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Channel identification and interference compensation for OFDM system in long multipath environment
ARTICLE IN PRESS Signal Processing 89 (2009) 1589–1601
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Channel identification and interference compensation for OFDM system in long multipath environment L. Sun a,, A. Sano b, W. Sun a, A. Kajiwara a a b
Faculty of Environmental Engineering, The University of Kitakyushu, 1-1 Hibikino, Wakamatsu-ku, Kitakyushu 808-0135, Japan Department of System Design Engineering, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan
a r t i c l e in fo
abstract
Article history: Received 27 June 2008 Received in revised form 26 November 2008 Accepted 22 February 2009 Available online 9 March 2009
Guard interval (GI) is usually applied to reduce the influence caused by multipath interferences in orthogonal frequency division multiplexing (OFDM) systems. However, the long multipath interferences will deteriorate the orthogonality of the sub-carriers if they have longer delay time than GI. It is illustrated that not only inter-symbol interference (ISI) but also inter-carrier interference (ICI) is caused by the collapse of orthogonality in the received signal. As a result, both the channel identification and equalization become difficult, and the communication performance cannot be guaranteed. In this work, a new channel identification algorithm is proposed to estimate the frequency response function from the spectral periodograms which are compensated by the replica of leakage error. Since most of the computation is performed in the frequency domain through fast Fourier transform, the algorithm has such low computational complexity that it can be implemented easily in applications. & 2009 Elsevier B.V. All rights reserved.
Keywords: OFDM channel identification Guard interval Orthogonality Leakage error Periodogram
1. Introduction Orthogonal frequency division multiplexing (OFDM), which adopts multi-carrier modulation, has been widely applied in the services of digital terrestrial broadcasting, asymmetric digital subscriber (ADSL) and 5 GHz wireless LAN. Since the spectra of orthogonal OFDM sub-carriers allow to be overlapped by each other to form an approximate rectangular spectrum, OFDM transmission has high frequency efficiency. Consequently, OFDM has been considered as a powerful candidate for the next generation of communications [1]. In OFDM communications, guard interval (GI) is attached at the header of every effective transmission symbol, consequently OFDM has a strong tolerance against the multipath interferences whose delay time
Corresponding author.
E-mail addresses: [email protected] (L. Sun), [email protected] (A. Sano). 0165-1684/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2009.02.017
does not exceed GI. However, there are some interferences with long delay time especially in high speed data transmission or mobile communications. Moreover, it is desired to squeeze the length of GI for higher transmission efficiency. If the long delay time of interference exceeds GI, the spectral leakage error, which yields inter-symbol interference (ISI) as well as inter-carrier interference (ICI), destroys the orthogonality of sub-carriers in the received signal, and makes both the channel identification and equalization very difficult. As a result, the communication performance degrades significantly [2]. The error correcting codes and frequency selective diversity can help to equalize the information symbols from the received signal when the multipath interferences do not exceed GI, nevertheless, their effectiveness becomes weak since the components of ISI and ICI interferences, which are caused by the long multipaths, contaminate all the sub-carriers. The methods dealing with long multipath interference can be classified into two main groups: (1) the direct methods that use specific criteria to design equalizer without channel model; (2) the indirect methods that use
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the estimated channel model to compensate or cancel the interferences. They can be implemented in the time domain, or the frequency domain, depending on the criterion and optimization procedures of the processing algorithms. The time-domain equalizers (TEQ) and the spatial equalizers using antenna diversity in the frequency domain are typical direct methods. Their criterion functions are composed of the received signals, desired performance indices and equalizer parameters, and are usually complicated nonlinear functions. For example, the MMSE criterion [3,4], the maximum bit rate criterion [5] in TEQs are solved by nonlinear optimization without explicit channel identification, however, they suffer from computational complexity for nonlinear optimization or slow convergence rate [6]. Differently, some spatial equalizers use antenna array to simplify the computational complexity. For instance, the spatial equalizers extract the interferences exceeding GI through a specific window function [7], or spectral envelope for PSK signals [8]. Nevertheless, the performance of spatial equalizers strongly depends on the capacity of antenna diversity or the signal arrival angle, and these equalizers may not work well in the severe multipath interference environments with restricted number of antenna elements. Though increasing the sampling rate may shorten the length of channel impulse response [9], the circuit complexity restricts its application to the practical systems. On the other hand, the indirect methods separate the problem into two stages of channel identification and model based equalization design. When the channel has been identified, the immediate ways to the second stage in the time domain are the compensation of the received signal by estimating the interference replica [10,11], and the length compression of impulse response of the combined communication channel, including transmission channel and equalizer [12]. Most of them identify the channel model by RLS or LMS algorithm in the time domain from preamble or training symbols. However, too long impulse response of a communication channel leads to heavy computational load for RLS or slow convergence for LMS, and the identification performance is fragile to observation noise and signal band limitations. Consequently, the higher equalization error occurs. Besides the time domain methods, some frequency domain methods are also developed. For example, an interpolation algorithm has been considered for the frequency response function of channel model between adjacent pilot subcarriers to reduce computational complexity [13]. However, the function varies dramatically due to the long interference hence the interpolation accuracy cannot be guaranteed when the pilot rate is not high enough. Although the frequency domain methods through Hanning window function to reduce ICI in channel identification [14], or using unused sub-carriers to compensate ISI and ICI [15] are also investigated, the high performance for strong long multipath interferences is not guaranteed. Channel identification and equalization of OFDM communication with long interferences are more difficult than those in processing of short multipath interferences.
To improve the equalization performance, or to simplify the equalizer design, they should consider the difficulties of a large number of parameters required for long impulse response, and a large bit error rate (BER) caused by the collapse of orthogonality in the received signal. In this paper, a novel algorithm of channel identification for long multipath interferences is presented to overcome these difficulties. Compared with other methods, the proposed algorithm has the following features: (1) most of computation is performed by fast Fourier transform (FFT) in the frequency domain, therefore, the complexity in computation can be moderated even though the channel impulse response is very long; (2) the frequency response function of the OFDM channel is estimated from the spectral periodograms, which are tolerant to noise and the symbol estimation errors; (3) the diversity of multiple antennas in their frequency response functions is used to choose the strong orthogonal components for high BER performance of symbol estimation as well as high convergence rate of channel identification; (4) the algorithm is performed iteratively and has the adaptability to the time-varying channels. The rest of the paper is arranged as follows. In Section 2, the OFDM signals and channel model are addressed, then the properties of interferences with long delay time are investigated in Section 3. Based on the investigated properties, the algorithms of identification and interference compensation are given in Section 4, and their effectiveness is demonstrated by some simulation examples in Section 5. 2. Problem statement 2.1. OFDM signals Let the period of OFDM information symbol be denoted as N. As shown in Fig. 1, GI attaches a copy of the tail part of effective symbol to its head as a cyclic prefix when the signal is transmitted. The length of GI is denoted as Ngi , then the practical transmission period denoted as Ntx becomes to N þ Ngi . In the transmission symbol period N tx , the transmitted signal in baseband is generated by N-point IFFT as follows: dðkÞ ¼
N=2 X
Dðn; lÞejno0 ðklNtx Þ
n¼N=2þ1
for lN tx Ngi pkolN tx þ N,
Fig. 1. Guard interval in OFDM signal.
(1)
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where the FFT size is N, which is an integer with power of 2. k and o0 ¼ 2p=N are the normalized sampling instant and angular frequency, respectively. Dðn; lÞ is the information symbol conveyed on the n-th sub-carrier in the l-th transmission symbol period, and it belongs to a modulation constellation with finite elements. Generally Dðn; lÞ can be treated as a random sequence with respect to the sub-carrier number n and symbol period l, i.e., lim
L!1
L 1X ¯ 2 dðn1 n2 Þdðl1 Þ D ðn1 ; lÞDðn2 ; l l1 Þ ¼ D L l¼1
(2)
holds true, where d is the delta function, * denotes the ¯ 2 is the mean square of the conjugate complex, D constellation, n1 and n2 are the sub-carrier numbers, l1 is an arbitrary integer. 2.2. Multipath channel model Assume that the received baseband signal under multipath environment can be approximated by yðkÞ ¼
M X m¼0
r m ðkÞ þ eðkÞ ¼
M X
hm dðk km Þ þ eðkÞ,
(3)
m¼0
where r m ðkÞ is the m-th multipath wave to the receiver, hm is its coefficient, km is the delay tap, and eðkÞ is the additive noise. Correspondingly the channel model can be expressed by z transform as HðzÞ ¼ h0 þ h1 zk1 þ h2 zk2 þ þ hM zkM ,
(4)
1
where z is a backward shift operator, kM is the longest effective delay tap of interference. Substituting z ¼ ejo0 into (4) also yields the frequency response function of the channel model. 2.3. Interference exceeding GI Consider the interference r m ðkÞ with long delay time exceeding GI. In the l-th effective symbol period for
1591
lN tx pkolN tx þ N, the component of interference with delay time km in Fig. 2 is given by r m ðkÞ ¼ hm dðk km Þ 8 N=2 > P > > hm Dðn; l 1Þejno0 ðkkm þNgi lNtx Þ > > > > n¼N=2þ1 > > > > > < for k km lN tx o Ngi ; ¼ N=2 > P > > > hm Dðn; lÞejno0 ðkkm lNtx Þ > > > n¼N=2þ1 > > > > > : for Ngi pk km lNtx oN:
(5)
Performing N-point FFT of r m ðkÞ within the FFT window for lN tx pkolN tx þN yields the frequency components of r m ðkÞ. For example, the component on the n-th sub-carrier can be given by þN1 1 lNtxX r m ðkÞejno0 ðklNtx Þ N k¼lN tx
1 1 NX ¼ r m ðk þ lN tx Þejno0 k N k¼0 0 1 km N gi 1 N1 X 1@ X jno0 k jno0 k A ¼ r m ðk þ lN tx Þe þ r m ðk þ lNtx Þe N k¼km N gi k¼0 00 1 1 km N 1 N=2 hm Xgi @@ X Dðn1 ; l 1Þejn1 o0 ðkkm þNgi Þ Aejno0 k A ¼ N n1 ¼N=2þ1 k¼0 00 1 1 N=2 N 1 hm X @@ X Dðn1 ; lÞejn1 o0 ðkkm Þ Aejno0 k A þ N k¼k N n1 ¼N=2þ1 m gi 00 1 1 km N gi 1 N=2 hm X @@ X Dðn1 ; l 1Þejn1 o0 ðkkm þNgi Þ Aejno0 k A ¼ N n1 ¼N=2þ1 k¼0 00 1 1 km N gi 1 N=2 hm X @@ X Dðn1 ; lÞejn1 o0 ðkkm Þ Aejno0 k A N n1 ¼N=2þ1 k¼0 00 1 1 N=2 N 1 hm X@@ X jn1 o0 ðkkm Þ A jno0 k A Dðn1 ; lÞe þ e . N k¼0 n ¼N=2þ1 1
Fig. 2. Multipath interference in OFDM system.
(6)
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Following the property of orthogonal basis function of ejno0 k , the last term in (6) can be written as hm ejno0 km Dðn; lÞ.
(7)
Since the frequency component in (7) is only on the n-th sub-carrier, clearly it still holds the carrier orthogonality. Nevertheless, the first and the second terms in (6), which are the summation within the interval 0pkpkm N gi 1 of an incomplete FFT window, yield leakage error whose frequency components contaminate all the subcarriers. Now consider frequency components of all the multipaths. The orthogonal term at n-th sub-carrier becomes to M X
hm ejno0 km Dðn; lÞ ¼ Hðejno0 ÞDðn; lÞ.
