EP4490870A1

EP4490870A1 - Transmitting device, receiving device, communication apparatus and methods for random-access communication

Info

Publication number: EP4490870A1
Application number: EP22723035.6A
Authority: EP
Inventors: Maxime Guillaud; Paul FERRAND; Alexis DECURNINGE
Original assignee: Huawei Technologies Co Ltd
Current assignee: Huawei Technologies Co Ltd
Priority date: 2022-04-12
Filing date: 2022-04-12
Publication date: 2025-01-15
Also published as: CN119096496A; WO2023198279A1

Abstract

A transmitting device for random access communication, includes an encoding circuit to encode an input message into a sequence of bits, split the sequence of bits into d blocks of bits and determine d vectors from the d blocks of bits. The transmitting device includes a Kronecker product circuit to construct a symbol vector by computing a Kronecker product of the d vectors to generate a rank-1 tensor structure of order d and a mapping circuit configured to map each element of the symbol vector into one of a plurality of time-frequency resources to generate a vector of time-frequency modulated symbols. The transmitting device includes at least one antenna configured to transmit each of the time-frequency modulated symbols over a radio frequency signal to a receiving device. The transmitting device manifests a reduced probability of decoding error and an improved communication reliability in a massive random-access scenario.

Description

TRANSMITTING DEVICE, RECEIVING DEVICE, COMMUNICATION

APPARATUS AND METHODS FOR RANDOM-ACCESS COMMUNICATION

TECHNICAL FIELD

The present disclosure relates generally to the field of wireless communication; and more specifically, to a transmitting device, a receiving device, a communication apparatus comprising the transmitting device and the receiving device, and methods for random-access communication.

BACKGROUND

With the rapid increase in the number of communication devices in a network, concerns about communication reliability have become prominent. Traditionally, multiple access methods with fixed resource assignments are used by conventional communication devices for communication of data in a network. In these multiple access methods, a given communication device is assigned dedicated time-frequency resources in order to send data via a communication channel. These conventional multiple access methods are less efficient in terms of channel utilization because the given communication device may not have any data to transmit in the dedicated time-frequency resources, resulting in a loss in terms of spectral efficiency.

In certain scenarios, instead of the conventional fixed access methods, conventional randomaccess methods are used for a comparatively better utilization of the communication channel. In the conventional random-access methods, a conventional communication device (e.g., a transmitter) is allowed to send data on the communication channel whenever it has some data to transmit. There is no need of a preassigned time slot or a fixed frequency for data transmission. The conventional random-access methods are typically classified into coherent and non-coherent methods. In coherent methods, the channel state is known to communication devices (i.e., both the transmitter and the receiver). In non-coherent methods, the channel state is supposed to be unknown to both the conventional transmitter and the conventional receiver. In an implementation scenario of the non-coherent methods, a tensor-based modulation is used for random-access communication between a number of the conventional transmitters and the conventional receiver. The tensor-based modulation is performed over an orthogonal frequency division multiplexing (OFDM) modulation technique. Generally, the OFDM modulation technique may be defined as a signal modulation technique in which a high data rate modulating stream is divided into a number of closely spaced orthogonal subcarriers that are transmitted in parallel. In said implementation scenario, the closely spaced orthogonal subcarriers have imperfect timing and carrier frequency synchronization which further induce timing and carrier frequency offsets in the transmitted data of each conventional transmitter. This further leads to an increase in the probability of decoding error and hence, an erroneous (or unreliable) recovery of data at the conventional receiver. Thus, there exists a technical problem of residual asynchronicity in a conventional OFDM system and hence, an increased probability of decoding error and inadequate communication reliability exists in the tensor-based modulation used along with the conventional OFDM system.

Therefore, in light of the foregoing discussion, there exists a need to overcome the aforementioned drawbacks associated with the tensor-based modulation used along with the conventional OFDM system.

SUMMARY

The present disclosure provides a transmitting device, a receiving device, a communication apparatus comprising the transmitting device and the receiving device, and methods for random-access communication. The present disclosure provides a solution to the existing problem of residual asynchronicity in a conventional OFDM system and hence, an increased probability of decoding error and inadequate communication reliability in the tensor-based modulation used along with the conventional OFDM system. An aim of the present disclosure is to provide a solution that overcomes at least partially the problems encountered in prior art, and provide an improved transmitting device, an improved receiving device, an improved communication apparatus comprising the improved transmitting device and the improved receiving device, and improved methods for random-access communication.

The object of the present disclosure is achieved by the solutions provided in the enclosed independent claims. Advantageous implementations of the present disclosure are further defined in the dependent claims. In one aspect, the present disclosure provides a transmitting device for random access communication, comprising an encoding circuit configured to encode an input message into a sequence of bits, split the sequence of bits into d blocks of bits, where d > 1, and determine, from the d blocks of bits, d vectors. The transmitting device further comprises a Kronecker product circuit configured to construct a symbol vector by computing the Kronecker product of the d vectors to generate a rank-1 tensor structure of order d. The transmitting device further comprises a mapping circuit configured to map the symbol vector in a time-frequency grid associated with a plurality of time-frequency resources according to a pre-defined permutation matrix, where each element of the symbol vector is mapped onto one of the time-frequency resources of the time-frequency resource grid to generate a vector of time-frequency modulated symbols and at least one antenna configured to transmit each of the time-frequency modulated symbols over a radio frequency signal to a receiving device.

The disclosed transmitting device manifests a reduced probability of decoding error and an improved communication reliability when used in a communication system where there is no perfect synchronization (or no block fading channel) between the transmitting device (or a plurality of transmitting devices) and the receiving device. Moreover, the transmitting device may be used in the communication system employing tensor-based modulation along with OFDM and hence, enables an enhanced spectral efficiency. The transmitting device includes time-frequency mapping between the tensor elements and the used physical resources that enables the receiving device to more accurately estimate and compensate the timing and carrier frequency offsets and hence, the improved communication reliability is obtained in the communication system comprising the transmitting device and the receiving device. Furthermore, the use of the time-frequency mapping preserves rank-1 property of the symbol vector under timing offsets. Additionally, the transmitting device enables an improved waveform design for the massive random-access communication with reduced probability of decoding error and an enhanced spectral efficiency.

In an implementation form, the Kronecker product circuit is configured to apply a rotation to each of the d vectors by multiplying each of the d vectors by a rotation matrix, prior to computing the Kronecker product.

The use of the rotation matrix provides an additional degree of freedom in the design of subconstellations. Moreover, an adequate choice of the rotation matrix enables the receiving device to accurately estimate and compensate the timing and carrier frequency offsets which further leads to the improved communication reliability.

In a further implementation form, the encoding circuit is configured to determine each of the d vectors by using a corresponding sub-constellation based on bits in a corresponding block of bits, and where each of the corresponding sub-constellations has a structure selected from at least a Grassmannian constellation, a cube-split constellation, or a constellation where each of the plurality of vectors is divided into a pilot part and a data part.

Each of the corresponding sub-constellations has a defined structure and each structure has its own characteristics, such as the structured Grassmannian constellation provides a dense constellation structure. The dense constellation structure (e.g., of the symbol vector) based on the structured Grassmannian constellation results in a compact form of the input data and promotes a low complexity decoding process at the receiving device.

In a further implementation form, the encoding circuit is configured to encode the input message into a sequence of bits according to a binary code.

By virtue of encoding the input message into the sequence of bits according to the binary code, redundancy is added to the input message which makes the input message robust to the noise and different kind of interferences.

In another aspect, the present disclosure provides a receiving device for random access communication, comprising at least one antenna configured to receive a plurality of radio frequency signals concurrently from a plurality of transmitting devices. The receiving device further comprises a separation circuit configured to arrange each of the radio frequency signals into a time-frequency grid associated with a plurality of time-frequency resources, according to a predefined permutation matrix, to generate a plurality of estimated symbol vectors from the received plurality of radio frequency signals, and separate the plurality of estimated symbol vectors using a rank-1 tensor structure of order d associated with each transmitting device, where d > 1. The receiving device further comprises a plurality of compensation circuits each configured to estimate a corresponding time offset and a corresponding frequency offset in one corresponding estimated symbol vector amongst the plurality of estimated symbol vectors, and generate a corresponding compensated symbol vector by applying a time offset compensation, based on the corresponding time offset, and a frequency offset compensation, based on the corresponding frequency offset, to the corresponding estimated symbol vector. The receiving device further comprises a plurality of decoders each configured to decode one compensated symbol vector generated by one of the compensation circuits, to generate a plurality of decoded messages, each decoded message comprising a sequence of bits that corresponds to data associated with a corresponding transmitting device amongst the plurality of transmitting devices.

The disclosed receiving device manifests a reduced probability of decoding error and an improved communication reliability when used in a communication system where there is no perfect synchronization (or no block fading channel) between the transmitting device (or the plurality of transmitting devices) and the receiving device. By virtue of the time-frequency mapping and the tensor elements rotated by use of the rotation matrix used at the transmitting device and the plurality of compensation circuits, an accurate estimation and compensation of timing and carrier frequency offsets can be achieved which further leads to the reduced probability of decoding error and the improved communication reliability. Moreover, the receiving device may be used in the communication system employing tensor-based modulation along with OFDM and hence, enables an enhanced spectral efficiency.

In an implementation form, each of the compensation circuits in the plurality of compensation circuits is configured to use a corresponding constellation comprising symbol vectors, and determine the time offset compensation and frequency offset compensation so as to minimize a distance between the corresponding estimated symbol vector and a symbol vector of the constellation.

The use of the corresponding constellation comprising symbol vectors in determination of the time offset compensation and frequency offset compensation leads to more accurate estimation and compensation of the timing and carrier frequency offsets.

In a yet another aspect, the present disclosure provides a communication apparatus, comprising the transmitting device and the receiving device.

The communication apparatus achieves all the advantages and technical effects of the transmitting device as well as the receiving device of the present disclosure.

