CHAPTER 2 CARRIER FREQUENCY OFFSET ESTIMATION IN OFDM SYSTEMS

Transcription

1 4 CHAPTER CARRIER FREQUECY OFFSET ESTIMATIO I OFDM SYSTEMS. ITRODUCTIO Orthogonal Frequency Division Multiplexing (OFDM) is multicarrier modulation scheme for combating channel impairments such as severe multipath fading and impulsive noise. However, the principal disadvantage of OFDM is that it is highly susceptible to carrier frequency offset (CFO). Carrier frequency offset occurs due to frequency discrepancies between transmitter and receiver and Doppler shift of the mobile channel. The impact of CFO are the loss of orthogonality among subcarriers, inter subcarrier interference, accumulation of phase error over successive symbols. These effects can degrade system performance to a significant extent. In order to rectify the aforementioned issues, a signal processing algorithm is to be developed to estimate frequency offset and correct it.. LITERATURE REVIEW The signal processing algorithms for CFO estimation in OFDM systems are grouped either as blind or data aided. In blind estimation algorithms, the periodic structure of Cyclic Prefix (CP) is used to estimate CFO (Beek et al 997). In literature, many blind algorithms have been reported (Bölcskei 00, Huang and Latif 006, Huang and Latif 006a, Lee and Kim 006). Traditionally, blind estimation algorithms are bandwidth efficient and do not require additional overhead (Zeng 008).

2 5 In data aided CFO estimation algorithms, a known preamble, or pilot symbol is inserted in front of each data packet such that it can easily be employed by the receiver to achieve synchronization, thereby making it suitable for applications involving packet based transmission. The major drawback of the data aided estimation algorithm is the overhead associated with the pilots or training in the OFDM symbols. In data aided algorithms, CFO estimation is carried out in two stages namely a coarse estimation or acquisition stage and fine estimation or tracking stage (Gao et al 008). This chapter addresses the CFO estimation in packet based transmission and proposes a novel data aided algorithm for fine estimation... Data Aided CFO Estimation The data aided method uses correlation between repetitive slots to achieve the CFO estimation.... Coarse CFO estimation methods A Maximum Likelihood (ML) frequency offset estimator based on the use of two consecutive and identical symbols was presented by Moose (Moose 994). The maximum frequency offset that can be handled is where f, f is the subcarrier spacing. When the training symbols are shortened by a factor of two, acquisition range of CFO is doubled. However, when symbols are shorter, there are fewer samples over which average has to be performed. The training symbols need to be kept longer than the guard interval so that channel impulse response does not cause distortion while estimating the frequency offset. To overcome this drawback, a null symbol based method was proposed by ogami and agashima(995). In this method, the CFO is estimated in the frequency domain after applying a Hanning window and taking the Fast Fourier Transform. However, this

3 6 approach requires an extra overhead for null symbol and increase in computational complexity. Schmidl s method (Schmidl and Cox 997) of CFO estimation investigates the usage of two training symbols. The first has two identical halves and is used to estimate a frequency offset less than the subcarrier spacing while the second symbol contains a pseudo noise sequence used to increase the range of estimation. The drawback of this method is that it consumes more overhead due to the usage of two training symbols. Improved frequency offset estimation was proposed by Morelli and Mengali based on the Best Linear Unbiased Estimation (BLUE) principle at the cost of increased complexity (Morelli and Mengali 999). In continuation, Minn et al has developed three methods based on the BLUE principle. The frequency offset estimation and MSE performance of the first method is almost same as in the method by Morelli (Morelli and Mengali 999). But the other two methods show better performance, especially at low SR values (Minn et al 00). This work also analyzes the effects of number of identical parts contained in the training symbol on the frequency offset estimation performance. This gives an insight on how the training symbols should be designed in order to achieve a better MSE performance with the same amount of training overhead. A burst format for OFDM transmission was initially proposed for frequency synchronization with a large estimation range and good accuracy (Lambrette et al 997). In the sequel, Averaged Decision-Directed Channel Estimation (ADDCE) technique for burst data was proposed to track time-variation of a wireless channel as well as to reduce noise effects at sub-channels (Song et al 000). Bang et al (00) proposed a coarse CFO estimation algorithm that is robust for any symbol timing offset that falls within an allowed range. The

