A novel simplified channel tracking method for MIMO–OFDM systems with null sub-carriers

ARTICLE IN PRESS

Signal Processing 88 (2008) 1002–1016 www.elsevier.com/locate/sigpro

A novel simplified channel tracking method for MIMO–OFDM systems with null sub-carriers Hyoung-Goo Jeona, Erchin Serpedinb, a

Department of Information Communication Engineering, Dongeui University, Busan, Republic of Korea Department of Electrical and Computer Engineering, Texas A&M University College Station, TX 77843-3128, USA

b

Received 10 April 2007; received in revised form 7 August 2007; accepted 19 October 2007 Available online 26 November 2007

Abstract This paper proposes an efficient scheme to track the time variant channel induced by multi-path Rayleigh fading in mobile wireless multiple input multiple output–orthogonal frequency division multiplexing (MIMO–OFDM) systems with null sub-carriers. In the proposed method, a blind channel response predictor is designed to cope with the time variant channel. The proposed channel tracking scheme consists of a frequency domain estimation approach that is coupled with a minimum mean square error (MMSE) time domain estimation method, and does not require any matrix inverse calculation during each OFDM symbol. The main attributes of the proposed scheme are its reduced computational complexity and good tracking performance of channel variations. The simulation results show that the proposed method exhibits superior performance than the conventional channel tracking method [Y.G. Li, N. Seshadri, S. Ariyavisitakul, Channel estimation for OFDM systems with transmitter diversity in mobile wireless channels, IEEE J. Sel. Areas Commun. 17 (March 1999) 461–471] in time varying channel environments. At a Doppler frequency of 100 Hz and bit error rates (BER) of 104 , signal-to-noise power ratio (E b =N 0 ) gains of about 2.5 dB are achieved relative to the conventional channel tracking method [Y.G. Li, N. Seshadri, S. Ariyavisitakul, Channel estimation for OFDM systems with transmitter diversity in mobile wireless channels, IEEE J. Sel. Areas Commun. 17 (March 1999) 461–471]. At a Doppler frequency of 200 Hz, the performance difference between the proposed method and conventional one becomes much larger. r 2007 Elsevier B.V. All rights reserved. Keywords: Channel; Estimation; MIMO; OFDM; Tracking; Fading; Doppler

1. Introduction The multiple input multiple output (MIMO) technique represents an efficient method to increase data transmission rate without increasing bandCorresponding author. Tel.: +1 979 458 2287;

fax: +1 979 862 4630. E-mail addresses: [email protected] (H.-G. Jeon), [email protected] (E. Serpedin).

width since different data streams are transmitted from each transmit antenna [1]. Recently, orthogonal frequency division multiplexing (OFDM) has been effectively used for transmitting high speed data in multi-path fading channel environments. In OFDM, the high speed data stream is processed in parallel and transmitted by N (in general, a power of 2) orthogonal sub-carriers. The high spectral efficiency of OFDM and its robustness to multipath fading channel environments are the main

0165-1684/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2007.10.017

ARTICLE IN PRESS H.-G. Jeon, E. Serpedin / Signal Processing 88 (2008) 1002–1016

reasons for its widespread usage in high bit-rate transmissions such as digital audio broadcasting (DAB), digital video broadcasting (DVB) and wireless local area networks (WLAN) [2]. The combined transmission method of MIMO–OFDM has attracted a lot of attention as a new data transmission method in high speed data rate systems. In MIMO–OFDM receivers, the estimated channel frequency response is used to separate the mixed signals received from multiple antennas. An important aspect is the fact that the performance of MIMO–OFDM receivers highly depend on the accuracy of the channel estimator. Thus far, numerous studies for channel estimation in MIMO–OFDM systems have been reported (see e.g., [3–10]). Among the most notable results, Li proposed a MMSE channel estimation method [3] that exhibits good accuracy. However, this method is computationally very complex due to the inverse matrix calculation. In [6], by exploiting the correlation of the subcarrier responses, Minn et al. proposed a low complexity channel estimation method which reduced the inverse matrix size by half. However, Minn et al. method may cause channel estimation errors in large delay spread environments. Li also proposed a simplified channel estimation method which required no matrix inversion [4]. However, as mentioned in [6], if null subcarriers are used, Qii ½n of [4] would not be the identity matrix, and there may be some performance degradation according to the number of null subcarriers. Since real OFDM systems have null subcarriers in the guard band, a low complexity channel estimation method considering null sub-carriers is still needed. As a possible solution to these problems, we are proposing a novel simplified channel tracking method that relies on a blind channel predictor. The proposed method does not require prior channel information or matrix inversion calculation at all. In addition, the proposed method can effectively track the nonlinear time varying channel by using a piecewise linear model. To reduce its computational complexity while maintaining a good tracking accuracy, the proposed channel tracking scheme is built by coupling a frequency domain estimation approach with an MMSE time domain channel estimation approach. The remainder of this paper is organized as follows. In Section 2, the MIMO–OFDM system and channel model are briefly described. Section 3 introduces the proposed channel tracking method.

