Chapter 15

Multiuser CFOs Estimation in OFDMA Uplink Systems Yik-Chung Wu, Jianwu Chen, Tung-Sang Ng, and Erchin Serpedin

Contents 15.1 15.2 15.3 15.4 15.5

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 CFO Estimation in Subband-Based CAS Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 CFO Estimation in Interleaved CAS Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 CFO Estimation in Generalized CAS Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 15.5.1 ML Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 15.5.1.1 Special Case: Asymptotic Decoupled ML Estimator. . . . . . . . . . . . . . 404 15.5.2 Iterative Algorithms for ML Estimator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 15.5.3 ML Estimation with Importance Sampling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 15.5.3.1 Choice of Importance Function g (ω). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 15.6 Complexity Analyses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 15.7 Performance Comparisons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 15.8 Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414

15.1 Introduction Since orthogonal frequency-division multiple access (OFDMA) is widely regarded as a promising technique for broadband wireless networks, it has attracted a lot of attention recently [1–4]. In OFDMA, all users transmit their data to the base station (BS) simultaneously by modulating 397

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exclusive sets of orthogonal subcarriers; thus the receiver at the BS can separate each user’s signal easily in frequency domain. Like other OFDM-based systems, the performance of OFDMA systems is very sensitive to carrier frequency offsets (CFOs). In particular, the CFO estimation problem is very challenging in uplink transmissions since different users have different CFOs, and thus in addition to intercarrier interference (ICI), multiple access interference (MAI) is also introduced in the received signal. There are three different carrier assignment schemes (CASs) for OFDMA systems. Depending on the CAS employed, different algorithms for frequency estimation have been proposed for OFDMA uplink in the literature.  



For systems with subband-based CAS, bandpass filters are used to separate users before synchronization [5,6]. To maximize frequency diversity and increase the capacity of OFDMA systems, interleaved CAS is required. For this kind of OFDMA systems, a frequency estimation scheme exploiting the periodic structure of the signal transmitted by each user was proposed in Ref. [7]. Coping with the requirements for dynamic resource allocation and scheduling in future wireless systems, generalized CAS provides more flexibility than subband-based or interleaved schemes. For OFDMA systems with generalized CAS, a number of frequency estimation algorithms were proposed in Refs. [8–15]. Synchronization on one new user while assuming all existing users have already been synchronized was discussed in Ref. [8]. The more general case where all users need to be synchronized was also addressed [9–15], in which iterative techniques (e.g., alternating projection [9] and expectation Maximization [10–13]), importance sampling-based techniques [14] and asymptotic estimators [14,15] were proposed.

In this chapter, the details of the above algorithms will be reviewed and their relationships will be identified. Furthermore, their complexities and performances will be compared. The following notations are used throughout this chapter. Superscripts (·)−1 , (·)T , and (·)H denote the inverse, the transpose, and the conjugate transpose operations, respectively. The k × k identity matrix is denoted by Ik and the zero matrix with dimension k1 × k2 is denoted by 0k1 ×k2 . The operator diag{x} denotes a diagonal matrix with elements of x located on the main diagonal. Symbol x represents the L2 norm of the vector x and ∠x is the angle of x. Notation E{·} is the expectation of a random variable. Symbol vec(Z) is the vec operator applied to the matrix Z.

15.2 System Model In the considered OFDMA system as shown in Figure 15.1, K users transmit different data streams simultaneously using exclusive sets of subcarriers to the BS. Before initiating the transmission, the timing for each user is acquired by using the downlink synchronization channel from BS. Consequently, the transmissions from all users can be regarded as quasi-synchronous. The total number of subcarriers is denoted as N and one block of frequency-domain symbols sent by the kth user is denoted as dk = [dk (0), . . . , dk (N − 1)]T , where dk (i), i = 0, . . . , N − 1 is nonzero if and only if the ith subcarrier is modulated by the kth user. The frequency-domain data are first modulated onto different subcarriers by left multiplying an N -point inverse discrete Fourier transform√(IDFT) matrix FH , where F is the discrete Fourier transform (DFT) matrix with F(p.q) = (1/ N ) exp(−j2πpq/N ). After inserting a cyclic prefix (CP) of length Lcp into each block of the time domain signal (denoted as FH dk ), the augmented block is serially transmitted through the multipath channel. Let the channel impulse

