# A Fast LMMSE Channel Estimation Method for OFDM Systems

- Wen Zhou
^{1}Email author and - Wong Hing Lam
^{1}

**2009**:752895

https://doi.org/10.1155/2009/752895

© W. Zhou and W.H. Lam. 2009

**Received: **20 July 2008

**Accepted: **20 March 2009

**Published: **4 May 2009

## Abstract

A fast linear minimum mean square error (LMMSE) channel estimation method has been proposed for Orthogonal Frequency Division Multiplexing (OFDM) systems. In comparison with the conventional LMMSE channel estimation, the proposed channel estimation method does not require the statistic knowledge of the channel in advance and avoids the inverse operation of a large dimension matrix by using the fast Fourier transform (FFT) operation. Therefore, the computational complexity can be reduced significantly. The normalized mean square errors (NMSEs) of the proposed method and the conventional LMMSE estimation have been derived. Numerical results show that the NMSE of the proposed method is very close to that of the conventional LMMSE method, which is also verified by computer simulation. In addition, computer simulation shows that the performance of the proposed method is almost the same with that of the conventional LMMSE method in terms of bit error rate (BER).

## 1. Introduction

Orthogonal frequency division multiplexing (OFDM) is an efficient high data rate transmission technique for wireless communication [1]. OFDM presents advantages of high spectrum efficiency, simple and efficient implementation by using the fast Fourier transform (FFT) and the inverse Fast Fourier Transform (IFFT), mitigation of intersymbol interference (ISI) by inserting cyclic prefix (CP), and robustness to frequency selective fading channel. Channel estimation plays an important part in OFDM systems. It can be employed for the purpose of detecting received signal, improving the capacity of orthogonal frequency division multiple access (OFDMA) systems by cross-layer design [2], and improving the system performance in terms of bit error rate (BER) [3–5].

### 1.1. Previous Work

The present channel estimation methods generally can be divided into two kinds. One kind is based on the pilots [6–9], and the other is blind channel estimation [10–12] which does not use pilots. Blind channel estimation methods avoid the use of pilots and have higher spectral efficiency. However, they often suffer from high computation complexity and low convergence speed since they often need a large amount of receiving data to obtain some statistical information such as cyclostationarity induced by the cyclic prefix. Therefore, blind channel estimation methods are not suitable for applications with fast varying fading channels. And most practical communication systems such as World Interoperability for Microwave Access (WIMAX) system adopt pilot assisted channel estimation, so this paper studies the first kind.

For the pilot-aided channel estimation methods, there are two classical pilot patterns, which are the block-type pattern and the comb-type pattern [4]. The block-type refers to that the pilots are inserted into all the subcarriers of one OFDM symbol with a certain period. The block-type can be adopted in slow fading channel, that is, the channel is stationary within a certain period of OFDM symbols. The comb-type refers to that the pilots are inserted at some specific subcarriers in each OFDM symbol. The comb-type is preferable in fast varying fading channels, that is, the channel varies over two adjacent OFDM symbols but remains stationary within one OFDM symbol. The comb-type pilot arrangement-based channel estimation has been shown as more applicable since it can track fast varying fading channels, compared with the block-type one [4, 13]. The channel estimation based on comb-type pilot arrangement is often performed by two steps. Firstly, it estimates the channel frequency response on all pilot subcarriers, by lease square (LS) method, LMMSE method, and so on. Secondly, it obtains the channel estimates on all subcarriers by interpolation, including data subcarriers and pilot subcarriers in one OFDM symbol. There are several interpolation methods including linear interpolation method, second-order polynomial interpolation method, and phase-compensated interpolation [4].

In [14], the linear minimum mean square error (LMMSE) channel estimation method based on channel autocorrelation matrix in frequency domain has been proposed. To reduce the computational complexity of LMMSE estimation, a low-rank approximation to LMMSE estimation has been proposed by singular value decomposition [6]. The drawback of LMMSE channel estimation [6, 14] is that it requires the knowledge of channel autocorrelation matrix in frequency domain and the signal to noise ratio (SNR). Though the system can be designed for fixed SNR and channel frequency autocorrelation matrix, the performance of the OFDM system will degrade significantly due to the mismatched system parameters. In [15], a channel estimation exploiting channel correlation both in time and frequency domain has been proposed. Similarly, it needs to know the channel autocorrelation matrix in frequency domain, the Doppler shift, and SNR in advance. Mismatched parameters of the Doppler shift and the delay spread will degrade the performance of the system [16]. It is noted that the channel estimation methods proposed in [6, 14–16] can be adopted in either the block-type pilot pattern or the comb-type pilot pattern.