1003

In Section 4, the mean square error (MSE) and computational complexity of the proposed channel tracking scheme are assessed. The performance of the proposed method is corroborated by computer simulations in Section 5. Finally, Section 6 concludes the paper. 2. Channel and MIMO–OFDM system description The channel impulse response of the mobile wireless channel [3,6] can be modeled by hðt; tÞ ¼

L1 X

ak ðtÞdðt  tk Þ,

(1)

k¼0

where ak ðtÞ denotes the complex gain of the kth path, tk represents the delay of the kth path, L is the number of the multi-paths in the channel and dðtÞ stands for the impulse function. The frequency response at time t is given by Z 1 L1 X Hðt; f Þ9 hðt; tÞ ej2pf t dt ¼ ak ðtÞ ej2pf tk . 1

k¼0

ð2Þ Considering the motion of the mobile station, the path gains ak ðtÞs are modeled to be independent wide-sense stationary, narrow band complex Gaussian processes and to have different average powers s2k . With tolerable leakage, the channel frequency response can be expressed as [3] H½l; k9HðlðT f þ T g Þ; kDf Þ ¼

L 0 1 X

h½l; nW kn N ,

(3)

n¼0

where h½l; n9hðlðT f þT g Þ; nts Þ, W N 9 expðj2p= NÞ, L0 stands for the channel length and depends on the time dispersion of the wireless channel, N is the number of tones and the fast Fourier transformation/inverse fast Fourier transformation (FFT/ IFFT) size, T f and Df denote the OFDM symbol period and sub-carrier spacing of the OFDM system, respectively, T g represents the guard time ðT g 9T f =4Þ and ts is the sample interval given by ts ¼ 1=ðNDf Þ. In a MIMO–OFDM system, the output signal at each Rx (receive) antenna is a mixed signal consisting of the data streams coming from all Tx (transmit) antennas. If the channel response does not change during one OFDM symbol and the cyclic prefix is longer than the channel response length, the receive signal at the jth Rx antenna can be expressed in the frequency domain

ARTICLE IN PRESS H.-G. Jeon, E. Serpedin / Signal Processing 88 (2008) 1002–1016

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the variable a, respectively. In this system model, time synchronization is assumed to be perfect, and the maximum likelihood (ML) detection method is used. Tx antennas transmit a long preamble consisting of two training symbols before data transmission mode, as WiBro and WLAN systems do [2,12]. It is assumed that in the data transmission mode, N d OFDM symbols are transmitted consecutively in each Tx antenna. For unbiased performance comparison of channel tracking algorithms, no channel coding is used.

as follows: Rj ½l; k ¼

Nt X

H ij ½l; kX i ½l; k þ W j ½l; k,

i¼1

j ¼ 1; . . . ; N r ; 0pkpN  1,

ð4Þ

where H ij ½l; k is the channel frequency response corresponding to the kth sub-carrier and the lth OFDM symbol transmitted between the ith Tx antenna ði ¼ 1; . . . ; N t Þ and the jth Rx antenna. Also, let N, N r and N t denote the number of subcarriers, the number of Rx antennas, and the number of Tx antennas, respectively. X i ½l; k denotes the data transmitted from the ith Tx antenna on the kth sub-carrier at the lth OFDM symbol. W j ½l; k represents the additive white Gaussian noise (AWGN) at the jth receiver antenna, with zero mean and variance s2n , and is assumed to be uncorrelated for different j’s, k’s, or l’s. Under the assumption that the channel stays constant within one OFDM symbol duration but the channel changes from symbol to symbol, we will develop a channel tracking scheme with improved performance relative to the conventional scheme [3]. The computer simulations, which assume realistic Rayleigh fading conditions (that are not limited to the block fading assumption) [11], corroborate the superior performance of the proposed channel tracking scheme. The indices n and k denote time and frequency-domain indices, respectively. The ~ a^ and a denote the temporally estimated symbols a, value, the estimated value and the predicted value of

3. Proposed channel estimation method Since the wireless channel is time-variant, it is necessary to track the channel response continuously. In addition, since the received signal at each Rx antenna in MIMO–OFDM systems is a multiple-input single-output (MISO) signal, a time domain channel estimation cannot be directly applied on the received signal. In this paper, we propose a low complexity adaptive channel estimation method based on a blind channel prediction scheme that is suitable for time variant channel environments. The conceptual block diagram of the proposed channel tracking scheme is shown in Fig. 1. Before channel estimation, the frequency domain MISO signal received at the jth Rx antenna (Rj ½l; k) is converted into the desired single-input single-output (SISO) signal (See Section 3.2 for the definition of desired SISO signal) by canceling the

Delay device

rj [l,n]

FFT

Rj [l,k]

MISO to SISO conversion

MISO

S˜ij [l,k]

Freq. domain Channel Est.

H ˜ ij [l,k]

Time domain Channel Estimation

IFFT

SISO

X˜i [l,k] Pre-ML detector

Hij [l,k] Channel predictor

Fig. 1. Block diagram of the proposed channel tracking method.

Hˆij [l,k] h ˆij [l,n]

FFT

Post ML detector

ARTICLE IN PRESS H.-G. Jeon, E. Serpedin / Signal Processing 88 (2008) 1002–1016

other interfering signals coming from the other Tx antennas. When pre-demodulating and converting the MISO signal Rj ½l; k into the desired SISO signal, the predicted channel response H ij ½l; k is used to cope with the time variant channel instead of using the previously estimated channel response H^ ij ½l  1; k. Once the SISO signal is obtained, temporal channel estimation in frequency domain is carried out to remove the inverse matrix calculation required in the following time domain channel estimation. In the time domain channel estimation block, the channel impulse response is obtained by minimizing a MSE cost function, considering the presence of null sub-carriers. 3.1. A blind channel response predictor design In this paper, we design a blind channel predictor by exploiting a piecewise linear model for the time varying channel. Fig. 2 shows an example of the time varying channel frequency response at the kth sub-carrier. The channel frequency response of each sub-carrier, H ij ½l; k, varies nonlinearly with time. However, the adjacent channel frequency responses H ij ½l; k and H ij ½l  1; k present a certain correlation with each other and this relation is expressed as H ij ½l; k ¼ H ij ½l  1; k þ Dij ½l; k,

(5)

where Dij ½l; k denotes the difference between the channel frequency responses corresponding to lth and ðl  1Þth symbols at the kth sub-carrier. If a piecewise linear model [13] is used during the short time of some OFDM symbol periods as shown in Fig. 2, the nonlinear time varying channel frequency response H ij ½l; k can be treated as a linear model. Using the piecewise linear model, let us assume that H ij ½l; k varies linearly during the time of the consecutive M OFDM symbols. The variable M can be set according to the channel response changing

Channel frequency response

Hij [l − 1,k]

Hij [l ,k]

Hij[l − 2,k] Hij [l − 3,k]

Dij [l − 2,k]

rate. For example, a channel with a short coherence time will have a small M, and vice versa. In the piecewise linear model, the condition Dij ½l; k ’ Dij ½l  1; k is assumed. Therefore, from (5) we infer that H ij ½l; k ’ H ij ½l  1; k þ Dij ½l  1; k. If we know H^ ij ½l  1; k and D^ ij ½l  1; k at the time instant corresponding to the lth symbol, then the predicted channel response H ij ½l; k can be obtained as H ij ½l; k9H^ ij ½l  1; k þ D^ ij ½l  1; k.

