Estimation of Time-Variant Channels for OFDM Systems Using Kalman and Wiener Filters Hye Mi Park and Jae Hong Lee School of Electrical Engineering and INMC, Seoul National University, Shillim-Dong, Gwanak-Gu, Seoul 151-742, Korea
[email protected] Abstract- In this paper, two channel estimation schemes for OFDM systems are investigated in a time-variant channel. The orthogonality of the subcarriers is destroyed in a time-variant channel, resulting in intercarrier interference (ICI). To mitigate the ICI, the 2-D Kalman filtering (2-D KF) scheme estimates the channel in time domain by using the Kalman filters in both time and frequency domains and the iterative Wiener filtering (IWF) scheme calculates the ICI through the iteration. To mitigate error propagation, comb-type pilots are adopted in the two proposed schemes. Simulation results show that the proposed schemes have better performance than the conventional schemes. Keywords-time-variant, OFDM, Wiener filter, Kalman filter
I.
for time-variant channels: the 2-dimensional Kalman filtering (2-D KF) and iterative Wiener filtering (IWF) schemes. In the 2-D KF scheme, the channels at comb type pilots are estimated by using the data not corrupted by the ICI. In the IWF scheme, iteration is used to calculate the ICI. The rest of the paper is organized as follows. In section II, we give system model and pilot arrangement. In section III, estimation schemes for the time-variant channels are proposed. Simulation results are presented in section IV and conclusions are drawn in section V. II.
SYSTEM MODEL AND PILOT ARRANGEMENT
INTRODUCTION
Rapid data transmission induces intersymbol interference (ISI). Orthogonal frequency division multiplexing (OFDM) is an effective system to reduce the ISI by transmitting high-rate serial data into low-rate parallel subcarriers. When the channel fading is time-variant, the performance of the OFDM system is severely degraded due to the intercarrier interference (ICI). In most papers, the channel estimation schemes for the OFDM systems are based on the assumption that the channel fading is quasi-static [3]. The estimators for the quasi-static fading channels have an error floor in time-variant channels. In order to mitigate the error floor, all channel variations within the OFDM symbol need to be estimated. However, it is impossible to estimate all channel variations even if the pilots are allocated to all subcarriers of the OFDM symbol, because there are more unknown variables than known equations [7]. To overcome this problem, the channel variations are tracked in time domain by using the Kalman filter in [1]. As another approach, the channel responses are interpolated between sample times by using the Wiener filter in [2]. In this paper, we propose two channel estimation schemes This work was supported in part by the ITRC Program and the Brain Korea 21 Project.
1-4244-0063-5/06/$2000 (c) 2006 IEEE
A. System Model Consider the OFDM system with N subcarriers and L guard intervals. The transmitted multicarrier signal at sample time n is given by N −1
s (n) = ∑ dk e i 2π nk / N ,
− L ≤ n ≤ N − 1 , (1)
k =0
where dk is the data for the k th subcarrier. The transmitted signal experiences the Rayleigh fading. The impulse response of the Rayleigh fading channel for the l th path at sample time n is described by h(n, l ) . Let H k (n) denote the Fourier transform of the channel L impulse response at sample time n , i.e., H k (n) = ∑ l = 0 h(n, l ) e − i 2π lk / N , then the k th subcarrier output from the FFT is given by Hk =
1 N
N −1
∑H n=0
k
( n) .
(2)
At the receiver, the received signal at sample time n is given by y ( n) =
1
P −1
∑d H N k =0
k
k
( n) e
i 2π nk / N
+ w(n ) , 0 ≤ n ≤ N − 1 , (3)
where w(n) is an additive white Gaussian noise (AWGN).
