 = eTl ,
l = 1, 2, · · · , 4LNr ,
(18)
where el is the lth column of I4LNr . The solution to (18) is determined as
w1l () = R−1 Φ()Γ1 Ψ−1 ()el ,
l = 1, 2, · · · , 4LNr .
(19)
To reach the minimum value of the receiver output power in (18), the weight vector w1l is essential H Φ(k )Γk Hk ≈ 0T4 such to lie in the complementary subspace of the MAI subspace. Thus, we have w1l (p)
that the output of the MCMOE receiver chiefly consists of the signal-of-interest d1 (i) and noise, and is
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approximately equal to H ()y(i) z1l (i) = w1l (p)
H H ≈ σ1 φi (1 )w1l ()Φ(1 )Γ1 H1 (:, p)d1 (i) + w1l ()v(i),
(20)
with the corresponding output power being defined by Pl (i) = E{|z1l (i)|2 } H 2 H ()Φ(1 )Γ1 H1 (:, p) + σn2 w1l ()w1l (). ≈ σ12 w1l
(21)
According to the inherent features of the MOE receiver [24], it can be found that the output power in (21) decreases as the difference between the normalized frequency and the true CFO 1 increases. In H (1 )Φ1 (1 )Γ1 = eTl in (18), the output power achieves the maximum particular, by using the constraints w1l H (1 )w1l (1 ), as = 1 . To alleviate the noise effect, the CFO estimate value, given by σ12 |H1 (l, p)|2 +σn2 w1l
is therefore determined by using the total output power of the MCMOE receiver, and defined by PM CM OE () =
4LN r
Pl (i).
(22)
l=1
Substituting (2), (20), and (21) into (22) and applying algebraic manipulations, we have PM CM OE () = = =
4LN r l=1 4LN r l=1 4LN r
H w1l ()Rw1l ()
eTl Ψ−1 ()Ψ()Ψ−1 ()el
Ψ−1 ()
ll
l=1
= tr Ψ−1 () .
(23)
By exploiting (23), the CFO estimate can be reached by searching the frequency where the output power has a maximum value. Thus, the searching algorithm is given by ˆ1 = arg max PM CM OE ()
≡ arg max tr Ψ−1 () .
(24)
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Because performance of this approach is inversely proportional to the value of searching granularity, the CFO estimate performance can be improved by choosing a granularity which is as small as possible. However, decreasing the searching granularity will bring about a great deal of increase in computational complexity. In the following, we present an interesting technique to reduce the computational complexity for the blind MCMOE CFO estimation. This technique uses the Taylor series expansion approximation to avoid the time-consuming calculations for the inverse of the CFO-dependent matrix Ψ() in (24), which requires the computational complexity of order O(43 L3 Nr3 ). To realize the characteristic of the proposed estimator, Fig. 2 shows the power spectrum obtained with (23) versus the error of the CFO estimation, Δ1 = ˆ1 − 1 , for different input SNR’s in a ten-user system over a four-path fading channel (K=10 and L=4). For comparison, the maximum value of the power spectrum is normalized to one for each input SNR. As expected, the output power decreases as the CFO values increase, and reaches the maximum at Δ1 = 0 for all values of the input SNR, which indicates that the correct CFO estimate can be obtained by searching the maximum value of the output power of the proposed blind MCMOE receiver. In addition, because the MCMOE scheme is very sensitive to the mismatch between the constraint matrix in (18) and the received data for high SNR, the power spectrum at SNR=20 dB possesses a narrower mainlobe. This confirms that excellent performance in the CFO estimation can be achieved by increasing the input SNR, which is similar to the phenomenon in the direction-of-arrival estimation of array signal processing [25]. Consequently, a reasonable criterion for determining the CFO estimate is to search the peak location of the convex power spectrum.
C. Search-Free CFO Estimation The CFO estimate in (24) is definitely the correct result by using the infinitesimal granularity. Nevertheless, searching every possible point is unreachable. In this subsection, we develop a low complexity CFO estimation, which avoids to find the inverse of the ”carrier frequency-dependent” matrix in (24). For a sufficiently small value of ( 1), the CFO-driven matrix S() can be expanded by a
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first-order Taylor series technique, and approximated by ˜ S() ≈ S()
dS() = IN + d =0
where
= IN + S˙ 0 ,
(25)
dS) . S˙ 0 = d =0
(26)
Substituting (25) into (3), we have ˜ Φ() ≈ Φ() = IN Nr + (INr ⊗ S˙ 0 ).
(27)
According to (27), the inverse matrix in (24) is approximated by H −1 −1 H −1 Ψ () ≈ IN Nr + (INr ⊗ S˙ 0 ) Γ1 Γ1 IN Nr + (INr ⊗ S˙ 0 ) R −1 −1 ˙ ˙ ≈ T − TΓH (I ⊗ S ) + (I ⊗ S )R R Γ1 T Nr 0 Nr 0 1 −1 ˙ ˙ −2 TΓH 1 (INr ⊗ S0 )R (INr ⊗ S0 )Γ1 T,
(28)
where T = Ψ−1 (0). It is noteworthy that we have employed the Taylor’s series expansion approximation in the last equality of (28). Approximating the inverse matrix, the total output power can be written as
TΓH 1
−1
−1 ˙ ˙ ⊗ S0 ) + (INr ⊗ S0 )R Γ1 T
PM CM OE () ≈ tr {T} − tr R (INr H −1 ˙ ˙ −tr TΓ1 (INr ⊗ S0 )R (INr ⊗ S0 )Γ1 T 2 .
(29)
The approximation of (29) allows us to simplify the searching of the peak position of the output power spectrum to a 1-D optimization problem, and the optimal solution of ˆ1 is derived by H −1 −1 ˙ ˙ tr TΓ1 R (INr ⊗ S0 ) + (INr ⊗ S0 )R Γ1 T ˆ1 = − H −1 ˙ ˙ 2 × tr TΓ1 (INr ⊗ S0 )R (INr ⊗ S0 )Γ1 T tr R−1 (INr ⊗ S˙ 0 )Γ1 T2 ΓH 1 . = − tr (INr ⊗ S˙ 0 )R−1 (INr ⊗ S˙ 0 )Γ1 T2 ΓH 1
(30)
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D. Iterative CFO Estimation For a small CFO, i.e., 1 ≈ 0, the CFO estimate can be reached accurately by using the approximated version of the output power in (29). However, a large CFO brings forth significant disparity between S() ˜ and S() in (25), and leads to substantial estimation error. This means that another step is required to improve performance of the search-free CFO estimation. To remedy such an estimation error, an iterative (0)
scheme is presented by using the CFO estimate in (30) as the initial guess, denoted as ˆ1 . We defined (j)
ˆ1 is the jth CFO iteration result. The corresponding CFO-compensated data is given by (j)
y(j) (i) = ΦH (ˆ1 )y(i)
(31)
such that the data correlation of the jth iteration is derived by (j)
R
=
3
(j) ˜ (4i + m)y(j)H (4i + m) E y
m=0 (j)
(j)
= ΦH (ˆ1 )RΦ(ˆ1 ).
(32)
By feeding the CFO-compensated data in (31) back to the input of the MCMOE receiver, the output power of the jth iteration can be rewritten as (j) PM CM OE () ≈ tr T(j) − 2 × tr [R(j) ]−1 S˙ 0 Γ1 T(j)2 ΓH 1 (j) −1 ˙ (j)2 H 2 , −tr S˙ H [R ] Γ T Γ S 0 1 0 1
(33)
where
−1 (j) −1 ΓH 1 [R ] Γ1 −1 (j) H (j) −1 = ΓH Φ (ˆ )R Φ(ˆ )Γ 1 1 1 1
T(j) =
(j)
= Ψ−1 (ˆ1 ).
(34)
The expression of the residual CFO estimate is identical to (30) except that R and T are replaced by the jth iteration results R(j) and T(j) , respectively. The CFO estimate of the (j + 1)th iteration is obtained
15
by adding the residual CFO estimate to the jth iteration result:
(j+1)
ˆ1
tr [R(j) ]−1 (INr ⊗ S˙ 0 )Γ1 [T(j) ]2 ΓH 1 (j) . = ˆ1 − tr (INr ⊗ S˙ 0 )[R(j) ]−1 (INr ⊗ S˙ 0 )Γ1 [T(j) ]2 ΓH 1
(35)
Note that the performance of the proposed estimator can be improved by increasing the iteration times for the above-mentioned procedure.
E. Convergence Characteristic of the Proposed Iterative CFO Estimation It can be shown that the power spectrum of the proposed scheme can be derived from the MCMOE criterion which is well-known to be used in the optimization of the antenna array system [25]. The MCMOE criterion has been verified not only to suppress the interference effectively but also achieve maximum output SINR while the array vectors of the constraints are correctly pre-determined [24], [28]. The pointing error of the MCMOE beamformer brings about the desired signal cancellation effect, and thus significantly degrades the output SINR. In general, the larger the pointing error is, the more the performance degradation will be. Recall that the power spectrum in (23) obtained with the MCMOE receiver is defined as the cost function, which is a convex function of the normalized frequency over the interval (−0.5, 0.5). One may exploit the convexity of the cost function to ensure the convergence of the iterations [29]. Because the first-order Taylor’s series expansion definitely converges within the CFO search interval, it is reasonable to assume that the approximation in (29) exists for the jth iteration. Given the jth CFO estimate, the (j + 1)th optimized result can be found based on maximizing the approximation (j)
of PM CM OE () in (33) such that we have PM CM OE ((j+1) ) ≥ PM CM OE ((j) ).