H ij ½l; k9

M1 X

om H^ ij ½l  m; k

m¼1

¼

M1 X

om ðH ij ½l  m; k þ Zij ½l  m; kÞ,

where om is a weight value and Zij ½l; k9H^ ij ½l; k  H ij ½l; k and denotes the random channel estimation error with zero mean and variance s2e (see Section 4.2). Let us define DH ij ½l; k9H ij ½l; k  H ij ½l; k as the channel response prediction random error. The weight values can be found by minimizing the following MSE cost function: x½l; k9EfjDH ij ½l; kj2 g 8 2 9  = < M1 X   om H^ ij ½l  m; k . ¼ E H ij ½l; k   ; : m¼1

ð8Þ

During the M OFDM symbols, H ij ½l; k ’ H ij ½l  m; k þ m  Dij ½l; k, 1pmoM. Therefore, ( x½l; k ’ E jH ij ½l  M þ 1; k þ ðM  1ÞDij ½l; k M1 X

om ðH ij ½l  M þ 1; k

m¼1

Fig. 2. Piecewise linear model for nonlinear time variant channel.

ð7Þ

m¼1



Piecewise linear model

(6)

Referring to Fig. 2 of the piecewise linear model, D^ ij ½l  1; k can be expressed by using the previously estimated channel responses, since 2Dij ½l  1; k ’ H ij ½l  1; k  H ij ½l  3; k and Dij ½l  1; k ’ H ij ½l  2; k  H ij ½l  3; k. In the case where the channel response varies linearly during the M consecutive OFDM symbols, therefore, (6) can be expressed by the linear combination of the ðM  1Þ previously estimated channel responses as follows:

2Dij [l − 1,k]

Time(l )

1005

þ ðM  1  mÞDij ½l; k ) þZij ½l  m; kÞj2

ARTICLE IN PRESS H.-G. Jeon, E. Serpedin / Signal Processing 88 (2008) 1002–1016

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( 1

¼

!

M1 X m¼1

þ M 1

the proposed predictor requires no additional training process and convergence time.

om H ij ½l  M þ 1; k M1 X

!

)2 3.2. Conversion of MISO Rx signal into SISO desired signal

om ðM  1  mÞ Dij ½l; k

m¼1

þ

s2e

M1 X

!

o2m

.

ð9Þ

m¼1

The weight values om can be determined by solving qx½l; k=qom ¼ 0. However, since H½l  M þ 1; k, D½l; k and s2e are not given due to the blind estimation property, we cannotP solve the equation M1 directly. From (9), when ð1  i¼1 om Þ ¼ 0 and PM1 ðM  1  m¼1 om ðM  1  mÞÞ ¼ 0, the cost function x½l; k can be minimized, and then P 2 x½l; k ’ s2e M1 m¼1 om . The surface of the cost function is convex with respect to om . Therefore, om can be numerically determined by finding the values minimizing the cost function of Cðo1 ; o2 ; . . . ; oM1 Þ P 2 while satisfying the conditions 9 M1 m¼1 om PM1 PM1 o ¼ 1 and m m¼1 m¼1 om ðM  1  mÞ ¼ M  1. In this paper, the weight values om are obtained for M ¼ 4, 5 and 6, and the results are given as follows: M ¼ 4;

o1 ¼ 43;

o2 ¼ 13;

o3 ¼ 23,

x½l; k ’ 2:3s2e , M ¼ 5;

o1 ¼ 1;

o2 ¼ 0:5;

o4 ¼ 0:5; x½l; k ’ 1:5s2e , M ¼ 6; o1 ¼ 0:8; o2 ¼ 0:5; o4 ¼ 0:2;

o5 ¼ 0:35;

o3 ¼ 0, o3 ¼ 0:25,

x½l; k ’ 1:11s2e .

ð10Þ

It is assumed that the initial channel response values H^ ij ½1; k and H^ ij ½2; k can be obtained by using the long preamble signal consisting of two consecutive training OFDM symbols, as is the case with WiBro and WLAN systems. After processing the long preamble signal, we have two initial channel response values H^ ij ½1; k and H^ ij ½2; k, but no access yet to H^ ij ½3; k; . . . ; H^ ij ½1  M; k. For this reason, the initial condition is given as H^ ij ½2; k ¼ H^ ij ½3; k ¼    ¼ H^ ij ½1  M; k. In the conventional channel estimation method [2], the long preamble signal was averaged to reduce noise variance. It should be noted that in this paper, however, the long preamble signal is used for channel response prediction in time varying channel environments. Since the proposed channel predictor uses piecewise linear model, it can adaptively track a nonlinearly varying channel response. In addition,

As mentioned in the previous section, it is efficient to estimate channel response after converting the MISO Rx signal into the SISO desired signal by canceling the interference signal coming from the other Tx antennas. The SISO desired signal between the ith Tx antenna and the jth Rx antenna is defined by S ij ½l; k9X i ½l; kH ij ½l; k ¼ X i ½l; kðH ij ½l  1; kþ Dij ½l; kÞ. The temporally estimated SISO signal S~ ij ½l  1; k is given by S~ ij ½l; k ¼ Rj ½l; k 

