Worst-Case Robust Training Design for Correlated MIMO Channels in the Presence of Colored Interference

Abstract

The covariance information at the transmitter side is often subject to mismatches due to various impairments. This paper considers a training design problem for multiple-input multiple-output (MIMO) systems when both channel and interference covariance matrices are imperfect at the transmitter side. We first derive the structure of the optimal training signal, minimizing the worst-case mean square error (MSE). With the training structure, the original problem becomes a simple power allocation problem. We propose a numerical optimal power allocation scheme and a closed-form suboptimal power allocation scheme. Simulation results show that the proposed schemes considerably outperform the conventional schemes in terms of the worst-case MSE and bit error rate (BER) performances, and the proposed closed-form training scheme has comparable performance to that of the optimal one. For example, the proposed schemes yield more than 2.5 dB signal-to-interference ratio (SIR) gains at a BER of ${10}^{-4}$.

1. Introduction

In the past decades, it has been shown that the use of multiple antennas at both ends of a communication link, so called multiple-input multiple-output (MIMO), can provide either diversity gain or multiplexing gain [1,2]. For this reason, MIMO techniques have been employed in many system standards such as IEEE 802.11 for wireless local area networks (WLANs), IEEE 802.16, and 3rd Generation Partnership Project Long Term Evolution Advanced (3GPP LTE-A) for wireless communications, as a solution to demands for high data rates or low error rates [3,4]. To fully exploit the advantages that MIMO systems offer, multiple channel elements need to be accurately estimated. For example, in IEEE 802.11ac [5], accurate channel state information (CSI) at both ends is typically required for both transmit beamforming and coherent detection. Therefore, accurate channel estimation is an important problem for practical MIMO systems.

1.1. Prior Works and Limitations

In most current standards such as WLAN and WiMAX, unknown channel parameters are usually estimated at the receiver by sending a priori known training (or pilot) symbols from the transmitter [5,6]. Even though an arbitrary training signal can be used for channel estimation, it is possible to significantly enhance the estimation performance by optimizing the training symbols based on some prior knowledge on the channel and noise (or possibly interference) statistics. For this reason, training design problems have received considerable interest in recent years [7,8,9,10,11,12]. Most of these works used the mean square error (MSE) of channel estimation as a performance metric to improve the estimation accuracy [7,8,9,10,11]. On the other hand, another criterion, maximization of the conditional channel entropy, was considered in [12]. In the traditional training schemes [7,8,9,10,11,12], the perfect knowledge of both the channel and noise covariance matrices at both ends was commonly required for optimal design.

In practice, the covariance information about the channel and noise should be estimated at the receiver from received samples in recent consecutive training blocks [9,10]. The covariance information at the receiver side is reasonably assumed to be perfect when there is a sufficient number of samples [9,10]. Whereas the covariance information at the transmitter side is usually acquired by means of limited feedback from the receiver, which is often called covariance feedback [13,14]. Due to feedback-related issues, e.g., quantization, feedback errors, and delay, the covariance information at the transmitter is imperfect in general [15,16,17]. For this reason, it might be more practical to take such imperfection into account for training designs. In the literature, there are few studies concerning this issue [18,19,20].

Except for the work by Shariati et al. [20], all previous works have considered the white noise case only. However, the assumption of white noise is not always true in some practical scenarios. For example, in multi-user or small-cell environments, the noise term is no longer white due to the presence of non-negligible interference [21]. Also, for several applications such as mesh networking in IEEE 802.11s [22] and relay-aided systems such as IEEE 802.11ah [23] and IEEE 802.16j [24], total noise is often colored because the relays broadcast background noise as well as received data streams. Although the work in [20] considered a colored interference scenario, this work assumed full knowledge of the interference covariance at the transmitter side. This assumption is suitable if the covariance mismatch of the interference is sufficiently small compared with that of the channel. When the interference covariance matrix is subject to substantial errors at the transmitter, however, the scheme in [20] may become ineffective since it cannot properly deal with the interference covariance uncertainty. Moreover, since the iterative algorithm in [20] was designed with a heuristic approach, it may require considerable complexity during the channel training phase.

1.2. Motivations and Contributions

Motivated by the aforementioned discussions and to break through the limitations of the existing techniques, in this paper, we develop a novel and high-performing training strategy for the estimation of correlated MIMO channels in the presence of colored interference based on the worst-case robustness philosophy. The detailed technical contributions of our work are described below:

• We propose a general framework for robust training optimization under imperfect channel and interference covariance information at the transmitter. Particularly, we design an optimal training signal for MIMO systems with interference, taking the imperfection of both the channel and interference covariance into account.
• In our proposed framework, the worst-case MSE criterion is used as a performance metric similar to previous works [18,19,20]. In contrast to the previous problems, however, the considered design problem is not convex–concave due to the uncertainty in the interference covariance, and consequently the design of the training signal is more complicated. To solve the problem, we take innovative approaches: we initially derive an optimal structure of the training signal, which includes the existing training structures as special cases, and then we solve the training power allocation problem.
• Two power allocation schemes are proposed. First, an optimal power allocation is determined numerically by finding an optimal solution. Next, to reduce the complexity required for optimal power allocation, a closed-form power allocation scheme is proposed by finding a suboptimal solution.
• Based on the latter power allocation strategy, we also propose a suboptimal, yet closed-form, training scheme with low complexity.
• We compare the performance of the proposed schemes with that of the conventional schemes by simulations. Through numerical results, we empirically demonstrate that the proposed schemes substantially surpass the existing schemes with remarkable performance improvements and the proposed suboptimal training scheme provides comparable performance to that of the optimal training scheme.

1.3. Organization and Notation

The organization of this paper is as follows. In Section 2, the system model considered is described and an optimization problem considered is formulated. The methods of the proposed training schemes are described in Section 3. The simulation results for the performance comparison are presented in Section 4. We conclude the paper in Section 5.

Notation:${(·)}^T$, ${(·)}^{*}$, ${(·)}^H$, ⊗, $\mathrm{vec}(·)$, and $\mathrm{Tr}(·)$ denote the transpose, conjugate, conjugate transpose, Kronecker product, vectorization, and trace operators, respectively. $\Vert ·{\Vert }_F$ denotes the Frobenius norm of a matrix. The notation $\mathbf{A}⪰\mathbf{0}$ means that a Hermitian matrix $\mathbf{A}$ is positive semi-definite. The Cartesian product of two sets A and B is denoted as $A\times B$. The notation $\mathrm{Diag}(a_1,\cdots ,a_m)$ and $\mathrm{Blkdiag}({\mathbf{A}}_1,\cdots ,{\mathbf{A}}_m)$ denotes a diagonal matrix whose diagonal elements are given by $a_1,\cdots ,a_m$ and a block diagonal matrix whose diagonal blocks are given by the matrices ${\mathbf{A}}_1,\cdots ,{\mathbf{A}}_m$, respectively. ${\mathbf{\nabla }}_{\mathbf{A}}$ and ${\mathbf{\nabla }}_{\mathbf{A}}^2$ represent the gradient and Hessian operators with respect to (w.r.t.) the variable $\mathbf{A}$. $\mathbb{E}(·)$ denotes the expectation operator, and a circular symmetric Gaussian random vector $\mathbf{a}$ with mean $\overline{\mathbf{a}}$ and covariance matrix $\mathbf{A}$ is denoted by $\mathbf{a}\sim CN(\overline{\mathbf{a}},\mathbf{A})$.