4 7 proposed algorithm uses the concept of the coherence phase bandwidth for reducing the effect of a symbol timing offset. Liu et al (004) proposed a CFO estimation algorithm using a multi-stage synchronization in time and frequency domain for OFDM. This algorithm uses all pilots including continual pilots and scattered pilots to estimate the CFO in frequency domain, and it can acquire more accurate estimation results than the conventional algorithm which uses continual pilots. Lottici et al (005) presented an algorithm considering the selectivity of the channel, leading to the use of a weighted window instead of a rectangular one. Shi et al (005) proposed a new Decision Directed (DD) post-fft CFO synchronization scheme without relying on pilots. It is shown that the proposed CFO estimator is approximately unbiased in both AWG as well as frequency selective channels. Lin (006) developed an effective technique for frequency acquisition based on ML detection for mobile OFDM. The proposed technique employs a frequency acquisition stage and a tracking stage. By exploiting the differential coherent detection of a single synchronization sequence, where a Pseudo-oise (P) sequence is used as a synchronization sequence. Data aided frequency acquisition with frequency directional P matched filters reduce probability of false alarm and probability of miss on a channel whose coherence bandwidth is sufficiently wide. Laourine et al (007) proposed a new data aided CFO scheme for OFDM communications suitable for frequency selective channels. It is based on the transmission of a specially designed synchronization symbol that generates a particular signal structure between the received observation samples at the receiver (Laourine et al 007). The proposed work offers a wide acquisition range with reduced computational complexity. Sevillano et

5 8 al (007) proposed ML based carrier frequency estimator for OFDM systems with the preambles formed by Short Training Symbols (STS). The FFT processor in the OFDM receiver is used for efficient implementation of the proposed algorithm. Ghogho et al (009) proposed a method to design optimal preamble using the CRB as a metric. This involves optimizing the number of repetitive slots and the power loading. They provided closed-form expressions illustrating the impact of multipath diversity on estimation performance... Data Aided Fine Frequency Estimation After the acquisition stage of CFO estimation, residue CFO is present either due to the insufficient accuracy during the coarse estimation, or the time varying nature of the surrounding environment. The residue CFO, if not compensated, may still lead to performance degradation. Hence, many existing standards reserve a limited number of scattered pilot symbols in each OFDM blocks to improve the system robustness in different aspects. For example, in IEEE 80.a WLA standards, four pilots are placed at the subcarriers with indices {7,, 43, 57} for the purpose of combating the residue CFO and the phase noise. Classen and Meyr s (994) proposed a method for fine CFO estimation assumed that the Channel Impulse Response (CIR) remains constant for two consecutive OFDM blocks over a slow fading channel. For a small CFO (much less than one subcarrier spacing) and a low SR, the Inter Carrier Interference (ICI) induced by the CFO can be ignored as opposed to the large additive noise. Therefore, the CFO can be estimated by comparing the received symbols on the pilot carriers from the two consecutive OFDM blocks.

6 9 Gao et al (008) proposed a novel CFO tracking algorithm using the scattered pilot carriers embedded in each OFDM block. Identifiability of this algorithm was studied for the noise free case, and a constellation rotation strategy was proposed to eliminate a major type of CFO ambiguity for widely used constellations. To improve the performance of the CFO estimation and enhance the robustness to the CFO ambiguity, the virtual carriers existing in practical OFDM standards were used. Later, merits of both the algorithms were combined by exploiting both scattered pilots and virtual carriers. In summary, there exist a large variety of algorithms for course CFO estimation and a few algorithms for fine CFO estimation. Each of the algorithm attempts to reduce the mean square error and increase the range of CFO estimation with reduced computational complexity. However there exists tradeoff between the MSE and computational complexity..3 ISSUES AD PROBLEM FORMULATIO Though much work has been done on coarse CFO estimation in OFDM, there exist very few algorithms for fine CFO estimation. Further these algorithms are computationally intensive due to the ML search operation. Hence, they can not be directly applied to develop low cost solutions to Wireless standards such as WLA and WiMAX standards (IEEE 004). Further, the algorithms in the literature assume the channel is uncorrelated, static and timing synchronization is perfect. However, the real time channels are correlated due to the mobility of the communication transceivers and the practical timing estimation algorithms results in a finite residual timing error. Hence, CFO estimation algorithms need to be developed for correlated fading channels, considering the timing errors.

