RECEIVER DEVICES AND METHODS FOR SINGLE-CARRIER MODULATION SCHEMES TECHNICAL FIELD The present disclosure relates generally to the field of single-carrier modulation systems, particularly to receiver devices of single-carrier modulation systems. To this end, receiver devices, a system, and methods are disclosed, e.g., for time-frequency-domain single-carrier Lagrange-Vandermonde Division Multiplexing (LVDM) equalization in doubly selective (time- and frequency-selective) channels. BACKGROUND Conventionally, in frequency-selective channels, single-carrier frequency-domain equalization (SC-FDE) has been used. For instance, a conventional single-carrier frequency-domain multiple access (SC-FDMA) scheme has been adapted for fourth generation (4G) uplink transmissions. However, an issue of conventional devices based on the SC-FDMA scheme is that a perfect recovery (PR) cannot be achieved in deep fading scenarios. Furthermore, conventional time-domain channel estimation and detection methods have also been suggested for single-carrier signals in fast and/or frequency-selective Rayleigh fading channels. However, an issue of such conventional methods is that, as a consequence of the time- varying channel, the orthogonality between subcarriers is destroyed, resulting in inter-carrier interference (ICI), which increases an irreducible error floor in proportion to a normalized Doppler frequency. Furthermore, conventional methods for cancelling inter-symbol interference (ISI) have also been suggested. Such cancellation methods attempt to remove ISI from the received signal, in order to isolate the desired data symbol. However, an issue of the conventional ISI cancellation methods is, for example, their substantial performance error. Another issue of conventional devices and methods is generally their complexity. For example, in order to prevent error propagation, reliable decisions should be fed to such conventional devices and methods. SUMMARY In view of the above-mentioned problems and disadvantages, embodiments of the present disclosure aim to improve receiver devices and receiving methods for single-carrier modulation schemes. An objective is to provide an advanced receiver for a single-carrier LVDM scheme that is able to deal with doubly selective channels (i.e., channels varying in time and frequency). One or more of the objectives is achieved by the embodiments of the disclosure as described in the enclosed independent claims. Advantageous implementations of the embodiments of the disclosure are further defined in the dependent claims. An approach to improve the conventional devices and methods, on which embodiments of the disclosure are based, is disclosed in an exemplary system 1700 comprising a transmitter device 1710 and a receiver device 1720, which are described in the following with reference to FIG. 17 and FIG.18. The exemplary system 1700 lays the basis for the embodiments of the present disclosure. FIG.17 depicts an exemplary scheme comprising a single-carrier transmitter device 1710, and a receiver device 1720 using a Vandermonde matrix for demodulation. The transmitter device 1710 and the receiver device 1720 communicate over a communication channel 1704. Moreover, a new waveform generalizing the SC-FDE, referred to as single-carrier LVDM, has been proposed in this approach, wherein the PR condition has been satisfied. The SC-LVDM relies on a one-tap equalization leading to a low-complexity implementation of the transmitter device 1710 and receiver device 1720. In the following, a discussion for the system 1700 is presented based on the SC-LVDM in frequency-selective channels. The transmitter device 1710 includes a precoder 1701, a modulator 1702, and a Zero Padding (ZP) block 1703. The transmitter device 1710 may also be compatible with a transmitter device of a multicarrier LVDM scheme. The transmitter 1710 of the SC-LVDM scheme uses the precoder 1701 and the modulator 1702 given by R and Ω, respectively, as follows: and where it can be verified that Furthermore, during the transmission process, the transmitter device 1710 may add a number of/. zeros to the K symbols, and may further transmit P symbols, where P =K + L, using a single carrier. Furthermore, the receiver device 1720 uses a number of K distinct nonzero complex points referred to as signature roots, which are chosen uniformly spread over a circle of radius a, such that

The receiver device 1720 includes a receiver filter 1705, pre-equalization 1706, a one-tap equalizer unit 1707, post-equalization 1708, and a decision block 1709.