Nt X

X~ m ½l; kH^ mj ½l  1; k

m¼1;mai

ð11Þ ¼ X i ½l; kH ij ½l; k þ N ij ½l; k, PN t 0 where N ij ½l; k9W j ½l; k þ m¼1;mai X m ½l; kDmj ½l; k and D0ij ½l; k is defined by D0ij ½l; k9H ij ½l; k  H^ ij ½l  1; k, and X~ i ½l; k denotes the demodulated data by using the previous channel response H^ ij ½l  1; k. For convenience, X i ½l; k ¼ X~ i ½l; k is assumed. Since D0ij ½l; k is proportional to the channel variation, it can be seen from (11) that the faster the channel variations are and the more Tx antennas are used, the higher the channel estimation errors are. Moreover, if the previous channel response H^ mj ½l  1; k is used in fast time varying channel environments, X i ½l; k ¼ X~ i ½l; k cannot be assumed anymore. In that case, the noise term N ij ½l; k may increase significantly. In this paper, in order to reduce the noise term in (11), the predicted channel response H ij ½l; k is used not only to cancel the interference signal in the MISO Rx signal but also to pre-demodulate X i ½l; k as shown in the block diagram of Fig. 1. Using H ij ½l; k and X i ½l; k, (11) is revised as S~ ij ½l; k ¼ Rj ½l; k 

Nt X

X m ½l; kH mj ½l; k

m¼1;mai

ð12Þ ¼ X i ½l; kH ij ½l; k þ N 0ij ½l; k, P t where N 0ij ½l; k9W j ½l; k þ N m¼1;mai X m ½l; kDH mj ½l; k. For convenience, X i ½l; k ¼ X i ½l; k is assumed. If there is no prediction error such that DH ij ½l; k ¼ 0, then S~ ij ½l; k ¼ X i ½l; kH ij ½l; k þ W j ½l; k, regardless of the channel response variation and the number of Tx antennas.

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3.3. Joint frequency and time domain MMSE channel estimator The channel estimation methods [3,6] required inverse matrix calculation in every OFDM symbol. As mentioned in [6], Qii ½n of [4] is not the identity matrix in the presence of null sub-carriers and nonconstant modulation. For this reason, Li’s simplified method [4], which requires no matrix inversion, cannot be used in general MIMO–OFDM systems with null sub-carriers and employing non-constant modulation. Therefore, our goal is to derive a novel low complexity channel estimation method which can be used in general MIMO–OFDM systems while maintaining good performance. In this section, we propose a frequency domain estimation approach coupled with an MMSE method that requires no inverse matrix calculation in every symbol. In the proposed method, frequency domain channel estimation is carried out first such that the subsequent MMSE estimator does not require a matrix inversion during any OFDM symbol. The proposed method can be used in any MIMO–OFDM system, regardless of null sub-carriers and non-constant modulation. Referring to (12), considering the null subcarriers and using the least squares (LS) method in frequency domain, the temporally estimated channel frequency response is given by Y ij ½l; k ¼ ðS~ ij ½l; k=X i ½l; kÞZ½k ¼ ðH ij ½l; k þ V ij ½l; kÞZ½k,

V ij ½l; k ¼

Nt X m¼1;mai

þW j ½l; k

guard band. Note that estimating the channel response in the frequency domain removes the need of calculating an inverse matrix in the next step of time domain channel estimation employing an MMSE technique. The time domain channel estimation is performed during the next step, by considering all the null sub-carriers used in the guard band. The time domain channel estimate h^ij ½l; n can be found by minimizing the following MSE cost function: Cðfh^ij ½l; ng; i ¼ 1; . . . ; N t Þ  2 L0  N 1 X X   nk ^ hij ½l; nW N Z½k . 9 Y ij ½l; k   n¼0 k¼0

( ) qCðfh^ij ½l; ngÞ 1 qCðfh^ij ½l; ngÞ qCðfh^ij ½l; ngÞ 9 j 2 qRðh^ij ½l; n0 Þ qh^ij ½l; n0  qIðh^ij ½l; n0 Þ ¼ 0,

ð17Þ

where RðÞ and IðÞ denote the real and imaginary parts of a complex number, respectively, and n0 ¼ 0; 1; . . . ; L0 . Direct solving (17) results in N1 X

Y ij ½l; k 

L0 X

! h^ij ½l; nW kn N Z½k

0 Z½kW kn ¼ 0. N

n¼0

(18)

ð13Þ Define q½n9

X m ½l; kDH ij ½l; k

ð16Þ

h^ij ½l; n is the estimated channel impulse response and can be determined by solving

k¼0

where V ij ½l; k denotes the random estimation error given by

1007

N 1 X

Z½kW kn N ,

(19)

k¼0

!, X i ½l; k.

ð14Þ

Z½k is a time invariant function to denote the pattern of null sub-carriers used in the guard band and is given by 8 1 if 1pkoN=2  g or > > > > < N=2 þ gokpN  1; (15) Z½k ¼ 0 if k ¼ 0 or > > > > : N=2  gpkpN=2 þ g; where index k ¼ 0 denotes the DC component and g stands for the number of null sub-carriers in the

yij ½l; n9

N 1 X

Y ij ½l; kZ½kW kn N .

(20)

k¼0

Then, (18) can be expressed as L 0 1 X

h^ij ½l; nq½n0  n ¼ yij ½l; n0 

(21)

n¼0

for i ¼ 1; . . . ; N t and n0 ¼ 0; 1; . . . ; L0 . Eq. (21) can be expressed in matrix form as Qh^ ij ½l ¼ yij ½l,

(22)

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1008

where 0 B B B Q9B B @

be expressed as q½0 q½1 q½1 q½0 .. .. . . q½L0  1 q½L0  2

   

1

h^ ij ½l ¼ Q1 yij ½l ¼ hij ½l þ Q1 vij ½l.

q½1  L0  C q½2  L0  C C C, .. C . A q½0

The MSE of the channel impulse response estimate can be given by MSE½l9Efkh^ ij ½l  hij ½lk2 g (23) (24)

yij ½l ¼ ðyij ½l; 0; yij ½l; 1; . . . ; yij ½l; L0  1ÞT .

