### Maximum Likelihood Timing and Carrier Frequency

### Offset Estimation for OFDM Systems

### With Periodic Preambles

*Hung-Tao Hsieh, Student Member, IEEE, and Wen-Rong Wu, Member, IEEE*

**Abstract—Symbol timing offset (STO) and carrier frequency**

**offset (CFO) estimation are two main synchronization **
**opera-tions in packet-based orthogonal frequency division multiplexing**
**(OFDM) systems. To facilitate these operations, a periodic **
**pream-ble is often placed at the beginning of a packet. CFO estimation**
**has been extensively studied for the case of two-period preambles.**
**In some applications, however, a preamble with more than two **
**pe-riods is available. A typical example is the IEEE802.11a/g wireless**
**local area network system, which features a ten-period preamble.**
**Recently, researchers have proposed a maximum likelihood (ML)**
**CFO estimation method for such systems. This approach first**
**estimates the received preamble using a least squares method and**
**then maximizes the corresponding likelihood function. In **
**addi-tion to the standard calculaaddi-tions, this method requires an extra**
**procedure to solve the roots of a polynomial function, which is**
**disadvantageous for real-world implementations. In this paper, we**
**propose a new ML method to solve the likelihood function directly**
**and thereby perform CFO estimation. Our method can obtain a**
**closed-form ML solution, without the need for the root-finding**
**step. We further extend the proposed method to address the**
**STO estimation problem as well as derive a lower bound on**
**the estimation performance. Our simulations show that while the**
**performance of the proposed method is either equal to or better**
**than the existing method, the computational complexity is lower.**

**Index****Terms—Frequency****offset,** **maximum** **likelihood**
**(ML), orthogonal frequency division multiplexing (OFDM),**
**synchronization.**

I. INTRODUCTION

**O**

RTHOGONAL frequency division multiplexing (OFDM)
is known as an efficient modulation technique [21], [22].
However, the performance of OFDM systems is sensitive to
both symbol timing offset (STO) [19], [20] and carrier
fre-quency offset (CFO). STO will reduce the effective cyclic prefix
(CP) length and induce intersymbol interference, while CFO
will damage the orthogonality among subcarriers and thereby
induce intercarrier interference. For typical OFDM receivers,
STO and CFO have to be estimated and compensated before
data detection can be conducted.
Manuscript received April 23, 2008; revised October 25, 2008, January 7, 2009, and March 4, 2009. First published April 3, 2009; current version published October 2, 2009. The review of this paper was coordinated by Dr. C. Cozzo.

The authors are with the Department of Communication Engineering, National Chiao Tung University, Hsinchu 300, Taiwan (e-mail: dow.cm93g@ nctu.edu.tw; wrwu@faculty.nctu.edu.tw).

Digital Object Identifier 10.1109/TVT.2009.2019820

Estimation methods for CFO and STO in OFDM systems can be classified into the following two categories: 1) data aided and 2) blind. The former is more suitable for packet-based transmission, while the latter is appropriate for continuous transmission such as broadcasting. Blind methods exploit the periodic structure of CPs to accomplish the estimation task [1]–[7]. Data-aided methods insert a known preamble, or pilot symbol, in front of each data packet such that it can easily be used by the receiver to achieve synchronization [9]–[18]. In this paper, we consider only the data-aided method.

CFO estimation usually consists of a fractional part and
an integer part. Most researchers focus on how to estimate
the fractional part, which is also the focus of this paper. For
integer part estimation, see [9] and [10]. It has been shown
that the performance of OFDM systems is greatly affected
by CFO [30], and an accurate CFO estimation is required
for real-world applications. A maximum likelihood (ML) CFO
estimator using a preamble with two identical pilot symbols was
first proposed in [11]. Using the same periodic preamble and
taking null subcarriers into consideration, Huang and Letaief
[12] propose a method that is able to estimate both fractional
and integer CFOs. To avoid the extra overhead required in [12],
Schmidl and Cox [13] introduce a preamble composed of two
OFDM symbols: The first one has two identical periods (to
estimate the fractional CFO and STO), and the second one has
a special correlation with the first one (to estimate the integer
CFO). To improve the performance, Morelli and Mengali [14]
extend this area of research to treat preambles with periodicities
of greater than two. Using the approach in [14], one can remove
the second pilot symbol as required in [13]. As an improved
*version, Minn et al. [15] propose a CFO estimation based on*
the best linear unbiased estimation principle. Note that Morelli
*and Mengali [14] and Minn et al. [15] still use the same STO*
estimator as that in [13]. When the number of periods is greater
than two, the method in [11] is no longer optimal. An ML
CFO estimator for this problem was proposed in [16]. However,
the required computational complexity is high. To alleviate this
problem, a low-complexity approach was then proposed in [17].
Another simplified algorithm was also proposed in [18].
However, due to excessive approximation in the likelihood
function, the performance of the CFO estimation in [18] does
not approach the Cramér–Rao lower bound (CRLB) [25].

In this paper, we focus on CFO and STO estimation in the OFDM system with a periodic preamble. Specifically, we consider a preamble with more than two periods. The ML CFO

estimation for the system has been considered in [17]. The method in [17] is essentially a two-step approach: it first es-timates the received preamble with a least squares (LS) method and then maximizes the corresponding likelihood function. In addition to regular computations, this method requires an extra procedure to solve for the roots of the derivative of the like-lihood function. Thus, its computational complexity is higher, and the cost for real-world implementations is also increased.

In this paper, we develop a new ML method that solves the likelihood function directly for the CFO-estimation problem. Our method generates a closed-form ML solution, and the root-finding procedure is not required. As a result, the computational complexity and the implementation cost are lower than those in [17], while the performance of the proposed method is either equal to or better than that in [17]. The proposed method is further extended to STO estimation, and a theoretical lower performance bound is derived. Note that the performance bound for STO estimation has not previously been addressed in the literature. This paper is organized as follows. In Section II, the CFO-estimation method in [17] is briefly reviewed. The proposed CFO- and STO-estimation procedures are described in Sections III and IV. A lower bound on STO estimation performance is presented in Section V. Our simulation results are reported and discussed in Section VI. Our conclusions are presented in Section VII.

II. EXISTINGAPPROACH

In this section, we briefly review the algorithm proposed in
[17]. Let the preamble in the OFDM system be periodic with
*period N and length QN . Denote the preamble signal as s(k),*
*where k = 0, 1, . . . , QN− 1. The preamble is placed at the*
beginning of a packet and is subsequently transmitted through
*a wireless channel. Denote the channel response as h(k) and*
*the output signal as x(k). Then, we have x(k) = s(k)∗ h(k),*
where *∗ denotes the convolution operation. Assume that the*
*maximum channel delay is N . Then, we can discard the first*
*received N samples and retain the periodic property of the*
*preamble x(k). Thus, the received preamble can be expressed*
as [1]

*y(k) = ej2πεkN* _{x(k) + w(k)}_{(1)}

*where k = N, N + 1, . . . , QN− 1, ε is CFO, and w(k) *
*repre-sents additive white Gaussian noise with a variance of σ _{w}*2. We

*can perform an index transformation by letting k = mN + n,*

*where m = 1, . . . , Q and n = 0, . . . , N− 1 such that x(k) =*

*x(mN + n). For notational simplicity, we further let x _{m}(n) =*

*x(mN + n), denoting the nth sample of the mth period of*
*x(k). Due to periodicity, we have x _{p}(n) = x_{q}(n) for p, q∈*

*{1, . . . , Q}. Similarly, we can define ym(n) = y(mN + n) =*

*y(k), and w _{m}(n) = w(mN + n) = w(k). Let K = Q−1, and*

* y(n) = [ y*1

*(n)*

*y*2

*(n)*

*· · · yK(n) ]T*

* x(n) = [ x*1

*(n)*

*x*2

*(n)*

*· · · xK(n) ]T*

* w(n) = [ w*1

*(n)*

*w*2

*(n)*

*· · · wK(n) ]T.*(2)

In addition, we define four matrices as follows:

**Y = [ y(0) y(1) · · · y(N − 1) ]**

**X =****x(0) x(1)e**j2πεN**· · · x(N − 1)e**j2πε(N−1)N**W = [ w(0) w(1) · · · w(N − 1) ]****A =**
⎡
⎢
⎢
⎣
*ej2πε* _{0} _{· · ·}_{0}
0 *ej2πε·2* _{· · ·}_{0}
..
. ... . .. ...
0 0 *· · · ej2πε·K*
⎤
⎥
⎥
*⎦ .* (3)

The received preamble in (1) can then be rewritten as

* Y = AX + W.* (4)

The method in [17] uses a two-step approach: it first estimates

**X using an LS method and then estimates CFO by maximizing**

the likelihood function. Since the noise is a Gaussian
random variable, **y(n) is a Gaussian random vector with***a covariance matrix of σ*2_{w}**I, where I denotes the identity**
matrix. For a given**A, the LS estimate of X can be expressed**
as **X***LS= (1/K)***A***H Y ≡ A*+

