This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2010 proceedings
An EM Algorithm-Based Channel Estimation for OFDM Amplify-andForward Relaying Systems Jeng-Shin Sheu* and Wern-Ho Sheen** *Department of Computer Science and Information Engineering National Yunlin University of Science & Technology Douliou, Yunlin, Taiwan
[email protected] ** Department of Information and Communication Engineering Chaoyang University of Technology 168, Jifong East Road, Wufong Township, Taichung County 41349, Taiwan Abstract—The issue of channel estimation is investigated for the orthogonal frequency-division multiplexing (OFDM) relaying system with amplify-and-forward (AF) operation mode. Channel estimation in the AF mode is unique in that both the source-torelay and relay-to-destination channels are needed at the destination in order to perform optimum combining of the received signals from the direct (source-to-destination) and relay paths if a cooperative diversity transmission is employed. Until now, however, there have been only literatures to study the estimation of the composite channel of the relay path. In this work, an iterative expectation-maximization (EM) algorithmbased channel estimation is proposed for respective estimations of the source-to-relay and relay-to-destination links at the destination. Computer simulations show that the iterative channel estimator performs satisfactorily and converges rapidly.
In a typical three-node, two-phase (timeslot) relaying system, the source transmits a packet in the first phase, and the relay forwards the packet to the destination in the second phase. The destination may or may not listen to the packet from the source in the first phase and the source may or may not transmit in the second phase, depending on the cooperative protocols [3]-[5]. Furthermore, relay can use different modes for forwarding the packet. Two modes (and their variants) have drawn much research interest: one is amplify-and-forward (AF), and the other is decode-and-forward (DF). In the AF mode, the relay amplifies the received signal and forwards it to the destination, whereas in the DF mode, the relay decodes the received signals, re-encodes and forwards it to the destination. Each mode of operations has merits against the other under different system setups as discussed in [3]-[5].
Keywords- Channel estimation, amplify-and-forward OFDM relaying systems, expectation maximization algorithm
Channel estimation in the DF relaying systems is done in each respective link, that is the source-to-destination link, the source-to-relay link, and the relay-to-destination link. Therefore, the methods developed for the traditional point-topoint systems apply in this case as well. For the AF relaying systems, however, the channel estimation of the relay path consisting of the source-to-relay and relay-to-destination links imposes much challenge because of the non-Gaussian nature of the cascaded channel gain and noise [6].
I.
INTRODUCTION
Using relays in a cellular network is an emerging technology to improve system coverage, user throughput, and save the transmit power of mobile stations by reducing the propagation loss between a mobile station and a base station [1]-[2]. Relays can also be used to facilitate cooperative diversity where the source and relay transmit the data for destination in a cooperative way to increase the diversity order. Cooperative diversity is an effective technique in combat of the shadowing and multi-path fading in wireless communication environments, especially if the source, relay and destination are equipped with one antenna [3]-[5]. OFDM (orthogonal frequency division multiplexing), on the other hand, is an efficient modulation scheme to combat inter-symbol interference (ISI) in high data-rate communications. By using parallel orthogonal sub-carriers along with cyclic-prefix, ISI can be removed completely as long as the cyclic-prefix is lager than the maximum delay spread of the channel. In addition, OFDM-based multiple access scheme such as OFDMA (orthogonal frequency division multiple access) has been widely considered as one promising multiple access scheme for broadband cellular systems. OFDMA has been adopted in the 3GPP-LTE (long term evolution) and the IEEE 802.16e specifications. In this work, we are concerned with the channel estimation in the OFDM relaying systems.
There have been a few works addressing the channel estimation issue for the AF relay path [6]-[11]. In [6]-[9], the composite channel of the relay path was estimated at the destination for the flat-fading channels. In particular, the composite channel was estimated under the linear minimum mean square error (LMMSE) criterion in [6]. (The optimal MMSE estimator is too complex because of the non-Gaussian nature of the cascaded channel gain and noise.) In [7], the authors extended the LMMSE channel estimator for relay networks with time division multiplexing (TDM). In [8], the authors proposed a low rank MMSE (LrMMSE) estimator based on the singular value decomposition (SVD) in order to avoid the inverse operation of channel correlation matrix in LMMSE estimator. In [9], by treating the noise at the destination as a Gaussian variable under a specific overall channel realization, channel estimators were derived based on the least squares (LS) and MMSE criteria. In [10]-[11], two channel estimators were developed for frequency-selective fading channels; in [10], a channel estimator was developed in the frequency-domain based on the Cholesky factorization of
This work is supported by the National Science Council, Taiwan, R.O.C., under Grant NSC 98-2220-E-224-010.
