Research  Open  Published:
A new blind algorithm for channel estimation in OFDMbased amplifyandforward twoway relay networks
EURASIP Journal on Wireless Communications and Networkingvolume 2018, Article number: 183 (2018)
Abstract
In this paper, we propose a blind channel estimation algorithm for the amplifyandforward (AF) twoway relay network (TWRN) which consists of two terminal nodes and one relay node. The orthogonal frequency division multiplexing (OFDM) modulation is adopted for frequency selective channel. Both cyclic prefix (CP) and zero padding (ZP) are considered. The two cascaded channels are estimated in two steps. First, the cascaded channel causing the selfinterference is estimated using a proposed power reduction method. Then, the other cascaded channel from source to destination is estimated by subspace method. Closedform formulas for channel estimates are derived. In addition, we also carry out the theoretical mean square error analysis and derive the approximated CramerRao bounds.
Introduction
Research on wireless relay networks became popular since the pioneering work [1] developed lowcomplexity cooperative diversity strategies. In [1], data streams flow unidirectionally from the source to the relay and then to the destination. This network structure is known as the oneway relay network (OWRN). However, since most communication systems are bidirectional, it is necessary to consider the situation when the source node and the destination node exchange their roles. Such a relay network is known as the twoway relay network (TWRN). In TWRN, the relay treats the received signals in a “network coding”like manner [2], and the terminals can recover the signal collision since they know their own transmitted signals. As a result, the overall communication rate between two source terminals in TWRN is approximately twice that achieved in OWRN [3].
Despite its throughput advantage, TWRN faces more challenges in terms of transceiver design, relay processing optimization, and transmission protocol development. In [4], the capacity analysis and the achievable rate region for amplifyandforward (AF) and decodeandforward (DF) TWRN are explored. In [5], the authors point out that the throughput of AFTWRN is 1.5 times of DFTWRN. The distributed spacetime code (STC) at relays for both AFTWRN and DFTWRN has been developed in [6]. Moreover, the optimal beamforming with full channel knowledge at the multiantenna relay that maximizes the overall system capacity of AFTWRN is derived in [7]. In [8], the authors address the problem of robust linear relay precoder and destination equalizer design for multipleinput multipleoutput relay systems. In [9], the authors compare several networkcoding AFTWRN and consider imperfect time synchronization. Most existing works on TWRN [2–9] have assumed perfect channel state information (CSI) at the relay node and/or the source terminals. While traditional channel estimation methods can be applied to DFTWRN, the channel estimation problem for AFTWRN is more challenging due to the selfinterfering signals.
In traditional channel estimation methods for pointtopoint systems, they can be divided into two groups: dataaided (DA) [10–17] and non dataaided (blind) [18–26]. In general, DA channel estimation methods differ in the way they interpolate or filter punctual DA least square (DALS) channel estimates over data subcarriers. This can be accomplished using timefrequency Wiener filtering [10, 11], which is optimal in the minimum mean square error (MMSE) sense if knowledge of the channel statistics (KCS) is available. On the other hand, channel estimation can be accomplished by elaborating raw estimates in the time domain using a discrete Fourier transform (DFT)based scheme. In [12], the MMSE channel estimator working in the time domain has been proposed. In order to reduce computational complexity, using the singular value decomposition and several lowrank approximations to the MMSE estimator has been proposed in [13] and [14]. Li et al. [12–14] also require complete KCS. In [15], the authors compare the MMSE approach with maximum likelihood (ML) channel estimation, where complete KCS is not required. This latter approach works well with dense multipath channels and quasiuniform profiles. In practice, after the inverse DFT (IDFT), not all the channel impulse response (CIR) samples are significant because many may correspond to delays where no propagation channel paths are actually present. Therefore, the authors in [16] exploit this idea to estimate channel. In [17], the authors propose a method to approach the MMSE channel estimation performance, while avoiding the need for a priori KCS.
For blind channel estimation methods, earlier works require either higher order statistics (HOS) of the received data [18] or oversampling at the receiver [19]. By exploiting linear redundant precoding, only secondorder statistics (SOS) of the received data is required and these methods are robust to channel order overestimation [20, 21]. Another popular blind algorithm is the socalled subspacebased algorithm which was originally developed in [19]. The subspace method has simple structure and achieves good performance. In [22], a blind channel identification method by exploiting virtual carriers (VC) is derived. In [23], a generalization in cyclic prefix (CP) systems is proposed. By arranging the received data appropriately, [23] generates a rankdeduction matrix, and thus, subspace method can work. In [24], the authors propose another simpler arrangement of the received data. Pan and Phoong [25] and [26] utilize the repetition method to reduce the number of required received data and consider the existence of VCs.
