New Estimation and Equalization Approach for OFDM ... - CiteSeerX

NEW ESTIMATION AND EQUALIZATION APPROACH FOR OFDM UNDER DOPPLER-SPREAD CHANNEL Mitsuru Nakamura1 , Tsutomu Seki1 , Makoto Itami1 , Kohji Itoh1 , A.Hamid Aghvami2 1 Department of Applied Electronics, Tokyo University of Science 2641 Yamazaki, Noda, Chiba, 278-8510 Japan

Tel:+81-4-7124-1501 ext.4232 Fax:+81-4-7122-9195 2

E-mail:[email protected],[email protected] Centre for Telecommunications Research King’s College London University of London Strand, London WC2R 2LS, UK

Tel:+44-20-7848-2898 Fax:+44-20-7848-2664 E-mail: [email protected]

Abstract In OFDM transmission, a loss of subcarrier orthogonality due to Doppler-spread of the channel leads to inter-carrier interference(ICI), especially, under mobile reception environments. ICI causes significant degradation of bit error rate characteristics and this influence becomes larger as the carrier frequency or moving velocity of the receiver increases. In this paper, a method to reduce the ICI caused by Doppler-spread of the channel is proposed. In the proposed method, the transmission channel is modeled by a combination of multiple Doppler-shifted propagation paths and their parameters such as attenuation, relative delay and Doppler-shift are estimated using scattered pilot symbols. Then, ICI is canceled by multiplying the inverse matrix of the estimated channel matrix to the received OFDM symbol vector. As the result of computer simulation, it is confirmed that the proposed method can improve the bit error rate characteristics under Dopplerspread channel. Keywords - OFDM, ICI, Doppler-spread, Channel Estimation, Equalization I. INTRODUCTION OFDM (Orthogonal Frequency Division Multiplexing) is widely known as a modulation scheme which can utilize the allocated frequency band very efficiently and realizes high speed data transmission under limited band width. The OFDM signal consists of many carriers which are orthogonally arranged in the frequency axis and the data symbols are transmitted by modulating each carrier using usual digital modulation schemes such as QPSK, QAM and so on. The OFDM signal is suitable for limited band width transmission because the shape of its power spectrum is almost rectangular. Moreover, it is possible to reduce the affection of inter-symbol interference by inserting a guard interval without much degrading data

0-7803-7589-0/02/$17.00 ©2002 IEEE

transmission rate[1][2][3]. Therefore, it is widely used and considered for high speed data transmission such as digital television broadcasting, wireless LAN, power-line communication, the next generation mobile communication systems and so on. In OFDM transmission, a loss of sub-carrier orthogonality due to Doppler-spread of the transmission channel leads to inter-carrier interference(ICI). Especially, the influence of ICI under mobile reception environments causes significant degradation of bit error rate characteristics and this becomes larger as the carrier frequency becomes higher or the moving velocity of the receiver becomes faster[4][5][6]. In this paper, a method to reduce the influence of ICI caused by Doppler-spread of the channel is proposed in OFDM transmission. In the proposed method, the transmission channel is modeled by a combination of multiple Doppler-shifted propagation paths and their parameters such as attenuation, relative delay and Doppler-shift are estimated using scattered pilot symbols. Then, ICI is cancelled by multiplying the inverse matrix of the channel matrix obtained from the estimated parameters of paths. Since the dimension of channel matrix is usually very large in OFDM transmission, it is not practical to calculate its complete inverse matrix. Therefore, simplification is necessary. In this paper, a simplification technique proposed in [7] is used.

II. SYSTEM MODEL AND PROPOSED METHOD

In this section, a method to reduce the influence of ICI is described. Fig.1. shows the block diagram of the proposed system.