978-1-4244-6404-3/10/$26.00 ©2010 IEEE
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2010 proceedings
channel matrices, and in [11] the LrMMSE estimator based on SVD was devised for the multiple AF relays-assisted broadband cooperative diversity systems. Note that except of frequency-domain method [10], the other time-domain methods mentioned above assumed the perfect knowledge of multipath intensity profile. So far, as discussed above, the channel estimation for the AF relay path has been focused on the estimation of the composite channel at the destination. In a cooperative diversity transmission, however, the respective channels of the sourcerelay and relay-destination links are needed at the destination in order to do optimal combination of signals received from the direct and relay paths [3]-[4]. In addition to the optimal combination, the respective link information also plays a crucial role in link adaptation, channel reciprocity exploitation, etc. In this paper, we address this important issue of the channel estimations for both the source-to-relay and relay-todestination links in the OFDM relaying systems. An iterative channel estimator based on the expectation-maximization (EM) algorithm is proposed and evaluated by computer simulations. Numerical results show that the iterative estimator performs satisfactorily in the signal-to-noise ratios (SNRs) of interest and converges rapidly by exploiting detected symbols in a decisiondirected mode. The rest of the paper is organized as follows. Section II describes the system model. Section III presents the EM algorithm-based channel estimator. Simulation results and conclusion are given in Sections IV, and V, respectively.
r
where the set of Lr path gains {hk } denote the channel impulse response (CIR) of the source-to-relay link and are modeled as zero-mean, uncorrelated, complex Gaussian variables, wnr is r the received white Gaussian noise, and Hk is the corresponding channel frequency response (CFR) over kth subcarrier. After DFT, the signal at kth sub-carrier takes the form Ykr = H kr d k + Wkr , r
where Wk is the AWGN noise with the variance of σ W2 . The r
relay amplifies Yk , takes IDFT and forwards it to the destination after insertion of a cyclic prefix. The signal coming out of IDFT is given by
(
xnr = 1
(
xn = 1
N
)∑
N −1 k =0
d k e j 2π kn N , 0 ≤ n ≤ N − 1,
(1)
where {dk} are zero-mean, i.i.d. data with an average power of P. A cyclic prefix is appended to the sequence {xn} before transmission. The cyclic prefix is assumed longer than the maximum delay spread of the overall channel, and therefore there is no inter-OFDM symbols interference. In the second phase, first, the relay performs DFT and amplification. Then the packet is forwarded to the destination after performing IDFT and insertion of a cyclic prefix. The operation of IDFT and DFT allows the relay to amplify the OFDM signal on a sub-carrier basis, as to be discussed. The received signal at the relay (after removal of cyclic prefix) is given by
ynr = ∑ l =r 0 hlr xn − l + wnr L −1
(
= 1
N
)∑
N −1 k =0
H kr d k e j 2π kn N + wnr , 0 ≤ n ≤ N − 1,
(2)
)∑
N −1 k =0
α k Ykr e j 2π kn N , 0 ≤ n ≤ N − 1,
(4)
(
)
2 α k = P P H kr + σ W2 . In case of α k = α , called per-OFDM
basis AF, the operations of DFT and IDFT at the relay can be omitted. At the destination during the second phase, the received signal takes the form ynd =
SYSTEM MODEL
We consider the relay path of a two-phase, half-duplexing OFDM relaying system, where the source transmits a packet of OFDM symbols in the first phase, and the relay forwards it to the destination in the second phase. At the source, the signal coming out of N-point IDFT (inverse discrete Fourier transform) for a particular OFDM symbol is
N
where α k is the amplifying factor for sub-carrier k which can be fixed or variable as in [6]. The variable gain is given by
Ld −1
∑h
xnr −l + wnd
1 N
∑⎜ ∑ h e
l =0
II.