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Transmitter of user k dk(0)

hk

dk(1)

IDFT

P/S

Add CP

Channel

dk(N–1)

exp(jωkm)

Receiver at base station

xk(m) x S/P

DFT

Remove CP

x2(m) n(m)

x1(m)

Figure 15.1 Baseband system diagram for OFDMA uplink. (From J. Chen, Y.-C.Wu, S. C. Chan, and T.-S. Ng, IEEE Trans. Veh. Technol., 57 (6), 3462–3470, 2008. © 2008 IEEE. With permission.)

response (including all transmit/receive filtering effects) between the kth user and the BS be denoted as ξk = [ξk (0), . . . , ξk (Lk − 1)]T , where Lk is the channel length. Denoting the timing offset caused by propagation delay as μk , the compound channel response can be written as T T 0T hk  [0T μk ×1 ξk (L−μk −Lk )×1 ] , where L is the upper bound on the compound channel length. For user k, the normalized CFO is denoted as εk . At the BS, after timing synchronization and removal of CP, the received signal component from user k is given by xk  [xk (0), . . . , xk (N − 1)]T = Γ(ωk )Ak hk ,

(15.1)

Γ(ωk )  diag{1, . . . , exp(j(N − 1)ωk )}, √ Ak  N FH Dk FL ,

(15.2)

where

Dk  diag{dk }, 2πεk , ωk  N

(15.3) (15.4) (15.5)

with FL denoting the first L columns of F. Since the received signal at the BS (denoted as x) is a superposition of the signals from all the users plus noise, we have K  Γ(ωk )Ak hk + n, (15.6) x= k=1

where n = [n(0), . . . , n(N − 1)]T is the complex white Gaussian noise vector with zero mean and covariance matrix Cn = E{nnH } = σ2 IN . In this chapter, only the case where the BS is equipped with one antenna is considered, but the above signal model can be easily generalized to the case where the BS is equipped with more than

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one antenna due to the fact that the multiple antennas equipped at the BS always share the same oscillator. Remark: Although there is an implicit assumption that the upper bound of the sum of channel delay spread and propagation delay for each user (L) is less than the length of CP (Lcp ), the considered system model is practical. The reason is that the timing offsets due to different propagation delays are limited to several samples only and in practical OFDM systems the CP is always longer than the exact channel order.

15.3 CFO Estimation in Subband-Based CAS Systems If subcarriers are assigned to users in nonoverlapping contiguous groups, bandpass filters can be used to separate different users before synchronization. Interference from neighboring groups can be mitigated by the use of guard band between blocks of subcarriers assigned to different users. If the frequency offset of each user is smaller than the guard band, then conventional single user synchronization algorithms [16,17] can be applied to the separated signal. The idea is illustrated in Figure 15.2, where the output of the bandpass filter for the kth user is given by rk = Γ(ωk )Ak hk + Ξk + ηk

(15.7)

with Ξk being the residual interference from other users to user k due to imperfect user separation, and ηk is the filtered noise. With rk , a number of conventional single user frequency estimation algorithms can be used. For example, in Ref. [5], the estimator exploiting CP correlation [16] is used: . / SNR k Φk (μ) , (15.8) μ ˆ k = arg max |γk (μ)| − μ 1 + SNR k 1 εˆ k = − ∠γk (μ ˆ k ), (15.9) 2π ...

Subcarriers assigned to user 2 user 1 ...

...

f r1

user K ...

CFO estimator

ε1

... f X

Bank of bandpass filters

...

Received signal

... f rk

CFO estimator

εk

^

Figure 15.2 User separations and parameter estimations for an OFDMA system with subbandbased CAS.

