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Lowcomplexity adaptive iteration algorithm for frequency tracking in OFDM systems
EURASIP Journal on Wireless Communications and Networking volume 2016, Article number: 57 (2016)
Abstract
The carrier frequency offset (CFO), destroying the orthogonality between subcarriers, greatly degrades the performance of an orthogonal frequencydivision multiplexing system. One of the most frequently used ways for a CFO estimator design is to adopt maximumlikelihood (ML) estimation, achieving high accuracy tightly close to the CramerRao lower bounds (CRLBs).
One of the MLbased algorithms, called linearlycombined CFO (LCCFO), evaluates all the singletimeslot CFO estimates first and then linearly combined these CFO estimates in the minimum meansquareerror sense. Its tracking range is quite wide up to half the carrier spacing, and convergence speed is very fast, costing only several tens of iterations; moreover, its meansquare error (MSE) performances are very much close to CRLBs at mediumtohigh signaltonoise ratio (SNR) values. However, a set of arctangent functions is needed to be evaluated for each iteration, which increases the computational complexity. In this article, a low complexity, called simplified LCCFO (SLCCFO), is proposed that the set of arctangent functions are replaced by lowcomplexity limiters, resulting in simplifying the receiver design and reducing the computational load while keeping nearly the same tracking range and MSE performances. With proper choice of a parameter, SLCCFO even shows faster convergence speed and lower MSE value at low SNR, compared with LCCFO. Simulation results demonstrate all these aforementioned properties.
Introduction
Orthogonal frequencydivision multiplexing (OFDM) transforms a frequencyselective multipath fading channel into multiple frequencyflat channels that makes appealing for designing a receiver using simple signal processing techniques [1]. The orthogonality between subcarriers, allowing overlapping between subbands without introducing interchannel interference, provides high spectral efficiency. However, the existence of carrier frequency offset (CFO) due to the Doppler spread and/or the instability of the local oscillators destructs the orthogonality, resulting in serious performance degradation [2–4]. Therefore, the demand for CFO estimation with high accuracy is essential to OFDM systems.
There are many correlationbased algorithms proposed for CFO estimation in [5–9]. They are proved useful for CFO acquisition, acquiring coarse CFO estimates. To secure CFO estimates with high accuracy, the technique using maximumlikelihood (ML) estimation perhaps is the best choice for CFO tracking, securing fine CFO estimates. In [10–13], the ML CFO estimators are developed. However, they can only work for systems employing repeated preambles. The others proposed in [14–20] have no such a limitation. Applying ML estimation for estimating CFO is usually cumbered by the nonlinear nature of the likelihood function. The problem of local extremum or multiple solutions arises. Many papers resort to making approximation by truncating high order terms of a Taylor series [14–16, 18–20]. Recently, some research papers consider the inclusion of phase noise when performing CFO estimation [21, 22]. The source of phase noise is the instability of local oscillators, which can be modeled as timevarying correlated noise [23]. In [21], the truncation of high order Taylor series is used for acquiring the approximate CFO estimate. One of the two proposed methods in [22] must resort to exhaustive search for CFO estimation, and the other is correlation based, which is applicable to the systems employing repeated preambles. The closedform formula for the ML CFO estimate without carrying out Taylor series truncation is reported in [17], where the closedform solutions to all the singletimeslot samples are first found and then those estimates are linearly combined in the sense of minimum meansquare error (MSE). Thus, we call it linearly combined CFO (LCCFO) estimator. The conducted simulations for it show that the MSE performances of the CFO estimator are very much close to the CramerRao lower bounds (CRLBs).
In this article, we develop a lowcomplexity CFO estimator, called simplified LCCFO (SLCCFO), adapted from LCCFO developed in [17]. The main idea of SLCCFO having a low complexity lies in replacing a set of arctangent functions in LCCFO by simple, lowcomplexity limiters. As for the implementation issue of an arctangent function, the use of lookup table stored in memory perhaps is attractive for highspeed arctangent computation. However, this simple method requires high amount of memory to provide accuracy. In [24], the rational approximation for an arctangent function is proposed. Dividers and multipliers are needed for the implementation. The circuit of coordinate rotation digital computer [25] is another choice for angle estimation. However, the quantization issue affecting the precision [26] must be considered. Since no arctangent functions are required in our proposed method, the receiver design is simplified and requires less computational loading. The effects on the replacement with limiters will be thoroughly discussed and examined by computer simulation. The tracking range, the convergent speed, and the MSE for CFO estimation as well as channel impulse response (CIR) estimation are included.
