A Blind Synchronizer for OFDM Systems Based on
SINR Maximization in Multipath Fading Channels
Wen-Long Chin and Sau-Gee Chen
Abstract—This paper presents a blind synchronizer for
or-thogonal frequency-division multiplexing (OFDM) systems based on signal-to-interference-and-noise ratio (SINR) maximization. Due to the incurred losses from intersymbol interference (ISI) and intercarrier interference (ICI) introduced by synchronization errors, the SINR of the received data drastically drops. By taking advantage of this characteristic, both the symbol time and carrier frequency offsets are intuitively estimated by maximizing the SINR metric. For the SINR metric, two blind SINR estimations are investigated. The estimations do not need prior knowledge of the channel profiles and transmitted data. As such, the pro-posed maximum SINR (MSINR) synchronization algorithm is nondata aided so that the transmission efficiency can be im-proved. Moreover, to reduce the computational complexity, the early–late gate technique is proposed for the implementation of the synchronizer. Simulation results exhibit better performance for the MSINR algorithm than conventional techniques in multipath fading channels.
Index Terms—Blind synchronization, early–late gate (ELG),
orthogonal frequency-division multiplexing (OFDM), signal-to-interference-and-noise-ratio (SINR) maximization.
I. INTRODUCTION
O
RTHOGONAL frequency-division multiplexing (OFDM) is a promising technology for broadband transmission due to its high-spectrum efficiency and its robustness to the effects of multipath fading channels and impulse noises [1]–[4]. OFDM has been adopted by many state-of-the-art communication standards such as digital video broadcasting–terrestrial, digital audio broadcasting, digital subscriber line, wireless local area network systems based on 802.11x standards, and fixed or mobile ad hoc network systems based on 802.16x standards. It has also become a key technol-ogy in mobile communication systems beyond third generation. However, OFDM systems are sensitive to synchronization errors. First, uncertain OFDM symbol arrival time introduces a symbol time offset (STO), and the coarse symbol time (CST) [5] and fine symbol time (FST) [6], [7] estimations are needed. Second, the mismatch of the oscillators in carrier frequenciesManuscript received September 17, 2007; revised March 6, 2008, May 3, 2008, and May 4, 2008. First published May 14, 2008; current version published February 17, 2009. This work was supported in part by Grant NSC 95-2219-E-009-004 and Grant MOEA 95-EC-17-A-01-S1-048, Taiwan. The review of this paper was coordinated by Dr. K. T. Wong.
W.-L. Chin is with the Department of Engineering Science, National Cheng Kung University, Tainan 701, Taiwan (e-mail: johnsonchin@pchome.com.tw). S.-G. Chen is with the Department of Electronics Engineering and Institute of Electronics, National Chiao Tung University, Hsinchu 30050, Taiwan.
Digital Object Identifier 10.1109/TVT.2008.925320
(CFs) between the transmitter and the receiver produces a CF offset (CFO) so that the resulting fractional CFO (FCFO) [5], integral CFO (ICFO) [8], and residual CFO (RCFO) [6], [7] have to be eliminated. Moreover, the mismatch of sampling clocks between the digital-to-analog converter and the analog-to-digital converter introduces a sampling CFO (SCFO) [7].
The symbol time (ST) estimation is usually the first step in an entire OFDM synchronization process, because it provides an estimated symbol boundary for the remaining synchronization steps. When the ST is located in the intersymbol interference (ISI)-free region, no ISI results, and only phase rotation in the demodulated frequency-domain data occurs, which can simply be compensated by using the single-tap equalizer. If channels are time variant, aside from ISI, extra channel-induced intercarrier interference (ICI) is inevitable. The CFO and SCFO introduce additional ICI.