In a yet another aspect, the present disclosure provides a method for a random-access communication. The method comprises encoding, by an encoder of a transmitting device, an input message into a sequence of bits. The method further comprises splitting, by the encoder, the sequence of bits in d blocks of bits, where d > 1 and determining, by the encoder, from the d blocks of bits, d vectors. The method further comprises constructing a symbol vector, by a Kronecker product circuit of the transmitting device, by computing the Kronecker product of the d vectors to generate a rank-1 tensor structure of order d. The method further comprises mapping, by a mapping circuit of the transmitting device, the symbol vector in a time-frequency grid associated with a plurality of time-frequency resources according to a pre-defined permutation matrix, where each element of the symbol vector is mapped onto one of the timefrequency resources of the time-frequency resource grid to generate a vector of time-frequency modulated symbols and transmitting, by at least one antenna of the transmitting device, each of the time-frequency modulated symbols over a radio frequency signal to a receiving device.

The method achieves all the advantages and technical effects of the transmitting device of the present disclosure.

In a yet another aspect, the present disclosure provides a method for a random-access communication. The method comprises receiving, by at least one antenna of a receiving device, a plurality of radio frequency signals concurrently from a plurality of transmitting devices. The method further comprises arranging, by a separation circuit of the receiving device, each of the radio frequency signals into a time-frequency grid associated with a plurality of time-frequency resources, according to a predefined matrix, to generate a plurality of estimated symbol vectors from the received plurality of radio frequency signals, and separating the plurality of estimated symbol vectors using a rank-1 tensor structure of order d associated with each transmitting device, where d > 1. The method further comprises estimating, by a plurality of compensation circuits of the receiving device, a time offset and a frequency offset in each of the plurality of estimated symbol vectors, and generating, by the compensation circuits, a plurality of compensated symbol vectors by applying time offset compensations, based on the time offsets, and frequency offset compensations, based on the frequency offsets, to the plurality of estimated symbol vectors. The method further comprises decoding, by a plurality of decoders in the receiving device, the plurality of compensated symbol vectors to generate a plurality of decoded messages, each decoded message comprising a sequence of bits that corresponds to data associated with a corresponding transmitting device of the plurality of transmitting devices.

The method achieves all the advantages and technical effects of the receiving device of the present disclosure. In a yet another aspect, the present disclosure provides a computer program product comprising program instructions for performing the methods for random access communication, when executed by one or more processors in a computer system.

In a yet another aspect, the present disclosure provides a computer system comprising one or more processors and one or more memories, the one or more memories storing program instructions which, when executed by the one or more processors, cause the one or more processors to execute the methods for random access communication.

The one or more processors of the computer system achieve all the advantages and technical effects of the methods after execution of the methods for random access communication.

It is to be appreciated that all the aforementioned implementation forms can be combined.

It has to be noted that all devices, elements, circuitry, units and means described in the present application could be implemented in the software or hardware elements or any kind of combination thereof. All steps which are performed by the various entities described in the present application as well as the functionalities described to be performed by the various entities are intended to mean that the respective entity is adapted to or configured to perform the respective steps and functionalities. Even if, in the following description of specific embodiments, a specific functionality or step to be performed by external entities is not reflected in the description of a specific detailed element of that entity which performs that specific step or functionality, it should be clear for a skilled person that these methods and functionalities can be implemented in respective software or hardware elements, or any kind of combination thereof. It will be appreciated that features of the present disclosure are susceptible to being combined in various combinations without departing from the scope of the present disclosure as defined by the appended claims.

Additional aspects, advantages, features and objects of the present disclosure would be made apparent from the drawings and the detailed description of the illustrative implementations construed in conjunction with the appended claims that follow. BRIEF DESCRIPTION OF THE DRAWINGS

The summary above, as well as the following detailed description of illustrative embodiments, is better understood when read in conjunction with the appended drawings. For the purpose of illustrating the present disclosure, exemplary constructions of the disclosure are shown in the drawings. However, the present disclosure is not limited to specific methods and instrumentalities disclosed herein. Moreover, those in the art will understand that the drawings are not to scale. Wherever possible, like elements have been indicated by identical numbers.

Embodiments of the present disclosure will now be described, by way of example only, with reference to the following diagrams wherein:

FIG. 1 is a network environment diagram of a system with a plurality of transmitting devices and a receiving device, in accordance with an embodiment of the present disclosure;

FIG. 2 is a block diagram that illustrates various exemplary components of a transmitting device, in accordance with an embodiment of the present disclosure;

FIG. 3 is an illustration of time-frequency mapping of a symbol vector in a time-frequency grid, in accordance with an embodiment of the present disclosure;

FIG. 4 is a flowchart of a method for a random-access communication, in accordance with an embodiment of the present disclosure;

FIG. 5 is a block diagram that illustrates various exemplary components of a receiving device, in accordance with an embodiment of the present disclosure;

FIG. 6 is an illustration of timing offsets for a symbol vector, in accordance with an embodiment of the present disclosure;

FIG. 7 is an illustration of a receiving device, in accordance with another embodiment of the present disclosure;

FIG. 8 is a flowchart of a method for a random-access communication, in accordance with another embodiment of the present disclosure;

FIG. 9 is a block diagram that illustrates various exemplary components of a communication apparatus, in accordance with an embodiment of the present disclosure;

FIG. 10 is a graphical representation that illustrates variation of block error rate (BLER) with respect to signal-to-noise ratio (SNR), in accordance with an embodiment of the present disclosure;

FIG. 11 is a graphical representation that illustrates variation of BLER with respect to SNR for a Cube Split constellation, in accordance with an embodiment of the present disclosure; and FIG. 12 is a graphical representation that illustrates variation of BLER with respect to SNR for a quadrature amplitude modulation (QAM) constellation, in accordance with an embodiment of the present disclosure.

In the accompanying drawings, an underlined number is employed to represent an item over which the underlined number is positioned or an item to which the underlined number is adjacent. A non-underlined number relates to an item identified by a line linking the nonunderlined number to the item. When a number is non-underlined and accompanied by an associated arrow, the non-underlined number is used to identify a general item at which the arrow is pointing.

DETAILED DESCRIPTION OF EMBODIMENTS

The following detailed description illustrates embodiments of the present disclosure and ways in which they can be implemented. Although some modes of carrying out the present disclosure have been disclosed, those skilled in the art would recognize that other embodiments for carrying out or practicing the present disclosure are also possible.

FIG. 1 is a network environment diagram of a system with a plurality of transmitting devices and a receiving device, in accordance with an embodiment of the present disclosure. With reference to FIG. 1, there is shown a network environment of a system 100 that includes a plurality of transmitting devices 102 and a receiving device 104. There is further shown a communication network 106. The plurality of transmitting devices 102 includes K transmitting devices, such as a first transmitting device 102A, a second transmitting device 102B, and up to a K-th transmitting device 102K.

Each of the plurality of transmitting devices 102 may include suitable logic, circuitry, interfaces and/or code that is configured to communicate with the receiving device 104 via the communication network 106 (e.g., a propagation channel). Examples of each of the plurality of transmitting devices 102 may include, but are not limited to, an Intemet-of- Things (loT) device, a smart phone, a machine type communication (MTC) device, a computing device, an evolved universal mobile telecommunications system (UMTS) terrestrial radio access (E-UTRAN) NR- dual connectivity (EN-DC) device, a server, an loT controller, a drone, a customized hardware for wireless telecommunication, a transmitter, or any other portable or non-portable electronic device. In the system 100, each of the plurality of transmitting devices 102 has a single antenna for communication with the receiving device 104. However, in another implementation of the system 100, each of the plurality of transmitting devices 102 may have more than one antenna for communication with the receiving device 104.

The receiving device 104 may include suitable logic, circuitry, interfaces and/or code that is configured to receive one or more radio frequency signals concurrently from the plurality of transmitting devices 102, via the communication network 106. Examples of the receiving device 104 may include, but are not limited to, an Internet-of- Things (loT) controller, a base station, a server, a smart phone, a customized hardware for wireless telecommunication, a receiver, or any other portable or non-portable electronic device. In the system 100, the receiving device 104 has more than one antenna for communication with the plurality of transmitting devices 102.

The communication network 106 includes a medium (e.g., a communication channel) through which the plurality of transmitting devices 102, potentially communicates with the receiving device 104. Examples of the communication network 106 may include, but are not limited to, a cellular network (e.g., a 2G, a 3G, long-term evolution (LTE) 4G, a 5G, or 5G New Radio (NR) network, such as sub 6 GHz, cmWave, or mmWave communication network), a wireless sensor network (WSN), a cloud network, a Local Area Network (LAN), a vehicle-to-network (V2N) network, a Metropolitan Area Network (MAN), and/or the Internet. Each of the plurality of transmitting devices 102 in the network environment is configured to connect to the receiving device 104, in accordance with various wireless communication protocols. Examples of such wireless communication protocols, communication standards, and technologies may include, but are not limited to, IEEE 802.11, 802.1 Ip, 802.15, 802.16, 1609, Worldwide Interoperability for Microwave Access (Wi-MAX), Transmission Control Protocol and Internet Protocol (TCP/IP), User Datagram Protocol (UDP), Hypertext Transfer Protocol (HTTP), Long-term Evolution (LTE), File Transfer Protocol (FTP), Enhanced Data GSM Environment (EDGE), Voice over Internet Protocol (VoIP), a protocol for email, instant messaging, and/or Short Message Service (SMS), and/or other cellular or loT communication protocols.

Additionally, a random number of the plurality of transmitting devices 102 may be active at a time and may be configured to transmit a plurality of radio frequency signals concurrently to the receiving device 104 via the communication network 106. Therefore, the system 100 may also be referred to as an exemplary implementation of a massive random-access communication in which the plurality of transmitting devices 102 transmits the plurality of radio frequency signals concurrently to the receiving device 104. Moreover, each of the plurality of transmitting devices 102 transmits the plurality of radio frequency signals concurrently to the receiving device 104 without any prior resource request (or grant).

