- Research Article
- Open Access
A Low-Complexity LMMSE Channel Estimation Method for OFDM-Based Cooperative Diversity Systems with Multiple Amplify-and-Forward Relays
EURASIP Journal on Wireless Communications and Networking volume 2008, Article number: 149803 (2008)
Orthogonal frequency division multiplexing- (OFDM-) based amplify-and-forward (AF) cooperative communication is an effective way for single-antenna systems to exploit the spatial diversity gains in frequency-selective fading channels, but the receiver usually requires the knowledge of the channel state information to recover the transmitted signals. In this paper, a training-sequences-aided linear minimum mean square error (LMMSE) channel estimation method is proposed for OFDM-based cooperative diversity systems with multiple AF relays over frequency-selective fading channels. The mean square error (MSE) bound on the proposed method is derived and the optimal training scheme with respect to this bound is also given. By exploiting the optimal training scheme, an optimal low-rank LMMSE channel estimator is introduced to reduce the computational complexity of the proposed method via singular value decomposition. Furthermore, the Chu sequence is employed as the training sequence to implement the optimal training scheme with easy realization at the source terminal and reduced computational complexity at the relay terminals. The performance of the proposed low-complexity channel estimation method and the superiority of the derived optimal training scheme are verified through simulation results.
Multiple-input multiple-output (MIMO) wireless communication systems have attracted considerable interest in the last few years for their advantages in improving the link reliability, as well as increasing the channel capacity [1, 2]. Unfortunately, it is not practical to equip multiple antennas at some terminals in wireless networks due to the cost and size limits. To overcome these limitations, the concept of cooperative diversity has been recently proposed for single-antenna systems to exploit the spatial diversity gains in wireless channels [3–6]. Utilizing the broadcasting nature of radio waves, the source terminal can cooperate with the relay terminals in information transport. In this manner, the spatial diversity gains can be obtained even when a local antenna array is not available.
Currently, several cooperative transmission protocols have been proposed and can be categorized into two principal classes: the amplify-and-forward (AF) scheme and the decode-and-forward (DF) scheme. In the AF scheme, the relay terminals amplify the signals from the source terminal and forward them to the destination terminal. In the DF scheme, the relay terminals first decode their received signals and then forward them to the destination terminal. Compared with the DF scheme, the AF scheme is more attractive for its low complexity since the cooperative terminals do not need to decode their received signals. Hence, we focus our attention on the AF relay scheme in this paper.
To take the advantages that cooperative transmission can offer, accurate channel state information (CSI) is usually required at the relay and/or destination terminal. For example, if distributed space-time coding (DSTC) is applied at the relays, then the accuracy of CSI of all links at the destination terminal is crucial for the improvement of the system performance. The training-sequences-aided method is one of the most widely used approaches to learn the channel in wireless communication systems due to its simplicity and reliability . However, there have been only a few literatures on training-based AF channel estimation, and research in this area is still in its infancy. Based on the assumption of flat-fading channels, [8, 9] propose training-sequences-aided least square (LS) and linear minimum mean square error (LMMSE) channel estimators for single-relay-assisted cooperative diversity systems in cellular networks. In [10, 11], minimum variance unbiased (MVU) and LS channel estimators are introduced respectively for orthogonal frequency division multiplexing (OFDM-) based single-relay-assisted cooperative diversity systems over frequency-selective fading channels. The channel estimators developed in these literatures only consider the single-relay-assisted cooperative communication scenario. Training designs that are optimal in the scenarios of multiple-relays-assisted cooperative communication have drawn relatively little attention. It was investigated for the case of multiple-relays-assisted AF cooperative networks over frequency-flat fading channels in  using the channel estimation performance bound as a metric for training design. It was found that the optimal training can be achieved from an arbitrary sequence and a set of well-designed precoding matrices for all relays. In this study, we are interested in the broadband cooperative communication scenarios, for example, the real-time video surveillance application in distributed sensor networks . As the broadband applications demand high-speed data transmission, the frequency-flat channels become time-dispersive when the transmission bandwidth increases beyond the coherence bandwidth of the channels. Thus, how to obtain the accurate CSI in a low-complexity manner for multiple AF-relays-assisted broadband cooperative diversity systems could be a challenge problem and has not been satisfactorily addressed, which motivates our present work.
