Blind CFO Estimation for Zero-Padded OFDM over Underwater Acoustic Channels

Transcription

1 Blind CFO Estimation for Zero-Padded OFDM over Underwater Acoustic Channels Wei Zhou, Zhaohui Wang,JieHuang, and Shengli Zhou Dept. of Electrical and Computer Engineering, University of Connecticut, Storrs, CT 6269 Wireless System R&D, Marvell Semiconductor, Santa Clara, CA 9554 Abstract In this paper, we compare the performance and complexity of three existing blind carrier-frequency-offset (CFO) estimators when they are applied to estimate the residual Doppler shift for zero-padded OFDM transmissions over underwater acoustic channels: the null subcarrier based method [1], the O-M algorithm [2], and the Y-G algorithm [3]. Performances of these three methods are evaluated by extensive numerical simulations and by data sets collected from the Mobile Acoustic Communication Experiment conducted off the coast of Martha s Vineyard, MA, July 23, 21. Simulation results show that the Y-G algorithm always outperforms the O-M algorithm. As far as the fractional part of CFO is concerned, the Y-G algorithm can perform as good as the null subcarrier based method yet with a much lower computational complexity. When working with real data, the Y-G algorithm needs to be modified and integer CFO estimation based on a few null subcarriers is incorporated. Experimental results show that the modified Y-G algorithm, having lower complexity, performs slightly worse than the null subcarrier based method for small and moderate-size constellations, and becomes ineffective for a large constellation. Index Terms Carrier frequency offset (CFO), zero-padded orthogonal frequency division multiplexing (ZP-OFDM), underwater acoustic communications (UWA) I. INTRODUCTION Multicarrier modulation in the form of orthogonal frequency division multiplexing (OFDM) is sensitive to carrier frequency offset (CFO). CFO compensation has been extensively studied in the broadband wireless communications literature. In addition to data-aided or preamble-aided estimators such as [4] [6], which rely on either training symbols or periodically transmitted pilots, there are a rich set of blind CFO estimators available, which exploit various underlying structures of the received signals. Cyclic-prefix (CP) structure. The existence of a CP segment for each OFDM block can be utilized for blind CFO estimation [7], [8]. s. Not all the available subcarriers are used in practical OFDM systems. Utilizing the fact that the energy leakage to the null subcarriers is minimized if CFO is correctly compensated, several blind CFO estimators [1], [9] [11] have been derived. The combined use of CP and null subcarriers is proposed in [12]. Second-order statistics. The relationship among the second-order statistics at different lags for CP-OFDM is exploited for blind CFO estimation [13]. This work was supported by the ONR grant N (PECASE) and by the NSF grant ECCS Nonlinear or higher order statistics. The O-M algorithm exploits the fact that the 1st, 3rd, and 4th order nonlinearities of oversampled frequency-domain observations lead to a spectral line which is dependent on the CFO [2], applying the idea from timing recovery in singlecarrier systems [14]. The Y-G algorithm utilizes the non- Gaussianity of frequency observations measured by a Kurtosis function to estimate the CFO. Time-domain oversampling. An explicit relationship between one set of baseband samples with another set of baseband samples lagging half of the sampling interval was formulated based on the time-domain waveform expression for CP-OFDM, which allows for blind CFO recovery [15]. A. The Objective of This Paper The methods aforementioned are developed in the context of wireless radio CP-OFDM. We are interested in applying them to zero-padded OFDM transmissions over underwater acoustic channels [16], [17], where zero-padding is adopted to save transmission power over long guard intervals. The difference between ZP and CP renders many existing methods not applicable to ZP-OFDM. The null subcarrier based method [1] has been adopted for ZP-OFDM in UWA channels and has been demonstrated effective to estimate the residual mean Doppler shift [16], [17]. The motivation of carrying out the research in this paper is to answer the following questions: What other blind CFO estimators can be applied for underwater ZP-OFDM? What would be their performance when decoding real data sets? Is there any complexity advantage relative to the null-subcarrier based method? We identified that the O-M algorithm and the Y-G algorithm are also suitable for ZP-OFDM, in addition to the null subcarrier based method. We carry out extensive numerical simulations and experimental studies to compare the performance of these three methods, and also provide analysis on the computational complexity. Simulation results show that the O-M method has the worst performance, while the Y- G algorithm performs similarly as the null subcarrier based method with a much lower complexity. When it comes to deal with real data sets, however, the Y-G algorithm has to be modified and coupled with an integer CFO estimator. With a lower complexity, the modified Y-G algorithm performs slightly worse than the null subcarrier based method for small to moderate-size constellations, and becomes ineffective for

