Time-domain channel estimator based on cyclic correlation for OFDM systems with guard interval

THE JOURNAL OF CHINA UNIVERSITIES OF POSTS AND TELECOMMUNICATIONS Volume 15, Issue 2, June 2008

JIA Min, GU Xue-mai, IM Se-bin, CHOI Hyung-jin

Time-domain channel estimator based on cyclic correlation for OFDM systems with guard interval CLC number

TN929.05

Document A

Article ID 1005-8885 (2008) 02-0026-06

Abstract Channel impulse response (CIR) can be estimated on the basis of cyclic correlation in time-domain for orthogonal frequency division multiplexing (OFDM) systems. This article proposes a generalized channel estimation method to reduce the estimation error by taking the average of different CIRs. Channel impulse responses are derived according to the different starting points of cyclic correlation. In addition, an effective CIR length estimation algorithm is also presented. The whole proposed methods are more effective to OFDM systems, especially to those with longer cyclic prefix. The analysis and the simulation results verify that the mean square error performance is 45 dB better than the conventional schemes under the same conditions.

the MMSE estimators. The blind channel estimation is limited to time-invariant channels because it requires a longer data record and entails higher complexity [1]. Because of the limitations of blind estimation, pilot symbol-aided methods of channel estimation are usually used in practical OFDM systems. There are two basic types of pilot pattern, block type and comb type. The comb type pilot is developed for fast fading channels and to satisfy the need for equalizing. It also has the advantage of avoiding advanced channel tracking, especially, with high velocity. The first category of channel estimation is based on pilot interpolation in the frequency domain (FD). The whole channel frequency response (CFR) is estimated by interpolation at pilot positions [2]. The second category is based on the time-domain channel estimation (CE). There are two main estimation methods for OFDM systems. One is the discrete Fourier transform (DFT) based on CE methods proposed in Ref. [3], and the other method is based on the correlation [4, 5], where the coarse channel impulse response (CIR) can be derived directly by the correlation between pilot sequence and receive signal. Correlation with the received signal is adopted only after the removal of CP at the receiver, but the coarse CIR is overestimated due to cyclic correlation error and the additive noise. The cancellation of cyclic correlation errors has been introduced in detail in Ref. [6]. With the usage of correlation, and on the basis that OFDM symbols generally have CP, an improved CIR estimation method based on time-domain processing is proposed. Unlike the method described in Ref. [7], it is unnecessary to attach the special orthogonal sequence as CP. The proposed method reduces the estimation error by taking the average of different CIRs, which are derived by the different starting points of cyclic correlation. Moreover, because the sliding range of starting points depends on maximum value of multipath fading CIR length, a CIR length estimation method, which is performed by reverse detection processing, is also proposed. In addition, the proposed method is more generalized. The whole process of this proposed method is discussed in this article.

Keywords CIR, OFDM, channel estimation, mean square error (MSE)

1



Introduction

OFDM has been applied widely in wireless communication systems recently because of its easy implementation by fast Fourier transform (FFT) and its robustness to multipath distortion and intersymbol interference (ISI). To eliminate the residual effect of ISI and preserve the orthogonality among subcarriers, guard interval (GI) called cyclic prefix (CP), which has been appended in front of each symbol is exactly a replica of the last portion of the original OFDM symbol. The channel has the time- and frequency-selective characteristics, and equalizer requires accurate channel estimation to achieve better performance at the receiver. Minimum mean square error (MMSE) estimators have gained better performance at the cost of adding complexity. However, considering the trade-off between complexity and performance in practice, the least square (LS) estimators might have more advantages over Received date: 2007-07-24 JIA Min ( ), GU Xue-mai, IM Se-bin, CHOI Hyung-jin Communication Research Center, Harbin Institute of Technology, Harbin 150001, China E-mail: [email protected]

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JIA Min, et al.: Time-domain channel estimator based on cyclic correlation for OFDM systems…

2.1

N 1

j

2ʌnk N

(1)

and the base band signal at the output of the inverse DFT block can be expressed as (2) x( n) d ( n)  p ( n) where 1 N 1 ¦ D(k )e j2ʌ nk / N ; D(k ) Nk 0

­data; k  SD ® ¯0; otherwise

(3)