The rest of the article is organized as follows. Section 2 introduces the system model and derives the singetimeslot CFO estimator. Section 3 reviews the LCCFO algorithm. In Section 4, SLCCFO, adapted from LCCFO, is developed. Simulation results are then presented in Section 5, showing the advantages of SLCCFO with detailed discussions. Finally, conclusions are drawn in Section 6.
System model and singletimeslot CFO estimator
Consider an OFDM system with N subcarriers. Let X _{ k } be the transmitted kth subcarrier symbol, H _{ k } be the kth subcarrier frequency channel response, k = 0, 1, ⋯, N − 1, and δ be the carrier frequency offset normalized to the carrier spacing. At the receiver, after discarding the cyclic prefix, the complex baseband received signal at the nth time slot, n = 0, 1, ⋯, N − 1, in an OFDM block can be expressed as
where w _{ n } is the additive white Gaussian noise with zero mean and variance \( {\sigma}_w^2 \) at the nth time slot sample and
is the noisefree, CFOfree received signal at the nth time slot.
The ML estimator of δ for each time slot sample in (1) can be obtained as [17]
where Re{⋅} and Im{⋅} mean real part and imaginary part, respectively, and ε _{ n } is defined as
Notice that the parameter ε _{ n } is not be included in [17]. We have slightly modified the singletimeslot CFO estimator developed in [17]. The functionality of ε _{ n } is explained as follows. The arctangent function tan^{− 1}{⋅} in (3) represents the angle of the complex product \( {r}_n{y}_n^{*} \); moreover, the range of tan^{− 1}{⋅} is from − π/2 to π/2. Aided by the additional parameter ε _{ n } following tan^{− 1}{⋅}, the angle of \( {r}_n{y}_n^{*} \) represented can be extended to the range from − π to π. Furthermore, for the N received signal samples {r _{ n }, n = 0, 1, ⋯, N − 1} of (1), δ exists only in the N − 1 received signal samples {r _{ n }, n = 1, ⋯, N − 1}. The number of the singletimeslot CFO estimators of (3) is thus N − 1.
Review of the LCCFO algorithm
Linearly combining these N − 1 singletimeslot estimators of (3), the LCCFO estimator can be expressed as [17]
where
is obtained from minimizing the estimate error \( \varDelta \delta ={\widehat{\delta}}_{LC}\delta \) in a linearminimumMSE sense.
The exact values of the noisefree, CFOfree received signals {y _{ n }, n = 1, 2, ⋯, N − 1} in Eqs. (3), (4), and (6) are unknown to the receiver due to the lack of the knowledge of CIR. By applying ML estimation to estimate CIR, the resulting estimator of y = [y _{0}, y _{1}, ⋯ y _{ N − 1}]^{T} with the superscript T denoting transpose can be readily obtained as
where
with
and
The symbol diag{⋅} represents a diagonal matrix with its diagonal entries being inside the braces.
The parameter δ in (7) is still unknown at this present. To solve this problem, (5) and (7) are updated alternatively, which is called Version A in [17]. Another algorithm, called Version B in [17] or LCCFO in this article, is to add an additional step to Version A after performing the CFO estimator of (5). Since the LCCFO algorithm is iterative such that it will eventually converge after enough rounds of iterations, we will add an additional subscript s to represent the sth iteration. r _{ s }, for example, represents the received signal vector r having been performing the sth iteration. The LCCFO (or Version B) algorithm, in our notations, are as the following.
Step 1)
Step 2)
where
Step 3)
r _{0} coming up at the first iteration is set equal to the received noisy signal r. Notice that, at Step 3, frequency offset is corrected, and an equivalent received noisy signal r _{ s } is produced. Then, the process goes back to Step 1 for the next iteration to estimate the residual CFO left in the equivalent received noisy signal produced in the previous iteration. In Step 1, no need to make correction of CFO as made in (7) since in the previous step, i.e., in Step 3, the CFO correction has been done. After several iterations, \( {\widehat{\delta}}_{LC,s} \) in Step 2 will approximately be zero. After a total of L iterations are performed, the CFO estimate is summation of L LCCFO estimates, i.e., \( {\displaystyle \sum_{p=1}^L{\widehat{\delta}}_{LC,p}} \). Then, the accumulated estimate error after s iterations is \( \varDelta {\delta}_s={\displaystyle \sum_{p=1}^s{\widehat{\delta}}_{LC,p}}\delta \), s = 1, 2, ⋯, L. In addition, the ML CIR estimator can be readily to be shown as
The proposed SLCCFO algorithm
From the LCCFO algorithm reviewed in the previous section, the CFO estimator \( {\widehat{\delta}}_{LC,s} \) of (14) estimates the residual CFO existing in r _{ s − 1}. Here, we propose the replacement of the N − 1 arctangent functions in (14) by N − 1 simple limiters since the residual CFO estimator \( {\widehat{\delta}}_{LC,s} \) of (14) will approach to zero when LCCFO converges and the fact that tan^{− 1} ϕ _{ n,s } ≈ ϕ _{ n,s } when ϕ _{ n,s } is small enough. The following describes the proposed SLCCFO algorithm.
Step 1)
Step 2)
Where