The pre-equalization 1706 (i.e., the demodulator) performs a demodulation based on constructing a matrix E, which is a Vandermonde matrix having a size of KxP as follows: The Vandermonde matrix is applied to a received signal to obtain a demodulated signal given by: where C is a propagation channel of order L, given by . Furthermore, the convolution of the transmitter filter, the parameter C, and the receiver filter may be given by a channel matrix H (frequency-selective channels). Moreover, in the above Eq. (4), E(:,1:K) is the K first columns of the Vandermonde matrix E. Moreover, the signature roots are chosen such that the can be obtained. The one-tap equalizer 1707 of the receiver device 1720 uses a KxK diagonal matrix given by D ^-1 , followed by the post-equalization unit (given by M which is a KxK matrix). Therefore, the demodulated signal may be obtained according to Eq. (5) as follows: where the following matrix M is a Vandermonde matrix and no inversion is needed. Here, for example, a perfect recovery of s is satisfied. Next, the decision block 1709 of the receiver device 1720 will be discussed. Reference is made to FIG.18, which is a schematic view of the system 1700 comprising the transmitter device 1710 and the receiver device 1720 using a radius of a circle for building its modules. For example, the radius a of the circle may be modified, e.g., optimized “a _opt ”, at the receiver device 1720 and the modified, optimized radius of the circle may further be used to build the modules of the receiver device, as shown in FIG.18. The receiver device 1720 may further comprise a channel estimation block 1801 and an optimization block 1802 that may obtain the channel state information (CSI), e.g., from the channel estimation block 1801, and may further compute the ^ _^^^ . The receiver device 1720 of FIG. 18 uses the modified, optimized radius ^ _^^^ to compute the pre-equalizer (in 1706), the one-tap equalizer (in 1707), and the post-equalizer (in 1708) modules. In some embodiments, optionally, a refinement block 1803 may be provided which may, for example, use a refinement algorithm to refine the plurality of signature roots. However, in time-selective channels, the orthogonality between subcarriers may be destroyed and the matrix D (e.g., in the one-tap equalizer 1707 of the receiver device 1720) is not diagonal anymore. Consequently, ISI appears and the one-tap equalization techniques may not be adequate for detecting the symbols. In comparison with the receiver device 1720, the receiver devices and methods of the present disclosure enable to better deal with doubly (time and frequency) selective channels, while maintaining the low complexity of the receiver device 1720. A first aspect of the present disclosure provides a receiver device for single-carrier modulation. The receiver device is configured to determine channel state information (CSI) of a communication channel between the receiver device and a transmitter device, determine a plurality of signature roots based on the CSI, wherein each signature root of the plurality of signature roots is a nonzero complex point, and wherein the plurality of signature roots are uniformly distributed on a circumference of a circle in a complex plane, construct a first Vandermonde matrix based on the plurality of signature roots, perform a demodulation of a single-carrier modulated signal received from the transmitter device, based on the first Vandermonde matrix, to obtain a demodulated signal, and obtain symbols based on performing a first iterative procedure on the demodulated signal, the first iterative procedure comprising a set of parallel interference cancellation (PIC) iterations.

The receiver device may determine the CSI. For example, the receiver device may comprise a channel estimation unit that may run a channel estimation algorithm for determining the CSI and/or an estimated channel matrix, for example, in order to equalize doubly selective channels.

Furthermore, the receiver device may determine the plurality of signature roots ρ _k based on the CSI. Each signature root of the plurality of signature roots is a nonzero complex point and may be geometrically represented in the form of a real part and an imaginary part in the complex plane, which has a real axis and an imaginary axis. Moreover, a geometric representation of the plurality of signature roots in the complex plane may indicate that the plurality of signature roots ρ _k are uniformly distributed on a circumference of a circle in the complex plane, e.g., the plurality of signature roots ρ _k may uniformly spread over a circle of radius a, such that ρ _k = Moreover, the receiver device may determine the radius a of the circle in the complex plane, based on the determined CSI. For instance, the estimated channel matrix may be used for determining the radius a of the circle in the complex plane. For example, the algorithm may compute an optimization metric (for instance, a mean square error (MSE)) and may then compute the radius of the circle based on the optimization metric. In some embodiments, the receiver device may estimate the radius of the circle and may further modify and/or optimize the radius of the circle, for instance, based on a gradient descent algorithm (GDA).

For example, the receiver device may construct one or more Vandermonde matrices. For example, the first Vandermonde matrix E may be constructed based on the plurality of signature roots and according to the Eq. (3) given above. The receiver device may use the constructed Vandermonde matrix E, which may have a size of KxP and may be used for performing the demodulation, and thus the demodulated signal may be obtained.