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where μ+Lcp −1

γk (μ) =



rk∗ (n)rk (n + N ),

(15.10)

n=μ

⎡ ⎤ μ+Lcp −1 1⎣  |rk (n)|2 + |rk (n + N )|2 ⎦ , Φk (μ) = 2 n=μ

(15.11)

with SNR k being the signal-to-noise ratio of user k [16], and rk (n) is the nth element of rk . Note that the above CP-based estimator was originally derived for the AWGN channel and considerable performance degradation is expected when applied to frequency-selective fading channel. To remedy this problem, other single user frequency estimators designed for frequency-selective fading channels can be applied instead. In Ref. [6], it was proposed that the estimator minimizing the received power at the virtual (null) subcarriers [17] of user k can be used: H 2 ω ˆ k = arg min WH −k Γ(ω) rk  , ω

(15.12)

where W−k consists of a subset of columns of FH that correspond to subcarriers that user k has not used. In fact, applications of other existing OFDM single user CFO estimators after signal separation are possible. Examples include the super-resolution subspace algorithm [18], cyclic-based estimator [19], and the algorithm exploiting the constant modulus property of data [20].

15.4 CFO Estimation in Interleaved CAS Systems Subband-based CAS facilitates user separation, and makes synchronization easy. However, frequency diversity in frequency-selective fading channel is not fully exploited in subband-based CAS. If the coherent bandwidth of the channel is large, several consecutive subcarriers may be subject to the same fading at the same time. A deep fading may destroy the whole subband, leading to total loss of data for a user. Interleaved CAS avoids the above problem by providing maximum separation among the subcarriers assigned to a single user. Unfortunately, by interleaving subcarriers from different users, synchronization becomes more complicated since signals from different users cannot be separated simply by a bank of bandpass filters. Without loss of generality, let N = MR where both M and R are integers. Further, assume that the subcarrier indices assigned to user k are {ik , ik + R, ik + 2R, . . . , ik + (M − 1)R} where ik is any integer within [0, R − 1]. Then, from Equation 15.1, the received samples from the user k can be written as M −1  dk (ik + mR)Hk (ik + mR)ej2π(ik +mR+εk )n/N , (15.13) xk (n) = m=0

where Hk = FL hk and Hk (n) is the nth element of Hk . Using Equation 15.13, it can be readily shown that (15.14) xk (n + νM ) = ej2πνθk xk (n), where ν is an integer and θk = (ik + εk )/R. This implies that the OFDMA time domain signal of user k has a periodic structure. Based on this periodic property, the received vector x in Equation 15.6

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can be re-expressed in an R × M matrix as Y = VS + Z,

(15.15)

where vec(Y) = x, vec(Z) = n, ⎡ ⎢ ⎢ V=⎢ ⎣

1 (1)

(2)

ej2πθ .. .

ej2πθ .. . (1)

⎡

··· ··· .. .

1

ej2π(R−1)θ

(2)

···

ej2π(R−1)θ

x1 (0) x1 (1) · · · ⎢ x2 (0) x2 (1) · · · ⎢ S=⎢ .. .. .. ⎣ . . . xK (0) xK (1) · · ·

x1 (M − 1) x2 (M − 1) .. .

⎤

1

⎥ ⎥ ⎥, ⎦

(K )

ej2πθ .. .

(15.16)

(K )

ej2π(R−1)θ ⎤ ⎥ ⎥ ⎥. ⎦

(15.17)

xK (M − 1)

In Ref. [7], a CFO estimator for interleaved CAS OFDMA systems was derived based on the equivalent system model in Equation 15.15. The idea of the estimator in Ref. [7] is as follows. First, the correlation matrix of Y is given by Ψ = E[YYH ] = V E[SSH ]VH + σ2 M IR .

(15.18)

On the other hand, when K < R, eigendecomposition on Ψ gives 2 H Ψ = Us Σs UH s + σ M Uz Uz ,

(15.19)

where Σs = diag{λ1 , λ2 , . . . , λK } contains the eigenvalues of the signal subspace, Us is composed of eigenvectors corresponding to λ1 , . . . , λK , and Uz is composed of eigenvectors corresponding to the noise subspace. Since Us and Uz are unitary and orthogonal to each other, multiplying Equations 15.18 and 15.19 by Uz from the right and subtracting the two resulting equations give V E[SSH ]VH Uz = 0R×(R−K ) .