Before removing the GI at the receiver, the mth received OFDM symbol represented by N + N g samples in time domain can be expressed as follows: rm , n r ((n  N g  mN )T )

(8)

l

Demodulation of the subcarriers through an FFT yields the mth received data symbols that can be expressed as N 1

¦r

Ym, k

m, n

e  j2ʌ( n / N ) k

(9)

n 0

Assuming the channel to be constant during one OFDM symbol, the demodulated data symbols, mth OFDM symbol, kth subcarrier, can be given as follows: Ym, k X m , k H m, k  Wm , k (10)

3 3.1

Principles of CIR estimation Coarse CIR estimation

The coarse CIR estimation can be derived by cyclic correlation of the time-domain pilot with the received signal (Eq. 11).

Crp (n) (4)

rm , n can be defined as r (n) (  N g İn  N ) when only one symbol is considered and the received signal r (n) obtained after the transmitted signal s (n) passing the multipath channel can be expressed as (5) r (n) h(n) … s (n)  w(n) where … denotes the convolution and w( n) is the additive white Gaussian noise (AWGN) with variance, V w2 . 2.2

nT

AWGN, and H m , k is the CFR.

The N-point complex transmitted modulation sequence s (n) with GI in CP-OFDM systems can be expressed as

­° x( N  n);  N g 1İn  0 ® °¯ x( n); 0İn  N

overall power. Sampling the signal at time instants t yields r ( n) ¦ hl ( nT ) s ( nT  W l )  w(nT )

where X m , k is the data symbol, Wm, k is the complex-valued

­r E ; k  SP p (n) ® ¯0; otherwise and n, k range from 0, 1,..., N  1 . 1 N 1 ¦ P(k )e j2ʌ nk / N ; P(k ) Nn 0

s ( n)

(7)

gains hl (t ) is zero mean stochastic process with normalized

n 0

d ( n)

l

In Eq. (7), W l is the propagation delay of path l, the path

In pilot-aided OFDM systems, the transmitted signal X ( k ) composed of the data sequence D ( k ) and the pilot sequence P ( k ) can be expressed as

¦ x(n)e

l

l

OFDM signal model

X (k )

¦ h (t )s(t  W )  w(t )

r (t )

OFDM system model

27

Channel model

According to the statistics of the channel model, which is wide sense stationary uncorrelated scattering (WSSUS) proposed in Ref. [8], the CIR is given by (6) h(W , t ) ¦ hl (t )į(W l  lT ) ; 1İlİL l

where T is the sampling period, hl (t ) is the corresponding complex amplitude at delay lT , į(˜) is the Dirac delta function, and L is the maximum length of channel. Assuming that the receiver filter is flat within the transmitter bandwidth, the receiver input signal is

N 1

¦ r (u ) p (u  n)

N

(11)

u 0

where Crp is the output of cyclic correlation, (˜) N denotes modulus, N, and denotes the conjugate operation. In accordance with the conclusion of Appendix A in Ref. [6], Eq. (11) becomes (12) Crp (n) Cpp (n) … h(n)  e(n) where e( n) is the error component, and Cpp is the output of cyclic correlation with pilot sequences in the time domain. ­ N 1 2 ° ¦ pn ; n W l (13) Cpp ( n) ® n 0 °C ( n  W ); n z W l l ¯ err where Cerr ( n  W l ) is the correlation error.

3.2

Proposed time-averaging CIR

The error component consists of two terms. The first one is the correlation error determined by Eq. (13), and the second one is the additive noise. These errors are distributed over the whole length of the coarse CIR. First, perfect timing synchronization is assumed here. Otherwise, the solving method is taken into account as proposed in Ref. [9]. The length of GI denoted as N g should

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The Journal of CHUPT

be sufficiently long and at least equal to or greater than the maximum channel length, L. The index of one symbol is defined as shown in Fig. 1. Assuming that L is known, the generalized cyclic correlation between r (n) (  N g İn  N ) and p ( n) ( 0İn  N ) for CIR estimation can be expressed as follows: hD (n)

Crp (n) Cpp

1 Cpp

N D

¦ r (u) p (u  n)

(14a)

N

u D

The first selected correlation output is Crp (n) 1 N hc (n) ¦ r (u ) p (u  n) N Cpp Cp p u 0

(14b)

D is chosen as the starting point of correlation sliding during the interval of  N g  LİD  0 in one OFDM symbol.

