WO2022258193A1 - Generation and reception of precoded signals based on codebook linearization - Google Patents

Abstract

Various embodiments relate to the generation and reception of precoded signals based on codebook linearization. A device may obtain an antipodal input sequence and a precoding matrix. The precoding matrix may be based on a multiplication of a codebook matrix and an antipodal matrix, wherein the codebook matrix comprises M codewords of complex symbols, and wherein the antipodal matrix comprises M different antipodal vectors. The device may generate a precoded codeword based on multiplying the antipodal input sequence by the precoding matrix. Another device may receive and demodulate the signal, and perform a linear equalization of the demodulated

Description

GENERATION AND RECEPTION OF PRECODED SIGNALS BASED ON CODEBOOK LINEARIZATION

TECHNICAL FIELD

The present disclosure generally relates to the field of wireless communications. In particular, some embodiments of the disclosure relate to the generation and reception of precoded signals based on codebook linearization.

BACKGROUND

In wireless communication systems, such as for example the fifth generation (5G) system specified by the 3^rd generation partnership project (3GPP), a large number of terminal devices may need to be connected together to support various applications and service categories operating in a broad range of frequencies and deployment scenarios. In order to address the requirement of a large number of users, non-orthogonal multiple access (NOMA) techniques comprising sharing of resource blocks between users may be applied to increase spectral efficiency.

SUMMARY

This summary is provided to introduce a selection of concepts in a simplified form that are further described below in the detailed description. This summary is not intended to identify key features or essential features of the claimed subject matter, nor is it intended to be used to limit the scope of the claimed subject matter.

It is an objective of the present disclosure to improve spectral efficiency, reduce receiver complexity, and thereby increase energy efficiency of signal transmission. The foregoing and other objectives may be achieved by the features of the independent claims. Further implementation forms are apparent from the dependent claims, the description, and the drawings.

According to a first aspect, a device for generating a signal is provided. The device may be configured to: obtain an antipodal input sequence; obtain a precoding matrix, wherein the precoding matrix is based on a multiplication of a codebook matrix and an antipodal matrix, wherein the codebook matrix comprises M codewords of complex symbols, and wherein the antipodal matrix comprises M different antipodal vectors; and generate a precoded codeword based on multiplying the antipodal input sequence by the precoding matrix. This solution improves the transmission performance and enables a reduced receiver complexity by linearization of the codebook matrix.

According to an implementation form of the first aspect, the antipodal matrix may comprise M different antipodal vectors corresponding to binary representations of values 0 to M — 1 in an ascending order. This solution provides an implementation for determining the antipodal matrix, to improve transmission performance and linearize the codebook matrix.

According to an implementation form of the first aspect, the precoding matrix S_fc of the k-th user may comprise S_fe =

X_fe B^T . where X_fc is the codebook matrix and B^Tis the antipodal matrix. This solution provides an implementation for generating the precoding matrix, to improve transmission performance and enable reduced complexity at a receiver by codebook linearization.

According to an implementation form of the first aspect, the device may be further configured to: obtain a plurality of the antipodal input sequences corresponding to a plurality of users associated with a plurality of the codebook matrices; obtain a plurality of the precoding matrices corresponding to the plurality of users, wherein a precoding matrix of a k- th user is based on multiplication of a codebook matrix of the k- th user and the antipodal matrix; and generate a plurality of precoded codewords for each of the plurality of users, wherein each precoded codeword of the k- th user is based on a multiplication of a different subset of an antipodal input sequence of the k- th user and the precoding matrix of the k- th user; and generate the signal based on a concatenation of the plurality of precoded codewords for each of the plurality of users, wherein a number of the plurality of users is higher than a number of complex symbols of codewords of the plurality of the codebook matrices. This solution enables generation of multiple signals, to improve transmission performance for multiple users sharing transmission resources according to a NOMA scheme. According to an implementation form of the first aspect, the device may be further configured to: generate a modulation symbol based on multiplying the concatenated plurality of precoded codewords of the k- th user by a modulation matrix G_fc of the k- th user, wherein

where signature roots p_{n. k} of the modulation matrix of the P-th user are p_n,k =

with radius α_k of the k- th user, and where K_k is a normalization factor of the k- th user; and insert zero-padding at an end of the modulation symbol. This solution improves transmission performance by applying Lagrange- Vandermonde division multiplexing (LVDM) with the codebook linearization. In combination with the codebook linearization, this solution improves the performance of a linear receiver in doubly-selective channels.

According to an implementation form of the first aspect, the device may be further configured to receive an indication of the radius α_k of the k- th user or an indication of the signature roots p_{n k} of the modulation matrix of the k- th user from a receiver. This solution enables dynamic optimization of the LVDM modulation for each user separately, to improve transmission performance.

According to an implementation form of the first aspect, the device may be further configured to determine the normalization factor K_k of the k- th user based on the radius α_k of the k-th user. The normalization factor enables to avoid increasing or decreasing the transmit symbol energy in practical implementations. This solution also enables an efficient implementation using unitary energy filters, where the coefficients of the filters include the columns of the LVDM modulation matrix G_fc.

According to an implementation form of the first aspect, the device may be further configured to: generate a plurality of the modulation symbols for the k- th user; insert a zero-padding at an end of each of the plurality of modulation symbols; and insert a plurality of training sequences periodically within the plurality of modulation symbols, wherein the plurality of training sequences comprises L zeros at an end of each training sequence. This solution enables an improved channel estimation for an LVDM signal at a receiver.

According to an implementation form of the first aspect, the plurality of training sequences may be different for each of the plurality of users. This solution improves the separation of channel impulse responses of different users at a receiver.

According to an implementation form of the first aspect, the device may be further configured to transmit the plurality of training sequences for each of the plurality of users with a time shift between training sequences of the plurality of users. This solution improves the separation of channel impulse responses of different users at a receiver.

According to a second aspect, a device for receiving a signal is provided. The device may be configured to: receive the signal, wherein the signal comprises at least one precoded codeword generated based on a multiplication of an antipodal input sequence by a precoding matrix, wherein the precoding matrix is based on a multiplication of a codebook matrix and an antipodal matrix, wherein the codebook matrix comprises M codewords of complex symbols, and wherein the antipodal matrix comprises M different antipodal vectors; and demodulate the signal and perform a linear equalization of the demodulated signal. This solution improves the reception performance and enables the use of linear equalization to reduce receiver complexity.

According to an implementation form of the second aspect, the antipodal matrix may comprise M different antipodal vectors corresponding to binary representations of values 0 to M — 1 in an ascending order. This solution provides an implementation for the antipodal matrix to improve the reception performance and enable linear equalization.

According to an implementation form of the second aspect, the device may be further configured to append a discrete Fourier transform matrix with L first columns of the discrete Fourier transform matrix, wherein the demodulation of the signal is based on the appended discrete Fourier transform matrix. This solution enables demodulation of the received signal to take advantage of zeros inserted at the end of LVDM symbols. According to an implementation form of the second aspect, the signal may comprise a plurality of modulation symbols of a k- th user of a plurality of users, the plurality of modulation symbols generated based on multiplication of a plurality of the precoded codewords by a modulation matrix G_fc of the k- th user, wherein

where signature roots p_{n k} of the modulation matrix of the P-th user are p_{n k} = i

with radius a_k of the P-th user, and K_k is a normalization factor of the P-th user. This solution further improves the reception performance by applying Lagrange-Vandermonde division multiplexing (LVDM) with the codebook linearization. In combination with codebook linearization, this solution improves the performance of a linear receiver in doubly-selective channels.

According to an implementation form of the second aspect, the device may be further configured to: receive a plurality of training sequences located periodically within the plurality of modulation symbols of the k- th user, wherein the plurality of training sequences comprises L zeros at an end of each training sequence; and determine an estimate of a radio channel for the plurality of modulation symbols based on the received plurality of training sequences, wherein the linear equalization of the demodulated signal is based on the estimate of the radio channel. This solution enables an improved channel estimation for an LVDM signal.

According to an implementation form of the second aspect, the device may be further configured to: stack the plurality of received training sequences into a vector of received training sequences r_m, wherein the vector of received training sequences is of the form

where

is additive noise, comprises a p-th

coefficient of a k- th delay tap of a Fourier basis expansion of the radio channel, and matrix f_ίh comprises

N_c is a coherence time of the radio channel, / is a length of the plurality of training sequences,

is a lower triangular Toeplitz matrix whose first column is

is a training sequence of the k- th user,

is a number of the plurality of training sequences, and M_s — 1 is a number of modulation symbols between training sequences; and determine the estimate of the radio channel based on a linear minimum mean square estimator based on the matrix This solution enables an efficient MMSE based channel

estimation based on the training sequences to improve the reception performance.