When the assumption that the channel is time-invariant within one OFDM symbol is not valid due to high Doppler shift or synchronization error, the intercarrier interference (ICI) has to be considered. Some channel estimation and signal detection methods have been proposed to compensate the ICI effect [17, 18]. In [17], a new equalization technique to suppress ICI in LMMSE sense has been proposed. Meanwhile, the authors reduced the complexity of channel estimator by using the energy distribution information of the channel frequency matrix. In [18], the authors proposed a new pilot pattern, that is, the grouped and equispaced pilot pattern and corresponding channel estimation and signal detection to suppress ICI.

### 1.2. Contributions

In this paper, the OFDM system framework based on comb-type pilot arrangement is adopted, and we assume that the channel remains stationary within one OFDM symbol, and therefore there is no ICI effect. We propose a fast LMMSE channel estimation method. The proposed method has three advantages over the conventional LMMSE method. Firstly, the proposed method does not require the knowledge of channel autocorrelation matrix and SNR in advance but can achieve almost the same performance with the conventional LMMSE channel estimation in terms of the normalized mean square error (NMSE) of channel estimation and bit error rate (BER). Secondly, the proposed method needs only fast Fourier transform (FFT) operation instead of the inversion operation of a large dimensional matrix. Therefore, the computational complexity can be reduced significantly, compared with the conventional LMMSE method. Thirdly, the proposed method can track the changes of channel parameters, that is, the channel autocorrelation matrix and SNR. However, the conventional LMMSE method cannot track the channel. Once the channel parameters change, the performance of the conventional LMMSE method will degrade due to the parameter mismatch.

### 1.3. Organization

The paper is organized as follows. Section 2 describes the OFDM system model. Section 3 describes the proposed fast LMMSE channel estimation. We analyze the mean square error (MSE) of the proposed fast LMMSE channel estimation and the MSE of the conventional LMMSE channel estimation in Section 4. The simulation results and numerical results of the proposed algorithm are discussed in Section 5 followed by conclusion in Section 6.

## 2. System Model

where denotes the transmitted signal in frequency domain at the th subcarrier in the th OFDM symbol. The comb-type pilot pattern [4] is adopted in this paper. The pilot subcarriers are equispaced inserted into each OFDM symbol. It is assumed that the number of the total pilot subcarriers is and the inserting gap is . Each OFDM symbol is composed of the pilot subcarriers and the data subcarriers. It is assumed that the index of the first pilot subcarrier is . Therefore, the set of the indeces of pilot subcarriers, , can be written as

where . The received signal in frequency domain after FFT can be written as

where
and *s* is the set containing all constellation points, which depends on modulation method, that is, the signal mapper. For instance, if QPSK modulation is adopted, the set
. Finally, the estimated frequency signal
passes through the signal demapper to obtain the received bit sequence.

## 3. The Proposed Fast LMMSE Algorithm

### 3.1. Properties of the Channel Correlation Matrix in Frequency Domain

The channel impulse response in time domain can be expressed as

where is the complex gain of the th path in the th OFDM symbol period, is the Kronecker delta function, is the delay of the th path in unit of sample point, and is the number of resolvable paths. Assume that different paths are independent from each other and the power of the th path is . The channel is normalized so that The channel response in frequency domain is the FFT of and it is given by

where denotes points FFT operation. The channel autocorrelation matrix in frequency domain can be expressed as

where denotes expectation. Denote the vector form of the channel autocorrelation matrix by and we have . It is easy to find that the matrix is a circulant matrix. Therefore, as in [20], the eigenvalues of are given by

The formula (8) can be equivalently written as

We can easily obtain from (7) and (9) that the number of nonzero eigenvalues of
is equal to the total number of resolvable paths,
(see Appendix A)*.* It is known by us that the rank of a square matrix is the number of its nonzero eigenvalues. Therefore the rank of
is
and
is a singular matrix since
. The matrix
does not have the inverse matrix and has only the Moore-Penrose inverse matrix. However, the rank of the matrix
is *N* (see Appendix A), where
is an
by
identity matrix. Therefore, the matrix
is not singular and has the inverse matrix.