(15.20)

Since V is a Vandermonde matrix, it is of full rank. Furthermore, assuming that signals from different users are independent, E[SSH ] is also of full rank. Therefore, we have VH Uz = 0K ×(R−K ) . Assume that the normalized CFO is in the range εk ∈ (−0.5, 0.5), the range of θk is ((ik − 0.5)/R, (ik + 0.5)/R). Since different users employ distinct values of ik , θk values for different users are in nonoverlapping ranges. Based on this fact and VH Uz = 0K ×(R−K ) , a CFO estimator was proposed in Ref. [7] as locating the largest K local maximums of 1 2 UH z a(θ)

,

(15.21)

where a(θ) = [1, ej2πθ , . . . , ej2π(R−1)θ ]T . As the ranges of θk for different k values are nonoverlapping, there will be one and only one θˆ from Equation 15.21 falling in the range ((ik − 0.5)/R,

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Multiuser CFOs Estimation in OFDMA Uplink Systems  403

(ik + 0.5)/R). Denoting that estimate as θˆ k , finally, the normalized CFO estimate for user k is given by (15.22) εˆ k = R θˆ k − ik . ˆ = YYH , and Uz is Note that, in practice, Ψ is approximated by its sample correlation matrix Ψ ˆ ˆ z obtained from the eigendecomposition of Ψ. approximated by U

15.5 CFO Estimation in Generalized CAS Systems Coping with the requirements of dynamic resource allocation and scheduling in future wireless systems, generalized CAS in OFDMA systems is very much desired. For generalized CAS, in addition to frequency diversity (also provided by interleaved CAS), multiuser diversity can also be obtained [4]. With the channel state information at the BS, users are assigned the best subchannels they experience, thus significant increases in throughput of OFDMA systems can be realized. Unfortunately, CFOs estimation in generalized CAS is the most challenging the three CAS schemes, as signals from different users cannot be separated easily (as opposed to subband-based CAS) and there is no fixed structure in the OFDMA signal (as opposed to interleaved CAS). In Ref. [8], a CFO estimation scheme was proposed for a new user entering the OFDMA system, with other existing users already synchronized. However, this scheme may not be general enough for a practical system. In the following, we will first present the maximum likelihood (ML) estimator which jointly estimates the CFOs of all users; then we will discuss several efficient methods for obtaining the optimal solution. While previously introduced estimators for subband-based and interleaved CASs do not require training, in the following, it is assumed that a training OFDM symbol is transmitted at the beginning of the data packet by each user, to aid the CFO estimations.

15.5.1 ML Estimator  Denoting h = h1T . . . hKT

T

, the signal model in Equation 15.6 can be rewritten as x = Q (ω)h + n,

(15.23)

where Q (ω) = [Γ(ω1 )A1 Γ(ω2 )A2 . . . Γ(ωK )AK ]. Based on the model in Equation 15.23, the ML estimate of parameters {h, ω} is given by maximizing [21] ˜ ω) ˜ = ψ(x; h,

. / 1 1 H ˜ ˜ ˜ h] [x − Q (ω) ˜ h] · exp − 2 [x − Q (ω) (πσ2 )N σ

(15.24)

or equivalently minimizing ˜ ˜ ω) ˜ H [x − Q (ω) ˜ h], ˜ = [x − Q (ω) ˜ h] Λ(x; h,

(15.25)

˜ are trial values of h and ω, respectively. Due to the linear dependence of parameter where h˜ and ω ˜ is fixed) is given by h in Equation 15.23, the ML estimate for the channel vector h (when ω ˜ (ω)) ˜ −1 Q H (ω)x. ˜ hˆ = (Q H (ω)Q

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(15.26)

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404  Orthogonal Frequency Division Multiple Access Fundamentals and Applications

Putting hˆ into Equation 15.25, the estimate of ω can be obtained as   2 ˜  PQ (ω)x ˜ ˆ = arg max φ (ω) , ω ˜ ω

(15.27)

H (ω)Q ˜ = Q (ω)(Q ˜ ˜ (ω)) ˜ −1 Q H (ω). ˜ where PQ (ω) A direct way maximizing Equation 15.27 is to use an exhaustive search over the multidimensional ˜ However, this approach is extremely computationally expensive and not suitable space spanned by ω. for implementation. In the next two subsections, several efficient methods for finding the maximum point in Equation 15.27 will be discussed. As a remark, the ML estimator in Ref. [8] can be regarded as a special case of Equation 15.27, in which only the entering user needs to be synchronized and all other existing users’ CFO and channel parameters are assumed to be known perfectly.