According to an implementation form of the second aspect, the device may be further configured to: determine a predicted estimate of the radio channel for at least one subsequent modulation symbol based on the plurality of received training sequences; determine updated signature roots p_{n k} of the modulation matrix of the k- th user based on the predicted estimate of the radio channel; and transmit an indication of the updated signature roots p_{n k} of the modulation matrix of the k-ih user to a transmitter. This solution provides an implementation for optimizing the LVDM modulation for each user separately, to improve reception performance.

According to an implementation form of the second aspect, the device may be further configured to: determine a predicted estimate of the radio channel for at least one subsequent modulation symbol based on the plurality of received training sequences; determine an updated radius a_{k opt} of the k- th user based on the predicted estimate of the radio channel; and transmit an indication of the updated radius a_{k opt} of the k- th user to a transmitter. This solution provides an implementation for optimizing the LVDM modulation for each user separately, to improve reception performance. According to an implementation form of the second aspect, the device may be further configured to determine the updated radius a of the k- th user

based on

is a frequency domain coefficient of the predicted estimate

of the radio channel of the k- th user at subcarrier n, and wherein PN is a size of the discrete Fourier transform matrix. This solution provides an implementation for determining the updated radius of the LVDM modulation for each user separately, to improve reception performance.

According to a third aspect, a method for generating a signal is provided. The method may comprise: obtaining an antipodal input sequence; obtaining a precoding matrix, wherein the precoding matrix is based on a multiplication of a codebook matrix and an antipodal matrix, wherein the codebook matrix comprises M codewords of complex symbols, and wherein the antipodal matrix comprises M different antipodal vectors; and generating a precoded codeword based on multiplying the antipodal input sequence by the precoding matrix. This solution improves the transmission performance and enables a reduced receiver complexity by linearization of the codebook matrix.

According to a fourth aspect, a method for receiving a signal is provided. The method may comprise: receiving the signal, wherein the signal comprises at least one precoded codeword generated based on a multiplication of an antipodal input sequence by a precoding matrix, wherein the precoding matrix is based on a multiplication of a codebook matrix and an antipodal matrix, wherein the codebook matrix comprises M codewords of complex symbols, and wherein the antipodal matrix comprises M different antipodal vectors; and demodulating the signal and performing a linear equalization of the demodulated signal. This solution improves the reception performance and enables the use of linear equalization to reduce the receiver complexity.

According to a fifth aspect, a computer program is provided. The computer program may comprise program code configured to cause performance of any implementation form of the method of the third aspect, when the computer program is executed on a computer.

According to a sixth aspect, a computer program is provided. The computer program may comprise program code configured to cause performance of any implementation form of the method of the fourth aspect, when the computer program is executed on a computer.

Any implementation form may be combined with one or more other implementation forms. Implementation forms of the present disclosure can thus provide devices, methods, and computer programs, for generating or receiving a chirp waveform. These and other aspects of the present disclosure will be apparent from the example embodiment(s) described below.

DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are included to provide a further understanding of the example embodiments and constitute a part of this specification, illustrate example embodiments and, together with the description, help to explain the example embodiments. In the drawings:

FIG. 1 illustrates an example of a communication system, according to an embodiment of the present disclosure;

FIG. 2 illustrates an example of a device configured to practice one or more embodiments of the present disclosure;

FIG. 3 illustrates an example of a sparse code multiple access (SCMA) scheme, according to an embodiment of the present disclosure;

FIG. 4 illustrates an example of a factor graph representation of a sparse code multiple access (SCMA) scheme, according to an embodiment of the present disclosure; FIG. 5 illustrates examples of an effective channel with Doppler frequencies f_D = 0 KHz and f_D = 1 KHz, according to an embodiment of the present disclosure;

FIG. 6 illustrates examples of an effective channel with Doppler frequencies f_D = 5 KHz and f_D = 2 KHz, according to an embodiment of the present disclosure;

FIG. 7 illustrates an example of multi-dimensional SCMA mapping with an 8-point codebook, according to an embodiment of the present disclosure;

FIG. 8 illustrates an example of multi-dimensional SCMA mapping with a 16-point codebook, according to an embodiment of the present disclosure;

FIG. 9 illustrates an example of an LVDM transmitter of a k-th user applying a linearized codebook, according to an embodiment of the present disclosure;

FIG. 10 illustrates an example of an LVDM-based NOMA system, according to an embodiment of the present disclosure;

FIG. 11 illustrates an example of a pilot pattern for use with LVDM-NOMA symbols, according to an embodiment of the present disclosure;

FIG. 12 illustrates an example of applying training sequences for channel estimation and prediction, according to an embodiment of the present disclosure;

FIG. 13 illustrates an example of average bit-error rate (BER) for SCMA MPA and LVDM-based NOMA in frequency selective channels, according to an embodiment of the present disclosure;

FIG. 14 illustrates an example of BER for OFDM- and LVDM-based NOMA in doubly selective channels, according to an embodiment of the present disclosure;

FIG. 15 illustrates an example of average BER for OFDM- and LVDM- based NOMA in a 3GPP Extended Vehicular (EVB) channels, according to an embodiment of the present disclosure;

FIG. 16 illustrates an example of average BER for OFDM- and LVDM- based NOMA in a 3GPP Tapped Delay Line C (TDL-C) channels, according to an embodiment of the present disclosure; FIG. 17 illustrates an example of pilot patterns for a plurality of users, according to an embodiment of the present disclosure;

FIG. 18 illustrates an example of the sensitivity to channel estimation and radius estimation errors for LVDM-based NOMA, according to an embodiment of the present disclosure;

FIG. 19 illustrates an example of a method for generating a signal, according to an embodiment of the present disclosure; and

FIG. 20 illustrates an example of a method for receiving a signal, according to an embodiment of the present disclosure.

Like references are used to designate like parts in the accompanying drawings.

DETAILED DESCRIPTION

Reference will now be made in detail to example embodiments, examples of which are illustrated in the accompanying drawings. The detailed description provided below in connection with the appended drawings is intended as a description of the present embodiments and is not intended to represent the only forms in which the present examples may be constructed or utilized. The description sets forth the functions of the examples and the sequence of operations for constructing and operating the examples. However, the same or equivalent functions and sequences may be accomplished by different examples.

The 5G system provides three service categories: enhanced mobile broadband (eMBB), ultra-reliable low-latency communications (URLLC), and massive machine type communications (mMTC). In mMTC scenario, there are a large number of terminal devices to be connected together, which means that a massive amount of connections may need to be supported. Orthogonal Frequency Division Multiple Access (OFDMA) as well as other orthogonal multiple access (OMA) techniques may not provide support for the massive connectivity requirement due to the spectrum scarcity.

NOMA techniques involve the concept of users’ overloading, which may comprise sharing resource blocks between users leading to the spectral efficiency increase. Different NOMA schemes may be designed considering the time-varying channels in 5G networks. However, maximum likelihood (ML) based solutions such as the message passing algorithm (MPA) may be used in an attempt to achieve sufficient performance while keeping the complexity low compared to the full ML solution. However, such solutions may be targeted for low velocity, such as for example 3 km/h, and therefore intercarrier interference caused by higher mobility may corrupt the MPA-based solutions.

One MPA-based approach is to jointly detect the superposed users’ data by using MPA. By leveraging the sparsity in SCMA, the decoder may benefit from the strong iterative decoding scheme that provides a near ML detection performance. However, such MPA algorithm may require excessive computational complexity even for small SCMA codes. For example, the computation of the soft information sent from the resource nodes to the user nodes may have an exponential complexity, 0(IR2^aί^m) where df is the threshold value for MPA layers, to enumerate all possible input combinations of colliding symbols.

MPA may be enhanced with the expectation propagation algorithm (EPA) to reduce the complexity of the receiver. EPA may comprise approximating a distribution with another distribution through a distribution projection into a family of simple distributions. By choosing the projection into complex Gaussian distributions, the message passing reduces to the update of mean and variance parameters. In addition, the messages between the nodes, which are complex vector Gaussian distributions, are simplified to scalar complex Gaussian distributions. However, while EPA-MPA reduces the complexity to 0(lNPdf 2^m), the error performance of an EPA-MPA receiver may be similar to a full MPA receiver. Therefore, under high mobility regime, the performance achievements may be lost.