7 30.4 SYSTEM MODEL This section introduces the system model that is considered for CFO estimation. A detailed derivation on the effect of much smaller CFO in terms of ICI is presented. This analysis motivates the development of a novel algorithm for the CFO estimation. The information bit stream is multiplexed into symbol streams, each with symbol period T, modulating a set of sub-carriers that are spaced by T. Using the Inverse Discrete Fourier Transform (IDFT), the OFDM transmitted symbol is given by m x n X ( m)exp j n n 0,,... m 0 (.) where X ( m ) are the discrete baseband symbols on each sub-carrier, that are derived from a modulation alphabet of size M. A cyclic prefix is added before transmission to combat Inter Symbol Interference (ISI). At the receiver, the cyclic prefix is removed followed by Discrete Fourier Transform (DFT) processing. After DFT processing the received signal is expressed as n Y k y n exp j k k 0,,..., n 0 (.) Due to the frequency offset, the received baseband signal is given by y n x n exp j n u n (.3)

8 3 fd f LO where f is the normalized frequency offset with respect to sub-carrier spacing f, fd f LO is the total frequency offset due to both the doppler frequency, transmitter and the receiver, f D, and the local oscillator mismatch between the f LO. u n represents the complex Gaussian noise with zero mean and variance Equation (.),. Substituting Equation (.3) into n n Y k x n exp j exp j k U k n 0 (.4) where U k is the DFT of the noise u n. Substituting Equation (.), Equation (.4) can now be written as, n n Y k X m exp j m exp j k U k n 0 m 0 n X m exp j m k U k m 0 n 0 (.5) Using the identity, n 0 a n a a (.6) The term n 0 n exp j m k can be expressed as, n exp j m k exp j m k n 0 exp j m k

9 3 Using the trigonometric identity sin x e jx e j jx, n exp j m k exp j m k n 0 sin sin m m k k (.7) Substituting Equation (.7) into Equation (.5), yields sin m k Y k X m exp j m k U k m 0 sin m k (.8) Thus, the received signal in Equation (.8) can then be decomposed as, Y k X k 0 X m m k U k (.9) m 0, m k where m k are the ICI coefficients between the which is given by, th m and th k sub-carriers sin m k m k exp j m k sin m k (.0) The first term in Equation (.0) denotes the desired signal, while the second term represents the ICI which appears as a result of the frequency offset and the third term is the noise. Ignoring the noise term and focusing only on the ICI as a source of impairment, the carrier-to-interference ratio is given by

10 33 C I E X k 0 E X m m k m 0, m k (.) where E. represent the expectation operator.if the transmitted data symbols are assumed to have zero mean and are statistically independent, Equation (.) becomes C I 0 0 m k m m 0, m k m (.) A plot of the carrier to interference ratio versus frequency offset for FFT size of 04 is shown in Figure. In the WiMAX system (IEEE 004), the subcarrier spacing f 0.94 khz, and fd f LO =00Hz, thus 0.0 (i.e. % of the sub-carrier spacing). It can be seen from Figure. that at 00Hz frequency offset the carrier to interference ratio due to the ICI is about 9dB. The impact of ICI on each sub-carrier is negligible even for the 00Hz frequency offset. Hence, the effect of ICI can be eliminated while deriving CFO estimation algorithms for applications such as WiMAX and LTE.