(4)
where y = [ y (0) y (1)... y ( N − 1)]T , d = [d0 d1 ... d N −1 ]T , w = [w(0) w(1)... w( N − 1)]T , and H1 (0) H 0 (0) H1 (1)e j 2π / N 1 H 0 (1) H= # # N ( 1) ( 1) H N H N e j 2π ( N −1) / N − − 1 0
. " # " H N −1 ( N − 1)e j 2π ( N −1)( N −1) / N "
H N −1 (0)
"
H N −1 (1)e j 2π ( N −1) / N
The received signal is demodulated using the FFT. In timevariant channels, the received signal is corrupted by the ICI through the FFT which increases an error floor in proportion to the normalized Doppler frequency. Hence, in order to mitigate the performance degradation, the channel needs to be estimated in time domain. Let α k denote the ICI, i.e., N −1 N −1 α k = 1/ N ∑ m = 0,m ≠ k d m ∑ n =0 H m (n) exp[ j 2π n(m − k ) / N ], then the demodulated signal for the k th subcarrier is given by Yk = dk H k + α k + Wk
(5)
where Wk = 1/ N ∑ n = 0 w(n)e− j 2π nk / N . B. Pilot Arrangement In the ‘data symbol,’ data are allocated to all subcarriers of the OFDM symbol. In the ‘pilot symbol,’ pilots are allocated to all subcarriers of the OFDM symbol. Consider three pilot arrangements shown in Fig. 1. Fig. 1(a) shows the block type pilot arrangement in which pilots are allocated to all subcarriers of the selected OFDM symbols [6]. Since the channel is estimated under the assumption that the channel does not change in the adjacent OFDM symbols, the block type pilot arrangement is especially suitable for the slow fading channel. Fig. 1(b) shows the comb type pilot arrangement in which pilots are allocated to equi-spaced subcarriers of the OFDM symbol. It has been shown that the comb type pilot arrangement is suitable for the relatively fast fading channel [6]. Fig.1(c) shows the grouped comb type pilot arrangement. N −1
III. TIME-VARIANT CHANNEL ESTIMATION SCHEMES In a time-variant channels, it is desired to estimate all elements of the channel matrix H from the received vector y . Since there are only N equations for N 2 unknowns, all elements of the channel matrix can not be estimated even if the pilots are allocated to all subcarriers of the OFDM symbol. We assume that the average channel response given H k is equal to the channel response of the N / 2th sample time given H k ( N / 2) . We propose, for the time-variant channels, two channel estimation schemes: 2-D Kalman filtering (2-D KF) scheme and iterative Wiener filtering (IWF) scheme. A. 2-D Kalman Filtering (2-D KF) scheme In the time domain Kalman filter, it is necessary to adopt the block type pilot arrangement to track channel variations within the OFDM symbol. In the previous scheme [1], in which channel variations are estimated by using the time domain Kalman filter, extrapolation using polynomial model is used for the channel estimation at the data symbol. Since the extrapolation is conducted under the assumption that the regression coefficient vector of the polynomial model does not change in the succeeding OFDM symbols, there is some performance degradation due to the error propagation. symbol subcarrier
y = Hd + w
It has been shown that the pilot grouping in frequency domain is suitable for the time-variant channel [2]. As shown in Fig. 1(c), the grouped pilots are divided into center and side pilots.
subcarrier
The vector form of the received signal is given by
n-th OFDM symbol
sample
time
time
(a) Block type,
(b) Comb type,
symbol
time pilot
Fig. 1. Pilot arrangement.
H k (n )
IFFT
frequency
frequency
frequency
Time-domain KF n-th OFDM symbol
Hk
Freq-domain KF
data
k-th subcarrier
side pilot
(c) Grouped comb type.
Block type pilot
Fig. 2. Pilot arrangement of the 2-D KF.
Comb type pilot
0 0 h(0, L − 1) " " h(0,0) h (1,1) h (1,0) 0 0 h(1, L − 1) " # # h( L − 1, 0) 0 0 " " h ( L − 1, L − 1) # # 0 0 0 h ( N − 1, L − 1) " "
" " 0 0 h(0, L − 1) h(0, 0) " h(1,1) h(1, 0) 0 0 h(1, L − 1) # # " " h ( N / 2, 0) 0 0 h( N / 2, L − 1) # # 0 " 0 " 0 h( N − 1, L − 1)
h(0,1) h(1, 2) # 0 " # " h( N − 1,0)
" "
h(0,1) h(1, 2) # " 0 # " h( N − 1,0) "
H1 (1) " H1 (n) " H1 ( N ) # % # H k (1) Hk (n) Hk ( N ) % # # H N (1) " H N (n) " H N ( N )
h(0, l )
"
h( N − 1, l )
H1 # Hk # H N
"
h(0, l )
h( N − 1, l )
h( N − 1, l )
h(0, l )
"
"
"
"
Fig. 3. Flow graph of the 2-D KF.