(36)
According to (36), the objective function in (23) must be monotonically increasing for j = 1, 2, · · ·. This means the iterations will converge, and eventually reach to a maximum solution.
16
F. Computational Complexity In this subsection, we investigate the computational complexity for the MOE as well as both our proposed CFO estimators, in which the complexity is evaluated by counting the numbers of their required (p)
complex multiplications (CMs). Because the construction of the post-FFT matrix Ck , sample correlation matrix in (8) and its inverse matrix is independent of the actual CFO value, the corresponding CM counts are discarded for simplicity. Owing to the block diagonal structure [21], the calculation for Φ() requires ˜ p) = Φ()Γ1 H1 (:, p) and H ˜ H (:, p)R−1 H ˜ 1 (:, p) for p = 1, 2, 3, about N 2 CMs. The CM counts of H(:, 1 and 4 are LN 2 + LN Nr2 and N 2 Nr2 + N Nr CMs, respectively. As a result, the MOE CFO estimator based on (17) requires a total count of CMs, which is expressed as CMM OE = μ N 2 + 4(LN 2 + LN Nr2 ) + 4(N 2 Nr2 + N Nr ) = μN 2 (4L + 4Nr2 + 1) + 4μN Nr (LNr + 1) ≈ μN 2 (4L + 4Nr2 + 1),
N LNr ,
(37)
where μ represents the search point number. In each search loop of , the proposed MCMOE CFO estimator has to compute Φ()Γ1 and Ψ() which involve 4LN 2 and 8(LN 2 + L2 N ) CMs, respectively. In addition, the computational complexity for computing the inverse of the 4LNr × 4LNr matrix Ψ() is approximately 32 [(4LNr )3 + (4LNr )2 ] CMs. Since there are totally μ search points, the overall computational complexity for the proposed MCMOE CFO estimator in (24) is written by CMM CM OE = μ N 2 + 4LN 2 + 8(LN 2 + L2 N ) + 24L2 Nr2 (4LNr + 1) = μN 2 (12L + 1) + 8μL2 N + 24μL2 Nr2 (4LNr + 1) ≈ μN 2 (12L + 1),
N LNr .
Similarly, the total CM count for the proposed iterative CFO estimator is expressed as CMIterative = J 24L2 Nr2 (4LNr + 1) + N 2 + 4LN 2 + 8LN (N + 2L) + 16L2 Nr2 (4LNr + 1)
(38)
17
= JN 2 (12L + 1) + 16JL2 N + 40JL2 Nr2 (4LNr + 1) ≈ JN 2 (12L + 1),
N LNr .
(39)
where J is the number of iteration times. Table I lists the computational complexities for the MOE and both proposed CFO estimation approaches. It can be shown that the proposed iterative estimation costs much less computational complexity than the non-blind MOE scheme in spite of sacrificing a reasonable performance loss in CFO estimate as μ J. TABLE I L IST FOR THE COMPUTATIONAL COMPLEXITIES AMONG THE MOE AND BOTH PROPOSED CFO
ESTIMATION SCHEMES .
Numbers of Complexity Multiplications (CMs) Algorithm
N subcarriers
N LNr
MOE
μN 2 (4L + 4Nr2 + 1) + 4μN Nr (LNr + 1)
μN 2 (4L + 4Nr2 + 1)
MCMOE
μN 2 (12L + 1) + 8μL2 N + 24μL2 Nr2 (4LNr + 1)
μN 2 (12L + 1)
Iterative
JN 2 (12L + 1) + 16JL2 N + 40JL2 Nr2 (4LNr + 1)
JN 2 (12L + 1)
IV. C OMPUTER S IMULATIONS In this section, we show the computer simulations to verify the advantages of the proposed CFO estimate technique. Consider a MIMO MC-CDMA uplink quasi-synchronous system deployed with two receive antennas in a four-path Rayleigh fading channel (L = 4), where the channel impulse response is (p)
normalized such that hk 2 = 1. The transmit symbols are drawn from the quadrature phase shift keying (p)
(QPSK) modulation scheme. In order to consider an asynchronous transmission, the spreading codes ck
of length 32 (N = 32) for each user are randomly generated and normalized to one, i.e., ck 2 = 1. For ISI suppression, a CP of length eight is inserted into the data stream. Suppose the number of active asynchronous users is K = 10, and all users have an identical power level which is normalized to one for simplicity. The input SNR is defined as 10 log10 (1/σn2 ) dB. The CFO’s of the interferers (MAI’s) are ˆ (m) for a randomly generated in (−0.5, 0.5). Ns is the number of symbols that are used to compute R
18
single trial of simulation. We consider a quasi-static environment, in which the channels are assumed to remain constant during per processing interval of Ns symbol periods, and vary independently between different intervals. To reach the characteristic of the proposed CFO estimator, the numerical results of the mean square errors (MSE’s), E{|Δ1 |2 } = E{|ˆ1 − 1 |2 }, are provided to evaluate the estimator’s performance. Throughout the simulations, the granularity is given by 10−4 for the proposed MCMOE CFO estimation. Denote J as the number of iterations for the proposed iterative CFO estimation. For performance comparison, the results obtained with the ideal MOE estimator and the MOE estimator were also included. The ideal MOE estimator was implemented by using the true channel response H1 . On the ˆ 1 = H1 + 0.1ΔH, other hand, the MOE estimator was constructed by an erroneous channel estimate H where ΔH is a random matrix with the entries being i.i.d. complex Gaussian random variables with unit variance. All the simulation results are obtained by averaging the results of 500 independent trials, in which each trial uses a different set of spreading codes, channel impulse response, and data/noise sequences. Finally, the typical values of the following parameters are given with SNR=10 dB, K = 10, 1 = 0.2 and Ns = 103 in this section. In Fig. 3, we investigate the number of iterations for convergence by using the MSE performance as function of J, with input SNR varied as a parameter. It presents that the MSE decreases as J increases for all SNR, and at high SNR the MSE converges only in a few iteration times (J ≥ 7). Fig. 4 examines the characteristic of the MSE versus input SNR for the proposed CFO estimator. The results shown in Fig. 4 indicate that the proposed MCMOE scheme effectively approaches the ideal MOE estimator. The MOE estimation performs poorly because of extreme sensitivity to errors in the channel estimation. Although Fig. 4 shows that the estimator’s performance can be improved by increasing the iteration times, the MSE appears to be significant while S() is substituted by its first-order Taylor series expansion. For instance, the proposed scheme with two iterations performs poorly due to the erroneous approximation of S(). On the other hand, the iterative CFO estimator with J = 10 has almost the same performance of the MCMOE scheme, confirming the assertion in Section III-D.
19
The MSE plot versus the normalized CFO, shown in Fig. 5, indicates that the proposed blind MCMOE CFO estimator is able to offer similar performance as that of the ideal MOE estimation. In addition, the proposed iterative scheme reaches a reliable performance just using two times of iteration when the CFO is smaller than 0.05. However, the estimate accuracy significantly degrades when the CFO becomes large. Fortunately, the performance degradation can be greatly alleviated by increasing the iteration times. Fig. 6 examines the effect of the system capacity with different values of K. It is presented that the expected advantage of the proposed iterative scheme can be achieved at the expense of a reasonable performance loss as compared to that of the ideal MOE CFO estimation even in the heavy loaded system (large value of K). In addition, the proposed scheme can improve the performance in the CFO estimation as iterations proceed, and the improvement can be achieved as the number of users, K, decreases. In a heavy-loaded system, the proposed CFO estimator is not able to achieve similar performance obtained in a MAI-free (K = 1) case. The reason is that the mutual correlation between users because of the finite sample effect makes the proposed MCMOE receiver fail to suppress each individual MAI, and the increment in the residual MAI buried in the receiver output cannot be negligible. The iterative CFO estimation, again, provides a comparable MSE performance as the MCMOE scheme. ˆ (m) in In the final set of simulation, we study the impact of finite samples, Ns . The correlation matrix R (8) is obtained by using Ns data samples. Fig. 7 plots the MSE versus Ns . As expected, the MSE values decrease as the number of received symbols increases for both cases of J=5 and 10. On the contrary, increasing Ns seems helpless for the proposed scheme with two iterations because the CFO brings about a significant disparity while the approximation in (25) is being used. V. C ONCLUSIONS In this paper, we presented a new blind CFO estimator especially suitable for the use of the uplink in a QO-STBC based MC-CDMA system. The major idea of the presented CFO estimator originates from the design criterion, which is derived by maximizing of the output power of the MCMOE receiver. In addition, two useful approaches were proposed in the paper where the first one was used to reduce the
20
implementation complexity of the proposed estimator, and the other one was developed to compensate the performance loss due to the complexity reduction. For the first approach, the complexity is reduced because we exploit the Taylor series expansion technique while solving the quadratic function so as to avoid the time-consuming CFO line search. The contribution of the second approach is to further enhance the estimate precision by taking advantage of feeding back the CFO-compensated data to the input of the MCMOE receiver. A number of numerical examples have been conducted to demonstrate the characteristics of estimate MSE versus various factors such as the iteration times, input SNR, normalized CFO, and the number of data samples. It was shown that the proposed blind scheme can provide not only a good CFO estimate performance but also the improvement of implementation complexity while we imposing the proposed two approaches on the considered QO-STBC based MC-CDMA scheme.
ACKNOWLEDGMENTS This work was sponsored in part by the National Science Council of Taiwan, R.O.C. under Grant NSC 101-2221-E-239 -018.