Moreover, in an implementation of the system 100, the communication network 106 may be configured to behave like a block fading channel between the plurality of transmitting devices 102 and the receiving device 104. Alternatively stated, the plurality of transmitting devices 102 and the receiving device 104 are considered to be perfectly synchronized with each other and therefore, no timing and carrier frequency offsets are present in the system 100. In another implementation of the system 100, where there is no block fading channel between the plurality of transmitting devices 102 and the receiving device 104 and therefore, timing and carrier frequency offsets are present at the plurality of transmitting devices 102 and required to be compensated at the receiving device 104 in order to reduce probability of decoding error and obtain an adequate communication reliability in the system 100. In this embodiment (i.e., the system 100), no block fading channel is assumed between the plurality of transmitting devices 102 and the receiving device 104.

FIG. 2 is a block diagram that illustrates various exemplary components of a transmitting device, in accordance with an embodiment of the present disclosure. FIG. 2 is described in conjunction with elements from FIG. 1. With reference to FIG. 2, there is shown a block diagram 200 of a transmitting device 202 that includes an encoding circuit 204, a Kronecker product circuit 206, a mapping circuit 208, an antenna 210, a processor 212 and a memory 214. In an implementation, each of the encoding circuit 204, the Kronecker product circuit 206, and the mapping circuit 208 may be a part of the processor 212. In another implementation, the encoding circuit 204, the Kronecker product circuit 206, and the mapping circuit 208, are separate circuits or modules (and may not be a part of the processor 212). The encoding circuit 204, the Kronecker product circuit 206, and the mapping circuit 208 are communicatively coupled to the antenna 210, the processor 212 and the memory 214. In this embodiment, the transmitting device 202 includes a single antenna, such as the antenna 210. However, in other embodiments, the transmitting device 202 may include multiple antennas.

The transmitting device 202 corresponds to one of the plurality of transmitting devices 102 (of FIG. 1), such as the K-th transmitting device 102K. The transmitting device 202 may be configured to communicate with the receiving device 104 (of FIG. 1). The encoding circuit 204 may include suitable logic, circuitry, and/or interfaces that is configured to encode an input message and into a sequence of bits. In an implementation, the encoding circuit 204 may be a binary encoder.

The Kronecker product circuit 206 may include suitable logic, circuitry, and/or interfaces that is configured to construct a symbol vector by computing a Kronecker product of d vectors to generate a rank-1 tensor structure of order d.

The mapping circuit 208 may include suitable logic, circuitry, and/or interfaces that is configured to map the symbol vector to a vector of time-frequency modulated symbols. The mapping circuit 208 may also be referred to as a time-frequency resource mapping circuit.

The antenna 210 may include suitable logic, circuitry, and/or interfaces that is configured to transmit each of the time-frequency modulated symbols over a radio frequency signal to the receiving device 104 (of FIG. 1). Examples of the antenna 210 may include, but are not limited to, a radio frequency transceiver, a network interface, a telematics unit, or any antenna suitable for use in an loT device, an loT controller, a user equipment, a repeater, a base station or other portable or non-portable communication devices. The antenna 210 may wirelessly communicate by use of various wireless communication protocols.

The processor 212 may include suitable logic, circuitry, and/or interfaces that is configured to execute instructions stored in the memory 214. Examples of the processor 212 may include, but are not limited to an integrated circuit, a co-processor, a microprocessor, a microcontroller, a complex instruction set computing (CISC) processor, an application-specific integrated circuit (ASIC) processor, a reduced instruction set (RISC) processor, a very long instruction word (VLIW) processor, a central processing unit (CPU), a state machine, a data processing unit, and other processors or circuits. Moreover, the processor 212 may refer to one or more individual processors, processing devices, a processing unit that is part of a machine.

The memory 214 may include suitable logic, circuitry, and/or interfaces that is configured to store machine code and/or instructions executable by the processor 212. The memory 214 may temporally store one or more time-frequency modulated symbols, which are then transmitted by the antenna 210 in form of one or more radio frequency signals to the receiving device 104. Examples of implementation of the memory 214 may include, but are not limited to, an Electrically Erasable Programmable Read-Only Memory (EEPROM), Random Access Memory (RAM), Read Only Memory (ROM), Hard Disk Drive (HDD), Flash memory, a Secure Digital (SD) card, Solid-State Drive (SSD), a computer readable storage medium, and/or CPU cache memory. The memory 214 may store an operating system and/or a computer program product to operate the transmitting device 202. A computer readable storage medium for providing a non-transient memory may include, but is not limited to, an electronic storage device, a magnetic storage device, an optical storage device, an electromagnetic storage device, a semiconductor storage device, or any suitable combination of the foregoing.

In operation, the transmitting device 202 is configured for random-access communication with the receiving device 104 (of FIG. 1). The transmitting device 202 comprises the encoding circuit 204 that is configured to encode an input message into a sequence of bits. In an implementation, the encoding circuit 204 is configured to encode the input message, for example a binary message, m_k of B bits, into the sequence of bits having the number of bits (i.e., B') greater than the bits (i.e., B) of the input message.

In accordance with an embodiment, the encoding circuit 204 is configured to encode the input message into the sequence of bits according to a binary code. In an implementation, the encoding circuit 204 may be configured to encode the input message (i.e., the binary message, m_k of B bits) into the sequence of bits (i.e., B') by use of the binary code. Examples of the binary code may include but are not limited to, source codes, channel codes, and the like. The channel codes may include low-density parity check (LDPC) codes, Turbo codes, Viterbi codes, and the like.

The encoding circuit 204 is further configured to split the sequence of bits into d blocks of bits, where d > 1, and determine, from the d blocks of bits, d vectors. Thereafter, the encoding circuit 204 is configured to split the sequence of bits (i.e., B') into the d blocks, where d is an arbitrary number and greater than one. Each block of the d blocks has B_L bits in such a way that = B'. The i-th block, where 1 < i < d is mapped into a T_t- dimensional vector x_{i k} G C^Ti generated from a sub-constellation Q <= C^Ti. The dimensions T_lt ... , T_d are such that nf=i 7) = T. The mapping of the d vectors from the d block of bits is performed by use of the encoding circuit 204 (more specifically, a vector symbol mapper). In this way, the d vectors may also be represented as x_{l k}, ■■■ , x_{d k}.

In accordance with an embodiment, the encoding circuit 204 is configured to determine each of the d vectors by using a corresponding sub-constellation based on bits in a corresponding block of bits, and where each of the corresponding sub-constellations has a structure selected from at least a Grassmannian constellation, a cube-split constellation, or a constellation wherein each of the plurality of vectors is divided into a pilot part and a data part. The encoding circuit 204 is further configured to determine each of the d vectors (i.e., x_{l k}, ••• , x_{d k}) by use of the corresponding sub-constellation (i.e., C₍ <= (C⁷*) based on the bits in the corresponding block of bits. Each of the corresponding sub-constellations belongs to a constellation and has two or more dimensions. The constellation has a definite and a fixed multi-dimensional data structure, which reduces the computational complexity for waveform design for data transmission. Moreover, in an implementation, the structure of each of the corresponding sub-constellations is selected from the Grassmannian constellation (e.g., a cube-split constellation). In another implementation, the structure of each of the corresponding sub-constellations is selected from the constellation where each of the plurality of vectors is divided into the pilot part of one or more scalar reference symbols and the data part with remaining scalar elements. The data part can be generated either by from a quadrature amplitude modulation (QAM) or from a phase shift keying (PSK) modulation.

The transmitting device 202 further comprises the Kronecker product circuit 206 that is configured to construct a symbol vector by computing the Kronecker product of the d vectors to generate a rank-1 tensor structure of order d. The Kronecker product circuit 206 is configured to construct the symbol vector (may also be represented as ii_k) by computing the Kronecker product of the d vectors (i.e., x_{l k}, ••• , x_{d k}) to generate the rank-1 tensor structure of order d (i.e., u_k = The Kronecker product is a form of matrix multiplication and is represented by a mathematical notation 0. The Kronecker product is also known as a tensor product (or a direct product). The use of the Kronecker product is beneficial as the dimensions of a plurality of matrices being multiplied together on the basis of the Kronecker product do not need to have any relation with each other. The Kronecker product of the d vectors (i.e., x_{l k}, ■■■ , x_{d k}) is represented as x_{l k} 0 ■■■ ® x_{d k}. The constructed symbol vector has a multi-dimensional data structure. Moreover, the constructed symbol vector carries the same information as the input message (i.e., the binary message, m_k) acquired by the transmitting device 202 and encoded by the encoding circuit 204. The constructed symbol vector may also be referred to as the rank-1 tensor structure of order d which means that the constructed symbol vector may be construed as a d-dimensional array of real and complex numbers. The d- dimensional array of real or complex numbers is of respective dimensions 7\ ... T_d (which can be denoted as "a tensor of order "d" and size "7\ X ... T_d"). The ? = nf=i 7) numbers contained in the tensor structure can also be stored sequentially (in a predefined order) in a vector of size Any tensor structure that can be potentially denoted as where , refers to vectors with respective dimensions 7\ ... T_d is usually called the rank- 1 tensor structure. In a case where a tensor structure can be denoted as a sum of at least n vectors, it is potentially deemed as a rank-n tensor. The use of the Kronecker product is advantageous as it results in a definite and fixed tensor structure of the symbol vector (i.e., a definite multi-dimensional data structure), which simplifies a waveform design for randomaccess communication.

