J. Cent. South Univ. Technol. (2010) 17: 1293-1299

DOI: 10.1007/s11771-010-0634-x

Cancellation for frequency offset in OFDM system based on TF-LMS algorithm

GUAN Qing-yang(关庆阳), ZHAO Hong-lin(赵洪林), GUO Qing(郭庆)

Communication Research Center, Harbin Institute of Technology, Harbin 150001, China

? Central South University Press and Springer-Verlag Berlin Heidelberg 2010

Abstract: In an orthogonal frequency division multiplexing (OFDM) system, a time and frequency domain least mean square algorithm (TF-LMS) was proposed to cancel the frequency offset (FO). TF-LMS algorithm is composed of two stages. Firstly, time domain least mean square (TD-LMS) scheme was selected to pre-cancel the frequency offset in the time domain, and then the interference induced by residual frequency offset was eliminated by the frequency domain mean square (FD-LMS) scheme in frequency domain. The results of bit error rate (BER) and quadrature phase shift keying (QPSK) constellation figures show that the performance of the proposed suppression algorithm is excellent.

Key words: orthogonal frequency division multiplexing (OFDM); frequency offset; least mean square algorithm; cancellation

1 Introduction

Orthogonal frequency division multiplexing (OFDM) is a strong scheme for future communication, which is a multi-carrier modulation system to achieve high data rates. OFDM is also selected as the physical layer technology in digital audio broadcasting system and terrestrial digit video broadcasting. But OFDM is sensitive to the frequency errors that are between the transmitter and the receiver and result in loss of orthogonality among subcarriers. Reasons for this loss are mainly due to oscillator instabilities, Doppler shifts or phase noise.

HUTTER et al [1] analyzed linear minimum mean-square error (LMMSE) estimator to reduce the frequency errors that are due to Doppler spread. Methods based on maximum likelihood (ML) [2], least-square  (LS) [3] and minimum mean-square error (MMSE) [4] were studied to cancel the interference that is due to the frequency offset. The previous algorithms [1-4] consider the interference induced by frequency offset as a part of additive noise, which could not perform well for canceling frequency offset.

WEERADDANA et al [5] proposed an exact analysis for OFDM systems with frequency offset. HUANG and LETAIEF [6] proposed a method based on null subcarrier, which used one complete OFDM symbol with all odd subcarriers, but most of the even subcarriers were null subcarriers. ZHU and LEE [7] proposed a similar technique with an overhead of two ODFM symbols. CHOI et al [8] proposed a potential estimation using the time-variant parameter of the channel, but this technique required knowledge of the channel statistic. SUN et al [9] proposed an expectation maximization (EM) based iterative scheme using multiple pilot slots, which offered only a limited frequency offset estimation range. Thus, this method is not adaptive for large frequency offset. Recently, based on receiver windowing, BEAULIEU and TAN [10] proposed a characteristic function to analyze the exact bit error rate (BER) of OFDM system with frequency offset.

There were also several low-complexity equalizers that adopted inter-carrier interference (ICI) properties due to frequency offset. HOU and CHEN [11] proposed an MMSE detector with a parallel interference cancellation (PIC), which had a weight to minimize the joint minimum square error, but had an error floor at the high signal-to-noise ratio (SNR). MOSTOFI and COX [12] proposed a method for restricting ICI, which assumed that the channel variation could be linear in an OFDM block, but had degradation in the presence of large frequency offsets. CHEN et al [13] proposed successive interference cancellation (SIC) to cancel the interference due to ICI in frequency domain. PARK et al [14] proposed an algorithm based on SIC method by taking the advantage of time-domain channel matrix, and HWANG et al [15] proposed a low complexity iterative cancellation with the MMSE-SIC scheme. But these scheme forms [13-15] need matrix inverse calculation and have a high complexity proportional to the subcarriers.