(3)
=
d l
N −1
⎛ Ld −1
k =0
⎝
1 N
+
d − j 2π kl N l
l =0
N −1
⎛ Ld −1
k =0
⎝
⎞ ⎛ Lr −1 r − j 2π kl ' ⎟ ⎜ ∑ hl ' e ⎠ ⎝ l' =0
∑⎜ ∑ h e
d − j 2π kl N l
l =0
⎞ r j 2π kn ⎟ α kWk e ⎠
N
N
⎞ j 2π kn N ⎟α k dk e ⎠ + wnd ,
0 ≤ n ≤ N −1
(5)
r
where the set of Ld path gains {hk }, the CIR of the relay-todestination link, are zero-mean, uncorrelated complex Gaussian d variables, and the set of noise samples {wn } denotes the i.i.d. 2 AWGN with variance σ d . w
Equation (5) can be written in the following vector-matrix form. y d = ⎡⎣ y0d , y1d ,…, y Nd −1 ⎤⎦
{
T
(6)
}
= U H ADdiag U Lr h r U Ld h d + U H AW r U Ld h d + w d ,
where U is the N-by-N matrix, whose (p, q) entry, [U]p,q, is
(1
)
N e
− j 2π ( p −1)( q −1) N
, p, q = 1, 2,…, N. U Ld is an N-by-Ld
matrix, whose (p, q) entry is e ( )( ) . The superscripts T, and H represents transposition and Hermitian transposition, respectively. Furthermore, we use diag{b} to stand for a diagonal matrix with the vector b along its main diagonal. Define m and a to be N-by-1 column vectors whose kth r element are Wk and α k , respectively. The others are defined as − j 2π p −1 q −1 N
r
follows: A = diag{a}, D = diag{d0,d1,…,dN–1}, W = diag{m},
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2010 proceedings
r
r
r
r
T
d
d
d
d
T
d
d
h = [h0 , h1 ,.., h Lr–1] , h = [h0 , h1 ,.., h Ld–1] , and w = [w0 , d T d w1 ,.., w N–1] . For later use, we further define y ds = U H ADdiag U Lr h r U Ld h d , and yWd = U H AW r U Ld h d .
{
}
THE EM ALGORITHM-BASED CHANNEL ESTIMATION In this section, an EM algorithm is developed to estimate the d r channel responses h and h and the scaling factor a at the destination. The EM algorithm is briefly reviewed first followed by the detailed derivation of the proposed method.
III.
A. EM Algorithm Suppose that Φ ∈Ω is the set of parameters to be estimated from some observed data vector y. Then, the maximumˆ of the parameter set Φ is given likelihood estimate (MLE) Φ ˆ = arg max by Φ , where p ( y Φ ) is the conditional Φ∈Ω p ( y Φ )
{
several computationally simpler conditional maximization (CM) steps. B. The Proposed Method To begin with, the observed IDFT sequence of y d given in (6) is considered as the incomplete data, the vector m as the missing data, {m, y d } as the complete data, and the parameter set Φ to be estimated is {h r , h d , a} . Now, the definition of EM algorithm is given by
{
}
In the EM algorithm, the observed data vector y is called incomplete data implying that some part of data (called the missing data) in the observation y is unobservable which makes the MLE problem intractable. The basic idea of the EM algorithm is to associate the observed data y (incomplete data) with a complete data vector x (not unique). With a well chosen complete data x, the original likelihood function p ( y Φ ) can be obtained through the likelihood function of the complete ˆ ( 0) , an EM data p ( x Φ ) . Starting from some initial estimate Φ
algorithm solves the MLE by the following iterative procedure between E-step and M-step:
(
)
}
)
(9)
It is obvious that the expectation above is done over the random vector of the missing data m. The solution of (9) can be obtained through the iteration between E-step and M-step:
(
)
(
)
ˆ ( j ) = E ⎡log p m, y d Φ y d , Φ ˆ ( i ) ⎤ ; (10) Compute Q ′ Φ Φ m ⎢ ⎥⎦ ⎣
density of the observed data vector, given the parameter set to be estimated. The MLE might be hard to find due to either the likelihood function p ( y Φ ) has no closed-form or the
calculation of MLE is prohibitively complex in practical implementation. The EM algorithm has been known as an efficient iterative procedure for solving the MLE problem [10].