Fig. 1

Symbol index in one symbol

 h(n) is defined by the effectively selected CIR coefficients after cyclic correlation. N is taken into account, and h (n) g

c

is taken as the effectively estimated CIR coefficient after the first cyclic correlation. Then, L is taken into account. The  effective part of CIR hD (n) is taken as the output after taking the second cyclic correlation and n is from the interval   L  D  1 to N g  L  D  1 . hD (n) is one part of the newly estimated channel coefficients. Let the effective part of  previous estimation h (n) be h (n) . Thus, the proposed c

c

time-averaging CIR can be denoted as    hc (n)  hD (n  D  N g  L  1) ; hpro (n) 2

3.3

used directly, that is, hc (n) ( N g İn  N ). Therefore, the threshold can be set as 2 * = E ª hc ( n) º ; N g İn  N ¬« ¼»

(15)

CIR length estimation

As the channel power only concentrates on a few CIR samples, only the effective samples of CIR in the length of N g are needed to determine the whole CFR. As the CIR is WSSUS and Gaussian, it can be extended into a periodic random process, and it is periodic uncorrelated. Its DFT is 2 periodic WSS. The power delay profile of the CIR, E{ hl } , can be regarded as the spectra of the CFR. It is known that the power of actual path is much larger than that caused by the noise. Therefore, a proper threshold is needed to be set to detect both the effective taps index and L. Fortunately, the remaining part derived after taking the first cyclic correlation can be

(16)

The noise variations can be reduced by time-averaging of the noisy signal frequency components. The same operations can be done as iteration steps after the continuous received symbols in one packet or block structure, and then the average of hc (n) is derived and can be expressed as hav (n) . In Eq. (16), h (n) can be replaced by h (n) . c

av

The reverse detection is used, and the detection point is defined starting from Ng. If the power of the ith tap is larger than the threshold and * (i+1) < * (i), then the tap index is recorded. Moreover, the effective path of CIR can be extracted by tap detection where ineffective taps are processed by using zero-padding. The CIR length, L, can be considered in making a decision about correlation starting points. The accurate CIR estimation vector can be expressed as     (17) hˆpro ( n) [ hpro0 hpro1 hpro2 ... hpro( L 1) ]T As the simple implementation of equalization in OFDM systems, CFR is needed to satisfy one-tap simple equalization in the FD. The CFR is the N-point DFT of the CIR. The extended CIR vector by zero-padding can be expressed as  °­hpro ( n); 0İn  L ˆ (18) hpro (n) ® Lİn  N °¯0 ;

The CFR can be obtained by FFT and applied to the CIR. Hˆ ( k )

DFT{hˆpro ( n)}

N 1

¦ hˆ

pro

( n)e

j

2ʌnk N

(19)

n 0

3.4

0İn  N g

2008

Processing and operation steps

The whole process of the proposed method in practical operations is summarized as follows: 1) Starting point selection: D is set as 0, proceeding as Eq. (11). 2) Level compensation of coarse CIR according to Eq. (13). 3) Effective CIR coefficients selection: the correlation output hc (n) ( 0İn  N g ) is selected as the effective CIR coefficients for the first time. 4) Threshold setting and length detection: the remaining part of the output after the first correlation is hc (n) ( N İn  N ). After power calculation of h for each tap, g

c

the threshold should be set as in Eq. (16) and be stored every time. Proceed to Eq. (1) after next symbol generation. Reverse detection is made use of and the first detection point is N g . If the power of ith tap is larger than the threshold and * (i+1) <

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JIA Min, et al.: Time-domain channel estimator based on cyclic correlation for OFDM systems…

* (i), i is recorded, and L= N g  i  1 ; or else, continue with step 4). 5) Make new selection of D : assume that the following correlation point is D , its sliding range is from  N g  L to 0. Set D =  N g  L as the second correlation starting point. Proceed to steps 2) and 3), and the effective CIR index n of hD (n) ranges from  L  D  1 to N g  L  D . 6) Average the CIRs derived from steps 3) and 5) as Eq. (15). 7) Accurate CIR can be expressed by Eq. (17).