In this paper, we propose a training-sequences-aided LMMSE channel estimation method for OFDM-based cooperative diversity systems with multiple AF relays over frequency-selective block-fading channels. First, the mean square error (MSE) bound on the proposed method is computed. Then, the optimal training scheme with respect to this bound is derived. By exploiting the inherent orthogonal characteristic of the optimal training scheme, we utilize the optimal training sequence as the singular vector to decompose the channel correlation matrix and then introduce an optimal low-rank channel estimator based on singular value decomposition (SVD) [14, 15]. Since we avoid the matrix inverse operation, the computational complexity at the destination terminal is reduced significantly. Furthermore, the Chu sequence is employed as the training sequence at the source terminal to achieve the minimum MSE estimation performance while avoid the complex matrix multiplication operation at the relay terminals. Simulation results verify the performance of the low-complexity channel estimation method in the multiple AF relays-assisted broadband cooperative communication scenario. And the superiority of the derived optimal training scheme is also confirmed.
This paper is organized as follows. Section 2 describes the channel and system model. We introduce the low-complexity LMMSE channel estimation method in Section 3. In Section 4, we design the optimal training scheme. Simulation results and discussions are given in Section 5, followed by our conclusions in Section 6.
, , , , , and denote inverse, transpose, Hermitian transpose, modulo-, element-wise production, and convolution operation, respectively. stands for a diagonal matrix with on its diagonal. denotes an arbitrary nonminus integer less than . denotes expectation, denotes the trace of a matrix, denotes the th entry of a vector. denotes the identity matrix of size , and denotes all-zero matrix of size . Bold uppercase letters denote matrices and bold lower-case letters denote vectors.
2. Channel and System Model
2.1. Channel Model
As shown in Figure 1, the wireless cooperative diversity systems we consider consist of terminals which are placed randomly. We assume that all the terminals are equipped with only one antenna and work in the half-duplex mode, that is, they cannot receive and transmit simultaneously. Introduce the variables, , , and , to depict the large-scale path loss of the links , , and . Let and be the geometric gains of the link and relative to the direct transmission link . The small-scale channel impulse response of each wireless link is modeled as a tapped delay line with tap spacing equal to the sample duration :
where represents the number of resolvable paths for the link and denotes the channel gain of the path of the link . is described by a zero-mean complex Gaussian random process, which is independent for different paths with variance . We normalize the channel by letting . Denote the channel power vector of link as . Since the spacing between each terminal is generally larger than the coherent distance, all the signals transmitted from different terminals and received at different terminals are assumed to undergo independent fades. We assume that the channel remains constant over the transmission of a frame but varies independently from frame to frame, and then drop the time index for brevity in the following sections.
2.2. System Model
In this paper, a simple bandwidth-efficient two-hop AF protocol is adopted for communications in the cooperation systems. Specifically, the source terminal broadcasts the blockwise information to the relay terminals , where , in the first time slot. Then these relays perform DSTC via multiplying their received blockwise signals with local matrix and forward the coded signals to the destination terminal simultaneously in the second time slot [16–20]. Since the channel between terminal and terminal is the conventional single-input single-output (SISO) one and can be separately estimated in the first time slot, the direct transmission link is omitted in our discussion. Later, it will be shown that the training sequence employed by this channel estimation method can also achieve the optimal estimation performance for this direct SISO link. For combating the intersymbol interference from multipath channels, cyclic prefixes (CPs) at the source terminal and relay terminals are added to the information and the length of CPs should be more than the maximum number of multipath to undergo in each time slot. As OFDM can turn frequency-selective fading channel into several parallel frequency-flat ones, cooperative communication in time-dispersive channels is applicable by extending some DSTC methods, for example, the work in , to corresponding subcarriers at each relay in a form of OFDM symbol blockwise transmission. Since multiplying OFDM symbol in the time domain is equal to multiplying each subcarrier in the frequency domain, the requirement of DFT and IDFT operation at the relay terminals can be relaxed. Then terminal requires the knowledge of channel frequency responses of concatenation links, , , to decode the received signals. Equivalently in the time domain, terminal needs to know , where , which will be discussed in the next section.