In [5], the STO and the FCFO are jointly estimated by a delayed-correlation algorithm. It is a maximum-likelihood (ML) estimation and is only good for the additive white Gaussian noise (AWGN) channels. In [9], a new method that uses training symbols in time domain was proposed. However, its correlation result exhibits an uncertain plateau in multipath fading channels [10]. A remedy for this ambiguity is pro-posed in [10]. Some techniques [6], [9]–[12] have good ST performances. However, extra time-domain training symbols are needed. Although the technique in [13] can identify the ISI-free region in multipath fading channels, for an accurate ST estimation, it may involve many symbols. In [6] and [7], the channel frequency response (CFR) must first be estimated. The inverse fast Fourier transform (IFFT) is then applied to obtain the channel impulse response (CIR), which is then used to adjust the symbol boundary. The work in [14] treats the CST estimation in multipath fading channels. Some works locate the symbol boundary at the sampling instance that has the minimum interference [15]–[17] or maximum correlation result of the cyclic prefix (CP) [5]. The interference is used as the metric in the minimum-interference ST estimators.
The proposed scheme aims to effectively reduce the synchro-nization errors, including the STO and the CFO, so that the signal-to-interference-and-noise-ratio (SINR) is maximized. In other words, we consider not only the interference power but also the signal power. We investigate two blind SINR estimations (without needing prior knowledge of the channel profiles and transmitted data) for the proposed maximum-SINR (MSINR) synchronization algorithm. The MSINR algorithm is nondata aided (NDA) so that the bandwidth efficiency can be improved. In addition, to reduce the computational complexity, an early–late gate (ELG) technique is also proposed for the implementation of the new MSINR algorithm.
Fig. 1. Simplified OFDM system model. The MSINR synchronizer operates in the frequency domain by utilizing the FFT operations. A detailed block diagram of the synchronizer can be found in Fig. 7.
Fig. 2. Three different ST regions.
The rest of this paper is organized as follows. The OFDM system and signal models in multipath fading (slow-fading, block-fading, flat fading) channels are introduced in Section II. The MSINR synchronization algorithm and the ELG technique are detailed and analyzed in Section III. The performance is analyzed in Section IV. Simulation results are provided in Section V. Finally, we conclude our paper in Section VI.
II. OFDM SYSTEM ANDSIGNALMODELS
In the following discussion, all the quantities indexed with l belong to the lth symbol. A simplified OFDM system model is shown in Fig. 1, where Xl, k/ ˜Xl, k is the transmitted/received
frequency-domain data at the kth subcarrier, xl,nis the
trans-mitted time-domain data after the IFFT, and ˜xl,nis the received
time-domain data, where −NGn< N , N is the number of
subcarriers, and NGis the CP length. 1/TSis the sampling
fre-quency, εf is the CFO (normalized by the subcarrier spacing),
fcis the CF, nΔis the estimated STO, and εfis the estimated
CFO. On the transmitter side, N complex data symbols are modulated onto N subcarriers by using the IFFT. The last NG IFFT output samples are copied to form the CP that is
inserted at the beginning of each OFDM symbol. By inserting the CP, a guard interval is created so that ISI can be avoided and the orthogonality among subcarriers can be sustained. The
receiver uses the fast Fourier transform (FFT) to demodulate received data.
As shown in Fig. 2, an estimated ST generally falls into one of the three regions of an OFDM symbol: 1) the Bad-ST1 region, 2) the Good-ST region (also known as the ISI-free region), and 3) the Bad-ST2 region in which the STO nΔ is
confined within the ranges of −NG ≤ nΔ≤ −NG+ τd− 1,
−NG+ τd≤ nΔ≤ 0, and 1 ≤ nΔ≤ N − 1, respectively,
where τdis the maximum delay spread of the channel. The
Bad-ST1 and Good-ST regions are in the guard interval of length NG. When the ST is located in the Good-ST region, no ISI
results; however, when the ST is located in the Bad-ST1 and Bad-ST2 regions, the lth symbol has ISI from the (l− 1)th and (l + 1)th symbols, respectively. There is freedom to select the ST in the Good-ST region. This region is obviously defined by the channel length. Hence, for good ST estimation, one should consider the channel effect to avoid the ISI such that the SINR is maximized. The STO is with reference to the ideal ST of the lth symbol, which is marked by the time index at zero. Detailed analysis of the received frequency-domain data (at the kth subcarrier) in these three regions can be found in [18], and the results are rewritten as follows (for only the Bad-ST2 region):