*denotes the Hermitian operation. Substituting*

**Y, where (·)**H**X**

*LS*back into (4), we can obtain the log-likelihood function as Λ(

**A) =**

_{N}*n=1 y(n) − AA*+

*2*

**y(n)***= N*+)

**· trace((I − AA****R**

*Y*), where

**R**

_{Y}*= E[*

**YY**

*H*

_{]. The (p, q)th entry of}

_{R}*Y* is
*(1/N )N −1 _{n=0}yp(n)yq∗(n), p, q∈ [1, K] [28]. The desired CFO*
estimation can then be derived as

ˆ
*ε = arg*
min
*ε* trace
(* I − AA*+)

**R**

*Y*= arg

*{max*

*ε*

**a**

*H*

_{R}*Y*(5)

**a}**where **a is a vector consisting of the diagonal elements of A.**
It was shown in [26] that

**a***H*_{R}*Y***a =**

*K−1*
*m=−(K−1)*

*b(m)ej2πmε* (6)

*where b(m) = _{q−p=m}(1/N )N −1_{n=0}yp(n)yq∗(n). Taking the*

*derivative of (6) with respect to ε and letting the result be zero,*we obtain

*K−1*

_{}

*m=1*

*mb(m)zm*=

*K−1*

_{}

*m=1*

*mb(−m)z−m*

_{(7)}

*where z = ej2πε*_{. Equation (7) can be rewritten as}

Im
_{K−1}*m=1*
*mb(m)zm*
= 0 (8)

where Im(*·) is an operator that isolates the imaginary part of*
a scalar value. Denote the set containing the roots of (8) by Ω.
The CFO can then be estimated as follows [17]:

ˆ

*ε =* 1

*j2π*ln(ˆ*z)* (9)

The procedure for CFO estimation in [17] can now be summarized as follows.

1) Construct the correlation matrix**R***Y*.
2) Calculate the coefficient of (8) using**R***Y*.
3) Find the nonzero roots of (8).

4) Substitute the roots into (6), find the maximum root, and
calculate ˆ*ε using (9).*

As we see, (8) requires a root-finding operation. Thus, a set of suboptimum algorithms to address this issue was proposed in [17]. Unfortunately, these suboptimum methods cannot effectively reduce the computational complexity while still maintaining good performance.

III. PROPOSEDML CFO ESTIMATION

In this section, we develop a new CFO estimation method
that solves the likelihood function directly. The signal model
we use is the same as that in (4). We assume that each
data packet is transmitted through a slow-fading channel
*with an impulse response of h(k), k = 0, . . . , L− 1. Here,*
*the h(k)s have Rayleigh distributions, and they are *
statisti-cally independent. Note that the time-domain preamble
sig-nal is obtained from the discrete Fourier transform of the
frequency-domain preamble signal, and the frequency-domain
preamble signal is generally a white sequence. From the
central limit theorem, the time-domain preamble signal can
then be approximated as a white Gaussian sequence. Thus,
*the channel output x(k), which equals* *L−1 _{l=0}*

*h(l)s(k− l),*

*and the received preamble y(k) in (1) can be *
approxi-mated as Gaussian sequences. Let the variance of the
*time-domain preamble signal, i.e., s(k) be σ _{s}*2. Then, the variance

*of x(n) equals σ*2

_{x}*= E{L−1*

_{j=0}L−1_{l=0}*h(j)s(k− j)h(l)∗s(k−*

*l)∗} = σ*2

_{s}*L−1*

_{l=0}*|h(l)|*2

*= σ*2

*2*

_{s}σ_{h}*, and that of y(k) equals σ*2

*+*

_{x}*σ _{w}*2

*. Note that s(k) can be a psuedonoise sequence. In such a*

*case, σ*2

_{s}*indicates the averaged preamble power of s(k).*

*Let f (·) be a probability density function. Then, we explicitly*
*write out the log-likelihood function of ε as follows [1]:*

*Λ(ε) = ln*
⎧
⎨
⎩
*n∈ ˜I*
*f ( y(n))*
⎫
⎬
⎭
= ln
⎧
⎪
⎨
⎪
⎩

*n∈ ˜I*

*f (*

**y(n))***m∈[1,K]*

*n∈ ˜I*

*f (y*

_{m}(n))*·*

*m∈[1,K]*

*n∈ ˜I*

*f (y*⎫ ⎪ ⎬ ⎪ ⎭ = ln ⎧ ⎨ ⎩

_{m}(n))*n∈ ˜I*

*f (*

**y(n))***f (y*

_{1}

*(n))· · · f (y*

_{K}(n))*·*

*m∈[1,K]*

*n∈ ˜I*

*f (y*⎫ ⎬ ⎭

_{m}(n))*.*(10)

It is clear that the last term in (10), i.e., _{m∈[1,K],n∈ ˜}_{I}×

*f (ym(n)), is independent of ε [1]. As a result, this term can*
be dropped. Let

**u(n) = e**j2πεnN_{[ x}_{1}_{(n)e}j2πε*· · · x _{K}_{(n)e}j2πε·K*

_{]}

*T*

_{.}_{(11)}We then rewrite (4) as

**Y = U + W, where U = AX =**[

*Define*

**u(0), u(1), . . . , u(N − 1)]. Then, y(n) = u(n) + w(n).****R**

*u= E[*

**u(n)u**H(n)] and**R**

*y= E[*we have

**y(n)y**H(n)]. Then,**R**

*u= σ*2

*x*⎡ ⎢ ⎢ ⎣ 1

*e−j2πε*

*· · · e−j2π(K−1)ε*

*ej2πε*

_{1}

*.. . ... . .. ...*

_{· · · e}−j2π(K−2)ε*ej2π(K−1)ε*

_{e}j2π(K−2)ε

_{· · ·}_{1}⎤ ⎥ ⎥ ⎦ (12)

and**R***y*=**R***u+ σw*2**I, where I is an identical matrix. Thus, we**
*can express f ( y(n)) as [23], [24]*

*f ( y(n)) =*

*πK*det(

**R**

*)*

_{y}*−1*exp

**−y(n)**H**R**

*−1*

_{y}

**y(n)***.*(13) According to the matrix inversion lemma [8], we derive the inverse of

**R**

*y*as

**R***−1*

*y* *= σw−2 I −*

*σ−4 _{w}*

**R**

*u*

*1 + σ _{w}−2E{uHu}.* (14)

*Note that for n∈ ˜I, we have*

*E**yp(n)y∗q(n)*
=
*σ _{x}*2

*+ σ*2

_{w}*,*

*if q− p = 0*

*σ*2

_{x}*e−j2πε(q−p),*

*if q− p = 0*(15)

*where p, q*

**∈ [1, K]. As a result, R**−1_{y}*= σ*4 +

_{w}−2**I − [R**u/(σw*Kσ*2

*2*

_{w}σ*)], and*

_{x}*f (y*exp

_{p}(n)) =*−yp(n)y∗p(n)*

*σ*2

*x+σw*2

*π (σ*2

_{x}*+ σ*2

*) (16)*

_{w}*where p∈ [1, K]. Thus, the exponential term in (13) becomes*

**y(n)**H_{R}*−1*
*y* **y(n)= σ**w−2*K*
*p=1*
*yp(n)yp∗(n)*
*−C*0
*K*
*p=1*
*K*
*q=1*
*yp(n)y∗q(n)ej2π(q−p)ε*
=*σ−2 _{w}−C*

_{0}

*K*

*p=1*

*y*

_{p}(n)y∗_{p}(n)*−2C*0Re

*K−1*

*p=1*

*K*

*q>p*

*y*! (17)

_{p}(n)y_{q}∗(n)ej2π(q−p)ε*where C*_{0}*= σ*2* _{x}/(σ_{w}*4

*+ Kσ*2

_{w}*σ*2

*), and Re*

_{x}*{·} denotes the*opera-tion that isolates the real part of the indicated complex variable.