(8)
m¼0
On the other hand, the first term in (6) for km 4Ngi yields the ISI denoted as Es ðn; lÞ in the frequency domain. It is the interference from the ðl 1Þ-th symbol period to the l-th period and can be expressed by 0 0 km N 1 N=2 M X hm @ Xgi @ X Es ðn; lÞ ¼ Dðn1 ; l 1Þ N m¼m1 n1 ¼N=2þ1 k¼0 1 1 jn1 o0 ðkkm þNgi Þ A jno0 k A
e
e
,
(9)
where m1 is the smallest integer such that km1 4N gi . Moreover, the effect of the second term in (6) leads to the ICI indicated by Ec ðn; lÞ as follows: 0 0 km N 1 N=2 M X hm @ Xgi @ X Ec ðn; lÞ ¼ Dðn1 ; lÞ N m¼m1 n1 ¼N=2þ1 k¼0 1 1 ejn1 o0 ðkkm Þ Aejno0 k A.
(10)
Let the sum of ISI and ICI be denoted as leakage error Eðn; lÞ. Therefore, the frequency domain expression of the received signal in the l-th symbol period is given by Yðn; lÞ ¼ Hðejno0 ÞDðn; lÞ þ Ec ðn; lÞ þ Es ðn; lÞ þVðn; lÞ, |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}
(11)
ceeding sub-model of the multipath channel be expressed by z transform as
GðzÞ ¼ hm1 zNgi þ1km1 þ hm1 þ1 zNgi þ1km1 þ1 þ þ hM zNgi þ1kM ,
(12)
where hm1 ; hm1 þ1 ; . . . are the coefficients of the multipaths exceeding GI. Then performing inverse Fourier transform of Es ðn; lÞ in (9) and Ec ðn; lÞ in (10) yields the signals of ISI and ICI in the time domain as
s ðk; lÞ ¼ GðzÞds ðk; lÞ, c ðk; lÞ ¼ GðzÞdc ðk; lÞ
(13) (14)
for k ¼ 0; 1; . . . ; N 1, where ds ðk; lÞ and dc ðk; lÞ are the corresponding transmitted signals included in (9) and (10). They can be given by ( dðk 1 þ ðl 1ÞNtx Þ for N þ Ngi þ 1pkp0; ds ðk; lÞ ¼ 0 for k40; ( dc ðk; lÞ ¼
(15) dðk N gi 1 þ lNtx Þ
for N þ N gi þ 1pkp0;
0
for k40:
(16) On the other hand, in the frequency domain, Es ðn; lÞ can be rewritten by N=2 X
Es ðn; lÞ ¼
Dðn1 ; l 1ÞHs ðn; n1 Þ,
(17)
n1 ¼N=2þ1
where Hs ðn; n1 Þ is Hs ðn; n1 Þ 8 M P hm ejn1 o0 ðkm Ngi Þ ejno0 ðkm Ngi Þ > > > > < m¼m1 N 1 ejðnn1 Þo0 ¼ M > > > P km N gi hm ejno0 ðkm Ngi Þ > : N m¼m1
for nan1 ; for n ¼ n1 :
Similarly, Ec ðn; lÞ is approximated by
Eðn;lÞ
where Yðn; lÞ and Vðn; lÞ are the frequency components of the received signal and noise on the n-th sub-carrier. It is clear that the leakage error Eðn; lÞ deteriorates the orthogonality of OFDM sub-carriers and will cause large demodulation error. It is important to remove the influence of Eðn; lÞ to guarantee high communication performance. 3. Properties of leakage error Some important properties of ICI, ISI and the leakage error are revealed statistically in this section. 3.1. Expressions of ICI and ISI It is seen that only the coefficients hm for km 4N gi remain in (9) and (10). Correspondingly, let the GI ex-
Ec ðn; lÞ ¼
N=2 X
Dðn1 ; lÞHc ðn; n1 Þ,
(18)
n1 ¼N=2þ1
where Hc ðn; n1 Þ is given by Hc ðn; n1 Þ 8 M P hm ejn1 o0 km ejno0 km ejðn1 nÞo0 Ngi > > > > < m¼m1 N 1 ejðnn1 Þo0 ¼ M > > > P km N gi hm ejno0 km > : N m¼m1
for nan1 ; for n ¼ n1 :
From (17) and (18), the data in the frequency domain fulfil the following expression: HDðlÞ ¼ YðlÞ EðlÞ VðlÞ ¼ YðlÞ Es ðlÞ Hc DðlÞ VðlÞ,
(19)
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where YðlÞ, DðlÞ, EðlÞ, Es ðlÞ and VðlÞ are the vectors of FFT coefficients of the received signal, the source information symbols, leakage error, ISI and noise in the l-th period, respectively, and 2
2 6 6 Hc ¼ 6 4
..
0
0
.
HðejN=2o0 Þ Hc ðN=2 þ 1; N=2 þ 1Þ
Hc ðN=2 þ 1; N=2Þ
3
.. .
..
.. .
Hc ðN=2; N=2 þ 1Þ
7 7 7. 5
.
Hc ðN=2; N=2Þ
L 1X D ðn; lÞEs ðn; lÞ L!1 L l¼1
lim
n1 ¼N=2þ1
¯ Hðe
1PL l¼1 ðD ðn; lÞYðn; lÞÞ Þ ¼ lim L L!1 1PL ðD ðn; lÞDðn; lÞÞ L l¼1 1PL l¼1 ðD ðn; lÞEc ðn; lÞÞ jno0 Þ þ lim L ¼ Hðe L!1 1PL ðD ðn; lÞDðn; lÞÞ L l¼1 m M 1 1 X X km N gi ¼ hm ejnkm o0 þ 1 hm ejnkm o0 . N m¼m1 m¼0 (23)
¯ jno0 Þ as h¯ 0 ; h¯ 1 ; h¯ 2 ; . . ., Denote the IFFT coefficients of Hðe then the coefficients hm can be obtained by
Since Es ðn; lÞ in (17) is only related to the information symbols in the ðl 1Þ-th symbol period, then from (2), Dðn; lÞ and Es ðn; lÞ are uncorrelated, i.e.,
N=2 X
jno0
7 7 7, 5
3.2. Statistical properties of ICI and ISI
¼
as follows:
3
HðejðN=2þ1Þo0 Þ
6 6 H¼6 4
1593
! L 1X D ðn; lÞDðn1 ; l 1Þ Hs ðn; n1 Þ ¼ 0. L!1 L l¼1
( hm ¼
h¯ m Nh¯ m =ðN Ngi þ km Þ
for 0pkm pNgi ; for km 4N gi :
(24)
From (23) and (24), it can be seen that it is possible to estimate Hðejno0 Þ by using the properties of the leakage error even when the channel has long multipath interferences.
lim
4. Channel identification
(20) Moreover, multiplying Ec ðn; lÞ in (18) by D ðn; lÞ and using the results in (2) lead to the following result: L 1X D ðn; lÞEc ðn; lÞ L!1 L l¼1
lim
¼
N=2 X n1 ¼N=2þ1
! L 1X lim D ðn; lÞDðn1 ; lÞ Hc ðn; n1 Þ L!1 L l¼1
¯2 ¯ 2 Hc ðn; nÞ ¼ D ¼ D
M X km Ngi hm ejnkm o0 . N m¼m1
(21)
Following (21), it is clear that the longer the delay tap km , the greater the leakage error is. Therefore, the symbol equalization or interference compensation becomes more difficult. Furthermore, from (11), the following equation: L L 1X 1X D ðn; lÞYðn; lÞ ¼ Hðejno0 Þ D ðn; lÞDðn; lÞ L l¼1 L l¼1
þ
L 1X D ðn; lÞðEs ðn; lÞ þ Ec ðn; lÞÞ L l¼1
þ
L 1X D ðn; lÞVðn; lÞ L l¼1
(22)
holds true. Then by using the results of (2), (20) ¯ jno0 Þ defined in (23) can be obtained by (22) and (21), Hðe
When several preamble or training symbols are available, (23) and (24) can give a batch channel identification only with computational complexity of OðNÞ. When the channel model is time-varying, or the successive training symbols are not available for channel identification, a new channel identification algorithm can be developed by making use of spectral periodograms whose ISI and ICI are compensated by the leakage error replica. If the training symbols are unavailable, except the pilot symbols at pilot carriers, the information symbols have to be estimated from the received signals for channel identification. From (19), the symbol vector DðlÞ in the lth period may be estimated by ^ ¼ ðH ^ þH ^ c Þ1 ðYðlÞ Es ðlÞÞ. DðlÞ
(25)
^ þH ^ c Þ and However, the computation of inverse of ðH ^ updating Hc using the latest channel estimation are timeconsuming. In the proposed algorithm a simple estimation method with low computational complexity is considered. As mentioned in Section 3, the components of ICI and ISI contaminate all the sub-carriers, and the frequency response function Hðejno0 Þ varies remarkably when the channel has long multipath interferences, as a result, neither the interpolation method nor equalization using the frequency selective diversity can yield satisfactory result. For example, at the n-th sub-carrier where jHðejno0 Þj is small, the orthogonal component in (6) attenuates to such a small value that symbol equalization becomes fragile to the noise and leakage error, even the error correction techniques might fail to correct the equalization errors at the sub-carriers with small magni-
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tude of frequency response. In order to overcome these difficulties, the diversity of multiple receiver antennas is used in the proposed identification algorithm. Let the total number of antenna elements be Q , then the received signal at the q-th antenna can be denoted as yq ðkÞ, where 1pqpQ . Correspondingly, the frequency response function from the transmitter to the q-th antenna is Hq ðejno0 Þ, and its GI exceeding part expressed by z transform is Gq ðzÞ. Channel identification is performed iteratively for every symbol period. In the ðl 1Þ-th iteration, denote ^ the estimates of information symbol as Dðn; l lÞ, the jno0 ^ frequency response as Hq ðe ; l 1Þ, and its GI exceeding ^ q ðz; l 1Þ, respectively. These estimates are used part as G in the next iteration, thus the mitigation of the influence caused by these estimation errors is considered in the proposed algorithm. In the l-th iteration, the information symbol is estimated first by removing the affection of ISI. 4.1. Initial estimation of information symbols In the l-th symbol period, let the frequency component of yq ðkÞ at the n-th sub-carrier be denoted by Y q ðn; lÞ. It is calculated from FFT of yq ðkÞ within the FFT window ^ lN tx pkolN tx þ N. Using the estimation of Dðn; l 1Þ in the ðl 1Þ-th symbol period, d^ s ðk; lÞ can be obtained by (15). Therefore, s;q ðk; lÞ can be estimated by (13) using G^ q ðz; l 1Þ and d^ s ðk; lÞ in the time domain. Furthermore, the ISI replica E^ s;q ðn; lÞ to the q-th receiver antenna is obtained by FFT of s;q ðk; lÞ. Thus the initial estimate of ^ Dðn; lÞ can be given by removing ISI Y q ðnÞ ðn; lÞ E^ s;qmax ðnÞ ðn; lÞ ^ , Dðn; lÞ ¼ max ^ q ðnÞ ðejno0 ; l 1Þ H max
(26)
where qmax ðnÞ is the antenna number determined by ^ q ðejno0 ; l 1ÞjÞ. qmax ðnÞ ¼ arg maxðjH
(27)
q
^ q ðnÞ ðejno0 ; l 1Þj takes the maximum of It is seen that jH max ^ q ðejno0 ; l 1Þj for 1pqpQ at the n-th sub-carrier so that jH the strongest orthogonal component is used to estimate the information symbol Dðn; lÞ. ^ On the other hand, the estimate Dðn; lÞ in (26) can ^ hd ðn; lÞ following further be modified by hard decision D the constellation of Dðn; lÞ [16]. 4.2. Estimation of leakage error ^ ^ hd ðn; lÞ, then d^ c ðk; lÞ dðkÞ is calculated through IFFT of D ^ q ðz; l can be obtained by (16). Using the estimation of G 1Þ and d^ c ðk; lÞ, c;q ðk; lÞ can be estimated by (14) in the time domain first. Consequently, E^ c;q ðn; lÞ can be calculated from FFT of c;q ðk; lÞ, and the leakage error E^ q ðn; lÞ can also be obtained by E^ s;q ðn; lÞ þ E^ c;q ðn; lÞ. 4.3. Re-estimation of information symbols It is seen that the ICI effect still remains in the initial ^ estimation of Dðn; lÞ in (26). By using the estimate E^ q ðn; lÞ obtained in Section 4.2, the more accurate estimation of
^ information symbols Dðn; lÞ is obtained as follows: Y q ðnÞ ðn; lÞ E^ qmax ðnÞ ðn; lÞ ^ Dðn; lÞ ¼ max . ^ q ðnÞ ðejno0 ; l 1Þ H
(28)
max
^ hd ðn; lÞ can also be updated. FurtherCorrespondingly D more, the new E^ q ðn; lÞ can be modified by using the ^ ^ hd ðn; lÞ similarly as Section 4.2. The updated Dðn; lÞ or D ^ estimation procedure of Eq ðn; lÞ and Dðn; lÞ can be repeated several times to reduce their estimation errors, where the strongest component is used for symbol estimation.