˜
where ˜ Xl, kd Δ =N− nΔ N HkXl, kW (−lNsεf−knΔ) N (2)
is the desired data, Hk is the CFR at the kth subcarrier, WN
is e−j2π/N, and NS = N + NG is the OFDM symbol length,
including the CP. In addition
˜ Nk Δ = 1 N m=k N−1 n=0 W−n(m−k+εf) N HmXl,m × W(−lNsεf−knΔ) N + vk− ˜Xl, kici + ˜Xl, kisi (3)
is the combined interference and the AWGN vk, where
˜ Xl, kici = 1 N m=k N−1 n=N−nΔ W−n(m−k+εf) N × HmXl,mW (−lNsεf−knΔ) N (4)
is the ICI term, and
˜ Xl, kisi = 1 N m N−1 n=N−nΔ W−n(m−k+εf) N HmXl+1,m × W(−lNsεf−k(nΔ+NG)) N (5)
is the ISI term. Note that the dependence of the received frequency-domain data on the STO and CFO is dropped for notational clarity. It can easily be shown that the desired signal power is σX2˜d l, k = E ˜Xd l, k 2 = N− nΔ N 2 m|X|2|Hk|2 (6)
and the combined interference and AWGN power is σN2˜ k= E | ˜Nk|2 =m|X|2 N2 ⎧ ⎨ ⎩ m=k N−1 n1=0 N−1 n2=0 W−(n1−n2)(m−k+εf) N |Hm| 2 + N−1 n1=N−nΔ N−1 n2=N−nΔ W−(n1−n2)εf N |Hk|2 + 2 m=k N−1 n1=N−nΔ N−1 n2=N−nΔ × W−(n1−n2)(m−k+εf) N |Hm|2+ σ2vk ⎫ ⎬ ⎭ (7) where m|X|2 = E[|Xl, k|2] = E[|Xl+1, k|2] is the transmitted
data power, and σ2
vk is the AWGN power. Finally, the
theoretical SINR can be written as
ηth(nΔ, εf) = 1 N k σ2 ˜ Xd l, k 1 N k σ2 ˜ Nk (8)
which is highly dependent on the STO and CFO.
III. PROPOSEDMSINR SYNCHRONIZATIONTECHNIQUE When the STO or CFO increases, the signal power de-creases, whereas the interference power increases [19], [20], as also suggested in (6) and (7), respectively. Consequently, it is evident that the SINR is a good metric for indicating the syn-chronization offsets. On the other hand, minimum-interference synchronizers [15]–[17] only consider the interference power. As such, they may have performance losses, because they do not simultaneously consider both the signal and interference power.
A. Blind SINR Estimations for the MSINR Synchronization Algorithm
Before introducing the MSINR synchronization technique, the SINR estimation for (8) is first investigated. It can be real-ized that a good SINR estimation for an effective synchronizer in practical environments should have the following proper-ties: 1) The estimated SINR profile is a unimodal function of the STO and CFO and has a maximum value at the ideal synchronization point, and 2) the variance of the estimation is small. In the following discussion, we will investigate two blind SINR estimations based on the second and fourth moments of the envelope and evaluate their conformity with these two properties.
1) Time-Oriented SINR Estimation: After an extensive sur-vey of the SINR estimation techniques, we found that in [21], the SINR estimation of a real-valued single-carrier signal is useful due to its simplicity and robustness. It was derived by utilizing the second and fourth moments of the envelope in slow-fading channels. We generalize and extend it to the complex-valued signals for the OFDM systems. It is omitted in this paper because the derivation is similar to [21]. The key parameters and results are written as follows. First, one can define the following second and fourth moments of | ˜Xl, k| at
the kth subcarrier based on time sample averages: Mk Δ =1 L l | ˜Xl, k|2 (9) Qk Δ =1 L l | ˜Xl, k|4 (10) Rk Δ =1 L l | ˜Xl, k|2| ˜Xl+1, k|2 (11)
where L is the number of OFDM symbols. Then, the signal power Sk and interference power Ik at the kth subcarrier can
be determined as Sk = Mk2+ Rk− Qk 1/2 (12) Ik = Mk− Sk (13)
respectively. Finally, the SINR can be obtained by averaging over all subcarriers (in the frequency domain)
η (n Δ, εf) = 1 N k Sk 1 N k Ik . (14)
2) Frequency-Oriented SINR Estimation: In the time-oriented method, a slow-fading channel condition is assumed. The channels are possibly fast fading; thus, it is inadequate to average over symbols in (9)–(11). To reflect dynamical environments and derive the SINR profile with enough data samples in a few symbols, we propose to average over all N subcarriers in the frequency domain first. Moreover, under the condition that only one OFDM symbol is available (i.e., L = 1), it can be shown that M2
k = Qk. Therefore, the signal power Sl
and interference power Ilcan be written as Sl= 1 N k ˜Xl, kX˜l+1, k∗ (15) Il= 1 N k | ˜Xl, k|2 1 N k | ˜Xl+1, k|2 − S l (16)
respectively. Note that the geometric average of the total power of two symbols is used to represent the total power. The SINR estimate ηlof the lth symbol can be written as
ηl= SlIl. (17) Moreover, the estimation (17) can be averaged over symbols to improve the accuracy.