Dropping the superfluous terms and substituting (13)–(17) into (10), we finally express the log-likelihood function as

*Λ(ε) =*
*N −1*_{}
*n=0*
ln
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
*σ _{x}*2

*+ σ*2

_{w}*K*exp

**−y(n)**H

_{R}*−1*

*y*

*det(*

**y(n)****R**

*) exp ⎡ ⎢ ⎢*

_{y}*⎣−*

*K*

*p=1*

*yp(n)y∗p(n)*

*σ*2

*x+σ*2

*w*⎤ ⎥ ⎥ ⎦ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (18)

*= C*1

*+ C*2

*φ + C*3

*K−1*

_{}

*p=1*

*K*

*q>p*

*|γpq| cos(ψpq*) (19) where

*γ*=

_{pq}*N −1*

_{}

*n=0*

*y*

_{p}(n)y∗_{q}(n)*(q≥ p and p ≥ 1)*(20)

*ψpq= 2πε(q− p) + ∠γpq,*

*φ =*

*K*

*p=1*

*γ*(21)

_{pp}*C*

_{1}

*= N· ln*

*σ*2

_{x}*+ σ*2

_{w}*K*det(

**R**

*y*) (22)

*C*

_{2}= (1

*− K)*

*ρ*2

*σ*2

_{w}*(1 + (K− 1)ρ)*(23)

*C*3=

*2C*2 (1

*− K)ρ*(24)

*ρ =*

*σ*2

*x*

*σ*2

_{x}*+ σ*2

_{w}*.*(25)

*Note that φ is the received signal energy and that det(***R***y*) is
*a constant, independent of ε. The detailed derivation of (19) is*
provided in Appendix A. Ignoring unrelated terms, we obtain
the log-likelihood as
*Λ(ε)∝*
*K−1*
*p=1*
*K*
*q>p*
*|γpq| cos(ψpq).* (26)
*To maximize the function, we first take a derivative of Λ(ε) with*
*respect to ε and obtain*

*∂*
*∂εΛ(ε) =−*
*K−1*
*p=1*
*K*
*q>p*
*2π(q− p)|γ _{pq}| sin(ψ_{pq}).* (27)

Thus, we have an alternative expression to that in (8). Now, the problem is how to solve (27). Since (27) involves a non-linear sine function, a closed-form solution will be difficult to

calculate. Here, we use a simple approximation method to overcome the problem. Using (20) and (1), we obtain

*γpq= ej2πε(p−q)*
*N −1*_{}
*n=0*
*|x*1*(n)|*2+
*N −1*_{}
*n=0*
*wp(n)w∗q(n)*
*+ ej2πε(pN +p)*
*N −1*
*n=0*
*x*_{1}*(n)w∗ _{p}(n)*

*+ ej2πε(pN −q)*

*N −1*

*n=0*

*x∗*

_{1}

*(n)wq(n).*(28)

*In (28), we have used the periodic property that x*_{1}*(n) =*

*x _{p}(n) = x_{q}(n). Now, if the noise level is low, the noise related*
terms in (28) can be ignored. We then have

*∠γpq≈ 2πε(p − q).* (29)

From (29), we write

*ψ _{pq}≈ 2πε(q − p) + 2πε(p − q) = 0.* (30)

*From (30), we can then assume that sin(ψpq*)

*≈ ψpq*and ap-proximate the expression in (27) by

*∂*
*∂εΛ(ε) −*
*K−1*
*p=1*
*K*
*q>p*
*2π(q− p)|γ _{pq}|(ψ_{pq}).* (31)
Setting the result in (31) to zero, we can estimate CFO as

ˆ
*ε =−*
*K−1*
*p=1*
*K*
*q>p|γpq|(q − p)∠γpq*
*2πK−1*
*p=1*
*K*
*q>p*
*|q − p|*2_{|γ}_{pq}_{|}*.* (32)

Note that the approximation in (30) will become exact if noise
*is not present and if ε is the true CFO. In other words, (27)*
and (31) will have the same zero-crossing point although the
two functions are different, indicating that (31) and (27) will
yield the same optimum solution. If noise is present, however,
(31) and (27) will not have the same optimum solution. The
accuracy of the solution in (31) depends on the signal-to-noise
ratio (SNR) in (28). We define the SNR in (28) as SNR*γ* and
*that in (1) as SNR. Then, SNR = σ*2* _{x}/σ_{w}*2, as typically defined.
From (28), it is simple to see that

SNR*γ* =

*N*2*σ*4_{x}

*N σ _{w}*4

*+ 2N σ*2

_{x}*σ*2 =

_{w}*N· SNR*2

1 + 2SNR*.* (33)

From (33), we can see that SNR*γ*can be much larger than SNR
*as long as N is reasonably large and SNR is not very low.*
Subsequently, the approximation in (31) will introduce only a
*small error for a wide SNR range. As a simple example, let N =*
16 and SNR = 0 dB. From (33), we obtain SNR*γ* *= 7.27 dB,*
which is much higher than SNR.

Note that the proposed estimate requires that we extract
*the phase from γpq*. It is simple to see that the result is only
unambiguous when *|∠γ _{pq}| < π. For a particular combination*

*of p and q, the estimation range for CFO is|ε| ≤ 1/[2(q − p)].*

TABLE I

COMPUTATIONALCOMPLEXITYCOMPARISON FOR THEALGORITHM IN[17]AND FOR THEPROPOSEDALGORITHMS

*Since the maximum value for q− p is K − 1, the estimation*
range for CFO is *|ε| ≤ 1/[2(K − 1)]. When K is large, the*
range becomes small. In the following, we propose a method
to remedy this problem. The basic idea is to apply the
phase-unwrapping procedure. We first calculate the phase angle for
*each γ _{pq}. Then, for each p, we calculate the phase difference of*

*∠γpq, q = p + 1, p + 2, . . . , K. Let dr,s*denote the phase

*dif-ference, i.e., d*=

_{r,s}*∠γ*

_{rs}− ∠γ_{r(s−1)}, r = 1, 2, . . . , K− 2 and*s = r + 2, r + 3, . . . , K. Since the maximum value of|d _{r,s}| is*

*π, whenever|dr,s| > π, the phase need to be unwrapped. This*can be performed with the following operation:

*d _{r,s}*=

*dr,s− 2π if dr,s> π*

*dr,s+ 2π* *if dr,s<−π.* (34)
*For a value of r, the dr,s*values should have the same signs. We
can use this property to further correct occasional errors. Let

*g be the sum of all dr,s* *values, i.e., g =*
_{K−2}

*r=1*
_{K}

*s=r+2dr,s*.
*Then, we use the sign of g to determine the sign of dr,s*and to
evaluate*∠γpk, k = p + 1, . . . , K. Finally, the unwrapped∠γpq*
*can be written (with q≥ p + 2) as*

*∠γpq*=*∠γp(p+1)*+
*q*
*s=p+2*

*dp,s.* (35)

Substituting (35) into (32), we can estimate CFO. Using our
proposed procedure, the CFO estimation range can be greatly
extended up to*|ε| < 1/2.*

Now, the procedure for our proposed ML CFO estimation can be summarized as follows.

*1) Construct all γ _{pq}’s, where p∈ [1, K − 1] and q ∈ [p +*

*1, K], and calculate their amplitude.*

2) Use the phase unwrapping scheme to estimate the phase
*of γ _{pq}*.

3) Substitute the results into (32), and calculate the ML estimate.

Clearly, the proposed estimate does not require the root-finding procedure, and this, in turn, effectively reduces the computa-tional complexity. Step 1) above is similar to the calculation of

**R in Section II. However, our method is easier since we only**

*have to compute γpqfor q > p.*

In this paragraph, we compare the computational complexity of the proposed ML estimate with that of the algorithm in [17]. Three algorithms are proposed in [17], which are referred to as algorithms A, A, and B. While algorithm A is optimal,

algorithms A and B are suboptimal. Table I summarizes this
result. In Table I, MUL, ADD, LN, ABS, PH, and DIV denote
the multiplication, addition, natural logarithm, absolute value,
phase derivation, and division operations, respectively. In
ad-dition, the algorithm proposed in this section is referred to as
proposed algorithm I, and the one in Section IV is termed
pro-posed algorithm II. For the propro-posed algorithms, we consider
*the worst case in which all the phase differences dr,s* need to
*be unwrapped. Fig. 1 shows several examples of how Q and N*
affect the complexity. Note that the computational complexity
for the root-finding procedure in [17] is not included here.
For convenience, we treat all operations other than addition as
multiplications. As we can see, the computational complexity
for the proposed algorithm is slightly lower than that for
algorithms A and B in [17], and algorithm A in [17] is the
lowest. However, algorithm Atruncates the polynomials with
*order higher than two in (6), i.e., Λ(z) =*2_{m=−2}b(m)zm_{.}
This impacts the estimation accuracy. Note that we can always
truncate the summation terms in (32) and thereby reduce the
computational complexity of proposed algorithm I. Since
sub-optimum approaches are not our focus, we will not consider the
details here. We will now discuss the computational complexity
of the root-finding procedure. As shown in [27] and [29], the
*root-finding procedure requires O(K*3) multiplications. Table I
shows that the computational complexity of algorithm A is

*O(N K*2). Thus, the computational complexity of the
*root-finding procedure will be high when K is large. Furthermore,*
its implementation cost will also be higher, since we may need
dedicated electronic circuitry to implement this function.