In order to reduce the complexity, a Gaussian approximation based MPA (GA-MPA) algorithm may be used. In GA-MPA, the discrete information exchanged between user and resource nodes may be approximated as continuous Gaussian functions, avoiding thus the high complex marginalization operation of MPA. GA-MPA exhibits complexity reduction compared to MPA, with a complexity order of However, GA-MPA may not bring any

additional performance gain in case of high mobility.

Another option is to apply successive interference cancellation (SIC) based MPA (SIC-MPA), where the features of SIC and those of MPA may be combined to strike a good balance between the performance gains and the implementation complexity. In SIC-MPA receiver, MPA may be first applied to a limited number of users, so that the number of colliding layers over each resource element (RE) does not exceed the MPA layer df . Then, the successfully decoded MPA layers may be removed by SIC and the procedure may be continued until all users are successfully decoded. In SIC-MPA receiver, the complexity order is which is comparable to MPA.

The additional gains are achieved at the cost of high complexity. Therefore, embodiments of the present disclosure provide NOMA communication schemes that improve performance of communication over doubly-selective channels, while enabling reduced complexity.

According to an embodiment, a device for generating a signal is disclosed. The device may obtain an antipodal input sequence and a precoding matrix. The precoding matrix may be based on multiplication of a codebook matrix and an antipodal matrix, wherein the codebook matrix comprises M codewords of complex symbols, and wherein the antipodal matrix comprises M different antipodal vectors. The device may generate a precoded codeword based on multiplying the antipodal input sequence by the precoding matrix. This enables linearization for the codebook such that linear equalization may be applied at a receiver. Another device may receive and demodulate the signal, and perform linear equalization of the demodulated signal. Computational complexity is thereby reduced.

FIG. 1 illustrates an example of a communication system 100, according to an embodiment of the present disclosure. The communication system 100 may comprise a transmitter 130, which may

communicate over a radio channel 120. The transmitter 110 may generate a transmitted signal based on a bit vector of a k- th user. There may be one or

a plurality of users nd therefore the number of users . The transmitter

110 may generate a signal based on applying a user-specific precoding matrix S_fc , as will be further described below. The transmitted signal may be fed through the radio channel 120, which may be modeled by a channel matrix H. Noise h may be modeled by additive white Gaussian noise added after the radio channel 120. Receiver 130 may determine an estimate b_k of the transmitted bit vector based on demodulation and linear equalization of the received signal y, as will be further described below.

FIG. 2 illustrates an example of a device configured to practice one or more embodiments. Device 200 may be, for example, configured to generate or receive signals according to a NOMA scheme. Device 200 may comprise at least one processor 202. The at least one processor 202 may comprise, for example, one or more of various processing devices, such as for example a coprocessor, a microprocessor, a controller, a digital signal processor (DSP), a processing circuitry with or without an accompanying DSP, or various other processing devices including integrated circuits such as, for example, an application specific integrated circuit (ASIC), a field programmable gate array (FPGA), a microcontroller unit (MCU), a hardware accelerator, a special- purpose computer chip, or the like.

The device 200 may further comprise at least one memory 204. The memory 204 may be configured to store, for example, computer program code or the like, for example operating system software and application software. The memory 204 may comprise one or more volatile memory devices, one or more non-volatile memory devices, and/or a combination thereof. For example, the memory may be embodied as magnetic storage devices (such as hard disk drives, magnetic tapes, etc.), optical magnetic storage devices, or semiconductor memories (such as mask ROM, PROM (programmable ROM), EPROM (erasable PROM), flash ROM, RAM (random access memory), etc.).

Device 200 may further comprise communication interface 208 configured to enable the device 200 to transmit and/or receive information. The communication interface 208 may comprise an internal communication interface such as for example an interface between baseband circuitry and radio frequency (RF) circuitry of a transmitter, receiver, or a transceiver device. Alternatively, or additionally, the communication interface 208 may be configured to provide at least one external wireless radio connection, such as for example a 3GPP mobile broadband connection (e.g. 3G, 4G, 5G, or future generations); a wireless local area network (WLAN) connection such as for example standardized by IEEE 802.11 series or Wi-Fi alliance; a short range wireless network connection such as for example a Bluetooth connection. The communication interface 208 may hence comprise one or more antennas to enable transmission and/or reception of radio frequency signals over the air.

The device 200 may further comprise other components and/or functions such as for example a user interface (not shown) comprising at least one input device and/or at least one output device. The input device may take various forms such a keyboard, a touch screen, or one or more embedded control buttons. The output device may for example comprise a display, a speaker, a vibration motor, or the like.

When the device 200 is configured to implement some functionality, some component and/or components of the device, such as for example the at least one processor 202 and/or the at least one memory 204, may be configured to implement this functionality. Furthermore, when the at least one processor 202 is configured to implement some functionality, this functionality may be implemented using program code 206 comprised, for example, in the at least one memory 204.

The functionality described herein may be performed, at least in part, by one or more computer program product components such as software components. According to an embodiment, the device 200 comprises a processor or processor circuitry, such as for example a microcontroller, configured by the program code 206, when executed, to execute the embodiments of the operations and functionality described herein. Alternatively, or in addition, the functionality described herein can be performed, at least in part, by one or more hardware logic components. For example, and without limitation, illustrative types of hardware logic components that can be used include field-programmable gate arrays (FPGAs), application-specific integrated circuits (ASICs), application-specific standard products (ASSPs), system-on-a-chip systems (SOCs), complex programmable logic devices (CPLDs), graphics processing units (GPUs), or the like. The device 200 may be configured to perform method(s) described herein or comprise means for performing method(s) described herein. In one example, the means comprises the at least one processor 202, the at least one memory 204 including program code 206 configured to, when executed by the at least one processor 202, cause the device 200 to perform the method(s).

The device 200 may comprise, for example, a computing device such as for example a modulator chip, a demodulator chip, a baseband chip, a mobile phone, a tablet, a laptop, an intemet-of-things device, a base station, or the like. Although the device 200 is illustrated as a single device, it is appreciated that, wherever applicable, functions of the device 200 may be distributed to a plurality of devices, for example between components of a transmitter, a receiver, or a transceiver.

FIG. 3 illustrates an example of a sparse code multiple access (SCMA) scheme, according to an embodiment of the present disclosure. Among the various NOMA techniques, the example embodiments have been described in context of SCMA. It is however noted that the example embodiments may be applied to any suitable transmission scheme. SCMA may be based on low- density spreading code division multiple access (LDS-CDMA) as depicted in FIG. 3. Considering K users, each of the users may have a user-specific codebook with M = 2^μ codewords (complex vectors) of size N . Herein, parameter M represents the modulation order while N is the number of resource elements (REs) or subcarriers that each user uses to transmit m = log₂ M bits. A resource block (RB) may comprise a group of N consecutive REs, either in time or frequency direction, or both.

A m-size binary vector b_fc of the k- th user may be mapped to a N- size codeword x_k . The codeword x_k may comprise N complex symbols from a constellation, for example phase-shift keying (PSK) or quadrature amplitude modulation (QAM) constellation of order M. In the example of FIG. 3, each user has M = 8 = 2³ codewords of size N = 4. Each user binary vector (m = 3 bits) may be mapped to a codeword of four complex symbols to be transmitted over four physical layer resource elements, for example subcarriers. The mapping function may map the binary vector b₁ = Oil of the User 1 into the fourth codeword in Layer 1 to be transmitted as x_x, and so on. Consequently, the codewords may be transmitted by using a multicarrier waveform such as an orthogonal frequency division multiplexing (OFDM) waveform. In orthogonal multiple access (OMA), six users would need at least six REs to transmit the same data. A user may be understood as a human user or alternatively a user may be an application, an information stream, a layer, or the like. In general, a user may be a data source and the system may comprise K data sources.