Based on the PIC algorithm, LAKSHMISH [16] proposed a hard decision based parallel interference cancellation (HDPIC) scheme to reduce the effect of this interference on the uplink of OFDM algorithm, but the spectrum of system model was inefficient, which also could cause SNR loss. There are also some ICI cancellation methods such as selective PIC (SPIC) proposed by MARABISSI et al [17], which has a good behavior in overcoming the interference induced by carrier frequency offset (CFO). The performances of the algorithms are not satisfactory because of inaccurate estimator. WANG and CHEN [18] proposed a training sequence-based CFO estimator based on an iterative PIC unit. However, the training sequence needs lots of symbols. MANOHAR et al [19] proposed a multistage linear parallel interference cancellation (LPIC) to mitigate the effect of this multi-user interference (MUI) on the uplink of OFDM algorithm, which needed a large complex.

The TF-LMS algorithm was proposed to cancel the interference due to frequency offset (FO). Compared with MMSE-SIC algorithm, TD-LMS scheme in the time domain improves the accuracy of cancellation for FOs, and degrades the complexity of the algorithms in frequency domain. Moreover, there are no requirements on the prior knowledge and the range of frequency offset in comparison with the MMSE algorithm.

2 System model

OFDM converts serial data stream into parallel blocks of size N and modulates these blocks using inverse discrete Fourier transform (IDFT). N input points are transformed into an N-point symbol frame X(k)= [X(0), X(1), …, X(N-1)]^T, which is then transformed into an N-point frame x(n)=[x(0), x(1), …, x(N-1)]^T in time domain. The nth frame can be expressed as

        (1)

where n=0, 1, 2, …, N-1.

The nth received frame can be written as

y(n)=h(n)*x(n)                               (2)

where h(n) represents the channel impulse response at instant n; and * denotes linear convolution. Assuming that the symbol satisfies the Nyquist criterion for samples taken at intervals T/N, where T represents the length of sampling time, the nth received sample frame (r(n)) by frequency offset can be written as

                 (3)

where Δf represents the frequency offset caused by Doppler and mismatch in the frequencies of the transmitter and the receiver; ξ=Δf T is defined as the channel frequency offset normalized by the subcarrier separation; and v(n) is the additive white Gaussian noise with variance σ².

Fig.1 shows the receiver structure for the proposed TF-LMS cancellation algorithm, which consists of two parts. The time domain least mean square (TD-LMS) scheme was used to eliminate the interference in the time domain in the first part, and then the interference due to residual frequency offset was canceled by the frequency domain least mean square (FD-LMS) scheme in the frequency domain. For the current analysis, the pre-cancellation scheme with TD-LMS algorithm is utilized with the aim of improving the performance of FD-LMS algorithm in the frequency domain.

3 Pre-cancellation for interference with TD- LMS scheme

It is possible to obtain the optimum result through some iterative computation with TD-LMS algorithm, which is based on the criteria of MSE and has a fast convergence. Fig.2 shows the scheme of pre-cancellation for frequency offset with LMS scheme. The Nth iterative output symbol d(n) is achieved by passing r(n) through an adaptive filter with weight H(n), which is given by x(n)=H^*(n)r(n), where superscript * represents transpose operator. Here, we use one filter, but do not use a more generally used tap filter, in order to reduce the processing delay time. For the current analysis, using a more generally used taps filter is not able to obviously improve the performance of the algorithm in the time domain. L is defined as the number of subcarriers that transmit the training symbols, which can be known by the transmitter and the receiver. When n＜L, d(n) is the reference symbol known by the receiver. After TD-LMS algorithm learns adaptively through these symbols, TD-LMS scheme can be converged. When n≥L, d(n) is considered as the reference symbol for TD-LMS algorithm, and in practice, LN.

Fig.1 Cancellation for frequency offset in OFDM system with TF-LMS scheme

Fig.2 Pre-cancellation for frequency offset with TD-LMS scheme in OFDM system

The adaptive scheme is derived based on MSE criterion, in which the time-domain cost function J(n) is minimized as:

   (4)

where E(?) represents the expectation operator; x(n)=  [x₁(n), x₂(n), …, x_N-1(n)] and H(n)=[H(0), H(1), …, H(N-1)].