The receiver device may be a receiver for a transceiver device of the single-carrier modulation scheme. The receiver device is an advanced receiver for SC-LVDM in that it is able to deal with doubly selective channels. The receiver device may be an electronic device comprising circuitry. The circuitry may comprise hardware and software. The hardware may comprise analog or digital circuitry, or both analog and digital circuitry. In some embodiments, the circuitry comprises one or more processors and a non-volatile memory connected to the one or more processors. The non- volatile memory may carry executable program code which, when executed by the one or more processors, causes the receiver device to perform the operations or methods described herein.

According to some embodiments, a two-stage time-frequency-domain equalization may be performed. For example, in some embodiments, a channel estimation algorithm may be introduced that may feed the detector to build the blocks for equalization.

In an implementation form of the first aspect, the receiver device is configured to determine the radius of the circle in the complex plane, based on the CSI.

In a further implementation form of the first aspect, for each PIC iteration, the receiver device is configured to construct a second Vandermonde matrix based on the plurality of signature roots, obtain a first matrix based on the first Vandermonde matrix, obtain a second matrix based on the first and the second Vandermonde matrices to estimate a first ISI in a current PIC iteration, perform a one-tap equalization by applying a diagonal matrix, which is obtained based on the first and the second Vandermonde matrices and an estimated channel matrix, on the demodulated signal or on a remainder of the demodulated signal obtained after removing a contribution of the estimated first ISI from estimated symbols in a previous PIC iteration, and obtain the symbols based on removing the contribution of the estimated first ISI in the current PIC iteration from the estimated symbols.

For example, the first matrix may be a matrix V that may be built from the demodulation (Vandermonde) matrix E that is constructed based on the plurality of signature roots and according to the Eq. (3) given above.

Moreover, the second matrix may be a matrix T that may be used for estimating the ISI, and may be given by:

Eq. (7) where Eq. (8) where H is an estimated channel matrix that may be determined through the CSI estimation. Moreover, E and M are the two Vandermonde matrices, wherein E is a Vandermonde matrix obtained based on the plurality of the signature roots and used for performing the demodulation. Moreover, M is a Vandermonde matrix obtained based on the plurality of the signature roots and may further be used for performing the PIC iterations. Furthermore, the receiver device may perform the one-tap equalization. For example, the receiver device may obtain the diagonal matrix based on the two Vandermonde matrices E and M and the estimated channel matrix H, according to Eq. (9) as follows: Eq. (9) where can be obtained according to Eq. (8). In a further implementation form of the first aspect, during each PIC iteration, the first matrix and the second matrix are respectively applied on the estimated symbols, wherein the first matrix is applied for performing a frequency-domain transformation of the estimated symbols, and wherein the second matrix is applied for estimating the first ISI of transformed symbols in the frequency domain. In a further implementation form of the first aspect, the first iterative procedure is performed until a converging of the set of PIC iterations meets a predefined criterion. In a further implementation form of the first aspect, the symbols are further obtained based on performing a second iterative procedure on the obtained symbols output from the first iterative procedure, the second iterative procedure comprising a set of interference cancellation refinement (ICR) iterations. In a further implementation form of the first aspect, the receiver device is further configured to obtain a cancelation filter based on the estimated channel matrix or the determined CSI of the communication channel. In a further implementation form of the first aspect, for each ICR iteration, the receiver device is configured to apply the cancelation filter on the output of the first iterative procedure or an output of a previous ICR iteration, for estimating a second ISI, and obtain the symbols based on applying an inversion of a diagonal matrix on the filtered received signal, after removing a contribution of the estimated second ISI.

In a further implementation form of the first aspect, during each subsequent ICR iteration, a subsequent cancelation filter is applied on the obtained symbols output from the first iterative procedure or a previous ICR iteration, to estimate the second ISI.

In a further implementation form of the first aspect, the second iterative procedure is performed until a converging of the set of ICR iterations meets a predefined criterion.

A second aspect of the disclosure provides a receiver device for single-carrier modulation. The receiver device is configured to determine channel state information (CSI) of a communication channel between the receiver device and a transmitter device, determine a plurality of signature roots, based on the CSI, wherein each signature root of the plurality of signature roots is a nonzero complex point, and wherein the plurality of signature roots are uniformly distributed on a circumference of a circle in a complex plane, construct a first Vandermonde matrix and a second Vandermonde matrix based on the plurality of signature roots, perform a demodulation of a single-carrier modulated signal received from the transmitter device, based on the first Vandermonde matrix, to obtain a demodulated signal, and obtain symbols based on performing an iterative procedure on estimated symbols obtained by a frequency-domain equalization comprising a one-tap equalization followed by a post-equalization performed based on the second Vandermonde matrix, the iterative procedure comprising a set of interference cancellation refinement (ICR) iterations.