2. System Model and Problem Formulation

2.1. System Model

As depicted in Figure 1, we consider a MIMO system consisting of a transmitter equipped with $M_t$ antennas, a receiver equipped with $M_r$ antennas, and a total of K interferers, where the kth interferer has $M_k$ antennas, $k=1,\cdots ,K$. It is assumed that the background additive noise is much weaker than the interference, and hence we ignore the additive noise term for convenience (this is valid in practice as the noise power ranges from −192.5 dBm/Hz to −174 dBm/Hz, whereas the interference signal power ranges from −100 dBm to −10 dBm). In the channel training procedure, multiple training symbols are sent from the transmitter during L symbol times. The received signal matrix is then given by

$$\mathbf{Y}=\mathbf{H}\mathbf{P}+\underset{=\mathbf{N}}{\underbrace{{\displaystyle \sum _{k=1}^K}{\mathbf{H}}_k{\mathbf{S}}_k}}=\mathbf{H}\mathbf{P}+\mathbf{N},$$

(1)

where $\mathbf{P}\in {\mathbb{C}}^{M_t\times L}$ and ${\mathbf{S}}_k\in {\mathbb{C}}^{M_k\times L}$ represent the training signal matrix sent from the transmitter and the interfering signal matrix sent from the kth interferer, respectively. We simply denote the total interference as $\mathbf{N}$. The channel matrix between the transmitter and receiver is denoted by $\mathbf{H}\in {\mathbb{C}}^{M_r\times M_t}$ and that between the kth interferer and receiver by ${\mathbf{H}}_k\in {\mathbb{C}}^{M_r\times M_k}$. Without loss of generality, it is assumed that there exists spatial correlation at all nodes. According to the well-known Kronecker model [25], we represent the channels and interfering signals as follows (the analysis in our work is not confined to a specific distribution of the channel matrices, but valid for any distribution, as the proposed method requires only the knowledge of the covariance information of the channel matrices, not their probability distribution):

$$\mathbf{H}={\mathbf{R}}^{1/2}{\mathbf{H}}_w{\mathbf{T}}^{1/2},$$

(2)

$${\mathbf{H}}_k={\mathbf{R}}^{1/2}{\mathbf{H}}_{w,k}{\mathbf{T}}_{I,k}^{1/2},k=1,2,\cdots ,K,$$

(3)

$${\mathbf{S}}_k={\mathbf{\Psi }}_{s,k}^{1/2}{\mathbf{S}}_{w,k}{\mathbf{\Psi }}_{\tau ,k}^{1/2},k=1,2,\cdots ,K,$$

(4)

where the elements of the matrices ${\mathbf{H}}_w$, ${\left\{{\mathbf{H}}_{w,k}\right\}}_{k=1}^K$, and ${\left\{{\mathbf{S}}_{w,k}\right\}}_{k=1}^K$ are independent and identically distributed (i.i.d.) as $\mathcal{CN}(0,1)$. $\mathbf{T}⪰\mathbf{0}$, $\mathbf{R}⪰\mathbf{0}$, and ${\mathbf{T}}_{I,k}⪰\mathbf{0}$ are the spatial correlation matrices at the transmitter, receiver, and kth interferer, respectively. ${\mathbf{\Psi }}_{\tau ,k}⪰\mathbf{0}$ and ${\mathbf{\Psi }}_{s,k}⪰\mathbf{0}$, respectively, represent the temporal and spatial correlations of the interfering signal. Taking the vectorizing operation on both sides of (1), the received signal can be rewritten in vector form as

$$\mathrm{vec}\left(\mathbf{Y}\right)=\mathbf{y}=\left({\mathbf{P}}^T\otimes {\mathbf{I}}_{M_r}\right)\mathbf{h}+\mathbf{n},$$

(5)

where

$$\mathbf{h}=\mathrm{vec}\left(\mathbf{H}\right)$$

and

$$\mathbf{n}=\mathrm{vec}\left(\mathbf{N}\right)=\sum _{k=1}^K\left({\mathbf{I}}_L\otimes {\mathbf{H}}_k\right)\mathrm{vec}\left({\mathbf{S}}_k\right).$$

For the design purpose, we assume that the channel and interference covariance matrices are perfectly known at the receiver side (The issue with imperfect covariance information at the receiver side was studied in [20]. In [20] (Remark 2), it was shown that when the covariance information is imperfect at the receiver side, the resulting channel estimation MSE has a similar mathematical expression to (12), with an additional (negligible) loss term. In [20], it was also empirically demonstrated that the imperfect covariance information at the receiver side has a negligible impact on the performance, whereas that at the transmitter side has a much more significant impact).

We consider the linear minimum MSE (LMMSE) channel estimation from the observation vector $\mathbf{y}$ in (5). In classical estimation theory [26], it is well known that the LMMSE estimate of the channel $\mathbf{h}$ is given by

$$\widehat{\mathbf{h}}={\mathbf{C}}_h\left({\mathbf{P}}^{*}\otimes {\mathbf{I}}_{M_r}\right){\left[{\mathbf{C}}_n+\left({\mathbf{P}}^T\otimes {\mathbf{I}}_{M_r}\right){\mathbf{C}}_h\left({\mathbf{P}}^{*}\otimes {\mathbf{I}}_{M_r}\right)\right]}^{-1}\mathbf{y},$$

(6)

where

$${\mathbf{C}}_h=\mathbb{E}\left[\mathbf{h}{\mathbf{h}}^H\right]={\mathbf{T}}^T\otimes \mathbf{R}$$

and

$${\mathbf{C}}_n=\mathbb{E}\left[\mathbf{n}{\mathbf{n}}^H\right]={\mathbf{Q}}^T\otimes \mathbf{R}$$

are the channel and interference covariance matrices, respectively. Also, we have

$$\mathbf{Q}=\sum _{k=1}^K\mathrm{Tr}\left({\mathbf{\Psi }}_{s,k}{\mathbf{T}}_{I,k}\right){\mathbf{\Psi }}_{\tau ,k}$$

to represent the interference correlation at the transmitter side. In practice, the matrix inversion for the LMMSE channel estimation in (6) can be approximated or replaced by several low-complexity alternatives suggested in [27] based on the matrix polynomial expansion with arbitrary degrees of freedom. This approach indeed significantly reduces the computational burden of the matrix inverse operation from a cubic complexity to a square complexity. The MSE of the LMMSE channel estimate $\widehat{\mathbf{h}}$ given the parameters $\mathbf{P}$, $\mathbf{T}$, $\mathbf{R}$, and $\mathbf{Q}$ can be obtained as [26]

$$\begin{array}{cc}\hfill f(\mathbf{P},\mathbf{T},\mathbf{R},\mathbf{Q})& =\mathbb{E}\left\{\left.\parallel \mathbf{h}-\widehat{\mathbf{h}}{\parallel }^2\right|\mathbf{P},\mathbf{T},\mathbf{R},\mathbf{Q}\right\}\hfill \\ \hfill & =\mathrm{Tr}\left[{({\mathbf{T}}^{-1}+\mathbf{P}{\mathbf{Q}}^{-1}{\mathbf{P}}^H)}^{-1}\otimes \mathbf{R}\right]\hfill \\ \hfill & =\mathrm{Tr}\left(\mathbf{R}\right)·\mathrm{Tr}\left[{({\mathbf{T}}^{-1}+\mathbf{P}{\mathbf{Q}}^{-1}{\mathbf{P}}^H)}^{-1}\right].\hfill \end{array}$$

(7)

Remark 1.

When the additive noise term $\mathbf{W}\in {\mathbb{C}}^{M_r\times L}$ is considered (i.e., not ignored), the total interference-plus-noise term can be written as

$$\mathbf{N}=\sum _{k=1}^K{\mathbf{H}}_k{\mathbf{S}}_k+\mathbf{W}.$$

Suppose that ${\mathbf{S}}_k$, $\forall k$, and $\mathbf{W}$ are independent of each other and that the covariance matrix of $\mathbf{W}$ takes the following form:

$$\mathbb{E}\left[\mathit{vec}\left(\mathbf{W}\right){\mathit{vec}}^H\left(\mathbf{W}\right)\right]={\mathbf{\Phi }}^T\times \mathbf{R}.$$

Then, it still follows that ${\mathbf{C}}_n=\mathbb{E}\left[\mathbf{n}{\mathbf{n}}^H\right]={\mathbf{Q}}^T\otimes \mathbf{R}$, with $\mathbf{Q}$ given by

$$\mathbf{Q}=\sum _{k=1}^K\mathit{Tr}\left({\mathbf{\Psi }}_{s,k}{\mathbf{T}}_{I,k}\right){\mathbf{\Psi }}_{\tau ,k}+\mathbf{\Phi }.$$

In a similar fashion, it is also possible to cover a multi-user or multi-cell scenario by treating $\mathbf{W}$ as the intra-cell or inter-cell interference, respectively.