3. Low-Complexity LMMSE Channel Estimation Method
3.1. LMMSE Channel Estimation Method
This subsection proposes a training-based method for channel estimation of multiple AF-relays-assisted cooperative diversity systems in the simple bandwidth-efficient two-hop protocol. Suppose the time-domain training sequence with unit power, which is transmitted from the source terminal in the first time slot, is denoted by the vector . Before transmission, this vector is preceded by a CP with length . We assume that , where . After removing the CP, the received vector by relay terminal can be written as
where is a circulant matrix with the first column given by ; is the complex additive white Gaussian noise (AWGN) at terminal with zero-mean and variance . As performing DSTC in the data transmission section, terminal is also assumed to forward a linear function of its received signal vector in the training section that is given by
where is a linear transformation unitary matrix to ensure channel identifiable, as explained later; is the relay amplification factor to meet the power constraint for each relay terminal and is given by
where is the transmission power at terminal . The factor considered in this paper does not depend on the instantaneous channel realization [21, 22], thus no channel estimation is required at the relay terminals. In the second time slot, each relay terminal appends a CP with length to and transmits it to the destination terminal . It is assumed that , where . Terminal collects signals from relay terminals, and the received vector after removing the CP can be written as
where is a circulant matrix with the first column given by ; is the complex AWGN at terminal with zero-mean and variance . Introduce the variable and let be the training sequence of terminal . Then, (5) becomes
where is a circulant matrix with the first column given by ; the effective noise term is denoted by . Denote the circulant training matrix of terminal as whose first column is equal to , then (6) can be rewritten as
where and .
Note that, the channel vector is identifiable if and only if has full column rank, which occurs when
If terminal only forwards a untransformed version of its received signals, or equivalently , the columns of are in proportion which would cause the column rank of deficient and then channel vector is unidentifiable. Consequently, the unitary transformation matrix is necessary for each relay terminal in the training section. This explains those channel estimators in [10, 11] designed for broadband AF cooperative communication cannot be extended straightforwardly to the multiple relays scenario in the two-hop protocol.
Each concatenation channel , where , has taps. Denote the channel tap number of all concatenation links as . It is found from (8) that the training length should not be less than the channel tap number of all concatenation links; otherwise, the channel vector would be unidentifiable. On the other hand, given a specific training length , we can use (8) to determine the maximum relay number that this channel estimator can supply.
The simplest algorithm for the channel estimation using (7) is the LS estimator, which does not exploit a priori knowledge of channel statistics and noise power and has worse estimation performance relative to the MMSE estimator. However, it is intractable to perform MMSE channel estimation for the AF channel because the total channel is non-Gaussian. Therefore, we focus our attention on the suboptimal LMMSE channel estimator. The analysis and simulation results shown in later sections indicate that our low-complexity channel estimation method provides satisfactory performance.
Exploiting the noncorrelation property of channels of a different link , we can obtain the autocorrelation matrix of the channel vector :
We assume that the relative distances among all terminals are far enough to ensure local noise and to be uncorrelated. Using , the statistical autocorrelation matrix of the effective noise term can be written as
Since are assumed to be uncorrelated for different paths , we can obtain
By substituting (11) into (10), the statistical can be rewritten as
The autocorrelation matrix of a received signal is
Based on the LMMSE criterion , the estimated channel can be written as
And the autocorrelation matrix of estimation error is
When is rank deficient, a small value can be added to the diagonal of . Therefore, the average MSE of the LMMSE channel estimator can be represented as
For positive definite matrix with its th diagonal element given by , the following inequality holds:
where equality holds if and only if is diagonal.
Based on this lemma, the minimum of (16) is achieved if and only if is diagonal. Therefore, the optimal training scheme is
By substituting (9), (12), and (18) into (16), we obtain the MSE bound of this channel estimation method
3.2. Low-Complexity LMMSE Channel Estimator
The LMMSE channel estimator (14) is of considerable complexity since a matrix inversion is involved. To simplify this estimator, we exploit the optimal training scheme (18) to get an optimal low-complexity LMMSE channel estimator based on SVD in this subsection [14, 15].
If has orthonormal columns, then there exists such that is orthogonal.
Based on this lemma, there exists matrix to make matrix ensure , because the training matrix shown in (18) has orthonormal columns. Denote the diagonal entry of , , and as , , and . Introduce the vector . Then, the Hermitian matrix can be rewritten as
where is a unitary matrix from . Substituting (20) into yields
Using (20) and (21), LMMSE channel estimator (14) can be rewritten as
See the appendix.
Since the optimal low-rank LMMSE channel estimator (22) avoids the matrix inverse calculation, the computation complexity is significantly reduced compared with (14). Building upon a similar deduction of minimizing MSE, we find condition (18) is also the optimal training scheme for LS channel estimator. Thus, we can see that the performance of the LMMSE channel estimator (22) is equal to the Wiener-filtered LS channel estimator. When the second-order channel statistics and the noise power are not available at the destination terminal, we can resort to the LS channel estimator to obtain initial channel estimates and then use these estimates to estimate and .