15.5.1.1 Special Case: Asymptotic Decoupled ML Estimator From the definition of Ak in Equation 15.3 and noting that for OFDMA systems, only one user is allowed to transmit on each subcarrier, it is straightforward to show that AiH Aj = 0L×L , ∀i = j. Thus, for the OFDMA system with a large number of subcarriers, we have limN →∞ AiH Γ(ωj − ωi )Aj /N = 0L×L , ∀i = j [22], and therefore, if N is large enough, we can approximate Q H (ω)Q (ω) with its block diagonal version: ⎡

A1H A1 .. .

⎢ Q H (ω)Q (ω) = ⎣ ⎡ ⎢ ≈⎣

AKH Γ(ω1 − ωK )A1 0 A1H A1 . . . .. . 0

⎤ . . . A1H Γ(ωK − ω1 )AK ⎥ .. .. ⎦ . . H ... AK AK ⎤

⎥ .. ⎦  B. . . H . . . A K AK ..

(15.28)

(15.29)

Using the approximation in Equation 15.29, the CFO estimates for all users in Equation 15.27 can be decoupled as (k = 1, . . . , K ):   ˜ k )Ak (AkH Ak )−1 AkH ΓH (ω ˜ k )x , ω ˆ k = arg max x H Γ(ω ω ˜k

(15.30)

and the solution can be found by K one-dimensional searches [14]. Note that Equations 15.29 and 15.30 hold only when the number of subcarriers is infinite. For a practical system with finite subcarriers, Equation 15.30 only offers approximate solutions to the original estimation problem. In the “Performance Comparisons” section, we will see that the decoupled estimator in Equation 15.30 suffers significant performance loss when the number of subcarriers is not large enough. Thus, efficient algorithms that can find the exact solution of Equation 15.27 with affordable complexity are needed. Recognizing that the approximation Q H (ω)Q (ω) ≈ B is too coarse and leads to significant performance loss, Sezginer and Bianchi [15] propose using a better approximation of Q H (ω)Q (ω). They write Q H (ω)Q (ω) = B + E, where E/N → 0KL×KL when N → ∞. Then, in general, (Q H (ω)Q (ω))−1 can be approximated by its Taylor series expansion truncated to order Ma , that

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 a m −1 m −1 is, (Q H (ω)Q (ω))−1 = (B + E)−1 ≈ M m=0 (−1) (B E) B . It was shown in Ref. [15] that with this approximation, the optimization problem (15.27) can be approximated as   ˜  (−1)Ma xH [Q (ω)B ˜ −1 Q (ω) ˜ H − IN ]Ma +1 x . ˆ = arg max J (Ma ) (ω) ω ˜ ω

(15.31)

Obviously, the larger the Ma , the better the approximation in Equation 15.31 with respect to Equation 15.27. Furthermore, when Ma = 0, Equation 15.31 reduces to the decoupled estimator in Equation 15.30. Note that although with better approximation, when Ma ≥ 1, the multiple CFOs in Equation 15.31 are coupled. Therefore, while computation of the inverse (Q H (ω)Q (ω))−1 can be avoided, we are still facing a multidimensional maximization problem. In Ref. [23], another decoupled estimator was proposed, not by asymptotic argument, but by constraining the minimum distance between any two active subcarriers from different users, such that the MAI can be controlled to a small value. Then the multidimensional optimization problem can be approximately transformed into K one dimensional optimization problems. However, this scheme requires careful placement of pilots of different users.

15.5.2 Iterative Algorithms for ML Estimator To reduce the computational burden brought by the multidimensional searches in the ML estimator, iterative methods for maximizing the likelihood function were proposed in Ref. [9] and Ref. [10]. In Ref. [9], the alternative projection algorithm is exploited. The alternative projection method reduces a K -dimensional maximization problem in Equation 15.27 into a series of one-dimensional maximization problems, by updating the CFO estimate of a single user at a time, while keeping the (i) other CFOs fixed at the previous estimated values. Let ω ˆ k the estimate of ωk at the ith iteration. Further, let (i+1) (i+1) (i) (i) ˆ (i) ˆ1 ...ω ˆ k−1 ω ˆ k+1 . . . ω ˆ K ]T . (15.32) ω −k = [ω (0)