3) Comparison of the SINR Estimations: The time-oriented (14) and frequency-oriented (17) SINR estimations under var-ious signal-to-noise ratios (SNRs) are qualitatively compared (using the estimated SINR profile) as follows and will quan-titatively be compared (using the estimations’ variances) in Section V-A.
We assume an OFDM system with the following specifi-cations: N = 256, NG = 32, fc= 2.4 GHz, Ts= 115.2 μs,
and the signal bandwidth is 2.5 MHz. The time-oriented and frequency-oriented SINR estimations against the STO in a time-invariant channel are demonstrated in Fig. 3(a) and (b). The channel is assumed to have NGpaths. Therefore, the
Good-ST region contains only one sample (i.e., the zeroth sample in Fig. 2). The channel taps are randomly generated by indepen-dent zero-mean, unit-variance complex Gaussian variables. As shown in the figure, both estimated SINR profiles are unimodal and have their maximum values in the Good-ST region (at the zeroth sample). The estimated SINR profiles in a time-variant channel are shown in Fig. 4. The normalized Doppler frequency (NDF) is 0.035. As shown in Fig. 4 and depicted in [21], the time-oriented estimation is more sensitive to the NDF than the frequency-oriented estimation.
The time-oriented and frequency-oriented SINR estimations against the CFO in a time-invariant channel are demonstrated in Fig. 5(a) and (b), where both estimated SINR profiles are unimodal and have their maximum values at CFO = 0. The estimated SINR profiles in a time-variant channel are
demon-Fig. 3. SINR estimations (14) and (17) against the STO in a time-invariant channel are shown in (a) and (b), respectively. CFO = 0. N = 256. L = 50 symbols.
Fig. 4. SINR estimations (14) and (17) against the STO in a time-variant channel are shown in (a) and (b), respectively. NDF = 0.035. CFO = 0. N = 256. L = 50 symbols.
strated in Fig. 6. The NDF is 0.035, and the time-oriented estimation is similarly more sensitive to the NDF than the frequency-oriented estimation.
In summary, the frequency-oriented estimation is more suit-able for the synchronization algorithm than the time-oriented estimation in terms of the estimated SINR profile (i.e., the mentioned first good property of an estimator).
Fig. 5. SINR estimations (14) and (17) against the CFO in a time-invariant channel are shown in (a) and (b), respectively. STO = 0. N = 256. L = 50 symbols.
Fig. 6. SINR estimations of (14) and (17) against various CFOs, under the time-variant channel, are shown in (a) and (b), respectively. STO = 0.
N = 256. L = 50 symbols.
B. MSINR Synchronization Algorithm
1) MSINR Global Search Algorithm (MSINR-GSA): As we have observed and concluded, the SINR drastically drops due to the incurred interferences from ICI and ISI. Hence, one can intuitively estimate the STO and CFO by maximizing (17) as
(nΔ,GSA(l), εf,GSA(l)) = arg max nΔ,εf{η
l(nΔ, εf)} (18)
where nΔ,GSA(l) and εf,GSA(l) are the lth symbol’s STO
and CFO estimates of the 2-D MSINR-GSA, respectively. This is a complicated 2-D search problem. The SINR profile is a unimodal function of the STO and CFO; therefore, without losing too much performance, we separate it into two 1-D processes, i.e., the STO search and the CFO search, as follows:
⎧ ⎨ ⎩ n
Δ,GSA(l) = arg maxn
Δ η l(nΔ, εf,GSA(l− 1)) ε
f,GSA(l) = arg maxε
f η l(nΔ,GSA(l− 1), εf) (19) where nΔ,GSAand εf,GSAare the lth symbol’s STO and CFO estimates of the 1-D MSINR-GSA, respectively.