It is well known that the performance of an unbiased
estima-tor is bounded by the CRLB [25]. If the variance of an unbiased
estimator reaches the CRLB, we consider the estimator
effi-cient. Following the procedure to derive performance bounds
in [25], we can calculate the CRLB for our CFO estimation
procedure. Let ˆ*ε be an estimate of ε. The CRLB for our CFO*

estimation is then
CRLB(ˆ*ε) =−* 1
*E*
"
*∂*2
*∂ε*2*Λ(ε)*
#
= *(8π*
2* _{ρ)}−1_{σ}*2

*w(1 + (K− 1)ρ)*

*E*$

*K−1*

*p=1*

*K*

*q>p*

*(q− p)*2Re

*γpqej2πε(q−p)*%

Fig. 1. Computational complexity comparison for the algorithm in [17] and proposed algorithm I. Note that the complexity of the root-finding procedure is not
considered in [17].
= *σw*2*(1 + (K− 1)ρ)*
*8π*2*ρN σ _{x}*2

*K−1*

*p=1*

*K*

*q>p*

*(q− p)*2 =

*1 + K· SNR*

*8π*2

*N· SNR*2 1

*K−1*

*p=1*

*K*

*q>p*

*(q− p)*2 (36)

*where E[·] denotes the expectation.*

IV. PROPOSEDJOINTML STOANDCFO ESTIMATION

In this section, we extend the method developed in Section III
to solve the STO-estimation problem. The core idea is to apply
*a sliding data window for the received (Q + 1)N samples;*
each window covers the preamble in the context of a particular
timing offset. We perform the ML CFO estimation for data
in each window and store the estimated CFO and the
corre-sponding maximum log-likelihood. Thereafter, the estimated
CFO with the largest log-likelihood is selected as the ML
CFO estimate. The corresponding window position is taken
*as the ML STO estimate. Let the window size be QN , and*
*define the set Vi*=*{y(i), y(i + 1), . . . , y(i + QN − 1)} to be*
*the received data in window i. Since the maximum delay is*
*shorter than N , it is clear that 0≤ i ≤ N − 1. If we let the*
*STO be θ, V _{θ}*will cover the complete preamble. In Appendix B,

*we show that the log-likelihood function for V _{i}* can be
ex-pressed by
Λ

*i(ε) = C*

_{1}

*i+ C*

_{2}

*iφi+ C*

_{3}

*i*

*Q−2*

_{}

*p=0*

*Q−1*

_{}

*q>p*&&

_{γ}i*pq*&&cos

*ψi*(37)

_{pq}*where the superscript i indicates that all the variables are*

*calculated within Vi, and C*1

*i, C*2

*i, and C*3

*i*can be treated as window independent. Thus, we can simplify the above log-likelihood function using

Λ*i(ε)≈ C*_{2}*φi+ C*_{3}
*Q−2*
*p=0*
*Q−1*
*q>p*

&&*γ _{pq}i* &&cos

*ψi*(38)

_{pq}*where C*2

*and C*3are the same as those in (23) and (24). Since

*the received signal power φi*is independent of CFO, we can estimate CFO using (32) as

ˆ
*εi*=*−*
*Q−2*
*p=0*
*Q−1*
*q>p*
&&*γi*
*pq*&&*(q− p)∠γpqi*
*2π*
*Q−2*
*p=0*
*Q−1*
*q>p|q − p|*
2&&_{γ}i*pq*&&
*.* (39)

*Note that the upper bound in the summation terms of (39) is Q*
*instead of K. The estimated STO is then*

ˆ
*θ = arg*
max
*i*
Λ*i*( ˆ*εi*_{)}_{= i}_{opt}_{.}_{(40)}

Now, the procedure for the proposed joint ML STO and CFO estimation can be summarized as follows.

*1) Calculate γi _{pq}*

*and its amplitude, where i∈ [1, N], p ∈*

*[0, Q− 2], and q ∈ [p + 1, Q − 1].*

2) Use the phase unwrapping procedure outlined above to
calculate*∠γi*

*pq*.

3) Substitute the results into (38) and (39), and calculate
Λ*i*_{(ˆ}_{ε}i_{) and ˆ}_{ε}i_{.}

*4) Find i*_{opt}such that Λ*i*opt_{(ˆ}* _{ε}i*opt

_{) > Λ}i_{(ˆ}

*opt.*

_{ε}i_{), i}_{= i}*5) The ML STO estimate is i*

_{opt}, and the ML CFO estimate

is then ˆ*εi*opt_{.}

As we can see from the above procedure, the computational
*complexity of the algorithm will be N times higher than that*
*in Section III. Note also that the upper limit of p is Q− 2*
*instead of K− 2. In other words, we have an extra period for*
CFO estimation. By leveraging the sliding window structure,
we can effectively reduce the computational complexity in
*calculating γi*

*pq. Similar to the definition of γpq*, we obtain

*γ _{pq}i* =

*i+N −1*

_{n=i}*yp(n)[yq(n)]∗*. Then, it is simple to show that

*γ _{pq}i*

*= γ*

_{pq}i−1+ yp(i + N− 1) [yq(i + N− 1)]∗*−yp(i− 1) [yq(i− 1)]∗.* (41)
*From (41), we can see that except for i = 0, the calculation of*

*γi*

*pq*requires only two complex multiplications and two complex
additions. This will greatly reduce the required computational
complexity in the scenario of joint STO and CFO estimation.
The required computational complexity has been summarized
in Table I.

We can also obtain the CRLB for the CFO estimate. All we
*have to do is to replace K with Q in (36). Since Q = K + 1, the*
CRLB is lower than that in (36). Note that the STO is a discrete
value. No performance lower bounds have been reported to date
in the literature. In Section V, we will derive a lower bound to
address this omission.

V. PERFORMANCEANALYSIS OFSTO ESTIMATION

In this section, we analyze the performance of the proposed
STO estimation method. We first redefine (38) as Λ*i _{(ε) =}*

*C*_{2}*φi _{+ C}*
3

*ξi*, where

*φi*=

*K*

*p=0*

*i+N −1*

_{}

*n=i*

*yp(n)yp∗(n)*=

*K*

*p=0*

*N −1*

_{}

*n=0*

*x*+ 2Re

_{p}(n)x∗_{p}(n) + w_{p}(n)w∗_{p}(n)*x*'

_{p}(n)w_{p}∗(n) exp*j2πεpN + n*

*N*() (42)

*ξi*=

*K−1*

_{}

*p=0*

*K*

*q>p*

&&*γ _{pq}i* &&cos

*ψ*

_{pq}i=
*K−1*
*p=0*
*K*
*q>p*
*N −1*
*n=0*
*x _{p}(n)w∗_{q}(n) exp*
'

*j2πεqN + n*

*N*(

*+ w*'

_{p}(n)x∗_{q}(n) exp*−j2πεpN + n*

*N*(

*+ w*(43)

_{p}(n)w∗_{q}(n) exp (j2πε(q− p)) + x_{p}(n)x∗_{q}(n).*Note here that φi _{and ξ}i*

_{are random variables. The mean value}of Λ

*i*

_{(ε), which is denoted by μ}iΛ*, is equal to C*2*μiφ+ C*3*μiξ*,
*where μi*

*φ* *and μiξ* *are the mean of φi* *and ξi*, respectively.
The variance of Λ*i _{can be expressed by ν}i*

Λ*= C*22*νφi* *+ C*32*νξi*+
*2C*_{2}*C*_{3}*κi*

*φξ, where νφi* *and νξi* *denote the variance of φi* *and ξi*,
*respectively, and κi _{φξ}*

*the covariance between φi*

*and ξi*. The

*whole set of Vi*, 0

*≤ i ≤ N − 1, has (Q + 1)N samples, and it*may cover three regions. The first region consists of the noise samples, the second region the periodic preamble samples, and the third region the data samples. We denote these regions

*by I*

_{N}, I_{P}, and I_{D}. Thus, the signal variance in I_{N}*is σ*2,

_{w}*that in IP*

*is σx*2

*+ σw*2

*, and that in ID*

*is σ*2

*d+ σ*2

*w, where σ*2

*d*

*represents the variance of data samples. Recall that θ is the*

*actual STO in the system. Using θ as a reference, we can have*

*the following three cases for the value of i: 1) i = θ; 2) i < θ;*

*and 3) i > θ (0≤ i ≤ N − 1). The statistics of φi*

_{and ξ}i_{are}different across these three cases. In Appendix C, we provide a

*detailed derivation of μi*.

_{φ}, μi_{ξ}, ν_{φ}i, ν_{ξ}i, and κi_{φξ}For the proposed STO-estimation algorithm, an error occurs
*when i*opt*= θ. Thus, we can define the error probability of*
*STO estimation as P (∪i,i=θ{Λθ< Λi}), where P (·) denotes*
the probability of a certain event. Note that the evaluation
*of P (Λθ _{< Λ}i*

_{) only requires 1-D integration. If the }

*log-likelihood functions for all i’s are independent and identically*

*distributed, we have P (∪*

_{i,i=θ}{Λθ_{< Λ}i_{}) =}*i,i=θP (Λθ<*
Λ*i*_{). Unfortunately, the log-likelihood functions are not }
inde-pendent. As a result, we have to conduct multidimensional
integration, which is both complex and difficult. Therefore, we
propose a simple alternative to overcome the problem. Instead
of the exact error probability, we attempt to derive a lower
bound.

As shown in [13], the likelihood function is approximately
Gaussian. We denote the distribution of Λ*i* _{using G(μ}i

Λ*, ν*Λ*i*),
*where G(·) denotes the Gaussian distribution. Consider the*
joint density function of Λ*i* and Λ*j*. Using the Gaussian
as-sumption, we write the bivariate Gaussian distribution as

*P (Λi, Λj*) = 1
*2π· νi*
Λ*· ν*Λ*j* *·*
*
1*− Cc(i, j)*
*· exp*
'
*−* *zij*
2 (1*− Cc(i, j))*
(
(44)
where 1*≤ i, j ≤ N, and*
*z _{ij}* =
Λ

*i*Λ

_{− μ}i_{2}

*νi*Λ + Λ

*j− μj*

_{Λ}

_{2}

*ν*

_{Λ}

*j*

*−2Cc(i, j)*Λ

*i− μi*

_{Λ}Λ

*j− μj*

_{Λ}+

*νi*Λ

*· ν*Λ

*j*(45)

*Cc(i, j) =*

*E*Λ

*i*

_{(Λ}

*j*

_{)}

*∗*

*Λ*

_{− μ}i*μj*

*∗*Λ +

*νi*Λ

*· ν*Λ

*j*

*.*(46)

Fig. 2. Comparison of simulated and theoretical*P (Λ _{θ}> Λ_{j}*).