(25)

Hence, the channel impulse response can be estimated by

4. MSE calculation and complexity comparison 4.1. MSE calculation In this section, we will derive the MSE of the proposed channel estimation scheme. From (20), one infers that ! N 1 L 0 1 X X mk yij ½l; n ¼ hij ½l; mW N þV ij ½l; k Z½kW nk N k¼0

¼

m¼0

hij ½l; mq½n  m þ vij ½n;

0pnpL0 ,

m¼0

ð27Þ PN1

nk k¼0 V ij ½l; kZ½kW N .

where vij ½n ¼ be expressed in matrix form as

Eq. (27) can

ð32Þ

H

A generic entry of Efvij ½lvij ½l g is given by ( ! N 1 X n1 k1  Efvij ½l; n1 vij ½l; n2 g ¼ E V ij ½l; k1 Z½k1 W N k1 ¼0



(26)

Since matrix Q is a time invariant constant matrix determined by the null sub-carrier pattern, the inverse matrix Q1 can be pre-calculated and stored in the memory before beginning the channel tracking. Therefore, no explicit calculation of Q1 is required for every OFDM symbol. However, [3] and [6] require inverse matrix calculation for every OFDM symbol.

L 0 1 X

¼ EfðQ1 vij ½lÞH Q1 vij ½lg ¼ TracefQ1 Efvij ½lvij ½lH gðQ1 ÞH g.

h^ ij ½l9ðh^ij ½l; 0; h^ij ½l; 1; . . . ; h^ij ½l; L0  1ÞT ,

h^ ij ½l ¼ Q1 yij ½l.

(31)

N1 X

!)

2 k2 V ij ½l; k2 Z½k2 W þn N

k1 ¼0

¼

N 1 X

EfV ij ½l; k1 V ij ½l; k2 g

k1 ;k2 ¼0 1 k 1 þn2 k 2 , Z½k1 Z½k2 W n N

ð33Þ

where n1 ; n2 ¼ 1; 2; . . . ; L0 . If we substitute (14) into (33), Eq. (33) can be rewritten as Efvij ½l; n1 vij ½l; n2 g (PN 2 t N 1 X m¼1;mai jX m ½l; kj ¼ s2p 2 jX ½l; kj i k¼0 ) s2n ðn1 n2 Þk þ , ð34Þ Z½kW N jX i ½l; kj2 P 2 where s2p 9EðjDH ij ½l; kj2 Þ ¼ s2e M1 m¼1 om and DH ij ½l; k and W j ½l; k are assumed to be independent of each other. Hence, if a constant modulus modulation is used,   s2n Efvij ½lvij ½lH g ¼ ðN t  1Þs2p þ Q: ð35Þ jX i ½l; kj2 If no null sub-carriers are used, since Q ¼ NI,   L0 1 2 ðN t  1Þsp þ . (36) MSE½l ¼ SNR N

(28)

Given a SNR, the MSE½l is dependent on the prediction error, the channel response length and the number of Tx antennas.

hij ½l ¼ ðhij ½l; 0; hij ½l; 1; . . . ; hij ½l; L0  1ÞT ,

(29)

4.2. Mean and variance of random variable Zij ½l; k

vij ½l ¼ ðvj ½l; 0; vj ½l; 1; . . . ; vij ½l; L0  1ÞT .

(30)

yij ½l ¼ Qhij ½l þ vij ½l, where

From (28), the channel impulse response estimate corresponding to Tx antenna i and Rx antenna j can

For convenient comprehension, let us assume that BPSK is used for X i ½l; k, then E½X i ½l; k ¼ 0. X m ½l; k, DH ij ½l; k, W j ½l; k and X i ½l; k are independent of each other. It is clear that E½X m ½l; k ¼ 0,

ARTICLE IN PRESS H.-G. Jeon, E. Serpedin / Signal Processing 88 (2008) 1002–1016

E½X i ½l; k ¼ 0 and E½W j ½l; k ¼ 0. Therefore, it is also clear that E½V ij ½l; k ¼ 0, where V ij ½l; k is given by (14). From (31), the channel estimation error in time domain can be expressed P as E½h^ ij ½l  hij ½l ¼ Q1 E½vij ½l. Since E½vij ½n ¼ N1 k¼0 E½V ij ½l; kZ½k 1 W nk ¼ 0, Q E½v ½l ¼ 0. It means that the mean ij N of the channel estimation error in time domain is zero. Since the channel frequency response estimate H^ ij ½l; k is obtained by performing Fourier transformation for the channel impulse response estimate h^ij ½l; n, the mean of the channel estimation error in frequency domain is also zero. That is, E½Zij ½l; k ¼ E½H^ ij ½l; k  H ij ½l; k ¼ 0. The MSE of the channel frequency response in the kth sub-carrier is defined by E½ðH^ ij ½l; k H ij ½l; kÞ2  ¼ s2e . The total MSE of the channel impulse response in time domain is equal to MSE½l given by (36). When the channel impulse response h^ij ½l; n is Fourier transformed to obtain the channel frequency response H^ ij ½l; k, the total MSE in time domain is equal to the total MSE in frequency domain. Therefore, if there are N subcarriers in the OFDM system, then s2e ¼ MSE½l=N. 4.3. Computational complexity comparison In this section, the computational complexities of the schemes proposed in [3,6] and the method proposed herein are compared briefly. The complexity comparison will be focused on the channel estimation based on the decision-directed estimation method, as [6] did. Since Q1 can be pre-calculated, Q1 calculation is not included in the complexity comparison. Since Li’s simplified method [4] may cause estimation error in MIMO–OFDM systems with null sub-carrier and employing non-constant