Fig. 2

4

29

Estimator structure and evaluation

4.1

Channel estimator structure

The estimator can be derived based on the system model and the theory presented in the previous section. There is a one-to-one correspondence between the mth received OFDM symbol r and CIR h m, n , there for, the estimator structure m,n

is presented in Fig. 2.

Estimator structure

The transmitted symbols are X and the received symbols are ˆ X appearing after the equalizer. With the linear model given in Eq. (10), frequency domain equalization (FDE) uses the simple one-tap equalizer. Zero forcing (ZF) equalizer is based on the channel estimation. ZF equalizer can be obtained by the following equation as Xˆ Hˆ 1Y (20)

4.2 Performance evaluation about common CE methods Comparison with the other CE methods, the difference of the proposed method is as follows. First, the normal pilot-aided channel estimation method is to estimate the CFR at the pilot positions. The chosen interpolation in the FD is to obtain the channel estimation for the whole subcarriers. The normal interpolation method is based on the FD signal, whereas the proposed method is based on the time-domain signal first, and CIRs are derived directly. Second, the direct DFT-CE method based on the inverse DFT of CFR has always caused serious error floor by spectral leakage. Third, the averaging of CIRs is derived according to the proper selection of different correlation starting points. Compared with frequency-domain pilots (FPTP) and time-domain processing method, the CIR length detection is used for the proposed method. In addition, the output of first cyclic correlation can be used for both the following CIR length detection and time-averaging of CIRs.

5

Simulation results

Simulation results are presented to show the performance achieved by the proposed method. The OFDM system is considered in the European Telecommunication Standards Institute Vehicular A channel environment with 6 taps delay path (Table 1) [10], and modeled as Rayleigh fading process, which is WSSUS as mentioned above with the Doppler frequency. The system with bandwidth, WB=10 MHz, carrier frequency, fc = 2 GHz, and the velocity of 60 km/h is considered. N = 1 024 and N g =128 is chosen as the length of CP. Table 1 International Mobile Telecommunications 2000 vehicular model A channel Taps 1 2 3 4 5 6

Relative time/ns 0.0 310 710 1 090 1 730 2 510

Relative power/dB 0.0  1.0  9.0  10.0  15.0  20.0

The pilot symbols are inserted equispaced arranging in frequency direction in the system The pilot spacing must satisfy the sampling theorem, thus, it is necessary that the pilot spacing 'f İN / N g , and N p =128 pilot subcarriers spaced 'f = 8 apart is chosen. ZF equalizer based on the channel estimation is used for all the methods.

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5.1

The Journal of CHUPT

MSE analysis of CIR

The MSE of CIR with subcarrier n, evaluated and averaged over all subcarriers, is given as 2 (21) M CIR  E ª h(n)  hˆ( n) º «¬ »¼ where hˆ(n) and h(n) are the estimated CIR and the ideal

CIR, respectively. h( n) is derived by using the ideal channel state information (CSI) without noise. Under the AWGN environment, the difference between the CIR estimation and the ideal CIR is the effect caused by AWGN component. The MSE performance of the proposed method is compared with the original FPTP method under this environment. MSE performance is expressed by M c and M pro . 2 M c  E ª h(n)  hˆc (n) º »¼ ¬«

1 ª N 1 E « ¦ w(l ) p (l  n) N  Cp2p «¬ l 0

M pro

2

º » »¼

E 2V w2

2 ª N 1 º E « ¦ w(l ) p (l  N  n) » «¬ l 0 »¼

(23a) 2

º » » ¼

2 N  N g  L 1 º N 1 1 ª«

»      E w l p l N n w l p l N n ( ) ( ) ( ) ( ) ¦ ¦ » 4 « l  Ng  L l 0 ¬ ¼ 1

¦

N  N g  L 1

w(l ) p (l  N  n) 

 Ng  L

l 0

¦

w(l ) p (l  N  n) 

l 0

N  N g  L 1

¦

Fig. 3 Comparison of mean square error performance under additive white Gaussian noise environment

5.2

N  N g  L 1 N 1 1 ª« E w(l ) p (l  n) N  ¦ w(l ) p (l  n) N ¦ 4 « l  Ng  L l 0 ¬

1 ª E« 4 «¬ l

simulation and the theoretical results with different values of N g is shown in Fig. 3.