a)
\( {\phi}_{n,s}=\mathrm{I}\mathrm{m}\left\{{r}_{n,s1}{\widehat{y}}_{n,s}^{*}\right\}/\mathrm{R}\mathrm{e}\left\{{r}_{n,s1}{\widehat{y}}_{n,s}^{*}\right\} \) when \( \mathrm{R}\mathrm{e}\left\{{r}_{n,s1}{\widehat{y}}_{n,s}^{*}\right\}>0 \) and \( \lambda \le \mathrm{I}\mathrm{m}\left\{{r}_{n,s1}{\widehat{y}}_{n,s}^{*}\right\}/\mathrm{R}\mathrm{e}\left\{{r}_{n,s1}{\widehat{y}}_{n,s}^{*}\right\}\le \lambda; \)

b)
ϕ_{ n,s } = − λ when \( \mathrm{R}\mathrm{e}\left\{{r}_{n,s1}{\widehat{y}}_{n,s}^{*}\right\}>0 \) and \( \mathrm{I}\mathrm{m}\left\{{r}_{n,s1}{\widehat{y}}_{n,s}^{*}\right\}/\mathrm{R}\mathrm{e}\left\{{r}_{n,s1}{\widehat{y}}_{n,s}^{*}\right\}<\lambda, \) or when \( \mathrm{R}\mathrm{e}\left\{{r}_{n,s1}{\widehat{y}}_{n,s}^{*}\right\}<0 \) and \( \mathrm{I}\mathrm{m}\left\{{r}_{n,s1}{\widehat{y}}_{n,s}^{*}\right\}<0; \)