2 our collected data with 64-QAM. The rest of this paper is organized as follows. In Section II, a brief overview of the ZP-OFDM system is given. In Section III, three blind CFO estimators are explicitly introduced. In Sections IV and V, simulations and experimental data are used for comparison of these three estimators, respectively. The conclusions are drawn in Section VI. II. FREQUENCY-DOMAIN REPRESENTATION OF THE SYSTEM Let T denote the OFDM symbol duration and T g the length of the guard interval between consecutive OFDM blocks. With a total of K subcarriers, the kth subcarrier is located at the frequency f k = f c + k, k = K/2,...,K/2 1 (1) T where f c is the center frequency and 1/T is the subcarrier spacing. The bandwidth is B = K/T. Consider one ZP-OFDM block. Let s[k] denote the information symbol on the kth subcarrier, and define S A and S N as the non-overlapping sets of active and null subcarriers, respectively, which satisfy S A S N = { K/2,...,K/2 1}. The passband signal can be expressed as ( ) s(t) =2Re s[k]e j2πfkt g(t), t [,T + T g ] (2) where g(t) is the rectangular window with nonzero support within [,T]. The Fourier transform of s(t) at the positive frequency range is S(f) = s[k]g (f f k ), f > (3) where G(f) is the Fourier transform of g(t). For a rectangular window, we have G(f) =e jπft sin(πft)/(πft), and thus G(k/T) =for any nonzero integer k. Assume that the channel consists of N p discrete paths. As in [16], we assume that all the channel paths share a common path delay variation rate, i.e., Doppler scale a, such that the channel within one OFDM block can be approximated as N p h(τ; t) = A p δ (τ (τ p at)), (4) where A p and τ p are the amplitude and initial delay of the pth path, respectively. As such, the received passband signal is N p y(t) = A p s((1 + a)t τ p )+ n(t), (5) where n(t) is the additive noise. In the preprocessing step, the receiver resamples the received OFDM block at passband to remove the main Doppler scaling effect using an estimated resampling factor ˆa [16], [19] z(t) = y ( t ) 1+ˆa N p = A p s ( 1+a 1+ˆa t τ p ) + n(t), (6) where we ignore the resampling effect on the noise. Downshifting on z(t) followed by low pass filtering (LPF) leads to a baseband signal z(t) =LPF[e j2πfct z(t)]. (7) After straightforward manipulations, the Fourier transform of z(t) within the frequency band [ B/2,B/2] is obtained as Z(f) =H(f) ( 1+ˆa s[k]g 1+a f k T a b ) 1+a f c + N (f), (8) where N(f) is the Fourier transform of the filtered noise, and H(f) = 1+ˆa 1+a Define the residual CFO as Assuming that 1+ˆa 1+a Z(f) H(f) ε = N p 1+ˆa j2π(f+fc) A p e 1+a τp. (9) (a ˆa) 1+a f c (1) f f when f <B/2, we obtain s[k]g (f kt ε ) + N (f). (11) If the CFO is known, evaluating Z(f) at (m/t + ε) leads to z m = Z(f) f= m T +ε H m s[m]+n m (12) where H m = H ( m T + ε) and n m = N( m T +ε). Hence, proper CFO estimation and compensation can minimize intercarrier interference [16]. III. THREE EXISTING BLIND CFO ESTIMATORS In this section, we overview three CFO estimation methods [1] [3], which are applicable to ZP-OFDM. The first method takes advantage of the fact that most OFDM systems have null (or virtual) subcarriers. The principle is to minimize energy of frequency observation at null subcarriers with respect to the tentative CFO estimates [1]. The second method proposed in [2], termed as the O-M algorithm, exploits the fact that the 1st, 3rd, and 4th order nonlinearities of the oversampled frequencydomain observations lead to a spectral line related to the CFO. The non-gaussianity of frequency observations measured by a Kurtosis function is utilized in the third method [3]. Different CFO estimators have different capabilities. Normalize the CFO by the subcarrier spacing as ε =(d + α)/t, (13) where d is the integer part and α (.5,.5] is the fractional part of the CFO. For brevity, we will term d as integer CFO and α as the fractional CFO thereafter. While the nullsubcarrier based method can estimate both the integer and fractional parts of the CFO, the O-M and Y-G algorithms can only estimate a fractional CFO.

