Fractionally Spaced Frequency-Domain MMSE ... - IEEE Xplore

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 59, NO. 9, NOVEMBER 2010

Fractionally Spaced Frequency-Domain MMSE Receiver for OFDM Systems Qinghua Shi, Liang Liu, Student Member, IEEE, Yong Liang Guan, Member, IEEE, and Yi Gong, Senior Member, IEEE

Abstract—Based on frequency-domain oversampling and the Bayesian Gauss–Markov theorem, we propose a fractionally spaced frequency-domain minimum mean-square error (FSFDMMSE) receiver for orthogonal frequency-division multiplexing systems. It is shown that frequency diversity inherent in a frequency-selective fading channel can be extracted and exploited by the proposed FSFD-MMSE receiver. This diversity advantage outweighs the effect of intercarrier interference generated by frequency-domain oversampling, due largely to the MMSE receiver’s interference suppression capability. Numerical results show that the FSFD-MMSE receiver outperforms the conventional MMSE receiver under both ideal and practical situations (i.e., with frequency offset, channel estimation errors, and even doubly selective channel fading). In addition, the FSFD-MMSE receiver only needs a fast Fourier transform size that is no larger than N + Q − 1 (N = number of data subcarriers, and Q = number of resolvable multipath components). Index Terms—Doubly selective fading, frequency-domain oversampling, minimum mean square error (MMSE) receiver, orthogonal frequency-division multiplexing (OFDM).

I. I NTRODUCTION

O

RTHOGONAL frequency-division multiplexing (OFDM) [1], [2] can effectively cope with frequency-selective fading channels by using simple frequency-domain equalization [3], [4]. This property of OFDM is particularly attractive for high-data-rate applications. Therefore, OFDM and its variants are now widely adopted for broadband wireline and wireless communications [5]–[7]. However, OFDM suffers greatly when there is a frequency offset between transmitter and receiver carriers. The frequency offset, which may be caused by oscillator instability, Doppler effect, and imperfect carrier synchronization, destroys the orthogonality among subcarriers, resulting in intercarrier interference (ICI) [5]. It has been shown in [8] that OFDM is Manuscript received March 22, 2010; revised July 24, 2010; accepted August 28, 2010. Date of publication September 27, 2010; date of current version November 12, 2010. The review of this paper was coordinated by Dr. T. Taniguchi. Q. Shi was with the Positioning and Wireless Technology Centre, Nanyang Technological University, Singapore 639798. He is now with the Department of Electronic Engineering, University of Electro-Communications, Tokyo 182-8585, Japan. L. Liu is with Nanyang Technological University, Singapore 639798, on leave from the School of Electronic and Information Engineering, Beihang University, Beijing 100083, China. Y. L. Guan and Y. Gong are with the Positioning and Wireless Technology Centre, School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2010.2078846

much more sensitive to frequency offset than its single-carrier counterpart. For this reason, frequency offset estimation and compensation have been subject to intensive research [9]. In this paper, we take a different approach, aiming at an OFDM receiver that is inherently robust against frequency offset. This is made possible by noting that a dual problem of frequency offset for multicarrier systems is timing jitter for single-carrier systems, which is traditionally solved via timedomain-oversampling techniques [10]. Therefore, it can be expected that frequency-domain-oversampling approaches should be robust to frequency offset. Frequency-domain oversampling can also be readily implemented through zero-padding fast Fourier transform (FFT). In [11], frequency-domain oversampling is used to estimate frequency offset for OFDM. In [12], a frequency-domain-oversampling-based receiver is proposed to suppress multiple access interference (MAI) for the uplink of multicarrier code-division multiple-access by exploiting the structure of MAI due to excess samples and frequency offset. This receiver relies particularly on the use of excess time samples, which leads to a reduction in the number of subcarriers and results in non-orthogonal subcarriers at the transmitter side. As a result, FFT algorithms, which are commonly employed in OFDM systems, cannot, in general, be used at the transmitter side [12]. We propose a novel fractionally spaced frequency-domain minimum mean-square error (FSFD-MMSE) receiver for OFDM systems with zero postfix (ZP). The basic idea is that the received signal is first projected onto a higher dimensional space via frequency-domain oversampling, and then, an optimal linear receiver in the minimum mean square error (MSE) sense is derived based on the Bayesian Gauss–Markov theorem [13]. As expected, the proposed FSFD-MMSE receiver shows robustness to frequency offset and doubly selective fading. More interestingly, it turns out that the proposed receiver can also exploit frequency diversity in a frequency-selective fading channel, which conventional uncoded OFDM receivers cannot achieve. We notice that a time-domain-oversamplingbased OFDM receiver [14] has been proposed to exploit frequency diversity. These two oversampling schemes, apart from the clear distinction in the signal-processing techniques used, are fundamentally different in terms of achievable frequencydiversity order. Specifically, the maximum diversity order in [14] is mainly determined by the time-domain-oversampling factor, whereas for our approach, the achievable diversity order is primarily determined by the number of resolvable multipath components of the channel. A more detailed comparison between them will be provided in Section IV-G.