As in the traditional pointtopoint systems, study of channel estimation algorithm is also demanded for AFTWRN systems [27–34]. DA channel estimation methods for AFTWRN are proposed in [27–30]. Gao et al. [27] develops an optimal training design for flatfading environment. The authors also combine their algorithm with orthogonal frequency division multiplexing (OFDM) to estimate the channel impulse responses for frequency selective environment in [28]. The case of multipleinput multipleoutput is considered in [29], and [30] provides two channel training algorithms for channel estimation.
On the other hand, [31–34] are blind channel estimation methods. In [31], the authors propose a ML approach to estimate the flatfading channels blindly, but the transmitted signals are limited to constant modulus modulation. Zhao et al. [32] find a closedform solution and thus provides a lowcomplexity ML algorithm. For nonconstant modulus modulation, [33] gives an iterative algorithm, which is based on the maximum a posteriori (MAP) approach, and it requires a large number of received blocks. In [34], the authors consider the frequency selective environment. They apply a nonunitary linear precoding at both terminals and derive a blind channel estimation algorithm from SOS of the received signals. However, the use of nonunitary linear precoding leads to degradation in bit error rate (BER) performance.
In this paper, we develop a blind channel estimation algorithm for AFTWRN under OFDM modulation. Our method consists of two steps. The first step is to estimate the cascaded channel causing the selfinterference. Since the terminal knows its own transmitted signal, we choose the method based on power reduction to estimate the channel, which is also named LS method. The selfinterference signal can be removed by using the estimated channel. The second step is to estimate the cascaded channel from source to destination. We utilize the rank reduction method, which is also known as subspacebased algorithm [23–26]. This is because subspace methods do not require complete KCS, work well with all multipath channels, and achieve good performance. Closedform formulas for these two cascaded channel estimates are derived. The theoretical performance analysis and approximated CramerRao bounds (ACRB) are given as well. The proposed method can be applied to both CPbased and zero padding (ZP)based OFDM systems. Simulation results will be provided to show the performance of the proposed method.
The rest of this paper is organized as follows. The system model for CPOFDM AFTWRN is introduced in Section 2. Section 3 describes the proposed algorithm for blind channel estimation. In Section 4, we analyze the performance of the proposed channel estimation methods and the ACRBs. Simulation results are presented in Section 5, and concluding remarks are made in Section 6. The results in Section 3.1 and 3.2 of this paper have appeared in a conference paper [35].
Notation In this paper, E{x} stands for the statistical expectation of the random variable x. The symbols A^{T}, A^{∗}, and A^{†} denote the transpose, the complex conjugate, and the conjugatetranspose of matrix A, respectively. ∥A∥_{F} is the Frobenius norm of matrix A. If A is a square, tr(A) denotes the trace of matrix A. I_{m} is the m×m identity matrix, whereas 0 represents an allzero matrix with appropriate dimension. $\jmath =\sqrt {1}$ is the imaginary unit. T_{m}(c) and $\tilde {\mathbf {T}}_{m}(\mathbf {c})$ are two Toeplitz matrices respectively defined as
and
where c=[c_{1},c_{2},…,c_{n}]^{T} is an arbitrary vector.
System model
Consider a TWRN with two terminal nodes $\mathbb {T}_{1}$ and $\mathbb {T}_{2}$, and one relay node $\mathbb {R}$, as shown in Fig. 1. Each node has one antenna which cannot transmit and receive simultaneously. The channel from $\mathbb {T}_{i}$ to $\mathbb {R}$ is denoted as $\mathbf {f}_{i}=[f_{i,0}^{},f_{i,1}^{},\ldots,f_{i,L}^{}]^{T}$, whereas the one from $\mathbb {R}$ back to $\mathbb {T}_{i}$ is denoted as $\mathbf {g}_{i}=\left [g_{i,0}^{},g_{i,1}^{},\ldots,g_{i,L}^{}\right ]^{T}$ for i=1 and 2. For notational simplicity, we assume that the lengths of f_{1}, f_{2}, g_{1}, and g_{2} do not exceed L+1.^{Footnote 1} Similar to most other algorithms, we assume that the channels do not change when the channel estimation is performed.
OFDM modulation at terminals
Denote the kth OFDM block from $\mathbb {T}_{i}$ as $\mathbf {s}_{k}^{(i)}=\left [s_{k,0}^{(i)},s_{k,1}^{(i)},\ldots,s_{k,N1}^{(i)}\right ]^{T}$, where N is the OFDM block length. The corresponding time domain signal block is obtained from the normalized IDFT as
where W is the N×N normalized DFT matrix with the (m,n)th entry given by $\frac {1}{\sqrt {N}}e^{\jmath 2\pi mn/N}$. To maintain the subcarrier orthogonality during the overall transmission, we propose to add a CP of length 2L.^{Footnote 2} This implicitly requires N≥2L which is nevertheless satisfied by most OFDM systems. Define $\mathbf {x}_{k,cp}^{(i)}=[x_{k,N2L}^{(i)},\ldots,x_{k,N1}^{(i)}]^{T}$. The signal sent out from $\mathbb {T}_{i}$ is expressed as $\left [\begin {array}{cc}\mathbf {x}_{k,cp}^{(i)T}&\mathbf {x}_{k}^{(i)T}\end {array}\right ]^{T}$ for i=1 and 2.
Relay processing
The relay $\mathbb {R}$ receives the signal [34]
where $\mathbf {x}_{k1,isi}^{(i)}$ is the term which causes the intersymbol interference (ISI):
Moreover, each element in the noise vector n_{k,r} is assumed to be independent and identically distributed (i.i.d.) zeromean complex white Gaussian.
We assume that the relay $\mathbb {R}$ employs the amplifyandforward scheme. It scales r_{k} by the factor of
where P_{r} is the average transmission power of $\mathbb {R}$. In the second equality, we have made the assumptions that the transmitted signals $\mathbf {x}_{k}^{(1)}$, $\mathbf {x}_{k}^{(2)}$, and the received noise n_{k,r} are uncorrelated with variances $\sigma _{1}^{2}$, $\sigma _{2}^{2}$, and $\sigma _{n_{r}}^{2}$, respectively. Then, the relay broadcasts αr_{k} to both terminals.