3 A. The Null Subcarrier based Method With a tentative fractional CFO α, one can draw K samples from Z(f) as z m ( α) =Z(f) f= m T + α, m = K T 2,...,K 2 1 (14) The null subcarrier based method [1] can be casted as (ˆα, ˆd) =arg min E α, d m SN z m+ d( α) 2. (15) The solution ˆα can be obtained by coarse initial search followed by fine bi-sectional search [16], [2]. For each tentative fractional CFO α, one K-point FFT is performed which allows for integer CFO estimation with little effort. B. The O-M Algorithm The frequency observation in (11) has a similar formulation as [2, eq. (2)] for CP-OFDM, and hence the O-M method [2] is applicable. The O-M method can be described as follows. Perform frequency-domain oversampling by a factor M: z m = Z(f) f= m MT, m = MK MK,, (16) Perform nonlinear (NL) operation over the oversampled frequency-domain measurements. The first-, third-, and fourth-order nonlinearities (NL) have been recommended with r m = z m, r m = z m 3, and r m = z m 4, respectively. Estimate the CFO as: { 1 ˆα = where 2π arg(x), absolute value NL, 1 2π arg( X), third- & four-order NL, (17) X = m m j2π r m e M. (18) The key advantage of the O-M algorithm is that instead of using an exhaustive search, an analytical solution is available. C. The Y-G Algorithm The key idea is that the distribution of z m ( α) in (14) is more non-gaussian when α = α than when α = α. The non- Gaussianity of a distribution can be measured by a Kurtosis function. The objective function of the Y-G algorithm [3] using only one OFDM symbol can be represented as: J( α) = K/2 1 m= K/2 z m( α) 4 ( K/2 1 m= K/2 z m( α) 2) 2. (19) With sufficiently large data points, the objective function J( α) resembles a cosine function [3] and α = α can be a unique global minimum or maximum solution of the objective function J( α). Instead of an exhaustive search, a closed-form estimate of ˆα is available. The extreme point of the objective function J( α) can be obtained analytically as follows: 1 2π tan 1 (B/A) if A>, 1 ˆα = π tan 1 (B/A) if A<,B, π tan 1 (B/A) if A<,B <. (2) where A := (J(1/4) + J( 1/4))/2 J(), and B := (J( 1/4) J(1/4))/2. Ifˆα leads to the global minimum of J( α), the global maximum of J( α) can be obtained by shifting ˆα by.5. Whether ˆα is an extreme point for global minimum or global maximum depends on the used constellation and some channel properties [3]. IV. SIMULATION RESULTS In this section, we perform numerical simulation to compare the three CFO estimators. Both linear time-invariant (LTI) and linear time-varying (LTV) channels are considered. A. Simulation Setup The system parameters are the same as in [17]. The bandwidth of the OFDM signal is B =9.77 khz, and the carrier frequency is f c =13kHz. Out of K = 124 subcarriers, there are 256 pilot subcarriers, 96 null subcarriers, and 672 data subcarriers. The OFDM symbol duration is T = K/B = ms, and the subcarrier spacing is 1/T =9.54 Hz. We generate sparse channels with 1 discrete paths, where the inter-arrival times are exponentially distributed with interarrival mean of 1 ms, leading to an average delay spread of 1 ms. The amplitudes are Rayleigh distributed with the average power decreasing exponentially with delay, where the difference between the beginning and the end of the guard time is 2 db. Each path is associated with a separate moving speed v which is drawn from a uniform distribution with mean m v m/s and standard deviation of σ v m/s. The corresponding Doppler scale is a = v/c with c = 15 m/s. We consider the following three setups. LTI channels are generated with m v =. m/s and σ v =.m/s (denoted as LTI ). With LTI channels, we artificially add a fractional CFO before receiver processing. The