2) MSINR-ELG-ST Recovery Loop: The computational complexity of (19) is still high, because the frequency-domain data and their SINRs have to be obtained over all possible synchronization points. To reduce the computational complex-ity, instead of searching the whole range of nΔ, it is assumed
that the signal is sampled early at nΔ,ELG(l)− τΔand late at
n
Δ,ELG(l) + τΔ, where nΔ,ELG(l) is the estimated STO of
the MSINR-ELG-ST, and τΔ is the time shift relative to the
estimated symbol boundary. The estimates (17) at these two symbol boundaries are subtracted and filtered to form an error signal eΔ(l) [16], [17]. If the samples are, indeed, taken at the
left and right samples with the same distance relative to the ideal ST, the error signal is zero for flat-fading channels. For frequency-selective channels, the SINRs at these two samples are generally not the same; however, they are the same under the assumption that the mean of the channel power E{|Hk|2}
is the same at all subcarriers. Therefore, the estimated symbol boundary is taken at the midpoint between the two samples. Otherwise, the nonzero error signal is produced to fine-tune the symbol boundary and iteratively make the error signal become zero. The S-curve of the timing discriminator is given by eΔ(l + 1) = ηl+1(nΔ,ELG(l)− τΔ, εf,ELG(l))
− η
l+1(nΔ,ELG(l) + τΔ, εf,ELG(l)) (20)
where εf,ELG(l) is the estimated CFO of the lth symbol.
The MSINR ST synchronizer implemented by the ELG is shown in Fig. 7(a). An approximated linear model of the ELG timing recovery loop is shown in Fig. 8, where
F (z) loop filter [22]; KF gain of the loop filter;
KI intrinsic gain of the timing detector;
KV gain of the voltage-controlled oscillator;
NΔ(z) z-transform of nΔ;
NΔ(z) z-transform of nΔ,ELG(l);
EΔ(z) z-transform of eΔ(l);
EΔ (z) z-transform of the filtered error signal eΔ(l). The ST is iteratively adjusted by the VCO as follows:
n
Δ,ELG(l) = nΔ,ELG(l− 1) + KVNSTSeΔ(l). (21)
The closed-loop system equation L(z)= NΔ Δ(z)/NΔ(z) can
be shown as
L(z) = KTz
−1(1− az−1)
1 + z−1(KT − 2) + z−2(1− KTa)
Fig. 7. Architecture of the MSINR-ELG recovery loop. (a) ST synchronizer. (b) CF synchronizer. Note that the symbol indices l in nΔ,ELG(l) and εf,ELG(l)
are dropped in this figure for notational simplicity.
Fig. 8. Approximated discrete-time model of the ELG recovery loop of the ST synchronizer.
where KT
Δ
= KFKIKVNSTS is the overall loop gain. The
loop is stable under the conditions of 0 < a < 1 and 0 < KT <
4
1 + a. (23) With (22), it can be shown that limz→1L(z) = 1. From the final
value theorem, this implies that the estimated ST converges toward the ideal ST.
3) MSINR-ELG-CF Recovery Loop: The MSINR CF chronizer is shown in Fig. 7(b) and is similar to the ST syn-chronizer, except that the signal is assumed to be shifted in frequency early by εf,ELG(l)− τfand late by εf,ELG(l) + τf,
where τf is the frequency shift relative to the estimated CFO.
The S-curve of the frequency discriminator is given by ef(l + 1) = ηl(nΔ,ELG(l), εf,ELG(l)− τf)
− η
l(nΔ,ELG(l), εf,ELG(l) + τf). (24)
The CF synchronizer utilizes a similar ELG recovery loop as the ST synchronizer; thus, it is omitted in this paper.
C. ELG Loop Analysis and Design
With (8), by excluding the AWGN, a typical plot of the S-curve (20) (of the timing discriminator) under various CFOs is shown in Fig. 9, which shows that a larger CFO results in a smaller S-curve slope than a smaller CFO, so it is a KT value.