The receiver device of the second aspect may be similar or identical to the receiver device of the first aspect, and may perform a similar or identical function. The receiver device is an advanced receiver device for SC-LVDM in that it is able to deal with doubly selective channels. For example, the receiver device of the second aspect may determine the CSI, the radius of the circle, the plurality of signature roots, etc., as it is described for the receiver device of the first aspect. In an implementation form of the second aspect, the receiver device is further configured to determine the radius of the circle in the complex plane, based on the CSI.

In a further implementation form of the second aspect, the receiver device is further configured to obtain a cancelation filter based on the determined CSI of the communication channel.

In a further implementation form of the second aspect, for each ICR iteration, the receiver device is configured to apply the cancelation filter on the estimated symbols during the first iteration or on an output of a previous ICR iteration, to estimate a second ISI, and obtain the symbols based on applying an inversion of a diagonal matrix on the filtered received signal, after removing a contribution of the estimated second ISI.

In a further implementation form of the second aspect, during each subsequent ICR iteration, a subsequent cancelation filter is applied on the symbols obtained from a previous ICR iteration to estimate the second ISI.

In a further implementation form of the second aspect, the iterative procedure is performed until a converging of the set of ICR iterations meets a predefined criterion.

In a further implementation form of the second aspect, the receiver device is further configured to compute a metric for determining the radius of the circle in the complex plane and/or evaluating the plurality of signature roots, based on the CSI of the communication channel.

In a further implementation form of the second aspect, the receiver device is further configured to modify individually each signature root from the plurality of signature roots based on a machine learning algorithm, in particular a gradient descent algorithm.

In a further implementation form of the second aspect, the receiver device is further configured to perform the demodulation of the single carrier modulated signal, considering the individual modification of each signature root.

According to some embodiments, the receiver device may perform a two-stage time-frequency domain equalization comprising the first iterative procedure and the second iterative procedure. Further, a channel estimation algorithm is disclosed that feeds the detector to build the blocks for equalization.

In summary, the receiver device of the first aspect and the receiver device of the second aspect are designed to deal with doubly selective channels, such as doubly selective fading channels, without limiting the present disclosure to specific channels.

A third aspect of the present disclosure provides a system for single-carrier modulation, the system comprising a transmitter device configured to generate a single-carrier modulated signal based on constructing a Lagrange matrix or a Vandermonde matrix, and a receiver device according to the first aspect or any of its implementation forms, or a receiver device according to the second aspect or any of its implementation forms.

A fourth aspect of the present disclosure provides a method for a receiver device for single- carrier modulation, wherein the method comprises determining channel state information (CSI) of a communication channel between the receiver device and a transmitter device, determining a plurality of signature roots based on the CSI, wherein each signature root of the plurality of signature roots is a nonzero complex point, and wherein the plurality of signature roots are uniformly distributed on a circumference of a circle in a complex plane, constructing a first Vandermonde matrix based on the plurality of signature roots, performing a demodulation of a single-carrier modulated signal received from the transmitter device, based on the first Vandermonde matrix, to obtain a demodulated signal, and obtaining symbols based on performing a first iterative procedure on the demodulated signal, the first iterative procedure comprising a set of PIC iterations.

In an implementation form of the fourth aspect, the method further comprises determining the radius of the circle in the complex plane, based on the CSI.

In a further implementation form of the fourth aspect, for each PIC iteration, the method further comprises constructing a second Vandermonde matrix based on the plurality of signature roots, obtaining a first matrix based on the first Vandermonde matrix, obtaining a second matrix based on the first and the second Vandermonde matrices to estimate a first ISI in a current PIC iteration, performing a one-tap equalization by applying a diagonal matrix obtained based on the first and the second Vandermonde matrices and an estimated channel matrix on the demodulated signal or on a remainder of the demodulated signal obtained after removing a contribution of the estimated first ISI from estimated symbols in the previous PIC iteration, and obtaining the symbols based on removing the contribution of the estimated first ISI in the current PIC iteration from the estimated symbols.

In a further implementation form of the fourth aspect, during each PIC iteration, the first matrix and the second matrix are respectively applied on the estimated symbols, wherein the first matrix is applied for performing a frequency domain transformation of the estimated symbols, and wherein the second matrix is applied to estimate the first ISI of transformed symbols in the frequency domain.