artificial CFO is generated uniformly and independently from [.4,.4] for each OFDM block. The channels are generated with discrete paths using m v =. m/s and σ v =.1 m/s (denoted as LTV1 ). The channels are generated with discrete paths using m v =.2 m/s and σ v =.1 m/s (denoted as LTV2 ). For channel estimation, we adopt the FFT based least-squares (LS) estimator [16]. For each point, 1 blocks are tested and each OFDM symbol is decoded separately with a rate-1/2 nonbinary low-density parity-check (LDPC) code [18]. We use the mean-square-error () and the decoding block-error-rate (BLER) as figures of merit.

4 O M algorithm, First O M algorithm, Third O M algorithm, Fourth O M algorithm, First O M algorithm, Third O M algorithm, Fourth O M algorithm, First O M algorithm, Third O M algorithm, Fourth (b) 16-QAM (c) 64-QAM Fig. 1. comparison, LTI channels with CFO B. Complexity Comparison For all these channels under consideration, only the estimate of the fractional part of CFO is needed. For the null subcarrier based algorithm, coarse initial search followed by bi-sectional search is performed [2]. The initial search range is from.4 to.4 with a step size.1. During the bi-sectional search, the step size decreases to.5,.25,.125, and.625, with two FFTs at each iteration searching the left and right points around the previously obtained CFO value. Thus the total computational complexity is 9+8=17 K-point FFTs. The computational complexity of the O-M algorithm is one MK-point FFT where frequency-domain oversampling with M =4is used. For the Y-G algorithm, the desired CFO is a global minimum of the objective function for all tested cases, thus the computational complexity is just three K-point FFTs. C. Comparison Fig. 1 shows the performance of the three CFO estimators over LTI channels with an artificially added CFO using QPSK, 16-QAM, and 64-QAM constellations. It can be seen that the null subcarrier based method is very stable regardless of the adopted constellation. In fact, the performance of the null subcarrier based method depends only on the desired resolution which is related to the computational complexity. The Y-G algorithm and the O-M algorithm degrade as the constellation size increases from QPSK to 64-QAM. Further, the Y-G algorithm always outperforms the O-M algorithm (we thus omit the results of the O-M algorithm in later discussions). For QPSK modulation, the Y-G algorithm can outperform the null subcarrier based method whereas for 16- QAM and 64-QAM modulations the null-subcarrier based method outperforms the Y-G algorithm, especially in the high range. However, the Y-G algorithm has the least computational complexity. D. Effect of Number of Null Subcarriers We also evaluate the effect of the number of null subcarriers for the null subcarrier based method. Fig. 2 shows this effect for the LTI channel with artificially added CFO using 16- QAM modulation. The energy on a portion of the total 96 null subcarriers is used as the objective function for this purpose. We can see from Fig. 2 that the performance of the null subcarrier based method degrades as the number of null subcarriers decreases. For this setting, the improvement after more than 6 null subcarriers is very slow. E. BLER Comparison Figs. 3 and 4 show the BLER performance of the Y-G algorithm with that of the null subcarrier based method. The decoding results without CFO compensation is also included. Three settings are evaluated and LS based channel estimator is adopted. We can see from Fig. 4 that slight performance improvement with CFO compensation compared with no CFO compensation is observed for the LTV1 channel with m v =. m/s. Significant performance improvement