Given the initial estimates {ω ˆ k }Kk=0 , the ith (i ≥ 1) iteration of the alternative projection algorithm for maximizing Equation 15.27 has the following form: For k = 1, . . . , K ,   (i) 2 ˆ (i−1) ˜ k, ω )x . (15.33) ω ˆ k = arg max PQ (ω −k ω ˜k

Multiple iterations on i are performed until the CFO estimates converge to a stable solution. For practical implementation, an equivalent but more computationally efficient scheme than Equation 15.33 was proposed in Ref. [9]. The idea is as follows. Observing that when updating (i) ˆ (i−1) ˜ k, ω ˜ k , thus there is no need ω ˆ k , most of the columns of Q(ω −k ) are fixed and not related to ω (i)

(i−1)

ˆ −1 ) is ˆ 1 , Q(ω ˜ 1, ω to recompute these columns for different ω ˜ k . For example, when estimating ω expressed as (i−1) (i−1) ˆ (i−1) ω ˜ 1 )A1 Γ(ω ˆ )A2 . . . Γ(ω ˆ K )AK ], (15.34) Q(ω ˜ 1, ω −1 ) = [Γ( 9 :; < 9 2 :; < Q 1 (ω ˜ 1)

(i−1)

ˆ −1 ) Q 2 (ω (i−1)

ˆ −1 ) can be considered fixed where Q1 (ω ˜ 1 ) is the submatrix that depends on ω ˜ 1 only, and Q2 (ω ˆ (i−1) ˜ k, ω ) can be decomposed into two parts: with respect to ω ˜ 1 . In general, for every k, Q(ω −k (i−1)

ˆ −k ) containing the columns ˜ k ) containing the columns related only to ω ˜ k and (2) Q2 (ω (1) Q1 (ω that are not related to ω ˜ k.

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(i−1)

ˆ −k ), it was shown in Ref. [9] that PQ (ω ˆ −k ) can With this special structure of Q(ω ˜ k, ω ˜ k, ω also be decomposed into two parts: (i−1)

(i−1)

(i−1)

ˆ −k ) = PQ2 (ω ˆ −k ) + PΠ (ω ˆ −k ), ˜ k, ω ˜ k, ω P Q (ω

(15.35)

ˆ −k ) = Q2 (ω ˆ −k )[QH ˆ −k )Q2 (ω ˆ −k )]−1 QH ˆ −k ), PQ2 (ω 2 (ω 2 (ω

(15.36)

where (i−1)

(i−1)

(i−1)

(i−1)

(i−1)

ˆ −k ) = Π(ω ˆ −k )[ΠH (ω ˆ −k )Π(ω ˆ −k )]−1 ΠH (ω ˆ −k ), (15.37) P Π (ω ˜ k, ω ˜ k, ω ˜ k, ω ˜ k, ω ˜ k, ω (i−1)

(i−1)

(i−1)

(i−1)

(i−1)

(i−1)

ˆ −k ) = (IN − PQ2 (ω ˆ −k ))Q1 (ω Π(ω ˜ k, ω ˜ k ).

(i−1)

(15.38)

(i−1)

ˆ −k ) is independent of ω Since PQ2 (ω ˜ k , Equation 15.33 can be rewritten as   (i) 2 ˆ (i−1) ˜ k, ω )x . ω ˆ k = arg max PΠ (ω −k ω ˜k

(15.39)

(i−1)

ˆ −k ) involves only an L × L matrix inversion, as opposed ˜ k, ω Note that the computation of PΠ (ω (i−1)

ˆ −k ). Therefore, computational complexity is ˜ k, ω to the the KL × KL matrix inversion in PQ (ω saved in Equation 15.39 with respect to Equation 15.33. An additional scheme that further reduces the complexity of Equation 15.39 was also proposed in Ref. [9] by approximating the inversion −1 using truncated Taylor series expansion. ˆ (i−1) ˆ (i−1) ˜ k, ω ˜ k, ω [ΠH (ω −k )Π(ω −k )] Apart from the alternating projection method, another iterative method for CFOs estimation was proposed based on space alternative generalized expectation-maximization (SAGE) algorithm [10,11], applied directly to the system model in Equation 15.6. Similar to alternative projection algorithm, SAGE algorithm also reduces the joint K -user multiple parameters estimation problem (i) into a series of single user parameter estimation problems. Denoting hˆ k as the estimated channel for the kth user at the ith iteration, the SAGE algorithm proceeds as follows. For k = 1, . . . , K , E-step: Compute an estimate of the kth user’s signal