4.4. Estimation of frequency response function The channel is identified from the frequency compo^ nent Y q ðn; lÞ, the symbol estimate Dðn; lÞ and the replica of ^ lÞ. Consequently, it is important to leakage error Eðn; remove the influence caused by the noise term in ^ lÞ and Dðn; ^ Y q ðn; lÞ, the estimation errors of Eðn; lÞ. The phases of noise and estimation errors are usually random, then their influence can be mitigated through the smoothing effect of spectral periodograms [17]. In the lth iteration, the spectral periodograms DDðejno0 ; lÞ and DYðejno0 ; lÞ are defined as follows:
^ ðn; lÞD ^ hd ðn; lÞ, DDðejno0 ; lÞ ¼ ll DDðejno0 ; l 1Þ þ D hd jno0
DY q ðe
jno0
; lÞ ¼ ll DY q ðe
(29)
; l 1Þ
^ ðn; lÞðY q ðn; lÞ E^ q ðn; lÞÞ, þD hd
(30)
where ll is a forgetting factor over the range of 0oll o1. It is seen that the effects of noise and estimation errors are reduced when l becomes large [17]. Using the estimates DDðejno0 ; l 1Þ and DY q ðejno0 ; l 1Þ in the ðl 1Þ-th itera^ hd ðn; lÞ, Y q ðn; lÞ and E^ q ðn; lÞ in tion, as well as the estimates D the l-th iteration, the estimates of (29) and (30) are obtained. Thus the estimate of frequency response func^ q ðejno0 ; lÞ can be given by tion H jno0 ; lÞ ^ q ðejno0 ; lÞ ¼ DY q ðe . H DDðejno0 ; lÞ
(31)
^ q ðz; lÞ can be updated by (12) using IFFT of Moreover, G ^ q ðejno0 ; lÞ. Then let l ¼ l þ 1 for the next iteration. H As mentioned previously, the frequency response function varies remarkably when the interferences have long delay taps. As a result, the sidelobes often occur in the impulse response h^ m of channel model due to noise, estimation errors of the information symbols and leakage error, etc. Through setting a threshold between mainlobes and sidelobes, the sidelobes can be removed to improve the convergence performance of channel identification and BER performance of symbol estimation [18]. In this paper, the algorithm is developed for the case where the number of sub-carriers in one symbol is the same as the FFT size. Since it is difficult to design a low pass filter with sharp cut-off in the communication circuit without disturbing the adjacent channels, the number of sub-carriers is often smaller than the FFT size. In such communication systems, the OFDM signals are restricted within a specified band, and make the channel identification difficult [19]. The details of channel identification
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techniques for the limited signal band had been discussed in [20,21]. 4.5. Procedure of channel identification In the identification algorithm, the initial estimation of ^ Dðn; lÞ is calculated from the received signal compensated ^ lÞ is estimated, by E^ s ðn; lÞ first, next the leakage error Eðn; ^ then Dðn; lÞ is updated by using the compensation of Ec ðn; lÞ. Finally the channel frequency response is estimated from the spectral periodograms of the transmitted and received signals. The procedure of proposed algorithm can be summarized as follows: Step 1: Let the initial values of DY q ðejno0 ; 0Þ, DDðejno0 ; 0Þ ^ q ðz; 0Þ, and ^ q ðejno0 ; 0Þ, G be 0. Choose the initial values of H let the iteration number be l ¼ 1. Step 2: Calculate Y q ðn; lÞ through FFT of the received signal yq ðkÞ within the FFT window lN tx pkolN tx þ N. Step 3: Calculate ds ðk; lÞ by (15), and Es;q ðn; lÞ by FFT of s;q ðk; lÞ in (13). ^ Step 4: Estimate Dðn; lÞ by (26), and hard decision ^ hd ðn; lÞ, respectively. D Step 5: Calculate dc ðk; lÞ in (16), furthermore estimate the replica of ICI through FFT of ^ c;q ðk; lÞ in (14). ^ ^ hd ðn; lÞ, Step 6: Update Dðn; lÞ by (28) and hard decision D respectively. Step 7: Calculate DY q ðejno0 ; lÞ and DDðejno0 ; lÞ by (29) and (30), respectively. ^ q ðejno0 ; lÞ from (31) and update the GI Step 8: Estimate H ^ q ðz; lÞ by (12). Let l ¼ l þ 1, then return to exceeding part G Step 2 to repeat the iterations. The procedure in the l-th iteration is illustrated in Fig. 3. 4.6. Features of the proposed algorithm The features of the proposed algorithm are summarized as follows. (1) Periodograms can smooth DY q ðejno0 ; lÞ and DDðejno0 ; lÞ before their division, so that the periodograms based identification algorithm can reduce the estimation ^ q ðejno0 ; lÞ caused by the estimation errors of error of H ^ Dðn; lÞ and E^ q ðn; lÞ, or the noise term in Y q ðn; lÞ [17]. (2) Unlike some other identification algorithms in the time domain whose performance depends on the total number of parameters to be estimated, the proposed algorithm estimates the frequency response function per sub-carrier from the spectral periodograms and shows good convergence performance even for long impulse response of channel. (3) By virtue of the diversity of multiple antennas, the strongest orthogonal component is chosen from Q received signals to estimate the information symbols, and it improves the performance of channel identification and symbol estimation significantly compared with the single antenna case. It is noticed that the purpose of using multiple antennas is just choosing the strongest orthogonal component at each sub-carrier, the performance of the
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proposed algorithm does not depend on the total number of antenna elements as much as the conventional spatial equalizers based on antenna diversity. Though some maximum ratio combining techniques (MRC) work well in many multi-antenna cases [22], their performance may be weakened due to the correlation of leakage error Eðn; lÞ with Dðn; lÞ. If the channel has severe leakage error, the proposed algorithm should be used to reduce the identification error so that the correlation of Eðn; lÞ with Dðn; lÞ is removed from the received signal, then the maximum ratio combination or some correcting codes can be applied to obtain better BER performance. (4) The forgetting factor is used in periodogram estimation so that the algorithm can also deal with slow time-varying channels. A small forgetting factor has adaptability to quick channel variation and large BER, whereas a large one is used in low SNR environment for spectral smoothing. (5) When the interference exceeding GI is strong, the convergence of channel identification and BER performance can be improved by appropriately choosing the sidelobe threshold. (6) The main computation only requires FFT, multiplication, division of periodograms, therefore, the algorithm has less computational complexity and can easily be utilized in the practical applications. In each iteration, besides the computation of FFT, the proposed algorithm needs the following calculations: 2ðkM N gi Þ2 q multiplications to estimate ICI and ISI, 2N divisions for initial estimation and re-estimation of information symbols, ðq þ 1ÞN multiplications to estimate periodograms, qN divisions to estimate the frequency response functions. By contrast with the computational complexity of RLS algorithm, besides the estimation of information symbols in RLS algorithm, the recursive 2 identification requires about OðkM Ntx Þ multiplications for one symbol period. It is clear that the proposed identification algorithm has less complexity than RLS, especially for large kM . Though LMS only requires Oð2kM Ntx Þ multiplications for channel identification, its convergence rate is much slower than RLS [3]. 5. Numerical simulation examples The 3GPP 2.5 MHz OFDM transmission with 16QAM modulation is used in the examples where the FFT size N ¼ 256, GI length N gi ¼ 64 [1]. Moreover, the number of sub-carriers is 256, the total number of receiver antennas is Q ¼ 2, and their distance is 12 of the wave length. The noise is assumed as an additive Gaussian white noise. 5.1. Example of a channel with three multipath interferences Besides the direct wave, there are three multipath interferences in the transmission channel [7,8]. The coefficients of the waves are illustrated in Table 1. Let the SNR ¼ 25 dB. Six cases are considered: the case where only one pilot sub-carrier is known to remove the
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Fig. 3. Block diagram of l-th iteration.
ambiguity of channel identification and symbol estima1 1 1 1 tion, the cases where the scattered pilot rates are 16 , 8, 4, 2, respectively, and the case where training symbols are available for identification. At the pilot carriers, the values ^ of Dðn; lÞ are given by the corresponding true values of Dðn; lÞ, while at the other sub-carriers, the values of Dðn; lÞ have to be estimated from the received signals. Channel identification is started from the initial values of
^ ðz; 0Þ ¼ 0. The square error of channel ^ jno0 ; 0Þ ¼ 1, G Hðe identification Er H , which is defined as Er H ¼
P PN1 ^ jno0 Þ Hq ðejno0 Þj2 n¼0 jH q ðe q P PN1 jno0 Þj2 n¼0 jH q ðe q
(32)
is illustrated in Fig. 4, and BER curves of the estimated symbols are illustrated in Fig. 5, respectively. They show
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Table 1 Simulation conditions. Wave
Power (dB)
Phase
DOA
Delay tap
Direct Interference 1 Interference 2 Interference 3
0 3 5 5
0 p=6 p=4 p=3
p=8 p=4 p=3 p=6
0 3N gi =4 5N gi =4 7N gi =4
Fig. 6. BER comparison of maximum ration combination and proposed algorithm (average of 30 simulation runs).
Fig. 4. Channel identification error (average of 30 simulation runs).
Fig. 5. BER of estimated symbols (average of 30 simulation runs).
that the algorithm works well even for few pilot subcarrier cases. In Fig. 5, BER of the estimated symbols decreases to 0 after several iterations, whereas it is larger than 0.5 at low pilot rate in the first iteration due to the initial value of channel identification is quite different from the true one. It implies that the algorithm converges even from the severe initial conditions. The BER curves are plotted in Fig. 5 when the perfect channel estimate is used for symbol estimation. It is seen that the good BER performance can be guaranteed if the influence of ISI and ICI caused by the long multipath interferences are compensated by the replica of leakage error.