2.2. Problem Formulation

In practice, the matrices $\mathbf{T}$, $\mathbf{R}$, and $\mathbf{Q}$ are imperfect at the transmitter side due to feedback-related issues [15,16,17]. Considering such imperfection, we mathematically model the correlation uncertainties as follows:

$$\mathbf{T}=\widehat{\mathbf{T}}+\tilde{\mathbf{T}}\in \mathcal{T},\mathbf{R}=\widehat{\mathbf{R}}+\tilde{\mathbf{R}}\in \mathcal{R}\mathrm{and}\mathbf{Q}=\widehat{\mathbf{Q}}+\tilde{\mathbf{Q}}\in \mathcal{Q},$$

(8)

respectively, where $\widehat{\mathbf{T}}$, $\widehat{\mathbf{R}}$, and $\widehat{\mathbf{Q}}$ are imperfect estimates of $\mathbf{T}$, $\mathbf{R}$, and $\mathbf{Q}$, respectively, and $\tilde{\mathbf{T}}$, $\tilde{\mathbf{R}}$, and $\tilde{\mathbf{Q}}$ represent the corresponding correlation error matrices. The uncertainty sets are denoted by $\mathcal{T}$, $\mathcal{R}$, and $\mathcal{Q}$, and they are, respectively, defined as

$$\mathcal{T}=\left\{\tilde{\mathbf{T}}:{\parallel \tilde{\mathbf{T}}\parallel }_F^2\le {ϵ}_T,\widehat{\mathbf{T}}+\tilde{\mathbf{T}}⪰\mathbf{0}\right\},$$

(9)

$$\mathcal{R}=\left\{\tilde{\mathbf{R}}:{\parallel \tilde{\mathbf{R}}\parallel }_F^2\le {ϵ}_R,\widehat{\mathbf{R}}+\tilde{\mathbf{R}}⪰\mathbf{0}\right\},$$

(10)

and

$$\mathcal{Q}=\left\{\tilde{\mathbf{Q}}:{\parallel \tilde{\mathbf{Q}}\parallel }_F^2\le {ϵ}_T,\widehat{\mathbf{Q}}+\tilde{\mathbf{Q}}⪰\mathbf{0}\right\},$$

(11)

where the parameters ${ϵ}_T$, ${ϵ}_R$, and ${ϵ}_Q$ denote the spherical radii of $\mathcal{T}$, $\mathcal{R}$, and $\mathcal{Q}$, respectively, and they are related to the quantization step size and equal error contour [15,16,17]. To guarantee certain performance of the channel estimation for any possible uncertainty on the covariance information, in this paper, we adopt the spherical uncertainty model for the covariance errors as this model is the most uncertain among the uncertainty models. Even when other uncertainty models (e.g., bounded spectral norms or element-wise errors) are adopted, similar results or conclusions to those for the spherical uncertainty model in this paper can still be drawn.

Using (8), we can rewrite (7) as

$$J(\mathbf{P},\tilde{\mathbf{T}},\tilde{\mathbf{R}},\tilde{\mathbf{Q}})=\mathrm{Tr}\left(\widehat{\mathbf{R}}+\tilde{\mathbf{R}}\right)·\mathrm{Tr}\left[{\left\{{(\widehat{\mathbf{T}}+\tilde{\mathbf{T}})}^{-1}+\mathbf{P}{(\widehat{\mathbf{Q}}+\tilde{\mathbf{Q}})}^{-1}{\mathbf{P}}^H\right\}}^{-1}\right].$$

(12)

Note from (12) that the estimation MSE is a function of the known parameter $\mathbf{P}$ as well as the unknown parameters $(\tilde{\mathbf{T}},\tilde{\mathbf{R}},\tilde{\mathbf{Q}})$. To deal with the unknown parameters, we follow the widely used concept of the worst-case robustness [15,16,17]. Specifically, we use the worst-case MSE,

$$J^{\star }\left(\mathbf{P}\right)=\underset{(\tilde{\mathbf{T}},\tilde{\mathbf{R}},\tilde{\mathbf{Q}})\in \mathcal{T}\times \mathcal{R}\times \mathcal{Q}}{\max }J(\mathbf{P},\tilde{\mathbf{T}},\tilde{\mathbf{R}},\tilde{\mathbf{Q}}),$$

(13)

as a design criterion. Considering the training power constraint, an optimization problem of interest can then be formulated as follows:

$$\begin{array}{cc}\underset{\mathbf{P}}{\min }& \underset{\mathbf{T},\mathbf{R},\mathbf{Q}}{\max }J(\mathbf{P},\tilde{\mathbf{T}},\tilde{\mathbf{R}},\tilde{\mathbf{Q}})\hfill \end{array}$$

(14a)

$$\begin{array}{cc}\mathrm{s}.\mathrm{t}.& \mathrm{Tr}\left(\mathbf{P}{\mathbf{P}}^H\right)\le {\mathcal{P}}_T,\hfill \end{array}$$

(14b)

$$\begin{array}{cc}& \tilde{\mathbf{T}}\in \mathcal{T},\tilde{\mathbf{R}}\in \mathcal{R},\tilde{\mathbf{Q}}\in \mathcal{Q},\hfill \end{array}$$

(14c)

where ${\mathcal{P}}_T$ denotes the total transmit power of the training signal. It can be shown that problem (14) is not convex–concave (the problem

$$\underset{\mathbf{a}\in \mathcal{A}}{\min }\underset{\mathbf{b}\in \mathcal{B}}{\max }\phi (\mathbf{a},\mathbf{b})$$

is convex–concave if the constraint sets $\mathcal{A}$ and $\mathcal{B}$ are all convex, and the objective function $\phi$ is a convex function of the minimization variable $\mathbf{a}$ and a concave function of the maximization variable $\mathbf{b}$ [28,29]) due to the non-convexity of the worst-case MSE $J^{\star }$ in (13) w.r.t. $\mathbf{P}$. For this reason, it is more complicated to tackle the problem (14) than the problems in [18,19,20], which are all convex–concave. For example, the conventional iterative scheme in [20] does not work well since it may converge to a local optimal solution.

3. Training Signal Optimization

In this section, we optimize the training signal by solving problem (14). Specifically, a closed-form structure of the worst-case MSE minimizing training signal is initially obtained by deriving the structure of the optimal matrix solution for the problem (14). From the structure, the original problem (14) involving the complex-valued matrices becomes a simple power allocation problem involving the real-valued scalar variables only. Thereafter, optimal and suboptimal power allocation schemes are proposed. The optimal power allocation is determined numerically and the suboptimal power allocation is obtained in a closed-form with low complexity.

3.1. Worst-Case MSE Minimizing Training Structure

Let $({\mathbf{P}}^{\star },{\tilde{\mathbf{T}}}^{\star },{\tilde{\mathbf{R}}}^{\star },{\tilde{\mathbf{Q}}}^{\star })$ be an optimal solution to the minimax problem (14). Then, the structures of the optimal training signal ${\mathbf{P}}^{\star }$ and the worst-case correlation errors $({\tilde{\mathbf{T}}}^{\star },{\tilde{\mathbf{R}}}^{\star },{\tilde{\mathbf{Q}}}^{\star })$ are derived in the following theorem.

Theorem 1.

Let $\widehat{\mathbf{T}}={\mathbf{U}}_T{\mathbf{\Lambda }}_T{\mathbf{U}}_T^H$ and $\widehat{\mathbf{Q}}={\mathbf{U}}_Q{\mathbf{\Lambda }}_Q{\mathbf{U}}_Q^H$ be the eigenvalue decompositions (EVDs) of the matrices $\widehat{\mathbf{T}}$ and $\widehat{\mathbf{Q}}$, respectively, where

$${\mathbf{\Lambda }}_T=\mathit{Diag}({\lambda }_{T,1},\cdots ,{\lambda }_{T,M_t})$$

and

$${\mathbf{\Lambda }}_Q=\mathit{Diag}({\lambda }_{Q,1},\cdots ,{\lambda }_{Q,L})$$

are diagonal matrices whose diagonal entries are the eigenvalues of $\widehat{\mathbf{T}}$ and $\widehat{\mathbf{Q}}$, respectively. The columns of the unitary matrices ${\mathbf{U}}_T$ and ${\mathbf{U}}_Q$ consist of the eigenvectors of $\widehat{\mathbf{T}}$ and $\widehat{\mathbf{Q}}$, respectively. For massive MIMO systems, there are several efficient methods or low-complexity approximations to perform the EVDs of covariance matrices with large sizes. These include the power iteration, Lanczos algorithm, Arnoldi iteration, randomized EVD, Jacobi–Davidson, locally optimal block preconditioned conjugate gradient (LOBPCG), etc. Other matrix-free computation approaches based on matrix–vector multiplications can also be used. The optimal training signal ${\mathbf{P}}^{\star }$, minimizing the worst-case MSE, has the following structure (this solution generally requires very low signaling overheads for the feedback transmission; specifically, only $M_t$ real numbers and $M_t$ integers need to be fed back to construct the training signal matrix at the transmitter [10]; accordingly, the limited feedback issue may not be a serious design concern in practice):

