MAP Estimation of CFO and STO for a Convolutionally Coded OFDM System

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1 MAP Estimation of CFO STO for a Convolutionally Coded OFDM System Anushree Neogi Department of Electronics Communication Engineering, Indian Institute of Technology (IIT Guwahati, Guwahati , India. anushree.iitg@gmail.com Abhijit Mitra Department of Electronics Communication Engineering, Indian Institute of Technology (IIT Guwahati, Guwahati , India. a.mitra@iitg.ernet.in Abstract We propose a maximum a-posteriori (MAP based blind algorithm for carrier frequency offset (CFO symbol timing offset (STO estimation, for a convolutionally coded orthogonal frequency division multiplexing (COFDM system, in an additive white Gaussian noise (AWGN channel. We derive the MAP estimators mathematically. It is observed that though the MAP metric for CFO estimation gives a good performance for an AWGN channel, the MAP metric for STO estimation is unable to provide satisfactory performance. I. INTRODUCTION Coded orthogonal frequency division multiplexing (COFDM is a channel coding multicarrier transmission scheme that finds application in digital audio broadcasting (DAB, high definition digital television broadcasting (HDTV, digital video broadcasting terrestrial TV (DVB-T etc. However due to fluctuations in the system parameters caused primarily due to loss of synchronization at the receiver, the COFDM system becomes highly sensitive to frequency timing jitters. These errors are popularly known as the synchronization errors. In practice STO CFO are caused due to incorrect start of a COFDM symbol due to differences in the transmitter receiver oscillator frequencies respectively. Apart from the loss of orthogonality among the subcarriers leading to ICI, these errors also cause rotation attenuation of the useful symbols. Various blind mitigation techniques have been proposed in [1]- [6] to deal with the adverse effects of synchronization errors. The blind estimation algorithms are more efficient in terms of achieving high data rate because they donot require an extra overhead/preamble just for the purpose of synchronization. They are therefore more suitable for the next generations of wireless devices. Blind estimation of CFO STO may be broadly classified into a-priori a-posteriori estimation techniques. [1] discusses a maximum likelihood (a-priori blind estimation technique for an AWGN channel while [2] deals with minimum mean square error estimation (a-posteriori for a frequency selective, time varying channel. Though in [3] a maximum a-posteriori estimation of CFO for a frequency selective, time invariant channel has been proposed, to the best of our knowledge the joint estimation of CFO STO for a simplistic channel as AWGN channel has not been derived so far. A marginal addition of complexity yields a better performance of the proposed scheme with respect to the existing techniques for estimation of CFO STO. This is one of the motivations behind the study undertaken. In this paper we develop the MAP estimators of CFO STO in an AWGN channel through simulations we demonstrate their efficiency performance through different performance measures for a convolutionally coded OFDM system. The organization of this paper is as follows. Section 2 deals with the general description of a COFDM receiver. Section 3 contains the mathematical modeling of the received COFDM signal. Sections 4 descibes how the selection of apriori probabilities of the unknown rom variables is made. Section 5 deals with the mathematical description of the proposed algorithm. Section 6 presents our results the concluding remarks are given in section 7. II. SYSTEM DESCRIPTION OF THE COFDM RECEIVER The received COFDM symbols are first passed through the synchronization block in order to counteract the effects of CFO STO. This block implements the blind CFO STO MAP estimation correction technique. After error compensation, the COFDM symbols which still contain residual errors are passed through the COFDM demodulator block from where the multiplexed data symbols present in a COFDM symbol are extracted by fast Fourier transform operation. The next block is a differential QPSK demodulator whose output is a stream of coded data bits. The message bits are then extracted by passing this stream through a decoder employing 1/2 rate convolutional code having a constraint length of K 3. Since our interest lies in the design of a low complexity receiver, hence we restrain the value of K at 3 it is seen to suffice our requirements. For the purpose of simulation we have multiplexed N 64 complex symbols to form a 64 point COFDM symbol. The cyclic prefix (CP length, L is kept at 1/4-th of N is therefore 16. III. THE SIGNAL MODEL The transmitted COFDM signal is given by s(n. After passing through an AWGN channel with the noise given by

