A Modified Algorithm for Joint Frequency Offset and Channel Estimation in OFDM Systems

Shi-Hao Chiu, Kuo-Ching Fu, and Yung-Fang Chen

Department of Communication Engineering, National Central University

Chung-Li City, Taiwan, R.O.C.

Email: 995203024@cc.ncu.edu.tw; 955403002@cc.ncu.edu.tw; yfchen@ce.ncu.edu.tw

Abstract—Previously, a joint estimation method [1] of carrier frequency offset and channel in orthogonal frequency division multiplexing (OFDM) systems was proposed based on the maximum likelihood (ML) criterion. However, the proposed processing structure may not be suitable for current OFDM regular implementations. In the paper, the processing structure is modified and simulations are conducted for the new proposed processing structure. The proposed algorithm can be divided into two major portions. The first part is to use the rotation concept to estimate the initial CFO and design the frequency domain equalizer. The second part is making use of iterative concept to find the true frequency peak for better estimate. The proposed scheme does not require initial random guess as regular iterative algorithms, in which may suffer from the problem of the convergence to the local maximum. The proposed approach can surely converge to the global maximum for achieving the solution. The computation complexity is much less than the grid-search method and easily be implemented in hardware. The MSE performance of the proposed algorithm is close to the Cramer-Rao Lower bound which is shown in the simulation result.

Keyword: joint maximum likelihood estimation; carrier frequency offset (CFO); Orthogonal frequency division multiplexing (OFDM).

I. INTRODUCTION

Orthogonal frequency division multiplexing (OFDM) is recognized as a main modulation technique for fourth-generation (4G) broadband wireless systems and it is easy to implement since Fast Fourier transform (FFT) algorithm can apply [2]. The OFDM technique has lots of advantage such as high frequency spectral efficiency, robustness against frequency selective multipath fading channel, and easy equalizer designing. OFDM has been adopted in wideband communication standards over DAB, DVB, ADSL and VHDSL. It has also been chosen as the modulation technique such as IEEE 802.11a and HIPERLAN/2 [3]-[6].

Despites of these advantages, OFDM system still have critical synchronization issue including frequency synchronization and timing synchronization. Especially in frequency synchronization, OFDM is sensitive to the carrier frequency offset (CFO) which appears due to the frequency difference of the local oscillators between the

transceiver and the receiver and Doppler spread. CFO damages the subcarrier orthogonality and causes inter-carrier interference which degrades the performance of the OFDM system. The SNR degradation analysis due to the CFO can be found in [7].

Many synchronization algorithms in literatures are discussed in recent years including data aided of different pilots [8]-[11] and blind methods. In order to increase bandwidth efficiency, the non-data aided blind estimation has gotten a lot of attention [12]-[15]. However, without data-aided help, the estimation of parameters is more difficult and the errors also increase compared to the pilot-aided method. The blind estimation also requires collection over lots of OFDM symbol, which may not fit the structure of packet-based OFDM transmission. The other method is to perform inter-carrier interference cancellation directly [16]-[19], but it is generally more complex than the other two methods.

Lots of algorithms are proposed by using training sequence structure to estimate CFO and channel. In [20], the authors utilize Viterbi algorithm to update the estimate CFO and channel. In [21], the authors use maximum likelihood (ML) and least-squares (LS)-based methods to update CFO estimates iteratively. However, there are some drawbacks in these two algorithms. In [20], the algorithm needs lots of symbols to update the estimation results and the authors in [21] assume the initial channel estimation is known.

The maximum likelihood-based algorithms usually provide good performance. In [22], the authors propose an algorithm which can jointly estimate CFOs and channels. They also derive another mathematical expression to estimate timing offset. Although the performance of CFO estimation in [22] could be near the Cramer-Rao bound (CRB) by using grid-search method, it is very difficult to implement in hardware since the computational complexity is extremely high.

In view of this, a new iterative algorithm for CFO estimation is proposed. The proposed algorithm can find the estimated CFO by a rotation concept and use the estimated result as an initial value. It then uses an iterative method to improve accuracy for approaching CRB. The computational complexity of the proposal algorithm is much less than the grid-search based method. This algorithm is suitable for hardware implementation.

2013 2nd IEEE/CIC International Conference on Communications in China (ICCC): SPC: Signal Processing forCommunications

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Notation: the superscript, T , 1 , H and

N Ndiag

represent the transpose, inverse, Hermitian

transpose and N N diagonal matrix. The N N

identity matrix is denoted by N NI

II. SYSYEM MODEL

The preamble OFDM system with N subcarriers is considered [22]. At the OFDM transmitter, the baseband time domain signal after inverse-discrete-Fourier-transform (IDFT) and CP insertion can be represented as.