Since RLS algorithm is often used for channel identification in the previous methods, the results of RLS algorithm are also shown in Figs. 4 and 5 for comparison with the proposed algorithm. They are obtained under the same simulation conditions. In RLS algorithm, the recursion is performed per sampling instant to estimate the parameters of hm in (3) by using the latest samples of yðkÞ and dðkÞ, thus RLS updates the estimates N tx ¼ 320 times during 1 iteration in the proposed algorithm. In Fig. 4, if several successive training periods are available, i.e., the true values of dðkÞ can be used for channel identification directly, it is seen that RLS algorithm can yield small error by using the true dðkÞ while its computational load is heavier than that of the proposed algorithm. ^ However, when the training symbols are unavailable, dðkÞ has to be estimated for channel identification, the error of ^ dðkÞ deteriorates the identification performance of RLS algorithm. For example, when the pilot rate is 12, the convergence of channel identification and the symbol estimation used RLS estimation is much slower than that of the proposed algorithm, and its performance becomes very poor when the pilot rate is 14, whereas the proposed algorithm works well since it uses spectral periodograms. The BER curves of symbol estimation are plotted in Fig. 6 by choosing the strongest orthogonal components, and by maximum ratio combining techniques. In the low pilot rate situations, less of the leakage error is removed due to the large error of channel identification in the first several iterations, hence the convergence rate of maximum ratio combination becomes slow due to the correlation of Eðn; lÞ and Dðn; lÞ.
5.2. Channel identification vs. interference power Let the pilot rate be 18, and the power of interference 3 be changed from 0 to 20 dB, the other conditions be the same as those in Section 5.1. Er H and BER vs. interference power are illustrated in Figs. 7 and 8, respectively. Though there is strong multipath interference exceeding GI, both
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Fig. 7. Channel estimation error vs. interference power.
0pnp20 is illustrated in Fig. 10. It is shown that at the sub-carriers marked by circle, the symbols are difficult to be estimated by frequency selective diversity of single antenna, since low magnitude of Hq ðejno0 Þ leads to weak orthogonal component in the received signal. The curves of Er H and BER are plotted in Fig. 11, where BER is large for single antenna case, whereas the errors for Q ¼ 2 and 4 are almost the same since the strong orthogonal components are used in both of the two cases. It also shows that though BER is large in the first several iterations, the error of channel identification based on periodograms has been decreased to such a low level as to provide an effective channel model for equalization. 5.5. Identification of time-varying channel Fig. 8. BER vs. interference power.
Er H and BER are decreased to a considerable low level after just about 10 iterations. 5.3. Channel identification vs. SNR Let the pilot rate be 18, and SNR be changed from 10 to 40 dB, the other conditions be the same as those in Section 5.1. Er H plotted in Fig. 9 shows that the proposed algorithm converges even for low SNR conditions. 5.4. Channel identification vs. total number of antennas Let the pilot rate be 18. In order to investigate the influence of Q on the algorithm performance, the total number Q of antennas is chosen as 1, 2 and 4, respectively. The other simulation conditions are the same as those in Section 5.1. The relative magnitude of Hq ðejno0 Þ for
Let the pilot rate be 18, the power, phase and DOA of Interference 3 be changed at every 10 symbol periods so that the channel is time-varying. The power profile is shown in Fig. 12. For the variation of channel, the forgetting factor ll ¼ 0:25 is used and the results of Er H , BER are illustrated in Fig. 13. Though the channel varies quickly, the prompt reduction of errors shows that the proposed algorithm can also work well for time-varying channels. 5.6. Channel identification of COST 207 model Let the pilot rate be 18. Assume that the delay profile of the multipath in a hill area is a COST 207 model [23]. There are 11 waves with delay time 0pkm p10 and power ekm =2:5 , 21 waves with delay time 40pkm p60 and power 0:7079eðkm 40Þ=4 , 15 waves exceeding GI with delay time 72pkm p96 and power 0:5623eðkm 72Þ=3:6, and m ¼ 0 denotes the direct wave. The total power of interferences
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Fig. 9. Channel estimation error vs. SNR.
Fig. 10. Illustration of jHq ðejno0 Þj vs. Q .
Fig. 11. Channel estimation error and BER vs. Q .
Fig. 12. Power variation at interference 3. : jump points.
exceeding GI is 1:29 dB. The DOA of multipath waves are generated randomly, and the other conditions are the same as in Section 5.1. As illustrated in Fig. 14, the received signal suffers from strong multipath interferences, as a result, BER is about 0.45 without interference compensation. At the first several iterations, small ll and large sidelobe threshold are used to mitigate the influences of high BER of estimated symbols and sidelobes. With BER decreasing ll is increased to smooth the periodograms whereas the sidelobe threshold is decreased to handle weak multipath interferences. In the simulation ll is assumed to be ll ¼ minf0:075 1:05l ; 0:75g, and the sidelobe threshold is assumed to be maxf0:1 0:98l ; 0:005g. The corresponding ErH and BER of estimated symbols are shown in Fig. 15. Though the channel has strong multipath interferences,
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Fig. 13. Channel estimation error and BER of recovered signal.
Fig. 14. Relative power of interferences.
Fig. 15. Channel estimation error and BER of estimated symbols (average of 30 simulation runs).
ErH decreases from 1.0 to 0.003, BER decreases from 0.45 to 0 in about 25 iterations. The comparisons of the proposed algorithm with the conventional identification algorithms of RLS, LMS are also examined by using training symbols. The results of ErH are shown in Fig. 16 where the transverse axis is the
Fig. 16. Comparison of estimation errors of proposed algorithm, RLS and LMS.
sample number. RLS and LMS are performed per sampling instant, while the iteration is executed once per symbol period using 256 samples in the proposed algorithm. Therefore, RLS and LMS update their estimates 320 times during 1 iteration of the proposed algorithm. Since the training symbols can be used for identification without error of symbol estimation, the sidelobe threshold can be chosen as a very small value. In this simulation, the sidelobe threshold is 0.005, ll ¼ 0:75, and the length kM of channel impulse response is chosen as 128 taps for identification. In Fig. 16, ErH of RLS is too large to be kept within the axis range of ordinate in the first 150 recursions due to its sensitivity to the initial conditions. Moreover, not only the computational load of RLS is heavy for long impulse response, but the condition number of covariance matrix of regression vector is so large that the estimation is fragile to the noise and round-off errors of numerical computation, while the convergence rate of LMS is slow, as shown in Fig. 16. In proposed algorithm, ErH is reduced to 104 in only four iterations. In each iteration, besides the estimation for information symbols, compensation of leakage error requires 2ðkM Ngi Þ2 ¼ 8192 multiplications, periodogram estimation and channel identification require about 3N ¼ 768 multiplications in the proposed algorithm, whereas Ntx recursions for one 2 symbol period in RLS require 3kM Ntx 107 multiplications. ErH decreases to 0.005 by just one iteration in the proposed algorithm with low computation load, whereas RLS requires about 200 recursions to obtain the similar accuracy. Though the computation load is low in LMS algorithm, its convergence rate is so slow that it takes about more than 640 iterations to reach ErH 0:005. Moreover, if the training symbols are unavailable, the performance of RLS and LMS becomes worse due to the errors of symbol estimation, as shown in Fig. 4. 6. Conclusions Channel identification and its applications to OFDM communications with interferences exceeding GI have been studied in this work. The proposed algorithm can be
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implemented in the frequency domain by FFT, consequently it has low computational complexity, which does not depend on the length of interference delay time too much. Different from other identification or equalization algorithms, the proposed algorithm utilizes diversity of multiple antennas for symbol estimation, and spectral periodograms to estimate the frequency response function. The former allows the algorithm to estimate information symbols from the strongest orthogonal components of multiple received signals at the receivers, and the latter reduces the effect of noise as well as errors of symbol estimation, therefore the proposed algorithm has high convergence rate and can work under the strong multipath interferences. References [1] 3GPP TR 25.814 (Release 7), 3rd Generation Partnership Project hhttp://www.3gpp.org/ftp/Specs/html-info/TSG-WG–r1.htmi, 2006. [2] Y. Karasawa, N. Gejoh, T. Izumi, Modeling and analysis of OFDM transmission characteristics in Rayleigh fading environment in which the delay profile exceeds the guard interval, IEICE Trans. Commun. E88-B (7) (2005) 3020–3027. [3] J. Balakrishnan, R.K. Martin, C.R. Johson Jr, Blind, adaptive channel shortening by sum-squared auto-correlation minimization (SAM), IEEE Trans. Signal Process. 51 (12) (2003) 3086–3093. [4] G. Ysebaert, K. Vanbleu, G. Cuypers, M. Moonen, T. Pollet, Combined RLS–LMS initialization for per tone equalizers in DMT-receivers, IEEE Trans. Signal Process. 51 (7) (2003) 1916–1927. [5] K. Vanbleu, G. Ysebasert, G. Cuypers, M. Moonen, Adaptive bit rate maximizing time-domain equalizer design for DMT-based systems, IEEE Trans. Signal Process. 54 (2) (2006) 483–498. [6] R.K. Martin, et al., Unification and evaluation of equalization structures and design algorithms for discrete multitone modulation systems, IEEE Trans. Signal Process. 53 (10) (2005) 3880–3894. [7] S. Hori, N. Kikuma, N. Inagaki, MMSE adaptive array suppressing only multipath waves with delay times beyond the guard interval for fixed reception in the OFDM systems, IEICE Trans. Commun. J86B (2003) 1934–1940. [8] K. Higuchi, H. Sasaoka, Adaptive array suppressing inter symbol interference based on frequency spectrum in OFDM systems, IEICE Trans. Commun. J87-B (2004) 1222–1229.
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[9] C. Lim, Y. Chang, J. Cho, P. Joo, H. Lee, Novel OFDM transmission scheme to overcome caused by multipath delay longer than cyclic prefix, in: Proceedings of the IEEE Vehicle Technology Conference, 2005. [10] N. Suzuki, H. Uehara, M. Yokoyama, A new OFDM demodulation method with variable-length effective symbol and ICI canceller, IEICE Trans. Fundam. E85-A (12) (2002) 2859–2867. [11] M. Uesugi, An interference cancellation scheme for OFDM using adaptive algorithm, IEICE Trans Commun. E86-B (11) (2003) 3182–3191. [12] R. Schur, Impulse compression for OFDM transmission over timevarying multipath channels, in: Proceedings of the IEEE Vehicle Technology Conference, vol. 2, 2002, pp. 1074–1076. [13] V. Nguyen, M. Winkler, C. Hansen, H. Kuchenbecker, Channel estimation for OFDM systems in case of insufficient guard interval length, in: Proceedings of the 15th International Conference on Wireless Communications, Alberta, Canada, 7–9 July, 2003. [14] S. Suyama, M. Ito, H. Suzuki, K. Fukawa, A scattered pilot OFDM receiver with equalization for multipath environments with delay difference greater than guard interval, IEICE Trans. Commun. E86-B (1) (2003) 275–282. [15] T. Karp, S. Trautmann, N.J. Fliege, Frequency domain equalization of DMT/OFDM systems with insufficient guard interval, in: Proceedings of the IEEE International Conference on Communications, vol. 3, New York, 2002, pp. 1646–1650. [16] I. Glover, P. Grant, Digital Communications, Prentice-Hall, Englewood Cliffs, NJ, 1998, pp. 329–363 (Chapter 10). [17] R. Pintelon, J. Schoukens, System Identification—A Frequency Domain Approach, IEEE Press, New York, 2001, pp. 53–57 (Chapter 2). [18] T. Hamazumi, K. Imamura, Coupling canceller, P2000-341238A, Publication of Patent Applications, 2000. [19] G. Ysebaert, F. Pisoni, M. Bonavetura, R. Hug, M. Moonen, Echo cancellation in DMT-receivers: circulant decomposition canceller, IEEE Trans. Signal Process. 52 (9) (2004) 2612–2624. [20] L. Sun, A. Sano, Channel identification for SFN relay station with coupling wave in OFDM systems, IEICE Trans. Fundam. J88-A (9) (2005) 1045–1054. [21] L. Sun, A. Sano, Channel identification and applications to OFDM communication systems with limited bandwidth, in: Proceedings of the 15th European Digital Signal Processing Conference, Poznan, Poland, September 2007. [22] J. Burke, J. Zeidler, B. Rao, CINR difference analysis of optimal combining versus maximal ratio combining, IEEE Trans. Wireless Commun. 4 (1) (2005) 1–5. [23] COST 207, Digital land mobile radio communications, Final Report, Commission of the European Communities, 1989.