### 3.2. The Proposed Fast LMMSE Channel Estimation Algorithm

Let

denote the channel frequency response at pilot subcarriers of the th OFDM symbol, and let

denote the vector of received signal at pilot subcarriers of the th OFDM symbol after FFT. Denote the pilot signal of the th OFDM symbol by . The channel estimate at pilot subcarriers based on least square (LS) criterion is given by

The LMMSE estimator at pilot subcarriers is given by [6]

where is channel autocorrelation matrix at pilot subcarriers and is defined by , where denotes Hermitian transpose. It is easy to verify that the matrix is circulant, the rank of is equal to and the rank of is equal to . The signal-to-noise ratio (SNR) is defined by and is a constant depending on the signal constellation. For 16QAM modulation and for QPSK and BPSK modulation If the channel autocorrelation matrix and SNR are known in advance, needs to be calculated only once. However, the autocorrelation matrix and SNR are often unknown in advance and time varying. Therefore the LMMSE channel estimator becomes unavailable in practice. To solve the problem, we propose the fast LMMSE channel estimation algorithm. The algorithm can be divided into three steps. The first step is to obtain the estimate of channel autocorrelation matrices and . Firstly, we obtain the least square (LS) channel estimation at pilot subcarriers in time domain, and it is given by

Secondly, the most significant taps (MSTs) algorithm [21] has been proposed to obtain the refined channel estimation in time domain. The MST algorithm deals with each OFDM symbol by reserving the most significant paths in terms of power and setting the other taps to be zero. The algorithm can reduce the influence of AWGN and other interference significantly, compared with the LS method. However, the algorithm may choose the wrong paths and omit the right paths because of the influence of AWGN and other interference. Thus, we will improve the algorithm of [21] by processing several adjacent OFDM symbols jointly. We calculate the average power of each tap for adjacent OFDM symbols, and it is given by

Then we choose the most significant taps from and reserve the indeces of them into a set . Finally, the refined channel estimation in time domain, , is given by

Denote the first row of the matrix by . Then can be given from (7) by

where is a 1 by vector with each entry

where denotes points IFFT operation. Therefore the estimated LMMSE matrix can be obtained from circle shift of . The channel estimation in frequency domain at pilot subcarriers for the th OFDM symbol can be given by

The proposed fast LMMSE algorithm avoids the matrix inverse operation and can be very efficient since the algorithm only uses the FFT and circle shift operation. The proposed fast LMMSE algorithm can be summarized as follows.

Step 1.

Obtain the LS channel estimation of pilot signal in time domain, , by formula (14).

Step 2.

Calculate the average power of each tap for OFDM symbols, , by formula (15). Then, we choose the most significant taps from and reserve it as , by formula (18).

Step 3.

Obtain the estimate of SNR, , by formula (19).

Step 4.

Obtain the estimate of the first row of the LMMSE matrix, , by formula (20).

Step 5.

Obtain the estimation of the LMMSE matrix, , by circle shift of . Then, the channel estimation in frequency domain at pilot subcarriers can be obtained by formula (21).

It is noted that the estimation of the LMMSE matrix requires only points FFT operation and circle shifting operation, which reduce the computational complexity significantly compared with the conventional LMMSE estimator since it requires the inverse operation of a large dimension matrix.

## 4. Analysis of the Mean Square Error of the Proposed Fast LMMSE Algorithm

In this section, we will present the mean square error (MSE) of the proposed fast LMMSE algorithm. Firstly, we present the MSE of LMMSE algorithm for comparison. We study two cases. One case is the MSE analysis for matched SNR, that is, the designed SNR is equal to the true SNR, and the other one is the MSE analysis for mismatched SNR. Secondly, we present the MSE of the proposed fast LMMSE algorithm. Similarly, we study two cases. One is for matched SNR, and the other is for mismatched SNR.

### 4.1. MSE Analysis of the Conventional LMMSE Algorithm

*SNR*is the true SNR, and

*SNR design*is the designed SNR.