PIMRC 2002

receved symbol ^ D

−1 H^ l . D^ l

^l D

∼ Dl

ˆ l, n) is written as h(k,

demodulation

^ −1

ˆh(k, l, n) =

Hl

calculation of matrix inversion

π(N −1)(n−l+αi )

N ×e−j ej2παi f0 Ts k ×ri e−j2π(fc +nf0 +αi f0 )τi ,

H^ l generation of the channel matrix d^ (k, l p) d^ (k − 1, l p)

estimation of attenuation , relative delay and Doppler-shift for every path

known symbol

B.

Equalization Method

The input-output relationship of the Doppler-spread channel is expressed in vector form as

Fig. 1. Proposed system

ˆ = HD ˆ +W D A.

Received Signal

In this paper, it is assumed that an OFDM signal represented in eq.(1) is transmitted.
s(t) =

∞ 

g(t − mTs )·

m=−∞ N −1 



d(m, n)e

j2πnf0 (t−mT s) j2πfc t

e

, (1)

n=0

where, N is the number of carriers, f0 is the carrier interval, fc is the lowest carrier frequency, Ts is the symbol length, d(m, n) is the data symbol on the carrier whose frequency is fc + nf0 in the m th OFDM symbol. If it is assumed that the transmission channel consists of Np differently Doppler-shifted propagation paths, the received OFDM signal can be written as

sr (t) =

Np 

ri s(t − τi )ej2π∆fi (t−τi ) ,

(2)

i=1

where, ri , τi and ∆fi are attenuation, relative delay and Doppler-shift of i th path respectively. In the receiver, after down converting and sampling the received signal in eq.(2), the block of samples are transformed by DFT ˆ l), that corresponds to to generate the data symbol, d(k, ˆ l) is expressed in eq.(3). each carrier. d(k, ˆ l) = h(k, ˆ l, l)d(k, l) d(k, +

N −1 

(4)

where, αi (αi = ∆fi /f0 ) is normalized frequency offset of i th path.

d (k, l p) d (k − 1, l p)

r i ,τ i ,α i

pilot symbol

Np  1 sin{π(n − l + αi )} i) N sin π(n−l+α i=1 N

ˆ l, n)d(k, n) + w(k, l), h(k,

(3)

n=0

n=l

where w(k, l) is additive noise that corresponds to the ˆ l, n) is l th carrier in the k  th OFDM symbol and h(k, the transfer function from symbol d(k, n) to the l  th carˆ l, n) represents the influence of ICI. rier. If l
= n, h(k,

(5)

ˆ 0), · · · , d(k, ˆ N − 1)]T , D = ˆ = [d(k, where, D T [d(k, 0), · · · , d(k, N − 1)] , W = [w(k, 0), · · · , w(k, N − 1)]T , and   ˆ 0, 0) ˆh(k, 0, N − 1) h(k, ···  .. .. .. ˆ = H  . (6)  . . . ˆ ˆ h(k, N − 1, 0) · · · h(k, N − 1, N − 1) ˆ it is necesIn order to solve eq.(5) and to obtain D, ˆ and to calculate sary to estimate the channel matrix H ˆ can often have large dimenits inverse matrix. Since H sion, it is necessary to simplify the calculation of matrix inversion. Fig.2. shows average interference power to the 100 th carrier from the others carriers and interference average power is normalized by the received power of 100 th symbol(power of the desired symbol). This figure shows that interference power from the nearest carrier is largest and interference power gradually decreases apart from the 100 th carrier. Therefore, since most energy is concentrated in the neighborhood of the diagonal line in eq.(6), the ICI terms which do not significantly affect ˆ l) in eq.(6) can be neglected and it is assumed as d(k, ˆ l, n) = 0 for |l − n| > q/2, h(k,

(7)

where q denotes the number of dominant ICI terms against l th symbol. Then, input-output relationship corresponds to the l’th carrier is written as ˆl = H ˆ l Dl + Wl , D

(8)

