A Tensor Framework for Multi-Linear Complex MMSE Estimation

Abstract

Tensors are higher order generalization of vectors and matrices which can be used to describe signals indexed by more than two indices. This paper introduces a tensor framework for minimum mean square error (MMSE) estimation for multi-domain signals and data using the Einstein Product. The framework addresses both proper and improper complex tensors. The multi-domain nature of tensors has been harnessed to provide an augmented representation of improper complex tensors to account for covariance and pseudo-covariance. The classical notions of linear and widely linear MMSE estimators are extended to tensor case leading to the notion of multi-linear and widely multi-linear MMSE estimation. The Tucker product based n-mode Wiener filtering approach more commonly used in tensor estimation has been shown to be a special case of the proposed multi-linear MMSE estimation. An application of the tensor based estimation in a multiple antenna Orthogonal Frequency Division Multiplexing (MIMO OFDM) system is presented where the tensor formulation allows a convenient treatment of inter-carrier interference. A comparison between the tensor estimation and per sub-carrier estimation used for MIMO OFDM is presented which shows a significant performance advantage of using the tensor framework.

Multi-Domain Communication Systems and Networks: A Tensor-Based Approach

“…where N ∈ C J 1 ו••×J M represents the order M received noise tensor. The conventional MIMO matrix model can be seen as a specific case of (3) where the input and output are order 1 tensors (vectors) and the channel is an order 2 tensor (matrix) [10]. In such a case, the Einstein product between the channel and the input by definition reduces to standard matrix multiplication.…”

“…A detailed derivation of ( 18) can be found in [19]. An application of such a tensor-based receiver in a MIMO OFDM system is considered in [10] where a comparison with per sub-carrier estimation which ignores the inter-carrier interference terms for estimation is presented. It is shown that as interference from other sub-carriers becomes dominant, the performance of per sub-carrier receiver deteriorates significantly, while the tensor receiver's is significantly better since it makes use of the interference terms for data estimation.…”

“…In the past decade, the use of tensors for modeling communication systems has gained much attention for analysis and the improvement of system performance, as it allows a consolidated representation of multiple signaling domains [9][10][11][12][13][14]. In this paper, we show that tensors can be used for signals and systems representation, and tools from multilinear algebra, such as the Einstein product, can be employed for developing a generic system model.…”

“…A tensor-based approach maintains the natural multi-domain structure of the system, and the inter-domain interactions are therefore retained and easily identified. Recently, a tensor framework for multi-linear complex MMSE estimation has been developed in [10], which bypasses the need of any restructuring of the tensor signals for estimation purposes.…”

“…Using the properties of the Einstein product, several well-known linear algebra relations can be extended to a multi-linear setting without the requirement of any tensor to vector/matrix transformation. The multi-linear algebra notions developed using the Einstein product preserves the natural tensor structure of the associated quantities and therefore has various applications in engineering disciplines [10,[15][16][17][18]. In our work, we consider a multi-linear tensor framework that reveals the latent trade-off that exists between multiple domains.…”