11 34 Figure. C/I as a function of frequency offset.5 PROPOSED ALGORITHM FOR FREQUECY OFFSET ESTIMATIO In this section, a novel algorithm for CFO estimation is proposed. At first the Goa algorithm (Gao 008) for fine CFO estimation is explained. Then the proposed algorithm is discussed. The Gao method uses a ML approach for the fine CFO estimation.it can be described as ˆ arg min,,,, k 0 X k l Y k l X k l Y k l (.a) where Y k, l y n, l exp j exp j k n 0 n (.b) The Gao algorithm searches for a CFO which compensates the additional CFO in a adjacent symbol as in Equation (.b) and minimizes the deferential metric defined in Equation (.a). The ML operation in (.a)

12 35 is computational intensive and hence new method based on a novel signal model which results due to very less CFO is proposed as below. When the CFO is very less, it has been shown in section. that the ICI can be ignored. By ignoring the ICI term, Equation (.9) is written as, Y k X k 0 U k (.3) Further, for a small frequency offset,, (0) can be written as, sin 0 exp j exp j sin (.4) For a small frequency offset 0.Considering the channel frequency response at the k th subcarrier of l th OFDM symbol, the received symbol after DFT is represented as the product of channel frequency response, transmitted symbol and the frequency offset accumulated up to l th symbol. It is given by, Y k, l exp j l H k, l X k, l U k, l (.5) where H k, l is the channel frequency response at symbol. The FFT output at th k subcarrier of th k subcarrier of th l OFDM th l symbol is the product of transmitted pilot, channel frequency response at respective subcarriers and a phase term due to frequency offset at the l th symbol. This motivates the idea to propose the estimation of CFO by cross correlating the same carriers in adjacent symbols. Since the same subcarriers at adjacent symbols carry different pilots an intermediary symbol is proposed as * * Z k, l Y k, l X k, l Y k, l X k, l (.6)

13 36 Assuming perfect timing synchronization, and the fact that the channel is almost same across two symbol durations, and assuming that the modulation symbols are known pilots, Equation (.6) can be written as, Z k, l exp j H k, l V k, l (.7) where the noise term V k, l is due to the cross-product in Equation (.6). The effect of noise is reduced by averaging over L pilot symbols and number of subcarriers. It is given by Z L L k 0 l 0 Z k, l (.8) Equation (.8) represent the averaging over continual pilots. A similar averaging can also be applied for scattered pilots which are defined for estimating smaller CFO. It is noted that if there are number of scattered pilots, the averaging can also be done for number of scattered pilots in L symbols. Hence this method can also be applied for tracking or fine CFO estimation. By applying the ML principle (Kay 993) the estimate of the normalized CFO is proposed as ˆ arg Z (.9) output, as This estimate can be used to compensate the CFO at the DFT Y k, l exp j l ˆ Y k, l (.0)

14 37.6 PERFORMACE AALYSIS OF THE PROPOSED ESTIMATOR In this section, the MSE performance of the proposed CFO estimation algorithm is analyzed. The DFT output at ( l ) th OFDM symbols are given by th k subcarrier of th l and Y ( k, l) exp j l H ( k, l) X ( k, l) U ( k, l ) (.) Y ( k, l ) exp j l H ( k, l ) X ( k, l ) U ( k, l ) (.) It can be assumed that H k, l H k, l and X k, l X k, l. Substituting Equation (.) and Equation (.), equation (.6) can be written as, * Z( k, l) exp j H ( k, l) exp j l H ( k, l) U ( k, l ) X ( k, l ) * * * * exp j l H( k, l) U( k, l) X ( k, l) U( k, l) X ( k, l) U ( k, l ) X ( k, l ) (.3) can be written as, At high SR, the last term can be ignored. Then, Equation (.3) * * exp j l H( k, l) U ( k, l) X ( k, l) ] * Z( k, l) exp j [ H ( k, l) exp j l H ( k, l) U ( k, l ) X ( k, l ) (.4) Define * v( k, l) : exp j l H( k, l) X ( k, l) U ( k, l ) (.5)