2 g(n, the received COFDM signal y(n is modeled as y(n s(n ηe j2πζn/n + g(n. (1 Here, η denotes the STO ζ the CFO. IV. SELECTION OF A-PRIORI PROBABILITIES The MAP blind algorithm treats the unknowns as rom variables having a prior probability density function. As we are interested in the fine synchronization of ζ η, while selecting the prior pdf of ζ, we assume the CFO to vary not more than the subcarrier spacing. The variation of ζ is then restricted to ζ 0.5 range. Considering each value of ζ to be equiprobable we assume ζ to be a uniformly distributed continuous rom variable having a pdf f ζ (ζ u(ζ +0.5 u(ζ 0.5. (2 In a similar way, we restrict the range of symbol timing error to 0 η M 1 as each value of η is equiprobable, its pdf is given by f η (η 1 δ(η m. (3 M If we assume the prior pdfs to be Gaussian or Rayleigh then the mathematical deductions become tedious. Therefore to keep the results mathematically trackable to arrive at a closed form expression of the estimators, we have chosen uniform prior pdfs which is a practical assumption as given in [3]. V. MAXIMUM A-POSTERIORI (MAP ESTIMATION OF CFO AND STO The details of the mathematical background of this algorithm are provided in the following. With the arrival time η the frequency offset ζ, as the unknown rom variables to be estimated, this algorithm aims at maximizing the a- posteriori probability density function f(η, ζ/y of say 2N +L observed samples of y(n. The observables form a vector y [y(0...y(2n + L 1] T. If the number of sub-carriers is sufficiently large, y(n approximates a complex Gaussian process (from the central limit theorem [1]. Two sets of indices are defined for ease of explanation as V {η,..., η + L 1} W {η + N,..., η + N + L 1}. The set V contains the indices of the samples copied into the cyclic prefix the set W contains the indices of this prefix. The samples in the cyclic prefix their copies in a COFDM symbol are pairwise correlated as, k V : σ E{y(ky s 2 + σn 2 (k + m} σ 2 se j2πζ mn 0 otherwise Since η ζ are independent rom variables, f(η, ζ/y can be written as, f(η, ζ/y f(η/yf(ζ/y. (4 Maximization of the expression on the LHS implies the individual maximization of the terms f(η/y w.r.t η f(ζ/y w.r.t ζ. This would yield the estimates of ˆη MAP ˆζ MAP respectively. A. Determination of f(η/y f(η/y may be expressed as f(η/y f(η, y f(y 1 f(y f(η, y,ζdζ. (5 The term in the denominator does not play a role in the maximization of f(η/y it can be neglected hence f(η/y f(η, y,ζdζ I 1. (6 Moreover, f(η, y,ζ f(y/η, ζf(η, ζ f(y/η, ζf(ηf(ζ. (7 hence I 1 f(η f(y/η, ζf(ζdζ. But, f(y/η, ζ may be expressed as follows [1]. f(y/η, ζ f(y(k,y(k + N f(y(k k V W f(y(k,y(k + N f(y(k. (8 f(y(kf(y(k + N k k f(y(k being independent of η ζ, it may be expressed as [ ] 0.5 f(y(k,y(k + N I 1 f(η f(ζdζ, (9 f(y(kf(y(k + N which can be termed as I 2. The 2-D complex Gaussian pdf f(y(ky(k + N is given by ( exp y(k 2 + y(k+n 2 2ρRe{e j2πζ y(ky (k+n} (σs 2+σ2 n (1 ρ2 π 2 (σs 2 + σn 2 2 (1 ρ 2, (10 while the 1-D complex Gaussian pdfs f(y(k f(y(k + N are of the form exp ( y(k 2 σs f(y(k 2+σ2 n π(σs 2 + σn 2 (11 exp ( y(k+n 2 σs f(y(k + N 2+σ2 n π(σs 2 + σn 2. (12 Substituting the values of eq. (10, (11 (12 in eq. (9 yields I 2 f(η e c 1a(k+c 2P (k f(ζdζ,