- (i)
MSE Analysis for Matched SNR

- (ii)
MSE Analysis for Mismatched SNR

where is the first row of the matrix and denotes Hermitian transpose.

### 4.2. MSE Analysis for the Proposed Fast LMMSE Algorithm

- (i)
MSE for Matched SNR

- (ii)
MSE for Mismatched SNR

where denotes the channel estimate at the th pilot subcarrier in the th OFDM symbol, obtained by LMMSE algorithm or the proposed fast LMMSE algorithm, and denotes the number of OFDM symbols in the simulation.

## 5. Numerical and Simulation Results

Channel Power Intensity Profile.

Tap | Delay (u s) | Gain (dB) | Doppler Spectrum |
---|---|---|---|

1 | 0 | 0.0 | Clarke [24] |

2 | 0.5 | -6.0 | Clarke |

3 | 1.0 | -12.0 | Clarke |

4 | 1.5 | -18.0 | Clarke |

5 | 2.0 | -24.0 | Clarke |

6 | 2.5 | -30.0 | Clarke |

### 5.1. Channel Autocorrelation Matrix under Different SNRs

### 5.2. Normalized Mean Square Error (NMSE) Comparison of Channel Estimation between LMMSE Algorithm and the Proposed Fast LMMSE Algorithm

### 5.3. Bit Error Rate (BER) Comparison between LMMSE Algorithm and the Proposed Fast LMMSE Algorithm

## 6. Conclusion

In this paper, a fast LMMSE channel estimation method has been proposed and thoroughly investigated for OFDM systems. Since the conventional LMMSE channel estimation requires the channel statistics, that is, the channel autocorrelation matrix in frequency domain and SNR, which are often unavailable in practical systems, the application of the conventional LMMSE channel estimation is limited. Our proposed method can efficiently estimate the channel autocorrelation matrix by the improved MST algorithm and calculate the LMMSE matrix by Kumar's fast algorithm and exploiting the property of the channel autocorrelation matrix so that the computation complexity can be reduced significantly. We present the MSE analysis for the proposed method and the conventional LMMSE method and investigate the MSE thoroughly under two cases, that is, the matched SNR and the mismatched SNR. Numerical results and computer simulation show that a design for higher SNR is preferable as for mismatch in SNR.