*Note that Cc(i, j) is the corresponding correlation coefficient.*
*The numerator of Cc(i, j) is expressed as*

*E*Λ*i*(Λ*j*)*∗**= μi*_{Λ}*μ*_{Λ}*j∗+ C*_{2}2*κij _{φφ}+ C*

_{3}2

*κij*

_{ξξ}*+ C*_{2}*C*_{3}*κij _{φξ}+ C*

_{2}

*C*

_{3}

*κij*(47)

_{ξφ}*where κij*

_{ab}denotes the covariance of ai*and bj∗*

*(ai, bj*

*∈ {φi,*

*φj, ξi, ξj}). The main idea here is only to calculate P (Λθ>*

Λ*i) for all i’s (except for i = θ) and then use the result to derive*
*a lower bound. Thus, we only have to consider Cc(i, θ) as*

*κiθ _{φφ}= 2σ_{x}*2

*σ*2

_{w}*(QN− |i − θ|)*(48)

*κiθ*2

_{ξξ}= Q(Q− 1)σ*2*

_{x}σ*,*

_{w}*N*3

*(2Q− 1) −*1 2

*|i − θ|*-+1 2

*QN (Q− 1)σ*4

*w*(49)

*κiθ*2

_{φξ}= κiθ_{ξφ}= (Q− 1)*N σ*2

*2*

_{x}σ

_{w}*+ (Q− 1) (N − |i − θ|) σ*2

_{x}*σ*2

_{w}*.*(50) Substituting (45)–(50) into (44), we can then evaluate

*P (Λθ> Λi). Given this definition, we have P (Λθ> Λi*) =
._{∞}

*−∞*
._{Λ}*θ*

*−∞P (Λi, Λθ)dΛidΛθ*. Simulations have been conducted
to evaluate the validity of our theoretical results. Using the
scenario depicted in Section VI, we compare the theoretical
*and simulated P (Λθ> Λi*) in Fig. 2. In the figure, we see that
*the theoretical P (Λθ> Λi*) is close to the simulated result. If
*we let P*min= min

*i=θ* *P (Λ*

*θ _{> Λ}i_{), we can then treat P}*

minas an upper bound for the correct probability of STO estimation (i.e.,

*i*_{opt}*= θ). Thus, we can then have a lower bound for the error*
probability of STO estimation (LBSTO) as 1*− P*_{min}.

VI. SIMULATIONS ANDDISCUSSIONS

In this section, we report our simulation results, which eval-uate the performance of the proposed algorithms. We adopt a Rayleigh multipath channel with an exponential power decay and five channel taps. The preamble, which is generated from a

Fig. 3. Performance comparison of CFO estimation, the algorithm in [17], and proposed algorithm I; SNR= 10 dB.

Fig. 4. BER comparison for systems with and without CFO.

frequency-domain binary-phase-shift-keying-modulated signal,
has ten periods, and each period has 16 samples. The data
following the preamble are transmitted using a
16-quadratic-amplitude-modulation scheme. The mean square error (MSE)
of the estimated CFO is used as a performance measure. We first
consider the CFO-only estimation problem. In this case, the first
*received N samples are discarded. As previously mentioned,*
we term the proposed approach for this scenario as algorithm I
(as described in Section III). We compare the proposed ML
estimator with that in [17]. One optimum algorithm (algorithm
A) and two suboptimum algorithms (algorithm A and B) in
[17] are simulated. Fig. 3 shows the simulation result for SNR
at 10 dB. In the figure, we can see that the performances of
algorithms Aand B are poorer. Algorithm A and the proposed
algorithm offer a similar level of performance that is very close
to the CRLB. To evaluate the impact of CFO on system
perfor-mance, we conduct simulations for systems with and without
CFO. For the system with CFO, we first use the proposed
method to estimate CFO and then conduct CFO compensation.
*Fig. 4 shows the BER comparison for ε = 0.2. As we can see in*

Fig. 5. Performance comparison for CFO estimation, the algorithm in [17], and proposed algorithm II; SNR= 10 dB.

Fig. 6. Performance comparison for CFO estimation, the algorithm in [17],
and proposed algorithms I and II;*N = 16, and Q = 10.*

the figure, the BER performance degrades slightly when CFO is present.

We then consider the case of the joint STO and CFO
*esti-mation process. In this case, discarding the first received N*
samples is not necessary. As a result, one additional
pream-ble is availapream-ble. This means that the proposed method may
offer better performance compared with the previous scenario.
However, the price we pay for the additional STO estimation
is the increase in computational complexity. As mentioned,
we name this approach proposed algorithm II (as explained in
Section IV). Using a similar approach, the method in [17] can
also be used to estimate STO. However, its computational
com-plexity increases much more than our method. Fig. 5 shows the
simulation result for the CFO estimate. The proposed method
offers good performance. Only when CFO is very close to

*±0.5 does the performance of the proposed algorithms degrade.*

Fig. 6 shows the CFO estimation result for various SNRs. In
the figure, we see that the proposed method still works well
for SNRs as low as *−5 dB. The algorithms in [17] perform*

Fig. 7. Error probability of STO estimation (proposed algorithm II).

Fig. 8. Performance comparison for STO estimation (SNR= 2 and 10 dB).

well until SNR reaches *−7 dB, which is somewhat better*
than the proposed algorithms. However, when SNR falls below

*−8 dB, the proposed algorithms again outperform those in*

[17]. This may be because the correlation matrix in (6) is very
noisy, and the roots therefore cannot be solved reliably. Fig. 7
shows the error probability for the STO estimation. We observe
that the derived lower bound for the STO estimation is tight
when the SNR is high. Note that the error probability we
de-fined is only relevant to performance evaluation. If the channel
response is shorter than the CP (which is the typical case),
we can always has some tolerance for the STO estimation.
Thus, there is no need to calculate the exact channel delay.
In real-world applications, it is a common practice to reduce
the estimated STO by a couple of samples when conducting
STO compensation. Another property is that STO estimation
performance is not particularly impacted when CFO is close to
0.5. In the literature, there exist a number of STO estimation
methods. We select the two algorithms proposed in [13] and
[18] for comparison. Fig. 8 shows the MSE curves for these
*approaches and for the proposed algorithms (θ = 8). The figure*
confirms that the proposed method performs best.

VII. CONCLUSION

In this paper, we have developed new algorithms for ML STO and CFO estimation in OFDM systems with periodic pream-bles. The proposed algorithms do not have to calculate the roots of the derivative of the likelihood function. The operations are simple, and the computational complexity is low. With the proposed method, we can simultaneously solve the STO and CFO estimation problems. We also derive a lower bound for the STO estimation error. Simulations show that the proposed methods offer good performance, and the derived lower bound is tight when the SNR is high.

APPENDIXA DERIVATION OF(19)

The likelihood function in (18) can be rewritten as

*Λ(ε) =*
*N −1*
*n=0*
ln
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
*σ _{x}*2

*+ σ*2

_{w}*K*exp

**−y(n)**H

_{R}*−1*

*y*

*det(*

**y(n)****R**

*y*) exp ⎡ ⎢ ⎢

*⎣−*

*K*

*p=1*

*yp(n)y∗p(n)*

*σ*2

*x+σ*2

*w*⎤ ⎥ ⎥ ⎦ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ =

*N −1*

_{}

*n=0*ln"

*σ*2

*2*

_{x}+ σ_{w}*K*(det(

**R**

*y*))

*−1*# +

*K*

*p=1*

*yp(n)yp∗(n)*

*σ*2

*2*

_{x}+ σ

_{w}

**− y(n)***H*

_{R}*−1*

*y*

*!*

**y(n)***.*(51)

Then, by substituting (17) into (51), we derive the
log-likelihood function as
*Λ(ε) =*
*N −1*
*n=0*
ln
$
*σ _{x}*2

*+ σ*2

_{w}*K*det(

**R**

*y*) % +

*K*

*p=1*

*yp(n)yp∗(n)*

*σ*2

_{x}*+ σ*2

_{w}*−*

*σ*

_{w}−2− C_{0}

*K*

*p=1*

*yp(n)yp∗(n)*

*+ 2C*

_{0}Re

*K−1*

_{}

*p=1*

*K*

*q>p*

*yp(n)y∗q(n)ej2π(q−p)ε*! ! =

*N −1*

_{}

*n=0*ln"

*σ*2

_{x}*+ σ*2

_{w}*K*(det(

**R**

*y*))

*−1*# + , 1

*σ*2

_{x}*+ σ*2

_{w}−*σ−2*

_{w}*− C*

_{0}

*-K*

*p=1*

*y*

_{p}(n)y_{p}∗(n)*+ 2C*

_{0}Re

*K−1*

_{}

*p=1*

*K*

*q>p*

*yp(n)y∗q(n)ej2π(q−p)ε*! ! = (1

*− K)σ*4

*x*

*σ*2

_{w}*(σ*2

*2*

_{x}+ σ_{w}*) (Kσ*2

_{x}*+ σ*2

*)*

_{w}*K*

*p=1*

*N −1*

*n=0*

*y*

_{p}(n)y∗_{p}(n)*+ 2C*

_{0}Re

*K−1*

_{}

*p=1*

*K*

*q>p*

*N −1*

_{}

*n=0*

*yp(n)y∗q(n)ej2π(q−p)ε*!

