###### 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  
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 [PERSON] 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 802.11 ah [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.