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SHI et al.: FRACTIONALLY SPACED FREQUENCY-DOMAIN MMSE RECEIVER FOR OFDM SYSTEMS

The rest of this paper is organized as follows: In Section II, we describe the system in the FSFD framework, which is characterized by a linear signal model. In Section III, the FSFDMMSE receiver is proposed and analyzed. Section IV provides numerical results and discussions, followed by our conclusions in Section V. Notation: We denote matrices by bold uppercase letters and column vectors by bold lowercase letters; Tr(·), (·)T , (·)H , and (·)† denote trace, transpose, conjugate transpose, and Moore–Penrose pseudoinverse, respectively; IN represents the N × N identity matrix and diag{·} is a diagonal matrix; E[·] denotes expectation; and A(n, n ) denotes the element on the nth row and n th column of matrix A.

Fig. 1. Block diagram of the FSFD-MMSE receiver.

We consider a quasi-static frequency-selective fading channel unless otherwise specified; the impulse response of the channel is modeled as

II. S YSTEM M ODEL Consider a basic OFDM system with N subcarriers. Due to the use of a guard interval,1 which eliminates inter-blockinterference (IBI), we only need to deal with one OFDM symbol. The transmitted continuous-time signal of one OFDM symbol over [0, T + TGI ] may be expressed as 1 N −1 n √ n=0 dn exp j2π T t , 0 ≤ t < T N s(t) = 0, T ≤ t < T + TGI (1)

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h(τ ) =

Q−1

hq δ(τ − τq ),

q=0

Δ

d = [d0 , d1 , . . . , dN −1 ]T

(2)

the data symbol vector d satisfies E[d] = 0,

Cd = E[ddH ] = σs2 IN

(3)

where 0 is an all-zero vector of length N , Cd is the covariance matrix of d, and σs2 represents the average power of the transmitted symbols. Define the inverse discrete Fourier transform (IDFT) matrix as 1 2π Δ H FN ×N = √ exp j np , N N N ×N n, p = 0, 1, . . . , N − 1 (4)

E |hq |2 = 1

(7)

q=0

where Q is the number of multipath components characterized Q−1 by delays {τq }Q−1 q=0 and complex gains {hq }q=0 . The frequency response of the channel h(τ ) is given by ∞ H(f ) =

h(τ )e−j2πf τ dτ =

0

where T and TGI denote the time duration of the desired signal and guard interval, respectively; and dn represents a data symbol modulating the nth subcarrier, which is uniformly selected from quadrature amplitude modulation (QAM) or M -ary phase-shift keying (MPSK) constellations. Letting

Q−1

Q−1

hq e−j2πf τq .

(8)

q=0

At the receiver shown in Fig. 1, the received continuoustime signal r(t) is first discretized by sampling at t = p(T /N ), p = 0, 1, . . . , P in the usual manner with P = N + Ng and zero padded to L = SN points with the purpose of frequencydomain oversampling, where S denotes the oversampling factor. Note that the frequency-domain oversampling can be viewed as projecting the received signal onto L non-orthogonal basis functions {exp(−j2π(t/T )(l/S))}, l = 0, 1, . . . , L − 1 or, equivalently, correlating with L non-orthogonal subcarriers spaced by 1/ST . [This will be observed in (16).] Here, we assume that time synchronization is perfect, but there is a frequency offset ε (normalized by subcarrier spacing 1/T ) between the transmitter and receiver carriers, which may result from imperfect carrier synchronization or Doppler effect. To account for the effects of frequency offset, by introducing the matrix 2π 2π Δ (9) D(ε) = diag 1, e−j N ε , . . . , e−j N (L−1)ε the received discrete-time signal may be expressed as

and let

T ΞGI = ITN 0TNg ×N .

The discrete-time form of the transmitted signal in (1) may be expressed as s = ΞGI FH N ×N d

˜ +n ˜r = D(ε)Ξzp Hs

(5)

(6)

where ΞGI models the insertion of zero postfix. 1 We consider a ZP with the length N as the guard interval and assume that g Ng is larger than the maximum delay spread of the channel.