$${\mathbf{P}}^{\star }={\mathbf{U}}_T{\mathbf{D}}_P^{\star }{\mathbf{U}}_Q^H,$$

(15)

and the worst-case correlation error matrices $({\tilde{\mathbf{T}}}^{\star },{\tilde{\mathbf{R}}}^{\star },{\tilde{\mathbf{Q}}}^{\star })$, respectively, have the following structures:

$${\tilde{\mathbf{T}}}^{\star }={\mathbf{U}}_T{\mathbf{D}}_T^{\star }{\mathbf{U}}_T^H,{\tilde{\mathbf{R}}}^{\star }=\sqrt{{\displaystyle \frac{{ϵ}_R}{M_r}}}{\mathbf{I}}_{M_r},{\tilde{\mathbf{Q}}}^{\star }={\mathbf{U}}_Q{\mathbf{D}}_Q^{\star }{\mathbf{U}}_Q^H,$$

(16)

where ${\mathbf{D}}_P^{\star }\in {\mathbb{R}}^{M_t\times L}$ is a rectangular diagonal matrix containing non-negative elements on its main diagonal, and ${\mathbf{D}}_T^{\star }\in {\mathbb{R}}^{M_t\times M_t}$ and ${\mathbf{D}}_Q^{\star }\in {\mathbb{R}}^{L\times L}$ are real diagonal matrices. It thus can be inferred that the worst case corresponds to a diagonal channel covariance matrix and the best case corresponds to a channel covariance matrix whose main diagonal values are close to zero and other elements are close to each other.

Proof. 

See Appendix A. □

The optimal training structure (15) means that the transmit directions of the training signal should be matched to the eigenvectors of the estimated correlation matrices at the transmitter side, whereas the training power has to be allocated according to the worst-case eigenvalues of the imperfect correlation matrices. This observation actually results from the fact that the uncertainty sets in (9)–(11) do not contain any directional information. Next, the worst-case value of the receiver correlation error seen by the transmitter is given by the scaled identity matrix, i.e., equal perturbation. This is due to the Kronecker model, in which the correlation information at the transmitter and receiver sides can be separable [25]. Finally, it is important to note that the number of real-valued variables to be optimized is reduced from $2(M_{t}L+M_t^2+M_r^2+L^2)$ to $\min \{M_t,L\}+M_t+L$ with the proposed structures in (15) and (16).

Remark 2.

When the interference is temporally white and its variance information is perfectly known, the result (15) becomes ${\mathbf{P}}^{\star }={\mathbf{U}}_T{\mathbf{D}}_P^{\star }$, or equivalently, ${\mathbf{P}}^{\star }{\mathbf{P}}^{\star H}={\mathbf{U}}_T{\mathbf{\Lambda }}_P^{\star }{\mathbf{U}}_T^H$, where ${\mathbf{\Lambda }}_P^{\star }={\mathbf{D}}_P^{\star }{\mathbf{D}}_P^{\star T}$, which is consistent with the result derived in [19].

Remark 3.

When $\mathbf{Q}$ is perfect, but $\mathbf{T}$ is imperfect, which is the case considered in [20], the training structure in (15) becomes ${\mathbf{P}}^{\star }={\mathbf{U}}_T{\mathbf{D}}_P^{\star }{\mathbf{V}}_Q^H$, where the columns of ${\mathbf{V}}_Q$ consist of the eigenvectors of $\mathbf{Q}$.

3.2. Optimal Power Allocation

In this section, we determine the matrices ${\mathbf{D}}_P^{\star }$, ${\mathbf{D}}_T^{\star }$, and ${\mathbf{D}}_Q^{\star }$. Substituting the structures $\mathbf{P}={\mathbf{U}}_T{\mathbf{D}}_P{\mathbf{U}}_Q^H$, $\tilde{\mathbf{T}}={\mathbf{U}}_T{\mathbf{D}}_T{\mathbf{U}}_T^H$, and $\tilde{\mathbf{Q}}={\mathbf{U}}_Q{\mathbf{D}}_Q{\mathbf{U}}_Q^H$ into (14), we can rewrite the estimated MSE J in (12) as

$$\zeta ({\mathbf{d}}_P,{\mathbf{d}}_T,{\mathbf{d}}_Q)=\beta \sum _{i=1}^{\nu }{\left({\displaystyle \frac{1}{{\lambda }_{T,i}+d_{T,i}}}+{\displaystyle \frac{d_{P,i}}{{\lambda }_{Q,i}+d_{Q,i}}}\right)}^{-1}+\beta \sum _{i=\nu +1}^{M_t}({\lambda }_{T,i}+d_{T,i}),$$

(17)

where $\beta =\mathrm{Tr}\left(\widehat{\mathbf{R}}\right)+\sqrt{M_r{ϵ}_R}$ and $\nu =\min \{M_t,L\}$ denotes the maximum rank of $\mathbf{P}$. Also,

$${\mathbf{d}}_P={[d_{P,1},\cdots ,d_{P,\nu }]}^T,$$

$${\mathbf{d}}_T={[d_{T,1},\cdots ,d_{T,M_t}]}^T,$$

and

$${\mathbf{d}}_Q={[d_{Q,1},\cdots ,d_{Q,L}]}^T$$

are the vectors of the diagonal elements of ${\mathbf{D}}_P{\mathbf{D}}_P^T$, ${\mathbf{D}}_T$, and ${\mathbf{D}}_Q$, respectively. From (17), the optimal training power allocation ${\mathbf{d}}_P^{\star }$ can be obtained by solving the following problem:

$$\begin{array}{cc}\underset{{\mathbf{d}}_P}{\min }& \underset{{\mathbf{d}}_T,{\mathbf{d}}_Q}{\max }\zeta ({\mathbf{d}}_P,{\mathbf{d}}_T,{\mathbf{d}}_Q)\hfill \end{array}$$

(18a)

$$\begin{array}{cc}\mathrm{s}.\mathrm{t}.& {\mathbf{d}}_P\in {\mathcal{D}}_{\mathcal{P}},{\mathbf{d}}_T\in {\mathcal{D}}_{\mathcal{T}},{\mathbf{d}}_Q\in {\mathcal{D}}_{\mathcal{Q}},\hfill \end{array}$$

(18b)

where the constraint sets ${\mathcal{D}}_{\mathcal{P}}$, ${\mathcal{D}}_{\mathcal{T}}$, and ${\mathcal{D}}_{\mathcal{Q}}$ are defined as

$${\mathcal{D}}_{\mathcal{P}}=\left\{{\mathbf{d}}_P:{\sum }_{i=1}^{\nu }{d}_{P,i}\le {\mathcal{P}}_T,d_{P,i}\ge 0,i=1,\cdots ,\nu \right\},$$

$${\mathcal{D}}_{\mathcal{T}}=\left\{{\mathbf{d}}_T:{\parallel {\mathbf{d}}_T\parallel }^2\le {ϵ}_T,{\lambda }_{T,i}+d_{T,i}\ge 0,i=1,\cdots ,M_t\right\}$$

and

$${\mathcal{D}}_{\mathcal{Q}}=\left\{{\mathbf{d}}_Q:{\parallel {\mathbf{d}}_Q\parallel }^2\le {ϵ}_Q,{\lambda }_{Q,i}+d_{Q,i}\ge 0,i=1,\cdots ,L\right\},$$

respectively. In the following lemma, we show that the cost function of the problem (18) is convex–concave.

Definition 1

(Convex–concave function). [28,29] We say the function $\phi (\mathbf{a},\mathbf{b}):\mathcal{A}\times \mathcal{B}\subset {\mathbb{R}}^n\times {\mathbb{R}}^m\to \mathbb{R}$ is convex–concave if $\phi (\mathbf{a},\mathbf{b})$ is a convex function of $\mathbf{a}\in \mathcal{A}$ for fixed $\mathbf{b}\in \mathcal{B}$ and a concave function of $\mathbf{b}\in \mathcal{B}$ for fixed $\mathbf{a}\in \mathcal{A}$.