15 38 simplified as Substituting Equation (.5) in Equation (.4), Z ( k, l ) can be * Z( k, l) exp j [ H ( k, l) v( k, l ) v( k, l ) ] (.6) Substituting Equation (.6) in Equation (.8), Z can be written as L exp j * Z [ H( k, l) v( k, l ) v( k, l) ] L k 0 l 0 (.7) Substituting Equation (.7) in Equation (.9), the normalized frequency offset estimate is given by L * ˆ arg [ H ( k, l) v( k, l ) v( k, l) ] L k 0 l 0 (.8) written as, Assuming that v( k, l) v ( k, l) jv( k, l ) the estimation error is ˆ P A Q j A (.9) where C represent angle of the complex number C, A L k L 0 l 0 H( k, l) (.30) L P ( v ( k, l ) v ( k, l )) (.3) k 0 l 0 L Q ( v ( k, l ) v ( k, l )) (.3) k 0 l 0

16 39 Since the term v (, ) k l is the product of signal and noise, at high SR its value is less and hence both P j Q A A can be approximated as Q A P and Q are much smaller, and (Tretter 985, Rosnes and Vahlin 006); substituting it in Equation (.9), the estimation error is given by ˆ Q A (.33) Further, if and L are large, A can be approximated as L A H( k, l) E[ H( k, l) ] L k 0 l 0 h (.34) given by Then, MSE of the normalized frequency offset estimation is E[ Q ] E [( ˆ) ] (.35) 4 4 h Using Equation (.3), E[ Q ] is written as L E Q E v k, l v k, l k 0 l 0 u h u h Since v( k, l) 0, u h ; v( k, l) 0, ; v( k, l) 0, ; v ( k, l ) and v ( k, l ) are uncorrelated, E[ Q ] is computed to be E Q [( )] u h L (.36)

17 40 Substituting Equation (.36) in Equation (.35), the MSE of the proposed frequency offset estimator is given by, E ˆ [( ) ] ( L ) u h (.37).7 EFFECT OF TIMIG ERROR O THE FREQUECY OFFSET ESTIMATE The analysis of MSE performance assumes that the received OFDM symbol is free from timing errors. But in practice, most of the timing synchronization algorithms for OFDM results in residual timing errors. In this section, the robustness of the proposed algorithm in the presence of residual timing error is analyzed. Consider the case of timing error of m sampling periods. Let cp be the length of cyclic prefix and the allowable range of the timing error be cp m cp. It is shown by Mastofi and Cox (Mastofi and Cox 006) that for m, the SIR is in the order of 0dB to 30 db and the cp corresponding ICI and ISI can be neglected. Hence, Equation (.5) is rewritten as mk j j l m Y k, l e e H k, l X k, l U k, l (.38) For ( l ) th symbol, Equation (.38) can be rewritten as mk j l m j Y k, l e e H k, l X k, l U k, l (.39)

18 4 written as, Substituting Equations (.38) and (.39) in Equation (.6), it is ( j ) * Z( k, l) e [ H( k, l) v ( k, l ) v ( k, l ) ] (.40) where mk j ( j l ) * v( k, l) : e e H ( k, l) X ( k, l) U ( k, l ) (.4) Since the statistics of v( k, l ) and v( k, l ) are similar, the MSE of the CFO estimate ˆ given by Equation (.37) is applicable for the cases with residual timing offset. This proves the robustness of the proposed algorithm for CFO estimation..8 CRAMER RAO LOWER BOUD (CRLB) FOR THE CFO ESTIMATE rewritten as The FFT output at th k subcarrier in the th l OFDM symbol is, j l Y k l e H ( k, l) X ( k, l) U ( k, l) 0 k,0 l L (.4) Let X be a L L diagonal matrix with elements X 0,, X 0,,..., X ( L ),, A be a L L diagonal matrix with elements,exp j,...exp ( L ) j ; h be a L channel frequency response vector with elements [ H0,, H0,,..., H ( L ), ], u be a L with noise vector elements [ U 0,, U0,,..., U ( L ), ].Then the L vector representation of Equation (.4) is given as y XAh u X (.43)