with CFO compensation compared with no CFO compensation is observed for the LTV2 channel with m v =.2 and the LTI channel with artificial added CFO, as shown in Fig. 3. More interestingly, we see that the Y-G algorithm performs similar as the null subcarrier based method in all three channels (we have also simulated the performance with the sparse channel estimator from [17]; the same conclusion holds). V. EXPERIMENTAL RESULTS: MACE1 The Mobile Acoustic Communication Experiment 21 (MACE1) was conducted off the coast of Martha s Vineyard, MA, July 23, 21. The bandwidth of the OFDM signal is B = 4.88 khz, and the carrier frequency is f c =13kHz. We have designed two sets of signals, one with K = 124 subcarriers and the other with K = 512 subcarriers. The subcarrier spacings are 4.77 Hz for K = 124 and 9.54 Hz for K = 512. The OFDM symbol durations are 29.7 ms for K = 124 and ms for K = 512. The guard intervals are T g = 4.3 ms for K = 124 and T g = 25.2 ms for K = 512. There are 96 null subcarriers when K = 124 and 48 null subcarriers when K = 512.

5 5 db 1 db 15 db LTI LTV2, σ v =.1 m v = Number of null subcarriers 1 4 No Compensation No Compensation LTV1, σ v =.1 m v = Fig. 2. performance with different numbers of null subcarriers. Fig. 3. Decoding performance in the LTI and LTV2 settings. Fig. 4. setting. Decoding performance in the LTV1 Estimated normalized CFO 1.5 Estimated normalized CFO.5 Estimated normalized CFO Block index Block index Block index (b) 8-QAM (c) 16-QAM Fig. 5. Sample plots of the estimated CFOs. File ID: 357, K = 124; (This is the file with largest CFO differences among all files tested) The signal was transmitted at a depth of about 8 meters and received by a 12-element array attached to a mooring line. An array of four ITC-17 transducers was used for transmissions and towed from the minimum range (about 5 m) out to the maximum range (about 7 m) and then back to the minimum range at relatively slow speeds (1-2 m/s). There were two tows in this experiment; on one tow the transducer array was oriented vertically and on the other, it was horizontal. The OFDM signal was transmitted continuously every four minutes for the duration of the two tows. In different settings, different constellations were used together with the corresponding rate- 1/2 nonbinary LDPC codes [18]. In this paper, we present the results of tow 2 data with one transmitter to illustrate the BLER performance of different CFO estimators. A. The Modified Y-G Algorithm In decoding real data, the desired CFO can be a global minimum or maximum solution of the objective function in the Y-G algorithm, and this fact is not known beforehand. Further, the Y-G algorithm can only estimate the fractional CFO, while there are situations where integer CFO shows up even after the resampling operation on the received blocks, e.g., the last five blocks in Fig. 5(a). We make the following adjustments when using the Y-G algorithm on real data. After obtaining ˆα with the original Y-G algorithm, perform an additional test whether ˆα or ˆα +.5 corresponds to the minimum of the cost function. Use a small number of null subcarriers to estimate the integer CFO. In the following decoding results, only 2 and 1 null subcarriers are used to estimate the integer CFO for the K = 124 and K = 512 settings, respectively. Fig. 5 shows the estimated CFOs for 2 consecutive OFDM blocks corresponding to each setting with a different constellation from one recorded file. The CFO estimates from the modified Y-G algorithm agree well with those from the null-subcarrier based method, while the CFO estimates from the original Y-G algorithm are often off by.5 due to the minimum/maximum uncertainty.