FIG. 4 illustrates an example of a factor graph representation of a sparse code multiple access (SCMA) scheme, according to an embodiment of the present disclosure. This example is also provided for K = 6 and N = 4. At the receiver 130, the N- size vector of received signals from the K users sharing the same RB may be expressed as

where h_{n k} is the channel coefficient at the n-th RE of the k- th user, h is the additive Gaussian noise vector with covariance matrix s²I_N, and I_N is the N x N identity matrix. In the factor graph representation, each codebook layer may be represented by a variable node u_k . Each RE may be represented by a function node c_n. Therein, a given variable node u_k may be connected to a function node c_n through a factor graph edge if and only if the corresponding element in the SCMA N x K signature matrix S is non-zero. The signature matrix may comprise

Defming as the set of user indices that contribute to the

n-th RE and ] as the set of RE indices that are used by the

k- th user, the received signal at the n-th RE may be expressed as

where x_{n k} is the k- th user transmitted symbol over the n-th RE. The sparsity of the SCMA signature matrix enables the use of the message passing algorithm (MPA) that provides a near-optimal solution of the joint optimum maximum a posteriori probability (MAP) detection given by

where X vector is stacking the users’ codewords as and

the k- th codebook.

An MPA detector may alternately compute the information vectors

sent from the function nodes

to the variable nodes u_k (/c-th user) and the information vectors sent from the variable nodes u_k to the

function nodes c_n . Both vectors

^may be of size M and contain the reliability values for each of possible codewords

, the i-th element of ^may computed based on

The i-th element of may computed based on

In the above expressions p

is the k- th user a priori probability for the i-th codeword. Furthermore, X_{n i} is the (n, 0 -th element of X and x_k is the k- th column vector of X.

FIG. 5 illustrates examples of an effective channel with Doppler frequencies f_D = 0 KHz and f_D = 1 KHz, according to an embodiment of the present disclosure. In this simulation scenario, each SCMA encoded data was mapped to a resource block and MPA was performed for detection while leveraging the channel sparsity. In a frequency selective channel, the MPA performs well especially if the effective channel is block diagonal and the MPA detector is able exploit this structure as depicted in the left figure (no mobility, f_D = 0 KHz). However, when the channel becomes time-varying due to mobility, the orthogonality of the subcarriers (REs) is corrupted and the intercarrier interference (ICI) occurs, as illustrated on the right with f_D = 1 KHz. It is worth noting in case of f_D = 1 KHz, off-block diagonal elements appear in the effective channel and these may not be considered by the MPA detector.

FIG. 6 illustrates examples of an effective channel with Doppler frequencies f_D = 5 KHz and f_D = 2 KHz, according to an embodiment of the present disclosure. The simulation scenario is similar to FIG. 5, but the Doppler frequency is higher. In this example of a high mobility regime, the Doppler frequency spread makes the MPA scheme inefficient. It is observed that the off- block diagonal elements become more accentuated due to ICI.

Example embodiments of the present disclosure may be applied in NOMA transmission and reception over fast time-varying channels. This is beneficial since a key performance indicator for the Beyond 5G networks is high mobility. Using a NOMA scheme, such as for example SCMA, each encoded data may mapped to a resource block, for which an MPA detector is applied at the received by leveraging the codebook sparsity. However, as noted above, Doppler frequency spread induced by the mobility may destroy the sparsity of SCMA and make MPA detection impractical in some applications. Moreover, ICI may occur among several resource blocks and therefore performing MPA detection over all subcarriers (by taking into account the whole effective channel matrix, not only the block-diagonal elements) may be highly expensive in terms of implementation complexity and energy efficiency. To overcome these issues, the embodiments of the present disclosure provide NOMA schemes that are suitable for doubly selective channels (both frequency and time selectivity) and maintain affordable receiver complexity. Furthermore, a flexible transmitter, receiver, and transceiver implementations are disclosed that boost the performance while keeping the implementation cost low compared to MPA-based solutions.

According to an embodiment, a NOMA scheme that relies on a linear codebook enabling the use of linear receiver is provided. This reduces drastically the implementation cost. Furthermore, flexible transmitter, receiver, or transceiver implementations based on LVDM are disclosed. This improves performance of the linear codebook based NOMA scheme, while keeping its implementation cost low compared to maximum likelihood (ML) based solutions such as MPA. Therefore, LVDM transmitters, receivers, and transceivers that deal with doubly selective channels are disclosed. The receiver 130 may compute, for example referring to a specific metric such as the mean squared error (MSE), an optimal or improved value of the LVDM signature roots radius, a_opt, for each user. The receiver 130 may feed the determined radius back to the transmitter 110 during the training phase to build the precoder and modulator blocks. However, in time varying channels, the following phenomena may degrade the performance: 1) outdated feedback signaling that breaks the signature roots’ optimization, and 2) ICI that makes the MPA inadequate. Therefore, it is desired to enable receiver implementation that takes into account these factors with affordable implementation complexity.

FIG. 7 illustrates an example of multi-dimensional SCMA mapping with an 8-point codebook, according to an embodiment of the present disclosure. Linearization of the codebook may be performed based on the symmetry property of the codebook. In a symmetrical codebook, the complementary (bitwise inverted) binary words, for example 100 and Oil, may be mapped to the opposite codewords. Two example of such codebooks, corresponding to first and second non-zero entries, are depicted in FIG. 7. Similarly, examples of symmetrical codebooks, mapping for example complementary binary words 1101 and 0010 to opposite codewords of a 16-point codebook are illustrated in FIG. 8.

A codebook of M codewords may be generated through a linear transformation (matrix multiplication) of M different log₂ (M) -vectors with antipodal entries (e.g. ±1 ). The transmitter 110 may therefore obtain an antipodal input sequence for the k- th user. The antipodal input sequence may be obtained for example based on mapping the k- th user’s binary input vector b_fc of size log₂(M) to an antipodal a binary shift phase keying (BPSK) modulation through a linear transformation giving where 1 is the all-ones

vector. The generated (precoded) codeword transmitted may be

determined based on

where

precoding matrix of the k- th user. The transmitter 110 may therefore generate the precoded codeword based on multiplying the antipodal input sequence by the precoding matrix

The precoding matrix S_fc, may be based on matrix B, which may be of size log₂ (M) x M . Columns of matrix B may comprise M different antipodal vectors. Matrix B may therefore be an antipodal matrix. For example, the columns of matrix B, given in ascending order, may comprise the BPSK vectors corresponding to the M binary representations of 0, 1, ... , M — 1. Herein, the first row of B may represent the least significant bit (LSB). For example, for M = 4, the corresponding matrix B may be expressed as

where the columns form the left to the right represent antipodal binary vectors corresponding to values 0, 1, 2, and 3, respectively.

Furthermore, let X_fc be the N x M matrix representing the codebook of the k- th user (codebook matrix), whose columns correspond to the M different antipodal vectors. The codebook matrix X_fc may comprises M codewords of complex symbols. For example, given in ascending order, the columns ofX_fc may comprise the codeword vectors x_fc corresponding to the M binary representations of 0, 1, ... , M — 1 . Consequently, we have

According to the definition of and it follows that

Hence, the precoding matrix S_fc may be determined based on

The precoding matrix S_fc may be therefore based on multiplication of the codebook matrix X_fc and an antipodal matrix B^T. Elements of the precoding matrix may be further inversely proportional to the number of codewords M in the codebook. The transmitter 110 may apply a preconfigured precoding matrix S_fc. The transmitter 110 may for example retrieve the precoding matrix S_fcfrom the memory of the transmitter 110. Alternatively, the transmitter 110 may receive the precoding matrix S_fc from another device, for example as part of signaling information. It is also possible that the transmitter 110 determines the precoding matrix S_fc based on a preconfigured or received codebook matrix X_fc. The examples may be generalized to applications with multiple users and corresponding codebook matrices and/o precoding matrices.

Since there may be a plurality of users, the transmitter 110 may obtain a plurality of the antipodal input sequences b_fc corresponding to a plurality of users. Different users may be associated with different codebook matrices. The transmitter 110 may further obtain a plurality of the precoding matrices S_fc corresponding to the plurality of users. The precoding matrix of a k- th user may be based on multiplication of the codebook matrix of the k- th user and the antipodal matrix. The antipodal matrix may be the same for the K users. A plurality of precoded codewords may be then generated for each of the plurality of users. Each precoded codeword of the k- th user may be based on multiplication of a different subset of an antipodal input sequence b_fc of the k- th user and the precoding matrix S_fc of the k-th user. The signal may be then generated based on a concatenation of the precoded codewords for each of the K users. It is further noted that the number of users may be higher than a number of complex symbols of the codewords of the plurality of the codebook matrices. This enables overloading of information to the available resource elements according to the NOMA scheme.