ˆ l − q ), · · · , d(k, ˆ l + q )]T , Dl = [d(k, l − ˆ l = [d(k, where, D 2 2 q q T q q T 2 ), · · · , d(k, l + 2 )] , Wl = [w(k, l − 2 ), · · · , w(k, l + 2 )] , and ˆl = H  ˆ h(k, l − q2 , l − q2 ) · · ·  .. ..  . . q q ˆ h(k, l + 2 , l − 2 ) · · ·

ˆ l − q,l + q)  h(k, 2 2  .. . . q q ˆ h(k, l + 2 , l + 2 ) (9)

Average Interference Power [dB]

0

α=0.05 α=0.1 α=0.2

-5

E(Np ) =

-10

Pk 

ˆ − 1, l, l)d(k, l) 2

x(k − 1, l) − h(k +

-15 -20 -25 -30 -35 -40 80

85

90

95 100 105 Number of Carrier

110

115

120

Fig. 2. Average Interference power to 100 th carrier from each carrier

Pk−1

2 Np





= f (αi )ri e−j2π(fc +αi f0 +lf0 )τi d(k, l)

x(k, l) −



i=1 Pk

Np 

 + f (αi )e−j2παi f0 Ts

x(k − 1, l) −

i=1 Pk−1

2 ×ri e−j2π(fc +αi f0 +lf0 )τi d(k − 1, l) , (11)  where Pk means that the summation is performed as long as d(k, l) is a pilot symbol. f (αi ) is written as

Compensation of both multiplicative distortion and ˆ l to ICI is accomplished by multiplying the inverse of H eq.(9). The resulting signal can be expressed as follows: ˆ −1 D ˜l = H ˆl . D l

f (αi ) =

1 sin παi −j N −1 παi 2παi f0 Ts k N · e e . i N sin πα N

(12)

(10)

The transmitted symbols d(k, l) are obtained by se˜ l . In order to lecting the elements in the middle of D generate of channel transfer matrix, it is necessary to estimate the channel characteristics and the method of channel estimation is shown in the following section. C.



2 ˆ

x(k, l) − h(k, l, l)d(k, l)

Proposed Channel Estimation Method

In the proposed method, scattered pilot symbols are used to estimate the channel parameters. Scattered pilot symbols are inserted among the data symbols as shown in Fig.3.. Scattered pilot symbols are used for channel estimation in the standard of digital terrestrial television broadcasting in Japan and EU [8]. In [9][10], the channel estimation technique using scattered pilot symbols is proposed. This technique is extended to estimate of Doppler-shift as shown in followings. In the proposed method, the parameters of channel transfer function is estimated so as to minimize the mean square error shown in eq.(11).

In in eq.(11), the received symbols and the locally generated replicas of the received symbols against pilot symbols are compared and channel parameters such as ri , τi and αi is estimated to minimize this criterion. In in eq.(11), replicas of ICI is not generated because the transmitted data symbols are not known in this stage. And this influence is treated additive noise. Since it is very complex to obtain the parameters to minimize the criterion in eq.(11) directly, an approximated procedure shown in the following section is used. C..1

Search the First Propagation Path

In order to simplify this problem, first the case when the channel model only contains one propagation path, ie. Np = 1, is considered. In this case, criterion is rewritten as follows. 



x(k, l) − f (α1 )r1 e−j2π(fc +α1 f0 +lf0 )τ1 d(k, l) 2 E(1) = Pk +



x(k − 1, l) − f (α1 )e−j2πα1 f0 Ts

Pk−1

2

×r1 e−j2π(fc +α1 f0 +lf0 )τ1 d(k − 1, l) . Frequency Domain

Time Domain

l

One of the necessary conditions to minimize E(1) with regard to r1 , τ1 and α1 is that its partial derivative of r1 has zero value. This condition is shown as r1 =

k

Pilot Symbols Data Symbols

Fig. 3. Arrangement of scattered pilot symbols

(13)