Lemma 1.

The MSE ζ in (17) is convex in ${\mathbf{d}}_P\in {\mathcal{D}}_{\mathcal{P}}$ for fixed $({\mathbf{d}}_T,{\mathbf{d}}_Q)\in {\mathcal{D}}_{\mathcal{T}}\times {\mathcal{D}}_{\mathcal{Q}}$, and concave in ${\mathbf{d}}_T\in {\mathcal{D}}_{\mathcal{T}}$ and ${\mathbf{d}}_Q\in {\mathcal{D}}_{\mathcal{Q}}$ for fixed ${\mathbf{d}}_P\in {\mathcal{D}}_{\mathcal{P}}$.

Proof. 

See Appendix B. □

Lemma 1 implies that the power allocation problem (18) is convex–concave since the constraint sets ${\mathcal{D}}_{\mathcal{P}}$, ${\mathcal{D}}_{\mathcal{T}}$, and ${\mathcal{D}}_{\mathcal{Q}}$ are all convex. Unfortunately, it is not possible to find a closed-form solution in general due to the nonlinearity of the objective function and norm constraints on ${\mathbf{d}}_T$ and ${\mathbf{d}}_Q$. However, the optimal power allocation ${\mathbf{d}}_P^{\star }$ can be computed numerically by using well-known methods such as the interior point method or barrier method [28,30].

3.3. Suboptimal Power Allocation in Closed-Form

The numerical approach for optimal power allocation may be undesirable in real applications due to a non-negligible complexity during the channel training phase. Considering this problem, we propose a suboptimal power allocation in a closed-form. Since the power allocation problem (18) is convex–concave, we can interchange the minimum and maximum operators of the problem (18) according to [31] (Lemma 36.2)

as follows:

$$\begin{array}{cc}\underset{{\mathbf{d}}_T,{\mathbf{d}}_Q}{\max }& \underset{{\mathbf{d}}_P}{\min }\zeta ({\mathbf{d}}_P,{\mathbf{d}}_T,{\mathbf{d}}_Q)\hfill \end{array}$$

(19a)

$$\begin{array}{cc}\mathrm{s}.\mathrm{t}.& {\mathbf{d}}_P\in {\mathcal{D}}_{\mathcal{P}},{\mathbf{d}}_T\in {\mathcal{D}}_{\mathcal{T}},{\mathbf{d}}_Q\in {\mathcal{D}}_{\mathcal{Q}}.\hfill \end{array}$$

(19b)

Defining

$${\mathbf{x}}_T={[{\lambda }_{T,1}+d_{T,1},\cdots ,{\lambda }_{T,M_t}+d_{T,M_t}]}^T$$

and

$${\mathbf{x}}_Q={[{\lambda }_{Q,1}+d_{Q,1},\cdots ,{\lambda }_{Q,L}+d_{Q,L}]}^T,$$

problem (19) can be equivalently reformulated as

$$\begin{array}{cc}\underset{{\mathbf{x}}_T,{\mathbf{x}}_Q}{\max }& \underset{{\mathbf{d}}_P}{\min }g({\mathbf{d}}_P,{\mathbf{x}}_T,{\mathbf{x}}_Q)\hfill \end{array}$$

(20a)

$$\begin{array}{cc}\mathrm{s}.\mathrm{t}.& {\mathbf{d}}_P\in {\mathcal{D}}_{\mathcal{P}},{\mathbf{x}}_T\in {\mathcal{X}}_{\mathcal{T}},{\mathbf{x}}_Q\in {\mathcal{X}}_{\mathcal{Q}},\hfill \end{array}$$

(20b)

where the cost function g in (23a) is

$$g({\mathbf{d}}_P,{\mathbf{x}}_T,{\mathbf{x}}_Q)=\beta \sum _{i=1}^{\nu }{\left({\displaystyle \frac{1}{x_{T,i}}}+{\displaystyle \frac{d_{P,i}}{x_{Q,i}}}\right)}^{-1}+\beta \sum _{i=\nu +1}^{M_t}{x}_{T,i},$$

(21)

and the constraint sets ${\mathcal{X}}_{\mathcal{T}}$ and ${\mathcal{X}}_{\mathcal{Q}}$ are defined as

$${\mathcal{X}}_{\mathcal{T}}=\left\{{\mathbf{x}}_T:{\parallel {\mathbf{x}}_T-{\mathit{\lambda }}_T\parallel }^2\le {ϵ}_T,x_{T,i}\ge 0,i=1,\cdots ,M_t\right\}$$

and

$${\mathcal{X}}_{\mathcal{Q}}=\left\{{\mathbf{x}}_Q:{\parallel {\mathbf{x}}_Q-{\mathit{\lambda }}_Q\parallel }^2\le {ϵ}_Q,x_{Q,i}\ge 0,i=1,\cdots ,L\right\},$$

respectively. It is still difficult to obtain the optimal solution to problem (20) in a closed-form. To overcome this difficulty, in the following, we instead find a suboptimal solution.

Definition 2.

[32] For any $\mathbf{a}\in {\mathbb{R}}^n$, let $a_{\left[1\right]}\ge \cdots \ge a_{\left[n\right]}$ denote the components of $\mathbf{a}$ in decreasing order.

Definition 3.

[32] (Ch.1.A.1) The vector $\mathbf{a}\in {\mathbb{R}}^n$ is majorized by $\mathbf{b}\in {\mathbb{R}}^n$, denoted by $\mathbf{a}\prec \mathbf{b}$, if ${\sum }_{m=1}^l{a}_{\left[m\right]}\le {\sum }_{m=1}^l{b}_{\left[m\right]}$, $1\le l\le n-1$, and ${\sum }_{m=1}^n{a}_{\left[m\right]}={\sum }_{m=1}^n{b}_{\left[m\right]}$.

Definition 4

(Schur-concave function). [32] (Ch.3.A.1) A real-valued function $\phi :\mathcal{A}\subset {\mathbb{R}}^n\to \mathbb{R}$ is said to be Schur-concave on $\mathcal{A}$ if $\mathbf{a}\prec \mathbf{b}$⇒$\phi \left(\mathbf{a}\right)\ge \phi \left(\mathbf{b}\right)$.

Lemma 2.

Let $g^{\star }({\mathbf{x}}_T,{\mathbf{x}}_Q)=\underset{{\mathbf{d}}_P\in {\mathcal{D}}_{\mathcal{P}}}{\min }g({\mathbf{d}}_P,{\mathbf{x}}_T,{\mathbf{x}}_Q)$ be the minimum MSE. Then, $g^{\star }({\mathbf{x}}_T,{\mathbf{x}}_Q)$ is Schur-concave in ${\mathbf{x}}_T$ and ${\mathbf{x}}_Q$. In other words, if there exists the vectors ${\tilde{\mathbf{x}}}_T$ and ${\tilde{\mathbf{x}}}_Q$ such that ${\tilde{\mathbf{x}}}_T\prec {\mathbf{x}}_T$ and ${\tilde{\mathbf{x}}}_Q\prec {\mathbf{x}}_Q$, then $g^{\star }({\tilde{\mathbf{x}}}_T,{\tilde{\mathbf{x}}}_Q)\ge g^{\star }({\mathbf{x}}_T,{\mathbf{x}}_Q)$.

Proof. 