This phenomenon reveals the degradation of the error-tracking performance. The S-curve of the CF synchronizer can similarly be shown and omitted here.
Before introducing the loop design, it must be emphasized that there is no unique optimum design that can be applied to all conditions to the well-known compromise problem between the loop jitter and acquisition time [23], [24]. The overall loop gain is affected by many factors, including all factors that influ-ence the SINR such as synchronization errors and impairments suffered from the channels. In addition, the time shift τΔof the
Fig. 9. Theoretical S-curve eΔof the ST synchronizer under various CFOs.
τΔ= N/4.
the overall loop gain. Before determining the overall loop gain, the intrinsic gain KIhas to be determined first.
By designing for the case of SNR = 20 dB and consider-ing the acquisition range, we set the τΔ value to 0.25. The
intrinsic gain KI is determined as 32 to best fit the S-curve.
The remaining parameters of the ELG loop are configured as follows: KF = 1, a = 0.9997, and KVNSTS= 2× 10−3.
Finally, KT and the normalized loop bandwidth BNSTS can
be shown to be 0.064 and 0.02, respectively, where B is the loop bandwidth. Therefore, the rise time of the loop is roughly around 50 symbols.
D. Computational Complexity
The computational complexity is dominated by the four FFTs in Fig. 7. An N -point radix-2 FFT roughly requires (N/2) log2N complex multiplications. The squared magni-tude of a complex number only costs two real multiplica-tions and one addition, so a squared magnitude operation is roughly counted as a 0.5 complex multiplication. Therefore, the estimation (17) requires 2N complex multiplications and N absolute operations. The total amount of complexity is N ((1/2) log2N + 2) complex multiplications and N absolute operations for each (early/late) branch. Moreover, one division and one square-root operations are also required in (17) and (16), respectively, which are much less than the FFT opera-tion. The computational complexities of the blind frequency-domain synchronization algorithms [15] and [16] in terms of complex multiplication are roughly N ((1/2) log2N + 1.5)
and N ((1/2) log2N + 0.75), respectively. As shown in this
paper, the computational complexity is similar to those in [15] and [16], because the complexity is dominated by the FFT operation, which is the same for all the compared frequency-domain algorithms.
IV. PERFORMANCEANALYSIS A. CRB of the SINR Estimation
To assess the performance of the SINR estimation, the Cramér–Rao lower bound is used as reference. The Cramér–
Rao Bound (CRB) of the SINR estimation, for a complex-valued signal embedded in the Gaussian noise, has been derived in [25]. The SINR estimation variance ση2is lower bounded by the CRB and is rewritten as
σ2 η η2 ≥ 2 2 N η + 1 N (25) where η is the true SINR. The CRB of the SINR estimation is used to assess the performance of the synchronizers in the following section.
B. Performance Analysis of the MSINR-ELG Synchronizers The lower bound of the mean-squared error (MSE) for the ST tracking loop is derived as follows. As depicted in Fig. 9, in the vicinity of the ideal synchronization point, the S-curve is roughly linear and can be approximated as
eΔ≈ KInΔ (26)
where KI is the slope of the S-curve at the ideal synchro-nization point. With (20) and (26), assuming that the SINR variances are equivalent for both the early and late branches in the vicinity of the ideal synchronization point, the ST variance σ2 nΔ can be determined as σn2Δ = 2σ2 η KI2. (27)
Note that the factor in (27) is 2 because there are two subtraction terms in the RHS of (20). Furthermore, if we assume that the ST estimation error is a wide-sense stationary white-noise process, after being filtered by L(z), as depicted in Fig. 8, the MSE of the synchronizer can be written as
E (n Δ,ELG(l)− nΔ)2 ≥ σ2 n Δ =σ 2 n Δ 2πj C L(z)L(z−1)z−1dz (28) where j =√−1. By inserting (27) into (28), we have
E (nΔ,ELG(l)− nΔ)2 ≥ 2 KI2 σ2 n Δ 2πj C L(z)L(z−1)z−1dz. (29) Next, the contour integral in (29) can explicitly be solved. With (22) and the parameters in Section III-C, the partial fraction expansion of L(z) can be shown as
L(z) = −1.43309 × 10
−6
z− 9.99699 × 10−1 +
6.40014× 10−2
z− 9.36301 × 10−1. (30) Consequently, the contour integral in (29) can be solved using the Cauchy integral theorem, and (29) can be reduced to
E (n Δ,ELG(l)− nΔ)2 ≥ 3.32077 × 10−2×2σ 2 η KI2. (31)
Fig. 10. SINR estimation variances against the SNR. NDF = 0.035. Finally, by inserting (25) into (31), the MSE lower bound of the tracking loop can be determined as
E (nΔ,ELG(l)−nΔ)2 ≥1.32831×10−1×η(2+η) KI2N . (32) Clearly, the bound depends on the SINR, the characteristic of the S-curve, and the closed-loop system equation. The MSE lower bound of the CF synchronizer uses the same inequality as (28), except that the characteristic of the S-curve and the closed-loop system equation are replaced with those used in the CF synchronizer.