1 2 /

0

1, if 1

1

0, otherwise

N j nk Ngn

d n e N n Nx n N

where d n is the frequency domain data symbol on

each subcarrier 0,..., 1n N and gN is cyclic prefix

samples of the OFDM symbol which is appended in front of OFDM symbol to resistance intersymbol interference. The discrete-time composite channel impulse response (including the pulse shaping filters at both the transmitter and the receiver) is denote as

0 1 1, ...,T

Lh h h h (2)

where lh is the complex Gaussian gain of thl multipath with the exponential power delay profile. We assume the complex channel impulse response is time-invariant over one OFDM symbol. So the receive signal with CFO and timing error can be easy written to a convolution format [23]

1

2 /

0

rL

j n N

l

n e h l x n l v n

(3)

where v n is complex Gaussian noise with zero mean

and variance 2v ; denotes normalized CFO to

subcarrier spacing; is integer timing offset of the receive OFDM symbol; assumes that the L plus is smaller than the CP length to avoid ISI. We assumed that the timing offset is estimated by the first part of preamble and it would be compensated by timing synchronization at the receiver so the integer timing offset equal to zero.

After CP removal, the received signal collects N consecutive samples and we can rewrite the receive signal in

r D Xh v (4)

where X is a N L circular matrix formed by

transmitted signal x n ; D is a diagonal CFO

matrix 2 1 /2 /1, ,..., j N Nj N

N Ndiag e e

; and v is

a 1N noise vector; its covariance matrix is 2v N N I .

Let D Xh s , [ (0),..., ( 1)]Ts s N s so the SNR is

defined as [8]

2 2SNR= /s v (5)

where

1

22

0

1 , 0,1,..., 1

N

sn

s n n NN

(6)

III. JOINT MAXIMUM LIKELIHOOD ESTIMATION

A. Receiver Design

Fig. 1. Receiver structure

The proposed receiver is illustrated in Fig. 1. The difference from other OFDM receive structures is to use the frequency-domain equalizer to eliminate multipath interference, then feedback to time-domain signal for obtaining more accurate estimate of CFO. The coefficients of the equalizer are designed by the estimated result from channel estimation. The method to calculate the coefficients for the design will be mentioned later.

B. Initial CFO Estimation

For convenience, we define S D X and (4)

become

r Sh v (7)

According the Gaussian probability density function, the maximum likelihood function can be written as

2

2

1

21

( , )2

Ne

x Sh

h (8)

By computing the log-likelihood function in (8), and removing the constant, the objective function becomes to minimize the expression as

2

1 , h r Sh (9)

In (9), there are two unknown parameters. We can expand (9) and fix in order to use a gradient concept to find h

1H Hh S S S r (10)

Substituting (10) into (9), the ML objective function becomes to maximize

1

1H H H H

r DX X X X D r (11)

can be found by

1arg max H H H H

r DX X X X D r (12)

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The projection matrix 1H HX X X X is formed by the

preamble signal. It can be evaluated in advance. Thus, receivers do not re-calculate this matrix including the inverse computation so that it can avoid the computation cost every time. The projection matrix can be represented as

1

1 1,2 1, 1 1,

2,1 2 2, 1 2,

,1 ,2 , 1

...

...

...N N

H H

N N

N N

N N N N N

R C C C

C R C C

C C C R

X X X X

(13)

where iR , 1,...,i N is real-valued

and ,i jC , 1,..., , 1,...,i N j N , i j is complex-valued

which is the conjugate of ,j iC . After some careful

derivation, by substituting (13) into (12), the can be written as a particular format as follow

2* 2 /

2, 10

32 2 /*

13, 12

010 1

2 2 /*1 , 1

0

2 1 /*,1

1

2 ...max 2 Re

2

1 0

Nj N

i ii

Nj N

Ni i

iii

j N NN i i

i

j N NN

C r i r i e

C r i r i eR r i

C r N i r i e

C r N r e

(14)

In order to find such that equation (14) has a maximum value, we may maximize this equation term by term. Because the first part in (14) is a fixed real value, we only consider the second part. For example, as the first item in second part is a complex value which is caused by the frequency offset, we can rotate its phase to the real axis to achieve the maximum real value. (in other word, the frequency offset is compensated). The frequency offset can be obtained by

1

*1, 1

0

/ 2N n

i n ii

n angle C r i n r i N n

1,..., 1n N (15)

After finding phase of each item of (15), we averaged the phase of each item and calculated the initial CFO estimated as.