In order to improve the performance, the comb type pilot arrangement needs to be adopted in addition to the block type pilot arrangement. In the 2-D KF scheme, by adopting the assumption stated above, the channels at data symbols can be estimated as if the time domain Kalman filter is used. Since the spacing between the pilot symbols is reduced in inverse proportion to the normalized Doppler frequency, the additional adoption of the comb type pilot arrangement does not give a significant effect to the pilot density. Fig. 2 shows the pilot arrangement of the 2-D KF scheme. The time domain KF is used for the block type pilot and the frequency domain KF is used for the comb type pilot. In the 2-D KF scheme, the channels at comb type pilots are estimated by using the data not corrupted by the ICI. The flow graph of the 2-D KF scheme is shown in Fig. 3. The procedure of the 2-D KF scheme is as follows: 1) Estimate the channel impulse response at the pilot symbol by using the time domain Kalman filter given h(n, l ) . 2) Transform the channel impulse response to the channel frequency response by using the FFT given H k (n) . 3) Take an average of the channel frequency responses over the symbol duration of the OFDM symbol in order to obtain H k . 4) Track the average channel frequency response at the comb type pilot of the succeeding OFDM symbols by using the frequency domain Kalman filter. The frequency domain Kalman filter takes the average channel frequency response estimated from the step 3) as the input. 5) Interpolate the average channel frequency response between pilot subcarriers. 6) Regard the average channel frequency response as the channel frequency response of the N / 2 th sample time
given H k ( N / 2) . 7) Transform the channel frequency response of the N / 2 th sample time to the channel impulse response of the N / 2 th sample time. 8) Interpolate the channel impulse response between the N / 2 th sample time of the current OFDM symbol and that of the succeeding OFDM symbol. To implement the 2-D KF scheme, the time domain and frequency domain Kalman filtering algorithms are needed. The Kalman filtering algorithm is given in Table I. The coefficients of the AR equation and the covariance matrix of the noise are determined through the Yule-Walker equation [5]. In order not to increase the computational complexity, only the first order AR model is considered. TABLE I KALMAN FILTERING ALGORITHM
Kalman Filtering yn = C(n)T xn + v2 (n) time − domain KF x = F(n + 1,n)xn-1 + v1 (n) − state space model: n Yn = Cn xn + v2 (n) x = F (n + 1, n) x + v (n) freq − domain KF n n −1 1 F (n + 1, n) = diag{J 0 (2π f d Ts )} time − domain KF −transition matrix : freq − domain KF F( n + 1, n) = J 0 (2π f d NTs ) T C(n) = [ sn sn −1 ... s n− L+1 ] time − domain KF −measurement vector : freq − domain KF Ck = dk
Q ( n) = (1 − F (n + 1, n)2 ) × diag{σ h2 (n)} time − domain KF −correlationof v1 (n) : 1 2 freq − domain KF Q1 (n) = 1 − F(n + 1, n) 2 −correlationof v2 (n) : Q2 (n) = σ n G (n) = F(n + 1, n)K (n, n − 1)CH (n)(C(n)K ( n, n − 1)CH ( n) + Q2 (n)) −1 α (n) = y (n) − C( n) x ( n | y ) n −1
x (n + 1| yn ) = F(n + 1, n)x (n | yn −1 ) + G (n)α (n) K ( n) = K (n, n − 1) − F (n + 1, n)G (n)C( n)K (n, n − 1) K ( n + 1, n) = F(n + 1, n)K ( n)F H ( n + 1, n) + Q1 (n) where xn = [hn,0 hn,1 hn,2 ...hn ,L−1 ]T and xn = H n
H1 (1) # H k (1) # H (1) N
h(0, l )
"
H 1 (n )
"
H k (n )
"
% "
% "
H N ( n)
"
h( N / 2, l )
h (0, 0) h (1,1) # h ( N / 2, L − 1) # 0
H1 ( N ) # Hk (N ) # HN (N )
h( N − 1, l ) h(0, l )
0
"
0
h(0, L − 1)
"
"
h (1, 0)
0
"
0
h (1, L − 1)
"
"
h( N / 2, 0)
0
"
0
"
h( N − 1, L − 1)
"
# # "
h( N / 2, l )
0
"
0
h( N − 1, l ) h(0, l )
# 0 # h( N − 1, 0) h (0,1)
h(1, 2)
h( N / 2, l )
h( N − 1, l )
Fig. 4. Flow graph of the IWF.