∗ Sk∗ + ej2πα1 f0 Ts Sk−1 , f (α1 )e−j2π(fc +α1 f0 )τ1 {Dk + Dk−1 }

(14)

where, Sk

=

 Pk

e−j2πlf0 τ1 d(k, l)x∗ (k, l),

(15)

Sk−1



=

Dk

=

Dk−1

=

e−j2πlf0 τ1 d(k − 1, l)x∗ (k − 1, l),(16)

Pk−1  |d(k, l)|2 , Pk  |d(k − 1, l)|2 . Pk−1

(17) (18)

When eq.(14) is satisfied, eq.(13) is rewritten as   E(1) = |x(k, l)|2 + |x(k − 1, l)|2 Pk Pk−1 2

− |f (α1 )| |r1 |2 {Dk + Dk−1 } .

C..2 (19)

Eq.(19) indicates that E(1) has the minimum value if |f(α1 )|2 |r1 |2 has the maximum value. |f (α1 )|2 |r1 |2 is written in eq.(20).



∗ 2 |f(α1 )|2 |r1 |2 = Sk∗ + ej2πα1 f0 Ts Sk−1 .

(20)

Since |f(α1 )|2 |r1 |2 is the function of τ1 and α1 , the necessary condition to maximize this is derived by partially differentiating this by τ1 and α1 and assuming the obtained partial derivatives to be zero. ∗ ∗ Sk−1 − Wk Sk∗ − Wk−1 Sk−1 Wk∗ Sk + Wk−1 ∗ +e−j2πα1 f0 Ts Wk∗ Sk−1 + ej2πα1 f0 Ts Wk−1 Sk ∗ −ej2πα1 f0 Ts Wk Sk−1 − e−j2πα1 f0 Ts Wk−1 Sk∗ = 0, j2πα1 f0 Ts

e

∗ Sk−1 Sk

−j2πα1 f0 Ts

−e

Sk−1 Sk∗

= 0,

(21) (22)

where, Wk Wk−1

= =



le−j2πlf0 τ1 d(k, l)x∗ (k, l),

Pk 

le−j2πlf0 τ1 d(k − 1, l)x∗ (k − 1, l).

Pk−1 (23) From eq.(22), α1 is calculated from τ1 as   S S∗   

k−1 k S∗ S



k S∗ S k−1 k

k  arctan   Sk−1 ∗ k−1 S

α1 =

4πf0 Ts

.

(24)

Because of the periodicity of the arctan function, the range of α1 where τ1 is uniquely obtained is restricted to the following range. −

1 1 < α1 < . 4f0 Ts 4f0 Ts

an appropriate step, τ1 that maximizes the value of |f (α1 )|2 |r|2 of derived and τ1 is regarded to be the estimation of the relative delay. If τ1 is determined r1 and α1 are easily obtained from eq.(14) and eq.(24) respectively. In the proposed method, τ1 is changed by 1/(2N f0 ) (twice of sampling frequency) and rough estimation is first obtained. And this rough estimation is modified to more precise value using Newton method using the conditions in eq.(21). Newton Method

In the preceding section, rough estimation of the parameters of the path is described. In this section, the method of obtain a fine estimation by using the Newton method is described. In the derivation of rough estimation of τi , only the optimal condition in eq.(22) is used. However, the optimal τ1 must satisfy eq.(21). In the Newton method, τ1 is modified by using eq.(26) iteratively. τ1,n+1 = τ1,n −

g(τ1,n ) , g  (τ1,n )

where g(τ1 ) is left part of eq.(21) which is obtained by substituting α1 in eq.(24) into eq.(21). By using eq.(26) the roughly estimated value can be made to be more precise value, and only several iterations are sufficient to converge. C..3