See Appendix C. □

From the above lemma, we can find a suboptimal solution to problem (20), with which the value of $g^{\star }$ increases, but is not maximized, as follows:

$${\tilde{\mathbf{x}}}_T={\mathit{\lambda }}_T+{\delta }_T{\mathbf{1}}_{M_t}\mathrm{and}{\tilde{\mathbf{x}}}_Q={\mathit{\lambda }}_Q+{\delta }_Q\left[{\mathbf{1}}_{\nu },{\mathbf{0}}_{(L-\nu )\times 1}\right],$$

(22)

where

$${\mathit{\lambda }}_T={[{\lambda }_{T,1},\cdots ,{\lambda }_{T,M_t}]}^T$$

and

$${\mathit{\lambda }}_Q={[{\lambda }_{Q,1},\cdots ,{\lambda }_{Q,L}]}^T.$$

Also, ${\mathbf{1}}_n$ denotes the $n\times 1$ vector whose elements are all 1, ${\delta }_T={\displaystyle \frac{1}{M_t}}{\sum }_{j=1}^{M_t}{d}_{T,j}$ and ${\delta }_Q={\displaystyle \frac{1}{\nu }}{\sum }_{j=1}^{\nu }{d}_{Q,j}$. To satisfy the constraints ${\tilde{\mathbf{x}}}_T\in {\mathcal{X}}_{\mathcal{T}}$ and ${\tilde{\mathbf{x}}}_Q\in {\mathcal{X}}_{\mathcal{Q}}$, the values of ${\delta }_T$ and ${\delta }_Q$ can be chosen as ${\delta }_T=\sqrt{{\displaystyle \frac{{ϵ}_T}{M_t}}}$ and ${\delta }_Q=\sqrt{{\displaystyle \frac{{ϵ}_Q}{\nu }}}$, respectively. By substituting (22) into (20), an optimization problem for suboptimal training power allocation can be formulated as

$$\begin{array}{cc}\underset{{\mathbf{d}}_P}{\min }& g({\mathbf{d}}_P,{\tilde{\mathbf{x}}}_T,{\tilde{\mathbf{x}}}_Q)\hfill \end{array}$$

(23a)

$$\begin{array}{cc}\mathrm{s}.\mathrm{t}.& {\mathbf{d}}_P\in {\mathcal{D}}_{\mathcal{P}}.\hfill \end{array}$$

(23b)

It is assumed that the elements of ${\mathit{\lambda }}_T$ and ${\mathit{\lambda }}_Q$ are arranged in descending and ascending orders, respectively. Then, the solution to problem (23) can be obtained in a closed form by the Lagrange multiplier method [28], as

$$d_{P,i}^{\left(so\right)}=\left\{\begin{array}{cc}\left({\mathcal{P}}_T+\sum _{j=1}^r{\displaystyle \frac{{\lambda }_{Q,j}+{\delta }_Q}{{\lambda }_{T,j}+{\delta }_T}}\right){\displaystyle \frac{\sqrt{{\lambda }_{Q,j}+{\delta }_Q}}{{\sum }_{j=1}^r\sqrt{{\lambda }_{Q,j}+{\delta }_Q}}}-{\displaystyle \frac{{\lambda }_{Q,j}+{\delta }_Q}{{\lambda }_{T,j}+{\delta }_T}},\hfill & i=1,\cdots ,r,\hfill \\ 0,\hfill & i=r+1,\cdots ,\nu ,\hfill \end{array}\right.$$

(24)

where $r=\max \{i\in \{1,\cdots ,\nu \}:d_{P,i}^{\left(so\right)}>0\}$ is the largest i such that $d_{P,i}^{\left(so\right)}>0$. The suboptimal solution in (24) follows the conventional water-filling strategy, i.e., it assigns more training power to the larger eigenvalues of $\widehat{\mathbf{T}}$ and smaller eigenvalues of $\widehat{\mathbf{Q}}$. One difference is that there exist the equal errors ${\delta }_T$ and ${\delta }_Q$ in the eigenvalues of $\widehat{\mathbf{T}}$ and $\widehat{\mathbf{Q}}$, respectively.

Remark 4.

For the case considered in Remark 1, we obtain the suboptimal power allocation from (24) by setting ${\delta }_Q=0$ and ${\lambda }_{Q,j}={\sigma }_Q$, $j=1,\cdots ,L$, where ${\sigma }_Q$ denotes the interference variance.

Remark 5.

In the case considered in Remark 2, the suboptimal power allocation can be obtained from (24) by setting ${\delta }_Q=0$ and ${\lambda }_{Q,j}={\sigma }_{Q,j}$, $j=1,\cdots ,L$, where ${\left\{{\sigma }_{Q,j}\right\}}_{j=1}^L$ denote the eigenvalues of $\mathbf{Q}$.

The suboptimal solution ${\mathbf{d}}_P^{\left(so\right)}$ in (24) becomes optimum if the minimum MSE $g^{\star }$ is an increasing function of both ${\delta }_T={\displaystyle \frac{1}{M_t}}{\sum }_{j=1}^{M_t}{d}_{T,j}$ and ${\delta }_Q={\displaystyle \frac{1}{\nu }}{\sum }_{j=1}^{\nu }{d}_{Q,j}$, which, however, does not hold as can be seen in (A16) in Appendix C. Hence, the suboptimal solution does not guarantee achieving optimal performance in general. However, $g^{\star }$ is tightly upper bounded by an increasing function of ${\delta }_T$ and ${\delta }_Q$. To show this, we consider the simple case of $M_t=L$. Then, we have

$$\begin{array}{cc}\hfill g^{\star }({\mathbf{x}}_T,{\mathbf{x}}_Q)& =\underset{{\mathbf{d}}_P\in {\mathcal{D}}_{\mathcal{P}}}{\min }g({\mathbf{d}}_P,{\mathbf{x}}_T,{\mathbf{x}}_Q)\hfill \end{array}$$

(25a)

$$\begin{array}{cc}\hfill & \le \beta \sum _{i=1}^{M_t}{\left({\displaystyle \frac{1}{{\lambda }_{T,i}+d_{T,i}}}+{\displaystyle \frac{{\mathcal{P}}_T/M_t}{{\lambda }_{Q,i}+d_{Q,i}}}\right)}^{-1}\hfill \end{array}$$

(25b)

$$\begin{array}{cc}\hfill & \le \beta M_t{\left({\displaystyle \frac{1}{{\overline{\lambda }}_T+{\delta }_T}}+{\displaystyle \frac{{\mathcal{P}}_T/M_t}{{\overline{\lambda }}_Q+{\delta }_Q}}\right)}^{-1}\hfill \end{array}$$

(25c)

where ${\overline{\lambda }}_T={\displaystyle \frac{1}{M_t}}{\sum }_{i=1}^{M_t}{\lambda }_{T,i}$ and ${\overline{\lambda }}_Q={\displaystyle \frac{1}{M_t}}{\sum }_{i=1}^{M_t}{\lambda }_{Q,i}$. The inequality (25b) follows from the fact that ${\mathbf{d}}_P=({\mathcal{P}}_T/M_t){\mathbf{1}}_{M_t}$ is a feasible solution to the problem in (25a). The inequality (25c) follows from the concavity. The upper bound (25) becomes tight for the following cases: (i) $M_t=1$, (ii) ${\mathcal{P}}_T=0$, and (iii) ${\mathcal{P}}_T\to \infty$. Therefore, we deduce that the suboptimal scheme may provide near-optimal performance at low and high training powers when the number of transmit antennas is not large enough. Otherwise, the performance of the suboptimal scheme may degrade because the bound (25) becomes loose. To clarify our discussion, in Figure 2, we plot the value of $g^{\star }$ and that of its upper bound versus the total training power ${\mathcal{P}}_T$ for various numbers of transmit antennas. In the figure, we set ${\left[\widehat{\mathbf{T}}\right]}_{m,n}={\left[\widehat{\mathbf{Q}}\right]}_{m,n}={\rho }^{|m-n|}$, $1\le m,n\le M_t$, $\rho =0.5$, $\beta =M_t$, ${ϵ}_T=0.3\Vert \widehat{\mathbf{T}}{\Vert }_F^2$, and ${ϵ}_Q=0.3\Vert \widehat{\mathbf{Q}}{\Vert }_F^2$. The real curve and dashed-dot curve indicate the value of $g^{\star }$ and that of its upper bound, respectively. From the figure, we can observe that the upper bound is tight to the actual value when ${\mathcal{P}}_T$ is sufficiently low or high. Also, for the range of ${\mathcal{P}}_T=-5$ dB to ${\mathcal{P}}_T=15$ dB, the gap between $g^{\star }$ and its upper bound becomes larger as the number of transmit antennas increases. In the following section, the performance of the suboptimal training scheme is concretely demonstrated by numerical simulations.