Contents lists available at ScienceDirect
Signal Processing journal homepage: www.elsevier.com/locate/sigpro
Channel identification and interference compensation for OFDM system in long multipath environment L. Sun a,, A. Sano b, W. Sun a, A. Kajiwara a a b
Faculty of Environmental Engineering, The University of Kitakyushu, 1-1 Hibikino, Wakamatsu-ku, Kitakyushu 808-0135, Japan Department of System Design Engineering, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan
a r t i c l e in fo
abstract
Article history: Received 27 June 2008 Received in revised form 26 November 2008 Accepted 22 February 2009 Available online 9 March 2009
Guard interval (GI) is usually applied to reduce the influence caused by multipath interferences in orthogonal frequency division multiplexing (OFDM) systems. However, the long multipath interferences will deteriorate the orthogonality of the sub-carriers if they have longer delay time than GI. It is illustrated that not only inter-symbol interference (ISI) but also inter-carrier interference (ICI) is caused by the collapse of orthogonality in the received signal. As a result, both the channel identification and equalization become difficult, and the communication performance cannot be guaranteed. In this work, a new channel identification algorithm is proposed to estimate the frequency response function from the spectral periodograms which are compensated by the replica of leakage error. Since most of the computation is performed in the frequency domain through fast Fourier transform, the algorithm has such low computational complexity that it can be implemented easily in applications. & 2009 Elsevier B.V. All rights reserved.
Keywords: OFDM channel identification Guard interval Orthogonality Leakage error Periodogram
1. Introduction Orthogonal frequency division multiplexing (OFDM), which adopts multi-carrier modulation, has been widely applied in the services of digital terrestrial broadcasting, asymmetric digital subscriber (ADSL) and 5 GHz wireless LAN. Since the spectra of orthogonal OFDM sub-carriers allow to be overlapped by each other to form an approximate rectangular spectrum, OFDM transmission has high frequency efficiency. Consequently, OFDM has been considered as a powerful candidate for the next generation of communications [1]. In OFDM communications, guard interval (GI) is attached at the header of every effective transmission symbol, consequently OFDM has a strong tolerance against the multipath interferences whose delay time
Corresponding author.
E-mail addresses: [email protected] (L. Sun), [email protected] (A. Sano). 0165-1684/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2009.02.017
does not exceed GI. However, there are some interferences with long delay time especially in high speed data transmission or mobile communications. Moreover, it is desired to squeeze the length of GI for higher transmission efficiency. If the long delay time of interference exceeds GI, the spectral leakage error, which yields inter-symbol interference (ISI) as well as inter-carrier interference (ICI), destroys the orthogonality of sub-carriers in the received signal, and makes both the channel identification and equalization very difficult. As a result, the communication performance degrades significantly [2]. The error correcting codes and frequency selective diversity can help to equalize the information symbols from the received signal when the multipath interferences do not exceed GI, nevertheless, their effectiveness becomes weak since the components of ISI and ICI interferences, which are caused by the long multipaths, contaminate all the sub-carriers. The methods dealing with long multipath interference can be classified into two main groups: (1) the direct methods that use specific criteria to design equalizer without channel model; (2) the indirect methods that use
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the estimated channel model to compensate or cancel the interferences. They can be implemented in the time domain, or the frequency domain, depending on the criterion and optimization procedures of the processing algorithms. The time-domain equalizers (TEQ) and the spatial equalizers using antenna diversity in the frequency domain are typical direct methods. Their criterion functions are composed of the received signals, desired performance indices and equalizer parameters, and are usually complicated nonlinear functions. For example, the MMSE criterion [3,4], the maximum bit rate criterion [5] in TEQs are solved by nonlinear optimization without explicit channel identification, however, they suffer from computational complexity for nonlinear optimization or slow convergence rate [6]. Differently, some spatial equalizers use antenna array to simplify the computational complexity. For instance, the spatial equalizers extract the interferences exceeding GI through a specific window function [7], or spectral envelope for PSK signals [8]. Nevertheless, the performance of spatial equalizers strongly depends on the capacity of antenna diversity or the signal arrival angle, and these equalizers may not work well in the severe multipath interference environments with restricted number of antenna elements. Though increasing the sampling rate may shorten the length of channel impulse response [9], the circuit complexity restricts its application to the practical systems. On the other hand, the indirect methods separate the problem into two stages of channel identification and model based equalization design. When the channel has been identified, the immediate ways to the second stage in the time domain are the compensation of the received signal by estimating the interference replica [10,11], and the length compression of impulse response of the combined communication channel, including transmission channel and equalizer [12]. Most of them identify the channel model by RLS or LMS algorithm in the time domain from preamble or training symbols. However, too long impulse response of a communication channel leads to heavy computational load for RLS or slow convergence for LMS, and the identification performance is fragile to observation noise and signal band limitations. Consequently, the higher equalization error occurs. Besides the time domain methods, some frequency domain methods are also developed. For example, an interpolation algorithm has been considered for the frequency response function of channel model between adjacent pilot subcarriers to reduce computational complexity [13]. However, the function varies dramatically due to the long interference hence the interpolation accuracy cannot be guaranteed when the pilot rate is not high enough. Although the frequency domain methods through Hanning window function to reduce ICI in channel identification [14], or using unused sub-carriers to compensate ISI and ICI [15] are also investigated, the high performance for strong long multipath interferences is not guaranteed. Channel identification and equalization of OFDM communication with long interferences are more difficult than those in processing of short multipath interferences.
To improve the equalization performance, or to simplify the equalizer design, they should consider the difficulties of a large number of parameters required for long impulse response, and a large bit error rate (BER) caused by the collapse of orthogonality in the received signal. In this paper, a novel algorithm of channel identification for long multipath interferences is presented to overcome these difficulties. Compared with other methods, the proposed algorithm has the following features: (1) most of computation is performed by fast Fourier transform (FFT) in the frequency domain, therefore, the complexity in computation can be moderated even though the channel impulse response is very long; (2) the frequency response function of the OFDM channel is estimated from the spectral periodograms, which are tolerant to noise and the symbol estimation errors; (3) the diversity of multiple antennas in their frequency response functions is used to choose the strong orthogonal components for high BER performance of symbol estimation as well as high convergence rate of channel identification; (4) the algorithm is performed iteratively and has the adaptability to the time-varying channels. The rest of the paper is arranged as follows. In Section 2, the OFDM signals and channel model are addressed, then the properties of interferences with long delay time are investigated in Section 3. Based on the investigated properties, the algorithms of identification and interference compensation are given in Section 4, and their effectiveness is demonstrated by some simulation examples in Section 5. 2. Problem statement 2.1. OFDM signals Let the period of OFDM information symbol be denoted as N. As shown in Fig. 1, GI attaches a copy of the tail part of effective symbol to its head as a cyclic prefix when the signal is transmitted. The length of GI is denoted as Ngi , then the practical transmission period denoted as Ntx becomes to N þ Ngi . In the transmission symbol period N tx , the transmitted signal in baseband is generated by N-point IFFT as follows: dðkÞ ¼
N=2 X
Dðn; lÞejno0 ðklNtx Þ
n¼N=2þ1
for lN tx Ngi pkolN tx þ N,
Fig. 1. Guard interval in OFDM signal.
(1)
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where the FFT size is N, which is an integer with power of 2. k and o0 ¼ 2p=N are the normalized sampling instant and angular frequency, respectively. Dðn; lÞ is the information symbol conveyed on the n-th sub-carrier in the l-th transmission symbol period, and it belongs to a modulation constellation with finite elements. Generally Dðn; lÞ can be treated as a random sequence with respect to the sub-carrier number n and symbol period l, i.e., lim
L!1
L 1X ¯ 2 dðn1 n2 Þdðl1 Þ D ðn1 ; lÞDðn2 ; l l1 Þ ¼ D L l¼1
(2)
holds true, where d is the delta function, * denotes the ¯ 2 is the mean square of the conjugate complex, D constellation, n1 and n2 are the sub-carrier numbers, l1 is an arbitrary integer. 2.2. Multipath channel model Assume that the received baseband signal under multipath environment can be approximated by yðkÞ ¼
M X m¼0
r m ðkÞ þ eðkÞ ¼
M X
hm dðk km Þ þ eðkÞ,
(3)
m¼0
where r m ðkÞ is the m-th multipath wave to the receiver, hm is its coefficient, km is the delay tap, and eðkÞ is the additive noise. Correspondingly the channel model can be expressed by z transform as HðzÞ ¼ h0 þ h1 zk1 þ h2 zk2 þ þ hM zkM ,
(4)
1
where z is a backward shift operator, kM is the longest effective delay tap of interference. Substituting z ¼ ejo0 into (4) also yields the frequency response function of the channel model. 2.3. Interference exceeding GI Consider the interference r m ðkÞ with long delay time exceeding GI. In the l-th effective symbol period for
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lN tx pkolN tx þ N, the component of interference with delay time km in Fig. 2 is given by r m ðkÞ ¼ hm dðk km Þ 8 N=2 > P > > hm Dðn; l 1Þejno0 ðkkm þNgi lNtx Þ > > > > n¼N=2þ1 > > > > > < for k km lN tx o Ngi ; ¼ N=2 > P > > > hm Dðn; lÞejno0 ðkkm lNtx Þ > > > n¼N=2þ1 > > > > > : for Ngi pk km lNtx oN:
(5)
Performing N-point FFT of r m ðkÞ within the FFT window for lN tx pkolN tx þN yields the frequency components of r m ðkÞ. For example, the component on the n-th sub-carrier can be given by þN1 1 lNtxX r m ðkÞejno0 ðklNtx Þ N k¼lN tx
1 1 NX ¼ r m ðk þ lN tx Þejno0 k N k¼0 0 1 km N gi 1 N1 X 1@ X jno0 k jno0 k A ¼ r m ðk þ lN tx Þe þ r m ðk þ lNtx Þe N k¼km N gi k¼0 00 1 1 km N 1 N=2 hm Xgi @@ X Dðn1 ; l 1Þejn1 o0 ðkkm þNgi Þ Aejno0 k A ¼ N n1 ¼N=2þ1 k¼0 00 1 1 N=2 N 1 hm X @@ X Dðn1 ; lÞejn1 o0 ðkkm Þ Aejno0 k A þ N k¼k N n1 ¼N=2þ1 m gi 00 1 1 km N gi 1 N=2 hm X @@ X Dðn1 ; l 1Þejn1 o0 ðkkm þNgi Þ Aejno0 k A ¼ N n1 ¼N=2þ1 k¼0 00 1 1 km N gi 1 N=2 hm X @@ X Dðn1 ; lÞejn1 o0 ðkkm Þ Aejno0 k A N n1 ¼N=2þ1 k¼0 00 1 1 N=2 N 1 hm X@@ X jn1 o0 ðkkm Þ A jno0 k A Dðn1 ; lÞe þ e . N k¼0 n ¼N=2þ1 1
Fig. 2. Multipath interference in OFDM system.