In a further implementation form of the fourth aspect, the first iterative procedure is performed until a converging of the set of PIC iterations meets a predefined criterion.

In a further implementation form of the fourth aspect, the symbols are further obtained based on performing a second iterative procedure on the obtained symbols output from the first iterative procedure, the second iterative procedure comprising a set of interference cancellation refinement, ICR, iterations.

In a further implementation form of the fourth aspect, the method further comprises obtaining a cancelation filter based on the estimated channel matrix or the determined CSI of the communication channel.

In a further implementation form of the fourth aspect, for each ICR iteration, the method further comprises applying the cancelation filter on the output of the first iterative procedure or an output of a previous ICR iteration to estimate a second ISI, and obtaining the symbols based on applying an inversion of a diagonal matrix on the filtered received signal, after removing a contribution of the estimated second ISI.

In a further implementation form of the fourth aspect, during each subsequent ICR iteration, a subsequent cancelation filter is applied on the obtained symbols output from the first iterative procedure or a previous ICR iteration, to estimate the second ISI. In a further implementation form of the fourth aspect, the second iterative procedure is performed until a converging of the set of ICR iterations meets a predefined criterion.

A fifth aspect of the disclosure provides a method for a receiver device for single-carrier modulation, wherein the method comprises determining channel state information (CSI) of a communication channel between the receiver device and a transmitter device, determining a plurality of signature roots, wherein each signature root of the plurality of signature roots is a nonzero complex point, and wherein the plurality of signature roots are uniformly distributed on a circumference of a circle in a complex plane, constructing a first Vandermonde matrix and a second Vandermonde matrix based on the plurality of signature roots, performing a demodulation of a single-carrier modulated signal received from the transmitter device, based on the first Vandermonde matrix, to obtain a demodulated signal, and obtaining symbols based on performing an iterative procedure on estimated symbols obtained by a frequency domain equalization comprising a one-tap equalization followed by a post-equalization performed based on the second Vandermonde matrix, the iterative procedure comprising a set of ICR iterations.

In an implementation form of the fifth aspect, the method further comprises determining the radius of the circle in the complex plane, based on the CSI.

In a further implementation form of the fifth aspect, the method further comprises obtaining a cancelation filter based on the determined CSI of the communication channel.

In a further implementation form of the fifth aspect, for each ICR iteration, the method further comprises applying the cancelation filter on the estimated symbols during the first iteration or on an output of a previous ICR iteration to estimate a second ISI, and obtaining the symbols based on applying an inversion of a diagonal matrix on the filtered received signal, after removing a contribution of the estimated second ISI.

In a further implementation form of the fifth aspect, during each subsequent ICR iteration, a subsequent cancelation filter is applied on the symbols obtained from a previous ICR iteration to estimate the second ISI. In a further implementation form of the fifth aspect, the iterative procedure is performed until a converging of the set of ICR iterations meets a predefined criterion.

In a further implementation form of the fifth aspect, the method further comprises computing a metric for determining the radius of the circle in the complex plane and/or evaluating the plurality of signature roots, based on the CSI of the communication channel.

In a further implementation form of the fifth aspect, the method further comprises modifying individually a signature root from the plurality of signature roots based on a machine learning algorithm, in particular a gradient descent algorithm.

In a further implementation form of the fifth aspect, the method further comprises performing the demodulation of the single-carrier modulated signal, based on the individually modified signature root.

A sixth aspect of the present disclosure provides a computer program comprising a program code for performing the method according to the fourth aspect or the fifth aspect or any of their implementation forms.

A seventh aspect of the present disclosure provides a non-transitory storage medium storing executable program code which, when executed by a processor, causes the method according to the fourth aspect or the fifth aspect or any of their implementation forms to be performed.

It has to be noted that all devices, elements, 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. BRIEF DESCRIPTION OF DRAWINGS

The above described aspects and implementation forms will be explained in the following description of specific embodiments in relation to the enclosed drawings, in which:

FIG. 1 depicts a schematic view of a receiver device for single-carrier modulation, according to an embodiment of the present disclosure;

FIG. 2 depicts a schematic view of another receiver device for single-carrier modulation, according to an embodiment of the present disclosure;

FIG. 3 depicts a schematic view of a system for single-carrier modulation, according to an embodiment of the present disclosure;

FIG. 4 depicts a schematic view of the system for a SC-LVDM for a doubly selective channel;