19 4 where [,,,,...,, ] T L ; H l, n l, n e j l n X l, n l, n (.44) The CRLB of the CFO estimation is given by (Key 993) CRLB J where the Jacobian J = E ln P y / The observation vector y conditioned on the normalized frequency offset is Gaussian distributed with zero mean and covariance matrix R yy. Its likelihood function can be expressed as P y exp y H R y L det R yy yy (.45) By neglecting the terms independent of function can be written as, the log likelihood ln P y H y R yyy (.46) where H H H H R E yy E ( XAh u)( XAh u) A( XR hx 0IL ) A (.47) yy Substituting Equation (.47) in Equation (.46), log likelihood function is written as

20 43 ln P y / AX ( XR X I ) XA (.48) H H H H h 0 L where 0 ( L ) diag( X, X..., X ). For simplicity, log likelihood function can be written as ln H H P y ABA (.49) where H H B X ( XR X I ) X ( R ). Substituting the h 0 L h 0 definitions of A & B, the log likelihood function can be written as ln P y / = L L Re B(( l l ) n,( l ) m) e 0 l 0 l l n n 0 * j l,n ( l l ), n l = L l 0 Re ( ) j l g l e (.50) L * where g( l ) B(( l l ) n,( l ) n) l l n n l,n ( l l ), n The first derivative of Equation (.50) is ln P y L 4 Im g( l ) e l 0 j l (.5) The second derivative of Equation (.50) is calculated as ln P y / L j 8 Re g( l ) e l 0 l (.5) The statistical average of Equation (.5) is expressed as

21 44 ln P y E E 8 Re g( l ) e L j l 0 l L L * j E l,n B l l n l n e ( l l ), n 0 l 0 l l n n 0 8 Re [ ] (( ),( ) ) The expectation of l,n is calculated to be (Hoeher 997) l (.53) E [ ] E sinc f n n J f T l e * j l,n ( l l ), n b max 0 D l (.54) where Eb is the average bit energy, max is the maximum delay spread. Using Equation (.53) and Equation (.54), the Jacobian operator is written as J E ln P y (.55) L L 8 Re E( l, n, n ) F( l, n, n ) l 0 l l n 0 n 0 where E( l, n, n ) E sinc F n n J f T l e b j max 0 d l F( l, n, n ) B(( l l ) n,( l ) n) e j l The CRLB for CFO estimate in frequency selective correlated fading channel is calculated ascrlb J. From Equation (.55) it is noted that the CRLB depends on the maximum delay spread, Doppler spread, subcarrier spacing, signal to noise ratio and the correlation matrix of the wireless channel.

22 45.9 RESULTS AD DISCUSSIO The MSE performance of the proposed CFO estimation algorithm is evaluated using Monte Carlo simulation in MATLAB. The simulation parameters are shown in Table.(Hoeher 997). Table. Simulation parameters for CFO estimation S.o Parameters Values umber of subcarriers 8 Modulation QPSK 3 ormalized offset 0.05=5% 4 umber of OFDM symbols 5 Maximum delay spread max µsec 6 Maximum Doppler spread fd 30Hz 7 OFDM symbol duration T 60 µsec 8 Subcarrier spacing f 0kHz 9 umber of Monte Carlo runs 0000 Figure. shows the MSE performance of the proposed CFO estimator in a static channel. It shows the simulated MSE, theoretical MSE and the CRLB for the proposed CFO estimation method. The SR required to attain a MSE of 0-5 is.8 db. The simulated MSE is in close agreement with theoretical MSE at high SR.