(6)
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Following the property of orthogonal basis function of ejno0 k , the last term in (6) can be written as hm ejno0 km Dðn; lÞ.
(7)
Since the frequency component in (7) is only on the n-th sub-carrier, clearly it still holds the carrier orthogonality. Nevertheless, the first and the second terms in (6), which are the summation within the interval 0pkpkm N gi 1 of an incomplete FFT window, yield leakage error whose frequency components contaminate all the subcarriers. Now consider frequency components of all the multipaths. The orthogonal term at n-th sub-carrier becomes to M X
hm ejno0 km Dðn; lÞ ¼ Hðejno0 ÞDðn; lÞ.
(8)
m¼0
On the other hand, the first term in (6) for km 4Ngi yields the ISI denoted as Es ðn; lÞ in the frequency domain. It is the interference from the ðl 1Þ-th symbol period to the l-th period and can be expressed by 0 0 km N 1 N=2 M X hm @ Xgi @ X Es ðn; lÞ ¼ Dðn1 ; l 1Þ N m¼m1 n1 ¼N=2þ1 k¼0 1 1 jn1 o0 ðkkm þNgi Þ A jno0 k A
e
e
,
(9)
where m1 is the smallest integer such that km1 4N gi . Moreover, the effect of the second term in (6) leads to the ICI indicated by Ec ðn; lÞ as follows: 0 0 km N 1 N=2 M X hm @ Xgi @ X Ec ðn; lÞ ¼ Dðn1 ; lÞ N m¼m1 n1 ¼N=2þ1 k¼0 1 1 ejn1 o0 ðkkm Þ Aejno0 k A.
(10)
Let the sum of ISI and ICI be denoted as leakage error Eðn; lÞ. Therefore, the frequency domain expression of the received signal in the l-th symbol period is given by Yðn; lÞ ¼ Hðejno0 ÞDðn; lÞ þ Ec ðn; lÞ þ Es ðn; lÞ þVðn; lÞ, |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}
(11)
ceeding sub-model of the multipath channel be expressed by z transform as
GðzÞ ¼ hm1 zNgi þ1km1 þ hm1 þ1 zNgi þ1km1 þ1 þ þ hM zNgi þ1kM ,
(12)
where hm1 ; hm1 þ1 ; . . . are the coefficients of the multipaths exceeding GI. Then performing inverse Fourier transform of Es ðn; lÞ in (9) and Ec ðn; lÞ in (10) yields the signals of ISI and ICI in the time domain as
s ðk; lÞ ¼ GðzÞds ðk; lÞ, c ðk; lÞ ¼ GðzÞdc ðk; lÞ
(13) (14)
for k ¼ 0; 1; . . . ; N 1, where ds ðk; lÞ and dc ðk; lÞ are the corresponding transmitted signals included in (9) and (10). They can be given by ( dðk 1 þ ðl 1ÞNtx Þ for N þ Ngi þ 1pkp0; ds ðk; lÞ ¼ 0 for k40; ( dc ðk; lÞ ¼
(15) dðk N gi 1 þ lNtx Þ
for N þ N gi þ 1pkp0;
0
for k40:
(16) On the other hand, in the frequency domain, Es ðn; lÞ can be rewritten by N=2 X
Es ðn; lÞ ¼
Dðn1 ; l 1ÞHs ðn; n1 Þ,
(17)
n1 ¼N=2þ1
where Hs ðn; n1 Þ is Hs ðn; n1 Þ 8 M P hm ejn1 o0 ðkm Ngi Þ ejno0 ðkm Ngi Þ > > > > < m¼m1 N 1 ejðnn1 Þo0 ¼ M > > > P km N gi hm ejno0 ðkm Ngi Þ > : N m¼m1
for nan1 ; for n ¼ n1 :
Similarly, Ec ðn; lÞ is approximated by
Eðn;lÞ
where Yðn; lÞ and Vðn; lÞ are the frequency components of the received signal and noise on the n-th sub-carrier. It is clear that the leakage error Eðn; lÞ deteriorates the orthogonality of OFDM sub-carriers and will cause large demodulation error. It is important to remove the influence of Eðn; lÞ to guarantee high communication performance. 3. Properties of leakage error Some important properties of ICI, ISI and the leakage error are revealed statistically in this section. 3.1. Expressions of ICI and ISI It is seen that only the coefficients hm for km 4N gi remain in (9) and (10). Correspondingly, let the GI ex-
Ec ðn; lÞ ¼
N=2 X
Dðn1 ; lÞHc ðn; n1 Þ,
(18)
n1 ¼N=2þ1
where Hc ðn; n1 Þ is given by Hc ðn; n1 Þ 8 M P hm ejn1 o0 km ejno0 km ejðn1 nÞo0 Ngi > > > > < m¼m1 N 1 ejðnn1 Þo0 ¼ M > > > P km N gi hm ejno0 km > : N m¼m1
for nan1 ; for n ¼ n1 :
From (17) and (18), the data in the frequency domain fulfil the following expression: HDðlÞ ¼ YðlÞ EðlÞ VðlÞ ¼ YðlÞ Es ðlÞ Hc DðlÞ VðlÞ,
(19)
ARTICLE IN PRESS L. Sun et al. / Signal Processing 89 (2009) 1589–1601
where YðlÞ, DðlÞ, EðlÞ, Es ðlÞ and VðlÞ are the vectors of FFT coefficients of the received signal, the source information symbols, leakage error, ISI and noise in the l-th period, respectively, and 2
2 6 6 Hc ¼ 6 4
..
0
0
.
HðejN=2o0 Þ Hc ðN=2 þ 1; N=2 þ 1Þ
Hc ðN=2 þ 1; N=2Þ
3
.. .
..
.. .
Hc ðN=2; N=2 þ 1Þ
7 7 7. 5
.
Hc ðN=2; N=2Þ
L 1X D ðn; lÞEs ðn; lÞ L!1 L l¼1
lim
n1 ¼N=2þ1
¯ Hðe
1PL l¼1 ðD ðn; lÞYðn; lÞÞ Þ ¼ lim L L!1 1PL ðD ðn; lÞDðn; lÞÞ L l¼1 1PL l¼1 ðD ðn; lÞEc ðn; lÞÞ jno0 Þ þ lim L ¼ Hðe L!1 1PL ðD ðn; lÞDðn; lÞÞ L l¼1 m M 1 1 X X km N gi ¼ hm ejnkm o0 þ 1 hm ejnkm o0 . N m¼m1 m¼0 (23)
¯ jno0 Þ as h¯ 0 ; h¯ 1 ; h¯ 2 ; . . ., Denote the IFFT coefficients of Hðe then the coefficients hm can be obtained by
Since Es ðn; lÞ in (17) is only related to the information symbols in the ðl 1Þ-th symbol period, then from (2), Dðn; lÞ and Es ðn; lÞ are uncorrelated, i.e.,
N=2 X
jno0
7 7 7, 5
3.2. Statistical properties of ICI and ISI
¼
as follows:
3
HðejðN=2þ1Þo0 Þ
6 6 H¼6 4
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! L 1X D ðn; lÞDðn1 ; l 1Þ Hs ðn; n1 Þ ¼ 0. L!1 L l¼1
( hm ¼
h¯ m Nh¯ m =ðN Ngi þ km Þ
for 0pkm pNgi ; for km 4N gi :
(24)
From (23) and (24), it can be seen that it is possible to estimate Hðejno0 Þ by using the properties of the leakage error even when the channel has long multipath interferences.