V. SIMULATIONS
A. Comparison of the Proposed SINR Estimations
The frequency-oriented (17) and the time-oriented (14) es-timations have qualitatively been compared by the estimated SINR profiles in Section III-A3. In this section, these two differ-ent estimations will quantitatively be evaluated by comparing their variances based on the same system in Section III-A3. The estimation variances against the SNR are plotted in Fig. 10. The CRB (25) is also shown. As it can be seen, the frequency-oriented estimation has a smaller variance and is closer to the CRB than the time-oriented estimation.
In summary, the frequency-oriented estimation is more suitable for synchronization error estimations than the time-oriented estimation in terms of the estimation’s variance (i.e., the mentioned second good property of an estimator in Section III-A).
B. Performance of the Proposed Synchronizer
We use Monte Carlo simulations to evaluate the perfor-mance of the synchronizers and consider an OFDM system of N = 256 subcarriers and a guard interval of NG= N/8 = 32
samples. Only data subcarriers are assumed. The simulated modulation scheme is quadrature phase-shift keying. The signal bandwidth is assumed to be 2.5 MHz, and the radio frequency is 2.4 GHz. The subcarrier spacing is 8.68 kHz. The OFDM symbol duration is 115.2 μs. The CFO is 10% of the
sub-Fig. 11. MSEs of the compared ST synchronizers against the SNR. NDF = 0.0. CFO = 0.1.
carrier spacing, i.e., 0.87 kHz, in all simulations. To verify the performance of the proposed technique, the channel is assumed to have NG paths. Therefore, the Good-ST region
contains only one sample. The channel taps hτ are randomly
generated by independent zero-mean unit-variance complex Gaussian variables withτE{|hτ|2} = 1 for each simulation
run, where τ is the path delay. In each simulation run, 10 000 OFDM symbols are tested. We evaluate the performances of the synchronizers by using the estimations’ normalized MSE. All the results are obtained by averaging more than 2000 independent channel realizations.
1) ST Estimation Performance: For the ST estimation, we compare the proposed technique with the following blind or NDA techniques by assuming the same system settings: 1) blind minimum-interference ST estimator (Min-ITF 1) [16], 2) blind minimum-interference ST estimator (Min-ITF 2) [15], and 3) NDA ML ST estimator (TD-NDA) [5]. The ST estimator in [17] is data aided, so it is not considered here.
The MSEs of the estimated ST against the SNR in time-invariant and time-variant channels are shown in Figs. 11 and 12, respectively. The NDF = 0.035 corresponds to a maxi-mum Doppler frequency of 303.8 Hz, i.e., a mobile speed of 117 km/hr. In addition to the MSEs of the MSINR-ELG-ST and MSINR-GSA (18), the CRB (29) is also shown in Fig. 11. Note that KI is obtained by simulation. As it can be seen, the MSINR synchronizer achieves a smaller MSE than the compared synchronizers. The performance of the TD-NDA is not as good as the compared techniques, because the correlation result of the CP is vulnerable to the incurred ISI. In addition, the performance of the GSA is better than the MSINR-ELG-ST. However, their performances are not too far apart. Therefore, the complexity can greatly be reduced by the ELG technique while not losing too much performance.
