A Tensor Based Framework for Multi-Domain Communication Systems

Abstract

The demand for mobile data is likely to grow at a pace more than envisaged in the coming years. Further, as applications such as the Internet of Things (IoT) come to fruition, there will be increased diversity in the types of devices demanding Internet connectivity and their requirements. Significant increase in data rate requirements is also expected due to services such as Ultra High Definition (UHD) video streaming and cloud computing. To meet all these demands, physical layer waveform candidates for future generations of communications need to be robust and inherently capable of extending into multiple domains (space, time, frequency, users, transmission media, code etc.) to ensure efficient utilization of resources. Multiple domains can be innately integrated into the design process of modulation schemes by using tensors, which are multi-way arrays. This paper introduces a unified tensor framework, providing a foundation for multi-domain communication systems that can be used to represent, design and analyse schemes that span several domains. Transmitted signals are represented by Nth order time function tensors which are coupled, using a system tensor of order N + M, with the received signals which are represented by another tensor of order M through the contracted convolution. We begin with the continuous time representation of the tensor system model and present both the strict multi-domain generalization of the Nyquist criterion for zero interference (inter-tensor and intra-tensor interference) as well as a relaxation. We present an equivalent discrete time system model, and as an example of using the tensor framework we derive tensor based linear equalization methods to combat multi-domain interference. An application to multi-user MIMO-GFDM illustrates the utility of this novel framework for derivation of joint domain signal processing techniques.

A Tensor Framework for Multi-Linear Complex MMSE Estimation

“…In this section, we only provide some relevant tensor definitions and properties needed for this paper. More details on tensor algebra can be found in [2], [12], [15], [29], [37], [38]. Definition 1.…”

“…A few examples of such communication systems along with the considered signal domains are presented in Table II. [15].…”

“…Hence matrices and vectors can be seen as order 2 and order 1 tensors respectively [2]. Tensors have found widespread applications in various engineering disciplines including computer vision [3], [4], signal processing [5]- [8], big data and machine learning [9]- [12], image processing [13], [14], communications [15]- [18], and multi-linear system theory [19], [20]. Our work considers the MMSE estimation problem in context of tensors, thereby extending the basic subject beyond the common vectors and matrices settings.…”

“…In this paper, we present a generic tensor framework for MMSE estimation based on the Einstein product, which is concerned with estimating a signal in tensor form from a tensor based noisy observation. Many situations in practical cases involve signals that naturally involve multiple indices such as MIMO OFDM, Generalized Frequency Division Multiplexing (GFDM), Filter-bank Multi-carrier (FBMC) systems [15]. Hence representing such signals by tensors is natural.…”

“…For instance, in a communication system different modes can represent different domains such as space, time, frequency, codes, channel taps and users. Such a multi-domain approach towards communication systems using tensors can be employed in many multiusers (MU), multi-antenna and multi-carrier schemes [15]. This paper is organized as follows : a brief review of tensor algebra is presented in section II.…”