˜ GI FH d + Ξzp n = D(ε)Ξzp HΞ N ×N

(10)

Ξzp = [ITP 0T(L−P )×P ]T reflects the zero-padding ˜ is a P × P lower triangular Toeplitz matrix process, and H with the first column [h0 , h1 , · · · , hQ−1 , 0, · · · , 0]T . n denotes the additive white Gaussian noise vector with variance σn2 , ˜ zp = Ξzp HΞ ˜ GI corresponds to the L × N equivalent and H channel matrix, which combines the effects of ZP insertion, wireless propagation channel, and zero-padding operations. where

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mode such that the transmitted data symbol vector d can be estimated by

By defining 1 2π Δ FL×L = √ exp −j np L L L×L

FL×N

n, p = 0, 1, . . . , L − 1 1 2π l Δ p = √ exp −j N S N L×N l = 0, 1, . . . , L − 1; p = 0, 1, . . . , N − 1

ˆ = wH y d (11)

where wH denotes an N × L weight matrix to be determined, depending on the performance criterion adopted.

(12)

A. MMSE Receiver

(13)

We are particularly interested in the widely used MMSE criterion. By using the Bayesian Gauss–Markov theorem [13, p. 391] and noting (3) and (17), our FSFD-MMSE receiver has the following N × L weight matrix:

˜ zp can be decomposed as the equivalent channel matrix H ˜ zp = FH HFL×N H L×L

where FH L×L is an L × L IDFT matrix, and FL×N denotes an L × N zero-padded DFT matrix. The channel effect can be described by H = diag{H0 , H1 , · · · , HL−1 }, with the element Hl sampled at fractional frequencies of channel frequency response H(f ), i.e., Q−1 l l τq Δ S Hl = H = hq exp −j2π T S T q=0 l = 0, 1, . . . , L − 1.

(14)

Furthermore, defining an L × P zero-padded DFT matrix 1 2π l Δ p FL×P = √ exp −j N S L L×P l = 0, 1, . . . , L − 1; p = 0, 1, . . . , P − 1

(15)

the received signal in the fractionally spaced frequency domain is given by ˜ GI FH d + FL×L Ξzp n y = FL×L D(ε)Ξzp HΞ N ×N H = FL×L D(ε)FH L×L HFL×N FN ×N d + FL×P n

= FL×L D(ε)FH L×L HΩd + η

(16)

Δ

where Ω = FL×N FH N ×N reflects the frequency-domainoversampling process, which characterizes the mapping from Δ N transmitted subcarriers to L received subcarriers. η = T [η0 , η1 , · · · , ηL−1 ] denotes a colored (due to oversampling) Gaussian noise vector with zero mean and covariance H

Cη = E[ηη ] =

σn2 Γ,

Γ=

FL×P FH L×P

(18)

(17)

where σn2 is the variance of the background noise. As expected, the system model (16) reduces to the conventional OFDM if we set L = N . III. FSFD-MMSE R ECEIVER The received signal vector y suggests that our system fits a typical linear model with correlated Gaussian noise vector, for which there are a variety of well-studied signal-processing techniques, including both linear and nonlinear algorithms implemented either in batch mode or adaptively [15]–[17]. In this paper, we focus on a linear receiver operating in batch

wH = E[dyH ] · E[yyH ]† † σn2 H ˆH H ˆ ˆ = (HΩ) HΩΩ H + 2 Γ . σs

(19)

ˆ is estimated by the Here, we assume that the channel matrix H receiver, and therefore, channel estimation errors may occur. Before proceeding further, we make the following remarks. 1) We use Moore–Penrose pseudoinverse in (19) because oversampling leads to an overdetermined system (L ≥ N ); hence, the matrix to be inverted is rank deficient, and pseudoinverse is necessary. 2) We assume that the frequency offset ε is unknown to the receiver. The received signal vector y used to derive the optimal weight matrix wH is given by (16), but we set ε = 0 (i.e., FL×L D(ε)FH L×L = IL ). This is to model the effect that the FSFD-MMSE receiver is blind to the frequency offset between the transmitter and receiver carriers. The motivation of this treatment is that, in practice, a residual frequency offset is unavoidable. 3) Due to the matrix inverse involved, the complexity of the FSFD-MMSE receiver (on the order of O(L3 )) heavily depends on L. We emphasize that L need not be a multiple of N (i.e., S can be non-integer), and this issue will be elaborated upon later.

B. MSE Performance The performance of the FSFD-MMSE receiver can be characterized by its MSE. To simplify our analysis, we assume that there is no frequency offset (ε = 0), and channel estimation is perfect. With the help of [13, p. 391, eq. (12.28)], the covariance ˆ is matrix of the estimator error d − d

Δ ˆ ˆ H − d) CMSE = E (d − d)(d † σn2 2 2 H H = σs IN − σs (HΩ) (HΩ)(HΩ) + 2 Γ σs · (HΩ).