B. Iterative Wiener Filtering (IWF) scheme Since the Wiener filter does not have tracking capability, the channel can not be estimated directly in time domain. In the previous work [2], the channel is estimated by using the interpolation between selected sample times. However, there is some performance degradation as the channels at selected sample times are estimated based on the sense of least squares. In the IWF scheme, the grouped comb-type pilot arrangement is adopted and the ICI is mitigated by using the iteration. In the first iteration, the ICI for the center pilot is calculated. In order to calculate the ICI, transmitted data of all subcarriers are needed. However, since the ICI has a significant effect to the adjacent subcarriers and the pilot grouping in frequency domain is suitable for the time-variant channels, the grouped pilots are divided into center and side pilots and only the side pilots are used to calculate the ICI for the center pilot. Then, the calculated ICI is subtracted from the received signal. In the second iteration, the received signal at the center pilot is used for the channel estimation. The flow graph of the IWF scheme is shown in Fig. 4. The procedure of the IWF scheme is as follows: 1) Estimate the channel frequency response in the sense of i k . By the central limit theorem, least squares given H the ICI is regarded as an additive noise. The channel frequency response is obtained as i k = Yk . H dk 2)
(6)
l k ( N / 2) . error (MMSE) given H 4) Interpolate the channel frequency response of the N / 2 th sample time between pilot subcarriers. 5) Transform the channel frequency response of the N / 2 th sample time to the channel impulse response of the N / 2 th sample time given h ( N / 2, l ) . Since most of the energy in the channel impulse response is contained in the first L taps, the computational complexity is reduced. In addition, since the time domain correlations are available, the performance is improved. 6) Interpolate the channel impulse response between the N / 2 th sample time of the current OFDM symbol and that of the succeeding OFDM symbol by using the time domain correlation. The channel impulse response at the remaining sample time is given by −1 h (n, l ) = Rh h Rhh h ( N / 2, l )
where h ( N / 2, l ) = [h ( N / 2, l )1stOFDM h ( N / 2, l ) 2ndOFDM h ( N / 2, l )3rdOFDM ]T , Rh h = E{h (n, l )h ( N / 2, l )}, and Rhh = E { h ( N / 2, l ) h ( N / 2, l )} . 7) Calculate the ICI using the side pilots. 8) Subtract the calculated ICI from the received signal at the center pilot. 9) Estimate the channel frequency response in the sense of least squares by using the received signal at the center pilot obtained from the step 8). The channel frequency response is given by
Regard the obtained channel frequency response as the
channel frequency response of the N / 2 th sample time i k ( N / 2) . given H 3) Estimate the channel frequency response of the N / 2 th sample time in the sense of minimum mean square
(7)
i k = Yk − α k . H dk
10) Go to the step 2).
(8)
C. Comparisons between IWF and 2D-KF The IWF scheme is compared with the 2D-KF scheme in Table II. TABLE II COMPARISONS BETWEEN IWF AND 2-D KF IWF
ICI mitigation
Channel estimation stage Pilot arrangement
Regard the ICI as an additive noise. → Calculate the ICI. → Subtract the ICI from the received signal. After the OFDM demodulation Grouped comb
2-D KF Estimate the channel in time domain. Before the OFDM demodulation Block+Comb (a)
IV. SIMULATION RESULTS Consider the 2-D KF and IWF schemes for the OFDM systems over time-variant channels. Assume the channel statistics are available at the receiver. Suppose the number of subcarriers is 1024, and the normalized Doppler frequency is 0.01. The SER of the 2-D KF and IWF schemes is compared with the conventional KF [1] and conventional WF [2] schemes in Fig. 5 (a) and Fig. 5 (b) for typical urban (TU) delay profile. Fig. 5(a) shows the performance of the 2-D KF scheme. It is shown that the 2-D KF scheme has the SNR gain of about 1.5dB over the conventional KF scheme for the TU delay profile at the SER of 10 −1 as the 2-D KF scheme mitigates error propagation. Fig. 5(b) shows the performance of the IWF scheme. It is shown that the IWF scheme has the SNR gain of about 1.5dB over the conventional WF scheme at the SER of 10 −1 as the IWF subtracts most of the ICI through the iteration. From the Fig. 5(a) and the Fig. 5(b), it is shown that the 2-D KF scheme has better performance than the IWF scheme. It is because the 2-D KF scheme estimates the channel in time domain. V. CONCLUSIONS In this paper, we proposed the 2-D KF and IWF schemes over the time-variant channels. The 2-D KF scheme mitigated error propagation by using the frequency domain Kalman filter and the IWF scheme subtracted the ICI by using the iteration. Since one dimensional filters are used for the 2-D KF and the IWF schemes, the performance is improved with little increase in complexity. It is shown that the 2-D KF and IWF schemes have better performance than the conventional schemes.
(b) Fig. 5. Performance comparisons over the channels with the TU and the HT delay profiles. (a) 2-D KF, (b) IWF.
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