lim
4. Channel identification
(20) Moreover, multiplying Ec ðn; lÞ in (18) by D ðn; lÞ and using the results in (2) lead to the following result: L 1X D ðn; lÞEc ðn; lÞ L!1 L l¼1
lim
¼
N=2 X n1 ¼N=2þ1
! L 1X lim D ðn; lÞDðn1 ; lÞ Hc ðn; n1 Þ L!1 L l¼1
¯2 ¯ 2 Hc ðn; nÞ ¼ D ¼ D
M X km Ngi hm ejnkm o0 . N m¼m1
(21)
Following (21), it is clear that the longer the delay tap km , the greater the leakage error is. Therefore, the symbol equalization or interference compensation becomes more difficult. Furthermore, from (11), the following equation: L L 1X 1X D ðn; lÞYðn; lÞ ¼ Hðejno0 Þ D ðn; lÞDðn; lÞ L l¼1 L l¼1
þ
L 1X D ðn; lÞðEs ðn; lÞ þ Ec ðn; lÞÞ L l¼1
þ
L 1X D ðn; lÞVðn; lÞ L l¼1
(22)
holds true. Then by using the results of (2), (20) ¯ jno0 Þ defined in (23) can be obtained by (22) and (21), Hðe
When several preamble or training symbols are available, (23) and (24) can give a batch channel identification only with computational complexity of OðNÞ. When the channel model is time-varying, or the successive training symbols are not available for channel identification, a new channel identification algorithm can be developed by making use of spectral periodograms whose ISI and ICI are compensated by the leakage error replica. If the training symbols are unavailable, except the pilot symbols at pilot carriers, the information symbols have to be estimated from the received signals for channel identification. From (19), the symbol vector DðlÞ in the lth period may be estimated by ^ ¼ ðH ^ þH ^ c Þ1 ðYðlÞ Es ðlÞÞ. DðlÞ
(25)
^ þH ^ c Þ and However, the computation of inverse of ðH ^ updating Hc using the latest channel estimation are timeconsuming. In the proposed algorithm a simple estimation method with low computational complexity is considered. As mentioned in Section 3, the components of ICI and ISI contaminate all the sub-carriers, and the frequency response function Hðejno0 Þ varies remarkably when the channel has long multipath interferences, as a result, neither the interpolation method nor equalization using the frequency selective diversity can yield satisfactory result. For example, at the n-th sub-carrier where jHðejno0 Þj is small, the orthogonal component in (6) attenuates to such a small value that symbol equalization becomes fragile to the noise and leakage error, even the error correction techniques might fail to correct the equalization errors at the sub-carriers with small magni-
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tude of frequency response. In order to overcome these difficulties, the diversity of multiple receiver antennas is used in the proposed identification algorithm. Let the total number of antenna elements be Q , then the received signal at the q-th antenna can be denoted as yq ðkÞ, where 1pqpQ . Correspondingly, the frequency response function from the transmitter to the q-th antenna is Hq ðejno0 Þ, and its GI exceeding part expressed by z transform is Gq ðzÞ. Channel identification is performed iteratively for every symbol period. In the ðl 1Þ-th iteration, denote ^ the estimates of information symbol as Dðn; l lÞ, the jno0 ^ frequency response as Hq ðe ; l 1Þ, and its GI exceeding ^ q ðz; l 1Þ, respectively. These estimates are used part as G in the next iteration, thus the mitigation of the influence caused by these estimation errors is considered in the proposed algorithm. In the l-th iteration, the information symbol is estimated first by removing the affection of ISI. 4.1. Initial estimation of information symbols In the l-th symbol period, let the frequency component of yq ðkÞ at the n-th sub-carrier be denoted by Y q ðn; lÞ. It is calculated from FFT of yq ðkÞ within the FFT window ^ lN tx pkolN tx þ N. Using the estimation of Dðn; l 1Þ in the ðl 1Þ-th symbol period, d^ s ðk; lÞ can be obtained by (15). Therefore, s;q ðk; lÞ can be estimated by (13) using G^ q ðz; l 1Þ and d^ s ðk; lÞ in the time domain. Furthermore, the ISI replica E^ s;q ðn; lÞ to the q-th receiver antenna is obtained by FFT of s;q ðk; lÞ. Thus the initial estimate of ^ Dðn; lÞ can be given by removing ISI Y q ðnÞ ðn; lÞ E^ s;qmax ðnÞ ðn; lÞ ^ , Dðn; lÞ ¼ max ^ q ðnÞ ðejno0 ; l 1Þ H max
(26)
where qmax ðnÞ is the antenna number determined by ^ q ðejno0 ; l 1ÞjÞ. qmax ðnÞ ¼ arg maxðjH
(27)
q
^ q ðnÞ ðejno0 ; l 1Þj takes the maximum of It is seen that jH max ^ q ðejno0 ; l 1Þj for 1pqpQ at the n-th sub-carrier so that jH the strongest orthogonal component is used to estimate the information symbol Dðn; lÞ. ^ On the other hand, the estimate Dðn; lÞ in (26) can ^ hd ðn; lÞ following further be modified by hard decision D the constellation of Dðn; lÞ [16]. 4.2. Estimation of leakage error ^ ^ hd ðn; lÞ, then d^ c ðk; lÞ dðkÞ is calculated through IFFT of D ^ q ðz; l can be obtained by (16). Using the estimation of G 1Þ and d^ c ðk; lÞ, c;q ðk; lÞ can be estimated by (14) in the time domain first. Consequently, E^ c;q ðn; lÞ can be calculated from FFT of c;q ðk; lÞ, and the leakage error E^ q ðn; lÞ can also be obtained by E^ s;q ðn; lÞ þ E^ c;q ðn; lÞ. 4.3. Re-estimation of information symbols It is seen that the ICI effect still remains in the initial ^ estimation of Dðn; lÞ in (26). By using the estimate E^ q ðn; lÞ obtained in Section 4.2, the more accurate estimation of
^ information symbols Dðn; lÞ is obtained as follows: Y q ðnÞ ðn; lÞ E^ qmax ðnÞ ðn; lÞ ^ Dðn; lÞ ¼ max . ^ q ðnÞ ðejno0 ; l 1Þ H
(28)
max
^ hd ðn; lÞ can also be updated. FurtherCorrespondingly D more, the new E^ q ðn; lÞ can be modified by using the ^ ^ hd ðn; lÞ similarly as Section 4.2. The updated Dðn; lÞ or D ^ estimation procedure of Eq ðn; lÞ and Dðn; lÞ can be repeated several times to reduce their estimation errors, where the strongest component is used for symbol estimation.
4.4. Estimation of frequency response function The channel is identified from the frequency compo^ nent Y q ðn; lÞ, the symbol estimate Dðn; lÞ and the replica of ^ lÞ. Consequently, it is important to leakage error Eðn; remove the influence caused by the noise term in ^ lÞ and Dðn; ^ Y q ðn; lÞ, the estimation errors of Eðn; lÞ. The phases of noise and estimation errors are usually random, then their influence can be mitigated through the smoothing effect of spectral periodograms [17]. In the lth iteration, the spectral periodograms DDðejno0 ; lÞ and DYðejno0 ; lÞ are defined as follows:
^ ðn; lÞD ^ hd ðn; lÞ, DDðejno0 ; lÞ ¼ ll DDðejno0 ; l 1Þ þ D hd jno0
DY q ðe
jno0
; lÞ ¼ ll DY q ðe
(29)
; l 1Þ
^ ðn; lÞðY q ðn; lÞ E^ q ðn; lÞÞ, þD hd
(30)
where ll is a forgetting factor over the range of 0oll o1. It is seen that the effects of noise and estimation errors are reduced when l becomes large [17]. Using the estimates DDðejno0 ; l 1Þ and DY q ðejno0 ; l 1Þ in the ðl 1Þ-th itera^ hd ðn; lÞ, Y q ðn; lÞ and E^ q ðn; lÞ in tion, as well as the estimates D the l-th iteration, the estimates of (29) and (30) are obtained. Thus the estimate of frequency response func^ q ðejno0 ; lÞ can be given by tion H jno0 ; lÞ ^ q ðejno0 ; lÞ ¼ DY q ðe . H DDðejno0 ; lÞ
(31)
^ q ðz; lÞ can be updated by (12) using IFFT of Moreover, G ^ q ðejno0 ; lÞ. Then let l ¼ l þ 1 for the next iteration. H As mentioned previously, the frequency response function varies remarkably when the interferences have long delay taps. As a result, the sidelobes often occur in the impulse response h^ m of channel model due to noise, estimation errors of the information symbols and leakage error, etc. Through setting a threshold between mainlobes and sidelobes, the sidelobes can be removed to improve the convergence performance of channel identification and BER performance of symbol estimation [18]. In this paper, the algorithm is developed for the case where the number of sub-carriers in one symbol is the same as the FFT size. Since it is difficult to design a low pass filter with sharp cut-off in the communication circuit without disturbing the adjacent channels, the number of sub-carriers is often smaller than the FFT size. In such communication systems, the OFDM signals are restricted within a specified band, and make the channel identification difficult [19]. The details of channel identification
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techniques for the limited signal band had been discussed in [20,21]. 4.5. Procedure of channel identification In the identification algorithm, the initial estimation of ^ Dðn; lÞ is calculated from the received signal compensated ^ lÞ is estimated, by E^ s ðn; lÞ first, next the leakage error Eðn; ^ then Dðn; lÞ is updated by using the compensation of Ec ðn; lÞ. Finally the channel frequency response is estimated from the spectral periodograms of the transmitted and received signals. The procedure of proposed algorithm can be summarized as follows: Step 1: Let the initial values of DY q ðejno0 ; 0Þ, DDðejno0 ; 0Þ ^ q ðz; 0Þ, and ^ q ðejno0 ; 0Þ, G be 0. Choose the initial values of H let the iteration number be l ¼ 1. Step 2: Calculate Y q ðn; lÞ through FFT of the received signal yq ðkÞ within the FFT window lN tx pkolN tx þ N. Step 3: Calculate ds ðk; lÞ by (15), and Es;q ðn; lÞ by FFT of s;q ðk; lÞ in (13). ^ Step 4: Estimate Dðn; lÞ by (26), and hard decision ^ hd ðn; lÞ, respectively. D Step 5: Calculate dc ðk; lÞ in (16), furthermore estimate the replica of ICI through FFT of ^ c;q ðk; lÞ in (14). ^ ^ hd ðn; lÞ, Step 6: Update Dðn; lÞ by (28) and hard decision D respectively. Step 7: Calculate DY q ðejno0 ; lÞ and DDðejno0 ; lÞ by (29) and (30), respectively. ^ q ðejno0 ; lÞ from (31) and update the GI Step 8: Estimate H ^ q ðz; lÞ by (12). Let l ¼ l þ 1, then return to exceeding part G Step 2 to repeat the iterations. The procedure in the l-th iteration is illustrated in Fig. 3. 4.6. Features of the proposed algorithm The features of the proposed algorithm are summarized as follows. (1) Periodograms can smooth DY q ðejno0 ; lÞ and DDðejno0 ; lÞ before their division, so that the periodograms based identification algorithm can reduce the estimation ^ q ðejno0 ; lÞ caused by the estimation errors of error of H ^ Dðn; lÞ and E^ q ðn; lÞ, or the noise term in Y q ðn; lÞ [17]. (2) Unlike some other identification algorithms in the time domain whose performance depends on the total number of parameters to be estimated, the proposed algorithm estimates the frequency response function per sub-carrier from the spectral periodograms and shows good convergence performance even for long impulse response of channel. (3) By virtue of the diversity of multiple antennas, the strongest orthogonal component is chosen from Q received signals to estimate the information symbols, and it improves the performance of channel identification and symbol estimation significantly compared with the single antenna case. It is noticed that the purpose of using multiple antennas is just choosing the strongest orthogonal component at each sub-carrier, the performance of the
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proposed algorithm does not depend on the total number of antenna elements as much as the conventional spatial equalizers based on antenna diversity. Though some maximum ratio combining techniques (MRC) work well in many multi-antenna cases [22], their performance may be weakened due to the correlation of leakage error Eðn; lÞ with Dðn; lÞ. If the channel has severe leakage error, the proposed algorithm should be used to reduce the identification error so that the correlation of Eðn; lÞ with Dðn; lÞ is removed from the received signal, then the maximum ratio combination or some correcting codes can be applied to obtain better BER performance. (4) The forgetting factor is used in periodogram estimation so that the algorithm can also deal with slow time-varying channels. A small forgetting factor has adaptability to quick channel variation and large BER, whereas a large one is used in low SNR environment for spectral smoothing. (5) When the interference exceeding GI is strong, the convergence of channel identification and BER performance can be improved by appropriately choosing the sidelobe threshold. (6) The main computation only requires FFT, multiplication, division of periodograms, therefore, the algorithm has less computational complexity and can easily be utilized in the practical applications. In each iteration, besides the computation of FFT, the proposed algorithm needs the following calculations: 2ðkM N gi Þ2 q multiplications to estimate ICI and ISI, 2N divisions for initial estimation and re-estimation of information symbols, ðq þ 1ÞN multiplications to estimate periodograms, qN divisions to estimate the frequency response functions. By contrast with the computational complexity of RLS algorithm, besides the estimation of information symbols in RLS algorithm, the recursive 2 identification requires about OðkM Ntx Þ multiplications for one symbol period. It is clear that the proposed identification algorithm has less complexity than RLS, especially for large kM . Though LMS only requires Oð2kM Ntx Þ multiplications for channel identification, its convergence rate is much slower than RLS [3]. 5. Numerical simulation examples The 3GPP 2.5 MHz OFDM transmission with 16QAM modulation is used in the examples where the FFT size N ¼ 256, GI length N gi ¼ 64 [1]. Moreover, the number of sub-carriers is 256, the total number of receiver antennas is Q ¼ 2, and their distance is 12 of the wave length. The noise is assumed as an additive Gaussian white noise. 5.1. Example of a channel with three multipath interferences Besides the direct wave, there are three multipath interferences in the transmission channel [7,8]. The coefficients of the waves are illustrated in Table 1. Let the SNR ¼ 25 dB. Six cases are considered: the case where only one pilot sub-carrier is known to remove the
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Fig. 3. Block diagram of l-th iteration.
ambiguity of channel identification and symbol estima1 1 1 1 tion, the cases where the scattered pilot rates are 16 , 8, 4, 2, respectively, and the case where training symbols are available for identification. At the pilot carriers, the values ^ of Dðn; lÞ are given by the corresponding true values of Dðn; lÞ, while at the other sub-carriers, the values of Dðn; lÞ have to be estimated from the received signals. Channel identification is started from the initial values of
^ ðz; 0Þ ¼ 0. The square error of channel ^ jno0 ; 0Þ ¼ 1, G Hðe identification Er H , which is defined as Er H ¼
P PN1 ^ jno0 Þ Hq ðejno0 Þj2 n¼0 jH q ðe q P PN1 jno0 Þj2 n¼0 jH q ðe q
(32)
is illustrated in Fig. 4, and BER curves of the estimated symbols are illustrated in Fig. 5, respectively. They show
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Table 1 Simulation conditions. Wave
Power (dB)
Phase
DOA
Delay tap
Direct Interference 1 Interference 2 Interference 3
0 3 5 5
0 p=6 p=4 p=3
p=8 p=4 p=3 p=6
0 3N gi =4 5N gi =4 7N gi =4
Fig. 6. BER comparison of maximum ration combination and proposed algorithm (average of 30 simulation runs).