(20)

The MSE averaged over all subcarriers is then given by MSE = tr{CMSE }/N.

(21)


Note that the MSE is independent of the modulation format in use, provided that (3) is satisfied. C. Semi-Analytical Performance Analysis Another important performance measure for an OFDM system is the bit error rate (BER). Based on the same assumptions as used in the MSE analysis (i.e., ε = 0, and channel estimation is perfect). The MMSE weight matrix becomes † σn2 H H H H (22) HΩΩ H + 2 Γ . w = (HΩ) σs Substituting (22) and (16) into (18) gives ˆ = wH HΩd + wH η d

(23)

which can be rewritten as ˆ = Ad + η d

(24)

with Δ

A = wH HΩ

T Δ η = η0 , η1 , . . . , ηN −1

(25)

(26)

Note that the N × N matrix A is, in general, not diagonal due to frequency-domain oversampling. This means that ICI is inherent to our proposed FSFD-MMSE receiver, making an exact performance analysis rather involved [18]. Here, we present a semi-analytical performance analysis by approximating the ICI as a Gaussian random variable conditioned on the channel matrix. This conditional Gaussian approximation of the ICI is based on the Lyapunov’s Central Limit Theorem [19]. Specifically, supposing that nth subcarrier is of interest, the −1 the ICI term is given by N n =0,n =n A(n, n )dn , which, conditioned on A (equivalently, on H), is a sum of independent random variables with zero mean and finite variance. Moreover, due to the symmetry property of the QAM and MPSK constellations, the third central moments of {dn } are zero. From the Lyapunov Theorem of [19, p. 126], the ICI term is asymptotically Gaussian distributed as N increases. As such, we can approximate the ICI for subcarrier n by a conditional Gaussian random variable with zero mean and variance ⎫2 ⎤ ⎡⎧ −1 ⎨ N ⎬ 2 =E ⎣ ⎦ A(n, n )d σICI,n n H ⎩ ⎭ n =0,n =n

N −1

|A(n, n )| . 2

(27)

n =0,n =n

It follows that the conditional SINR for subcarrier n is given by |A(n, n)|2

. 2 + C (n, n)/σ 2 η s n =0,n =n |A(n, n )|

γn |H = N −1

Invoking the well-established BER expressions [20, p. 202, 8.31] and [20, p. 196, 8.14], the conditional BER for the nth subcarrier with MPSK and QAM constellations can be calculated, respectively, by Pb,n |H ∼ =

2 max(log2 M, 2) max(M/4,1) (2i−1)π × Q 2γn |H sin M i=1

(MPSK) (29)

√

1 M −1 ∼ √ Pb,n |H = 4 log2 M M √ ! M /2 3γn |H Q (2i − 1) × M −1 i=1

(QAM)

(30)

"∞ where Q(x) = (1/2π) x exp(−(u2 /2))du. The conditioning on the channel matrix H can be removed by Monte Carlo integration. IV. N UMERICAL R ESULTS AND D ISCUSSION

where η has a covariance matrix

Cη = E η (η )H = σn2 wH Γw.

= σs2

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(28)

Monte Carlo simulation and semi-analytical results are provided to compare the error performance of the proposed FSFDMMSE receiver with an oversampling factor S > 1 and the conventional MMSE (Conv. MMSE) receiver using cyclic prefix without oversampling (i.e., with S = 1) under various conditions. We consider uncoded error probability averaged over all subcarriers. Unless otherwise stated, the system parameters used are given here (mimicking the setup in HYPERLAN/2 [21] and 802.11a [22] standards): 1) numbers of subcarriers: N = 64; 2) modulation: QPSK; 3) oversampling factor: S = 2; 4) channel: quasi-static frequency-selective Rayleigh fading channel with multipath components spaced by (T /N ). For illustration, a 16-path (Q = 16) channel having a uniform power delay profile (PDP) is adopted; 5) bit-energy-to-noise-power-spectrum-density ratio (Eb/N0), which is defined as σs2 /(log2 M · σn2 ). The pseudoinverse in (19) is calculated via singular value decomposition by neglecting singular values smaller than 10−4 . A. Extracting Frequency Diversity In Fig. 2, we compare the BER performance of the proposed FSFD-MMSE receiver with the Conv. MMSE receiver operating in an ideal condition, i.e., there is no frequency offset, and channel estimation is perfect. Numerical results obtained via (29) are also plotted to check the accuracy of the semi-analytical approach. We can see that the FSFD-MMSE receiver with S = 2 performs significantly better than the Conv. MMSE. This performance advantage is in the form of a steeper BER slope, signifying that the FSFD-MMSE receiver enjoys a frequency diversity gain, which has not been seen before in conventional uncoded or non-adaptive OFDM systems. Furthermore, the results show that the proposed FSFD-MMSE

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Fig. 2. BER performance of the FSFD-MMSE receiver under different multipath fading channels.