(
ˆ (i +1) = arg max E ⎡log p m, y d Φ y d , Φ ˆ (i ) ⎤ . Φ Φ∈Ω ⎣ ⎦
(
)
ˆ ( j +1) = arg max Q ′ Φ Φ ˆ ( j) . Solve Φ Φ∈Ω
(11)
In (10), after applying the chain rule of probability and dropping the term independent of Φ , the E-step can be reduced ˆ ( i ) Em ⎡log ⎡ p y d Φ, m ⎤ ⋅ p y d Φ ˆ (i ) , m ⎤ . We now as Q Φ Φ ⎦ ⎣ ⎣ ⎦ ˆ (i ) by substituting the respective pdfs. expand Q Φ Φ
(
)
(
) (
(
)
)
Dropping the additive and multiplicative terms independent ˆ (i ) can be further replaced by of Φ , the E-step function Q Φ Φ
(
(
)
)
ˆ (i ) = Q1 Φ Φ
∫
{2 Re ⎡⎣⎢( y − y (Φ )) d
Ωm
d s
2 yWd ( Φ ) ⎤⎥ − yWd ( Φ ) − y d − y ds ( Φ ) ⎦
H
⎧ ˆ (i ) − y d Φ ˆ (i ) exp ⎨ − ⎜⎛ y d − y ds Φ W ⎩ ⎝
( )
( )
2
(
σ w2 ⎟⎞ − m d
⎠
( j)
Φ∈Ω
(
)
(8)
ˆ is the estimate of Φ generated by the jth iteration. It is Φ ˆ ( j ) obtained in the EM algorithm shown that the sequence Φ
{ }
above monotonically increases the incomplete-data likelihood function [10]. The maximization step in the EM algorithm is performed with respect to all unknown parameters simultaneously, which results in a slow maximization process since it requires search over a space with many dimensions. The expectationconditional maximization (ECM) algorithm [10] is a natural extension of the EM algorithm in situations where the maximization process on the M-step is relatively simple if the maximization is undertaken conditional on some function of the parameters. That is, the ECM algorithm takes advantage of the simplicity of complete-data conditional maximization by replacing a complicated M-step of the EM algorithm with
}×
) ⎭⎫
2
σ W2 ⎬ dm,
ˆ ( j ) = E ⎡ log p ( x Φ ) y , Φ ˆ ( j ) ⎤ (7) ● E-step: Compute Q Φ Φ ⎣ ⎦
ˆ ( j +1) = arg max Q Φ Φ ˆ ( j) ● M-step: Solve Φ
2
(12) where Ωm denotes the space of m. It is not straightforward to get the closed form for the above integration. However, we make careful observations on (12), and rewrite the exponent of the exponential function. It is shown that the Eˆ ( i ) can be equivalently replaced by another more step Q1 Φ Φ
(
)
(
)
ˆ ( i ) for the maximization in M-step with concise form Q2 Φ Φ respect to Φ (see Appendix):
(
)
( )
ˆ (i ) = − y d − y d ( Φ ) − Z ( Φ ) m Φ ˆ (i ) Q2 Φ Φ s
{
(
( )) ⋅ Z
ˆ − tr Z ( Φ ) ⋅ σ K Φ 2 wd
(i )
H
2
(13)
( Φ )},
where we define the N-by-N matrix Z(Φ ) = U H Adiag{U L hd } , d
the N-by-1 mean-value vector m(Φˆ (i ) ) = K (Φˆ (i ) ) × Z H (Φˆ (i ) ) ×
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ˆ (i ) ) = ˆ ( i ) )) , and the N-by-N covariance matrix K (Φ ( y d − y ds (Φ ˆ ( i ) ) Z( Φ ˆ (i ) ) + (σ 2 d σ 2 ) I]−1 . [ Z H (Φ w
W
The M-step aims at maximizing (13) with respect to all vectors in Φ . However, the multi-dimensional maximization in (13) has a prohibitive computational complexity. For the Mstep, we thus resort to the ECM algorithm, as already mentioned. At each stage, only one parameter is updated while the others are kept constant at their most updated values. Thus r d the solutions to the CIR vectors h and h in maximizing (13) at iteration (i+1) are respectively given by
( ( ) ( ))
ˆ (i) S Φ ˆ (i) hˆ r ,( i +1) = S rH Φ r
−1
( )(
( )) (14)
ˆ (i) y d − Z ( Φ ) m Φ ˆ (i ) S rH Φ
and ˆ (i )H (σ 2 d K (Φ ˆ (i ) U ]−1 × (15) ˆ (i ) )S (Φ ˆ (i ) ) + U H A ˆ (i ) )) A hˆ d ,(i +1) = [S dH (Φ d L L w H ˆ (i ) d (S (Φ ) y ), d
ˆ (i ) ) are where both N-by-L matrices S r (Φˆ (i ) ) and S d (Φ
V.