Signal reformulation at terminals
Due to symmetry, we only illustrate the processing at $\mathbb {T}_{1}$. The (N+2L)×1 vector received at $\mathbb {T}_{1}$ can be expressed as
where r_{k−1,isi} is similar to (5)
and each element in the noise vector n_{k,t} is assumed to be i.i.d. zeromean complex white Gaussian, with variance $\sigma _{n_{t}}^{2}$. Substituting (4) into (7), we have
where h_{1}=α(g_{1}∗f_{1}) and h_{2}=α(g_{1}∗f_{2}) with ∗ being the linear convolution between two vectors by the fact that the multiplication of two Toeplitz matrices is still a Toeplitz matrix. The last term n_{k,e} denotes the equivalent noise
When N≫L, n_{k,e} can be approximated as white noise.
Data detection at terminals
After removing the first 2L elements of y_{k} in (9), we obtain a vector of size N:
where $\bar {\mathbf {n}}_{k,e}$ is the last N elements of n_{k,e}. If the cascaded channel h_{1} is known to $\mathbb {T}_{1}$, then the first term on the righthand side of (11) can be removed since $\mathbb {T}_{1}$ knows its own signal $\mathbf {x}_{k}^{(1)}$. If h_{2} is known, the regular OFDM detection can be efficiently performed using fast Fourier transform. So $\mathbb {T}_{1}$ can recover the data from $\mathbb {T}_{2}$ if both h_{1} and h_{2} are available. Hence, our goal is to estimate h_{1} and h_{2}. Below, we will show how to blindly estimate these two cascaded channels from the received signal y_{k}.
Proposed method for channel estimation
In this paper, we assume that $\mathbf {x}_{k}^{(1)}$ and $\mathbf {x}_{k}^{(2)}$ are uncorrelated. Moreover, the transmitted signals and the noises are uncorrelated as well. Under these two assumptions, we propose an algorithm to estimate h_{1} and h_{2} blindly. Though our derivations are based on CPOFDM system, the results can be also extended to ZPOFDM system. The details will be discussed later.
The estimation of h _{1}
Let us look at the received vector $\bar {\mathbf {y}}_{k}$ in (11). Notice that $\mathbf {x}_{k}^{(1)}$ is known at $\mathbb {T}_{1}$. If we have a perfect estimate of h_{1}, then the first term at the righthand side of (11) can be eliminated completely from $\bar {\mathbf {y}}_{k}$. Due to uncorrelatedness of $\mathbf {x}_{k}^{(1)}$, $\mathbf {x}_{k}^{(2)}$, and $\bar {\mathbf {n}}_{k,e}$, the power of $\bar {\mathbf {y}}_{k}$ will be reduced when $\mathbf {x}_{k}^{(1)}$ is eliminated from $\bar {\mathbf {y}}_{k}$. Based on this power reduction, we are able to derive a closedform formula for an estimate of the (2L+1)×1 vector h_{1}, as shown below.
Define a cost function
where $\bar {\mathbf {y}}_{k}$ is the N×1 vector in (11) and $\hat {\mathbf {h}}_{1}$ is an estimate of h_{1}. Substituting (11) into (12), we get
Using the assumptions mentioned above to simplify the expression, we have
Obviously, the cost function has the minimum if and only if $\\mathbf {h}_{1}\hat {\mathbf {h}}_{1}\{~}_{F}^{2}=0$, or equivalently, $\hat {\mathbf {h}}_{1}=\mathbf {h}_{1}$. Assume that $\mathbb {T}_{1}$ has collected K blocks. For meanergodic processes, the ensemble average (or statistical average) can be well approximated by the time average:
where $\mathbf {D}\left (\mathbf {s}_{k}^{(1)}\right)$ is a diagonal matrix with the elements of $\mathbf {s}_{k}^{(1)}$ on the main diagonal, and W_{2L+1} is the first 2L+1 columns of the DFT matrix W. Let
and
Then, (15) can be rewritten as
where the symbol ⊗ denotes the Kronecker product. The least squares solution of (18) can be calculated as
When K>>1, we have $\mathbf {S}^{\dag }\mathbf {S}\approx K\sigma _{1}^{2}\mathbf {I}_{N}$ as the modulation symbols are statistically independent. In this case, (19) can be approximated as
where the symbol ⊙ denotes the Hadamard product. Notice that there is no scalar ambiguity in the estimation of h_{1} since $\mathbf {s}_{k}^{(1)}$ and $\bar {\mathbf {y}}_{k}$ are known at $\mathbb {T}_{1}$.
The estimation of h _{2}
In order to estimate the (2L+1)×1 vector h_{2}, we first remove the selfinterfering signal from the received vector. Define
Assuming that the estimation of h_{1} is perfect (i.e., $\hat {\mathbf {h}}_{1}=\mathbf {h}_{1}$), from (9) and (21), we have
Note that the vector z_{k} is simply the received vector in a usual CPOFDM system with channel h_{2} and transmitted vector $\mathbf {x}_{k}^{(2)}$. Many blind estimation methods have been proposed for the estimation of h_{2} from z_{k}. Below, we will adopt the subspacebased algorithm in [24]. Define the remodulated vector
where $z_{k,i}^{}$ is the ith entry of z_{k}. That is, $\tilde {\mathbf {z}}_{k}$ is a (N+2L)×1 vector formed by the last N entries of z_{k−1} and the first 2L entries of z_{k}. Next, we construct the vector
Substituting (9) and (21) into (24), we have
where $\tilde {\mathbf {T}}_{N}(\cdot)$ is defined in (2) and η_{k} is colored noise. The covariance matrix of η_{k} is [24]
where $\sigma _{n_{e}}^{2}$ is the average power of n_{k,e}. It can be verified that
with
Carrying out the whitening process on v_{k}, we get the whitened vector $\mathbf {v}_{k}^{(w)}=\mathbf {R}_{w}^{1/2}\mathbf {v}_{k}$ and its covariance matrix is
where $\mathbf {R}_{d}=\mathrm {E}\left \{\mathbf {d}_{k}\mathbf {d}_{k}^{\dag }\right \}$ is the covariance matrix of d_{k} defined in (25). A necessary condition that R_{d} has full rank is that $\mathbb {T}_{1}$ collects K≥N blocks. Utilizing eigenvalue decomposition, (27) can be computed as
where Σ is an N×N diagonal matrix and the (N+2L)×N matrix U_{s} spans the signal subspace. On the other hand, the (N+2L)×2L matrix U_{o} spans the noise subspace. That is,
Let
Then, (29) can be rewritten as
Hence, we can estimate h_{2} (up to a scalar ambiguity) by calculating the eigenvector corresponding to the smallest eigenvalue of U^{†}U.
In summary, our algorithm is as follows.