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modulus modulations, Li’s simplified method will not be discussed in this comparison. In the case of two Tx and Rx antennas and N sub-carriers, the channel estimation complexity for each method is given in Table 1 (refer also to [6]). In Table 1, FFTN denotes the number of multiplications required for the FFT operation with size N. invðL0  L0 Þ stands for the number of multiplications required for L0  L0 matrix inversion. N u denotes the number of the sub-carriers used. When N u ¼ N, no null subcarrier is used. The number of FFT operations for Li method [3] can be easily obtained by referring to Fig. 3 in [3]. Considering Eqs. (15) and (16) in [3], the number of multiplications required for calculating hij ½l can be derived straightforwardly. The calculation amount for Minn method [6] can be obtained by considering the similarity with Li method. Note that when N u ¼ N and constant modulation is used, Qii ½n is a identity matrix. In that case, Li’s method needs calculating only invðL0  L0 Þ instead of calculating invð2L0  2L0 Þ. From Table 1, we can see that the proposed method has the lowest complexity among these methods, regardless of the non-constant modulation and the presence of null sub-carriers. 5. Performance evaluation Computer simulations are carried out to evaluate the performance of the proposed method. Two Tx and two Rx antennas are used for the MIMO–OFDM system. There are a total of 128 subcarriers so that the FFT/IFFT size is 128. The DC component sub-carrier is not used, and 10 and 9 sub-carriers on each end of the spectral band, respectively, are used as guard band. The rest of 108

Table 1 Evaluation of computational complexity for each method Condition

Method

No. of complex multiplications and divisions

Constant modulus with N u ¼ N

Ref. [3] Ref. [6]

3N þ ð2L0 Þ2 þ 5FFTN þ invðL0  L0 Þ 4:5N þ 2ðL0 Þ2 þ 3FFTN=2 þ invðL0  L0 Þ

Proposed

3N þ ð2L0 Þ2 þ 5FFTN

Ref. [3] Ref. [6]

3N þ ð2L0 Þ2 þ 5FFTN þ invð2L0  2L0 Þ 4:5N þ 2ðL0 Þ2 þ 3FFTN=2 þ invðL0  L0 Þ

Proposed

3N þ ð2L0 Þ2 þ 5FFTN

Ref. [3] Ref. [6]

5N þ ð2L0 Þ2 þ 5FFTN þ invð2L0  2L0 Þ 6N þ 2ðL0 Þ2 þ 4FFTN=2 þ 2  invðL0  L0 Þ

Proposed

3N þ ð2L0 Þ2 þ 5FFTN

Constant modulus with N u aN

Non-constant modulus

ARTICLE IN PRESS H.-G. Jeon, E. Serpedin / Signal Processing 88 (2008) 1002–1016

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2

1 Eb/No = 15 dB fd = 200 Hz

Channel response Estimated channel response

0.5 Image {H11(l,10)}

Real {H11(l,10)}

1.5

Channel response Estimated channel response

1 0.5 0

Eb/No = 15 dB fd = 200 Hz

0 −0.5 −1

−0.5 −1

−1.5 0

10

20

30 40 50 60 70 OFDM symbol index

80

90 100

0

10

20

30 40 50 60 70 OFDM symbol index

80

90 100

Fig. 3. An example of channel frequency response H ij ½l; k and tracking at f d ¼ 200 Hz, k ¼ 10 and E b =N 0 ¼ 15 dB.

sub-carriers are used to transmit data. The OFDM symbol rate is 10 Ksps, and the symbol period is 100 ms, including the guard time of 20 ms. The channel length L0 is assumed to be 18. Modulation in sub-carriers is QPSK. The carrier frequency is 2.4 GHz. The multi-path Rayleigh fading channel assumes two rays with equal gain, and each ray has six multi-path delay taps. Each signal path is assumed to undergo an independent Rayleigh fading. The rms delay spread is 1:82 ms. We used the Rayleigh fading channel simulator (Jake’s sinusoid sum method) openly published in Ref. [11]. The Doppler frequencies 40, 100 and 200 Hz are used to represent different mobile environments. After completion of channel estimation by using the training signal, the system state is in data transmission mode. In data transmission mode, channel tracking for 20 consecutive OFDM symbols is carried out continuously, using a decision directed method in which the demodulated data is used as the reference data. The performance of the system is measured by the estimator’s MSE and bit error rates (BER), each averaged over 100,000 OFDM blocks. For unbiased comparison, no channel coding is used in this simulation. In order to track the time varying channel response, Kalman filter method may be used for MIMO–OFDM systems as Komninakis did [14]. However, the Komninakis method should calculate the inverse matrix in every OFDM symbol to obtain Kalman gain matrix and thereby the calculation amount increases significantly. For this reason, we compare the proposed method with Kalman filter estimator, using scalar Kalman filter in each subcarrier [15].

Fig. 3 shows an example of the proposed channel tracking and the nonlinear time variant channel frequency response H 11 ½l; k simulated at the given multi-path channel parameters, l ¼ ð1; 2; . . . ; 100Þ, k ¼ 10, and maximum Doppler frequency f d ¼ 200 Hz. In Fig. 3, solid line is the channel response tracked by the proposed method at E b =N 0 ¼ 15 dB. Fig. 3 shows that the nonlinear channel response is well tracked by the proposed method. The performance simulation results are shown in Figs. 4–11. Fig. 4 shows the MSE of the proposed method at the conditions of M ¼ 4, 5 and 6, and at the fixed SNR of E b =N 0 ¼ 25 dB. The range of the normalized Doppler frequency (f d T s ) is given from 0 to 0:03. In the case of f d T s o0:02, the performance of the proposed scheme (M ¼ 4) is the worst among all the investigated channel tracking methods. The reason is that in decision directed mode, the performance of the proposed channel response estimator is very highly affected by the prediction error caused by the first channel prediction just after two training OFDM symbols. The prediction error causes the demodulation error which results in MSE performance degradation and propagates into the next channel estimation. As mentioned before, when two training OFDM symbols are received, only two channel responses H ij ½1; k and H ij ½2; k are obtained, and H ij ½3; k; . . . ; H ij ½1  M; k are assumed to be equal to H ij ½2; k. Therefore, from (10), the MSEs for the first channel prediction are proportional to x½l; k / 1:8s2e when M ¼ 4, x½l; k / s2e when M ¼ 5 and x½l; k / 0:68s2e when M ¼ 6, respectively. There is a larger difference in MSE between M ¼ 4 and 5 than between M ¼ 5 and 6.