23 MSE vs SR over static channel simulation theoretical CRLB SR(dB) Figure. Performance analysis of proposed CFO estimate in static channel Figure.3 shows the MSE performance of the proposed CFO estimator in a frequency selective fading channel. It shows the simulated MSE, theoretical MSE and the CRLB for the proposed CFO estimation method. The SR required to attain a MSE of 0-5 is 3 db. The simulated MSE is in close agreement with theoretical MSE at high SR. There exists consistent 0.4 db difference with that of the CRLB. This is due to the high SR assumption made in deriving the MSE equation. Figure.4 shows the performance comparison with recent data aided fine frequency offset estimation method by Gao et al(008). The method by Gao et al(008) is a ML method and is computationally intensive due to search. The accuracy of the algorithm depends on the search interval. Here, it is chosen as 0.0 to maintain marginal computational complexity. Since the proposed method is a closed form solution, it requires less computation. With the given search interval, the proposed method gives 0.3dB gain over method by Gao et al(008) at a MSE of 0-4. Moreover at

24 47 5 db of SR the proposed method gives an MSE of However at very high SR the method by Gao et al (008) outperforms the proposed method since assumption of negligible ICI is no longer valid. Figure.5 shows the performance comparison with Gao method with normalized CFO of 0.3. With the given search interval of 0.0 for CFO estimation, the proposed method gives 0.6dB gain over method by Gao et al (008) at a MSE of 0-4. The performance of the proposed algorithm is similar to the performance with normalized CFO of However the gain over Goa method is 0.04dB lesser than that of the performance at a CFO of Moreover at 5 db of SR the proposed method gives an MSE of This reduction in the performance at higher normalized CFO is due to the fact that the proposed method is derived with low CFO assumption. At higher values of CFO the validity of the approximation reduces causing minor performance degradation. 0-3 MSE vs SR over correlated frequency selective channel simulation theoretical CRLB SR(dB) Figure.3 Performance analysis of proposed CFO estimate in frequency selective fading channel

25 MSE vs SR over fading channel Proposed-Simulation Gao Proposed-Theoretical SR(dB) Figure.4 Performance comparison of proposed algorithm with Gao method 0-3 MSE vs SR over fading channel Proposed-Simulation Gao Proposed-Theoretical SR(dB) Figure.5 Performance comparison of proposed algorithm with Gao method with normalized CFO of 0.3

26 49 Figure.6 shows the effect of timing offset on the MSE performance of proposed algorithms. The timing offset considered is 0,, and 5 samples. It is observed that there exists 0.dB loss at the MSE of 0-6 when the timing error is 5. This loss is negligible and thus the proposed algorithm is robust against timing error timing error=0 timing error= timing error= timing error= SR(dB) Figure.6 Effect of timing error on the performance of CFO estimate Figure.7 shows the range of frequency offset that can be estimated at 0 db of SR. When the normalized frequency offset is -0.5 to 0.5 the estimated frequency offset is equal to the true frequency offset with an error of 0.9*0-5.When the range is extended there exists larger error between the true and the estimated CFO. Hence, the CFO estimation range is found to be -0.5 to 0.5 of the subcarrier spacing. The breakdown region of the proposed algorithm for the normalized CFO is greater than +0.5 and less than -0.5.

27 true frequency offset Figure.7 Estimation range of the proposed estimator.0 SUMMARY In this chapter, low complex data aided fine CFO estimation algorithm with improved accuracy is developed. The proposed algorithm is based on continual pilots and can also be used for CFO estimation based on scattered pilots. It is assumed that the CFO is very less when compared to subcarrier spacing and the channel response is constant for two consecutive OFDM symbols. The impact of the very less CFO on OFDM in terms of carrier to interference ratio is characterized. A frequency domain CFO estimation algorithm with improved performance is proposed. An analytical expression for MSE performance of the algorithm is obtained. Cramer-Rao Lower Bound (CRLB) on the CFO estimation in frequency selective fading channel is also derived and it is shown that simulation results are matching with the analytical expressions. For a specific search interval of 0.0l, the proposed method gives 0.3dB gain over method by Gao et al (008) at a MSE of 0-4.