3.4. Complexity Comparison

To validate the efficiency of the proposed designs, in this section, we compare the computational complexity of the proposed schemes with that of the iterative algorithm in [20]. In each iteration of the conventional algorithm, the arithmetic complexity of $\mathcal{O}\left((M_t^{6.5}+L^{6.5})\log (1/\psi )\right)$ is required for computing $\tilde{\mathbf{T}}$ and $\tilde{\mathbf{Q}}$ for a fixed $\mathbf{P}$ [33], where $\psi$ is the solution accuracy for the interior point method, and the arithmetic complexity of $\mathcal{O}\left(M_t^3+L^3\right)$ is needed to compute $\mathbf{P}$ for fixed $\tilde{\mathbf{T}}$ and $\tilde{\mathbf{Q}}$ [34]. Thus, the conventional algorithm in [20] requires the complexity $\mathcal{O}\left(\left\{M_t^3+L^3+(M_t^{6.5}+L^{6.5})\log (1/\psi )\right\}\right)$ per iteration. Let $N_{\mathrm{iter}}$ denote the total number of iterations required for the algorithm in [20]. Then, it requires the total computational complexity of $\mathcal{O}\left(\left\{M_t^3+L^3+(M_t^{6.5}+L^{6.5})N_{\mathrm{iter}}\log (1/\psi )\right\}\right)$. Next, we consider the complexity of the proposed training schemes. To implement the training structure in (15), the arithmetic operation of $\mathcal{O}\left(M_t^3+L^3\right)$ is required for computing the EVDs of $\widehat{\mathbf{T}}$ and $\widehat{\mathbf{Q}}$ [34]. Also, we require the computational complexity of $\mathcal{O}\left((M_t^{3.5}+L^{3.5})\log (1/\psi )\right)$ to compute the optimal power allocation numerically [33]. Therefore, the overall complexity for the optimal training scheme is $\mathcal{O}\left(M_t^3+L^3+(M_t^{3.5}+L^{3.5})\log (1/\psi )\right)$. On the other hand, since the complexity of computing the suboptimal power allocation in (24) is insignificant compared with the complexity $\mathcal{O}\left(M_t^3+L^3\right)$ [34], the overall complexity for the suboptimal training scheme is $\mathcal{O}\left(M_t^3+L^3\right)$. In

Table 1. Computational complexity comparison.

Method	Computational Complexity	Processing Time (s)
Iterative algorithm in [20]	$\mathcal{O}\left(M_t^3+L^3+(M_t^{6.5}+L^{6.5})N_{\mathrm{iter}}\log (1/\psi )\right)$	$8.8098\times {10}^5$
Proposed optimal training scheme	Total $\mathcal{O}\left(M_t^3+L^3+(M_t^{3.5}+L^{3.5})\log (1/\psi )\right)$	206.4642
Proposed suboptimal training scheme	Total $\mathcal{O}\left(M_t^3+L^3\right)$	68.8214

, we summarize the analytical complexity results and processing time measured on the 12th Gen Intel(R) Core(TM) i9-12900K CPU when $M_t=M_r=L=4$.

4. Simulation Results

In this section, the performance of the proposed schemes is illustrated and compared with that of existing schemes by computer simulations.

4.1. Simulation Setup

In the simulations, we generate the matrices $\widehat{\mathbf{T}}$ and $\widehat{\mathbf{R}}$ according to the inverse Wishart distribution (some discussions on the use of the inverse Wishart distribution can be found in [35,36]; the inverse Wishart distribution is the conjugate prior to the actual correlation matrices $\mathbf{T}$ and $\mathbf{R}$ when $\mathbf{h}$ is Gaussian-distributed; nevertheless, our proposed scheme is applicable to any distributions or types of covariance matrices) as

$$\widehat{\mathbf{T}}\sim {\mathcal{W}}^{-1}(({\kappa }_T-M_t-1){\mathbf{C}}_T,{\kappa }_T)$$

and

$$\widehat{\mathbf{R}}\sim {\mathcal{W}}^{-1}(({\kappa }_R-M_r-1){\mathbf{C}}_R,{\kappa }_R),$$

respectively, where ${\kappa }_T$ and ${\kappa }_R$ denote the degrees-of-freedom parameters of the inverse Wishart distribution. The values of ${\kappa }_T$ and ${\kappa }_R$ are set to ${\kappa }_T=M_t+2$ and ${\kappa }_R=M_r+2$, respectively. The elements of ${\mathbf{C}}_T$ and ${\mathbf{C}}_R$ are generated by the one-ring model [37]:

$${\left[{\mathbf{C}}_T\right]}_{m,n}={\mathcal{J}}_0(2\pi {\Theta }_T|m-n|s_T/\lambda ),1\le m,n\le M_t$$

and

$${\left[{\mathbf{C}}_R\right]}_{m,n}={\mathcal{J}}_0\left(2\pi \right|m-n|s_R/\lambda ),1\le m,n\le M_r,$$

respectively, where ${\mathcal{J}}_0(·)$ is the zeroth-order Bessel function of the first kind, $\lambda$ the carrier wavelength, ${\Theta }_T$ the angular spread, and $s_T$ and $s_R$ the transmit and receive antenna spacings, respectively. The values of $s_T/\lambda$ and $s_R/\lambda$ are set to $s_T/\lambda =0.5$ and $s_R/\lambda =0.25$, respectively. We set $\widehat{\mathbf{Q}}={\sum }_{k=1}^K\mathrm{Tr}\left({\mathbf{\Psi }}_{s,k}{\mathbf{T}}_{I,k}\right){\mathbf{\Psi }}_{\tau ,k}$, where the matrices ${\mathbf{T}}_{I,k}$, ${\mathbf{\Psi }}_{\tau ,k}$, and ${\mathbf{\Psi }}_{s,k}$ are, respectively, generated by

$${\mathbf{T}}_{I,k}\sim {\mathcal{W}}^{-1}(({\kappa }_{I,k}-M_k-1){\mathbf{C}}_{I,k},{\kappa }_{I,k}),$$

$${\mathbf{\Psi }}_{\tau ,k}\sim {\mathcal{W}}^{-1}(({\kappa }_{\tau ,k}-L-1){\mathbf{C}}_{\tau ,k},{\kappa }_{\tau ,k}),$$

and

$${\mathbf{\Psi }}_{s,k}\sim {\mathcal{W}}^{-1}(({\kappa }_R-M_r-1){\mathbf{C}}_{s,k},{\kappa }_R)$$

for $k=1,\cdots ,K$. The parameters ${\kappa }_{I,k}$ and ${\kappa }_{\tau ,k}$ are, respectively, set to ${\kappa }_{I,k}=M_k+2$ and ${\kappa }_{\tau ,k}=L+2$, $k=1,\cdots ,K$. The elements of ${\mathbf{C}}_{I,k}$ are generated by

$${\left[{\mathbf{C}}_{I,k}\right]}_{m,n}={\mathcal{J}}_0(2\pi {\Theta }_{I,k}|m-n|s_{I,k}/\lambda ),1\le m,n\le M_k,$$

with the choice of $s_{I,k}/\lambda =0.5$, $k=1,\cdots ,K$, where ${\left\{{\Theta }_{I,k}\right\}}_{k=1}^K$ denotes the angular spreads. The elements of ${\mathbf{C}}_{\tau ,k}$ and ${\mathbf{C}}_{s,k}$ are generated by the exponential model [38]:

$${\left[{\mathbf{C}}_{\tau ,k}\right]}_{m,n}={\rho }_{\tau ,k}^{|m-n|},1\le m,n\le L$$

and

$${\left[{\mathbf{C}}_{s,k}\right]}_{m,n}={\rho }_{s,k}^{|m-n|},1\le m,n\le M_r,$$

respectively, for $k=1,\cdots ,K$, where ${\rho }_{\tau ,k}$ is the correlation coefficient of the elements of ${\mathbf{C}}_{\tau ,k}$ and ${\rho }_{s,k}$ the correlation coefficient of the elements of ${\mathbf{C}}_{s,k}$. The number of interfering users K is set to $K=3$ and that of the antennas at the kth interferer $M_k$ is set to $M_k=M_t$, $k=1,\cdots ,K$. The relative uncertainty parameters ${\alpha }_T$, ${\alpha }_R$, and ${\alpha }_Q$ are defined such that

$$({ϵ}_T,{ϵ}_R,{ϵ}_Q)=({\alpha }_T\parallel \widehat{\mathbf{T}}{\parallel }_F^2,{\alpha }_R\parallel \widehat{\mathbf{R}}{\parallel }_F^2,{\alpha }_Q\parallel \widehat{\mathbf{Q}}{\parallel }_F^2).$$

The system parameters used in the simulations are summarized in

Table 2. Simulation parameter setup.