Fig. 4.

BER performance with practical channel estimation errors.

N = 64 is large enough to ensure that the Gaussian approximation has good accuracy. B. Channel Estimation In practical OFDM systems, channel estimation is obtained through two steps: 1) channel state information (CSI) acquisition and 2) CSI tracking. CSI acquisition is usually implemented by using block-type pilot symbols (also called preamble), whereas CSI tracking is achieved by sending the pilot tones between the data subcarriers [21], [22]. In our FSFD framework, traditional CSI acquisition and tracking methods can also be applied, except that signals now become L-dimensional. Assuming that a known preamble x = [x0 , x1 , · · · , xN −1 ]T is sent at the transmitter, the corresponding received frequencydomain signal can be described as Fig. 3. Sinc spectrum of a typical OFDM subcarrier with frequency-domain oversampling (S = 2).

receiver can extract more frequency diversity from a channel with more paths (means a higher inherent diversity order). This feature can be intuitively explained by Fig. 3, which shows the sinc spectrum for a typical FSFD-MMSE receiver with oversampling factor S = 2. It can be seen that, in the FSFD framework, the signal sent on one subcarrier is received at many other fractional subcarrier locations by frequency-domain oversampling at the receiver side. This spectrum spreading effect gives rise to frequency diversity at the output of the proposed FSFD-MMSE receiver. However, the FSFD-MMSE scheme may not be able to achieve full frequency diversity because, in the frequency domain, each subcarrier possesses a sinc spectrum shape as shown in Fig. 3; therefore, the samples at non-orthogonal frequency locations have smaller powers than that of the peak sample, and adjacent frequency samples can be correlated. In addition, it can be seen that the simulation and semianalytical results agree very well. This is because the conditional Gaussian approximation to the ICI is guaranteed by the Lyapunovs Central Limit Theorem as the number of subcarriers

y = HΩx + η

(31)

˜ = Ωx where H = diag{H0 , H1 , · · · , HL−1 }. Letting x ˜ = [H0 , H1 , · · · , HL−1 ]T , (31) becomes y = and h ˜ +η =X ˜ + η. Note that X ˜h ˜ is diag{˜ x0 , x ˜1 , · · · , x ˜L−1 }h known to the receiver and can be calculated in advance. In practice, a least-square estimator is often adopted to estimate ˆ˜ ˜ −1 y. Fig. 4 shows the BER the channel, yielding h =X comparison of the proposed FSFD-MMSE receiver with conventional MMSE receiver under practical least-square channel estimation, which is obtained by using the same block-type pilot symbols as in HYPERLAN/2 [21]. It can be seen that the proposed FSFD-MMSE receiver still outperforms the conventional MMSE receiver in the presence of channel estimation errors. C. Effect of Frequency Offset Next, we consider a situation particularly relevant to OFDM systems, where there is a frequency offset ε between the transmit and receive carriers. We can see from Fig. 5 that the proposed FSFD-MMSE receiver outperforms the Conv. MMSE receiver at various frequency offsets. It has been shown in


Fig. 5. BER performance of the FSFD-MMSE receiver suffering from different frequency offsets ε = [0, 0.02, 0.05, 0.1].

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Fig. 6. Effect of doubly selective fading on the BER performance of the FSFD-MMSE receiver.

[23] that, to have an acceptable performance for a conventional OFDM system, the residual frequency offset should be about two orders of magnitude smaller than the subcarrier spacing. To make a clear comparison, let us assume 1.5-dB Eb /N0 degradation (due to frequency offset) at BER = 10−3 as the criterion. It can be seen that Conv. MMSE can only support a frequency offset of ε = 0.02, whereas FSFD-MMSE can tolerate a frequency offset up to at least ε = 0.05. The superiority of FSFD-MMSE over Conv. MMSE would be greater as the BER level decreases. D. Doubly Selective Fading The results obtained so far are all based on the frequencyselective fading channel assumption. Let us consider a more rigorous situation, where mobility between the transmitter and receiver introduces Doppler spread, and therefore, the channel is time varying within each block, which destroys the orthogonality among subcarriers. The resulting ICI is much more difficult to handle than that caused by a single fixed frequency offset (addressed in Section IV-C). To investigate the effect of doubly selective fading on the FSFD-MMSE receiver, we assume a symbol-spaced tap-delay-line channel model with 16 paths, where each channel tap is a complex Gaussian random process independently generated with a Doppler spectrum based on Jakes’ model. The results in Fig. 6 show that the proposed FSFD-MMSE receiver is robust to doubly selective fading. Even with a normalized Doppler shift fd T = 0.02, the proposed FSFD-MMSE receiver outperforms Conv. MMSE without Doppler spread. This result can be intuitively explained by viewing Doppler effect as a superposition of multiple frequency offsets. E. Effect of Oversampling Factor Fig. 7 shows the effect of the oversampling factor S = L/N on the BER performance of the proposed FSFD-MMSE receiver. It can be clearly seen that the FSFD-MMSE receiver performs better as the oversampling factor increases, and finally, an optimal performance is reached.