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10−1 Proposed(M = 4) Proposed(M = 5) Proposed(M = 6) Li original

MSE

10−2

10−3

10−4 0

0.005

0.01

0.015 fd*Ts

0.02

0.025

0.03

Fig. 4. MSE performance as a function of f d T s at a given M.

101 Proposed CE No predict FDE Li original

100

Eb/No = 25 dB

MSE

10−1

10−2

10−3

10−4 0

0.005

0.01

0.015 fd*Ts

0.02

0.025

0.03

Fig. 5. MSE performance as a function of f d T s for each channel tracking method.

The noise effect in channel prediction can be reduced by increasing M, as shown in Fig. 4. However, increasing M beyond a certain limit may result in performance degradation due to the

increased sensitivity to channel time variations. Note that MSEðM ¼ 6Þ is larger than MSEðM ¼ 5Þ at f d T s 40:015. We can see that when M ¼ 5, the proposed method shows the best performance in

ARTICLE IN PRESS H.-G. Jeon, E. Serpedin / Signal Processing 88 (2008) 1002–1016

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102 Proposed CE No predict FDE FED + Km filter Li original

101

MSE

100

10−1

10−2

10−3

10−4 0

5

10

15 Eb/No [dB]

20

25

30

Fig. 6. MSE performance at f d ¼ 40 Hz.

100 Proposed CE No predict FDE FED + Km filter Li original Perfect CE

10−1 10−2

BER

10−3 10−4 10−5 10−6 10−7 0

5

10

15 Eb/No [dB]

20

25

30

Fig. 7. BER performance at f d ¼ 40 Hz.

Fig. 4. Hereafter, M is set to 5 in all the simulations for performance evaluation. When M ¼ 5 and the OFDM symbol period is 100 ms, the time duration for which a piecewise linear model is assumed is

500 ms. Fig. 5 shows the MSE for each method as a function of f d T s at the conditions: E b =N 0 ¼ 25 and M ¼ 5. As we can see from Fig. 5, the MSE of the proposed method increases more slowly than the

ARTICLE IN PRESS H.-G. Jeon, E. Serpedin / Signal Processing 88 (2008) 1002–1016

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103 Proposed CE No predict FDE FED + Km filter Li original

102

101

MSE

100 10−1 10−2 10−3 10−4 0

5

10

15 Eb/No [dB]

20

25

30

25

30

Fig. 8. MSE performance at f d ¼ 100 Hz.

100

10−1

BER

10−2

10−3

10−4 Proposed CE No predict FDE FED + Km filter Li original Perfect CE

10−5

10−6 0

5

10

15 Eb/No[dB]

20

Fig. 9. BER performance at f d ¼ 100 Hz.

other methods as f d T s increases. On the other hand, MSE of Li’s original method increases rapidly when f d T s 40:015.

Figs. 6, 8 and 10 show MSE performances at Doppler frequencies of 40, 100 and 200 Hz, respectively. Figs. 7, 9 and 11 show BER performances at

ARTICLE IN PRESS H.-G. Jeon, E. Serpedin / Signal Processing 88 (2008) 1002–1016

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102 Proposed CE No predict FDE FED + Km filter Li original

101

MSE

100

10−1

10−2

10−3

10−4 0

5

10

15 Eb/No [dB]

20

25

30

25

30

Fig. 10. MSE performance at f d ¼ 200 Hz.

100

10−1

BER

10−2

10−3

10−4 Proposed CE No predict FDE FED + Km filter Li original Perfect CE

10−5

10−6 0

5

10

15 Eb/No [dB]

20

Fig. 11. BER performance at f d ¼ 200 Hz.

Doppler frequencies of 40, 100 and 200 Hz, respectively. In these figures, BER performance curves for perfect channel estimation are given to show the performance in the ideal channel estimation case. In

these figures, ‘FDE’ denotes the frequency domain channel tracking method of Ref. [10]. In order to compare the effect of channel prediction, BER and MSE curves for the ‘no predict’ tracking method are