R EFERENCES [1] K. Fazel, S. Kaiser, Multi-Carrier and Spread Spectrum Systems, John Wiley & Sons Inc., 2003. [2] H. Hsinsuke, P. Ramjee, Multicarrier Techniques for 4G Mobile Communications, Artech House, Boston London, 2003. [3] V. Tarokh, H. Jafarkhani, A.R. Calderbank, Space-time block codes from orthogonal designs, IEEE Trans. Inf. Theory 45 (7) (1999) 1456-1467. [4] S. Alamouti, A simple transmit diversity technique for wireless communications, IEEE J. Sel. Areas Commun. 16 (10) (1998) 14511458. [5] D. Gesbert, M. Shafi, D. Shiu, P.J. Smith, A. Naguib, From theory to practice: An overview of MIMO space-time coded wireless systems, IEEE J. Sel. Areas Commun. 21 (3) (2003) 281-302. [6] H. Li, X. Lu, G.B. Giannakis, Capon multiuser receiver for CDMA systems with space-time coding, IEEE Trans. Signal Process. 50 (5) (2002) 1193-1204. [7] M. Marey, O.A. Dobre, R. Inkol, Classification of space-time block codes based on second-order cyclostationarity with transmission impairments, IEEE Trans. Wireless Commun. 11 (7) (2012) 2574-2584.
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[8] J.L. Yu, I.T. Lee, MIMO Capon receiver and channel estimation for space-time coded CDMA systems, IEEE Trans. Wireless Commun. 5 (11) (2006) 3023-3028. [9] B. Seo, SINR lower bound based multiuser detector for uplink MC-CDMA systems with residual frequency offset, IEEE Commun. Lett. 16 (10) (2012) 1612-1615. [10] H. Wang, X.G. Xia, Q. Yin, Distributed space-frequency codes for cooperative communication systems with multiple carrier frequency offsets, IEEE Trans. Wireless Commun. 8 (2) (2009) 1045-1055. [11] L.B. Thiagarajan, S. Attallah, K. Abed-Meraim, Y.C. Liang, H. Fu, Non-data-aided joint carrier frequency offset and channel estimator for uplink MC-CDMA systems, IEEE Trans. Signal Process. 56 (9) (2008) 4398-4408. [12] Y. Ma, R. Tafazolli, Estimation of carrier frequency offset for multicarrier CDMA uplink, IEEE Trans. Signal Process. 55 (6) (2007) 2671-2627. [13] K.H. Kim, H.M. Kim, An ICI suppression scheme based on the correlative coding for Alamouti SFBC-OFDM system with phase noise, IEEE Trans. Wireless Commun. 10 (7) (2011) 2023-2027. [14] Q. Zou, A. Tarighat, A.H. Sayed, Compensation of phase noise in OFDM wireless systems, IEEE Trans. Signal Process. 55 (11) (2007) 5407-5424. [15] T.T. Lin, F.H. Hwang, MCMOE-based CFO Estimator aided with the correlation matrix approach for Alamouti’s STBC MC-CDMA downlink systems, IEEE Trans. Veh. Technol., 61 (8) (2012) 3790-3795. [16] J.H. Deng, T.S. Lee, An iterative maximum SINR receiver for multicarrier CDMA systems over a multipath fading channel with frequency offset, IEEE Trans. Wireless Commun. 2 (3) (2003) 560-569. [17] H. Jafarkhani, A quasi-orthogonal space-time block code, IEEE Trans. Commun. 49 (1) (2001) 1-4. [18] C. Yuen, Y.L. Guan, T.T. Tjhung, Quasi-orthogonal STBC with minimum decoding complexity, IEEE Trans. Wireless Commun. 4 (5) (2005) 2089-2094. [19] M. Greenberg, Advanced Engineering Mathematics, second ed., Prentice Hall, 1997. [20] A. Kong, C. Wan, A blind space-time constrained minimum output energy detector for DS-CDMA communication system, IEEE Trans. Veh. Technol. 56 (3) (2007) 1187-1196. [21] G.H. Golub, C.F. Van Loan, Matrix computation, third ed., Johns Hopkins University Press, Baltimore, MD, 1996. [22] B.D. Van Veen, K.M. Buckly, Beamforming: A versatile approach to spatial filtering, IEEE ASSP Mag. 5 (4) (1998) 4-24. [23] B. Seo, H.M. Kim, Frequency offset estimation and multiuser detection for MC-CDMA systems, in Proceeding of IEEE MILCOM, 2002, pp. 804-807. [24] T.S. Lee, T.T. Lin, Coherent interference suppression with complemtally transformed adaptive beamformer, IEEE Trans. Antennas Propagat. 46 (5) (1998) 609-617. [25] R.A. Monzingo, T.W. Miller, Introduction to Adaptive Arrays, New York, NY: John Wiley & Sons Inc., 1980. [26] T.T. Lin, F.H. Hwang, A novel CFO estimator with joint bisection-searching and complexity-reduction technique for uplink MC-CDMA systems, Expert Systems with Applications 39 (3) (2012) 3145-3152.
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[27] C.C. Yeh, W.D. Wang, Coherent interference suppression by an antenna array of arbitrary geometry, IEEE Trans. Antennas Propagat. 37 (10) (1989) 1317-1322. [28] B. Widrow, K.M. Duvall, R.P. Gooch, W.C. Newman, Signal cancellation phenomena in adaptive antennas: Cause and cares, IEEE Trans. Antennas Propagat. 30 (3) (1982) 469-478. [29] V. Solo, X. Kong, Adaptive Signal Processing Algorithms Stability and Performance, Englwood Cliffs, NJ: Prentice-Hall, 1994.
23
Φ Γ
1
= ITQ (:,1)
Φ Γ
1
= ITQ (:, 2)
w11H (ε )
w12H (ε )
{
H P1 (ε ) = E w11 y (i )
2
}
max P (ε )
εˆ1(0)
y (i )
εˆ1( j −1)
Q = 4 LN r
Φ Γ
w1HQ (ε )
Fig. 1.
1
εˆ1( J )
ε
{
PQ (ε ) = E w1HQ y (i)
= I QT (:, Q )
2
}
Block diagram of the proposed iterative CFO estimation.
0
Normalized Output Power (dB)
−5
−10
−15
−20
−25
−30
−35
−40 −0.5
Fig. 2.
SNR=0 dB SNR=10 dB SNR=20 dB −0.4
−0.3
−0.2
−0.1
Normalized power spectrum for K = 10 and Ns = 103 .
0 Δε1
0.1
0.2
0.3
0.4
0.5
24
−1
10
SNR=0 dB SNR=10 dB SNR=20 dB
−2
10
−3
Mean Square Error (MSE)
10
−4
10
−5
10
−6
10
−7
10
−8
10
0
5
10
15
Number of Iterations, J
Fig. 3.
The MSE versus the number of iterations, J, for the proposed CFO estimator given 1 = 0.2, K = 10 and Ns = 103 .
−1
10
−2
10
−3
Mean Square Error (MSE)
10
−4
10
−5
10
−6
10
−7
10
−8
10
−9
10
Fig. 4.
0
MOE (Ideal) MOE Proposed J=1 Proposed J=2 Proposed J=5 Proposed J=10 MCMOE 2
4
6
8
10 Input SNR (dB)
12
14
16
The MSE versus input SNR for the proposed CFO estimator given 1 = 0.2, K = 10 and Ns = 103 .
18
20
25
−1
10
MOE (Ideal) MOE Proposed J=1 Proposed J=2 Proposed J=5 Proposed J=10 MCMOE
−2
Mean Square Error (MSE)
10
−3
10
−4
10
−5
10
−6
10
−7
10 −0.3
Fig. 5.
−0.25
−0.2
−0.15
−0.1
−0.05 0 0.05 Carrier Frequency Offset
0.1
0.15
0.2
0.25
0.3
The MSE versus the normalized CFO for the proposed CFO estimator given input SNR=10 dB, K = 10 and Ns = 103 .
−1
10
−2
10
Mean Square Error (MSE)
−3
10
−4
10
−5
10
−6
10
MOE (Ideal) MOE Proposed J=1 Proposed J=2 Proposed J=5 Proposed J=10 MCMOE
−7
10
−8
10
Fig. 6.
1
2
4
6
8 Number of Users, K
10
12
14
15
The MSE versus the number of users, K, for the proposed CFO estimator given input SNR=10 dB, 1 = 0.2 and Ns = 103 .
26
−1
10
−2
Mean Square Error (MSE)
10
−3
10
−4
10
−5
10
−6
10
−7
10
20
MOE (Ideal) MOE Proposed J=1 Proposed J=2 Proposed J=5 Proposed J=10 MCMOE 100
1000
1500
Number of Samples, Ns
Fig. 7.
The MSE versus the data sample size, Ns , for the proposed CFO estimator given input SNR=10 dB, 1 = 0.2 and K = 10.
Tsui-Tsai Lin received the B.S., M.S., and Ph.D. degrees from National Chiao Tung University, Hsinchu, Taiwan, in 1991, 1993, and 1997, respectively, all in communication engineering. From 1997 to 2000, he was in the Computer & Communications Research Laboratories, Industrial Technology Research Institute. In 2001, he joined the faculty of National United University, Miaoli, Taiwan, where he currently holds a position as Professor in the Department of Electronics Engineering. His current research interests include communications signal processing, smart antenna for wireless communications, and geo-location.
Fuh-Hisn Hwang received the B.S. and M.S. degrees, both in communication engineering, from National Chiao Tung University at Hsinchu, Taiwan in 1991 and 1993, respectively. He received the Ph.D. degree in electrical engineering, from National Central University at JhongLi, Taiwan in 2003. In 2004, he joined the faculty of the National Kaohsiung Normal University, Kaohsiung, Taiwan, where he is a professor of the department of optoelectronics and communication engineering. His research interest is in the area of wireless digital communication techniques, including coded modulation, diversity schemes for MIMO channels, multicarrier modulation and channel estimation.