The gradient of Eq.(4) with respect to weight vector H(n) is given by

              (5)

where R_r,r(n)=E[r(n)r(n)] is the correlation matrix of the ith received symbol; and R_r,d(n)=E[r(n)d^*(n)] is the cross-correlation vector between the ith received reference and the symbol at the receiver. An adaptive solution that minimizes the cost function could be expressed as

        (6)

where μ represents the exponential weighting factor with a positive constant value less than unity.

To update the weight vector, the following equation can be obtained from Eq.(6):

                   (7)

where e(n) represents the error between the ith received reference symbol and ith received symbol in time domain.

                 (8)

Substituting Eq.(8) into Eq.(5), there is

            (9)

where .

Updating H, there are

               (10)

      (11)

where n≥0.

      (12)

Theoretically speaking, when the SNR is so large that the noise can be ignored, the output signal will be

         (13)

where

                       (14)

From Eq.(13), it can be seen that the output has the product of r(n) and  and input symbol r(n) can be compensated with the TD-LMS scheme. When G=1. So, G can be ignored when the frequency offset ξ is small. But if the frequency offset ξ is large, G will induce the residual frequency offset, which could also destroy the output signal of y(n).

The performance of TD-LMS in eliminating the interference due to the frequency offset could be obtained by the cancellation. It is supposed that SNR is 20 dB and there are 500 packets for each subcarrier. Figs.3 and 4 give the base band signal constellation with the frequency offset of ξ=0.03 and ξ=0.35, respectively. Figs.5 and 6 give the base band signal constellation after the cancellation with the TD-LMS scheme in time domain. From Figs.3-6, it can be seen that TD-LMS scheme may eliminate the disturbance completely when frequency offset value is small. But for a larger frequency offset value, there also exists residual frequency offset.

4 System model with residual frequency offsets ζ is defined as the residual frequency offset after pre-cancellation with TD-LMS scheme, which is caused by disturbance part G. Because the frequency offset |ξ| is fractional, the disturbance part mainly induces the fixed phase deviation for ideal signal and has little interference for the amplitude. We could also get this result from constellation simulation. This kind of disturbance also causes the phase rotation of ideal signal, so the signal after pre-cancellation could be expressed as

                    (15)

where  could only induce the phase rotation for s(n). After the discrete Fourier transform (DFT) transformation, the signal received in the frequency domain is expressed as

 (16)

where n=0, 1, 2, …, N-1.

After manipulation, there is

                   (17)

where S(k) is the vector with residual frequency offset after DFT operating on s(n); Y(k) represents the ideal signal with non-frequency offset after DFT operating on y(n); and V(k) represents the additive white Gaussian noise in frequency domain.

Fig.3 Signal constellation with frequency offset of ξ =0.03

Fig.4 Signal constellation with frequency offset of ξ =0.35

Fig.5 Signal pre-constellation with TD-LMS scheme

Fig.6 Signal pre-constellation with TD-LMS scheme

5  Cancellation for residual frequency offset with FD-LMS scheme

The FD-LMS scheme was used to eliminate the interference induced by residual frequency offset. Fig.7 shows the structure of FD-LMS algorithm, where S(k)=[S(0), S(1), …, S(N-1)]^T represents the input vector; U(k)=[U(0), U(1), …, U(N-1)]^T represents the output vector in frequency domain; and E(k) represents the error between the kth received reference symbol in the frequency domain and the kth received symbol in frequency domain. Output signal U(k) is achieved by passing S(k) through an adaptive filter with weight V(k). When k＜L, D(k) is the reference symbol after DFT operator, which is also known by the receiver. And L is also the number of subcarriers that can be known by  the transmitter and the receiver. After FD-LMS algorithm

Fig.7 Cancellation for frequency offset in frequency domain with FD-LMS scheme

learns adaptively through these symbols in the frequency domain, the FD-LMS scheme can be converged in this process. When k≥L, D(k) is considered as the reference symbol for FD-LMS algorithm in the frequency domain. The cost function could be defined as

(18)

The gradient vector of the cost function with respect to W(k) can be derived as

                                 (19)

After some simple mathematic manipulation, there is

      (20)

where f represents the exponential weighting factor with a positive constant value less than unity; R_{S, S}(k)= S(k)S^*(k) and R_{S, D}(n)=S(k)D^*(k).