FIG. 5 depicts pilot patterns using dedicated LVDM symbols;

FIG. 6 depicts NP stacked pilot vectors, where (NP — 1) are new pilot vectors with one buffered pilot vector;

FIG. 7 depicts a diagram illustrating the receiver device obtaining symbols based on performing the first iterative procedure comprising a set of PIC iterations;

FIG. 8 depicts a diagram illustrating performing a first iterative procedure comprising a set of PIC iterations followed by a second iterative procedure comprising a set of ICR iterations;

FIG. 9 depicts exemplary performance results of the receiver device, when performing only the first iterative procedure comprising the set of PIC iterations; FIG. 10 depicts exemplary performance results of the receiver device, when performing only the second iterative procedure comprising the set of ICR iterations;

FIG. 11 depicts exemplary performance results of the receiver device, when performing the first iterative procedure and the second iterative procedure;

FIG. 12 depicts exemplary performance results of the receiver device, in 3 GPP

ETU channels, using L = 10;

FIG. 13 depicts exemplary performance results of the receiver device, in 3 GPP

EVB channels, using L = 39;

FIGS. 14A-14B depict a sensitivity to the channel estimation and the error in the estimation of the radius a _opt of the circle;

FIG. 15 depicts a flowchart of a method for a receiver device for single-carrier modulation, according to an embodiment of the present disclosure;

FIG. 16 depicts a flowchart of a method for another receiver device for single- carrier modulation, according to an embodiment of the present disclosure;

FIG. 17 depicts an exemplary embodiment of a SC-LVDM system comprising a single-carrier transmitter device and a receiver device using a Vandermonde matrix for demodulation; and

FIG. 18 depicts a schematic view for the SC-LVDM system of FIG. 17 comprising a receiver device using a radius of a circle for building its modules. DETAILED DESCRIPTION OF EMBODIMENTS

FIG. 1 shows a schematic view of a receiver device 100 for single-carrier modulation, according to an embodiment of the present disclosure.

The receiver device 100 is configured to determine a channel state information (CSI) 101 of a communication channel 111 between the receiver device 100 and a transmitter device 110. For example, the receiver device 100 comprises a channel estimation unit 112 which may determine the CSI 101.

The receiver device 100 is further configured to determine a plurality of signature roots based on the CSI 101, wherein each signature root of the plurality of signature roots is a nonzero complex point, and wherein the plurality of signature roots are uniformly distributed on a circumference of a circle in a complex plane.

For example, the receiver device 100 comprises a determination unit 113 that may determine the plurality of signature roots based on the CSI 101. Each signature root of the plurality of signature roots is a nonzero complex point, and may be geometrically represented in the form of a real part and an imaginary part in the complex plane, which has a real axis and an imaginary axis. Moreover, a geometric representation of the plurality of signature roots, i.e., the non-zero complex points, in the complex plane may indicate that the plurality of signature roots ρ _k are uniformly distributed on a circumference of a circle in the complex plane, e.g., the plurality of signature roots ρ _k may uniformly spread over a circle of radius a, such that

Moreover, the receiver device may determine the radius a of the circle in the complex plane based on the CSI 101, and may further modify and/or optimize the determined radius of the circle in the complex plane, e.g., using a gradient descent algorithm (GDA).

The receiver device 100 is further configured to construct a first Vandermonde matrix based on the plurality of signature roots. For example, the receiver device 100 may construct the first Vandermonde matrix E based on the plurality of signature roots, according to Eq. (3), as described above. Moreover, in some embodiments, the plurality of signature roots may be used for constructing at least one other Vandermonde matrix. For example, the receiver device 100 may construct a second Vandermonde matrix M according to Eq. (6), which may be used for performing a post- equalization.

Moreover, by determining the plurality of signature roots ρ _k such that they are uniformly distributed on a circumference of a circle, it may be possible to reduce an optimization problem. For instance, a first matrix V may be determined based on the first Vandermonde matrix E. Moreover, the second Vandermonde matrix M may be the inverse of the first matrix V, i.e., the second Vandermonde matrix M may be the inverse of E(:, 1 :K). Therefore, by determining the signature roots ρ _k such that they are uniformly distributed on a circumference of a circle and/or by further optimizing the radius of the circle, it may be possible to reduce the optimization problem, for example, from a K-dimensional filed optimization to a one dimension filed optimization problem.

Moreover, by determining the plurality of signature roots ρ _k such that they are uniformly distributed on a circumference of a circle, a cost related to construction of the Vandermonde matrices may be reduced.