c)
ϕ_{ n,s } = λ when \( \mathrm{R}\mathrm{e}\left\{{r}_{n,s1}{\widehat{y}}_{n,s}^{*}\right\}>0 \) and \( \mathrm{I}\mathrm{m}\left\{{r}_{n,s1}{\widehat{y}}_{n,s}^{*}\right\}/\mathrm{R}\mathrm{e}\left\{{r}_{n,s1}{\widehat{y}}_{n,s}^{*}\right\}>\lambda \), or when \( \mathrm{R}\mathrm{e}\left\{{r}_{n,s1}{\widehat{y}}_{n,s}^{*}\right\}<0 \) and \( \mathrm{I}\mathrm{m}\left\{{r}_{n,s1}{\widehat{y}}_{n,s}^{*}\right\}>0. \)
Step 3)
The modifications to the LCCFO algorithm lie in Step 2, resulting in the SLCCFO algorithm with much low computational requirements. An additional parameter λ, acting as a threshold value for the N limiter functions {ϕ _{ n,s }, n = 1, 2, ⋯, N − 1}, is introduced since the discrepancy between tan^{− 1} ϕ _{ n,s } and ϕ _{ n,s } gets larger when the absolute value of ϕ _{ n,s } grows larger. This is illustrated in Fig. 1. Thus, we introduce the parameter λ, limiting the maximum value change at every iteration, to prevent the algorithm from divergence. In addition, during the first several iterations or at a low signaltonoise ratio (SNR) environment, the noisefree, CFOfree received estimator ŷ _{ s } is still unreliable, making what follows, the CFO estimator \( {\widehat{\delta}}_{LC,s} \), departs form the residual CFO existing in r _{ s − 1}. This is why we need a limiter with two threshold values of the upper limit λ and the lower limit − λ. As for the choice of λ, it will be thoroughly discussed in the next section.
Regarding the implementation issues of the arctangent function in (14), many researchers have suggested the CORDICrelated algorithms [1, 27], which is far more complex than the limiter that we replace with.
The CIR estimator remains the same as described in (17) without modifying. When the ZadoffChu sequences [28, 29] with constant amplitudes are used for training symbols, (17) can be further simplified to
The facts that \( {\mathbf{U}}_X^H{\mathbf{U}}_X={\mathbf{I}}_N \) and \( {\mathbf{F}}_v^H{\mathbf{F}}_v=N{\mathbf{I}}_N \), with I _{ N } denoting an identity matrix, have been used for the CIR estimator of (21).
Notice that the proposed SLCCFO as well as the original LCCFO is in fact the joint estimation of CFO and CIR. The CIR information is hidden in the CFOfree, noisefree received signal vector y such that \( \mathbf{y}=\frac{1}{\sqrt{N}}{\mathbf{F}}_N^H{\mathbf{U}}_X{\mathbf{F}}_v\mathbf{h} \). In each of the iterative round of (18) as well as (13), the temporary CIR estimate is implicitly evaluated. The reason why we do not explicitly show CIR in both the SLCCFO and the LCCFO is that we focus on the simplification of the iterative CFO estimator instead of the CIR estimator. The transient behavior of CIR estimates is of no importance. Only the final (or steadystate) CIR estimate is inspected using (21) in our simulations.
Simulation results and discussions
We evaluate the OFDM systems with the number of subcarriers N = 64. The CIR of length 9 is defined as h _{ n }, n = 0, 1, ⋯, 8. Both a static and a Rayleigh fading channels are considered. The two simulated channel models have the same power delay profile of E{h _{ n }^{2}} = ae ^{− n/4}, where E{⋅} denotes mathematical expectation, a is for power normalization to unity and n = 0, 1, ⋯, 8. All the simulated results are averaged over 2000 runs for the static channel and 20000 runs for the Rayleigh fading one. One of the ZadoffChu sequences {\( {X}_k={e}^{j\pi 7{k}^2/N} \), k = 0, 1, ⋯, N − 1} is chosen as the training symbols for both LCCFO and SLCCFO.
The static channel is first used to demonstrate the merits of SLCCFO over LCCFO and explore the properties of convergence speed and the estimator MSE. Figure 2 shows the learning curves of CFO estimators for LCCFO and SLCCFO at various CFO values. The threshold value of λ is set 0.5 for SCLCFO. The results show us that the larger the CFO values, the slower the convergence speed as expected since more iterations are required for approaching the estimated CFO values. Also notice that SLCCFO represents slightly faster convergence speed especially at larger CFO than LCCFO. More details are unveiled in Figs. 3 and 4 for SLCCFO at various threshold values and different CFOs. For δ = 0.2, SLCCFO and LCCFO have the same convergence speed when λ is greater than or equal to 1, shown in Fig. 3. For a larger CFO of δ = 0.5, SLCCFO shows faster convergence than LCCFO when λ is slightly greater than 1 as shown in Fig. 4. These results demonstrate that the larger the threshold value of λ, the faster the convergence speed, which is obvious for large CFO values.