In accordance with an embodiment, the Kronecker product circuit 206 is configured to apply a rotation to each of the d vectors by multiplying each of the d vectors by a rotation matrix, prior to computing the Kronecker product. In an implementation, the Kronecker product circuit 206 is configured to apply the rotation to each of the d vectors (i.e., x_{l k}, ••• , x_{d k}) by multiplying each of the d vectors by the rotation matrix (may also be represented as I7j) prior to computation of the Kronecker product. For any sub-constellation (i.e., C₍ ), a rotated sub-constellation (may also be represented as Q') can be obtained by the multiplying the sub-constellation (i.e., C₍ ) to the rotation matrix (i.e., I7j). Alternatively stated, if the elements of the sub-constellation (i.e., C; ) are represented as then, the elements of the rotated sub-constellation (i.e., Q') for j = 1, ... , 2^Bi are represented as c'_{i 7}, where c'_{i 7} = UiCij. The rotation matrix (i.e., I7j) provides an additional degree of freedom in the design of sub-constellation. In particular, using the structured sub-constellation together with the rotation matrix (i.e., I7j) enables a low- complexity at the receiving device 104 along with an improved performance with respect to timing and carrier frequency offsets estimation and compensation. The choice of the rotation matrix (i.e., I7j) has an impact on the performance of the time and carrier frequency offset estimation. Examples for the choice of the rotation matrix (i.e., I7j) are Fourier matrices, compositions of given rotations, normalized Hadamard matrices, and the like. The timing and carrier frequency offsets can also be optimized by considering the rotation matrix (i.e., I7j) that maximizes the minimum distance between all pairs of codewords affected by all possible offsets. Alternatively stated, a matrix that solves the optimization problem of the equation (1) can also be chosen as the rotation matrix (i.e., I7j). argmin

U[

The transmitting device 202 further comprises the mapping circuit 208 that is configured to map the symbol vector in a time-frequency grid associated with a plurality of time-frequency resources according to a pre-defined permutation matrix, where each element of the symbol vector is mapped onto one of the time-frequency resources of the time-frequency resource grid to generate a vector of time-frequency modulated symbols. In case of unsynchronized transmitting devices (e.g., the transmitting device 202) and the receiving device 104, time and carrier frequency offsets incur at the transmitting device 202 and required to be compensated at the receiving device 104. Moreover, a time-frequency mapping is required to perform at the transmitting device 202 and the corresponding time-frequency de-mapping is also required to perform at the receiving device 104 along with the time and carrier frequency offset estimation and compensation. The mapping circuit 208 is configured to map the symbol vector (i.e., ii_k) in the time-frequency grid of an OFDM modulation with F frequency subcarriers and S time symbols, described in detail, for example, in FIG. 3. The symbol vector (i.e., ii_k) constructed as the rank-1 tensor structure of order d>l is further parameterized by the pre-defined permutation matrix (may also be represented as A G in order to generate the vector of time-frequency modulated symbols (may also be represented as v_fc) according to the equation (2) where, each x_{k i} is a vector of dimension T_t for (i = 1, ... , d) and 7\ ... T_d are such that I]f=i Ti = T = SF. The pre-defined permutation matrix (i.e., A) can be seen as a way to map the symbol vector (i.e., ii_k) on the time-frequency grid associated with the plurality of timefrequency resources. The pre-defined permutation matrix (i.e., A) defines a bijective mapping between elements of the symbol vector (i.e., ii_k) and the time-frequency grid. For each mode, i = 1, the information bearing vector x_{i k} G C^Ti belongs to a known vector codebook, denoted by C₍. The resulting modulation codebook for the time-frequency modulated symbol (i.e., v_fc) is represented as C<= At each transmitting device (e.g., the transmitting device 202), the frequency-domain signal representation of each of the time-frequency modulated symbols with N subcarriers undergo an inverse discrete Fourier transform (IDFT) to form N time-domain samples. Furthermore, in order to avoid any inter-symbol interference, a cyclic prefix (CP) of length N_CP samples, is added to each OFDM symbol (i.e., the N time-domain samples) at the transmitting device 202 through a “cyclic prefix” module. The resulting signal is then converter to an analog domain using a digital-to-analog converter and further upconverted to radio frequencies and amplified by use of an up-conversion and power amplification module, not shown here for sake of brevity. The transmitting device 202 further comprises the at least one antenna (e.g., the antenna 210) configured to transmit each of the time-frequency modulated symbols over a radio frequency signal to the receiving device 104. The radio frequency signal refers to an electromagnetic wave used to transmit each of the time-frequency modulated symbols (i.e., v_fc) over the air. Each of the time-frequency modulated symbol vector (i.e., v_fc) is linearly mapped to a corresponding signal frequency (e.g., a carrier wave), which is transmitted to the receiving device 104 (of FIG. 1).

Thus, the transmitting device 202 manifests a reduced probability of decoding error and an improved communication reliability when used in a communication system where there is no perfect synchronization (or no block fading channel) between the transmitting device 202 (or the plurality of transmitting devices 102) and the receiving device 104. Moreover, the transmitting device 202 may be used in the communication system employing tensor-based modulation along with OFDM and hence, enables an enhanced spectral efficiency. The transmitting device 202 includes time-frequency mapping between the tensor elements and the used physical resources and a sub-constellation used within each mode of the tensor by adding a rotation parameter (i.e., the rotation matrix [/;). This enables an accurate estimation and compensation of timing and carrier frequency offsets at the receiving device 104 and hence, the improved communication reliability is obtained in the communication system comprising the transmitting device 202 and the receiving device 104.

FIG. 3 is an illustration of time-frequency mapping of a symbol vector in a time-frequency grid, in accordance with an embodiment of the present disclosure. FIG. 3 is described in conjunction with elements from FIGs. 1 and 2. With reference to FIG. 3, there is shown a time-frequency grid 300 that represents a plurality of time-frequency resources. The time-frequency grid 300 includes an X-axis 302 and a Y-axis 304. The X-axis 302 represents time-domain representation and the Y-axis 304 represents frequency-domain representation.

The mapping circuit 208 of the transmitting device 202 is configured to map the symbol vector (i.e., ii_k) in the time-frequency grid 300 of an OFDM modulation with F frequency subcarriers and S time symbols. The plurality of time-frequency resources of the time-frequency grid 300 are defined according to the pre-defined permutation matrix (i.e., d), where each element of the symbol vector (i.e., ii_k) is mapped onto one of the time-frequency resources of the timefrequency grid 300 to generate the vector of time-frequency modulated symbols (i.e., v_fc). For example, a complex value (i.e., v_fc(s, /)) corresponds to an element of the time-frequency modulated symbol vector (i.e., v_k) which is transmitted at a s-th time resource and a /-th frequency resource. Therefore, the vector of time-frequency modulated symbols (i.e., v_fc) may also be referred to as aSF dimensional vector. To be consistent with the vectorization operation related to the tensor structure, the SF dimensional vector may be represented as a column-first order vectorization of the time-frequency matrix as depicted by the equation (3).

FIG. 4 is a flowchart of a method for a random-access communication, in accordance with an embodiment of the present disclosure. FIG. 4 is described in conjunction with elements from FIGs. 1, 2, and 3. With reference to FIG. 4, there is shown a method 400 that includes steps 404 to 414. The method 400 is executed by each of the plurality of transmitting devices 102 (of FIG. 1) or by the transmitting device 202 (of FIG. 2).

The method 400 is provided for random-access communication in which a random number of the plurality of transmitting devices 102 (or the transmitting device 202) may be active at a time and may be configured to transmit a plurality of radio frequency signals concurrently to the receiving device 104 without any prior resource request (or grant). Moreover, each of the plurality of transmitting devices 102 (or the transmitting device 202) is not in perfect synchronization with the receiving device 104 and therefore, timing and carrier frequency offsets incur during transmission from each of the plurality of transmitting devices 102 (or the transmitting device 202) to the receiving device 104 and required to be compensated at the receiving device 104.

At step 404, the method 400 comprises encoding, by an encoder of a transmitting device (e.g., the encoding circuit 204 of the transmitting device 202), an input message acquired by the transmitting device, into a sequence of bits. In an implementation, the input message corresponds to a binary message comprising a certain number of bits . The input message is further encoded into an encoded message comprising the sequence of bits either by use of a binary encoder or a polar encoder, and the like. The encoded message has more number of bits than the input message for bits redundancy.

In accordance with an embodiment, encoding, by the encoder, the input message into a sequence of bits, uses a binary code. The input message is encoded into the encoded message using the binary code, such as excess-3 codes, Gray codes, reflective codes, sequential codes, and the like.

At step 406, the method 400 further comprises splitting, by the encoder, the sequence of bits in d blocks of bits, where d > 1. The encoded message with the sequence of bits is split into d block of bits, where d is an arbitrary number and greater than 1.

At step 408, the method 400 further comprises determining, by the encoder, from the d blocks of bits, d vectors. The d block of bits are further encoded into the d vectors based on the number of bits in each block of the d blocks.

In accordance with an embodiment, each of the d vectors is determined by using a corresponding sub-constellation based on bits in a corresponding block of bits, and each of the corresponding sub-constellations has a structure selected from at least a Grassmannian constellation, a cube-split constellation, or a constellation wherein each of the plurality of vectors is divided into a pilot part and a data part. In an implementation, each of the d vectors are obtained from the d block of bits by use of the corresponding sub-constellations. The use of the corresponding sub-constellations is based on the number of bits each block of the d blocks. Each of the corresponding sub-constellations has a definite and a fixed multi-dimensional data structure. Moreover, each of the corresponding sub-constellations has the structure selected from at least one of the Grassmannian constellation, the cube-split constellation, or the constellation where each of the plurality of vectors is divided into the pilot part and the data part, described earlier, for example, in FIG. 2.

At step 410, the method 400 further comprises constructing a symbol vector, by the Kronecker product circuit 206 of the transmitting device 202, by computing the Kronecker product of the d vectors to generate a rank-1 tensor structure of order d. The Kronecker product circuit 206 is configured to construct the symbol vector by computing the Kronecker product of the d vectors obtained from the d block of bits in order to generate the rank-1 tensor structure of order d. The constructed symbol vector may also be referred to as the rank-1 tensor structure of order d which means that the constructed symbol vector may be construed as a d-dimensional array of real and complex numbers.

In accordance with an embodiment, a rotation is applied to each of the d vectors by multiplying each of the d vectors by a rotation matrix prior to computing the Kronecker product. In an implementation, the Kronecker product circuit 206 is configured to apply the rotation to each of the d vectors by multiplying each of the d vectors by the rotation matrix prior to computation of the Kronecker product.