1.
Estimate h_{1} by (20).

2.
Eliminate the interference from $\mathbb {T}_{1}$ by (21).

3.
Calculate $\mathbf {v}_{k}^{(w)}=\mathbf {R}_{w}^{1/2}\mathbf {v}_{k}$ by (24) and (26) and obtain the (N+2L)×2L matrix U_{o} spanning the noise subspace by eigenvalue decomposition.

4.
Estimate h_{2} (up to a scalar ambiguity) by calculating the eigenvector corresponding to the smallest eigenvalue of U^{†}U.
A note on the identifiability issue
Note that the estimate of h_{1} is unique because the cost function in (14) has a unique minimum at $\hat {\mathbf {h}}_{1}=\mathbf {h}_{1}$. The second channel h_{2} is estimated by the subspace method. Let us look at the vector z_{k} in (22). When the selfinterfering signal is completely eliminated, the remaining part z_{k} is identical to the case of singleinput singleoutput (SISO) CPOFDM system in [24]. The identifiability issue of this method has been studied in [24]. It has been shown that if h_{2,0}≠0, then the vector h_{2} is uniquely determined (up to a scalar ambiguity).
Comparison with an existing work
A blind channel estimation algorithm in OFDMbased TWRN was proposed in [34]. Comparing our method with that in [34], there are two major differences. One is that [34] requires a precoding matrix P, where
A necessary condition on θ is $\frac {1}{N1}\leq \theta \leq 1$. In other words, the kth transmitted vector from $\mathbb {T}_{i}$ is the precoded vector $\mathbf {P}\mathbf {s}_{k}^{(i)}$ instead of $\mathbf {s}_{k}^{(i)}$. Notice that for θ≠0, P is not a unitary matrix. The channel noise can be amplified when the receiver performs the operation P^{−1}. It was shown in [34] that when θ increases from 0 to 1, the mean square error (MSE) of channel estimate decreases. Due to noise amplification, larger θ does not necessarily yield smaller BER, so there exists a compromise between channel estimation error and BER. Another difference between our method and [34] is that there is a 2×2 ambiguity matrix in [34], or equivalently, there are four ambiguity scalars. On the other hand, there is only one ambiguity scalar in our algorithm. In terms of complexity, we can see that the main complexity of our method is the computation of the eigenvalue decomposition of an N×N matrix in (28), whereas the eigenvalue decomposition in [34] is for a (2L+1)×(2L+1) matrix. Hence, our method is more complicated than [34].
Repeated use of the remodulated vector v _{k}
To obtain U_{o} in (28), $\mathbb {T}_{1}$ has to collect K≥N blocks. In OFDM systems, N is usually large. The number of blocks, K, needed for the channel estimation is large. In order to reduce the required block number K, we can use the repetition method proposed in [23, 25, 26]. Define the repetition parameter Q and form the matrix $\tilde {\mathbf {T}}_{Q}(\mathbf {v}_{k})$, where v_{k} is defined in (24). According to (25), $\tilde {\mathbf {T}}_{Q}(\mathbf {v}_{k})$ can be represented as
It was shown in [25] that $\tilde {\mathbf {T}}_{Q}(\boldsymbol {\eta }_{k})$ is colored noise, and its covariance matrix can be calculated as
where we have applied the eigenvalue decomposition in the second equality. Therefore, we need to whiten the matrix $\tilde {\mathbf {T}}_{Q}(\mathbf {v}_{k})$ by EΛ^{−1/2}E^{†}. Since each vector v_{k} is repeated Q times in (32), the required number of blocks becomes $K\geq \frac {N1}{Q}+1$ blocks [23]. Collecting these K blocks, we can follow the procedure in (28)–(31) to estimate h_{2} (up to a scalar ambiguity).
Multiple relay nodes
The extension to the case of multiple relay nodes is straight forward as shown in Fig. 2. Suppose that we have M relay nodes $\mathbb {R}_{1},\mathbb {R}_{2},\ldots,\mathbb {R}_{M}$. Let the channels from $\mathbb {T}_{i}$ to $\mathbb {R}_{m}$ be denoted as $\mathbf {f}_{i}^{(m)}$ and the channels from $\mathbb {R}_{m}$ to $\mathbb {T}_{i}$ be denoted as $\mathbf {g}_{i}^{(m)}$. Then, (4) becomes
where $\mathbf {r}_{k}^{(m)}$ is the signal received by relay node $\mathbb {R}_{m}$ and $\mathbf {n}_{k,r}^{(m)}$ is the noise at $\mathbb {R}_{m}$. When $\mathbb {T}_{1}$ receives the signal, (7) becomes
where α_{m} is the amplification scalar in the relay node $\mathbb {R}_{m}$. Combining (34) with (35), the received vector at $\mathbb {T}_{1}$ continues to have the form given in (9), but now the cascaded channels are $\mathbf {h}_{1}=\sum _{m=1}^{M}\alpha _{m}\left (\mathbf {g}_{1}^{(m)}\ast \mathbf {f}_{1}^{(m)}\right)$ and $\mathbf {h}_{2}=\sum _{m=1}^{M}\alpha _{m}\left (\mathbf {g}_{1}^{(m)}\ast \mathbf {f}_{2}^{(m)}\right)$, and the equivalent noise n_{k,e} becomes
Hence, the above methods can be applied to the case of multiple relay nodes.
The case of ZPOFDM systems
The proposed method can be also applied to TWRN ZPOFDM system. In this case, 2L zeros are padded at the end of $\mathbf {x}_{k}^{(i)}$ in (3) instead of adding the cyclic prefix of length 2L. Due to the padded zeros, the received vector does not suffer from ISI. Therefore, (9) can be rewritten as
To estimate h_{1}, we modify the cost function in (12) as
Following a procedure similar to (12)–(20), an estimate of h_{1} can be obtained by
where $\tilde {\mathbf {W}}$ is the (N+2L)×(N+2L) normalized DFT matrix with the (m,n)th entry given by $\frac {1}{\sqrt {N+2L}}e^{\jmath 2\pi mn/(N+2L)}$, whereas $\tilde {\mathbf {W}}_{2L+1}$ and $\tilde {\mathbf {W}}_{N}$ are respectively the first 2L+1 and N columns of $\tilde {\mathbf {W}}$.
Assume that the estimation of h_{1} is perfect so that we can eliminate the interference from $\mathbb {T}_{1}$. Similar to (21), define