Fig. 7. Effect of oversampling factor on the BER performance of the FSFDMMSE receiver.

The results indicate that S = 1 + (Q − 1/N ) (equivalently, L = N + Q − 1 = 79) is the minimum oversampling factor that enables the FSFD-MMSE receiver to achieve near optimal performance. Since the number of channel paths Q is usually much smaller than the number of subcarriers N , the FSFDMMSE receiver can use an oversampling factor that is much smaller than 2. This is a nice property for the FSFD-MMSE receiver from the complexity reduction point of view2 because L and S directly affect the dimension of (16) and the size of the matrix to be inverted in (19). An intuitive explanation for this finding is given as follows: For an OFDM system with N subcarriers in a Q-path channel, the number of degrees of freedom is N [due to N independent and identically distributed (i.i.d.) input data symbols] +(Q − 1) (due to Q i.i.d. fading paths). Oversampling beyond this number only gives additional correlated samples, which do not help much in the MMSE equalization. 2 Efficient fast algorithms for FFT with non-power-of-two lengths have been reported in [24].

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F. Effect of Timing Error In practice, timing error is also an important nonideal factor responsible for performance degradation. It can be classified into two categories: 1) sampling timing error, which is expressed as a fraction of the sampling interval T /N , and 2) symbol timing error, which is a multiple integer of T /N [23]. In the following, the effect of timing error on the performance of the proposed FSFD-MMSE receiver is qualitatively studied. Supposing that there is a sampling timing error δf , the received frequency-domain signal vector (16) becomes (omitting the noise and frequency offset for the moment) ˜ f )d . y = HΩD(δf )d = HΩd + HΩD(δ # $% &

(32)

additional ICI

The underlined part in (32) is not diagonal since Ω is not Δ diagonal (Ω = FL×N FH N ×N models the effect of frequencydomain oversampling), resulting in additional ICI, as shown in (32). This implies that the FSFD-MMSE receiver can be more sensitive to the sampling timing error than the conventional OFDM receiver. On the other hand, it is found that the late symbol timing error [23] results in the violation of ZP, thus giving rise to additional phase rotation ICI and IBI. However, for an early symbol timing error [23], as long as the sum of the timing error and multipath delay spread is smaller than the guard, no IBI will occur, and the equivalent channel matrix becomes the cyclic shift version of H in (13). Hence, the only effect of an early sampling timing error in the frequency domain is a phase rotation, which can be easily compensated by the frequency domain equalizer. It should be noted that the aforementioned effect is also faced by the conventional OFDM (see [23, p. 96–98]). Therefore, we believe that the proposed FSFD-OFDM shows a similar sensitivity to symbol timing error as the conventional OFDM. G. Comparison With Time-Domain Oversampling in [14] A low-complexity receiver using time-domain oversampling is proposed in [14] to exploit multipath diversity, which is fundamentally different from our proposed FSFD-MMSE receiver. The remarks given here highlight the differences between them. 1) In [14], time-domain oversampling is used, and the oversampling factor is an integer that is usually larger than 2. Our proposed FSFD-MMSE receiver uses frequency-domain oversampling, and the oversampling factor can be a fractional number that is much smaller than 2, as discussed in Section IV-E. The time-domain oversampling requires the sampler and analog-to-digital converter (ADC) to operate at a higher rate than before, whereas frequency-domain oversampling only needs zero-padding FFT. 2) The receiver in [14] employs maximum ratio combining (MRC), which is simple but may not effectively handle ICI caused, for example, by frequency offset. (Frequency offset is not considered in [14].) In contrast, FSFD-

Fig. 8. BER comparison between the FSFD-MMSE and TDFS-MRC receivers (QPSK, N = 64).