4. Optimal Training
4.1. Design of the Optimal Training Scheme
In this subsection, we employ the Chu sequence to implement the optimal training scheme (18). The Chu sequence is a kind of perfect N-phase sequences which have a constant magnitude in both the time domain and the frequency domain . The constant time-domain magnitude property of the Chu sequence precludes peak-to-average power ratio (PAPR) problem in implementation while the constant frequency-domain magnitude property makes the Chu sequence invaluable in the design of the optimal training scheme of many communication systems. A length- Chu sequence is defined as
where and are relatively prime to . It should be noted that the Chu sequence can be realized with compact direct digital synthesis (DDS) devices.
To implement the optimal training scheme (18), a length- Chu sequence is employed by the source terminal as the training sequence . Terminal appends a length- CP to and then broadcasts it to -relay terminals in the first hop. Define a vector , where , with the entry to be 1 and other entries to be 0. Let , where , be a circulant matrix with the first column to be and let be . After discarding CP, terminal , where , multiplies their received signal vectors with local unitary matrix to get the signal vector . Then, these relay terminals forward their signal vectors preceded with length- CPs to the destination terminal simultaneously in the next hop. Finally, terminal receives signal vector after removing the CP and obtains CSI via the low-complexity LMMSE channel estimator (22).
4.2. Optimality of the Proposed Training Scheme
This subsection will prove the optimality of the training scheme proposed in the last subsection. For the direct SISO link , the Chu sequence employed by this training scheme can achieve the optimal estimation performance in the first time slot, owing to its constant magnitude in the frequency domain. In the following, the optimality for the concatenation links will be proved.
Since both and , where , are circulant matrices, the following relation holds:
With this relation, the training sequence of terminal can be rewritten as
Using and perfect impulse-like autocorrelation property of , to prove that satisfies the first condition of (18) is straightforward. The proposed ensures that and are orthogonal, and and are orthogonal, where , and . Thus, the second condition of (18) can be satisfied. Besides, according to the definition of and the above discussion, to make sure that exists for all relay terminals, the following inequality is required for the extreme case or :
which is equivalent to (8). Therefore, under the premise that is identifiable, exist for all relays. Moreover, since the Chu sequence exists for any finite length, we conclude that for any finite number of total channel taps , this training scheme can always achieve the minimum MSE estimation performance.
5. Simulation Results and Discussion
5.1. System Parameters
The performance of the proposed LMMSE channel estimation method and the superiority of the derived optimal training scheme in the multiple AF-relays-assisted cooperative communication scenario are evaluated by computer simulations. We consider an OFDM cooperation system where each relay terminal utilizes the coding method as in  to perform DSTC in data transmission section. This type of DSTC is chosen because it obtains the optimal diversity-multiplexing gain (D-MG) performance of the considered orthogonal AF protocol, but other types of DSTC are also applicable since we are only interested in the performance of the proposed channel estimation method. The modulation mode is set 4-QAM and the maximum-likelihood decoder is applied for each subcarrier at the destination terminal. The MSE bound of the proposed channel estimation method shown in (19) is related with the power delay profile of the channel, thus it varies with different channel models. However, to verify that our optimal training scheme indeed attains the MSE bound deduced in theory, selecting a typical channel model through the Monte Carlo simulation is enough. Here, the typical urban (TU) twelve-path channel model , which is widely used in the community, is adopted to generate the multipath Rayleigh fading channels between each two terminals. The power delay profile of the channel model is set with tap mean power 4, 3, 0, 2, 3, 5, 7, 5, 6, 9, 11, and 10 dB at tap delays 0.0, 0.2, 0.4, 0.6, 0.8, 1.2, 1.4, 1.8, 2.4, 3.0, 3.2, and 5.0 μ s. The entire channel bandwidth is 5 MHz and is divided into 256 tones. CPs of 6.4- μ s duration are appended in source terminal and relay terminals to eliminate the effect of multipath fading. Perfect synchronization among relay terminals is assumed to observe the channel estimation performance alone. The transmission power at the source terminal is normalized to unity. The unitary matrices of relay terminals in the training section, mentioned in Section 4, are adopted to ensure channel identifiable in the simulation.