*+ N*ln"

*σ*2

_{x}*+ σ*2

_{w}*K*(det(

**R**

*y*))

*−1*# (52)

*where C*0*= σ _{x}*2

*/(σ*4

*2*

_{w}+ Kσ*2). By substituting (21) and (25) into (52), we can express (52) as*

_{w}σ_{x}*Λ(ε) = C*_{1}*+ C*_{2}
*K*
*p=1*
*γpp*
*+ C*_{3}Re
*K−1*
*p=1*
*K*
*q>p*
*γ _{pq}ej2π(q−p)ε*
!

*= C*1

*+ C*2

*K*

*p=1*

*γpp*

*+ C*

_{3}Re

*K−1*

_{}

*p=1*

*K*

*q>p*

*|γpq|ej∠γpq*

*ej2π(q−p)ε*!

*= C*

_{1}

*+ C*

_{2}

*φ + C*

_{3}

*K−1*

*p=1*

*K*

*q>p*

*|γpq| cos(ψpq*) (53)

*where ψpq, φ, C*1

*, C*2

*, and C*3are defined as (22)–(24).

APPENDIXB DERIVATION OF(38)

We assume that the channel noise, the received preamble,
and the received data are statistically uncorrelated with one
another. We define three column vectors**y**_{1}*(n) = [y*_{0}*(n), . . . ,*

*y _{Q−1}(n)]T*

_{,}

_{y}2*(n) = [y*1*(n), . . . , yQ−1(n)]T*, and **y**3*(n) =*
*[y*_{0}*(n), . . . , yQ−2(n)]T* and their autocorrelation matrix as**R***yk*
*for k = 1, 2, and 3. Note that i is the window index of (37),*
*and θ is the real STO. Therefore, (19) can be derived for the*
*following two cases: 1) i≤ θ and 2) i > θ. Using the approach*
taken to derive (18), we obtain the log-likelihood function for
the first case as

Λ*i≤θ(ε) = ln*
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
*n=i+N −1*_{}
*n=i*
*f ( y(n))*

*Q*

*k=1*

*f (yk−1(n))*⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ =

*θ−1*

*n=i*ln

*f (*

**y**

_{2}

*(n))*

*f (y*1

*(n))· · · f (yQ−1(n))*) +

*i+N −1*

_{}

*n=θ*ln

*f (*

**y**1

*(n))*

*f (y*

_{0}

*(n))· · · f (yQ−1(n))*) (54) =

*θ− i*

*N*

*C*12+

*i + N*

*− θ*

*N*

*C*1 +

*θ−1*

*n=i*

*C*

_{2}

*Q−1*

_{}

*p=1*

*yp(n)yp∗(n)*

+
*θ−1*
*n=i*
*C*_{3}Re
*Q−2*_{}
*p=1*
*Q−1*_{}
*q>p*
*yp(n)yq∗(n)ej2π(q−p)ε*
!
+
*i+N −1*_{}
*n=θ*
*C*_{3}Re
*Q−2*_{}
*p=0*
*Q−1*_{}
*q>p*
*yp(n)y∗q(n)ej2π(q−p)ε*
!
+
*i+N −1*_{}
*n=θ*
*C*_{2}
*Q−1*_{}
*p=0*
*yp(n)yp∗(n)* (55)
where
*C*_{12}*= N· ln*
*σ _{x}*2

*+ σ*2

_{w}*K*det(

**R**

*y2*) (56)

*C*

_{1}

*= N· ln*

*σ*2

_{x}*+ σ*2

_{w}*Q*det(

**R**

*y1*) (57)

*C*

_{2}= (1

*− Q)*

*ρ*2

*σ*2

_{w}*(1 + (Q− 1)ρ)*(58)

*C*

_{3}

*= =,*

*2C*2 (1

*− Q)ρ.*(59)

Similarly, we can derive the log-likelihood function for

*i > θ as*
Λ*i>θ(ε) =*
*i+N −1*_{}
*n=θ+N*
ln
*f (***y**3*(n))*
*f (y*_{0}*(n))· · · f (yQ−2(n))*
)
+
*θ+N −1*
*n=i*
ln
*f (***y**_{1}*(n))*
*f (y*0*(n))· · · f (yQ−1(n))*
)
(60)
=*i− θ*
*N* *C*13+
*θ + N− i*
*N* *C*
1
+
*i+N −1*
*n=θ+N*
*C*_{2}
*Q−2*_{}
*p=0*
*y _{p}(n)y∗_{p}(n)*
+

*i+N −1*

_{}

*n=θ+N*

*C*3Re

*Q−3*

_{}

*p=0*

*Q−2*

_{}

*q>p*

*yp(n)yq∗(n)ej2π(q−p)ε*! +

*θ+N −1*

*n=i*

*C*

_{2}

*Q−1*

_{}

*p=0*

*y*+

_{p}(n)y_{p}∗(n)*θ+N −1*

_{}

*n=i*

*C*

_{3}Re

*Q−2*

_{}

*p=0*

*Q−1*

_{}

*q>p*

*yp(n)yq∗(n)ej2π(q−p)ε*! (61) where

*C*

_{13}

*= N· ln*

*σ*2

*2*

_{x}+ σ

_{w}*K*det(

**R**

*y3*)

*.*Since

**y**2

*(n), i*3

**≤ n ≤ θ − 1 in (54) and y***(n), θ + N*

*≤ n ≤*

*i + N− 1 in (60) contain Q − 1 periods of the preamble,*

det(**R*** _{y2}*) and det(

**R**

*) will be the same as det(*

_{y3}**R**

*) [see (22)].*

_{y}*Consequently, C*

_{12}

*= C*

_{13}

*= C*

_{1}. From (22)–(24), we see that

*C*_{1}*, C*_{2}*, and C*_{3}*can be calculated by replacing Q and***R*** _{y1}*with

*K and***R**_{y}, respectively, in (57)–(59). When Q is reasonably*large, we obtain C*_{1} *≈ C*_{1}*, C*_{2} *≈ C*_{2}*, and C*_{3} *≈ C*_{3}. Thus, we
rewrite (55) and (61) as
Λ*i≤θ(ε)*
*θ−1*
*n=i*
*C*_{3}Re
*Q−2*_{}
*p=1*
*Q−1*_{}
*q>p*
*y _{p}(n)y∗_{q}(n)ej2π(q−p)ε*
!
+

*i+N −1*

_{}

*n=θ*

*C*

_{3}Re

*Q−2*

_{}

*p=0*

*Q−1*

_{}

*q>p*

*yp(n)y∗q(n)ej2π(q−p)ε*!

*+ C*

_{1}+

*θ−1*

*n=i*

*C*

_{2}

*Q−1*

_{}

*p=1*

*y*+

_{p}(n)y_{p}∗(n)*i+N −1*

_{}

*n=θ*

*C*

_{2}

*Q−1*

_{}

*p=0*

*yp(n)yp∗(n)*(62) Λ

*i>θ(ε)*

*θ+N −1*

*n=i*

*C*

_{3}Re

*Q−2*

_{}

*p=0*

*Q−1*

_{}

*q>p*

*y*! +

_{p}(n)y_{q}∗(n)ej2π(q−p)ε*i+N −1*

_{}

*n=θ+N*

*C*

_{3}Re

*Q−3*

_{}

*p=0*

*Q−2*

_{}

*q>p*

*yp(n)y∗q(n)ej2π(q−p)ε*!