Substituting (36) into (39), we have
This form is similar to (25), so we can follow the procedure in (25)–(31) to estimate h_{2}. Note that the noise n_{k,e} is (almost) white. Similar to the previous discussion, a necessary condition is K≥N. To reduce the limitation of a large K, we exploit the repetition method in [26]. That is, we utilize $\tilde {\mathbf {T}}_{Q}(\mathbf {z}_{k})$ instead of z_{k} to estimate h_{2}, and the necessary condition becomes $K\geq \frac {N1}{Q}+1$. In this case, the noise term $\tilde {\mathbf {T}}_{Q}(\mathbf {n}_{k,e})$ is colored (though n_{k,e} is white) and the covariance matrix is [26]
where D(·) is defined in (15) and Q^{′}= min{Q,N+2L}. Therefore, we need to whiten the matrix $\tilde {\mathbf {T}}_{Q}(\mathbf {z}_{k})$ by
Following a procedure similar to Section 3.5, one can obtain a blind estimate of h_{2} (up to a scalar ambiguity).
Analysis of MSE performance and CramerRao bound
In this section, we will derive the theoretical MSE about channel estimation for h_{1} and h_{2} respectively. In the following analysis, we assume that the channel taps are uncorrelated and the transmitted vectors $\mathbf {x}_{k}^{(i)}$ are also uncorrelated for different k or i.
The analysis of h _{1} estimate
In the estimation of h_{1}, we regard the signal from $\mathbb {T}_{2}$ as interference. Since (20) is the least squares solution of (18), the difference between $\hat {\mathbf {h}}_{1}$ and h_{1} can be calculated as
where ξ_{k} denotes the interference and noise. From (11), we have
where C(h_{2}) is an N×N circulant matrix having $\left [\begin {array}{cc}\mathbf {h}_{2}^{T}&\mathbf {0}_{1\times (N2L1)}\end {array}\right ]^{T}$as its first column. Assuming that $\mathbf {x}_{k}^{(2)}$ and n_{k,e} are uncorrelated, the covariance matrix of ξ_{k} can be computed as
where D(h_{2,f}) is the N×N diagonal matrix with diagonal entries from the N×1 frequency response vector $\mathbf {h}_{2,f}=\sqrt {N}\mathbf {W}_{2L+1}\mathbf {h}_{2}$. Then, the covariance matrix of Δh_{1} can be computed as
where A is a (2L+1)×(2L+1) Toeplitz and Hermitian matrix with $A_{m,n}=\sum _{l=0}^{2L+mn}h_{2,l}h_{2,lm+n}^{*}$ if m≤n and $A_{m,n}=\sum _{l=mn}^{2L}h_{2,l}h_{2,lm+n}^{*}$ if m≥n. Therefore, the theoretical MSE can be calculated as
where $tr\left \{\mathbf {R}_{\Delta \mathbf {h}_{1}}\right \}$ is the sum of the diagonal elements of $\mathbf {R}_{\Delta \mathbf {h}_{1}}\phantom {\dot {i}\!}$. Define the signaltonoise ratio (SNR) as
where the second equality is obtained by using (10). Then, (45) can be written as
Note from the above equation that the MSE is proportional to the signal power from $\mathbb {T}_{2}$ but inversely proportional to the signal power from $\mathbb {T}_{1}$ and the number of the received signal blocks. Moreover, for high SNR, the MSE floors at the value of $\frac {2L+1}{NK}\frac {\sigma _{2}^{2}}{\sigma _{1}^{2}}\\mathbf {h}_{2}\{~}_{F}^{2}$.
The analysis of h _{2} estimate
During the estimation of h_{2} in Section 3.2, it is assumed that the estimate of h_{1} is perfect. However, the estimation error Δh_{1} will affect the accuracy of the estimation of h_{2}. From (21), if Δh_{1}≠0, the interference and noise terms can be written as
Next, we look at v_{k} in (25). Following the procedure (21)–(26), the whitened vector $\mathbf {v}_{k}^{(w)}$ now becomes
where
Recall from the subspace method in Section 3.2 that the estimate $\hat {\mathbf {h}}_{2}$ is obtained from the noise subspace U_{o} in (28). Let λ_{1}≤λ_{2}≤⋯≤λ_{N+2L} be the eigenvalues of $\mathbf {R}_{v}^{(w)}$. The noise subspace U_{o} is the eigenspace corresponding to the smallest 2L eigenvalues λ_{1},λ_{2},…,λ_{2L}. The error vector ζ_{k} can cause two effects: (i) it perturbs the noise subspace U_{o} and (ii) it also perturbs the eigenvalues, i.e., λ_{1}+Δλ_{1},λ_{2}+Δλ_{2},…,λ_{N+2L}+Δλ_{N+2L}. Note that λ_{2L} belongs to the noise subspace U_{o} and λ_{2L+1} belongs to the signal subspace U_{s}. Their difference λ_{2L+1}−λ_{2L} is usually large. Nevertheless, the perturbation on eigenvalues may lead to the case λ_{2L}+Δλ_{2L}>λ_{2L+1}+Δλ_{2L+1}, especially when the SNR is low. In this case, the noise subspace will be polluted by the signal subspace and this will cause a large error in the estimation of h_{2}. Below, we derive the MSE by studying the following two cases separately.
Case I: λ_{2L}+Δλ_{2L}<λ_{2L+1}+Δλ_{2L+1}
In this case, we can exploit the firstorder approximation of the perturbation to U_{o}. In [24], the channel estimation error has been derived. However, the theoretical MSE derived in [24] is based on white noise. As the noise ζ_{k} is colored, the formula derived in [24] is not applicable. For the case of colored noise ζ_{k}, we have derived a new formula and the theoretical MSE of the h_{2} estimate can be calculated as
where $\Delta \mathbf {h}_{2}\triangleq \hat {\mathbf {h}}_{2}\mathbf {h}_{2}$, U^{♯} is the MoorePenrose pseudoinverse matrix of U defined in (31), and $\mathbf {R}_{\zeta }\triangleq \mathrm {E}\{\boldsymbol {\zeta }_{k}\boldsymbol {\zeta }_{k}^{\dag }\}$ is the covariance matrix of ζ_{k} and it can be written as
Notice that $\tilde {\mathbf {T}}_{N}(\Delta \mathbf {h}_{1})$ can be rewritten as
where J_{i} is defined in (30). Hence, (51) can be rewritten as
where $\mathbf {R}_{\Delta \mathbf {h}_{1}}\phantom {\dot {i}\!}$ is defined in (44).
Case II: λ_{2L}+Δλ_{2L}≥λ_{2L+1}+Δλ_{2L+1}