MMSE is based on the MMSE criterion, which makes FSFD-MMSE much more effective in coping with ICI. 3) The FSFD-MMSE receiver operating in batch mode is, in general, more complex than the time-domain counterpart of [14] due mainly to the matrix inversion involved. However, when FSFD-MMSE is adaptively implemented, this disadvantage disappears since the matrix inversion is no longer needed. The BER comparison between the FSFD-MMSE receiver and the time-domain oversampling-approach in [14] (which is termed as TDFS-MRC) is plotted in Fig. 8. The channel is adopted from [14]. Specifically, we assume that the channel has 18 multipath components and a uniform PDP, and all path arrival times are modeled as random variables uniformly distributed in [0, 14Ts ], where Ts is the sampling period. In addition, we assume that QPSK is employed, the number of subcarriers is N = 64, and the oversampling factor is S = 2 for TDFS-MRC and S = 82/64 for FSFD-MMSE. We can see from Fig. 8 that the proposed FSFD-MMSE receiver outperforms the TDFSMRC receiver with a higher diversity gain (steeper BER slope). V. C ONCLUSION Starting from the duality between the time jittering and frequency offset, we have proposed a frequency-domainoversampling-based FSFD-MMSE receiver for OFDM systems. One nice feature of the proposed FSFD-MMSE receiver is that it can extract frequency diversity from the OFDM signal, which typically enjoys no diversity gain with the conventional OFDM detection scheme. Numerical results show that the proposed FSFD-MMSE receiver outperforms the conventional (no oversampling) MMSE receiver under frequency-selective fading, frequency offset, channel estimation errors, and even doubly selective fading. It was also observed that the frequencydomain-oversampling factor can be reduced to a number given by Smin = 1 + (Q − 1/N ) (usually much smaller than 2) to minimize the receiver complexity while achieving the optimal performance promised by the frequency-domain-oversampling approach.


ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their constructive comments and suggestions, which led to a great improvement of our original paper. R EFERENCES [1] R. W. Chang, “Synthesis of band–limited orthogonal signals for multichannel data transmission,” Bell Syst. Tech. J., vol. 45, pp. 1775–1796, Dec. 1966. [2] S. B. Weinstein and P. M. Ebert, “Data transmission by frequency-division multiplexing using the discrete Fourier transform,” IEEE Trans. Commun. Technol., vol. COM-19, no. 5, pp. 628–634, Oct. 1971. [3] J. A. C. Bingham, “Multicarrier modulation for data transmission: An idea whose time has come,” IEEE Commun. Mag., vol. 28, no. 5, pp. 5–14, May 1990. [4] J. Peled and A. Ruiz, “Frequency domain data transmission using reduced computational complexity algorithms,” in Proc. IEEE ICASSP, Apr. 1980, vol. 5, pp. 964–967. [5] A. R. S. Bahai and B. R. Saltzberg, Multi-Carrier Digital Communications: Theory and Applications of OFDM. New York: Kluwer, 1999. [6] L. Hanzo, W. T. Webb, and T. Keller, Single- and Multi-Carrier Quadrature Amplitude Modulation: Principles and Applications for Personal Communications, WLANs and Broadcasting. New York: Wiley, 2000. [7] I. Koffman and V. Roman, “Broadband wireless access solutions based on OFDM access in IEEE 802.16,” IEEE Commun. Mag., vol. 40, no. 4, pp. 96–103, Apr. 2002. [8] T. Pollet, M. Van Bladel, and M. Moeneclaey, “BER sensitivity of OFDM systems to carrier frequency offset and Wiener phase noise,” IEEE Trans. Commun., vol. 43, no. 2–4, pp. 191–193, Feb.–Apr. 1995. [9] S. Patel, L. J. Cimini, and B. McNair, “Comparison of frequency offset estimation for burst OFDM,” in Proc. IEEE VTC—Spring, May 2002, vol. 2, pp. 772–776. [10] J. R. Treichler, I. Fijalkow, and C. R. Johnson, “Fractionally spaced equalizers,” IEEE Signal Process. Mag., vol. 13, no. 3, pp. 65–81, May 1996. [11] M. Luise, M. Marselli, and R. Reggiannini, “Blind subcarrier frequency ambiguity resolution for OFDM signals over selective channels,” IEEE Trans. Commun., vol. 52, no. 9, pp. 1532–1537, Sep. 2004. [12] B. Hombs and J. S. Lehnert, “Multiple-access interference suppression for MC-CDMA by frequency-domain oversampling,” IEEE Trans. Commun., vol. 53, no. 4, pp. 677–686, Apr. 2005. [13] S. M. Kay, Fundamentals of Statistical Signal Processing, Volume I: Estimation Theory. Englewood Cliffs, NJ: Prentice-Hall, 1993. [14] C. Tepedelenlioglu and R. Challagulla, “Low-complexity multipath diversity through fractional sampling in OFDM,” IEEE Trans. Signal Process., vol. 52, no. 11, pp. 3104–3116, Nov. 2004. [15] J. G. Proakis, Digital Communications. New York: McGraw-Hill, 1995. [16] S. Haykin, Adaptive Filter Theory. Englewood Cliffs, NJ: Prentice-Hall, 1995. [17] N. Benvenuto and G. Cherubini, Algorithms for Communications Systems and Their Applications. New York: Wiley, 2002. [18] T. Wang, J. G. Proakis, E. Masry, and J. R. Zeidler, “Performance degradation of OFDM systems due to Doppler spreading,” IEEE Trans. Wireless Commun., vol. 5, no. 6, pp. 1422–1432, Jun. 2006. [19] V. V. Petrov, Limit Theorems of Probability Theory: Sequences of Independent Random Variables. Oxford, U.K.: Oxford Univ. Press, 1995. [20] M. K. Simon and M.-S. Alouini, Digital Communication Over Fading Channels: A Unified Approach to Performance Analysis. New York: Wiley, 2000. [21] ETSI Normalization Committee, Doc. ETR0 230 002 Broadband Radio Access Networks (BRAN); High Performance Radio Local Area Networks (HIPERLAN) Type 2: System overview, Sophia-Antipolis, France, Apr. 1999. [22] Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications: High-Speed Physical Layer in the 5 GHz Band, IEEE Std. 802.11a, 1999. [23] M. Engels, Wireless OFDM Systems: How to Make Them Work? Norwell, MA: Kluwer, 2002. [24] G. Bi and Y. Q. Chen, “Fast DFT algorithms for length N = q ∗ 2m ,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 45, no. 6, pp. 685–690, Jun. 1998.