PII: DOI: Reference:
S1051-2004(13)00223-6 10.1016/j.dsp.2013.10.004 YDSPR 1499
To appear in:
Digital Signal Processing
Please cite this article in press as: T.-T. Lin, F.-H. Hwang, A simple blind CFO estimate technique for a QO-STBC based MC-CDMA uplink system, Digital Signal Process. (2013), http://dx.doi.org/10.1016/j.dsp.2013.10.004
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1
A simple blind CFO estimate technique for a QO-STBC based MC-CDMA uplink system Tsui-Tsai Lin and Fuh-Hsin Hwang
Abstract We propose a simple technique for estimating the carrier frequency offset (CFO) of a quasi-orthogonal spacetime block coding (QO-STBC) multicarrier code-division multiple-access (MC-CDMA) uplink system without the knowledge of channel state information in a multipath fading channel. Based on the intrinsic properties of the considered system, a useful multiply constrained minimum output energy (MCMOE) receiver is developed, and the CFO estimate is achieved by using a one-dimensional linear search, whose objective is to maximize the output power of the developed receiver. In addition, an alternative based on the Taylor series expansion technique is designed to avoid the exhaustive search. By feeding the CFO-compensated data back to the input of the MCMOE receiver, we use an iterative approach to further enhance the estimator’s performance. Numerical results have been conducted to demonstrate that the proposed CFO estimator provides an accurate estimate within a few times of iteration.
I. I NTRODUCTION The multicarrier code-division multiple-access (MC-CDMA) scheme [1]-[2] is known to achieve frequency diversity as well as combat against the frequency-selective channel fading, and the space-time block coding (STBC) technique [3]-[4] is widely used to provide reliable data-transfer for multiple-input Tsui-Tsai Lin is with Department of Electronics Engineering, National United University, No. 1 Lien-Da, Kung-Ching Li, Miaoli 360, Taiwan; Email: [email protected]. Fuh-Hsin Hwang is with Department of Optoelectronics and Communication Engineering, National Kaohsiung Normal University, No.62, Shenjhong Rd., Yanchao, Kaohsiung 824, Taiwan; Email: [email protected].
2
multiple-output (MIMO) communication systems [5]-[6]. Thereupon, the joint application of STBC and MC-CDMA, termed STBC MC-CDMA [7]-[8], achieves benefits from both the spatial and frequency diversity, and can be regarded as a practical solution of the physical layer for the air interface of the mobile communication systems. In spite of providing the inherent advantages, the STBC MC-CDMA scheme suffers from certain unfavorable effects which possibly exist in an MC-CDMA system, such as the carrier frequency offset (CFO) [9]-[10]. In a mobile environment, CFO may result from the mismatched frequency of local oscillators at transceivers or Doppler shift owing to user’s mobility. It prevents the subcarriers of an MC-CDMA system from being orthogonal, and thus degrades the system performance in case no countermeasure is applied. In the past, a number of effective algorithms regarding CFO estimation for MC-CDMA systems have been proposed. Thiagarajan [11] proposed a subspacebased CFO estimator, which is derived from the minimization of the determinant with respect to the noise subspace matrix. [12] presented a pilot-aided CFO estimation at the expense of spectral efficiency loss. By exploiting the correlative coding (CC), Kim and Kim [13] devised an intercarrier interference (ICI) suppressor to alleviate the transmission impairment, e.g. the phase noise (PHN) [14] and CFO, for Alamouti space-frequency block code (SFBC) OFDM system. However, this approach relied on the assumption that autocorrelation functions of PHN and the channel frequency responses were precisely known. These constraints are impractical and limit its applications in blind signal processing. A two-level scheme [15] was presented for blind CFO estimation using the fact that all the users’ CFOs were aligned, but the assumption only holds in downlink transmission. Despite providing efficient CFO estimation for downlink systems with aligned users’ CFO, this approach cannot be applied directly to uplink transmission, in which the CFOs for all users are quite different. In [16], Deng proposed an iterative MC-CDMA receiver by using the generalized sidelobe cancellation technique where the CFO estimate was determined by way of an exhaustive search. Despite gaining remarkable estimation performance, these aforementioned schemes cannot be directly applied to STBC MC-CDMA uplink systems owing to limitations of requirement in pilot symbols and heavy computational complexity (quite time-consuming). To the best of the authors’
3
knowledge, no work has been devoted to the development of a low complexity blind CFO estimator for an MC-CDMA STBC uplink system. In this paper, we revisit the STBC MC-CDMA scheme while considering an outdoor realistic uplink scenario. Recall that all users are supposed to have a common CFO in the related downlink system such that the estimate algorithm can be simplified efficaciously, but the assumption is no longer practical for the uplink systems. Consequently, the proposed CFO estimate approach in [15] is not suitable for the considered uplink system here. Specifically, the intent of the present paper is to solve the CFO problem for the uplink system of the hybrid quasi-orthogonal STBC (QO-STBC) MC-CDMA scheme [17]-[18], in which we assume each user suffers from an individual CFO [12]. Besides, we also present an improved iterative blind CFO estimator for the considered system. The proposed CFO estimation algorithm is primarily originated from the multiply constrained minimum output energy (MCMOE) receiver [15], which employs a set of correlators to collect the multipath signals emitted from the user-of-interest as well as suppress the multiple access interference (MAI). The principle of the MCMOE CFO estimate is based on the maximization of the output power of the MCMOE receiver. In the proposed technique, the peak location of the output power spectrum should be found. A possible way is to apply exhaustive searching, but it is very time-consuming. To conveniently achieve the maximization, we approximate the output power to a quadratic function in terms of CFO by using the first-order Taylor series expansion technique [19]. In this paper, a keypoint is demonstrated that the CFO estimate can be determined as the solution to the quadratic function instead of the time-consuming peak search. Finally, we introduce an iterative scheme to further enhance the estimate performance by feeding back the CFOcompensated data to the input of the MCMOE receiver during the next iteration period. Numerical results have been conducted to verify that the proposed blind estimator reaches a precision CFO estimation only within a few iterations. The remainder of this paper is organized as follows. Signal model and the MOE receiver are described in Section II. Section III develops our proposed blind iterative CFO estimation. Simulation results are given in Section IV, followed by conclusions in Section V.
4
The notations used in this paper are defined as follows. Vectors and matrices are typed with boldface lower and capital letters. In and 0n represent an n × n identity matrix and an n × 1 all zero vector, respectively. diag{a1 , · · · , an } is a block-diagonal matrix with diagonal blocks taking from a1 , · · · , an . A(:, k) and [A]ij denote the kth column and (i, j)th entry of matrix A. The Kronecker product, complex conjugate, transpose and Hermitian transpose are denoted by ⊗, (·)∗ , (·)T and (·)H , respectively. Besides, {·}, tr{·}, E{·}, | · | and · denote the operations of taking the real part, trace of a matrix, ensemble average, absolute value, and the matrix/vector Frobenius norm, respectively.
II. S IGNAL M ODEL AND MOE R ECEIVER A. Signal Model The framework of the considered scheme is a quasi-synchronous1 K-user MIMO MC-CDMA uplink system with four transmit antennas and Nr receive antennas over a multipath fading channel of the maximum excess delay spread L. The Jafarkhani’s QO-STBC coding scheme [17] is employed to achieve diversity gain and coding advantages. For each user, the modulated signal at the pth transmit antenna is (p)
spread with an exclusive spreading code ck of length N where CFO exists at the receiver end. The cyclic prefix (CP) of length Ng is inserted into the data stream so as to reduce the intersymbol interference (ISI) where Ng ≥ L. In the uplink transmission, the transmitted data from each user with independent RF oscillator propagates through an independent multipath fading channel to the base station. Therefore, each user’s signal has different CFO and channel impulse response, which require to estimate before decoding. At the receiver end, after CP removal and FFT operation, the N × 1 data vector collected at the qth receive antenna corresponding to the ith time slot is given by y(q) (i) =
4 K
(p)
(p,q) (p) dk (i)
σk φi (k )S(k )diag{ck }F(:, 1 : L)hk
+ v(q) (i)
k=1 p=1 1
We assume proper time-synchronization is achieved for the desired user and all active users are asynchronous. In addition, multiple antennas
at the transmitter front end share a single radio frequency oscillator, achieving frequency-synchronization among collocated antennas at the transmitter.