                (21)

Substituting Eq.(21) into Eq.(19), there is

         (22)

where .

Updating W, there are

                 (23)

         (24)

where k≥0.

    (25)

Theoretically speaking, when the SNR is so large that the noise can be ignored, the output signal will be

  (26)

From Eq.(26), it can be seen that the output has the product of S(k) and , signal Y(k) can be got with non-frequency offset. The symbol S(k) with residual frequency offset can be canceled with the FD-LMS scheme completely.

The comparison of the TD-LMS and FD-LMS is given in Table 1, and the complexity of TF-LMS is given in Table 2.

Table 1 Comparison of the TD-LMS scheme and FD-LMS scheme

Table 2 Complexity of TF-LMS algorithm

6 Numerical results and analysis

The performance of the proposed TF-LMS algorithm could be tested with Matlab software. The total number of the subcarriers is 512 and the number of DFT is also 512 for OFDM system. The transmission bandwidth is set to 20 MHz. Quadrature phase shift keying (QPSK) is used to modulate uncoded information bits. The frequency offsets are distributed within the range [-ξ_max, ξ_max], where ξ_max is the maximum allowed value of frequency offsets.

The uncoded bit error rate versus average SNR was simulated. All the subcarriers are assumed to have independent channels with the same average power. Figs.8-10 show the bit-error-rates (BER) versus average SNR at ξ_max=0.35, ξ_max=0.40 and ξ_max=0.45, respectively. The results with no frequency offset, no-compensation, MMSE-SIC algorithm, MMSE equalization algorithm, and the proposed compensation method are presented. From Figs.8-10, it can be seen that no cancellation suffers high BER because of ICI; MMSE algorithm performs better since ICI can be eliminated completely; MMSE-SIC algorithm performs better than MMSE algorithm since the noise enhancement can be  suppressed. TF-LMS gives better performance for higher SNR, since TF-LMS can not reduce the noise effect at lower SNR. More importantly, unlike MMSE and MMSE-SIC algorithms, the performance of TF-LMS algorithm becomes better when the frequency offset becomes larger.

The ability for the TF-LMS algorithm to cancel the frequency offset could be simulated with constellation. The QPSK mapping as the baseband signal was also selected. Suppose the SNR=20 dB in order to get comparison under different conditions. Assuming that the packet for every subcarrier is 1 000, and the fixed frequency ξ=0.35. Fig.11 shows the constellation after cancellation with the TF-LMS algorithm. Compared with Fig.4 and Fig.6, FD-LMS scheme is robust to the residual frequency offset after pre-cancellation with TD-LMS scheme. The TF-LMS algorithm also has the ability to suppress large frequency offset.

Fig.8  BER curves under different average SNR at ξ_max=0.35

Fig.9 BER curves under different average SNR at ξ_max=0.40

Fig.12 shows the leaning curve of TF-LMS algorithm, where SNR is 15 dB and the fixed frequency offset is ξ=0.35. After some packets learning, TF-LMS algorithm can be converged about 180 symbols, and the mean square error is reduced to 1×10^-3. But under the same conditions, the TD-LMS needs longer learning time, which is due to the larger frequency offset.

Fig.10  BER curves under different average SNR at ξ_max=0.45

Fig.11 Signal constellation after cancellation with TF-LMS algorithm

Fig.12 Learning curves under different iterative symbol numbers

7 Conclusions

(1) The TF-LMS algorithm is proposed to mitigate the interference due to the frequency offset in OFDM algorithm system.

(2) The TF-LMS scheme can cancel the frequency offsets with low complexity. Compared with the convention cancellation scheme, the proposed method has good behavior for the offset cancellation.

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(Edited by LIU Hua-sen)

Corresponding author: GUAN Qing-yang, Doctoral candidate; Tel: +86-13936457619; E-mail: guan_qy@163.com