*+ C*

_{1}+

*i+N −1*

*n=θ+N*

*C*

_{2}

*Q−2*

_{}

*p=0*

*y*+

_{p}(n)y∗_{p}(n)*θ+N −1*

_{}

*n=i*

*C*

_{2}

*Q−1*

_{}

*p=0*

*yp(n)y∗p(n).*(63)

We now approximate *Q−1 _{p=1}yp(n)yp∗(n) and*

_{Q−2}*p=1*
_{Q−1}

*q>p* *×*

*yp(n)yq∗(n)ej2π(q−p)ε* in (62) with

_{Q−1}

*p=0* *yp(n)yp∗(n)*
and *Q−2 _{p=0}*

*Q−1*

_{q>p}*yp(n)yq∗(n)ej2π(q−p)ε*, respectively. Similarly, we also approximate

*Q−2*

_{p=0}*yp(n)yp∗(n)*and

*Q−3*

_{p=0}Q−2_{q>p}*yp(n)y∗q(n)ej2π(q−p)ε*in (63) with

_{Q−1}*p=0*

*yp(n)y∗p(n) and*

_{Q−2}*p=0*

_{Q−1}*q>p*

*yp(n)yq∗(n)ej2π(q−p)ε*, respectively. Given these approximations, Λ

*i≤θ(ε) and Λi>θ(ε)*can be identically written as

Λ*i(ε) C*1*+ C*2
*i+N −1*_{}
*n=i*
*Q−1*_{}
*p=0*
*yp(n)yp∗(n)*
*+C*_{3}
*i+N −1*
*n=i*
Re
*Q−2*_{}
*p=0*
*Q−1*_{}
*q>p*
*yp(n)yq∗(n)ej2π(q−p)ε*
!
*.* (64)

Using the approach that is similar to that in Appendix A, we
finally obtain
Λ*i(ε) C*_{1}*+ C*_{2}*φi+ C*_{3}Re
*Q−2*_{}
*p=0*
*Q−1*_{}
*q>p*
&&_{γ}i*pq*&&cos
*ψ _{pq}i*
!
(65)
where

*γi*

*pq*=

_{i+N −1}*n=i*

*yp(n)yq∗(n),*

*φi*=

_{Q−1}*p=0*

*γppi*, and

*ψi*

*pq= 2πε(q− p) + ∠γipq*. Note that the approximations we
made are equivalent to adding*|θ − i| samples (noise or data)*

in calculating the likelihood functions. Since the number
*of samples in the ith sliding data window QN is usually*
much larger than the number of added samples *|θ − i|, the*
added samples will not change the likelihood functions too
much. The approximation errors also depend on the distance
between the window position and the actual STO, i.e.,*|θ − i|.*
When the distance is larger, the error is also larger. However,
if the distance is larger, the likelihood function tends to be
smaller, and a larger error is then tolerable. Finally, we note
that the added samples, either noise or data, are uncorrelated
with the preamble samples.

APPENDIXC
DERIVATIONS OF*μi*

*φ, μiξ, νφi, νξi*,AND*κiφξ*

*We first note that θ is the real STO in the system. Using θ as*
*a reference, we have the following three cases for the value of i:*
*1) i = θ; 2) i < θ; and 3) i > θ (0≤ i ≤ N − 1). For the first*
*case, the window covers the preamble data only (IP*). Thus,
(42) and (43) can be simplified to

*φθ*=
*Q−1*_{}
*p=0*
*θ+N −1*_{}
*n=θ*
*xp(n)x∗p(n) + wp(n)w∗p(n)*
+ 2Re
*xp(n)wp∗(n) exp*
'
*j2πεpN + n*
*N*
()
(66)
*ξθ*=
*Q−2*_{}
*p=0*
*Q−1*_{}
*q>p*
*θ+N −1*_{}
*n=θ*
*xp(n)x∗q(n)*
*+ xp(n)wq∗(n) exp*
'
*j2πεqN + n*
*N*
(
*+ w _{p}(n)x∗_{q}(n) exp*
'

*−j2πεpN + n*

*N*(

*+ wp(n)w∗q(n) exp (j2πε(q− p)) .*(67)

*The mean values of φi*

_{and ξ}i_{for the first case are then}

*μθ _{φ,1}= QN*

*σ*2

*2 (68)*

_{x}+ σ_{w}*μθ _{ξ,1}*=

*QN (Q− 1)*

2 *σ*

2

*x.* (69)

The corresponding variance values are

*ν _{φ,1}θ*

*= 2QN*

*σ*2

_{x}*σ*2 (70)

_{w}*ν*=

_{ξ,1}θ*QN (Q− 1)*2

*σ*4

*w*

*+ σ*2

*2*

_{x}σ*3*

_{w}QN (Q− 1)(2Q − 1)*.*(71)

The corresponding covariance value is

*κθ _{φξ,1}= QN (Q− 1)*

*σ*2

*2*

_{x}σ

_{w}*.*(72)

*Here, κi*

*φξ,j* *denotes κiφξ* *in the jth case discussed. For the*
*second case, the window covers the sets I _{N}*

*and I*

_{P}. Thus, φi*and ξi*_{can be expressed as}

*φi*=
*θ−1*
*n=i*
*w*_{0}*(n)w∗*_{0}*(n)*
+
*i+N −1*
*n=θ*
*x*_{0}*(n)x∗*_{0}*(n) + w*_{0}*(n)w∗*_{0}*(n)*
+ 2Re
*x*_{0}*(n)w*_{0}*∗(n) exp*
*j2πεn*
*N*
+
*Q−1*_{}
*p=1*
*i+N −1*_{}
*n=i*
*xp(n)x∗p(n) + wp(n)wp∗(n)*
+ 2Re
*xp(n)w∗p(n) exp*
'
*j2πεpN + n*
*N*
()
(73)
*ξi*=
*Q−1*_{}
*q>0*
*θ−1*
*n=i*
*w*_{0}*(n)x∗ _{q}(n) exp*

*−j2πεn*

*N*

*+ w*

_{0}

*(n)w∗*+

_{q}(n) exp(j2πεq)*Q−1*

_{}

*q>0*

*N +i−1*

_{}

*n=θ*

*x*

_{0}

*(n)x∗*

_{q}(n)*+ x*

_{0}

*(n)w∗*'

_{q}(n) exp*j2πεqN + n*

*N*(

*+ w*

_{0}

*(n)x∗*

_{q}(n) exp*−j2πεn*

*N*

*+ w*

_{0}

*(n)w∗*+

_{q}(n) exp(j2πεq)*Q−2*

_{}

*p=1*

*Q−1*

_{}

*q>p*

*i+N −1*

_{}

*n=i*

*xp(n)x∗q(n)*

*+ xp(n)w∗q(n) exp*'

*j2πεqN + n*

*N*(

*+ w*'

_{p}(n)x∗_{q}(n) exp*−j2πεpN + n*

*N*(

*+ wp(n)wq∗(n) exp (j2πε(q− p)) .*(74) Their mean values are

*μi _{φ,2}= (QN + i− θ)*

*σ*2

_{x}*+ σ*2

_{w}*+ (i− θ)σ*2 (75)

_{x}*μi*2 +

_{ξ,2}= (Q− 1)(N + i − θ)σ_{x}*(Q− 1)(Q − 2)*2

*N σ*2

*x*(76)

the corresponding variance values are

*ν _{φ,2}i*

*= 2σ*2

_{x}*σ*2

*(77)*

_{w}[QN + i− θ]*ν*=

_{ξ,2}i*QN (Q− 1)*2

*σ*4

*w+ σ*2

*xσ*2

*wN (Q− 1)*

*·*,

*(Q− 1)*' 2 +

*i− θ*

*N*( +

*(Q− 2)(2Q − 3)*3 -(78)

and the covariance value is

*For the third case, the window covers sets I _{P}*

*and I*, and we

_{D}*write φi*

_{and ξ}i_{as}

*φi*=

*Q−2*

_{}

*p=0*

*i+N −1*

*n=i*

*x*

_{p}(n)x∗_{p}(n) + w_{p}(n)w∗_{p}(n) + 2Re*×*,

*x*'

_{p}(n)w_{p}∗(n) exp*j2πεpN + n*

*N*(- ! +

*θ+N −1*

*n=i*

*x*+ 2Re $

_{Q−1}(n)x∗_{Q−1}(n) + w_{Q−1}(n)w∗_{Q−1}(n)*xQ−1(n)wQ−1∗*

*(n)*

*× exp*'

*j2πε(Q− 1)N + n*

*N*( %! +

*i+N −1*

*n=θ+N*

*x*

_{Q−1}(n)x∗_{Q−1}(n) + w_{Q−1}(n)w_{Q−1}∗*(n)*+ 2Re $

*x*

_{Q−1}(n)w_{Q−1}∗*(n)*

*× exp*'

*j2πε(Q− 1)N + n*

*N*( %! (80)

*ξi*=

*Q−3*

_{}

*p=0*

*Q−2*

_{}

*q>p*

*i+N −1*

_{}

*n=i*

*x*

_{p}(n)x∗_{q}(n)*+ xp(n)w∗q(n) exp*'

*j2πεqN + n*

*N*(

*+ wp(n)x∗q(n) exp*'

*−j2πεpN + n*

*N*(

*+ w*+

_{p}(n)w∗_{q}(n) exp (j2πε(q− p))*Q−2*

_{}

*p=0*

*θ+N −1*

_{}

*n=i*

*xp(n)x∗Q−1(n)*

*+ xp(n)w∗Q−1(n) exp*'

*j2πε(Q− 1)N + n*

*N*(

*+ w*'

_{p}(n)x∗_{Q−1}(n) exp*−j2πεpN + n*

*N*(

*+ wp(n)w∗Q−1(n) exp (j2πε(Q− p − 1))*+

*Q−2*

_{}

*p=0*

*i+N −1*

_{}

*n=θ+N*

*x*

_{p}(n)x∗_{Q−1}(n)*+ x*'