5
=
K
(1) (1,q) (2) (2,q) (3) (3,q) (4) (4,q) σk φi (k )S(k ) Ck hk , Ck hk , Ck hk , Ck hk dk (i) + v(q) (i),
(1)
k=1 (p)
where σk2 is the signal power of user k out of the total K users, and dk (i) is the symbol transmitted from (p)
the pth transmit antenna. Suppose that dk (i)’s are independent and identically distributed (i.i.d.) random variables with zero mean and unit variance. φ(k ) = ej2πk (N +Ng )/N is defined as the phase difference between two consecutive symbols where k is the corresponding frequency offset normalized with respect to the subcarrier spacing. The N × N Toeplitz unitary matrix S(k ) = FE(k )FH presents the impact of frequency offset in (1) among the subcarriers, where E(k ) = diag{1, ej2πk /N , · · · , ej2(N −1)πk /N }. The √ N × N matrix F is the fast Fourier transform (FFT) matrix with the (m, n)th entry ej2π(m−1)(n−1)/N / N . (p,q)
The L×1 vector hk
denotes the channel impulse response from the pth transmit antenna to the qth receive
antenna. Suppose the channel parameters keep constant during the time interval for calculating the time(p)
(p)
averaged correlation matrix, but vary from one interval to another. Furthermore, Ck = diag{ck }F(:, 1 : (1)
(2)
(3)
(4)
L) and dk (i) = [dk (i), dk (i), dk (i), dk (i)]T denote the post-FFT spreading matrix and the transmitted symbol vector associated to the kth user at the ith time slot, respectively. Finally, v(q) (i) is the complex white Gaussian noise with zero mean and variance σn2 IN , which is assumed to be independent of user symbols. Stacking Nr received data vectors, we have the N Nr × 1 received array data vector y(i) of the ith time slot T y(1)T (i), y(2)T (i), · · · , y(Nr )T (i) K (1) (1) (4) (4) = σk φi (k ) [INr ⊗ S(k )] (INr ⊗ Ck )hk , · · · , (INr ⊗ Ck )hk dk (i) + v(i)
y(i) =
= =
k=1 K k=1 K
i
σk φ (k )Φ(k )
(1) (2) (3) (4) Γk , Γ k , Γ k , Γ k
diag
(1) (2) (3) (4) hk , hk , hk , hk
σk φi (k )Φ(k )Γk Hk dk (i) + v(i),
where (p)
=
(p,1)T
hk
(p,2)T
, hk
dk (i) + v(i) (2)
k=1
hk
(p,Nr )T
, · · · , hk
T
6
v(i) =
T
v(1)T (i), v(2)T (i), · · · , v(Nr )T (i)
Φ(k ) = INr ⊗ S(k ) (p)
(p)
= I Nr ⊗ C k (1) (2) (3) (4) = Γk , Γk , Γk , Γk
Γk
Γk
(1) (2) (3) (4) Hk = diag hk , hk , hk , hk .
(3)
Without loss of generality, throughout the paper we assume that the desired user is the first one, and the other users’ signals are referred to as the MAI. Then the problem of interest here is to design a simple blind scheme to estimate the CFO 1 from y(i). Using the intrinsic structure of Jafarkhani’s QO-STBC, the coding (transmitted) data vector dk (i) and the information data vector bk (i) is characterized by the following relations: dk (4i) = bk (i); dk (4i + 1) = Pb∗k (i); dk (4i + 2) = JPb∗k (i); dk (4i + 3) = −Jbk (i),
(4)
where bk (i) = [bk (4i), bk (4i + 1), bk (4i + 2), bk (4i + 3)]T , with bk (i) being the ith information symbol for user k, and
⎡
⎤
⎡
⎤
⎡
⎤
⎢ ˜I O ⎥ ⎢ O ˜I ⎥ ⎢ 0 −1 ⎥ ⎥, J = ⎢ ⎥ and ˜I = ⎢ ⎥. P=⎢ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ O ˜I −˜I O 1 0
(5)
According to (4), the four consecutively received blocks are given by
y(4i) =
K
σk φ(4i) (k )Φ(k )Γk Hk bk (4i) + v(4i),
k=1
y(4i + 1) =
K
σk φ(4i+1) (k )Φ(k )Γk Hk Pb∗k (4i) + v(4i + 1),
k=1
y(4i + 2) =
K
σk φ(4i+2) (k )Φ(k )Γk Hk JPb∗k (4i) + v(4i + 2), and
k=1 K
y(4i + 3) = −
k=1
σk φ(4i+3) (k )Φ(k )Γk Hk Jbk (4i) + v(4i + 3).
(6)
7
The ensemble correlation matrix of each block of the received signal data can be derived and written by R(m) = E{y(4i + m)yH (4i + m)} =
K
H H 2 σk2 Φ(k )Γk Hk HH k Γk Φ (k ) + σn IN Nr ,
m=1, 2, 3 and 4.
(7)
k=1 (m) Note that we have used the fact that E{bk (4i)bH k (4i)} = I4 . Since the ensemble correlation matrix R
cannot be obtained practically, we alternatively calculate the time-average to approximate R(m) . Given ˆ (m) constructed by Ns data vectors, the sample correlation matrix R Ns −1 1 (m) ˆ R = y(4i + m)yH (4i + m) Ns i=0
(8)
is a consistent estimate of R(m) under the assumption of ergodicity.
B. MOE Receiver (p)
The objective of the MOE receiver [20] is to extract d1 (i) for p = 1, 2, 3 and 4 while MAI is suppressed from the received signal y(i). When the channel state information matrix H1 is available at the receiver, the design of the MOE receiver for the above scenario is based on the criterion of minimizing the receiver (p)
output power, which is subject to the constraint that the desired signal d1 (i) receives a unit gain in the presumed vector H1 (:, p). Specifically, the weight vector wp can be determined by solving the following constrained optimization problem: min wp
subject to:
3
E{|wpH y(4i + m)|2 } = wpH Rwp
m=0
wpH Γ1 H1 (:, p) = 1,
p = 1, 2, 3 and 4,
(9)
where R=
3
R(m) = 4R(m) .
(10)
m=0
Applying the method of Lagrange multipliers [21], the solution to (9) is given by wp =
R−1 Γ1 H1 (:, p) , H −1 HH 1 (:, p)Γ1 R Γ1 H1 (:, p)
p = 1, 2, 3 and 4.
(11)
8
III. P ROPOSED B LIND I TERATIVE CFO E STIMATOR It has been shown that the MOE receiver can achieve high output signal-to-interference-plus-noise ratio (SINR) as long as the carrier frequency is correctly synchronized, i.e., 1 = 0 [22]. However, in presence of CFO, the MOE receiver exhibits severe degradation in performance because of the desired signal cancellation effect. To provide reliable performance, CFO estimation is an imperious demand before decoding. In this section, we propose a blind CFO estimator depicted in Fig. 1 for a QO-STBC based MIMO system.
A. Nonblind MOE CFO Estimation In the past, the min-max based CFO estimation has been shown to provide reliable performance for MC-CDMA systems while considering an AWGN channel only [23]. This method with modification of the constraint equation can be applied to the considered hybrid MC-CDMA uplink system, in which the channel matrix H1 is pre-known. In particular, by using the frequency offset estimate of the desired user, which is to be determined next, the constraint equation in (9) can be modified to wpH ()Φ()Γ1 H1 (:, p) = 1,
p = 1, 2, 3 and 4.
(12)
The weight vector can be completely determined by R−1 Φ()Γ1 H1 (:, p) H H −1 HH 1 (:, p)Γ1 Φ ()R Φ()Γ1 H1 (:, p) −1 −1 R Φ()Γ1 H1 (:, p), = HH 1 (:, p)Ψ()H1 (:, p)
wp () =
p = 1,2, 3 and 4.
(13)
H −1 where Ψ() = ΓH 1 Φ ()R Φ()Γ1 . Since the MOE receiver can effectively suppress the MAI, and
block the noise power as much as it can, the receiver output, which is chiefly composed of the desired signal and noise [20], is approximately expressed by (p)
wpH ()y(i) ≈ σ1 φi (1 )wpH ()Φ(1 )Γ1 H1 (:, p)d1 (i) + wpH ()v(i).
(14)
9
According to (14), the total output power of the MOE receiver is given by PM OE () =
4 3 p=1 m=0
≈
4 3
2 E wpH ()y(4i + m) 2 H (p) 4i+m E wp ()[σ1 φ (1 )Φ(1 )Γ1 H1 (:, p)d1 (4i + m) + v(4i + m)]
p=1 m=0
=
4σn2
4 σ2 1
p=1
σn2
H H H × wpH ()Φ(1 )Γ1 H1 (:, p)HH 1 (:, p)Γ1 Φ (1 )wp () + wp ()wp ().
(15)
For high input signal-to-noise ratio (SNR), i.e., σ12 /σn2 1, (15) reduces to PM OE () ≈
4
H H 4σ12 wpH ()Φ(1 )Γ1 H1 (:, p)HH 1 (:, p)Γ1 Φ (1 )wp (),
(16)
p=1
which achieves a maximum value as = 1 [24]. It, therefore, provides a convenient solution for the CFO estimation problem, and the CFO can be effectively estimated by finding the frequency at which the maximum output power of the MOE receiver is reached. This principle is analogous to that of the direction-of-arrival (DOA) estimation for the antenna array system [25]. Following this principle, the CFO estimate is efficiently determined by: ˆ1 = arg max PM OE () =
= arg max
= arg max
= arg max
4 3
2 E wpH ()y(4i + m)
p=1 m=0 4
wpH ()Rwp ()
p=1 4
H H −1 HH 1 (:, p)Γ1 Φ ()R Φ()Γ1 H1 (:, p)
−1
p=1 4
HH 1 (:, p)Ψ()H1 (:, p)
−1
,
(17)
p=1
which can be regarded as a one-dimensional search problem. We refer this approach to as the MOE CFO estimation. Because the value of the normalized CFO is taken out of the interval (−0.5, 0.5), we divide the interval into nu sections with a uniform spacing μ = n−1 u , termed ”granularity”. Given nu , the output power expression (17) should be computed for nu + 1 times, and thus the computational complexity of this approach is inversely proportional to the value of μ [26]. Since the precision of estimation depends on the amount of μ, there is a trade-off between the estimate accuracy and computational complexity for the above-mentioned CFO estimation.