In this case, our algorithm cannot find the accurate noise subspace U_{o} because it has been polluted by signal subspace U_{s}. Thus, we assume that the eigenvector of U corresponding to the smallest eigenvalue is random and uncorrelated to the true cascaded channel h_{2}. Define this unitnorm eigenvector as $\tilde {\mathbf {h}}_{2}$. The scalar ambiguity can be calculated as $\alpha =\left (\tilde {\mathbf {h}}_{2}^{\dag }\mathbf {h}_{2}\right)/\left (\tilde {\mathbf {h}}_{2}^{\dag }\tilde {\mathbf {h}}_{2}\right) =\tilde {\mathbf {h}}_{2}^{\dag }\mathbf {h}_{2}$. That is,
The estimation error is $\Delta \mathbf {h}_{2}=\hat {\mathbf {h}}_{2}\mathbf {h}_{2}=\left (\tilde {\mathbf {h}}_{2}\tilde {\mathbf {h}}_{2}^{\dag }\mathbf {I}_{2L+1}\right)\mathbf {h}_{2}$. Hence, the theoretical MSE of the h_{2} estimate can be calculated as
Since the unitnorm vector $\tilde {\mathbf {h}}_{2}$ is assumed to be random, $\mathrm {E}\left \{\tilde {\mathbf {h}}_{2}\tilde {\mathbf {h}}_{2}^{\dag }\right \}$ can be approximated as $\frac {1}{2L+1}\mathbf {I}_{2L+1}$. Therefore, (54) can be written as
Overall MSE: Utilizing Bayes’ theorem, the theoretical MSE of the h_{2} estimate can be written as
where P_{err} is the probability of λ_{2L}+Δλ_{2L}≥λ_{2L+1}+Δλ_{2L+1} and it can be expressed by
where Q(·) is the Qfunction:
The derivation of P_{err} is given in Appendix Appendix A. Substituting (50) and (55) into (56), the theoretical MSE of the h_{2} estimate can be represented as
where R_{ζ} is given in (52).
Approximated CramerRao bound
When we estimate h_{1}, the signal from $\mathbb {T}_{2}$ can be viewed as interference, and the signal from $\mathbb {T}_{1}$ can be seen as pilot. To simplify the derivation, we assume that ξ_{k} in (42) is white. Hence, an ACRB of h_{1} estimation is [36]
where W_{2L+1} and S are defined in (15) and (17), respectively, and $\sigma _{\xi }^{2}$ is the average power of ξ_{k}. From (43), we have $\sigma _{\xi }^{2}=\sigma _{2}^{2}\\mathbf {h}_{2}\{~}_{F}^{2}+\sigma _{n_{e}}^{2}$. Therefore, (59) can be simplified as
Notice that this form is the same as (45).
Next, we consider the ACRB of h_{2}. In [24], the authors have derived an ACRB and concluded that the ACRB is the same as the channel estimation MSE. Hence, from (50), an ACRB of h_{2} estimation is
In the derivations of the ACRBs, the noises are assumed to be white even though they are actually colored. Therefore, the ACRBs in (60) and (61) are in general larger than or equal to the true CramerRao bounds.
Simulation results
In the simulation, we consider a TWRN with one relay node. The channel taps $f_{i,l}^{}$ and $g_{i,l}^{}$ are generated as independent and identically distributed zeromean complex Gaussian random variables with variances equal to 1/9. The order of these channels is L=8, so the order of the cascaded channels is 2L=16. The channels are normalized so that $\\mathbf {f}_{1}\{~}_{F}^{2}=\\mathbf {f}_{2}\{~}_{F}^{2}=\\mathbf {g}_{1}\{~}_{F}^{2}=\\mathbf {g}_{2}\{~}_{F}^{2}=1$. The channel does not change while the channel estimation is performed. The channel noise is additive white Gaussian noise (AWGN), and the transmission symbols are 16QAM with gray code. The size of the DFT matrix is N=64, and the length of CP is 2L=16. In all plots, we set $\sigma _{1}^{2}=\sigma _{2}^{2}$ and $\sigma _{n_{r}}^{2}=\sigma _{n_{t}}^{2}$. The SNR is defined in (46), and the normalized MSE is defined as
where $\hat {\mathbf {h}}_{i}^{(m)}$ represents the estimated h_{i} in the mth trial. M_{c}=2000 denotes the total number of MonteCarlo trials.
First, we look at the MSE performance of the proposed methods. The number of received blocks is K=500. In Fig. 3, we plot the normalized MSEs for h_{1} and h_{2}. The “simulation” curves of h_{1} and h_{2} are obtained by (20) and (31), respectively, whereas the “theory” curves of h_{1} and h_{2} are calculated by (47) and (58), respectively. Moreover, we also display the ACRBs of h_{1} and h_{2} according to (60) and (61). From Fig. 3, it can be seen that the simulated result, the theoretical MSE, and the ACRB of h_{1} is close. Moreover, the proposed method can give a good estimate of h_{1}, even at very low SNR of 0 dB. One can see that the MSE floors at $\frac {2L+1}{NK}=5.3\times 10^{4}$ at high SNR, and this confirms our analysis in (47). For h_{2}, the MSE performance is worse than that of h_{1} for SNR< 25 dB, but the MSE of h_{2} floors at a much smaller value of 1.2×10^{−5}. This flooring happens at very high SNR, and the estimation error of h_{1} affects the accuracy of h_{2} estimate. The gap between numerical and theoretical results is small at high SNR. At low SNR, the gap between simulation result and ACRB becomes very large, but the theoretical curve is still close to the numerical curve. Recall that the theoretical MSE value is a combination of two cases in Section 4.2, and the ACRB of h_{2} in (61) is equal to the theoretical MSE when we do not consider the perturbation on eigenvalues, i.e., case II in Section 4.2 (case II usually happens at low SNR). Therefore, the difference between the theoretical MSE and ACRB of h_{2} at low SNR is caused by the perturbation of eigenvalues. From Fig. 3, we conclude that the change of eigenvalue sequence dominates the performance degradation at low SNR. In addition, the assumption of white noise in the derivation of ACRB also affects the accuracy, especially when the SNR is low.