At step 412, the method 400 further comprises mapping, by the mapping circuit 208 of the transmitting device 202, the symbol vector in a time-frequency grid associated with a plurality of time-frequency resources according to a pre-defined permutation matrix, where each element of the symbol vector is mapped onto one of the time-frequency resources of the time-frequency resource grid to generate a vector of time-frequency modulated symbols. The mapping circuit 208 is configured to map the symbol vector in the time-frequency grid of an OFDM modulation with F frequency subcarriers and S time symbols, described earlier, for example, in FIG. 3. The symbol vector constructed as the rank-1 tensor structure of order d>l is further parameterized by the pre-defined permutation matrix in order to generate the vector of time-frequency modulated symbols.

At step 414, the method 400 further comprises transmitting, by at least one antenna 210 of the transmitting device 202, each of the time-frequency modulated symbols over a radio frequency signal to the receiving device 104. Each of the time-frequency modulated symbol vector is linearly mapped to a corresponding signal frequency (e.g., a carrier wave), which is transmitted to the receiving device 104 through the antenna 210 of the transmitting device 202.

The steps 402 to 414 are only illustrative and other alternatives can also be provided where one or more steps are added, one or more steps are removed, or one or more steps are provided in a different sequence without departing from the scope of the claims herein.

In one aspect, a computer program product is provided performing the method 400 when executed by one or more processors (e.g., the processor 212 of the transmitting device 202) in a computer system. In another aspect, a computer system is provided comprising one or more processors (e.g., the processor 212) and one or more memories (e.g., the memory 214), the one or more memories (i.e., the memory 214) storing program instructions which, when executed by the one or more processors (i.e., the processor 212), cause the one or more processors (i.e., the processor 212) to execute the method 400. In yet another aspect, the present disclosure provides a non-transitory computer-readable medium having stored thereon, computer- implemented instructions that, when executed by a computer, causes the computer to execute operations of the method 400. FIG. 5 is a block diagram that illustrates various exemplary components of a receiving device, in accordance with an embodiment of the present disclosure. FIG. 5 is described in conjunction with elements from FIGs. 1, and 2. With reference to FIG. 5, there is shown a block diagram 500 of the receiving device 104 (of FIG. 1) that includes at least one antenna 502, a separation circuit 504, a plurality of compensation circuits 506, a plurality of decoders 508, a memory 510 and a processor 512. In an implementation, each of the separation circuit 504, the plurality of compensation circuits 506, the plurality of decoders 508 may be a part of the processor 512. In another implementation, each of the separation circuit 504, the plurality of compensation circuits 506, the plurality of decoders 508, are separate circuits or modules (and may not be a part of the processor 512). The separation circuit 504, the plurality of compensation circuits 506, the plurality of decoders 508 are communicatively coupled to the memory 510 and the antenna 502. The receiving device 104 includes at least one antenna, such as the antenna 502 (or 1, 2, 3..., M number of antennas).

The at least one antenna 502 may include suitable logic, circuitry, and/or interfaces that is configured to receive a plurality of radio frequency signals concurrently from a plurality of transmitting devices, such as the plurality of transmitting devices 102 (of FIG. 1). Beneficially, the number of receiving antennas at the receiving device 104 is potentially less than number of transmitting signals. Examples of implementation of the antenna 502 is similar to that of the antenna 210 (FIG. 2).

The separation circuit 504 may include suitable logic, circuitry, and/or interfaces that is configured to arrange each of the radio frequency signals into a time-frequency grid associated with a plurality of time-frequency resources.

The plurality of compensation circuits 506 may include suitable logic, circuitry, and/or interfaces that is configured to estimate a corresponding time offset and a corresponding frequency offset in one corresponding estimated symbol vector amongst a plurality of estimated symbol vectors.

The plurality of decoders 508 may include suitable logic, circuitry, and/or interfaces that is configured to decode one compensated symbol vector generated by one of the compensation circuits, to generate a plurality of decoded messages. Each of the plurality of decoders 508 is a single-user decoder. The memory 510 may include suitable logic, circuitry, and/or interfaces that is configured to store machine code and/or instructions executable by the processor 512. The memory 510 may temporally store one or more decoded messages. Examples of implementation of the memory 510 is similar to that of the memory 214 (of FIG. 2). The memory 510 may store an operating system and/or a computer program product to operate the receiving device 104. A computer readable storage medium for providing a non-transient memory may include, but is not limited to, an electronic storage device, a magnetic storage device, an optical storage device, an electromagnetic storage device, a semiconductor storage device, or any suitable combination of the foregoing.

The processor 512 may include suitable logic, circuitry, and/or interfaces that is configured to execute instructions stored in the memory 510. Examples of implementation of the processor 512 is similar to that of the processor 212 (of FIG. 2).

In operation, the receiving device 104 comprises the at least one antenna 502 that is configured to receive a plurality of radio frequency signals concurrently from a plurality of transmitting devices. In a massive random-access scenario, the at least one antenna 502 of the receiving device 104 is configured to receive the plurality of radio frequency signals concurrently from the plurality of transmitting devices, such as the plurality of transmitting devices 102 (of FIG. 1).

The receiving device 104 further comprises the separation circuit 504 that is configured to arrange each of the radio frequency signals into a time-frequency grid associated with a plurality of time-frequency resources, according to a pre-defined permutation matrix, to generate a plurality of estimated symbol vectors from the received plurality of radio frequency signals, and separate the plurality of estimated symbol vectors using a rank-1 tensor structure of order d associated with each transmitting device, where d > 1. The separation circuit 504 is configured to arrange each of the received radio frequency signals into the time-frequency grid associated with the plurality of time-frequency resources using an inverse permutation. The inverse permutation corresponds to reverting the effect of the pre-defined permutation matrix (i.e., A) used at the transmitting device 202 (of FIG. 2) by applying the pre-defined permutation matrix (may also be represented as to each of the received radio frequency signals at the receiving device 104. After applying the pre-defined permutation matrix (i.e., to each of the received radio frequency signals, the plurality of estimated symbol vectors is generated at the receiving device 104. The plurality of estimated symbol vectors is subjected to a tensor decomposition by use of a canonical polyadic decomposition corresponding to each of the plurality of transmitting devices 102. The canonical polyadic decomposition is used to separate the plurality of estimated symbol vectors into single-user components, and hence, outputs an estimation of K vector symbols v_k for k = 1, The integer K denotes the estimated number of transmitted messages.

The receiving device 104 further comprises the plurality of compensation circuits 506 each configured to estimate a corresponding time offset and a corresponding frequency offset in one corresponding estimated symbol vector amongst the plurality of estimated symbol vectors, and generate a corresponding compensated symbol vector by applying a time offset compensation, based on the corresponding time offset, and a frequency offset compensation, based on the corresponding frequency offset, to the corresponding estimated symbol vector. In a case when there is no block fading channel between each of the plurality of transmitting devices 102 and the receiving device 104, timing and carrier frequency offsets occur at each of the plurality of transmitting devices 102 and required to be compensated at the receiving device 104. Therefore, the plurality of compensation circuits 506 is used to estimate as well as compensate the timing and carrier frequency offsets at the receiving device 104, described in detail, for example, in FIGs. 6 and 7.

In accordance with an embodiment, each of the compensation circuits in the plurality of compensation circuits 506 is configured to use a corresponding constellation comprising symbol vectors, and determine the time offset compensation and frequency offset compensation so as to minimize a distance between the corresponding estimated symbol vector and a symbol vector of the constellation. Each of the plurality of compensation circuits 506 is configured to use the corresponding constellation comprising the symbol vectors, and determine the time and carrier frequency offset compensation in order to minimize the distance between the corresponding symbol vector and the symbol vector of constellation, described in detail, for example, in FIG. 7.

The receiving device 104 further comprises the plurality of decoders 508 each configured to decode one compensated symbol vector generated by one of the plurality of compensation circuits 506, to generate a plurality of decoded messages, each decoded message comprising a sequence of bits that corresponds to data associated with a corresponding transmitting device amongst the plurality of transmitting devices 102. Each of the plurality of decoders 508 is configured to decode the compensated symbol vector into the plurality of decoded messages. Each of the plurality of decoded messages has the sequence of bits that corresponds to the data associated with the corresponding transmitting device amongst the plurality of transmitting devices 102.

Thus, the receiving device 104 manifests a reduced probability of decoding error and an improved communication reliability when used in a communication system where there is no perfect synchronization (or no block fading channel) between the transmitting device 202 (or the plurality of transmitting devices 102) and the receiving device 104. By virtue of the timefrequency mapping and the tensor elements rotated by use of the rotation matrix (i.e., t/j) used at the transmitting device 202 and the plurality of compensation circuits 506, an accurate estimation and compensation of timing and carrier frequency offsets can be achieved which further leads to the reduced probability of decoding error and the improved communication reliability. Moreover, the receiving device 104 may be used in the communication system employing tensor-based modulation along with OFDM and hence, enables an enhanced spectral efficiency.

FIG. 6 is an illustration of timing offsets for a symbol vector, in accordance with an embodiment of the present disclosure. FIG. 6 is described in conjunction with elements from FIGs. 1, 2, 3, 4, and 5. With reference to FIG. 6, there is shown a graphical representation 600 that illustrates timing offsets for a symbol vector (e.g., an OFDM symbol) with three transmitting devices, represented as UE1, UE2, and UE3. The graphical representation 600 includes an X-axis 602 that represents timing parameters. There is further shown a first line 604 that represents from where of the symbol vector (i.e., OFDM symbol) sampling starts and where sampling ends.