10
B. Blind MCMOE CFO Estimation The MOE CFO estimation is nonblind and explicitly requires the channel state information matrix H1 , which is typically obtained by the training sequence or pilot symbols. To reach correct estimate of H1 , the pilot symbol should be sufficiently long in the presence of strong MAI and CFO, especially, but this may not be practical owing to the limitation of channel resources. In addition, the MOE receiver is sensitive to the mismatch between the true channel matrix H1 of the received signal in (6) and the hard constraints in (12). In the presence of slight errors in the channel estimation, the MOE receiver exhibits severe degradation in performance. This makes the MOE CFO estimation fail to provide a reliable performance. To avoid estimating H1 in the constraint equation, the proposed CFO estimate can be reached while we introduce multiple constraints with the to-be-determined CFO into the optimization problem of the MOE receiver. In MOE processing, the adverse phenomenon of mutual cancellation usually occurs because of the coherence between the multipath signals of the desired user [27]. It is necessary to decouple the multipath signals by forcing a unit gain on one of the multipaths while putting hard null constraints on the others. Applying these linear constraints to the minimization of the MOE receiver, the optimization problem is equivalent to find the weight vectors w1l ’s for l = 1, 2, · · · , 4LNr :
min
w1l ()
Subject to:
3
H H E |w1l ()y(4i + m)|2 ≡ w1l ()Rw1l (),
m=0 H w1l ()Φ()Γ1 = eTl ,
l = 1, 2, · · · , 4LNr ,
(18)
where el is the lth column of I4LNr . The solution to (18) is determined as
w1l () = R−1 Φ()Γ1 Ψ−1 ()el ,
l = 1, 2, · · · , 4LNr .
(19)
To reach the minimum value of the receiver output power in (18), the weight vector w1l is essential H Φ(k )Γk Hk ≈ 0T4 such to lie in the complementary subspace of the MAI subspace. Thus, we have w1l (p)
that the output of the MCMOE receiver chiefly consists of the signal-of-interest d1 (i) and noise, and is
11
approximately equal to H ()y(i) z1l (i) = w1l (p)
H H ≈ σ1 φi (1 )w1l ()Φ(1 )Γ1 H1 (:, p)d1 (i) + w1l ()v(i),
(20)
with the corresponding output power being defined by Pl (i) = E{|z1l (i)|2 } H 2 H ()Φ(1 )Γ1 H1 (:, p) + σn2 w1l ()w1l (). ≈ σ12 w1l
(21)
According to the inherent features of the MOE receiver [24], it can be found that the output power in (21) decreases as the difference between the normalized frequency and the true CFO 1 increases. In H (1 )Φ1 (1 )Γ1 = eTl in (18), the output power achieves the maximum particular, by using the constraints w1l H (1 )w1l (1 ), as = 1 . To alleviate the noise effect, the CFO estimate value, given by σ12 |H1 (l, p)|2 +σn2 w1l
is therefore determined by using the total output power of the MCMOE receiver, and defined by PM CM OE () =
4LN r
Pl (i).
(22)
l=1
Substituting (2), (20), and (21) into (22) and applying algebraic manipulations, we have PM CM OE () = = =
4LN r l=1 4LN r l=1 4LN r
H w1l ()Rw1l ()
eTl Ψ−1 ()Ψ()Ψ−1 ()el
Ψ−1 ()
ll
l=1
= tr Ψ−1 () .
(23)
By exploiting (23), the CFO estimate can be reached by searching the frequency where the output power has a maximum value. Thus, the searching algorithm is given by ˆ1 = arg max PM CM OE ()
≡ arg max tr Ψ−1 () .
(24)
12
Because performance of this approach is inversely proportional to the value of searching granularity, the CFO estimate performance can be improved by choosing a granularity which is as small as possible. However, decreasing the searching granularity will bring about a great deal of increase in computational complexity. In the following, we present an interesting technique to reduce the computational complexity for the blind MCMOE CFO estimation. This technique uses the Taylor series expansion approximation to avoid the time-consuming calculations for the inverse of the CFO-dependent matrix Ψ() in (24), which requires the computational complexity of order O(43 L3 Nr3 ). To realize the characteristic of the proposed estimator, Fig. 2 shows the power spectrum obtained with (23) versus the error of the CFO estimation, Δ1 = ˆ1 − 1 , for different input SNR’s in a ten-user system over a four-path fading channel (K=10 and L=4). For comparison, the maximum value of the power spectrum is normalized to one for each input SNR. As expected, the output power decreases as the CFO values increase, and reaches the maximum at Δ1 = 0 for all values of the input SNR, which indicates that the correct CFO estimate can be obtained by searching the maximum value of the output power of the proposed blind MCMOE receiver. In addition, because the MCMOE scheme is very sensitive to the mismatch between the constraint matrix in (18) and the received data for high SNR, the power spectrum at SNR=20 dB possesses a narrower mainlobe. This confirms that excellent performance in the CFO estimation can be achieved by increasing the input SNR, which is similar to the phenomenon in the direction-of-arrival estimation of array signal processing [25]. Consequently, a reasonable criterion for determining the CFO estimate is to search the peak location of the convex power spectrum.
C. Search-Free CFO Estimation The CFO estimate in (24) is definitely the correct result by using the infinitesimal granularity. Nevertheless, searching every possible point is unreachable. In this subsection, we develop a low complexity CFO estimation, which avoids to find the inverse of the ”carrier frequency-dependent” matrix in (24). For a sufficiently small value of ( 1), the CFO-driven matrix S() can be expanded by a
13
first-order Taylor series technique, and approximated by ˜ S() ≈ S()
dS() = IN + d =0
where
= IN + S˙ 0 ,
(25)
dS) . S˙ 0 = d =0
(26)
Substituting (25) into (3), we have ˜ Φ() ≈ Φ() = IN Nr + (INr ⊗ S˙ 0 ).
(27)
According to (27), the inverse matrix in (24) is approximated by H −1 −1 H −1 Ψ () ≈ IN Nr + (INr ⊗ S˙ 0 ) Γ1 Γ1 IN Nr + (INr ⊗ S˙ 0 ) R −1 −1 ˙ ˙ ≈ T − TΓH (I ⊗ S ) + (I ⊗ S )R R Γ1 T Nr 0 Nr 0 1 −1 ˙ ˙ −2 TΓH 1 (INr ⊗ S0 )R (INr ⊗ S0 )Γ1 T,
(28)
where T = Ψ−1 (0). It is noteworthy that we have employed the Taylor’s series expansion approximation in the last equality of (28). Approximating the inverse matrix, the total output power can be written as
TΓH 1
−1
−1 ˙ ˙ ⊗ S0 ) + (INr ⊗ S0 )R Γ1 T
PM CM OE () ≈ tr {T} − tr R (INr H −1 ˙ ˙ −tr TΓ1 (INr ⊗ S0 )R (INr ⊗ S0 )Γ1 T 2 .
(29)
The approximation of (29) allows us to simplify the searching of the peak position of the output power spectrum to a 1-D optimization problem, and the optimal solution of ˆ1 is derived by H −1 −1 ˙ ˙ tr TΓ1 R (INr ⊗ S0 ) + (INr ⊗ S0 )R Γ1 T ˆ1 = − H −1 ˙ ˙ 2 × tr TΓ1 (INr ⊗ S0 )R (INr ⊗ S0 )Γ1 T tr R−1 (INr ⊗ S˙ 0 )Γ1 T2 ΓH 1 . = − tr (INr ⊗ S˙ 0 )R−1 (INr ⊗ S˙ 0 )Γ1 T2 ΓH 1
(30)
14
D. Iterative CFO Estimation For a small CFO, i.e., 1 ≈ 0, the CFO estimate can be reached accurately by using the approximated version of the output power in (29). However, a large CFO brings forth significant disparity between S() ˜ and S() in (25), and leads to substantial estimation error. This means that another step is required to improve performance of the search-free CFO estimation. To remedy such an estimation error, an iterative (0)
scheme is presented by using the CFO estimate in (30) as the initial guess, denoted as ˆ1 . We defined (j)
ˆ1 is the jth CFO iteration result. The corresponding CFO-compensated data is given by (j)
y(j) (i) = ΦH (ˆ1 )y(i)
(31)
such that the data correlation of the jth iteration is derived by (j)
R
=
3
(j) ˜ (4i + m)y(j)H (4i + m) E y
m=0 (j)
(j)
= ΦH (ˆ1 )RΦ(ˆ1 ).
(32)
By feeding the CFO-compensated data in (31) back to the input of the MCMOE receiver, the output power of the jth iteration can be rewritten as (j) PM CM OE () ≈ tr T(j) − 2 × tr [R(j) ]−1 S˙ 0 Γ1 T(j)2 ΓH 1 (j) −1 ˙ (j)2 H 2 , −tr S˙ H [R ] Γ T Γ S 0 1 0 1
(33)
where
−1 (j) −1 ΓH 1 [R ] Γ1 −1 (j) H (j) −1 = ΓH Φ (ˆ )R Φ(ˆ )Γ 1 1 1 1
T(j) =
(j)
= Ψ−1 (ˆ1 ).
(34)
The expression of the residual CFO estimate is identical to (30) except that R and T are replaced by the jth iteration results R(j) and T(j) , respectively. The CFO estimate of the (j + 1)th iteration is obtained
15
by adding the residual CFO estimate to the jth iteration result:
(j+1)
ˆ1
tr [R(j) ]−1 (INr ⊗ S˙ 0 )Γ1 [T(j) ]2 ΓH 1 (j) . = ˆ1 − tr (INr ⊗ S˙ 0 )[R(j) ]−1 (INr ⊗ S˙ 0 )Γ1 [T(j) ]2 ΓH 1
(35)
Note that the performance of the proposed estimator can be improved by increasing the iteration times for the above-mentioned procedure.