The received signal y may be generally expressed as

where b_k e (—1, 1}^1OS2^(m) comprises the k-th user’s BPSK word(s) and S_fc is k-th user precoding matrix. The k-th user diagonal channel matrix diag [h_{l k}, ··· , h_{N k} }, may be denoted by H_fc. The receiver 130 may concatenate the vectors b_fc into a K log₂ (M) -size vector b. The received signal may be therefore rewritten as

The overall N x K log₂(M) channel matrix H, may comprise

The K users may be configured to transmit, for example continuously, over P RBs of size N . Consequently, becomes of size while each

) symbols in may be transformed into SCMA codewords of size N

through to form together the concatenated vector x_fc of size NP, comprising

where <8) denotes the Kronecker product, I_P is the P x P identity matrix, and S_fc is the k- th block diagonal precoding matrix. The linearization of the codebook enables the receiver 130 to perform linear equalization of the received signal. This enables a significantly reduced complexity at the receiver 130.

FIG. 9 illustrates an example of an LVDM transmitter of a k- th user applying a linearized codebook, according to an embodiment of the present disclosure. LVDM provides a generalization of the OFDM waveform, yet offering a more flexible implementation. The LVDM transmitter 900 may receive as input the binary (antipodal) vector b_fc. The binary vector b_fc may be optionally divided into a plurality of subsets (subvectors) that may be provided as input to corresponding precoder(s) 902. Each precoded codeword of the k-th user may be therefore obtained based on multiplication of a different subset of the binary vector b_fc and the precoding matrix S_fc of the k-t user. The LVDM transmitter 900 may further comprise an LVDM modulator 904, which may apply a modulation matrix G_fc to the output(s) of the precoder(s) 902. An output of a precoder 902 may comprise a precoded codeword. An output of the LVDM modulator 904 may comprise an LVDM modulation symbol. The LVDM transmitter 900 may therefore concatenate the precoded codewords of the k-t user and generate a modulation symbol based on multiplying the concatenated precoded codewords by the modulation matrix G_fc. The modulation matrix may be user-specific. One or more of the K users may be therefore associated with different modulation matrices.

The modulation matrix G_fc may comprise

where the signature roots p_{n k} of the modulation matrix of the /c-th user are

with radius a_k of the /c-th user, and where K_k is a normalization factor of the k- th user. The normalization coefficient K_k may be applied to comply with the k- th user transmit power constraint given by Trace

NP. The normalization coefficient K_k may be determined based the radius a_k of the k- th user, for example based on

Each modulation symbol s_k may be entailed by L zeros before transmission. For example, zero-padding may be inserted at the end of the modulation symbol, as illustrated in FIG. 9. The LVDM transmitter 900 may further apply a parallel- to-serial transform 906 to the generated modulation symbol.

FIG. 10 illustrates an example of an LVDM-based NOMA system, according to an embodiment of the present disclosure. The system includes K transmitters 110 and a receiver 130. The transmitters 110 may be located within a single device 1010, which may be alternatively embodied as a system comprising multiple transmitters 110. The transmitters 110 may be also embodied at separate devices, for example at multiple UEs (user equipment). The receiver 130 may be also located at a separate device, for example a base station. However, a transceiver device or system may comprise both the transmitter(s) 110 and the receiver 130. The transmitters 110 may include, for example in their memories 204, NOMA codebooks 1 ... K , for example codebook matrices X_fc, for corresponding users 1 ... K. The transmitters 110 may apply the NOMA codebook(s) at corresponding precoders 1012 to precode the binary input sequences b_k of the K users. The precoded codeword(s) of each user may be processed at corresponding LVDM modulators 1014 based on the user-specific modulation matrices G_fc , as described above, and transmitted separately, for example via multiple antennas. Considering that each user is equipped with a LVDM modulator 1014, the modulated signal of the k- th user may be obtained by

received signal at the receiver 130, for example a base station, may comprise

where matrix with at most L + 1 nonzero first sub

diagonals (including the main diagonal) that represents the convolution with the k- th user time-varying channel impulse response, including the transmit-receiver filters, and h is the corresponding AWGN noise as described above. The received signal may be processed by receive filter 1030.

If the receiver 130 comprises an OFDM demodulator, the frequency domain received signal at the receiver 130 may be expressed as

where

matrix obtained by appending the NP x NP DFT matrix F with its L first column as

The receiver 130 may therefore comprise a DFT module 1032 for calculating the a DFT of the received signal with the appended DFT matrix. The frequency domain received signal at the base station may be rewritten as

Since the codebook has been linearized, low-complex linear based detection may be applied at the receiver 130. However, linear equalizers such zero-forcing (ZF) and MMSE may not be optimal in case of NOMA overloading, for example because the number of unknown variables may be higher than observations in the system. It is however possible to improve detection performance by applying linear based successive interference cancellation receivers, such as for example ordered successive interference cancellation (OSIC), in an LVDM/ OFDM based NOMA system. In the following, an overall frequency domain channel matrix including the NOMA precoding matrices is derived.

Furthermore, an minimum mean squared error (MMSE) based OSIC receiver that enables to simultaneously address the NOMA coding and ICI terms is disclosed. The frequency domain received signal may comprise

By stacking the BPSK vectors b _k, k = 1, ···, K, in b = [b^, bf, •••,bg ]^T of size PK log ₂(M), the frequency domain received signal y may be rewritten as

where the matrix Q is the effective overall frequency

domain channel matrix. be the order of the indices of the

detected symbols through MMSE-OSIC equalization in the vector b after (Z — 1) iterations. At the Z-th iteration

, the linear system to be solved may comprise

Herein, b_z. may comprise the binary (BPSK) vector b after removing the entries indexed by that have been detected in the (i — 1) previous iterations.

Matrix Q_z. may be determined by removing the corresponding columns of Q. The vector y_z. is the received vector y after removing the contribution the previously detected symbols b_z , ··· , b_z.-i where b, is the Z-th element of b.

It is worth noting that to be estimated may be a real-valued vector and

the widely linear (WL) MMSE may be applied, which reduces to the linear MMSE equalization when b_z. satisfies Linear equalization may

therefore comprise linear MMSE equalization. In the following example, and the WL-MMSE equalization processing may

comprise one or more of the following: a) At the Z-th iteration, may be carried

out using W_z. given by

Matrix may comprise the noise covariance matrix in the frequency

domain. b) The index Z_j of the selected symbol to be detected during the i -th iteration may be determined based on

where t comprises

Herein, H_{j i}nay comprise

may be given by

and q₍ may be given by

c) a hard decision b_z. may be determined based on

Complexity order of the disclosed MMSE-OSIC detector performing a WL- MMSE equalization may be assessed based on complexity during each iteration. The receiver 130 may perform PK log₂ (M) times the matrix W_z.. Computation of the noise covariance matrix, given by

, and its inverse R

computation may be performed in offline and therefore computations of R and R^_1are omitted the complexity analysis.

At the i-th iteration, the size of the matrix

1)) and the WL-MMSE operation cost is

. Since the number of OSIC iterations is the complexity order of the

disclosed MMSE-OSIC detector may be determined by

Therefore, the complexity order of the disclosed MMSE-OSIC detection is given by is the binary word size.

The following table summarizes the complexities of different types of receivers. Complexity orders have been derived as functions of M = 2^m, the codebook size, rather than m. To provide a fair comparison, the complexity orders of the non-LVDM based receivers have been multiplied by the number of SCMA codes P.

In combination with the codebook linearization, LVDM enables to improve performance of a linear receiver in doubly-selective channels. The linear receiver, unlike an MPA-based receiver, will scan the whole effective channel matrix and thus provide better performance than MPA in doubly selective channels. Furthermore, LVDM enables to boost the performance and overcome its losses by enabling to avoid use of a high-complexity maximum likelihood (ML) receiver and enabling use of a receiver that is has lower complexity but better energy efficiency.

FIG. 11 illustrates an example of a pilot pattern for use with LVDM-NOMA symbols, according to an embodiment of the present disclosure. The pilot pattern may be used to estimate the transmission channel 120 for LVDM- NOMA symbols. An LVDM-NOMA frame may comprise one or more LVDM- NOMA symbols 1101, illustrated with white color. The LVDM-NOMA frame may further comprise one or more training sequences (pilot vectors) 1102, illustrated with diagonal dashes, inserted within the LVDM-NOMA symbols 1101. The training sequences 1102 may be inserted periodically within the LVDM-NOMA symbols 1101, for example with a period of M_s symbols. FIG.