In a case, when there is no block fading channel between the transmitting device 202 and the receiving device 104 and therefore, timing and carrier frequency offsets occur during data transmission from the transmitting device 202 to the receiving device 104. It is also assumed that the sample rate of the analog-to-digital converter at the receiving device 104 is 1/A_S and the channel (i.e., the communication network 106, of FIG. 1) between a K-th transmitting device (e.g., the K-th transmitting device 102K) and the receiving device 104 (e.g., a base station) is comprised of a single path. The baseband frequency-domain representation is therefore, according to the equation (4) The delay r_k represents the propagation delay between the K-th transmitting device 102K and the receiving device 104, while g_k is a M x 1 vector, known as a steering vector, corresponding to the response of the antenna array (e.g., the antenna 502) to the angle of arrival of the considered path. The parameter a>_k represents time related effects, due to both carrier frequency offsets and timing offsets (or Doppler offsets). At the receiving device 104, the receiving device 104 samples the signal over a duration equal to the symbol vector (e.g., the OFDM symbol) with its cyclic prefix, removes the cyclic prefix and applies a discrete Fourier transform (DFT) operation to the received samples. The parameter (p_k is defined as (p_k = 27rr_k/(lVA_s). The M- dimensional vector y(s, f") of the one or more radio-frequency signals received by all the antennas (e.g., the antenna 502), on subcarrier f of the s-th OFDM symbol, can be written according to the equation (5)

Where, vector representing a random noise term. This illustrates that a conventional tensor-based modulation (TBM) is not suitable for the OFDM system since the conventional TBM assumes that the channel is block fading in nature. Indeed, in context of the OFDM system, the block fading assumption holds in the case where the plurality of transmitting devices 102 and the receiving device 104 are perfectly synchronized with each other and therefore, (p_k = a>_k = 0, where (p_k denotes timing offsets and a>_k denotes carrier frequency offsets. In the case of the block fading channel, the channel is flat in time and frequency since the received signal involves channel component independent from the frequency subcarrier f and the index of the OFDM symbol s, as represented by the equation (6)

Thus, the conventional TBM is suitable for use in the OFDM system only when the channel is block fading. Alternatively stated, each of the plurality of transmitting devices 102 and the receiving device 104 are considered to be perfectly synchronized with each other. In another case, when there is no block fading channel between each of the plurality of transmitting devices 102 and the receiving device 104, the conventional TBM seems unsuitable for use in the OFDM system due to presence of timing and carrier frequency offsets. However, the receiving device 104 performs estimation and compensation of the timing and carrier frequency offsets and hence, provides a reduced probability of decoding error and an improved communication reliability. If the transmission is properly synchronized so that <p_k = a)_k = 0 for all k, this yields to the equation (7)

If it is assumed that (p_k #= 0 and a>_k The timing and carrier frequency offsets affect the received signal y by adding a phase ramp in both the time and frequency domain with respect to each of the plurality of transmitting devices 102 (or the users). In other words, the equation (7) becomes like the equation (8) with D(a>_k, (p_k~), a diagonal matrix which may be defined according to the equation (9)

D(a)_k, (p_k~) = diag( diag(

The vector D(m_k, <p_k)d(x_{l k} ® ••• ® x_{d k}) is not in general the vectorization of a rank-one tensor. On the other hand, if the pre-defined permutation matrix (i.e., A) and the dimensions of the d vectors (i.e., T_lt ... , T_d) satisfy the following property, ® ••• ® x_{d k}) is a rank-one tensor, thus allowing the separation circuit 504 of the receiving device 104 to separate each of the plurality of transmitting devices 102 blindly using the tensor decomposition. The pre-defined permutation matrix (i.e., A) and the diagonal matrix D(a>_k, (p_k~) satisfies a property P. For any (p_k and m_k, and for the diagonal matrix D(a>_k, (p_k\ there exists diagonal matrices D₁(m_k, <p_k) G <C^T1XT1, ... , D_d(a)_k, (p_k~) G (C^Td^xTd _such as represented by the equation (10)

The property P is satisfied for example in the following cases:

- A is the identity matrix and either /?; divides F or F divides /?; with T_t for i = 1, ... , d.

- A is a Kronecker product of d permutation matrices A = A_t ® ... ® A_d and either /?; divides F or F divides If the property P is satisfied, the received radio frequency signal (or the plurality of radio frequency signals) can then be written as the sum of rank-one tensors, according to the equation (H)

FIG. 7 is an illustration of a receiving device, in accordance with another embodiment of the present disclosure. FIG. 7 is described in conjunction with elements from FIGs. 1, 2, 3, 4, 5 and 6. With reference to FIG. 7, there is shown the receiving device 104 that includes the separation circuit 504, the plurality of compensation circuits 506 and the plurality of decoders 508. There is further shown a processing of a radio frequency signal (or a plurality of radio frequency signals) in order to obtain a plurality of decoded messages.

The antenna 502 of the receiving device 104 is configured to receive the radio frequency signal (or the plurality of radio frequency signals) concurrently from each of the plurality of transmitting devices 102. Thereafter, the separation circuit 504 is configured to process the received radio frequency signal (i.e., y) through the inverse permutation in order to further perform the multi-user separation. The inverse permutation corresponds to reverting the effect of the pre-defined permutation matrix (i.e., A) used at the transmitting device 202 (of FIG. 2) by applying the pre-defined permutation matrix (i.e., (.4 0 IM)^-1) to the received radio frequency signal (i.e., y) at the receiving device 104. After applying the pre-defined permutation matrix (i.e., G4 0 /_M)^-1) to each of the received radio frequency signals, each signal behaves like a rank-one tensor structure of order d, which is then subjected to the tensor decomposition. The tensor decomposition can be done by use of the canonical polyadic decomposition to separate the plurality of estimated symbol vectors, such as ... , x_d jwith respect to a first transmitting device (e.g., the first transmitting device 102A), x_{1 K}, ... , x_{d K} with respect to the K-th transmitting device 102K of the plurality of transmitting devices 102. In this way, the separation circuit 504 outputs an estimation of K vector symbols v_k for k = 1, The integer K denotes the estimated number of transmitted messages. Although it does not impede the separation of each of the plurality of transmitting devices 102 (or the user separation), assuming that the property P (described earlier, for example, in FIG. 6) is satisfied, the timing and carrier frequency offsets still affect the information-bearing vectors through the multiplicative terms (i.e., Di(a>_k, (p_ky). On the other hand, in the presence of the timing and carrier frequency offsets, each of the plurality of estimated symbol vectors experiences a different channel in the form of matrices (i.e., D₍(m_k, <p_k)) explicating the dependency in (p_k that is compensated by use of the plurality of compensation circuits 506. Then, the timing and carrier frequency offsets are estimated and compensated independently for each transmitting device (or the user). The compensated symbol vectors may be represented as x_{l lt} ... , x_{d l}with respect to the first transmitting device 102A, up to the x_{1 K}, ... , x_{d K} with respect to the K-th transmitting device 102K of the plurality of transmitting devices 102. Finally, the compensated symbol vectors are decoded to the plurality of decoded messages by use of the plurality of decoders 508 (or single user decoders). Each decoded message comprises the sequence of bits that corresponds to data associated with the corresponding transmitting device amongst the plurality of transmitting devices 102. The decoded messages with respect to each of the plurality of transmitting devices 102 may be represented, such as a decoded message corresponds to the first transmitting device 102A and a decoded message corresponds to the K-th transmitting device 102K of the plurality of transmitting devices 102. In a case, where a single compensation circuit is used and the dependency on the user index k is dropped and noting as the output of the separation circuit 504 for any transmitting device, the estimation of the timing and carrier frequency offset parameters can be performed by the optimization problem of the equation (12) (12)

The offset correction parameters (p, a> are selected in such a way to minimize the distance between the corresponding estimated symbol vector and the symbol vector of the constellation (i.e., C). The vector value x_t = (p)^HXi corresponds to the symbol where the timing and carrier frequency offsets are compensated and it is used as the input to the plurality of decoders 508.

Moreover, in a case when each of the plurality of transmitting devices 102 has N antennas with N > 1, the operations performed on each of the plurality of transmitting devices 102 and the receiving device 104 can be adapted in the following manner.

Option 1 : The option 1 consists in modifying the constellation C by C" whose members are T x N matrices, defined by Cja^T for all c_y G C with a as a fixed vector of size N. The non- coherent multi-user equalizer and the plurality of decoders 508 (or single-user decoder) are then unchanged.

Option 2: The option 2 considers another modification of the constellation mapper (i.e., the mapping circuit 208) where the symbol vector sent by the K-th transmitting device 102K can be further described according to the equation (13) where, each X_{k i} is a matrix of size T_t X N_t (for i = 1 ... d) and T_lt ... , T_d and N_lt ... , N_d are such = N. The sub-constellations then consist in constellations C_L = being a matrix of dimension T_t X N_t. The receiving device 104 can then be adapted accordingly.

FIG. 8 is a flowchart of a method for a random-access communication, in accordance with another embodiment of the present disclosure. FIG. 8 is described in conjunction with elements from FIGs. 1, 2, 3, 4, 5, 6, and 7. With reference to FIG. 8, there is shown a method 800 that includes steps 802 to 808. The method 800 is executed by the receiving device 104 (of FIG. 1).

The method 800 is provided for random-access communication in which a random number of the plurality of transmitting devices 102 (or the transmitting device 202) may be active at a time and may be configured to transmit a plurality of radio frequency signals concurrently to the receiving device 104 without any prior resource request (or grant). Moreover, each of the plurality of transmitting devices 102 (or the transmitting device 202) is not in perfect synchronization with the receiving device 104 and therefore, timing and carrier frequency offsets incur during transmission from each of the plurality of transmitting devices 102 (or the transmitting device 202) to the receiving device 104 and are compensated at the receiving device 104 in the following way.

At step 802, the method 800 comprises receiving, by at least one antenna 502 of the receiving device 104, a plurality of radio frequency signals concurrently from the plurality of transmitting devices 102 (of FIG. 1). In the massive random-access scenario, the at least one antenna 502 of the receiving device 104 is configured to receive the plurality of radio frequency signals concurrently from the plurality of transmitting devices 102 (of FIG. 1). At step 804, the method 800 further comprises arranging, by the separation circuit 504 of the receiving device 104, each of the radio frequency signals into a time-frequency grid associated with a plurality of time-frequency resources, according to a predefined matrix, to generate a plurality of estimated symbol vectors from the received plurality of radio frequency signals, and separating the plurality of estimated symbol vectors using a rank-1 tensor structure of order d associated with each transmitting device, where d > 1. The separation circuit 504 is configured to arrange each of the received radio frequency signals into the time-frequency grid associated with the plurality of time-frequency resources using the predefined matrix, described earlier, for example, in FIG. 5. After applying the pre-defined matrix to each of the received radio frequency signals, the plurality of estimated symbol vectors is generated at the receiving device 104. The plurality of estimated symbol vectors is subjected to a tensor decomposition by use of a canonical polyadic decomposition corresponding to each of the plurality of transmitting devices 102. The canonical polyadic decomposition is used to separate the plurality of estimated symbol vectors into single-user components.