E. Convergence Characteristic of the Proposed Iterative CFO Estimation It can be shown that the power spectrum of the proposed scheme can be derived from the MCMOE criterion which is well-known to be used in the optimization of the antenna array system [25]. The MCMOE criterion has been verified not only to suppress the interference effectively but also achieve maximum output SINR while the array vectors of the constraints are correctly pre-determined [24], [28]. The pointing error of the MCMOE beamformer brings about the desired signal cancellation effect, and thus significantly degrades the output SINR. In general, the larger the pointing error is, the more the performance degradation will be. Recall that the power spectrum in (23) obtained with the MCMOE receiver is defined as the cost function, which is a convex function of the normalized frequency over the interval (−0.5, 0.5). One may exploit the convexity of the cost function to ensure the convergence of the iterations [29]. Because the first-order Taylor’s series expansion definitely converges within the CFO search interval, it is reasonable to assume that the approximation in (29) exists for the jth iteration. Given the jth CFO estimate, the (j + 1)th optimized result can be found based on maximizing the approximation (j)
of PM CM OE () in (33) such that we have PM CM OE ((j+1) ) ≥ PM CM OE ((j) ).
(36)
According to (36), the objective function in (23) must be monotonically increasing for j = 1, 2, · · ·. This means the iterations will converge, and eventually reach to a maximum solution.
16
F. Computational Complexity In this subsection, we investigate the computational complexity for the MOE as well as both our proposed CFO estimators, in which the complexity is evaluated by counting the numbers of their required (p)
complex multiplications (CMs). Because the construction of the post-FFT matrix Ck , sample correlation matrix in (8) and its inverse matrix is independent of the actual CFO value, the corresponding CM counts are discarded for simplicity. Owing to the block diagonal structure [21], the calculation for Φ() requires ˜ p) = Φ()Γ1 H1 (:, p) and H ˜ H (:, p)R−1 H ˜ 1 (:, p) for p = 1, 2, 3, about N 2 CMs. The CM counts of H(:, 1 and 4 are LN 2 + LN Nr2 and N 2 Nr2 + N Nr CMs, respectively. As a result, the MOE CFO estimator based on (17) requires a total count of CMs, which is expressed as CMM OE = μ N 2 + 4(LN 2 + LN Nr2 ) + 4(N 2 Nr2 + N Nr ) = μN 2 (4L + 4Nr2 + 1) + 4μN Nr (LNr + 1) ≈ μN 2 (4L + 4Nr2 + 1),
N LNr ,
(37)
where μ represents the search point number. In each search loop of , the proposed MCMOE CFO estimator has to compute Φ()Γ1 and Ψ() which involve 4LN 2 and 8(LN 2 + L2 N ) CMs, respectively. In addition, the computational complexity for computing the inverse of the 4LNr × 4LNr matrix Ψ() is approximately 32 [(4LNr )3 + (4LNr )2 ] CMs. Since there are totally μ search points, the overall computational complexity for the proposed MCMOE CFO estimator in (24) is written by CMM CM OE = μ N 2 + 4LN 2 + 8(LN 2 + L2 N ) + 24L2 Nr2 (4LNr + 1) = μN 2 (12L + 1) + 8μL2 N + 24μL2 Nr2 (4LNr + 1) ≈ μN 2 (12L + 1),
N LNr .
Similarly, the total CM count for the proposed iterative CFO estimator is expressed as CMIterative = J 24L2 Nr2 (4LNr + 1) + N 2 + 4LN 2 + 8LN (N + 2L) + 16L2 Nr2 (4LNr + 1)
(38)
17
= JN 2 (12L + 1) + 16JL2 N + 40JL2 Nr2 (4LNr + 1) ≈ JN 2 (12L + 1),
N LNr .
(39)
where J is the number of iteration times. Table I lists the computational complexities for the MOE and both proposed CFO estimation approaches. It can be shown that the proposed iterative estimation costs much less computational complexity than the non-blind MOE scheme in spite of sacrificing a reasonable performance loss in CFO estimate as μ J. TABLE I L IST FOR THE COMPUTATIONAL COMPLEXITIES AMONG THE MOE AND BOTH PROPOSED CFO
ESTIMATION SCHEMES .
Numbers of Complexity Multiplications (CMs) Algorithm
N subcarriers
N LNr
MOE
μN 2 (4L + 4Nr2 + 1) + 4μN Nr (LNr + 1)
μN 2 (4L + 4Nr2 + 1)
MCMOE
μN 2 (12L + 1) + 8μL2 N + 24μL2 Nr2 (4LNr + 1)
μN 2 (12L + 1)
Iterative
JN 2 (12L + 1) + 16JL2 N + 40JL2 Nr2 (4LNr + 1)
JN 2 (12L + 1)
IV. C OMPUTER S IMULATIONS In this section, we show the computer simulations to verify the advantages of the proposed CFO estimate technique. Consider a MIMO MC-CDMA uplink quasi-synchronous system deployed with two receive antennas in a four-path Rayleigh fading channel (L = 4), where the channel impulse response is (p)
normalized such that hk 2 = 1. The transmit symbols are drawn from the quadrature phase shift keying (p)
(QPSK) modulation scheme. In order to consider an asynchronous transmission, the spreading codes ck
of length 32 (N = 32) for each user are randomly generated and normalized to one, i.e., ck 2 = 1. For ISI suppression, a CP of length eight is inserted into the data stream. Suppose the number of active asynchronous users is K = 10, and all users have an identical power level which is normalized to one for simplicity. The input SNR is defined as 10 log10 (1/σn2 ) dB. The CFO’s of the interferers (MAI’s) are ˆ (m) for a randomly generated in (−0.5, 0.5). Ns is the number of symbols that are used to compute R
18
single trial of simulation. We consider a quasi-static environment, in which the channels are assumed to remain constant during per processing interval of Ns symbol periods, and vary independently between different intervals. To reach the characteristic of the proposed CFO estimator, the numerical results of the mean square errors (MSE’s), E{|Δ1 |2 } = E{|ˆ1 − 1 |2 }, are provided to evaluate the estimator’s performance. Throughout the simulations, the granularity is given by 10−4 for the proposed MCMOE CFO estimation. Denote J as the number of iterations for the proposed iterative CFO estimation. For performance comparison, the results obtained with the ideal MOE estimator and the MOE estimator were also included. The ideal MOE estimator was implemented by using the true channel response H1 . On the ˆ 1 = H1 + 0.1ΔH, other hand, the MOE estimator was constructed by an erroneous channel estimate H where ΔH is a random matrix with the entries being i.i.d. complex Gaussian random variables with unit variance. All the simulation results are obtained by averaging the results of 500 independent trials, in which each trial uses a different set of spreading codes, channel impulse response, and data/noise sequences. Finally, the typical values of the following parameters are given with SNR=10 dB, K = 10, 1 = 0.2 and Ns = 103 in this section. In Fig. 3, we investigate the number of iterations for convergence by using the MSE performance as function of J, with input SNR varied as a parameter. It presents that the MSE decreases as J increases for all SNR, and at high SNR the MSE converges only in a few iteration times (J ≥ 7). Fig. 4 examines the characteristic of the MSE versus input SNR for the proposed CFO estimator. The results shown in Fig. 4 indicate that the proposed MCMOE scheme effectively approaches the ideal MOE estimator. The MOE estimation performs poorly because of extreme sensitivity to errors in the channel estimation. Although Fig. 4 shows that the estimator’s performance can be improved by increasing the iteration times, the MSE appears to be significant while S() is substituted by its first-order Taylor series expansion. For instance, the proposed scheme with two iterations performs poorly due to the erroneous approximation of S(). On the other hand, the iterative CFO estimator with J = 10 has almost the same performance of the MCMOE scheme, confirming the assertion in Section III-D.
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The MSE plot versus the normalized CFO, shown in Fig. 5, indicates that the proposed blind MCMOE CFO estimator is able to offer similar performance as that of the ideal MOE estimation. In addition, the proposed iterative scheme reaches a reliable performance just using two times of iteration when the CFO is smaller than 0.05. However, the estimate accuracy significantly degrades when the CFO becomes large. Fortunately, the performance degradation can be greatly alleviated by increasing the iteration times. Fig. 6 examines the effect of the system capacity with different values of K. It is presented that the expected advantage of the proposed iterative scheme can be achieved at the expense of a reasonable performance loss as compared to that of the ideal MOE CFO estimation even in the heavy loaded system (large value of K). In addition, the proposed scheme can improve the performance in the CFO estimation as iterations proceed, and the improvement can be achieved as the number of users, K, decreases. In a heavy-loaded system, the proposed CFO estimator is not able to achieve similar performance obtained in a MAI-free (K = 1) case. The reason is that the mutual correlation between users because of the finite sample effect makes the proposed MCMOE receiver fail to suppress each individual MAI, and the increment in the residual MAI buried in the receiver output cannot be negligible. The iterative CFO estimation, again, provides a comparable MSE performance as the MCMOE scheme. ˆ (m) in In the final set of simulation, we study the impact of finite samples, Ns . The correlation matrix R (8) is obtained by using Ns data samples. Fig. 7 plots the MSE versus Ns . As expected, the MSE values decrease as the number of received symbols increases for both cases of J=5 and 10. On the contrary, increasing Ns seems helpless for the proposed scheme with two iterations because the CFO brings about a significant disparity while the approximation in (25) is being used. V. C ONCLUSIONS In this paper, we presented a new blind CFO estimator especially suitable for the use of the uplink in a QO-STBC based MC-CDMA system. The major idea of the presented CFO estimator originates from the design criterion, which is derived by maximizing of the output power of the MCMOE receiver. In addition, two useful approaches were proposed in the paper where the first one was used to reduce the
20
implementation complexity of the proposed estimator, and the other one was developed to compensate the performance loss due to the complexity reduction. For the first approach, the complexity is reduced because we exploit the Taylor series expansion technique while solving the quadratic function so as to avoid the time-consuming CFO line search. The contribution of the second approach is to further enhance the estimate precision by taking advantage of feeding back the CFO-compensated data to the input of the MCMOE receiver. A number of numerical examples have been conducted to demonstrate the characteristics of estimate MSE versus various factors such as the iteration times, input SNR, normalized CFO, and the number of data samples. It was shown that the proposed blind scheme can provide not only a good CFO estimate performance but also the improvement of implementation complexity while we imposing the proposed two approaches on the considered QO-STBC based MC-CDMA scheme.