Estimation when Multiple Paths Exist

Here the case when Np ≥ 2 is considered. In this method, it is assumed that the influence to the mean square error from each path is approximately independent. Therefore, if the case of Np = 1 is assumed and the parameters of this path is estimated by previously mentioned method, the influence of this path can be removed from the mean square error E(1). After this, there remains the influence of other paths in the mean square error. Actually, because of the approximation and estimation error, the influence of the first estimated path still remains in E(1). In this method, this is neglected. By removing the influence of the first estimated path from the mean square error, the following criterion, E(2) is obtained. Since E(2) contains the influence of other paths, the second path is estimated from E(2) using the same procedure in the previous section. 



x(1, k, l) − f (α2 )r2 e−j2π(fc +α2 f0 +lf0 )τ2 d(k, l) 2 E(2) = Pk +



x(1, k − 1, l) − f (α2 )e−j2πα2 f0 Ts

Pk−1

(25)

By changing τ1 within the assumed delay spread range (this is usually guard interval length), it is possible to calculate the value of |f (α1 )|2 |r|2 . By changing τ1 with

(26)

2

×r2 e−j2π(fc +α2 f0 +lf0 )τ2 d(k − 1, l) ,

where

 ˆ k, l, l)d(k, l) x(1, k, l) = x(k, l) − h(1,

(27)

+



 ˆ k, l, n)d(k, n) , h(1,

(28) 1e+00

Pk

Conventional Proposed system(4) Proposed system (10) Static

n=l 1e-01

1 sin{π(n − l + α1 )} 1) N sin π(n−l+α N −j

π(N −1)(n−l+α1 ) N

×e e −j2π(fc +nf0 +α1 f0 )τ1 ×r1 e ,

j2πα1 f0 Ts k

(29)

By repeating this operation by the number of paths in the channel model, all rNp , τNp and αNp can be obtained. In this method, it is considered that the estimated path whose |rNp |2 is very small doesn’t much contributes and such a path is not used in equalization. This reduce the influence of additive noise and estimation error. Actually, the estimated paths whose |rNp |2 is smaller than the predefined threshold value is not used.

Bit Error Rate

ˆh(1, k, l, n) =

1e-03

1e-04

1e-05 0

5

10 15 20 Carrier to Noise Ratio [dB]

1e+00

Table 1 Simulation Parameters

30

Conventional Proposed system (4) Proposed system (10)

1e-01

Bit Error Rate

In this section, the performance of the proposed system is analyzed by the computer simulation. The parameters used in the simulations are shown Table 1. In this simulation, the channel parameters don’t change within the successive two OFDM symbols.

25

Fig. 4. Bit error rate characteristics against carrier to noise ratio

III. PERFORMANCE ANALYSIS

Number of carrier (N ) Carrier interval (f0 ) Minimum frequency (fc ) Length of guard interval (τg ) Maximum path (Np ) Iteration number of newton Threshold level Carrier modulation scheme Channel Two path environment D/U ratio Delay time Normalized frequency offset

1e-02

1e-02 1e-03 1e-04 1e-05

1024 1.0[kHz] 600.0[MHz] 1/8f0 10 5 0.01 QPSK 5dB 5µs 0.15

In Fig.4., the bit error rate characteristic against carrier to noise ratio is shown. In this case, two path multipath environment where one direct path and one delayed path exist. The power of the delayed path is 5dB smaller than the direct path and relative delay against the direct path is 5µsec. The normalized frequency offset of the direct path and the delayed path are 0 and 0.15 respectively. Therefore, Doppler-spread of the channel is 150Hz. In Fig.4., ’Conventional’ denotes the characteristic of the conventional equalizer that doesn’t perform ICI cancellation and ’Proposed system’ denotes the characteristic of the proposed method and following number indicates the matrix size used for ICI cancellation. ’Static’