(i)

xˆ k = x −

k−1 

ˆ (i) Γ(ω ˆ (i) m )Am hm −

m=1

K 

(i−1) Γ(ω ˆ (i−1) )Am hˆ m m

(15.40)

m=k+1

 with the notation ba is zero when b < a. M-step: Compute an estimate of the CFO and channel for the user k   (i) (i) (i) (ω ˆ k , hˆ k ) = arg min ˆxk − Γ(ω ˜ k )Ak h˜ k 2 . ω ˜ k ,h˜ k

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(15.41)

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Using a similar derivation as the joint ML estimator in Section 15.5.1, the solution to the above minimization problem is given by   (i) (i) (i) ˜ k )Ak (AkH Ak )−1 AkH ΓH (ω ˜ k )ˆxk , ω ˆ k = arg max (ˆxk )H Γ(ω

(15.42)

(i) (i) (i) ˆ k )ˆxk . hˆ k = (AkH Ak )−1 AkH ΓH (ω

(15.43)

ω ˜k

Intuitively, the E-step is the MAI cancellation process, and the M-step is the estimation of CFO and channel for the kth user with the MAI-canceled signal. If the CFO and channel estimates of other users are accurate, then Equation 15.41 is in fact the ML estimator of CFO and channel for the kth user. Therefore, the SAGE algorithm can be viewed as a recursive approximation to the joint ML estimator in Equation 15.27. Interestingly, the estimation of CFO in Equation 15.42 is in the same form as the decoupled estimator in Equation 15.30 under asymptotic assumption. The only difference is that Equation 15.42 uses the MAI-reduced signal for estimation, while Equation 15.30 uses the original received data. Variants of the above SAGE algorithms have also been proposed in Refs. [12,13]. Realizing that the CAS for the preamble and for the data symbols are not necessary to be the same, Fu et al. [12] propose a modification of SAGE by using an interleaved CAS at the preamble and incorporating the MAI cancellation in both time and frequency domains. The resulting algorithm is shown to have better convergence rate, lower complexity, and better robustness against number of users than the conventional SAGE. On the other hand, in Ref. [13], joint frequency synchronization and data detection based on EM-type algorithms are demonstrated. Although the alternating projection and SAGE algorithms appear differently, their essences are actually the same. In particular, both algorithms are recursive and estimating the CFO of one user at a time. Also, both algorithms take care of the effect of MAI on the estimation process using the previous estimates of unknown parameters. Further discussion on the relationship between alternative projection and EM algorithms is given in Ref. [24]. (0) In general, for iterative type algorithms, an initial estimate of {ω ˆ k }Kk=0 close to the optimal solution is required in order to avoid the algorithms locking into a local maximum. A simple method (0) for obtaining the initial estimate is by setting ω ˆ k = 0, since ωk is typically a zero-mean random variable. Other possible initialization methods include the single user algorithm in Ref. [8] and the asymptotic decoupled ML estimator in Equation 15.30. However, note that no matter what initialization method is used, there is still no guarantee that an estimate obtained iteratively will be the global maximum. In the next subsection, we will review a computational method that guarantees is to obtain the global optimum of Equation 15.27 without the need of an initial estimate.

15.5.3 ML Estimation with Importance Sampling In Ref. [25], Pincus showed that it is possible to obtain a closed-form solution for Equation 15.27 that guarantees to be the global optimum. Based on the theorem given by Pincus, the ω that yields ˜ is given by the global maximum of φ (ω) 0 ω ˆ k = lim

ρ1 →∞

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0 ··· ωk exp(ρ1 φ (ω)) dω 0 0 , ··· exp(ρ1 φ (ω)) dω

k = 1, . . . , K

(15.44)

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408  Orthogonal Frequency Division Multiple Access Fundamentals and Applications

where ρ1 is a design parameter. If we denote φ(ω) = exp(ρ1 φ (ω)), the normalized version of φ(ω) can be obtained as ¯ φ(ω) =0