The estimated STs of the MSINR-ELG-ST against symbol index under various SNRs in time-invariant and time-variant channels are shown in Figs. 13 and 14, respectively. As shown in these figures, the estimated ST is around the Good-ST region (at the zeroth sample). However, the estimated ST gradually drifts away from the Good-ST region when the SNR decreases.
Fig. 12. MSEs of the compared ST synchronizers against the SNR. NDF = 0.035. CFO = 0.1.
Fig. 13. Acquisition time of the MSINR ST synchronizer under various SNRs. NDF = 0.0. CFO = 0.1.
2) CFO Estimation Performance: For the CFO estimation, we compare the proposed technique with the TD-NDA CFO estimation in [5]. The MSEs of the estimated CFO against the SNR in time-invariant and time-variant channels are shown in Figs. 15 and 16, respectively. In addition to the MSEs of the MSINR-ELG-CF and MSINR-GSA (18), the CRB is also shown in Fig. 15, where the MSINR synchronizer achieves a smaller MSE than the compared synchronizer. In addition, the ELG technique greatly reduces the complexity of the MSINR-GSA with a close performance to it.
The estimated CFOs of the MSINR-ELG-CF against sym-bol index under various SNRs in invariant and time-variant channels are shown in Figs. 17 and 18, respectively, where the estimated CFO is around the true CFO = 0.87 kHz. When the SNR decreases, the acquisition time apparently in-creases. The acquisition time is affected by the closed-loop bandwidth of the ELG recovery loop. When the SNR decreases, the closed-loop gain and bandwidth accordingly decrease. This slows down the acquisition time.
Fig. 14. Acquisition time of the MSINR ST synchronizer under various SNRs. NDF = 0.035. CFO = 0.1.
Fig. 15. MSEs of the compared CF synchronizers against the SNR. NDF = 0.0. CFO = 0.1.
Fig. 16. MSEs of the compared CF synchronizers against the SNR. NDF = 0.035. CFO = 0.1.
Fig. 17. Acquisition time of the MSINR CF synchronizer under various SNRs. NDF = 0.0. CFO = 0.1.
Fig. 18. Acquisition time of the MSINR CF synchronizer under various SNRs. NDF = 0.035. CFO = 0.1.
VI. CONCLUSION
In the literature, the synchronization problem has been treated by the viewpoint of the estimation theory, for example, the ML approach. In this paper, we have intuitively dealt with this problem by maximizing the SINR. The proposed technique is NDA and independent of the signal structure. For the MSINR synchronization algorithm, two blind SINR estimations have been investigated. The frequency-oriented SINR estimation has been found to be more suitable for the MSINR algorithm than the time-oriented SINR estimation. Moreover, it has been shown that the proposed ELG tracking loop is a good compro-mise between the performance and complexity.
ACKNOWLEDGMENT
The authors would like to thank the Editor and the anony-mous Reviewers for their helpful comments and suggestions, which have improved the quality of this paper.
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Wen-Long Chin received the B.S. degree in
elec-tronics engineering from National Chiao Tung Uni-versity, Hsinchu, Taiwan, in 1994, the M.S. degree in electrical engineering from the National Taiwan Uni-versity, Taipei, Taiwan, in 1996, and the Ph.D. degree in electronics engineering from National Chiao Tung University, Hsinchu, in 2008.
In February 2009, he joined National Cheng Kung University, Tainan, Taiwan, as an Assistant Profes-sor. His research interests lie in the areas of (VLSI) signal processing, communications, and networks. Before holding the faculty position, he had been with the Hsinchu Science Park for over 11 years, where he was in charge of communications and IC designs.
Sau-Gee Chen received the B.S. degree from the
National Tsing Hua University, Hsinchu, Taiwan, in 1978 and the M.S. and Ph.D. degrees in electrical engineering from the State University of New York, Buffalo, in 1984 and 1988, respectively.
From 2003 to 2006, he was the Director of the Institute of Electronics, Department of Electronics Engineering, National Chiao Tung University, Hsinchu, where he is currently a Professor. His research interests include digital communication, multimedia computing, digital signal processing, and VLSI signal processing. He has published more than 70 conference proceedings and journal papers and is the holder of several U.S. and Taiwan patents.