3 where So, where ρ 2 c 1 (σs 2 + σn(1 2 ρ 2, (13 2ρ c 2 (σs 2 + σn(1 2 ρ 2, (14 a(k y(k 2 + y(k + N 2, (15 b(k y(ky (k + N, (16 P (k b(k cos(2πζ + b(k. (17 I 2 A(η e c 2P (k f(ζdζ, (18 A(η f(ηexp I 2 may be further expressed as I 2 A(η A(η 2π q(η+π q(η π In the above equation, p(η c 2 P (k where q(η arctan c 1 a(k. (19 e {p(ηcos(2πζ+q(η} f(ζdζ (20 2 e p(ηcosx f(ζdζ. (21 + ( Q(k P (k Q(k (22 (23 P (k b(k cos( b(k (24 Q(k b(k sin( b(k. (25 The function e p(ηcosx is a periodic function of x with a period of 2π. Hence the integral of eq. (21 is nothing but the modified Bessel function of the first kind of zeroth order, I 0 (p(η. From eq. (2 we know that the value of f(ζ is unity in the range of [ 0.5] since the integral is over the same range, we can replace f(ζ with unity. Thus we are required to maximize I 2 which is now given by I 2 A(ηI 0 (p(η (26 δ(η me c 1a(k I 0 (p(η. (27 We know that the variation of η is limited in the range [0, M 1], so for a particular value of η, δ(η m takes a value of unity when m η. Moreover the variation of m is also in the same range as that of η. Thus ˆη MAP becomes ˆη MAP1 argmax exp c 1 a(k I 0 (p(η (28 η From the above equation we find that the complexity of the above metric is very high. We therefore employ trapezoidal rule of numerical integration as an approximation to the integral of eq. (21. The trapezoidal rule is b { } f(a+f(b f(xdx (b a. (29 a 2 Consider the integral of eq. (20 which can be written as e c 2P (k {u(ζ +0.5 u(ζ 0.5}dζ. (30 maybetermedasi 3. Taking u(0 1, wehavebythe trapezoidal rule, I 3 0.5exp c 2 P (k. (31 So, I 2 of eq. (27 can be written as I 2 δ(η mexp c 2 P (k. (32 According to eqn. (3 the variation of η is limited in the range [0,], for a particular value of η, δ(η m takes a value of unity when m η. Since the variation of m is also in the same range as that of η, after taking natural logarithm we can find ˆη MAP as ˆη MAP2 argmax η ρa(k+2p (k. (33 We find that the complexity of finding ˆη MAP2 is less than that of ˆη MAP1 hence may be used to find the MAP estimate of the STO. B. Determination of f(ζ/y f(ζ/y may be expressed as f(ζ/y f(ζ,y f(y 1 f(η, y,ζ. (34 f(y The term in the denominator does not play a role in the maximization of f(ζ/y, it can be neglected. Thus, f(ζ/y f(η, y,ζi 4. (35 Evaluating I 4 leads to the following expression, I 4 f(y/ζ, ηf(ζf(η

4 10 5 STO estimation CFO estimation Mean Square Error 10 0 Bit Error Rate E /N in db b STO estimation CFO estimation E b /N 0 in db Fig. 1. MSE performance of STO CFO estimation for the MAP algorithm in an AWGN channel. {e {e c 1a(k c 2P (k δ(η mf(ζ} c 1a(k c 2P (k δ(η m}. (36 The range of values of ζ for the above equation gets restricted to ζ 0.5 due to the multiplicative factor f(ζ. Hence, ˆζ MAP is found to be ˆζ MAP argmax ζ [ e c 1a(k c 2P (k VI. RESULTS AND DISCUSSIONS ]. (37 A. Performance of the MAP Synchronization Algorithm The mean square error (MSE curves of STO CFO estimation are obtained through 1000 Monte Carlo simulations. From fig. 1 we observe that the frequency estimator performs better than the time estimator. The MSE of CFO estimation is seen to be around 10 4 at 12dB SNR while the MSE of STO estimation is 10 3 throughout the entire range of SNR values. B. Performance of the Receiver System after Decoding The bit error rate (BER curves of STO CFO estimation are also obtained through 1000 Monte Carlo simulations. From fig. 2 we observe that the BER performance during CFO estimation corresponds to the BER curve obtained without any synchronization errors which implies that the residual CFO error after CFO estimation subsequent decoding is negligible. The unsatisfactory performance of STO estimation is reflected in its BER performance too as it saturates at a value of 0.5. Fig. 2. BER performance with STO CFO estimation for the MAP algorithm in an AWGN channel. TABLE I COMPUTATIONAL COMPLEXITY COUNT Parameters Computations MAP Algo. STO Complex Additions 3L-1 Complex Multiplications 8L+1 CFO Complex Additions 6L-1 Complex Multiplications 15L+1 C. Computational Complexity Count Table 1 gives a comparison of the computational complexity count for CFO STO estimation for the MAP algorithm in terms of the length of CP (L. From the table we observe that the number of complex operations during STO estimation is approximately half as compared to the operations required during CFO estimation. Implementation of eq. (33 (37 results in the following computational count table. VII. CONCLUSION For a COFDM system, we have presented the joint MAP estimator of time frequency offset that uses the redundant information contained within the cyclic prefix. Through a performance based study we have observed that though the CFO estimator gives a satisfactory performance, the STO estimator is unable to track the timing offset well that leads to a severe degradation in the BER of the receiver system. We have not compared the performance of MAP estimators to ML estimators in this paper primarily because both the techniques are entirely different, the former being an a- posteriori technique while the latter is an a-priori technique of estimation. As a future extension different prior pobabilities may be chosen which might provide greater accuracy at the cost of increased complexity. REFERENCES [1] J. J. V. Beek, M. Sell P.O. Borjesson, ML Estimation of Time Frequency Offset in OFDM Systems, IEEE Trans. Signal Process., vol. 45, no. 7, pp , July 1997.

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