ARTICLE IN PRESS H.-G. Jeon, E. Serpedin / Signal Processing 88 (2008) 1002–1016

also drawn in these figures. ‘No predict’ means the proposed method without the channel prediction function. If ‘no prediction’ is used, the previously estimated channel value H ij ½l  1; k is used as the current channel value H ij ½l; k. In this case, the channel estimation error is given by H ij ½l; k H^ ij ½l  1; k ¼ Dij ½l; k þ Zij ½l; k. When the channel prediction is used, the channel estimation error is given by DH ij ½l; k ¼ H ij ½l; k  H ij ½l; k. The MSE of ‘no prediction’ is given by E½ðH ij ½l; k H^ ij ½l  1; kÞ2  ’ Var½Dij ½l; k þ s2e . The MSE of ‘prediction’ is E½jDH ij ½l; kj2  ’ 1:5s2e as we can see from (10). Therefore, the performance of ‘prediction’ is better than that of ‘no prediction’ as long as Var½Dij ½l; k40:5s2e is satisfied. Var½Dij ½l; k  0 at a low Doppler frequency ands2e is inversely proportional to the E b =N 0 such that s2e  0 at a high E b =N 0 . Therefore, the lower the Doppler frequency is in the wireless channel, the higher E b =N 0 is required to satisfy the condition of Var½Dij ½l; k 40:5s2e . That is the reason why the performance of ‘no prediction’ is better than that of ’prediction’ at f d ¼ 40 Hz. Note that in Fig. 6, the performance gap in E b =N 0 between ‘no prediction’ and ‘prediction’ is getting narrow with increase of E b =N 0 . If we can know the information about E b =N 0 and the Doppler frequency, either ‘no prediction’ or ‘prediction’ can be selected to improve the performance of channel estimator, based on such information. At a Doppler frequency of 40 Hz, there is a little difference in the MSE and BER performance between the proposed method and Li’s original method. At f d ¼ 100 Hz and BER of 104 , the performance improvement provided by the proposed method is about 2.5 dB in E b =N 0 compared with that of Li’s original method. However, at a Doppler frequency of 200 Hz, the BER performance difference becomes much larger when compared with those of other methods. As we can see from Figs. 9 and 11, due to the inter sub-carrier interference (ICI) [16], BER performance at a Doppler frequency of 200 Hz is worse than that of 100 Hz. From these figures, we can observe that the channel estimation error of Li’s original method is very large at a given low E b =N 0 . At a given low E b =N 0 , BER is high and the demodulated data X^ ½l; k is erroneous. Since Li’s original method calculates the inverse of a matrix made of erroneous demodulated data X^ ½l; k, the channel estimation error is amplified by noise and the erroneous inverse matrix. In the worst case of low E b =N 0 (less than

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5 dB), the channel estimation error may diverge when tracking the channel response. On the other hand, since the proposed method uses the known inverse matrix Q1 which is a constant time-invariant matrix, the effect of the erroneous demodulated data is much less significant than that of Li’s original method. The simulation results show that as expected the proposed method does not diverge. 6. Conclusions This paper proposed a novel simplified channel tracking method to reduce the computational complexity and improve the tracking performance in time varying channel environments. In the proposed method, a blind channel response predictor is designed to cope with the time variant channel. The proposed channel tracking scheme consists of a frequency domain estimation approach that is coupled with an MMSE time domain estimation method, and does not require any matrix inverse calculation during each OFDM symbol. By converting the MISO signal into a SISO signal and performing temporal channel estimation in the frequency domain before beginning time domain channel estimation, no matrix inversion is required anymore. The simulation results show that the proposed method exhibits superior performance than Li’s original method in time varying channel environments. At a Doppler frequency of 100 Hz and BER of 104 , signal-to-noise power ratio (E b =N 0 ) gains of about 2.5 dB are achieved relative to Li’s original method. At a Doppler frequency of 200 Hz, the performance difference between the proposed method and conventional one becomes much larger. Acknowledgment This work was partially supported by research project 07-03 funded by Electronics and Telecommunications Research Institute (ETRI) in Korea. References [1] G.J. Foschini, Layered space–time architecture for wireless communication in a fading environment using multiple antennas, Bell Labs Tech. J. 1 (2) (Autumn 1996) 41–59. [2] J. Terry, J. Heiskala, OFDM Wireless LANs: A Theoretical and Practical Guide, Sams Publishing, 2002 [3] Y.G. Li, N. Seshadri, S. Ariyavisitakul, Channel estimation for OFDM systems with transmitter diversity in mobile

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H.-G. Jeon, E. Serpedin / Signal Processing 88 (2008) 1002–1016 wireless channels, IEEE J. Sel. Areas Commun. 17 (March 1999) 461–471. Y.G. Li, Simplified channel estimation for OFDM systems with transmit antenna, IEEE Trans. Commun. 1 (January 2002) 67–75. Y. Gong, K.B. Letaief, Low complexity channel estimation for space–time coded widehand OFDM systems, IEEE Trans. Wireless Commun. 2 (September 2003) 876–882. H. Minn, D.I. Kim, V.K. Bhargava, A reduced complexity channel estimation for OFDM systems with Transmit diversity in mobile wireless channels, IEEE Trans. Commun. 50 (5) (May 2002) 799–807. H. Minn, N. Al-Dhahir, Optimal training signals for MIMO OFDM channel estimation, Globecom 2004, November 2004. D. Schafhuber, G. Matz, F. Hlawatsch, Kalman tracking of time-varying channels in wireless MIMO–OFDM systems, in: Asilomar Conference on Signals, Systems and Computers 2003, vol. 2 (November 2003) pp. 1261–1265. K.A.G. Teo, S. Ohno, T. Hinamoto, Kalman channel estimation based on oversampled polynomial model for OFDM over doubly-selective channels, in: IEEE SPAWC 2005, June 2005, pp. 116–120.

[10] Y. Chen, D. Jayalath, A. Thushara, Low complexity decision directed channel tracking for MIMO WLAN system, in: ISISPCS, Hong Kong. December 2005, pp. 629–632. [11] H. Harada, R. Prasad, Simulation and Software Radio for Mobile Communications, Artech House, 2002. [12] Telecommunication Technology Association (TTA), Specification for 2.3 GHz Band Portable Internet Service— Physical Layer, June 2004. [13] S. Haykin, Adaptive filter Theory, third ed., Prentice-Hall, Englewood Cliffs, NJ, 1996. [14] C. Komninakis, C. Fragouli, A. Sayed, R. Wesel, Multiinput multi-output fading channel tracking and equalization using Kalman estimation, IEEE Trans. Signal Process. 50 (5) (May 2002) 1065–1076. [15] Z. Yuanjin, A novel channel estimation and tracking method for wireless OFDM systems based on pilots and Kalman filter, IEEE Trans. Consum. Electron. 49 (2) (May 2003) 275–283. [16] X. Cai, G.B. Giannakis, Bounding performance and suppressing intercarrier interference in wireless mobile OFDM, IEEE Trans. Commun. 51 (12) (Dec 2003) 2047–2056.