φ(ω) 0 . ··· φ(ω) dω

(15.45)

¯ Here, the function φ(ω) has all the properties of a probability density function (PDF), so it is referred to as the pseudo-PDF in ω. With this definition and Equation 15.44, the optimal solution of ω for Equation 15.27 is  ω ˆk =

 ···

¯ dω, ωk φ(ω)

k = 1, . . . , K

(15.46)

for some large value of ρ1 . Note that the multiple integral in Equation 15.46 is of the same form as ¯ Suppose we can generate realizations of ω according to the expectation of ωk with respect to φ(ω). ¯ φ(ω), the value of the integral in Equation 45.46 can be found by the Monte Carlo approximation as [26] T 1  i ωk , k = 1, . . . , K , (15.47) ω ˆk ≈ T i=1

where T is the number of realizations and ωik is the ith realization of ωk generated according ¯ ¯ to the pseudo-PDF φ(ω). Unfortunately, generating samples from φ(ω) is difficult since it is a multidimensional PDF. In Ref. [27], importance sampling is used to compute the multidimensional in Equa0 integral 0 ¯ tion 15.46. This approach is based on the observation that integrals of the type ··· h(ω)φ(ω) dω can be expressed as 

 ···

¯ h(ω)φ(ω) dω =

where g¯ (ω) = 0



 ···

h(ω)

¯ φ(ω) g¯ (ω) dω, g¯ (ω)

g (ω) 0 ··· g (ω) dω

(15.48)

(15.49)

with g (ω) > 0. The function g (ω) is called the importance function and its normalized version g¯ (ω) has all the properties of a PDF. Then, the right-hand side of Equation 15.48 can be expressed ¯ as the expected value of h(ω)(φ(ω)/¯ g (ω)) with respect to the pseudo-PDF g¯ (ω). Supposing that generation of samples ωi according to g¯ (ω) is relatively easy, the value of the integral in Equation 15.48 can be approximated as 

 ···

¯ h(ω)φ(ω) dω ≈

T ¯ i) φ(ω 1  h(ωi ) , T g¯ (ωi )

(15.50)

i=1

where T is the number of realizations. Now recall the estimate of ωk in Equation 15.46. By the importance sampling technique discussed above, and the fact that the frequency offset ωk has the properties of a circular random

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variable, it was shown that Equation 15.46 can be approximated as [14] ω ˆk ≈

T ¯ i) 1 1  φ(ω exp(j2πωik ), ∠ 2π T g¯ (ωi )

k = 1, . . . , K ,

(15.51)

i=1

where samples ωi are generated according to g¯ (ω). Note that ω ˆ k is only related to the angle of the complex value in Equation 15.51, the equivalent but simplified estimator is given by ω ˆk =

T 1 1  φ(ωi ) ∠ exp(j2πωik ), 2π T g (ωi )

k = 1, . . . , K

(15.52)

i=1

¯ in which the integrals needed for computing φ(ω) and g¯ (ω) can be avoided.

15.5.3.1 Choice of Importance Function g(ω) In general, the estimator in Equation 15.52 converges to Equation 15.44 by the Strong Law of Large Number regardless of the choice of the function g (ω). However, there are obviously some choices of g (ω) that are better than others in terms of computational complexity. For the problem under consideration, it was shown in Ref. [14] that the optimal choice of g (ω), which minimizes the variance of the estimator for a fixed number of realizations T , is proportional to φ(ω). However, this choice is not practical, since if we can generate samples from φ(ω), we do not need the importance function g (ω) to facilitate samples generation. In practice, the importance function g (ω) should be chosen according to the following two criteria [28]: 1. g (ω) should be a close approximation of φ(ω), 2. g (ω) should be as simple as possible to facilitate sample generation. From the asymptotic analysis in Equation 15.29, if we choose the importance function g (ω) as g (ω) = exp(ρ2 x H Q (ω)B−1 Q H (ω)x), 8 7K    H H −1 H H ρ2 x Γ(ωk )Ak (Ak Ak ) Ak Γ (ωk ) x , = exp

(15.53) (15.54)

k=1

=

K 3 k=1

exp(ρ2 x H Γ(ωk )Ak (AkH Ak )−1 AkH ΓH (ωk )x), 9 :;