11 illustrates an example of pilot patterns for K = 6 users using dedicated LVDM-NOMA symbols, where M_s = N_p = 4 and every pilot symbol vector is followed by (M_s - 1) LVDM-NOMA symbols (vectors). Training sequences 1102 may be inserted every M_s transmitted symbol vectors. For example, one training sequence vector 1102 may be followed by ( M_s — 1) LVDM-NOMA symbols 1101. Hence, each user k may periodically transmit a training sequence u_fc of PN log ₂(M) samples. The training sequences of the users may be transmitted simultaneously, as in the example of FIG. 11, or such that the pilot sequences 1102 of the users do not overlap in time, as in the example of FIG. 17. The length of the LVDM-NOMA frame, or in general a sequence of training sequences and modulation symbols used for channel estimation and prediction (CEP), may be therefore N_pM_s. A training sequence may be tailed by L zeros. The training sequences 1102 may therefore comprise L zeros at an end of each training sequence 1102. The training sequences 1102 may be different for different users. This enables separation of the users at the receiver 130.

With mobility the channel impulse response (CIR) of each user may vary even within one LVDM symbol, making the feedback signaling exchange as well as the detection processing challenging. The receiver 130 may therefore apply a joint channel estimation and prediction (CEP) algorithm, where some values of the radius a_{k opt} may be predicted at the receiver 130. The receiver 130 may send the predicted values to the transmitter(s) 110. A transmitter 110 may use the predicted radius to generate subsequent LVDM symbol(s) for the /e-th user. The receiver 130 may further use the channel estimation to detect the received LVDM users’ symbols with advanced processing to overcome ICI, as will be further described below.

Let T_max and f_D be the delay spread and the Doppler spread of the radio channel 120, respectively. The sampling period (sampling time) of the receiver 130 may be denoted by T_s. It is noted that that both T_max and f_D may be measured by the receiver 130 experimentally in practice.

Channel estimation may be based on a basis expansion channel model (BEM), where the /c-th user’s channel impulse response h_{p k}(t, t) is presented for t e tfN_cT_s, (z + 1 )N_CT_S) (in the z^ίH channel coherence time) using that remain invariant per block but are allowed

to change with z, and that capture the time variation but are

common for all z, where

, and [-1 represents the integer ceiling operator, u>_q = — . Consequently, )

may be approximated as

and where \ and L- J represents the integer floor operator. In this

example, the number of channel taps is L + 1. The number of samples in each channel impulse response may be equal to N_c (spanning in time domain

the whole frame duration).

The receiver 130 may receive the training sequences 1102. The receiver 130 may then perform the CEP algorithms over N_p received training sequences. The number of training sequences may satisfy where N_c is the

coherence period of the radio channel 120. When the p-th received LVDM- NOMA symbol corresponds to the m-th training sequence (p = mM_s, m e

the CEP module may utilize the N_p received training sequences

, to predict the channel time evolution over

the next ( M_s — 1) LVDM-NOMA symbols, to determine which the

receiver 130 may feed back to the transmitter 110. The transmitter 110 may configure the precoding and modulation blocks based on the received feedback. The receiver 130 may further estimate the channel taps during the last received ( M_s — 1) LVDM-NOMA symbols. The estimate may be provided to feed the equalizer for detection.

FIG. 12 illustrates an example of applying training sequences for channel estimation and prediction, according to an embodiment of the present disclosure. The channel estimation and prediction may be performed at the CEP module 1034 of the receiver 130. The received training sequence may be expressed as

where comprises the / x C matrix that captures the k- th time-varying CIR

evolution during the n-th transmit symbol, given by

where

where with N_c representing the coherence time of the radio

channel 120. Matrix may comprise a lower triangular Toeplitz matrix

whose first column comprises . Based on algebraic

manipulation, the received sequence may be expressed as

Matrix ower triangular Toeplitz matrix whose first

column

Vector

T

may comprise the m-th training sequence of the k- th user. Vectors c

may comprise the coefficients Parameter J may be equal to the

length of the training sequences. Each training sequence may therefore have the same length J.

The CEP module 1034 may estimate and predict the radio channel 120 using for example the BEM approximation. The CEP module 1034 may stack the iV_preceived training sequences into a vector of received training sequences F_m, which may of size and expressed by

In the example of FIG. 12, the receiver 130 may stack the four training sequences 1102 indicated to be involved in channel estimation/prediction. The stacked vector may comprise the current received multicarrier (MC) symbol and previously received multicarrier symbols. The received signal may be

of the form

where comprises additive noise. The additive noise vector may

comprise

Matrix _h may

compnse

The estimate of the radio channel 120 may be then estimated using a linear MMSE estimator based on matrix The linear MMSE estimator may be

applied to f_m to get

where R_c = E{cc^w] is the channel covariance matrix where Trace(R_c] = K. It noted that R_c depends on the power delay profile (PDP) of the channel model, which is known at the receiver 130.

Based on the coefficients

1)], estimated for example through MMSE as above, the CEP module 1034 may predict the radio channel 120 for the subsequent (for example next) LVDM- NOMA symbol(s) as illustrated in FIG. 12. The predicted time-varying CIRs of the users, which may be provided to the transmitter 110 for building the subsequent LVDM-NOMA symbol(s), may be determined for ( l, k ) e based on

with The estimated time-varying

CIRs of the users , which may be provided to an equalizer of the detector 1038 to detect the

(M_s 1) last received LVDM-NOMA symbols, may be determined

based on

with The CEP module 1034 may

therefore determine an estimate and a predicted estimate of the radio channel

120 for subsequent modulation symbol(s) based on the received training sequences. The determined channel estimate (denoted in FIG. 10 by nd

the predicted estimate of the radio channel 120 may be provided to the optimization block 1036.

The optimization block 1036 may use the estimate and/or prediction of the radio channel 120 for determining LVDM parameter(s) for subsequent symbol(s). For example, the optimization block 1036 may determine an updated radius of the k- th user based on the predicted estimate of the radio

channel 120. The optimization block, or in general the receiver 130, may transmit an indication of the updated radius a_{k opt} of the k- th user to the transmitter 110.

According to an embodiment, the updated (optimized) radius may be determined using a metric such as for example erfc. Hence, the updated radius may be determined for example based on

where

and is the k -th user frequency domain channel coefficient at

subcarrier n. Parameter PN may be equal to the size of the DFT matrix. Alternatively, a machine learning algorithm may be used for determining the updated radius for example based on a stochastic gradient descent

method.

Hence, upon reception of the training sequences, the channel estimation and prediction (CEP) module 1034 may determine an estimate of the radio channel 120 for the modulation symbols based on the received training sequences. The detector 1038 may then demodulate the received signal and perform linear equalization of the demodulated signal based on the determined channel estimate. The linear equalization is enabled by linearization of the codebook.

Embodiments of the present disclosure therefore provide a NOMA scheme that uses linear codebooks to enable linear receivers to deal with doubly selective channels in high mobility. In one embodiment, LVDM may be used at the transmitter side to allow a flexible implementation while boosting the performance compared to OFDM-based solutions. Frequency domain equalization may be used to reduce receiver complexity. Therein, a joint channel estimation and prediction may be applied. The channel estimation part may feed the detector (equalization) while the channel prediction output may be used to configure the transmitter blocks for the next LVDM symbols for each user. The complexity of transmitter, receiver, or transceiver implementation is however maintained at an affordable level.

In a first step, joint channel estimation and prediction (CEP) may be performed, for example as illustrated in FIG. 10, where the dashed arrows illustrate providing the updated radius values a_{k opt} as feedback to each transmitter 110. A joint channel estimation and prediction approach may be applied, where the channel estimation entity feeds the frequency domain equalizer to detect the (actual) received LVDM users’ symbols and a prediction entity is exploited to configure the transmitter blocks (e.g. precoder and modulator) for the next transmission slots. The optimization of the radius values ^a _k,opt may be performed based on the channel state information (from channel estimation) and an optimization metric (e.g. MSE as provided above). Furthermore, refinement of the determined radius values may be applied.

The signaling feedback or refined signature roots), derived from the

prediction entity, to the k- th transmitter 110 enables adaptation of the modulation and precoding blocks to transmitter-specific radio channel conditions. Furthermore, the disclosed training sequences enable separation of the users’ channel impulse responses at the receiver 130.