6 Y G, combined (b) 8-QAM (c) 16-QAM Fig. 6. Decoding performance comparison using MACE1 data, K = 124 Y G, combined (b) 8-QAM (c) 16-QAM Fig. 7. Decoding performance comparison using MACE1 data, K = 512 B. Comparison of Computational Complexity For the null subcarrier based method, a coarse initial search followed by bi-sectional search is performed for fractional CFO estimation [2]. The initial search uses a step size.2 within the range (.5,.5], and the fine search decreases the step size to.1,.5,.25, and.125. Notice that an integer CFO is equivalent to tune shift in the frequency domain, so integer CFO estimation does not require additional FFTs. The total computational complexity is 5+8=13 K-point FFTs for the null-subcarrier based algorithm. The computational complexity of the original Y-G algorithm is three K-point FFTs. For the modified Y-G algorithm, two additional K-point FFTs are performed to resolve the ambiguity between the global minimum and maximum. Again, integer CFO estimation requires little extra cost, the total computational complexity of the modified Y-G algorithm is about five K-point FFTs. C. BLER Performance Fig. 6 shows the decoding results for K = 124 with QPSK, 8QAM, and 16QAM constellations, where 3 files are decoded with each file containing 2 OFDM blocks, thus 6 OFDM blocks are decoded in total. Fig. 7 shows the decoding results for K = 512 with QPSK, 8QAM, and 16QAM constellations, where 54 files are decoded with each file containing 2 OFDM blocks, thus a total of 18 OFDM blocks are decoded. Fig. 8 shows the performance results for K = 512 with 64- QAM constellation, where 27 files are decoded with each file containing 2 OFDM blocks, thus a total of 54 OFDM blocks are decoded. (Due to the lack of time slots, 64-QAM with K = 124 was not transmitted.) Each file is resampled once using the estimated Doppler scaling factor from a preamble [19] preceding the 2 OFDM blocks. The block-by-block receiver [16] works on each OFDM block separately. We have the following observations. The modified Y-G algorithm significantly outperforms the original Y-G algorithm, except for the 64-QAM case, as the ambiguity between global minimum and maximum of the objective function and an integer CFO do occur with some small percentage of data blocks. The null subcarrier based method is robust for all constellations. For small size constellations such as QPSK and 8-QAM, the modified Y-G algorithm is only slightly worse

7 Fig. 8. Decoding performance comparison using MACE1 data with K = 512 and 64-QAM modulation. than the null subcarrier based method. For a moderate-size constellation such as 16-QAM, the performance gap is bigger, although the performance of the modified Y-G algorithm might be still acceptable. For the 64-QAM data set, the modified Y-G algorithm becomes not effective. With considerations on the performance-complexity tradeoff and the overhead spent on null subcarriers, we make the following recommendations. When the implementation complexity is not a concern for real time decoding and there are enough null subcarriers [2], the null subcarrier based method is preferred, having robust performance for all constellations. When the implementation complexity needs to be kept minimum, or, when there are no or only a few null subcarriers embedded in the transmissions, the modified Y-G algorithm is a good choice for small to moderatesize constellations. The modified Y-G needs to be coupled with an integer CFO estimator, which is usually easy to construct and is not necessarily based on null subcarriers. VI. CONCLUSIONS In this paper we studied the use of three existing blind CFO estimation algorithms for ZP-OFDM transmissions in underwater acoustic channels. We provided both numerical and experimental results. Simulations showed that the Y-G algorithm can perform as well as the null subcarrier based method, while having lower complexity, as far as the fractional part of CFO is concerned. When dealing with real data sets, the Y-G algorithm has to be modified and we further coupled it with an integer CFO estimator based on a few null subcarriers. Experimental results showed that the modified Y-G algorithm is a slightly worse than the null subcarrier based method for small to moderate-size constellations and becomes not effective for large constellations. 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