1

1

1

1

N

n

nN

(16)

C. Frequency-Domain Equalizer Design

Because of the multipath interference, the performance of the initially estimated CFO would be affected. To overcome this problem, a frequency-domain equalizer is used to cancel the multipath interference for obtaining more accurate estimate of CFO. At first, we initially estimate CFO i from (15)

and compute the coefficients of the channel response (10). Then, we use the result of channel response estimation to design the coefficients of equalizer. After that, we use the equalizer to equalize the receive signal and transform to time domain signal to estimate CFO again using the rotate concept as stated before. After the process of equalizer, the performance of the estimated CFO e after going through the equalization is close to the global maximum frequency and has lower computational complexity compared with the grid search-based method.

Although the estimation of CFO would approach to the global maximum, however, due to the short of computation samples (refer to (15) and the second part of (14), where we use only small portion of terms in each n rather than all of items). These may degrade

the performance. In next section, we propose a method that refining the estimate such that the performance may be close to CRB.

D. Small Step Iterative Searching

In this section we use the iteration method to refine estimate. At first, we let the estimated CFO e that is from the output of the equalizer to be the initial main frequency. Then, we chose the adjacent frequency of the initial main frequency as the candidate frequency. By putting the main and the candidate frequency into equation (11) and comparing the two values, we can find the searching direction by choosing the largest value. (Refer to Fig. 2. Step 1) After that, we make the chosen frequency to be the new main frequency m and add the

fixed step frequency f to be its adjacent frequency. Then, we compare the magnitudes as before. (Refer to Fig. 2. Step 2 ~ Step i ) The iterative method continues until the result of the main frequency is larger than its adjacent frequency. (Refer to Fig. 2. Step -1I , I is the total iteration number ) Finally, we search the frequency peak in the region between the last main frequency and the last adjacent frequency to find the maximum frequency peak f (Refer to Fig. 2. Step I )

However, if the selected small step frequency is not appropriate, the global maximum frequency peak would not be in the searching range. The estimated CFO with the iterative method is not the true frequency peak. For instance, the true global maximum frequency is in the previous region between the iterative main frequency and the iterative adjacent frequency. (Refer Fig. 3. Step

1I ) In this circumstance, the iterative adjacent frequency

im f has the value of (11) is larger than

that of iterative main frequency im . So it would judge

the global maximum frequency located at the right hand side of the iterative adjacent frequency (Refer Fig. 3. Step 1I ). The iteration will go on until the result of the main frequency larger than that of the adjacent frequency. (Refer Fig. 3. Step I ) However, it induces the maximum frequency peak out of the final searching range.

For the case of the global maximum frequency out of

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the searching region, the method of the reverse iterative searching is adopted. We measure the magnitudes between the two frequencies, and turn the searching range into the reverse direction if the maximum magnitude value at the boundary frequency. (Refer Fig. 3. Step I ). It means that the maximum peak frequency is at the other side of the boundary frequency. This process is called reverse iterative searching. Finally, the iterative reverse searching method could find the global maximum value. (Refer Fig. 3. Step 1I ) The diagram of the iterative small step frequency and reverse iterative searching method are figured in Fig. 2 & Fig. 3.

Main frequency

Estimated CFO

frequency

CFO peak

frequency

frequency

frequency

search

frequency

Adjacent frequency

Right hand side peak

Left hand side peak

Step 1

Step 2

Step i

Step I-1

Step I

Determine the searching direction

Iterative the main frequency

Iterative the main frequency

Stop the iteration

Searching the frequency peak in the range

e

m m f

1m 1m f

2m 2m f

2m 2m f f

f

f

f

f

Fig. 2. Iterative small step frequency

e

m m f

1m 1m f

1Im

1Im f

Im Im f f

f

f

f

f

f

ImIm rf

Fig. 3. Reverse iterative search

E. Computational Complexity & Procedure of the Proposed Method

The computational complexity and the procedure of the proposed algorithm are shown in Table I. and Table II.

Table I. The procedure for the proposed algorithm

Algorithm procedure Proposed algorithm Step 1 Using rotate concept (14) to

estimate each item of the second part and average the results to be the initially estimated CFO i

Step 2 Make use of the result of step 1 to estimate channel by (10) and design the coefficient of equalizer taps.

Step 3 Equalize the receive signal and estimate CFO e as step 1.

Step 4 Set the main freq. and the adjacent freq. form step 3 and find out the frequency peak range by the iterative method.

Step 5 Searching the frequency peak

f in the range form step 4.

Table II. Analysis of Computational Complexity

Algorithm Multiplications

Propose Algorithm

Step1 22 8 7N N Step2 3 22 2 2 log 6L N L L p p Step3 22 2 logN N N N Step4 2 3 1N N i Step5 2 3 1N N

Grid search in (11) 2 3N N M

where L is the number of multipaths; p is the tap number of the equalizer, i is the iteration number for convergence of the step frequency search; is the summation of the numbers of the searching range points and reverse searching points; and M is the number of the grid search points in possible CFO range, M N , M

IV. SIMULATION RESULT

In this section, the simulations are performed to demonstrate the efficacy of the proposed scheme.