In a second step, detection of the transmitted signal may be performed. The receiver 130 may for example apply the fast Fourier transform (FFT) for demodulation to keep the implementation cost low. A frequency domain equalization (method) may be used to deal with the doubly selective radio channel. Therein, a linear detector may be used and optionally enhanced with iterative processing.

Performance results for the disclosed embodiments are discussed below. First, performance results in frequency and doubly selective channel are provided. Performance of LVDM- and OFDM-based NOMA are discussed in terms of average bit error rate (BER), since the analysis is provided for a multiuser scheme, as a function of the signal-to-noise ratio (SNR). Sensitivity to the channel estimation and a_opt estimation errors is also presented.

FIG. 13 illustrates an example of average bit-error rate (BER) for SCMA MPA and LVDM-based NOMA in frequency selective channels, according to an embodiment of the present disclosure. The comparison between SCMA and LVDM-based NOMA is provided in a frequency selective channel (f_D = 0 Hz). LVDM-based NOMA scheme clearly outperforms SCMA, which gets saturated (BER floor) in high SNRs.

FIG. 14 illustrates an example of average BER for OFDM- and LVDM- based NOMA in doubly selective channels, according to an embodiment of the present disclosure. It is observed that LVDM-based NOMA still clearly outperforms the OFDM-based NOMA at high SNR. Furthermore, by using the proposed WL-MMSE-OSIC detector, OFDM-based NOMA (SCMA) performance got ameliorated even in the doubly selective channel. However, LVDM-based NOMA still harvests the frequency diversity in the channel and outperforms the OFDM-based NOMA. For example, the performance results show that LVDM outperforms OFDM by 5 dB at BER = 10^-5.

In the simulations of FIG. 13 and FIG. 14, the number of resource elements (subcarriers) is C = 64 and L = 16. Each resource block (RB) encloses four resource elements and therefore the number of resource blocks is 16. Consequently, there are also 16 NOMA codewords. For the doubly selective channels of FIG. 14, the maximum Doppler spread was f_D = 1 KHz. The subcarrier spacing was Af = 30 KHz. For both simulations quadratic phase- shift keying (QPSK) modulation was used and therefore M = 4.

FIG. 15 illustrates an example of average BER for OFDM- and LVDM- based NOMA in a 3GPP Extended Vehicular (EVB) channels, according to an embodiment of the present disclosure. It is shown that LVDM-based NOMA outperforms the OFDM-based NOMA. For example, almost 8 dB of SNR gains have been achieved at BER = 1CT⁵. In this simulation, L = 39.

FIG. 16 illustrates an example of average BER for OFDM- and LVDM- based NOMA in a 3GPP Tapped Delay Line C (TDL-C) channels, according to an embodiment of the present disclosure. It is shown that LVDM-based NOMA again outperforms the OFDM-based NOMA. For example, almost 7 dB of SNR gains have been achieved at BER = lO^-5. In this simulation, L = 20.

In the simulations of FIG. 15 and FIG. 16, the following set of simulation parameters was used: C = 64, carrier frequency f_c = 3.5 GHz, velocity v = 200 Km/h, Doppler spread f_D = 648 Hz, and subcarrier spacing Af = 30 KHz. QPSK modulation and same NOMA configurations were considered.

FIG. 17 illustrates an example of pilot patterns for a plurality of users, according to an embodiment of the present disclosure. As discussed above, a plurality of training sequences may be inserted periodically within the plurality of modulation symbols. Furthermore, as illustrated in FIG. 17 the training sequences for each of the users may be transmitted with a time shift between the training sequences of the users. A gap of zero symbols may be provided for each of the k users between successive bursts of non-zero symbols of any two of the users. In this example, N_p = 6 and M_s = 5. In order to enable the CEP algorithm 1034 to separate the users' channels, the ( K = 6) users may transmit different training sequences, for example given by the pattern of FIG. 17. Each user may therefore have a contiguous portion of non-zero symbols within the training sequence. Furthermore, the non-zero symbols of the different users may not overlap in time. This may be implemented for example by shifting the locations of the non-zero symbols for each user by a predetermined (user- specific) amount of symbols within their training sequences. In the example of FIG. 17, each of the Users 1 to 6 has four non-zero contiguous symbols among the 64 symbols of the training sequence in the time domain. The values of the non-zero samples may be determined for example as follows:

In general, the values of the non-zero samples may be selected such that the non-zero samples are orthogonal or have low cross-correlation among the set of users. The values of the non-zero samples may be for example drawn from Hadamard sequences, which provide code orthogonality, or Zadoff-Chu sequences, which have good correlation properties.

FIG. 18 illustrates an example of sensitivity to channel estimation and radius estimation errors for LVDM-based NOMA, according to an embodiment of the present disclosure. It is noted that while the simulation results of FIGs. 13 to 16 have been obtained using perfect channel state information (CSI) at the receiver 130, this figure provides performance results using the disclosed CEP algorithm at the receiver 130. Performance results with the pilot pattern of FIG. 17 are provided for the Extended Typical Urban (ETU) channels. In this simulation, C = 64, carrier frequency f_c = 3.5 GHz, velocity v = 200 Km/h, Doppler spread f_D = 648 Hz, and subcarrier spacing Af = 30 KHz. Again, QPSK modulation and the same NOMA configurations are considered. It is worth reminding that OFDM performance depends on the channel estimation quality only while LVDM performance depends on both the channel estimation and prediction qualities. However, FIG. 18 shows that LVDM-based NOMA keeps outperforming OFDM-based NOMA even under imperfect channel estimation. The LVDM scheme leverages the flexibility that helps the equalizer to overcome the mismatches. Therefore, the simulation results show that the embodiments of the present disclosure improve transmission performance in both frequency and doubly-selective channels.

FIG. 19 illustrates an example of a method 1900 for generating a signal, according to an embodiment of the present disclosure. At 1901, the method may comprise obtaining an antipodal input sequence.

At 1902, the method may comprise obtaining a precoding matrix. The precoding matrix may be based on multiplication of a codebook matrix and an antipodal matrix, wherein the codebook matrix comprises M codewords of complex symbols, and wherein the antipodal matrix comprises M different antipodal vectors.

At 1903, the method may comprise generating a precoded codeword based on multiplying the antipodal input sequence by the precoding matrix.

FIG. 20 illustrates an example of a method 2000 for receiving a signal, according to an embodiment of the present disclosure.

At 2001, the method may comprise receiving the signal. The signal may comprise at least one precoded codeword generated based on multiplication of an antipodal input sequence by a precoding matrix. The precoding matrix may be based on multiplication of a codebook matrix and an antipodal matrix, wherein the codebook matrix comprises M codewords of complex symbols, and wherein the antipodal matrix comprises M different antipodal vectors.

At 2002, the method may comprise demodulating the signal and performing linear equalization of the demodulated signal.

Further features of the methods directly result from the functionalities and parameters of the methods and devices, for example the transmitters 110, 900 the receiver 130, or device 200, as described in the appended claims and throughout the specification and are therefore not repeated here.

A device or a system may be configured to perform or cause performance of any aspect of the method(s) described herein. Further, a computer program may comprise program code configured to cause performance of an aspect of the method(s) described herein, when the computer program is executed on a computer. Further, the computer program product may comprise a computer readable storage medium storing program code thereon, the program code comprising instruction for performing any aspect of the method(s) described herein. Further, a device may comprise means for performing any aspect of the method(s) described herein. According to an example embodiment, the means comprises at least one processor, and at least one memory including program code, the at least one processor, and program code configured to, when executed by the at least one processor, cause performance of any aspect of the method(s).

Any range or device value given herein may be extended or altered without losing the effect sought. Also, any embodiment may be combined with another embodiment unless explicitly disallowed.

Although the subject matter has been described in language specific to structural features and/or acts, it is to be understood that the subject matter defined in the appended claims is not necessarily limited to the specific features or acts described above. Rather, the specific features and acts described above are disclosed as examples of implementing the claims and other equivalent features and acts are intended to be within the scope of the claims.

It will be understood that the benefits and advantages described above may relate to one embodiment or may relate to several embodiments. The embodiments are not limited to those that solve any or all of the stated problems or those that have any or all of the stated benefits and advantages. It will further be understood that reference to 'an' item may refer to one or more of those items. Furthermore, references to ‘at least one’ item or ‘one or more’ items may refer to one or a plurality of those items.