## Declarations

## Authors’ Affiliations

## References

- Weinstein SB, Ebert PM: Data transmission by frequency-division multiplexing using the discrete Fourier transform.
*IEEE Transactions on Communications*1971, 19(5, part 1):628-634. 10.1109/TCOM.1971.1090705View ArticleGoogle Scholar - Hui DSW, Lau VKN, Lam WH: Cross-layer design for OFDMA wireless systems with heterogeneous delay requirements.
*IEEE Transactions on Wireless Communications*2007, 6(8):2872-2880.View ArticleGoogle Scholar - Coleri S, Ergen M, Puri A, Bahai A: Channel estimation techniques based on pilot arrangement in OFDM systems.
*IEEE Transactions on Broadcasting*2002, 48(3):223-229. 10.1109/TBC.2002.804034View ArticleGoogle Scholar - Hsieh M-H, Wei C-H: Channel estimation for OFDM systems based on comb-type pilot arrangement in frequency selective fading channels.
*IEEE Transactions on Consumer Electronics*1998, 44(1):217-225. 10.1109/30.663750View ArticleGoogle Scholar - Zeng Y, Lam WH, Ng TS: Semiblind channel estimation and equalization for MIMO space-time coded OFDM.
*IEEE Transactions on Circuits and Systems I*2006, 53(2):463-473.MathSciNetView ArticleGoogle Scholar - Edfors O, Sandell M, van de Beek J-J, Wilson SK, Börjesson PO: OFDM channel estimation by singular value decomposition.
*IEEE Transactions on Communications*1998, 46(7):931-939. 10.1109/26.701321View ArticleGoogle Scholar - Simeone O, Bar-Ness Y, Spagnolini U: Pilot-based channel estimation for OFDM systems by tracking the delay-subspace.
*IEEE Transactions on Wireless Communications*2004, 3(1):315-325. 10.1109/TWC.2003.819022View ArticleGoogle Scholar - Zhao Y, Huang A: A novel channel estimation method for OFDM mobile communication systems based on pilot signals and transform-domain processing.
*Proceedings of the 47th IEEE Vehicular Technology Conference (VTC '97), May 1997, Phoenix, Ariz, USA*3: 2089-2093.Google Scholar - Lin J-C: Least-squares channel estimation for mobile OFDM communication on time-varying frequency-selective fading channels.
*IEEE Transactions on Vehicular Technology*2008, 57(6):3538-3550.View ArticleGoogle Scholar - Lin R, Petropulu AP: Linear precoding assisted blind channel estimation for OFDM systems.
*IEEE Transactions on Vehicular Technology*2005, 54(3):983-995. 10.1109/TVT.2005.844681View ArticleGoogle Scholar - Cai XD, Akansu AN: A subspace method for blind channel identification in OFDM systems.
*Proceedings of the IEEE International Conference on Communications (ICC '00), June 2000, New Orleans, La, USA*2: 929-933.View ArticleGoogle Scholar - Doukopoulos XG, Moustakides GV: Blind adaptive channel estimation in OFDM systems.
*IEEE Transactions on Wireless Communications*2006, 5(7):1716-1725.View ArticleGoogle Scholar - Coleri S, Ergen M, Puri A, Bahai A: A study of channel estimation in OFDM systems.
*Proceedings of the 56th IEEE Vehicular Technology Conference (VTC '02), September 2002, Vancouver, Canada*2: 894-898.View ArticleGoogle Scholar - van de Beek J-J, Edfors O, Sandell M, Wilson SK, Börjesson PO: On channel estimation in OFDM systems.
*Proceedings of the 45th IEEE Vehicular Technology Conference (VTC '95), July 1995, Chicago, Ill, USA*2: 815-819.Google Scholar - Li Y, Cimini LJ Jr., Sollenberger NR: Robust channel estimation for OFDM systems with rapid dispersive fading channels.
*IEEE Transactions on Communications*1998, 46(7):902-915. 10.1109/26.701317View ArticleGoogle Scholar - Morelli M, Mengali U: A comparison of pilot-aided channel estimation methods for OFDM systems.
*IEEE Transactions on Signal Processing*2001, 49(12):3065-3073. 10.1109/78.969514View ArticleGoogle Scholar - Kuo C, Chang J-F: Equalization and channel estimation for OFDM systems in time-varying multipath channels.
*Proceedings of the 15th IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC '04), September 2004, Barcelona, Spain*1: 474-478.Google Scholar - Song W-G, Lim J-T: Channel estimation and signal detection for MIMO-OFDM with time varying channels.
*IEEE Communications Letters*2006, 10(7):540-542.Google Scholar - Fernández-Getino García MJ, Páez-Borrallo JM, Zazo S: DFT-based channel estimation in 2D-pilot-symbol-aided OFDM wireless systems.
*Proceedings of the 53rd IEEE Vehicular Technology Conference (VTC '01), May 2001, Rhodes, Greece*2: 810-814.View ArticleGoogle Scholar - Böttcher A, Grudsky SM:
*Spectral Properties of Banded Toeplitz Matrices*. SIAM, Philadelphia, Pa, USA; 2005.View ArticleMATHGoogle Scholar - Minn H, Bhargava VK: An investigation into time-domain approach for OFDM channel estimation.
*IEEE Transactions on Broadcasting*2000, 46(4):240-248. 10.1109/11.898744View ArticleGoogle Scholar - Haykin S:
*Adaptive Filter Theory*. 4th edition. Publishing House of Electronics Industry, Beijing, China; 2002.MATHGoogle Scholar - Failli M:
*Digital land mobile radio communications—COST 207.*Commission of the European Communities, Brussels, Belgium; September 1988.Google Scholar - Rappaport TS:
*Wireless Communications Principles and Practice*. Publishing House of Electronics Industry, Beijing, China; 2002.MATHGoogle Scholar - Kumar R: A fast algorithm for solving a Toeplitz system of equations.
*IEEE Transactions on Acoustics, Speech, and Signal Processing*1985, 33(1):254-267. 10.1109/TASSP.1985.1164492View ArticleMathSciNetMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.