A. Algorithm Performance

In the OFDM system simulation, the parameters are as follow: N = 64, CP size = 16. The OFDM data symbols are modulated by QPSK technique through three Rayleigh fading multipaths and the equalizer tap number are eight. The normalized CFO is generated independently from the random variable uniformly distribution in the range [-0.5, 0.5]. The simulation result is averaged by 10000 Monte Carlo tests. The MSE performance of the proposal algorithm in different SNRs is illustrated in Fig. 4. In Fig. 4, the performance of the proposal algorithm is very good and it is close to the CRB. The lower bound is derived by the Fisher information matrix in [24].

2

1

2H HvCRB

Xh X M MXh (17)

where 1H H X I X X X X

2 12 4(0, , ,..., )

Ndiag

N N N

M

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The symbol error rates of different simulations are shown in Fig. 5. It includes (1) “ideal channel” condition which has perfect synchronization and perfect channel estimation, (2) “perfect CFO estimation and joint channel estimation”, and (3) “the proposal joint CFO and channel estimation”. The result can indicate that the joint estimation is very close to the others which have some ideal assumptions. So the proposed efficient algorithm has the advantage of much less computational complexity compared with the grid search-based method and provides very competitive performance of the CFO estimation.

Fig. 4. The MSE performance of the proposal algorithm compared with CRB

Fig. 5. The symbol error rate of the joint channel estimation

B. Step Frequency Analysis

In the proposed algorithm, the size of the small step frequency has to determine. We select five different frequencies step sizes to compare the total number of searching points. The simulation result of the iteration number with different small step frequencies versus SNR is shown in Fig. 6. In Fig. 6, we can find that the iteration number is larger while the step frequency is smaller. The average reverse iteration number versus SNR is shown in Fig. 7. After that, we can calculate the total searching numbers compared with different step frequencies in Table III & Table IV.

From Table III & Table IV we can find that the small step frequency equal to 0.0005 has fewest searching points in low SNR compared with others step

frequencies. Although the largest step frequency can speed up to find the searching range, it has to search too many points. On the other side, the smallest step frequency has fewest searching points in high SNR, but it iterates too many times in low SNR. If the receiver can detect the SNR, it can adjust suitable step frequency to achieve fast convergence and has lower computational complexity. This concept is the same as Least Mean

Fig. 6. The comparison of the iteration number for convergence with different small step frequencies

Fig. 7. The iteration number of reverse search for convergence

Table III. Numbers of total searching points in different step frequencies in three multipath channels

SNR=0(dB) SNR=5(dB) SNR=25(dB) SNR=30(dB)

0.01f 1125 1125 1080 1043

0.005f 575 567 555 550

0.001f 185 146 118 117

0.0005f 201 123 67 66

0.0001f 729 337 54 45

Table IV. Numbers of total searching points in different step frequencies in five multipath channels

SNR=0(dB) SNR=5(dB) SNR=25(dB) SNR=30(dB)

0.01f 1129 1121 1094 1075

0.005f 575 575 554 554

0.001f 197 149 120 118

0.0005f 223 126 70 68

0.0001f 831 348 68 62

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Square (LMS) algorithm in choosing a feasible step size to achieve convergence. In Fig. 8, the iteration number of different number of subcarriers versus SNR is displayed. Under the same step frequency, the iteration numbers in lager number of subcarriers is less than the smaller number of subcarriers. It means that the estimated result of the receive signal with the time domain equalizer is more close to the global maximum frequency. This result also can verify that the performance would be enhanced while the number of samples in each item (refer to equation (14)) increases.

Fig. 8. The iteration number with different subcarrier numbers for convergence

V. CONCLUSION

In this paper, we proposed a joint maximum likelihood method for CFO and channel estimation with a new processing structure compared with [1]. It provides good performance which is close to CRB and has a much less computational complexity compared with the grid search-based method. The proposed scheme has two major portions. In the first part, we use the rotation concept to find an initial coarse CFO and make use of a time domain equalizer to eliminate the multipath interference for getting the frequency peak which is close to the global maximum frequency. After that, we use the iterative method to find true CFO peak and analyze different step frequencies versus SNR. The simulation results indicate the performance similar to those of [1] while the structure is more suitable for current OFDM regular implementation.

ACKNOWLEDGMENT The authors would like to thank the National Science Council of the Republic of China, Taiwan, for financially supporting this research under Contract nos. NSC 101-2221-E-008 -069 and 102-2221-E-008 -005.

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