Fig. 4. Channel identification error (average of 30 simulation runs).
Fig. 5. BER of estimated symbols (average of 30 simulation runs).
that the algorithm works well even for few pilot subcarrier cases. In Fig. 5, BER of the estimated symbols decreases to 0 after several iterations, whereas it is larger than 0.5 at low pilot rate in the first iteration due to the initial value of channel identification is quite different from the true one. It implies that the algorithm converges even from the severe initial conditions. The BER curves are plotted in Fig. 5 when the perfect channel estimate is used for symbol estimation. It is seen that the good BER performance can be guaranteed if the influence of ISI and ICI caused by the long multipath interferences are compensated by the replica of leakage error.
Since RLS algorithm is often used for channel identification in the previous methods, the results of RLS algorithm are also shown in Figs. 4 and 5 for comparison with the proposed algorithm. They are obtained under the same simulation conditions. In RLS algorithm, the recursion is performed per sampling instant to estimate the parameters of hm in (3) by using the latest samples of yðkÞ and dðkÞ, thus RLS updates the estimates N tx ¼ 320 times during 1 iteration in the proposed algorithm. In Fig. 4, if several successive training periods are available, i.e., the true values of dðkÞ can be used for channel identification directly, it is seen that RLS algorithm can yield small error by using the true dðkÞ while its computational load is heavier than that of the proposed algorithm. ^ However, when the training symbols are unavailable, dðkÞ has to be estimated for channel identification, the error of ^ dðkÞ deteriorates the identification performance of RLS algorithm. For example, when the pilot rate is 12, the convergence of channel identification and the symbol estimation used RLS estimation is much slower than that of the proposed algorithm, and its performance becomes very poor when the pilot rate is 14, whereas the proposed algorithm works well since it uses spectral periodograms. The BER curves of symbol estimation are plotted in Fig. 6 by choosing the strongest orthogonal components, and by maximum ratio combining techniques. In the low pilot rate situations, less of the leakage error is removed due to the large error of channel identification in the first several iterations, hence the convergence rate of maximum ratio combination becomes slow due to the correlation of Eðn; lÞ and Dðn; lÞ.
5.2. Channel identification vs. interference power Let the pilot rate be 18, and the power of interference 3 be changed from 0 to 20 dB, the other conditions be the same as those in Section 5.1. Er H and BER vs. interference power are illustrated in Figs. 7 and 8, respectively. Though there is strong multipath interference exceeding GI, both
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Fig. 7. Channel estimation error vs. interference power.
0pnp20 is illustrated in Fig. 10. It is shown that at the sub-carriers marked by circle, the symbols are difficult to be estimated by frequency selective diversity of single antenna, since low magnitude of Hq ðejno0 Þ leads to weak orthogonal component in the received signal. The curves of Er H and BER are plotted in Fig. 11, where BER is large for single antenna case, whereas the errors for Q ¼ 2 and 4 are almost the same since the strong orthogonal components are used in both of the two cases. It also shows that though BER is large in the first several iterations, the error of channel identification based on periodograms has been decreased to such a low level as to provide an effective channel model for equalization. 5.5. Identification of time-varying channel Fig. 8. BER vs. interference power.
Er H and BER are decreased to a considerable low level after just about 10 iterations. 5.3. Channel identification vs. SNR Let the pilot rate be 18, and SNR be changed from 10 to 40 dB, the other conditions be the same as those in Section 5.1. Er H plotted in Fig. 9 shows that the proposed algorithm converges even for low SNR conditions. 5.4. Channel identification vs. total number of antennas Let the pilot rate be 18. In order to investigate the influence of Q on the algorithm performance, the total number Q of antennas is chosen as 1, 2 and 4, respectively. The other simulation conditions are the same as those in Section 5.1. The relative magnitude of Hq ðejno0 Þ for
Let the pilot rate be 18, the power, phase and DOA of Interference 3 be changed at every 10 symbol periods so that the channel is time-varying. The power profile is shown in Fig. 12. For the variation of channel, the forgetting factor ll ¼ 0:25 is used and the results of Er H , BER are illustrated in Fig. 13. Though the channel varies quickly, the prompt reduction of errors shows that the proposed algorithm can also work well for time-varying channels. 5.6. Channel identification of COST 207 model Let the pilot rate be 18. Assume that the delay profile of the multipath in a hill area is a COST 207 model [23]. There are 11 waves with delay time 0pkm p10 and power ekm =2:5 , 21 waves with delay time 40pkm p60 and power 0:7079eðkm 40Þ=4 , 15 waves exceeding GI with delay time 72pkm p96 and power 0:5623eðkm 72Þ=3:6, and m ¼ 0 denotes the direct wave. The total power of interferences
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Fig. 9. Channel estimation error vs. SNR.
Fig. 10. Illustration of jHq ðejno0 Þj vs. Q .
Fig. 11. Channel estimation error and BER vs. Q .
Fig. 12. Power variation at interference 3. : jump points.
exceeding GI is 1:29 dB. The DOA of multipath waves are generated randomly, and the other conditions are the same as in Section 5.1. As illustrated in Fig. 14, the received signal suffers from strong multipath interferences, as a result, BER is about 0.45 without interference compensation. At the first several iterations, small ll and large sidelobe threshold are used to mitigate the influences of high BER of estimated symbols and sidelobes. With BER decreasing ll is increased to smooth the periodograms whereas the sidelobe threshold is decreased to handle weak multipath interferences. In the simulation ll is assumed to be ll ¼ minf0:075 1:05l ; 0:75g, and the sidelobe threshold is assumed to be maxf0:1 0:98l ; 0:005g. The corresponding ErH and BER of estimated symbols are shown in Fig. 15. Though the channel has strong multipath interferences,
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Fig. 13. Channel estimation error and BER of recovered signal.
Fig. 14. Relative power of interferences.
Fig. 15. Channel estimation error and BER of estimated symbols (average of 30 simulation runs).
ErH decreases from 1.0 to 0.003, BER decreases from 0.45 to 0 in about 25 iterations. The comparisons of the proposed algorithm with the conventional identification algorithms of RLS, LMS are also examined by using training symbols. The results of ErH are shown in Fig. 16 where the transverse axis is the
Fig. 16. Comparison of estimation errors of proposed algorithm, RLS and LMS.
sample number. RLS and LMS are performed per sampling instant, while the iteration is executed once per symbol period using 256 samples in the proposed algorithm. Therefore, RLS and LMS update their estimates 320 times during 1 iteration of the proposed algorithm. Since the training symbols can be used for identification without error of symbol estimation, the sidelobe threshold can be chosen as a very small value. In this simulation, the sidelobe threshold is 0.005, ll ¼ 0:75, and the length kM of channel impulse response is chosen as 128 taps for identification. In Fig. 16, ErH of RLS is too large to be kept within the axis range of ordinate in the first 150 recursions due to its sensitivity to the initial conditions. Moreover, not only the computational load of RLS is heavy for long impulse response, but the condition number of covariance matrix of regression vector is so large that the estimation is fragile to the noise and round-off errors of numerical computation, while the convergence rate of LMS is slow, as shown in Fig. 16. In proposed algorithm, ErH is reduced to 104 in only four iterations. In each iteration, besides the estimation for information symbols, compensation of leakage error requires 2ðkM Ngi Þ2 ¼ 8192 multiplications, periodogram estimation and channel identification require about 3N ¼ 768 multiplications in the proposed algorithm, whereas Ntx recursions for one 2 symbol period in RLS require 3kM Ntx 107 multiplications. ErH decreases to 0.005 by just one iteration in the proposed algorithm with low computation load, whereas RLS requires about 200 recursions to obtain the similar accuracy. Though the computation load is low in LMS algorithm, its convergence rate is so slow that it takes about more than 640 iterations to reach ErH 0:005. Moreover, if the training symbols are unavailable, the performance of RLS and LMS becomes worse due to the errors of symbol estimation, as shown in Fig. 4. 6. Conclusions Channel identification and its applications to OFDM communications with interferences exceeding GI have been studied in this work. The proposed algorithm can be
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implemented in the frequency domain by FFT, consequently it has low computational complexity, which does not depend on the length of interference delay time too much. Different from other identification or equalization algorithms, the proposed algorithm utilizes diversity of multiple antennas for symbol estimation, and spectral periodograms to estimate the frequency response function. The former allows the algorithm to estimate information symbols from the strongest orthogonal components of multiple received signals at the receivers, and the latter reduces the effect of noise as well as errors of symbol estimation, therefore the proposed algorithm has high convergence rate and can work under the strong multipath interferences. References [1] 3GPP TR 25.814 (Release 7), 3rd Generation Partnership Project hhttp://www.3gpp.org/ftp/Specs/html-info/TSG-WG–r1.htmi, 2006. [2] Y. Karasawa, N. Gejoh, T. Izumi, Modeling and analysis of OFDM transmission characteristics in Rayleigh fading environment in which the delay profile exceeds the guard interval, IEICE Trans. Commun. E88-B (7) (2005) 3020–3027. [3] J. Balakrishnan, R.K. Martin, C.R. Johson Jr, Blind, adaptive channel shortening by sum-squared auto-correlation minimization (SAM), IEEE Trans. Signal Process. 51 (12) (2003) 3086–3093. [4] G. Ysebaert, K. Vanbleu, G. Cuypers, M. Moonen, T. Pollet, Combined RLS–LMS initialization for per tone equalizers in DMT-receivers, IEEE Trans. Signal Process. 51 (7) (2003) 1916–1927. [5] K. Vanbleu, G. Ysebasert, G. Cuypers, M. Moonen, Adaptive bit rate maximizing time-domain equalizer design for DMT-based systems, IEEE Trans. Signal Process. 54 (2) (2006) 483–498. [6] R.K. Martin, et al., Unification and evaluation of equalization structures and design algorithms for discrete multitone modulation systems, IEEE Trans. Signal Process. 53 (10) (2005) 3880–3894. [7] S. Hori, N. Kikuma, N. Inagaki, MMSE adaptive array suppressing only multipath waves with delay times beyond the guard interval for fixed reception in the OFDM systems, IEICE Trans. Commun. J86B (2003) 1934–1940. [8] K. Higuchi, H. Sasaoka, Adaptive array suppressing inter symbol interference based on frequency spectrum in OFDM systems, IEICE Trans. Commun. J87-B (2004) 1222–1229.
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