System Parameter	Values
Number of transmit antennas, $M_t$	$\{3,4,6,8\}$
Number of receive antennas, $M_r$	$\{3,4,6,8\}$
Training length, L	$\{3,4,6,8\}$
Number of interferers, K	3
Number of antennas at the kth interferer, $M_k$	$\{3,4,6,8\}$
Uncertainty parameters, ${\alpha }_T$, ${\alpha }_R$, and ${\alpha }_Q$	$\{0.1,0.3,0.6\}$
Angular spreads, ${\Theta }_T$ and ${\Theta }_{I,k}$, $\forall k$	$\{{10}^{\circ },{30}^{\circ }\}$
Correlation coefficients, ${\rho }_{\tau ,k}$ and ${\rho }_{s,k}$, $\forall k$	0.9

.

We compare the performance of the proposed schemes with that of two different existing schemes. One is the nonrobust scheme considered in the traditional work [9], where the values of $(\widehat{\mathbf{T}},\widehat{\mathbf{R}},\widehat{\mathbf{Q}})$ are used as perfect values. The other is the semi-robust scheme considered in the work [20], where only the value of $\widehat{\mathbf{Q}}$ is used as the actual value. In the figures, “Proposed optimal”, “Proposed suboptimal”, “Nonrobust”, “Semi-robust”, and “Semi-robust (suboptimal)” indicate the proposed scheme with the optimal power allocation, that with the suboptimal power allocation, the nonrobust scheme, the semi-robust scheme, and the semi-robust scheme with the proposed suboptimal power allocation, respectively.

4.2. Performance Comparison

In Figure 3 and Figure 4, we illustrate the worst-case MSE performance as a function of the effective training signal-to-interference ratio (SIR) for the systems with $M_t=M_r=L=4$ and $M_t=M_r=L=8$. The effective training SIR is defined as $\mathrm{SIR}={\mathcal{P}}_T\mathbb{E}\left[\mathrm{Tr}\left(\widehat{\mathbf{T}}\right)\right]/\mathbb{E}\left[\mathrm{Tr}\left(\widehat{\mathbf{Q}}\right)\right]={\mathcal{P}}_T\mathrm{Tr}\left({\mathbf{C}}_T\right)/{\sum }_{k=1}^3\mathrm{Tr}\left({\mathbf{C}}_{s,k}{\mathbf{C}}_{I,k}\right){\mathbf{C}}_{\tau ,k}$. In Figure 3, we set ${\Theta }_T={10}^{\circ }$, ${\Theta }_{I,k}={10}^{\circ }$ and ${\rho }_{\tau ,k}={\rho }_{s,k}=0.9$, $k=1,2,3$, which represents a strongly correlated environment. On the other hand, in Figure 4, we set ${\Theta }_T={30}^{\circ }$, ${\Theta }_{I,k}={30}^{\circ }$, and ${\rho }_{\tau ,k}={\rho }_{s,k}=0.3$, $k=1,2,3$, which represents a weakly correlated environment. The values of ${\alpha }_T$, ${\alpha }_R$, and ${\alpha }_Q$ are set to ${\alpha }_T={\alpha }_R={\alpha }_Q=0.3$. In the figures, the results are averaged over 500 realizations.

It can be observed from Figure 3 and Figure 4 that the proposed schemes outperform the other schemes. For example, in Figure 3, the proposed schemes have roughly 5 dB improvement in terms of the effective training SIR for the system with $M_t=M_r=L=8$ when the effective training SIR is higher than 10 dB (similar trends can still be observed even for other performance metrics, e.g., the ratio of the standard deviation to the mean value).

Even though the semi-robust scheme outperforms the nonrobust one, its performance degrades due to imperfect knowledge of the interference covariance matrix when the effective training SIR increases or the correlation becomes stronger. The nonrobust scheme provides the worst performance due to imperfect knowledge of both the channel and interference covariance matrices. In the proposed schemes, the suboptimal power allocation provides comparable performance to that of the optimal power allocation. The semi-robust scheme with the proposed suboptimal power allocation works well for the weakly correlated case, but it shows noticeable performance loss for the strongly correlated case.

The worst-case MSE performance is compared in Figure 5 and Figure 6 for various values of $\alpha ={\alpha }_T={\alpha }_R={\alpha }_Q$ when ${\Theta }_T={10}^{\circ }$, ${\Theta }_{I,k}={10}^{\circ }$ and ${\rho }_{\tau ,k}={\rho }_{s,k}=0.9$, $k=1,2,3$, where the parameter $\alpha$ is introduced to set $({ϵ}_T,{ϵ}_R,{ϵ}_Q)=(\alpha \Vert \widehat{\mathbf{T}}{\Vert }_F^2,\alpha \Vert \widehat{\mathbf{R}}{\Vert }_F^2,\alpha \Vert \widehat{\mathbf{Q}}{\Vert }_F^2)$. Figure 5 and Figure 6 show the performance comparison for the systems with $M_t=M_r=L=4$ and $M_t=M_r=L=6$, respectively. When the value of $\alpha$ decreases from 0.6 to 0.1, the performances of all the schemes are improved, and the gap between the proposed and semi-robust schemes increases, but the gap between the semi-robust and nonrobust schemes decreases. This means that the performance is dominated by the uncertainty in the interference covariance matrix when the value of $\alpha$ equals 0.1, i.e., the error is small, but the performance is dominated by the uncertainty in the channel covariance matrix when the value of $\alpha$ equals 0.6, i.e., the error is large. The suboptimal power allocation shows almost the same performance as that of the optimal one in the proposed scheme, but shows notable performance loss in the semi-robust scheme when $\alpha =0.6$.

To inspect the effect of the proposed designs on the quality of the communication systems, the bit error rate (BER) performance is compared in Figure 7 and Figure 8 for $3\times 3$ and $4\times 4$ MIMO systems, respectively (as convention, we here refer to a MIMO system with $M_t$ transmit antennas and $M_r$ transmit antennas as an $M_t\times M_r$ MIMO system). The training SIR is set to 15 dB and the uncertainty parameters are chosen as ${\alpha }_T={\alpha }_R={\alpha }_Q=0.3$. The orthogonal space–time block code in [39] is used to encode the QPSK-modulated symbols and the well-known minimum MSE (MMSE) receiver in [40] is employed to recover the transmitted symbols before detection. For the case of the proposed, nonrobust, and semi-robust training schemes, the imperfect CSI estimated from the training signal is used to implement the MMSE receiver. Additionally, we present the BER performance of the MMSE receiver with the perfect CSI as a benchmark for the BER performance and it is denoted by “Perfect CSI” in the figures. The results are averaged over 500 channel realizations, where $4\times {10}^6$ symbols are transmitted for each channel realization. From the figures, it can be observed that the proposed schemes provide better BER performance than the semi-robust and nonrobust schemes. As the symbol SIR increases, the BER performance of the nonrobust scheme saturates due to the uncertainties in both the channel and interference covariance matrices, and the gap between the proposed and semi-robust schemes increases due to the uncertainty in the interference covariance matrix. In both figures, the proposed suboptimal training scheme provides almost the same performance as that of the optimal one in terms of BER. Therefore, the proposed suboptimal training scheme is more practically useful than the optimal training scheme.

5. Conclusions

We designed an optimal training signal for MIMO systems in the presence of colored interference considering the imperfection of both the channel and interference covariance. From the solution on the structure, it was observed that the optimal training strategy was to allocate the training power according to the worst-case eigenvalues of the imperfect covariance matrices. It was also shown that the proposed training structure can cover the cases considered in the previous works [18,19,20]. The optimal training power allocation scheme was obtained numerically. An efficient suboptimal power allocation scheme was also proposed in a closed form. Simulation results show that the proposed schemes considerably outperform the semi-robust and nonrobust schemes in terms of the worst-case MSE and BER performances. In particular, the proposed closed-form training scheme shows near-optimal performance, and hence it is more practically useful than the optimal training scheme.

An important conclusion from our work is that the performance of training signal design for MIMO systems is sensitive to the channel and interference covariance uncertainties, and thus one should carefully select or determine the covariance uncertainty-based training signals in practice for reliable performance according to the system requirements and operating conditions.