At step 806, the method 800 further comprises estimating, by the plurality of compensation circuits 506 of the receiving device 104, a time offset and a frequency offset in each of the plurality of estimated symbol vectors, and generating, by the compensation circuits, a plurality of compensated symbol vectors by applying time offset compensations, based on the time offsets, and frequency offset compensations, based on the frequency offsets, to the plurality of estimated symbol vectors. The plurality of compensation circuits 506 is used to estimate as well as compensate the timing and carrier frequency offsets in each of the plurality of estimated symbol vectors and generate the plurality of compensated symbol vectors by applying the corresponding time and carrier offset compensation in each of the plurality of estimated symbol vectors, at the receiving device 104.

In accordance with an embodiment, each of the compensation circuit uses a corresponding constellation comprising symbol vectors, and where the corresponding time offset compensation and frequency offset compensation are determined so as to minimize the distance between the estimated symbol vector and a symbol vector of the constellation. Each of the plurality of compensation circuits 506 is configured to use the corresponding constellation comprising the symbol vectors, and determine the time and frequency offset compensation in order to minimize the distance between the corresponding symbol vector and the symbol vector of constellation, described earlier, for example, in FIG. 7. At step 808, the method 800 further comprises decoding, by the plurality of decoders 508 in the receiving device 104, the plurality of compensated symbol vectors to generate a plurality of decoded messages, each decoded message comprising a sequence of bits that corresponds to data associated with a corresponding transmitting device of the plurality of transmitting devices 102. Each of the plurality of decoders 508 is configured to decode the compensated symbol vector into the plurality of decoded messages. Each of the plurality of decoded messages has the sequence of bits that corresponds to the data associated with the corresponding transmitting device amongst the plurality of transmitting devices 102.

The steps 802 to 808 are only illustrative and other alternatives can also be provided where one or more steps are added, one or more steps are removed, or one or more steps are provided in a different sequence without departing from the scope of the claims herein.

In one aspect, a computer program product is provided performing the method 800 when executed by one or more processors (e.g., the processor 512 of the receiving device 104) in a computer system. In another aspect, a computer system is provided comprising one or more processors (e.g., the processor 512) and one or more memories (e.g., the memory 510), the one or more memories (i.e., the memory 510) storing program instructions which, when executed by the one or more processors (i.e., the processor 512), cause the one or more processors (i.e., the processor 512) to execute the method 800. In yet another aspect, the present disclosure provides a non-transitory computer-readable medium having stored thereon, computer- implemented instructions that, when executed by a computer, causes the computer to execute operations of the method 800.

FIG. 9 is a block diagram that illustrates various exemplary components of a communication apparatus, in accordance with an embodiment of the present disclosure. FIG. 9 is described in conjunction with elements from FIGs. 1, 2, and 5. With reference to FIG. 9, there is shown a block diagram 900 of a communication apparatus 902 that includes the transmitting device 202 (of FIG. 2) and the receiving device 104 (of FIG. 1). In another embodiment, the communication apparatus 902 may include the plurality of transmitting devices 102 and the receiving device 104.

The communication apparatus 902 comprising the transmitting device 202 and the receiving device 104 manifests a reduced probability of decoding error rate and an improved communication reliability. Moreover, the communication apparatus 902 may be used in the massive random-access scenario with the improved communication reliability and spectral efficiency. Examples of the communication apparatus 902 may include, but are not limited to, a transceiver, a base station, a user equipment, and the like. The communication apparatus 902 comprising the transmitting device 202 (or the plurality of transmitting devices 102) and the receiving device 104 can be used in Intemet-of- Things (loT), massive random-access scenario, uplink multi-input-multi-output (MIMO) random access with the plurality of transmitting devices 102, and the like.

FIG. 10 is a graphical representation that illustrates variation of block error rate (BLER) with respect to signal-to-noise ratio (SNR), in accordance with an embodiment of the present disclosure. FIG. 10 is described in conjunction with elements from FIGs. 1, 2, 3, 4, 5, 6, 7, 8, and 9. With reference to FIG. 10, there is shown a graphical representation 1000 that illustrates variation of block error rate (BLER) with respect to signal-to-noise ratio (SNR). The graphical representation 1000 includes an X-axis 1002 that represents SNR in decibels (dB) and a Y-axis 1004 that represents BLER.

The graphical representation 1000 illustrates a mixed discrete-continuous low-complexity optimization by using an approximate Cube Split decoder and a grid search over that = 200 points and complexity of the optimization criterion is aphical representation 1000, there is shown a first line 1006 that represents the variation of the BLER with respect to SNR with no time offset and no phase correction and for the Cube Split constellation and two active user equipments (UE) or transmitting devices. Further, a second line 1008 represents the variation of the BLER with respect to the SNR with a time offset of 0.5 cyclic prefix (CP) for a rotated Cube Split constellation and offset compensation and two active user equipments (UE) or transmitting devices. Further, a third line 1010 represents the variation of the BLER with respect to the SNR with a time offset of 0.5 CP for the original Cube Split modulation (i.e., without rotation) and the offset compensation and two active user equipments (UE) or transmitting devices. Moreover, there is a difference of 2dB SNR between the first line 1006 and the third line 1010. However, the second line 1008 is closer to the first line 1006 in comparison to the third line 1010 and shows a difference of 0.5dB SNR with the first line 1006. This means that the variation of the BLER versus SNR obtained with the rotated Cube-Split constellation along with the timing and phase offset compensation is equal to the case where no time offset and phase correction is used. In an implementation, the graphical representation 1000 can be obtained by considering a packet size of 10 bytes, two active user equipments or transmitting devices, two hundred and forty potential user equipments (UEs), a carrier frequency of 700MHz, a bandwidth of 6 resource blocks (RBs), polar code for channel coding, 8 base station antennas, one UE antenna, channel model of TDL-A 30ns and 3Km/h, timing offset as no offset for the first line 1006 and uniformly distributed between 0 to 0.5 CP for the second line 1008 and the third line 1010 and with a numerology of SCS 15kHz and 14 operating systems (OS).

FIG. 11 is a graphical representation that illustrates variation of BLER with respect to SNR for a Cube Split constellation, in accordance with an embodiment of the present disclosure. FIG. 11 is described in conjunction with elements from FIGs. 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10. With reference to FIG. 11, there is shown a graphical representation 1100 with an X-axis 1102 that represents the SNR in decibel (dB) and a Y-axis 1104 that represents BLER.

The graphical representation 1100 is obtained in an implementation scenario, where each transmitting device (e.g., each of the plurality of transmitting devices 102) is configured to transmit a payload of eighty bits within an OFDM of F = 72 frequency subcarriers and S = 14 time slots. Moreover, the graphical representation 1100 illustrates the variation of BLER with respect to SNR in an ideal case where no timing and carrier frequency offsets are considered and hence, the plurality of compensation circuits 506 is not present in the receiving device 104 as well as a case in which timing offsets are randomly distributed between 0 and 0.5 N_CPA_S and the plurality of compensation circuits 506 is present in the receiving device 104. The graphical representation 1100 is obtained by using same parameters as used for the obtaining the graphical representation 1000.

The graphical representation 1100 compares three different cases, such as a first case when the rotation matrix (i.e., Ui) is an identity matrix, a second case where the rotation matrix (i.e., Ui) is a random Fourier rotation matrix, and a third case where the rotation matrix (i.e., Ui) is a random cubic rotation matrix. Moreover, the three different cases of the rotation matrix are used in comparison of two systems, a first system (e.g., an ideal case) is where no timing and carrier frequency offsets are considered and a second system is subjected to timing offsets which are estimated and compensated by use of the plurality of compensation circuits 506 in the receiving device 104. In the graphical representation 1100, a first line 1106 represents the BLER for norotation (i.e., for Ui as the identity matrix), a second line 1108 represents the BLER for Fourier rotation (i.e., for Ui as the random Fourier rotation matrix), and a third line 1110 represents the BLER for cubic rotation (i.e., for Ui as the random cubic rotation matrix) in the first system where no timing and carrier frequency offsets are present. There is further shown a fourth line 1112 that represents the BLER for no-rotation (i.e., for Ui as the identity matrix), a fifth line 1114 represents the BLER for Fourier rotation (i.e., for Ui as the random Fourier rotation matrix), and a sixth line 1116 represents the BLER for cubic rotation (i.e., for Ui as the random cubic rotation matrix) in the second system where timing offsets are present which are estimated and compensated by use of the plurality of compensation circuits 506 in the receiving device 104. Moreover, the graphical representation 1100 represents that the BLER illustrated by the fifth line 1114 and the sixth line 1116 is close to the BLER illustrated by the first line 1106, the second line 1108 and the third line 1110. Alternatively stated, the performance of the second system with timing offsets estimation and compensation is close to the first system (or the ideal case) with no timing and carrier frequency offsets. Moreover, the use of the rotation matrix (i.e., Ui) improves the performance of the second system in the presence of timing offsets.

FIG. 12 is a graphical representation that illustrates variation of BLER with respect to SNR for a quadrature amplitude modulation (QAM) constellation, in accordance with an embodiment of the present disclosure. FIG. 12 is described in conjunction with elements from FIGs. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11. With reference to FIG. 12, there is shown a graphical representation 1200 with an X-axis 1202 that represents the SNR in decibel (dB) and a Y-axis 1204 that represents BLER.