_{p}(n)w∗_{Q−1}(n) exp*j2πε(Q− 1)N + n*

*N*(

*+ wp(n)x∗Q−1(n) exp*'

*−j2πεpN + n*

*N*(

*+ w*(81)

_{p}(n)w∗_{Q−1}(n) exp (j2πε(Q− p − 1)) .Thus, the mean values are

*μi _{φ,3}= (QN + θ− i)*

*σ*2

_{x}*+ σ*2

_{w}*+ (i− θ)*

*σ*2

*2*

_{d}+ σ*(82)*

_{w}*μi _{ξ,3}= (Q− 1)(N − i + θ)σ_{x}*2+

*(Q− 1)(Q − 2)*

2 *N σ*

2
*x* (83)

the variance values are

*ν _{φ,3}i*

*= 2σ*2

*2*

_{x}σ_{w}*(QN− i + θ) + 2(i − θ)σ*2

*2 (84)*

_{d}σ_{w}*ν*

_{ξ,3}i*= σ*2

_{x}*σ*2

_{w}N (Q− 1)*×*,

*(2Q− 3)*' 1 +

*θ− i*

*N*( +

*(Q− 2)(2Q − 3)*3 + 1 -+

*QN (Q− 1)*2

*σ*4

*w+ (Q− 1)*2

*(i− θ)(σd*2

*+ σw*2) (85) and the covariance value is

*κi _{φξ,3}= (Q− 1)σ*2

*2*

_{x}σ*(86)*

_{w}[QN + 2(θ− i)] .REFERENCES

[1] J. J. van de Beek, M. Sandell, and P. O. Borjesson, “ML estimation of time
*and frequency offset in OFDM systems,” IEEE Trans. Signal Process.,*
vol. 45, no. 7, pp. 1800–1805, Jul. 1997.

[2] N. Lashkarian, “Class of cyclic-based estimators for frequency-offset
*estimation of OFDM systems,” IEEE Trans. Commun., vol. 48, no. 12,*
pp. 2139–2149, Dec. 2000.

[3] Y.-C. Wu and E. Serpedin, “Comments on ‘Class of cyclic-based
*esti-mators for frequency-offset estimation of OFDM systems,”’ IEEE Trans.*

*Commun., vol. 53, no. 3, pp. 413–414, Mar. 2005.*

[4] M. Tanda, “Blind symbol-timing and frequency-offset estimation in
*OFDM systems with real data symbols,” IEEE Trans. Commun., vol. 52,*
no. 10, pp. 1609–1612, Oct. 2004.

*[5] H. Meyr, M. Moeneclaey, and S. A. Fechtel, Digital Communication*

*Receivers: Synchronization, Channel Estimation, and Signal Processing.*

New York: Wiley, 1998.

[6] E. Chiavaccini and G. M. Vitetta, “Maximum-likelihood frequency re-covery for OFDM signals transmitted over multipath fading channels,”

*IEEE Trans. Commun., vol. 52, no. 2, pp. 244–251, Feb. 2004.*

[7] D. Lee and K. Cheun, “Coarse symbol synchronization algorithms for
*OFDM systems in multipath channels,” IEEE Commun. Lett., vol. 6,*
no. 10, pp. 446–448, Oct. 2002.

*[8] S. Haykin, Adaptive Filter Theory 4E. Englewood Cliffs, NJ:*
Prentice–Hall, 2002, p. 440.

[9] M. Morelli, A. N. D’Andrea, and U. Mengali, “Frequency ambiguity
*resolution in OFDM systems,” IEEE Commun. Lett., vol. 4, no. 4,*
pp. 134–136, Apr. 2000.

[10] K.-T. Lee and J.-S. Seo, “Pilot-aided iterative frequency offset estimation
*for digital video broadcasting (DVB) systems,” IEEE Trans. Consum.*

*Electron., vol. 53, no. 1, pp. 11–16, Feb. 2007.*

[11] P. H. Moose, “A technique of orthogonal frequency division multiplexing
*frequency offset correlation,” IEEE Trans. Commun., vol. 42, no. 10,*
pp. 2908–2914, Oct. 1994.

[12] D. Huang and K. B. Letaief, “Carrier frequency offset estimation for
*OFDM systems using null subcarriers,” IEEE Trans. Commun., vol. 54,*
no. 5, pp. 813–823, May 2006.

[13] T. M. Schmidl and D. C. Cox, “Robust frequency and timing
*synchroniza-tion for OFDM,” IEEE Trans. Commun., vol. 45, no. 12, pp. 1613–1621,*
Dec. 1997.

[14] M. Morelli and U. Mengali, “An improved frequency offset estimator
*for OFDM applications,” IEEE Commun. Lett., vol. 3, no. 3, pp. 75–77,*
Mar. 1999.

[15] H. Minn, P. Tarasak, and V. K. Bhargava, “OFDM frequency offset
*esti-mation methods based on BLUE principle,” in Proc. IEEE Veh. Technol.*

*Conf., Vancouver, BC, Canada, Sep. 2002, pp. 1230–1234.*

[16] J. Li, G. Liu, and G. B. Giannakis, “Carrier frequency offset estimation
*for OFDM-based WLANs,” IEEE Signal Process. Lett., vol. 8, no. 3,*
pp. 80–82, Mar. 2001.

[17] J. H. Yu and Y. T. Su, “Pilot-assisted maximum-likelihood
*frequency-offset estimation for OFDM systems,” IEEE Trans. Commun., vol. 52,*
no. 11, pp. 1997–2008, Nov. 2004.

[18] M. H. Cheng and C. C. Chou, “Maximum-likelihood estimation of
fre-quency and time offsets in OFDM systems with multiple sets of
*identi-cal data,” IEEE Trans. Signal Process., vol. 54, no. 7, pp. 2848–2852,*
Jul. 2006.

[19] W. D. Warner and C. Leung, “OFDM/FM frame synchronization for
*mobile radio data communication,” IEEE Trans. Veh. Technol., vol. 42,*
no. 3, pp. 302–313, Aug. 1993.

[20] L. Hanzo, “Bandwidth-efficient wireless multimedia communications,”

*Proc. IEEE, vol. 86, no. 7, pp. 1342–1382, Jul. 1998.*

*[21] R. van Nee and R. Prasad, OFDM for Wireless Multimedia *

*Commu-nications. Boston, MA: Artech House, 2002.*

*[22] J. Heiskala and J. Terry, OFDM Wireless LANs: A Theoretical and *

*Prac-tical Guide. Indianapolis, IN: Sams, 2002.*

[23] Y. S. Choi, P. J. Voltz, and F. A. Cassara, “ML estimation of carrier frequency offset for multicarrier signals in Rayleigh fading channels,”

*IEEE Trans. Veh. Technol., vol. 50, no. 2, pp. 644–655, Mar. 2001.*

*[24] S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation*

*Theory. Englewood Cliffs, NJ: Prentice–Hall, 1993.*

*[25] H. V. Poor, An Introduction to Signal Detection and Estimation 2E.*
Berlin, Germany: Springer-Verlag, 1994.

[26] P. Stoica and A. Nehorai, “MUSIC, maximum likelihood, and Cramer–
*Rao bound,” IEEE Trans. Acoust., Speech, Signal Process., vol. 37, no. 5,*
pp. 720–741, May 1989.

[27] Y. Jian, H. Minn, X. Gao, X. You, and Y. Li, “Frequency offset estimation
*and training sequence design for MIMO OFDM,” IEEE Trans. Wireless*

*Commun., vol. 7, no. 4, pp. 1244–1254, Apr. 2008.*

[28] S. Tretter, “Estimating the frequency of a noisy sinusoid by linear
*re-gression,” IEEE Trans. Inf. Theory, vol. IT-31, no. 6, pp. 832–835,*
Nov. 1985.

[29] W. S. McCormick and L. L. James, “Efficient parallel rooting of complex
*polynomials on the unit circle,” IEEE Trans. Signal Process., vol. 39,*
no. 10, pp. 2347–2351, Oct. 1991.

[30] L. Rugini and P. Banelli, “BER of OFDM systems imparied by carrier
*frequency offset in multipath fading channels,” IEEE Trans. Wireless*

*Commun., vol. 4, no. 5, pp. 2279–2288, Sep. 2008.*

**Hung-Tao Hsieh (S’08) received the B.S. degree**

in physics in 2000 from the National Central Uni-versity, Taoyuan, Taiwan, and the M.S. degree in electro-physics in 2002 from the National Chiao Tung University (NCTU), Hsinchu, Taiwan, where he is currently working toward the Ph.D. degree with the Department of Communication Engineering.

His research interests include detection/estimation theories and communication signal processing.

**Wen-Rong Wu (M’89) received the B.S. degree**

in mechanical engineering from Tatung Institute of Technology, Taipei, Taiwan, in 1980 and the M.S. degrees in mechanical and electrical engineer-ing and the Ph.D. degree in electrical engineerengineer-ing from the State University of New York, Buffalo, in 1985, 1986, and 1989, respectively.

Since August 1989, he has been a faculty member with the Department of Communication Engineer-ing, National Chiao Tung University, Hsinchu, Taiwan. His research interests include statistical signal processing and digital communication.