1e-06 -0.3

-0.2

-0.1

0

0.1

0.2

0.3

Normalized Frequency Offset α

Fig. 5. Bit error rate characteristics against normalized frequency offset

denotes the characteristic when the no Doppler-spread exists. In Fig.5., bit error rate against normalized frequency offset under the two path channel when CNR is 20dB. In this case, the normalized frequency offset of the delayed path is changed. As shown in Fig.4., the BER characteristic of the conventional equalizer degrades because of ICI. On the contrary the proposed ICI canceller can improve the BER characteristic near to the case when no Doppler-spread exists. In the proposed method, the BER characteristic more improved as the dimension of the matrix is larger. However, the difference is small. In Fig.5., the BER characteristic of the conventional equalizer gradually degrades as α increases. In the proposed ICI canceller, better BER can be kept within the wide range of α. In the range of |α| > 0.2, the performance of the proposed method much degrades. This is because of the limitation in eq.(25) in the channel estimation algorithm. Therefore, the proposed ICI canceller must be used in

the range of |α| < 0.2. In order to expand this range, it is necessary to use more previous time symbols. However in many applications, this range is considered to be sufficient.

[8] V. Mignone, A. Morello, M. Visuntin: “CD-3: A New Channel Estimation Method to Improve the Spectrum Efficiency in Digital Terrestrial Television System”, IEE Conf. Publ., Vol.413, No.1, pp.122128, 1995

IV. CONCLUSION

[9] M.Itami, M.Kuwabara, M.Yamashita, H.Ohta, K.Itoh: “Equalization of Orthogonal Frequency Division Multiplexed Signal by Pilot Symbol Assisted Multipath Estimation”, Proc. of Globecom’98, CT5.6, pp.368-373, 1998

In this paper, in order to improve the influence of ICI caused by Doppler-spread of the transmission channel, the method of channel estimation and ICI cancellation is proposed. As the result of simulation, the proposed ICI canceller can well improve the bit error rate characteristic in the wide range of Doppler-spread as compared to the conventional equalizer. In the further researches, more detailed characteristics are examined under more actual time varying channel. Expansion of the range of applicable Doppler-spread and simplification of the algorithm are also investigated. REFERENCES [1] R. W. Chang and R. A. Gabby: “A Theoretical Study of Performance of an Orthogonal Multiplexing Data Transmission Scheme”, IEEE Trans. Comm., COM-16, pp.529-540, 1968 [2] S. B. Weinstein and P. W. Ebert: “Data Transmission by Frequency-Division Multiplexing using the Discrete Fourier Transform”, IEEE Trans. Comm., COM-19, pp.628-634, 1971 [3] B.Hirosaki: “An Analysis of Automatic Equalizer for Orthogonally Multiplexed QAM Systems”, IEEE Trans. Comm.,COM-28, pp.73-83, 1980 [4] A.Tsuzuku, H.Ohta, R.Nakamura: “A Study of Multipath Interferences on OFDM Transmission”, Journal of ITE, VOL.51, NO.9, pp.1493-1503, 1997 [5] H.Ohta, M.Itami, A.Tsuzuku: “A Study on Multipath Channel Modeling for Mobile Reception with an OFDM and Equalization Method for Reducing Interference by Using Estimated Propagation Path Characteristics”, Journal of ITE, VOL.52,NO.11, pp.1667-1675, 1998 [6] M.Okada, H.Takayanagi, H.Yamamoto: “ArrayAntenna-Assisited Doppler Spread Compensator for Mobile Reception of Terrestrial Television Broadcasting”, Journal of ITE, VOL.56, NO.2, pp.237244, 2002 [7] W. G. Jeon, K. H. Chang and Y. S. Cho: “An Equalization Technique for Orthogonal FrequencyDivision Multiplexing Systems in Time-Variant Multipath Channels”, IEEE Trans. Comm.,VOL.47, NO.1, pp.27-32, 1999

[10] M.Itami, M.Yamashita, M.Kuwabare, K.Itoh: “A Study on Equalization of OFDM Signal using Scattered Pilot Symbols”, Journal of ITE, VOL.52, NO.11, pp.1650-1657, 1998