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Qinghua Shi received the Ph.D. degree in wireless communications from Southeast University, Nanjing, China, in 2000. From September 2000 to January 2003, he was a Research Scientist with the Centre for Wireless Communications, University of Oulu, Oulu, Finland. From February 2003 to January 2008, he was a Research Fellow, first with the City University of Hong Kong, Kowloon, Hong Kong, and then with Nanyang Technological University, Singapore. In February 2008, he joined the Department of Electronic Engineering, University of Electro-Communications, Tokyo, Japan, as an Assistant Professor. His research interests include wireless communications and related signal processing techniques.

Liang Liu (S’08) received the B.E. degree from the National University of Defense Technology, Chang’Sha, China, in 2005. Since 2006, he has been working toward the Ph.D. degree with the School of Electronics and Information Engineering, Beijing University of Aeronautics and Astronautics (now Beihang University), Beijing, China. From September 2008 to October 2010, he was an exchange student at the Positioning and Wireless Technology Centre, Nanyang Technological University, Singapore. His research interests include modulation, equalization, and signal processing for communication systems.

Yong Liang Guan (M’99) received the B.Eng. degree (with first-class honors) from the National University of Singapore, Singapore, in 1991 and Ph.D. degree from Imperial College of Science, Technology and Medicine, University of London, London, U.K., in 1997. He is currently an Associate Professor with the School of Electrical and Electronic Engineering and the Director of the Positioning and Wireless Technology Centre, School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. He has also been appointed Adjunct Professor with the University of Electronic Science and Technology of China, Chengdu, China, and a Faculty Associate with the Institute of Infocomm Research, Agency of Science, Technology and Research, Singapore. His research interests include modulation, coding, and signal processing for communication systems and information security systems. Dr. Guan is an Associate Editor for the IEEE S IGNAL P ROCESSING L ETTERS.

Yi Gong (S’99–M’03–SM’07) received the B.E. and M.E. degrees from Southeast University, Nanjing, China, in 1995 and 1998, respectively, and the Ph.D. degree from Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, in 2002, all in electrical engineering. From 1998 to 2002, he was a Research and Teaching Assistant with the Department of Electrical and Electronic Engineering, Hong Kong University of Science and Technology. He then joined the Hong Kong Applied Science and Technology Research Institute as a Member of Professional Staff. He is now with the Positioning and Wireless Technology Centre, School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, as an Assistant Professor. His research interests include cognitive radio, cooperative communications, multiple-input multiple-output, orthogonal frequency-division multiplexing, and cross-layer design for wireless systems. Dr. Gong has been an Editor for the IEEE T RANSACTIONS ON W IRELESS C OMMUNICATIONS and an Associate Editor for the IEEE T RANSACTIONS ON V EHICULAR T ECHNOLOGY.