respectively given by
( )
ˆ (i ) Sr Φ
(
})
{
ˆ (i )Ddiag U hˆ d ,(i ) U UH A Ld Lr
(16)
and
( )
ˆ (i ) Sd Φ
(
In the following simulation results, the SNR is defined on a per-hop basis, and the detected symbols are exploited in a decision-directed mode. Fig. 1 illustrates the mean square error (MSE) performance as a function of SNR. At low SNRs, the MSE of source-to-relay link is apparently worse than that of the relay-to-destination link. However, the difference decreases rapidly as the SNR increases. The MSE performance exhibits an error floor at about 4×10–3. This may be attributed to that the proposed algorithm works in the absence of the knowledge of channel statistics. Nevertheless, the MSE floor is small enough to have a satisfactory bit error rate (BER) performance, as to be discussed in Fig. 3. In Fig. 2, we show the impact of iteration number on the average MSE performance. It is observed that the proposed algorithm converges after 3 iterations. Fig. 3 illustrates the BER performance. Clearly, the proposed algorithm performs very closely to the perfect one; the degradation is less than 0.5 dB at BER 10–3 as seen in the figure. From Figs. 2 and 3, it was found that the iterative estimator performs satisfactorily in the SNRs of interest and converges rapidly by exploiting detected symbols in a decision-directed mode.
{
{ ( )}) U
}
ˆ (i )Ddiag U hˆ r ,(i ) + A ˆ (i )diag m Φ ˆ (i ) UH A Lr
Ld
. (17)
ˆ (i ) depends on the estimation of the Note that the estimate A vector a, and it is only necessary for variable-gain case. r Following the estimate of the CIR vector h , a heuristic estimate of the vector a at iteration i is in the form whose kth
CONCLUSIONS
An EM-based channel estimation is proposed for the OFDMbased relaying systems with AF strategy. The proposed channel estimator provides an iterative method to separately estimate CIRs in a two-hop relay path. This is unique among the existing channel estimators for AF relaying systems which directly estimate the composite CIR at the destination. Moreover, the proposed iterative method works well in the absence of the channel statistics, such as multipath intensity profile (MIP) which is needed in the existing time-domain methods.
2
entry is given by αˆ k(i ) = P ( P H kr ,( i ) + σ W2 ), where Hˆ kr ,( i ) is the kth entry of U L hˆ r ,(i ) .
IV. SIMULATION RESULTS We consider a two-hop path, where a source node communicates with a destination node through a relay station that uses fixed-gain amplification. The BPSK modulation is applied to both the data symbols and the preamble (one OFDM symbol) that is placed at the beginning of each frame of 11 OFDM symbols. Both the source-to-relay and relay–todestination channels are Rayleigh-faded with Lr = Ld = 4. We assume equal average SNR on both hops. The channels are generated independently from frame to frame. A convolutional code with rate 1/2, memory 4 and random interleaving is applied to generate the coded symbols. In all our results, each simulation point comes from 20 simulation runs, each of which consists of 66 frames.