ACKNOWLEDGMENTS This work was sponsored in part by the National Science Council of Taiwan, R.O.C. under Grant NSC 101-2221-E-239 -018.
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[8] J.L. Yu, I.T. Lee, MIMO Capon receiver and channel estimation for space-time coded CDMA systems, IEEE Trans. Wireless Commun. 5 (11) (2006) 3023-3028. [9] B. Seo, SINR lower bound based multiuser detector for uplink MC-CDMA systems with residual frequency offset, IEEE Commun. Lett. 16 (10) (2012) 1612-1615. [10] H. Wang, X.G. Xia, Q. Yin, Distributed space-frequency codes for cooperative communication systems with multiple carrier frequency offsets, IEEE Trans. Wireless Commun. 8 (2) (2009) 1045-1055. [11] L.B. Thiagarajan, S. Attallah, K. Abed-Meraim, Y.C. Liang, H. Fu, Non-data-aided joint carrier frequency offset and channel estimator for uplink MC-CDMA systems, IEEE Trans. Signal Process. 56 (9) (2008) 4398-4408. [12] Y. Ma, R. Tafazolli, Estimation of carrier frequency offset for multicarrier CDMA uplink, IEEE Trans. Signal Process. 55 (6) (2007) 2671-2627. [13] K.H. Kim, H.M. Kim, An ICI suppression scheme based on the correlative coding for Alamouti SFBC-OFDM system with phase noise, IEEE Trans. Wireless Commun. 10 (7) (2011) 2023-2027. [14] Q. Zou, A. Tarighat, A.H. Sayed, Compensation of phase noise in OFDM wireless systems, IEEE Trans. Signal Process. 55 (11) (2007) 5407-5424. [15] T.T. Lin, F.H. Hwang, MCMOE-based CFO Estimator aided with the correlation matrix approach for Alamouti’s STBC MC-CDMA downlink systems, IEEE Trans. Veh. Technol., 61 (8) (2012) 3790-3795. [16] J.H. Deng, T.S. Lee, An iterative maximum SINR receiver for multicarrier CDMA systems over a multipath fading channel with frequency offset, IEEE Trans. Wireless Commun. 2 (3) (2003) 560-569. [17] H. Jafarkhani, A quasi-orthogonal space-time block code, IEEE Trans. Commun. 49 (1) (2001) 1-4. [18] C. Yuen, Y.L. Guan, T.T. Tjhung, Quasi-orthogonal STBC with minimum decoding complexity, IEEE Trans. Wireless Commun. 4 (5) (2005) 2089-2094. [19] M. Greenberg, Advanced Engineering Mathematics, second ed., Prentice Hall, 1997. [20] A. Kong, C. Wan, A blind space-time constrained minimum output energy detector for DS-CDMA communication system, IEEE Trans. Veh. Technol. 56 (3) (2007) 1187-1196. [21] G.H. Golub, C.F. Van Loan, Matrix computation, third ed., Johns Hopkins University Press, Baltimore, MD, 1996. [22] B.D. Van Veen, K.M. Buckly, Beamforming: A versatile approach to spatial filtering, IEEE ASSP Mag. 5 (4) (1998) 4-24. [23] B. Seo, H.M. Kim, Frequency offset estimation and multiuser detection for MC-CDMA systems, in Proceeding of IEEE MILCOM, 2002, pp. 804-807. [24] T.S. Lee, T.T. Lin, Coherent interference suppression with complemtally transformed adaptive beamformer, IEEE Trans. Antennas Propagat. 46 (5) (1998) 609-617. [25] R.A. Monzingo, T.W. Miller, Introduction to Adaptive Arrays, New York, NY: John Wiley & Sons Inc., 1980. [26] T.T. Lin, F.H. Hwang, A novel CFO estimator with joint bisection-searching and complexity-reduction technique for uplink MC-CDMA systems, Expert Systems with Applications 39 (3) (2012) 3145-3152.
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[27] C.C. Yeh, W.D. Wang, Coherent interference suppression by an antenna array of arbitrary geometry, IEEE Trans. Antennas Propagat. 37 (10) (1989) 1317-1322. [28] B. Widrow, K.M. Duvall, R.P. Gooch, W.C. Newman, Signal cancellation phenomena in adaptive antennas: Cause and cares, IEEE Trans. Antennas Propagat. 30 (3) (1982) 469-478. [29] V. Solo, X. Kong, Adaptive Signal Processing Algorithms Stability and Performance, Englwood Cliffs, NJ: Prentice-Hall, 1994.
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Φ Γ
1
= ITQ (:,1)
Φ Γ
1
= ITQ (:, 2)
w11H (ε )
w12H (ε )
{
H P1 (ε ) = E w11 y (i )
2
}
max P (ε )
εˆ1(0)
y (i )
εˆ1( j −1)
Q = 4 LN r
Φ Γ
w1HQ (ε )
Fig. 1.
1
εˆ1( J )
ε
{
PQ (ε ) = E w1HQ y (i)
= I QT (:, Q )
2
}
Block diagram of the proposed iterative CFO estimation.
0
Normalized Output Power (dB)
−5
−10
−15
−20
−25
−30
−35
−40 −0.5
Fig. 2.
SNR=0 dB SNR=10 dB SNR=20 dB −0.4
−0.3
−0.2
−0.1
Normalized power spectrum for K = 10 and Ns = 103 .
0 Δε1
0.1
0.2
0.3
0.4
0.5
24
−1
10
SNR=0 dB SNR=10 dB SNR=20 dB
−2
10
−3
Mean Square Error (MSE)
10
−4
10
−5
10
−6
10
−7
10
−8
10
0
5
10
15
Number of Iterations, J
Fig. 3.
The MSE versus the number of iterations, J, for the proposed CFO estimator given 1 = 0.2, K = 10 and Ns = 103 .
−1
10
−2
10
−3
Mean Square Error (MSE)
10
−4
10
−5
10
−6
10
−7
10
−8
10
−9
10
Fig. 4.
0
MOE (Ideal) MOE Proposed J=1 Proposed J=2 Proposed J=5 Proposed J=10 MCMOE 2
4
6
8
10 Input SNR (dB)
12
14
16
The MSE versus input SNR for the proposed CFO estimator given 1 = 0.2, K = 10 and Ns = 103 .
18
20
25
−1
10
MOE (Ideal) MOE Proposed J=1 Proposed J=2 Proposed J=5 Proposed J=10 MCMOE
−2
Mean Square Error (MSE)
10
−3
10
−4
10
−5
10
−6
10
−7
10 −0.3
Fig. 5.
−0.25
−0.2
−0.15
−0.1
−0.05 0 0.05 Carrier Frequency Offset
0.1
0.15
0.2
0.25
0.3
The MSE versus the normalized CFO for the proposed CFO estimator given input SNR=10 dB, K = 10 and Ns = 103 .
−1
10
−2
10
Mean Square Error (MSE)
−3
10
−4
10
−5
10
−6
10
MOE (Ideal) MOE Proposed J=1 Proposed J=2 Proposed J=5 Proposed J=10 MCMOE
−7
10
−8
10
Fig. 6.
1
2
4
6
8 Number of Users, K
10
12
14
15
The MSE versus the number of users, K, for the proposed CFO estimator given input SNR=10 dB, 1 = 0.2 and Ns = 103 .
26
−1
10
−2
Mean Square Error (MSE)
10
−3
10
−4
10
−5
10
−6
10
−7
10
20
MOE (Ideal) MOE Proposed J=1 Proposed J=2 Proposed J=5 Proposed J=10 MCMOE 100
1000
1500
Number of Samples, Ns
Fig. 7.
The MSE versus the data sample size, Ns , for the proposed CFO estimator given input SNR=10 dB, 1 = 0.2 and K = 10.
Tsui-Tsai Lin received the B.S., M.S., and Ph.D. degrees from National Chiao Tung University, Hsinchu, Taiwan, in 1991, 1993, and 1997, respectively, all in communication engineering. From 1997 to 2000, he was in the Computer & Communications Research Laboratories, Industrial Technology Research Institute. In 2001, he joined the faculty of National United University, Miaoli, Taiwan, where he currently holds a position as Professor in the Department of Electronics Engineering. His current research interests include communications signal processing, smart antenna for wireless communications, and geo-location.
Fuh-Hisn Hwang received the B.S. and M.S. degrees, both in communication engineering, from National Chiao Tung University at Hsinchu, Taiwan in 1991 and 1993, respectively. He received the Ph.D. degree in electrical engineering, from National Central University at JhongLi, Taiwan in 2003. In 2004, he joined the faculty of the National Kaohsiung Normal University, Kaohsiung, Taiwan, where he is a professor of the department of optoelectronics and communication engineering. His research interest is in the area of wireless digital communication techniques, including coded modulation, diversity schemes for MIMO channels, multicarrier modulation and channel estimation.