For Figs. 3 and 4, the SNR is set at 20 dB. The results of CFO estimator MSE as well as CIR estimator MSE for various SNR values are shown in Figs. 5 and 6 for δ = 0.2 and Figs. 7 and 8 for δ = 0.5, where the CramerRao lower bound (CRLB) is added for each of the figures for comparisons. The derivations for CRLB can be referred to [17]. We can observe that the curves of SLCCFO and LCCFO are tightly close at mediumtohigh SNR values. At low SNR, choosing large λ, e.g., λ = 10 or 50, results in worse MSE performance than LCCFO. A smaller threshold value of λ, e.g., λ = 1, presents better MSE performance than LCCFO. The property is much more obvious in CFO estimator MSE than in CIR estimator MSE. The reason that the large λ brings about the large MSE value especially at low SNR is quite apparent that a fast tracking estimator will follow closely to the random noise in a noisedominant environment, greatly degrading estimation accuracy.
Simulations for the channel with frequencyselective Rayleigh fading are shown in Figs. 9 and 10. We observe that λ = 1 has the best performance among others, and λ = 50 presents the worst case. With proper choice of λ, all simulations show that the lowcomplexity SLCCFO presents fast convergence speed and low MSE performance.
After several more simulations are conducted, we conclude that the best choice of λ with both faster convergence speed and lower MSE than LCCFO is somewhere between 1 and 3. The number of iterations required for convergence is around 20 for CFO up to 0.5. When the value of λ chosen is several tens, the convergence rate may be faster. However, its MSE will depart from the CRLB when the signaltonoise ratio (SNR) is lower than 20 dB.
Finally, we will compare SLCCFO with the existing method proposed by Salim et al. [21] because the Salim’s method is one of the most competing methods. In [21], a joint estimation of channel, phase noise, and frequency offset is considered. To make fair comparisons, no phase noise resulting from the instability of local oscillators is assumed, and the initial value of the CFO estimate is set zero. In Fig. 11, the learning curves of SLCCFO and Salim are plotted for \( \delta =0.2 \) and 0.5 at SNR = 20 dB. The same static channel as adopted in the previous simulations is used. The results show that SLCCFO has faster convergence rate and lower steady state MSEs, compared to the Salim’s method. For \( \delta =0.5 \), the convergence rate of SLCCFO is 10 iterations faster than that of Salim’s method, while, for \( \delta =0.2 \), SLCCFO is 5 iterations faster than the Salim’s method. In Figs. 12 and 13, the steady state CFO estimator MSEs and CIR estimator MSEs for SNR values from 0 to 40 dB are shown, respectively. Apparently, SLCCFO has lower steady state CFO estimator MSEs than the Salim’s method, while the CIR estimator MSEs are almost the same for both the methods except at low SNR values.
Conclusions
High accuracy for CFO estimation is essential for OFDM systems. Low complexity is one of the critical issues considered for implementation. We have proposed a lowcomplexity SLCCFO algorithm adapted from LCCFO with high accuracy. N − 1 simple limiters are adopted for each iteration instead of N − 1 arctangent functions used in LCCFO. The computational complexity is thus reduced. By proper choice of a parameter for the limiters, not only the MSE values of the both CFO and CIR estimators at mediumtohigh SNR values maintain the same as those of LCCFO, but the convergence speed and the MSE values at low SNR values can also be improved. A static channel and a Rayleigh fading one are employed for demonstrating those properties. In brief, without sacrificing the estimator performances, a new efficient algorithm is proposed.
Abbreviations
 CFO:

carrier frequency offset
 CIR:

channel impulse response
 CRLB:

CramerRao lower bounds
 LCCFO:

linearlycombined carrier frequency offset
 ML:

maximum likelihood
 MSE:

meansquare error
 OFDM:

Orthogonal frequencydivision multiplexing
 SLCCFO:

simplified linearlycombined carrier frequency offset
 SNR:

signaltonoise ratio
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Acknowledgements
The authors would like to thank the Ministry of Science and Technology, Taiwan, for the support of the research grant (MOST 1042221E030005).
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Liu, H., Yin, Y. Lowcomplexity adaptive iteration algorithm for frequency tracking in OFDM systems. J Wireless Com Network 2016, 57 (2016). https://doi.org/10.1186/s1363801605597
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Keywords
 Orthogonal frequencydivision multiplexing
 Carrier frequency offset
 Maximumlikelihood estimation
 Channel estimation
 Linear minimum meansquare error