The operations of the methods described herein may be carried out in any suitable order, or simultaneously where appropriate. Additionally, individual blocks may be deleted from any of the methods without departing from the scope of the subject matter described herein. Aspects of any of the embodiments described above may be combined with aspects of any of the other embodiments described to form further embodiments without losing the effect sought.

The term 'comprising' is used herein to mean including the method, blocks, or elements identified, but that such blocks or elements do not comprise an exclusive list and a method or device may contain additional blocks or elements.

It will be understood that the above description is given by way of example only and that various modifications may be made by those skilled in the art. The above specification, examples and data provide a complete description of the structure and use of exemplary embodiments. Although various embodiments have been described above with a certain degree of particularity, or with reference to one or more individual embodiments, those skilled in the art could make numerous alterations to the disclosed embodiments without departing from the scope of this specification.

Claims

1. A device (110, 200, 900, 1010) for generating a signal, the device (110, 200, 900, 1010) configured to: obtain an antipodal input sequence; obtain a precoding matrix, wherein the precoding matrix is based on a multiplication of a codebook matrix and an antipodal matrix, wherein the codebook matrix comprises M codewords of complex symbols, and wherein the antipodal matrix comprises M different antipodal vectors; and generate a precoded codeword based on multiplying the antipodal input sequence by the precoding matrix.

2. The device (110, 200, 900, 1010) according to claim 1, wherein the antipodal matrix comprises M different antipodal vectors corresponding to binary representations of values 0 to M — 1 in an ascending order.

3. The device (110, 200, 900, 1010) according to claim 1 or 2, wherein the precoding matrix S_fc of the k- th user comprises

where X_fc is the codebook matrix and B^Tis the antipodal matrix.

4. The device (200, 1010) according to any of claims 1 to 3, further configured to: obtain a plurality of the antipodal input sequences corresponding to a plurality of users associated with a plurality of the codebook matrices; obtain a plurality of the precoding matrices corresponding to the plurality of users, wherein a precoding matrix of a k- th user is based on a multiplication of a codebook matrix of the k- th user and the antipodal matrix; and generate a plurality of precoded codewords for each of the plurality of users, wherein each precoded codeword of the k- th user is based on a multiplication of a different subset of an antipodal input sequence of the k- th user and the precoding matrix of the k- th user; and generate the signal based on a concatenation of the plurality of precoded codewords for each of the plurality of users, wherein a number of the plurality of users is higher than a number of complex symbols of codewords of the plurality of the codebook matrices.

5. The device (200, 1010) according to claim 4, further configured to: generate a modulation symbol (1101) based on multiplying the concatenated plurality of precoded codewords of the k- th user by a modulation matrix G_fc of the k- th user, wherein

where signature roots p_{n k} of the modulation matrix of the P-th user are

with radius a_k of the P-th user, and where K_k is a normalization factor of

the k- th user; and insert a zero-padding at an end of the modulation symbol (1101).

6. The device (200, 1010) according to claim 5, further configured to receive an indication of the radius a_k of the P-th user or an indication of the signature roots p_{n k} of the modulation matrix of the P-th user from a receiver (130).

7. The device (200, 1010) according to claim 6, further configured to determine the normalization factor K_k of the P-th user based on the radius a_k of the k- th user.

8. The device (200, 1010) according to any of claims 5 to 7, further configured to: generate a plurality of the modulation symbols (1101) for the k- th user; and insert the zero-padding at an end of each of the plurality of modulation symbols; (1101) and insert a plurality of training sequences (1102) periodically within the plurality of modulation symbols (1101), wherein the plurality of training sequences (1102) comprises L zeros at an end of each training sequence (1102).

9. The device (200, 1010) according to claim 8, wherein the plurality of training sequences (1102) are different for each of the plurality of users.

10. The device (200, 1010) according to claim 8 or 9, further configured to transmit the plurality of training sequences (1102) for each of the plurality of users with a time shift between training sequences (1102) of the plurality of users.

11. A device (130, 200) for receiving a signal, the device configured to: receive the signal, wherein the signal comprises at least one precoded codeword generated based on a multiplication of an antipodal input sequence by a precoding matrix, wherein the precoding matrix is based on a multiplication of a codebook matrix and an antipodal matrix, wherein the codebook matrix comprises M codewords of complex symbols, and wherein the antipodal matrix comprises M different antipodal vectors; and demodulate the signal and perform a linear equalization of the demodulated signal.

12. The device (130, 200) according to claim 11, wherein the antipodal matrix comprises M different antipodal vectors corresponding to binary representations of values 0 to M — 1 in an ascending order.

13. The device (130, 200) according to claim 11 or 12, further configured to append a discrete Fourier transform matrix with L first columns of the discrete Fourier transform matrix, wherein the demodulation of the signal is based on the appended discrete Fourier transform matrix.

14. The device (130, 200) according to claim 13, wherein the signal comprises a plurality of modulation symbols (1101) of a k- th user of a plurality of users, the plurality of modulation symbols (1101) generated based on a multiplication of a plurality of the precoded codewords by a modulation matrix G_k of the k- th user, wherein

where signature roots p_{n k} of the modulation matrix of the k-th user are

_k with radius a_k of the k-th user, and K_k is a normalization factor of the k-th user.

15. The device (130, 200) according to claim 14, further configured to: receive a plurality of training sequences (1102) located periodically within the plurality of modulation symbols (1101) of the k- th user, wherein the plurality of training sequences (1102) comprises L zeros at an end of each training sequence (1102); and determine an estimate of a radio channel for the plurality of modulation symbols (1101) based on the received plurality of training sequences (1102), wherein the linear equalization of the demodulated signal is based on the estimate of the radio channel.

16. The device (130, 200) according to claim 15, further configured to: stack the plurality of received training sequences (1102) into a vector of received training sequences r_m, wherein the vector of received training sequences is of the form

where is additive noise, c =

Q comprises a q- th coefficient of

a k- th delay tap of a Fourier basis expansion of the radio channel, and matrix

comprises

a coherence time of the radio channel, / is a length of the plurality of training triangular Toeplitz matrix whose first column is is a training sequence (1102) of the /c-th user, N_p is

a number of the plurality of training sequences (1102), and is a number of

modulation symbols (1101) between training sequences (1102); and determine the estimate of the radio channel based on a linear minimum mean square estimator based on the matrix

17. The device (130, 200) according to claim 15 or 16, further configured to: determine a predicted estimate of the radio channel for at least one subsequent modulation symbol based on the plurality of received training sequences (1102); determine updated signature roots of the modulation matrix of the k- th

user based on the predicted estimate of the radio channel; and transmit an indication of the updated signature roots p_{n k} of the modulation matrix of the k- th user to a transmitter (110, 1010).

18. The device (130, 200) according to any of claims 15 to 17, further configured to: determine a predicted estimate of the radio channel for at least one subsequent modulation symbol based on the plurality of received training sequences (1102); determine an updated radius a of the k- th user based on the predicted estimate of the radio channel; and

transmit an indication of the updated radius a_{k opt} of the k- th user to a transmitter (110, 1010).

19. The device(130, 200) according to claim 18, further configured to determine the updated radius a_{k opt} of the k- th user based on

where

is a frequency -domain coefficient of the predicted estimate of the

radio channel of the k- th user at subcarrier n, and wherein PN is a size of the discrete Fourier transform matrix.

20. A method (1900) for generating a signal, the method comprising: obtaining an antipodal input sequence; obtaining a precoding matrix, wherein the precoding matrix is based on a multiplication of a codebook matrix and an antipodal matrix, wherein the codebook matrix comprises M codewords of complex symbols, and wherein the antipodal matrix comprises M different antipodal vectors; and generating a precoded codeword based on multiplying the antipodal input sequence by the precoding matrix.

21. A method (2000) for receiving a signal, the method comprising: receiving the signal, wherein the signal comprises at least one precoded codeword generated based on a multiplication of an antipodal input sequence by a precoding matrix, wherein the precoding matrix is based on a multiplication of a codebook matrix and an antipodal matrix, wherein the codebook matrix comprises M codewords of complex symbols, and wherein the antipodal matrix comprises M different antipodal vectors; and demodulating the signal and performing a linear equalization of the demodulated signal.

22. A computer program comprising program code (206) configured to cause performance of the method according to claim 20 or 21, when the computer program is executed on a computer.