MSE
Initially, the diagonal data matrix D in (6) is composed of training symbols in the preamble. After the initial channel estimation, the proposed algorithm can exploit the detected data symbols and perform in a decision-directed mode.
source-to-relay link relay-to-destination link average
0.1
r
0.01
1E-3 0
2
4
6
8
10
12
14
16
18
20
SNR (dB)
Fig. 1.
MSE performance as a function of SNR
22
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2010 proceedings
incomplete data yd can be re-expressed in terms of missing data m, as yWd = Z ( Φ ) m . Then we have
0.1
{
yWd ( Φ ) = tr Z ( Φ ) m ( Z ( Φ ) m ) 2
H
}.
(A2)
Average MSE
Substituting (A1) and (A2) into (12), we are led to iteration increasing
∫
Ωm
0.01
0
2
4
6
d
× e
1 iteration 2 iterations 3 iterations 4 iterations 5 iterations 6 iterations
1E-3
{2 Re{( y − y (Φ )) d s
} {
Z ( Φ ) m − tr Z ( Φ ) m ( Z ( Φ ) m )
H ⎛⎧ ˆ (i ) ⎤ K −1 Φ ˆ (i ) ⎡ m − m Φ ˆ (i ) ⎤ + y d − y d Φ ˆ (i) −⎜⎜ ⎨⎡ m −m Φ s ⎢ ⎦⎥ ⎣⎢ ⎦⎥ ⎝ ⎩⎣
( )
( )
( )
( )
2
−m
H
H
}− y
d
ˆ (i ) K −1 Φ ˆ (i) m Φ ˆ ( i ) ⎫⎬ ⎞⎟ σ 2 Φ d ⎟ ⎭⎠ w
( ) ( ) ( )
− y ds ( Φ )
2
}
dm.
(A3) The norm of effective disturbance noise in (A2) can be rewritten in the form: 8
10
12
14
16
18
20
{ } ˆ )) Z ( Φ ) Z ( Φ ) (m − m (Φ ˆ )) + = (m − m (Φ ˆ ) Z (Φ ) Z (Φ ) (m − m (Φ ˆ )) + 2 Re {m ( Φ } tr Z ( Φ ) m ( Z ( Φ ) m )
22
SNR (dB)
H
(i )
H
Fig. 2.
H
Impact of iteration number on the average MSE performance
( )
(i )
H
H
(i )
H
(i )
(A4)
( )
ˆ (i) Z H ( Φ ) Z ( Φ ) m Φ ˆ (i) . mH Φ
Using (A4) and keeping only the terms dependent on Φ , (A3) is led to (13).
1 iteration 2 iterations 3 iterations 4 iterations perfect
0.1
REFERENCES
BER
[1] 0.01
1E-3
0
2
4
6
8
10
12
14
16
18
20
22
SNR (dB)
Fig. 3.
BER Performance comparisons
APPENDIX In this Appendix, we show that the simplified E-step function defined in (13) is equivalent to that defined in (12). Using the definitions of Z(Φ ), m(Φˆ (i ) ), and K (Φˆ (i ) ), the exponent in (12) can be rewritten as
( ) ( ) + (1 σ ) m ){⎡⎣m − m ( Φˆ ( ) )⎤⎦ K ( Φˆ ( ) ) ⎡⎣m − m ( Φˆ ( ) )⎤⎦ + y ˆ ( ) ) K (Φ ˆ ( ) ) m (Φ ˆ ( ) )}. − m (Φ
(1 σ ) y 2 wd
= (1 σ w2 d
d
ˆ (i ) − y d Φ ˆ (i ) − y ds Φ W i
H
i
H
−1
−1
i
2
2 W
i
2
i
d
( )
ˆ (i ) − y ds Φ
2
i
(A1) According to (A1), it is obvious that the integration in (12) involves a pdf of an N-dimensional complex Gaussian random vector with mean vector m(Φˆ (i ) ) and covariance matrix ˆ (i ) ) . The effective disturbance noise from relay, y d , in the K (Φ W
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