The graphical representation 1200 is obtained in an implementation scenario, where each transmitting device (e.g., each of the plurality of transmitting devices 102) is configured to transmit a payload of eighty bits within an OFDM of F = 72 frequency subcarriers and S = 14 time slots. Moreover, the graphical representation 1200 illustrates the variation of BLER with respect to SNR in an ideal case where no timing and carrier frequency offsets are present and hence, the plurality of compensation circuits 506 is not present in the receiving device 104 compared to a case in which timing offsets are randomly distributed between 0 and 0.5 N_CPA_S and the plurality of compensation circuits 506 is present in the receiving device 104. The graphical representation 1200 is obtained by using same parameters as used for the obtaining the graphical representation 1000 (of FIG. 10) and the graphical representation 1100 (of FIG. H).

The graphical representation 1200 compares three different cases, such as a first case when the rotation matrix (i.e., Ui) is an identity matrix, a second case where the rotation matrix (i.e., Ui) is a random Fourier rotation matrix, and a third case where the rotation matrix (i.e., Ui) is a random cubic rotation matrix. Moreover, the three different cases of the rotation matrix are used in comparison of two systems, a first system (e.g., an ideal case) is where no timing and carrier frequency offsets are considered and a second system is subjected to timing offsets which are estimated and compensated by use of the plurality of compensation circuits 506 in the receiving device 104. In the graphical representation 1200, a first line 1206 represents the BLER for norotation (i.e., for Ui as the identity matrix), a second line 1208 represents the BLER for Fourier rotation (i.e., for Ui as the random Fourier rotation matrix), and a third line 1210 represents the BLER for cubic rotation (i.e., for Ui as the random cubic rotation matrix) in the first system where no timing and carrier frequency offsets are present. There is further shown a fourth line 1212 that represents the BLER for no-rotation (i.e., for Ui as the identity matrix), a fifth line 1214 represents the BLER for Fourier rotation (i.e., for Ui as the random Fourier rotation matrix), and a sixth line 1216 represents the BLER for cubic rotation (i.e., for Ui as the random cubic rotation matrix) in the second system where timing offsets are present which are estimated and compensated by use of the plurality of compensation circuits 506 in the receiving device 104. Moreover, the graphical representation 1200 represents that the BLER illustrated by the fifth line 1214 and the sixth line 1216 is close to the BLER illustrated by the first line 1206, the second line 1208 and the third line 1210. Alternatively stated, the performance of the second system with timing offsets estimation and compensation is close to the first system (or the ideal case) with no timing and carrier frequency offsets. Moreover, the use of the rotation matrix (i.e., Ui) improves the performance of the second system in the presence of timing offsets. From the graphical representation 1100 (of FIG. 11) and the graphical representation 1200, this can be analysed that the Cube-Split constellation manifests more benefits from the offset compensation in comparison to the QAM constellation compared using (T₁₍ T₂, T₃, T₄) = (4,3,6,14).

Modifications to embodiments of the present disclosure described in the foregoing are possible without departing from the scope of the present disclosure as defined by the accompanying claims. Expressions such as "including", "comprising", "incorporating", "have", "is" used to describe and claim the present disclosure are intended to be construed in a non-exclusive manner, namely allowing for items, components or elements not explicitly described also to be present. Reference to the singular is also to be construed to relate to the plural. The word "exemplary" is used herein to mean "serving as an example, instance or illustration". Any embodiment described as “exemplary” is not necessarily to be construed as preferred or advantageous over other embodiments and/or to exclude the incorporation of features from other embodiments. The word "optionally" is used herein to mean "is provided in some embodiments and not provided in other embodiments". It is appreciated that certain features of the present disclosure, which are, for clarity, described in the context of separate embodiments, may also be provided in combination in a single embodiment. Conversely, various features of the present disclosure, which are, for brevity, described in the context of a single embodiment, may also be provided separately or in any suitable combination or as suitable in any other described embodiment of the disclosure.

Claims

1. A transmitting device (202) for random access communication, comprising: an encoding circuit (204) configured to encode an input message into a sequence of bits, split the sequence of bits into d blocks of bits, where d > 1, and determine, from the d blocks of bits, d vectors; a Kronecker product circuit (206) configured to construct a symbol vector by computing the Kronecker product of the d vectors to generate a rank-1 tensor structure of order d; a mapping circuit (208) configured to map the symbol vector in a time-frequency grid associated with a plurality of time-frequency resources according to a pre-defined permutation matrix, wherein each element of the symbol vector is mapped onto one of the time-frequency resources of the time-frequency resource grid to generate a vector of time-frequency modulated symbols; and at least one antenna (210) configured to transmit each of the time-frequency modulated symbols over a radio frequency signal to a receiving device.

2. The transmitting device (202) according to claim 1, wherein the Kronecker product circuit (206) is configured to apply a rotation to each of the d vectors by multiplying each of the d vectors by a rotation matrix, prior to computing the Kronecker product.

3. The transmitting device (202) according to claim 1 or 2, wherein the encoding circuit (204) is configured to determine each of the d vectors by using a corresponding sub-constellation based on bits in a corresponding block of bits, and where each of the corresponding subconstellations has a structure selected from at least a Grassmannian constellation, a cube-split constellation, or a constellation wherein each of the plurality of vectors is divided into a pilot part and a data part.

4. The transmitting device (202) according to any of claims 1 to 3, wherein the encoding circuit (204) is configured to encode the input message into a sequence of bits according to a binary code.

5. A receiving device (104) for random access communication, comprising: at least one antenna (502) configured to receive a plurality of radio frequency signals concurrently from a plurality of transmitting devices (102); a separation circuit (504) configured to arrange each of the radio frequency signals into a time-frequency grid associated with a plurality of time-frequency resources, according to a predefined permutation matrix, to generate a plurality of estimated symbol vectors from the received plurality of radio frequency signals, and separate the plurality of estimated symbol vectors using a rank-1 tensor structure of order d associated with each transmitting device, where d > 1 ; a plurality of compensation circuits (506) each configured to estimate a corresponding time offset and a corresponding frequency offset in one corresponding estimated symbol vector amongst the plurality of estimated symbol vectors, and generate a corresponding compensated symbol vector by applying a time offset compensation, based on the corresponding time offset, and a frequency offset compensation, based on the corresponding frequency offset, to the corresponding estimated symbol vector; and a plurality of decoders (508) each configured to decode one compensated symbol vector generated by one of the compensation circuits, to generate a plurality of decoded messages, each decoded message comprising a sequence of bits that corresponds to data associated with a corresponding transmitting device amongst the plurality of transmitting devices (102).

6. The receiving device (104) according to claim 5, wherein each of the compensation circuits in the plurality of compensation circuits (506) is configured to use a corresponding constellation comprising symbol vectors, and determine the time offset compensation and frequency offset compensation so as to minimize a distance between the corresponding estimated symbol vector and a symbol vector of the constellation.

7. A communication apparatus (902), comprising a transmitting device (202) according to any of claims 1 to 4, and a receiving device (104) according to claim 5 or 6.

8. A method (400) for a random-access communication, comprising: encoding, by an encoder of a transmitting device (202), an input message into a sequence of bits; splitting, by the encoder, the sequence of bits in d blocks of bits, where d > 1; determining, by the encoder, from the d blocks of bits, d vectors; constructing a symbol vector, by a Kronecker product circuit (206) of the transmitting device (202), by computing the Kronecker product of the d vectors to generate a rank-1 tensor structure of order d; mapping, by a mapping circuit (208) of the transmitting device (202), the symbol vector in a time-frequency grid associated with a plurality of time-frequency resources according to a pre-defined permutation matrix, wherein each element of the symbol vector is mapped onto one of the time-frequency resources of the time-frequency resource grid to generate a vector of time-frequency modulated symbols; and transmitting, by at least one antenna (210) of the transmitting device (202), each of the time-frequency modulated symbols over a radio frequency signal to a receiving device (104).

9. The method (400) according to claim 8, wherein a rotation is applied to each of the d vectors by multiplying each of the d vectors by a rotation matrix prior to computing the Kronecker product.

10. The method (400) according to claim 8 or 9, wherein each of the d vectors is determined by using a corresponding sub-constellation based on bits in a corresponding block of bits, and each of the corresponding sub-constellations has a structure selected from at least a Grassmannian constellation, a cube-split constellation, or a constellation wherein each of the plurality of vectors is divided into a pilot part and a data part.

11. The method (400) according to any of claims 8 to 10, wherein encoding, by the encoder, the input message into a sequence of bits, uses a binary code.

12. A method (800) for a random-access communication, comprising: receiving, by at least one antenna (502) of a receiving device (104), a plurality of radio frequency signals concurrently from a plurality of transmitting devices (102); arranging, by a separation circuit (504) of the receiving device (104), each of the radio frequency signals into a time-frequency grid associated with a plurality of time-frequency resources, according to a predefined matrix, to generate a plurality of estimated symbol vectors from the received plurality of radio frequency signals, and separating the plurality of estimated symbol vectors using a rank-1 tensor structure of order d associated with each transmitting device, where d > 1; estimating, by a plurality of compensation circuits (506) of the receiving device (104), a time offset and a frequency offset in each of the plurality of estimated symbol vectors, and generating, by the compensation circuits, a plurality of compensated symbol vectors by applying time offset compensations, based on the time offsets, and frequency offset compensations, based on the frequency offsets, to the plurality of estimated symbol vectors; and decoding, by a plurality of decoders (508) in the receiving device (104), the plurality of compensated symbol vectors to generate a plurality of decoded messages, each decoded message comprising a sequence of bits that corresponds to data associated with a corresponding transmitting device of the plurality of transmitting devices (102).

13. The method (800) according to claim 12, wherein each of the compensation circuits uses a corresponding constellation comprising symbol vectors, and wherein the corresponding time offset compensation and frequency offset compensation are determined so as to minimize the distance between the estimated symbol vector and a symbol vector of the constellation.

14. A computer program product comprising program instructions for performing the method (400) according to any of claims 8 to 11 and/or the method (800) according to claim 12 or 13, when executed by one or more processors in a computer system.

15. A computer system comprising one or more processors and one or more memories, the one or more memories storing program instructions which, when executed by the one or more processors, cause the one or more processors to execute the method (400) according to any of claims 8 to 11 and/or the method (800) according to claim 12 or 13.