Next, we compare the performances of our method with the method proposed by Liao et al. in [34]. As mentioned in Section 3.4, Liao’s algorithm has a compromise between channel estimation error and BER. The parameter θ in Liao’s algorithm is set to 0.2, 0.4, 0.6, and 0.8. From [34], it is found that θ=0.4 yields a good BER performance when SNR=25 dB. In Figs. 4 and 5, the number of received blocks is K=500. Figure 4 shows the MSE performances. Since the MSEs of h_{1} and h_{2} by Liao’s algorithm are the same, we plot one MSE curve only. From the figure, we see that as θ increases from 0.2 to 0.8, the MSE of Liao’s algorithm decreases. For the estimation of h_{1}, our method is better than Liao’s methods for θ=0.2 and 0.4, but worse than that for θ=0.6 and θ=0.8. As we will see in Fig. 5, the BER performance for θ=0.8 is not good due to severe noise amplification. For h_{2}, Liao’s method is better at low SNR whereas our method is better at high SNR. In Fig. 5, we show BER performances. Zeroforcing equalizers are used at the receiver. The “perfect compensation” represents the case that the channel taps are perfectly known at the receiver. It is seen that among the four curves of θ=0.2, 0.4, 0.6, 0.8, Liao’s method has the best BER performance when θ is set as 0.4 for SNR=25 dB. Though the MSE of Liao’s method is the smallest when θ=0.8, its BER performance is not good due to the noise amplification problem of the precoding matrix. These results are matched with [34]. From Fig. 5, we see that the proposed algorithm outperforms Liao’s methods when SNR≥15 dB, and the performance of our method is close to the perfect compensation.
Figures 6 and 7 show the simulation results when the number of blocks is K=50. In this case, K<N, and thus, the estimation of h_{2} by (31) does not work. We exploit the repetition method discussed in Section 3.5 to solve this issue. We set the repetition parameter Q=10, and the necessary condition $K\geq \frac {N1}{Q}+1$ is satisfied. In Fig. 6, the MSE performance is shown. We can observe that the repetition method is extremely useful when the terminal receives few blocks. On the other hand, h_{1} estimation by (20) and Liao’s algorithm are based on the power reduction, so there is no limitation on the number of blocks K. From the figure, we see that the proposed method outperforms Liao’s algorithms with θ=0.2 and 0.4 for all SNR. In Fig. 7, the performance is measured by BER. It can be seen that the proposed algorithm performs better than Liao’s method when SNR≥15 dB. Comparing Fig. 7 with Fig. 5, we find that the BER performance degrades when K reduces from 500 to 50. This is due to the larger channel estimation errors for K=50 and imperfect interference cancelation by h_{1} using (21).
Finally, we compare the proposed algorithm for CPOFDM and ZPOFDM systems. In Fig. 8, the solid curves and the dashed curves represent the MSEs for CPOFDM and ZPOFDM, respectively. We can find that the performances are almost the same. In other words, our method works well for both CP and ZP systems.
Conclusions
In this paper, we propose a blind channel estimation method in OFDMbased amplifyandforward twoway relay networks. The first cascaded channel h_{1} is estimated by the power reduction method whereas the second cascaded channel h_{2} is estimated by the subspace method. Closeform formulas are derived. We also analyze the theoretical performance and derive the ACRBs for channel estimation. Our algorithm can be applied to both CPOFDM and ZPOFDM systems, and it can use repetition method to handle the case of few received blocks. Simulation results verify our analysis.
Appendix A
A proof of (57)
To simplify our derivation, we utilize the fact that this condition usually occurs at low SNR. From (47) and the simulation in Section 5, it can be seen that the estimate of h_{1} is still quite accurate at low SNR, so the second term n_{k,e} in (48) is dominant. Let λ_{1}≤λ_{2}≤⋯≤λ_{N+2L} be the eigenvalues of $\mathbf {R}_{v}^{(w)}$ and the corresponding unitnorm eigenvectors are respectively b_{1},b_{2},…,b_{N+2L}. By (27) and (52), $\mathbf {R}_{v}^{(w)}$ can be expressed by
Since the received signals are finite and the second term in (48) is dominant, we have the following approximation:
and the corresponding eigenvalues become λ_{1}+Δλ_{1},λ_{2}+Δλ_{2},…,λ_{N+2L}+Δλ_{N+2L}. Notice that N is a Hermitian matrix with mean 0. For large K, the central limit theorem indicates that the diagonal entries of N are real normal distributed and the other entries are circularly symmetric complex normal distributed [37]. According to the result in [38], all entries of N have the same variance $\frac {1}{K}\sigma _{n_{e}}^{4}$.
From matrix theory [39], the eigenvalue perturbation Δλ_{i} can be approximated as $\mathbf {b}_{i}^{\dag }\mathbf {N}\mathbf {b}_{i}$. Then, the mean is
and the variance is
Because $\mathbf {R}_{w}^{1/2}\boldsymbol {\eta }_{k}$ is white, all entries of N are uncorrelated, so the last term E{N(j,l)N^{∗}(m,n)} is equal to $\frac {1}{K}\sigma _{n_{e}}^{4}\delta (jm)\delta (ln)$, where δ(·) is the Kronecker delta function. Thus, (65) can be rewritten as
The last equality holds since $\mathbf {b}_{i}^{\dag }\mathbf {b}_{i}=1$ for i=1,2,…,N+2L.
Notice that N is normal distributed and b_{i} is constant, so the random variable Δλ_{i} is normal distributed as well. It means that the probability of λ_{2L}+Δλ_{2L}≥λ_{2L+1}+Δλ_{2L+1} can be computed as
Notes
 1.
The proposed method can be applied to the more general case of different channel lengths by simply using an appropriate cyclic prefix length.
 2.
If $\mathbb {T}_{1}$ and $\mathbb {T}_{2}$ add CP of length L, then the relay needs to carry out the operations of OFDM symbol timing synchronization, CP removal, and CP insertion. In order to simplify the tasks of the relay, $\mathbb {T}_{1}$ and $\mathbb {T}_{2}$ add CP of length 2L.
Abbreviations
 ACRB:

Approximated cramerrao bound
 AF:

Amplifyandforward
 AWGN:

Additive white Gaussian noise
 BER:

Bit error rate
 CIR:

Channel impulse response
 CP:

Cyclic prefix
 CSI:

Channel state information
 DA:

Dataaided
 DF:

Decodeandforward
 DFT:

Discrete fourier transform
 HOS:

Higher order statistics
 IDFT:

Inverse discrete fourier transform
 i.i.d.:

Independent and identically distributed
 ISI:

Intersymbol interference
 KCS:

Knowledge of the channel statistics
 LS:

Least squares
 MAP:

Maximum a posteriori
 ML:

Maximum likelihood
 MMSE:

Minimum mean square error
 MSE:

Mean square error
 OFDM:

Orthogonal frequency division multiplexing
 OWRN:

Oneway relay network
 SISO:

Singleinput singleoutput
 SNR:

Signaltonoise ratio
 SOS:

Secondorder statistics
 STC:

Spacetime code
 TWRN:

Twoway relay network
 VC:

Virtual carrier
 ZP:

Zero padding
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Funding
This work was supported by the Ministry of Science and Technology, Taiwan, R.O.C., under grant no. 1062221E002033.
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TCL and SMP constructed the theory. TCL performed simulations and wrote a draft. SMP modified the paper. Both authors read and approved the final manuscript.
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Correspondence to TzuChiao Lin.
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Keywords
 Blind channel estimation
 Orthogonal frequency division multiplexing (OFDM)
 Twoway relay network (TWRN)