5.2. Simulation Results
Figure 3 illustrates the MSE of the proposed LMMSE channel estimation method for different numbers of relay terminals when both length-256 Chu and random sequences are used. To observe the effect of the number of relays on the MSE and bit error rate (BER) performance alone, these relays are assumed to be distributed in a symmetrical way, for example, the geometric gains and for all relays are set 5 dB and 0 dB, respectively. The effect of the geometric gains on the MSE performance will be shown later. The unit transmission power in the second hop is equally divided among these relays. In the case of the unequal geometric gains, the Matlab function "fmincon" can be used for optimizing the power allocation of these relay terminals with respect to the MSE bound given by (19). Figure 3 also illustrates the MSE bound. We can see that the optimal training scheme mentioned in Section 4 indeed attains the MSE bound and outperforms substantially random training sequences. Besides, the MSE bound is below in moderate to high SNR ( dB), indicating good channel estimation performance.
Figure 4 plots the BER performance corresponding to the length-256 optimal and suboptimal training schemes when different numbers of relays are employed. As expected from the MSE performance comparison results, a substantial BER performance gain of the optimal training scheme over the suboptimal one is observed. The BER performance of perfect CSI is also given as a benchmark. From the figure, we can see that the BER performance of the optimal training scheme is very close to the perfect CSI case when only two relays are employed, which confirms the accuracy of the proposed channel estimation method, while the performance gap increases when another two relays are involved. This can be explained by the fact depicted in Figure 3 that the MSE performance decreases as the number of relays increases. However, since spatial diversity is dominant in the BER performance relative to the channel estimation error, four relays provide better BER performance than two relays in moderate to high SNR ( dB).
Figure 5 displays the impact of the relay number on the MSE performance of the length-256 and length-512 optimal training. Note that the longer training sequences lead to the higher MSE performance for the same relay number. This is expected from (19) since the transmitting energy in the training section is linear with the training length . From this figure, it is seen that increasing the relay number would degrade the MSE performance though the training energy in the cooperation system remains the same. This is because the apportioned training energy for each relay decreases while the variance of the effective noise at the destination remains unaltered. It is also seen from this figure that the length-256 channel estimator would not work when the relay number increases beyond 5. The reason for this phenomenon is because the relay number that can be supplied by this channel estimator is bounded by (8). Thus, to avoid this phenomenon, it is crucial to make the training length not less than the channel tap number of all concatenation links.
Figure 2 shows roughly the impact of geometric gains on the MSE performance bound with length-256 training. The geometric gains for all two relay terminals are set equal but varied from 0 dB to 15 dB with a step of 5 dB, while the geometric gains are fixed to 0 dB. Note that the larger geometric gains lead to the higher accuracy of channel estimation, resulting in a higher performance of the cooperation system. Numerical results show that is larger enough to achieve the best channel estimation performance with negligible loss compared to the case of larger .
5.3. Complexity Analysis
The description of the proposed channel estimation method in Section 3 shows that the overall complexity comes from complex matrix operations in the relay terminals and the destination terminal. Since multiplication operation of the unitary matrices of relay terminals given in the optimal training scheme is equivalent to circular shifting operation, the complex matrix multiplication operation in the relay terminals can be avoided. Besides, we exploit the optimal training scheme to derive a low-rank LMMSE channel estimator (22) based on SVD, where the performance is essentially preserved. Therefore, the complex matrix inverse calculation in the destination terminal can be avoided. To conclude, only complex multiplications and complex additions are required to obtain the accurate time-domain CSI in the cooperation system with multiple AF relays.
In this paper, a training-sequences-aided LMMSE channel estimation method has been proposed for OFDM-based cooperative diversity systems with multiple AF relays over frequency-selective block-fading channels. To obtain the minimum MSE of the proposed channel estimation method in the simple bandwidth-efficient two-hop AF protocol, the circulant training matrices of relay terminals must be orthogonal. Then, we exploit the inherent orthogonal characteristic of the optimal training scheme to simplify the LMMSE channel estimator based on SVD and introduce a low-complexity one where the performance is essentially preserved. In addition, the Chu sequence is employed as the training sequence to achieve the minimum MSE estimation performance while avoid the complex matrix multiplication operation at the relay terminals. The simulation results have verified the performance of the proposed low-complexity channel estimation method in the multiple AF-relays-assisted broadband cooperative communication scenario.
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This work was supported by the National High Technology Research and Development Program of China under Grant no. 2006AA01Z216.
Proof of Lemma 3
Substituting (20) and (21) into (14) yields
This completes the proof.