(22b)

and to be simple, assuming C pp2 =1 here, Mc

2 · N  L  Ng p (l  N  n) ¸ 4 Mc ¸ N ¹ Therefore, it can be concluded that N  Nu · §N M pro ¨ u + (25) ¸ Mc N ¹ © 2N where N u N g  L , and 0İL  N g . The comparison of the

(22a)

2 M pro  E ª h(n)  hˆpro ( n) º «¬ »¼ As Eqs. (4), (20) and (25a) becomes

Mc

2008

2 º

»   w ( l ) p ( l N n ) ¦ » N  N g  L 1 ¼ N 1

w(l ) p (l  N  n)  l

(23b) In Eq. (23b), the second term and the third term are the same component and can be simplified to be one term after taking the mean operation. Next, the first term and the last term can be transformed into each other. Variables A and B are used instead of the two parts, thus Eq. (23b) becomes 1 (24) ( A  B) M pro 4 Here, 1 § · N L 2 A E ¨ 2 ¦ w2 (l )  p (l  N  n) ¸ 2 g Mc ¨ l  Ng  L ¸ N © ¹ 2 § N  Ng  L 1 · § N  Ng  L 1 B E ¨ 2 ¦ w(l ) p (l  N  n) ¸ 4 E ¨ ¦ w(l ) ˜ ¨ l 0 ¨ ¸ l 0 © © ¹

Performance of proposed method

The MSE of CFR for subcarrier n evaluated and averaged over all subcarriers is given as follows: 2 (26) M CFR  E ª H k  Hˆ k º «¬ »¼ where H k denotes the CFR of ideal CSI. The MSE performance with quadrature phase shift keying modulation schemes for the CFR of the subcarriers is shown in Fig. 4. DFT-CE denotes the directly based on LSCE with interpolation, and MMSE denotes the MMSE equalizer.

Fig. 4 MSE versus SNR under vehicular A environment

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JIA Min, et al.: Time-domain channel estimator based on cyclic correlation for OFDM systems…

With the increase in the signal-to-noise ratio (SNR), the MSE values decrease as expected. For the same SNR, the performance of the proposed method is more efficient than other methods, and with the increase in SNR, the performance of the proposed method is close to that of the ideal CSI.

6

Conclusions

In this article, CIR length detection and time-averaging of CIRs are used to estimate the channel in time domain. In addition, a CIR length estimation method is proposed, which is performed by reverse detection processing. On the basis of a number of simulations, the results show that the performance of the proposed channel estimation method is more efficient than the normal time-domain processing channel estimation methods, especially, the improvement of MSE performance about CFR. Furthermore, the proposed method has many advantages when it is applied to the OFDM systems with sufficiently long CP.

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on Consumer Electronics, 2003, 49(4): 949957 6. Li Ming-qi, Tan Jian-guo, Zhang Wen-jun. A channel estimation method based on frequency-domain pilots and time-domain processing for OFDM systems. IEEE Transactions on Consumer Electronics, 2004, 50(4): 10491057 7. Wang C L, Wang H C. A low-complexity joint time synchronization and channel estimation scheme for orthogonal frequency division multiplexing systems. Proceedings of the IEEE International Conference on Communications (ICC’06): Vol 12, Jun 1115, 2006, Istanbul, Turkey. Piscataway, NJ, USA: IEEE, 2006: 56705675 8. Rosa Z Y, Shan X C. Simulation models with correct statistical properties for rayleigh fading channels. IEEE Transactions on Communications, 2003, 51(6): 920928 9. Auer G . Efficient Implementation of robust OFDM channel estimation. Proceedings of the IEEE Personal, Indoor and Mobile Radio Communications (PIMRC’05), Sep 1114, 2005, Berlin, Germany. Piscataway, NJ, USA: IEEE, 2005: 629633 10. Recommendation ITU-R M.1225. Guidelines for evaluation of radio transmission technology for IMT-2000. 1997

Acknowledgements This work is supported by the National Natural Science Foundation of China (60572039) and Commsys. Laborcotory

Biographies: JIA Min, Ph. D. Candidate from

of Sungkyunkwan University in Korea.

Communication Research Center in Harbin Institute Technology, interested in advanced

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