For example, the second Vandermonde matrix M may have a form of a simple Vandermonde matrix when it is constructed based on the plurality of signature roots ρ _k that are uniformly distributed on a circumference of a circle compared to when the signature roots ρ _k are distributed in a more complex manner. Moreover, since the second Vandermonde matrix M represents a simple Vandermonde matrix, therefore, a cost related to determining the inverse of the Vandermonde matrix E(:, 1 :K) may also be reduced.

Moreover, the receiver device 100 may optionally comprise a demodulator block 114. Moreover, the receiver device 100, e.g., its demodulator block 114, may determine the plurality of signature roots ρ _k that are uniformly distributed on the circumference of the circle as, for instance, determined by the determination unit 113. Furthermore, the demodulator block 114 may construct the first Vandermonde matrix (i.e., the matrix E according to Eq. (3)) based on the plurality of signature roots ρ _k . The receiver device 100 is further configured to perform a demodulation of a single-carrier modulated signal 120 received from the transmitter device 110, based on the first Vandermonde matrix E, to obtain a demodulated signal 121.

In particular, the receiver device 100 may receive the single-carrier modulated signal 120 from the transmitter device 110 via the communication channel 111. Moreover, the demodulator block 114 of the receiver device 100 may perform the demodulation on the single-carrier modulated signal 120 using the constructed Vandermonde matrix E, and may further obtain the demodulated signal 121. The modulation operation may be performed according to Eq. (4), as described above.

For example, the receiver device 100 may apply the first Vandermonde matrix E to a received signal in time domain r to obtain a demodulated signal y, according to Eq. (4), as described above.

The receiver device 100 is further configured to obtain symbols 130 based on performing a first iterative procedure on the demodulated signal 121, the first iterative procedure comprising a set of PIC iterations.

For example, the receiver device 100 may optionally comprise a PIC block 115. Moreover, the PIC block 115 of receiver device 100 may perform the first iterative procedure comprising the set of PIC iterations on the demodulated signal 121. The PIC block 115 may further obtain the symbols 130 by performing the first iterative procedure.

The receiver device 100 may comprise processing circuitry (not shown in FIG. 1) configured to perform, conduct or initiate the various operations of the receiver device 100 described herein. The processing circuitry may comprise hardware and software. The hardware may comprise analog circuitry or digital circuitry, or both analog and digital circuitry. The digital circuitry may comprise components such as application-specific integrated circuits (ASICs), field- programmable arrays (FPGAs), digital signal processors (DSPs), or multi-purpose processors. In one embodiment, the processing circuitry comprises one or more processors and a non- transitory memory connected to the one or more processors. The non-transitory memory may carry executable program code which, when executed by the one or more processors, causes the receiver device 100 to perform, conduct or initiate the operations or methods described herein. FIG. 2 shows a schematic view of another receiver device 200 for single-carrier modulation, according to an embodiment of the present disclosure.

The receiver device 200 is configured to determine CSI 201 of a communication channel 111 between the receiver device 200 and a transmitter device 110. For example, the receiver device 200 comprises a channel estimation unit 212 which may determine the CSI 201.

The receiver device 200 is further configured to determine a plurality of signature roots based on the CSI 201, wherein each signature root of the plurality of signature roots is a nonzero complex point, and wherein the plurality of signature roots are uniformly distributed on a circumference of a circle in a complex plane.

For example, the receiver device 200 comprises a determination unit 213 that may determine the plurality of signature roots based on the CSI 201. Each signature root of the plurality of signature roots is a nonzero complex point, and may be geometrically represented in the form of a real part and an imaginary part in the complex plane, which has a real axis and an imaginary axis. Moreover, a geometric representation of the plurality of signature roots, i.e., the non-zero complex points, in the complex plane may indicate that the plurality of signature roots ρ _k are uniformly distributed on a circumference of a circle in the complex plane, e.g., the plurality of signature roots ρ _k may uniformly spread over a circle of radius a, such that

Moreover, the receiver device 200 may determine the radius a of the circle in the complex plane based on the CSI 201, and may optionally modify and/or optimize the determined radius of the circle in the complex plane, e.g., based on a GDA.

The receiver device 200 is further configured to construct a first Vandermonde matrix and a second Vandermonde matrix based on the plurality of signature roots. For example, the receiver device 200 may construct the first Vandermonde matrix E based on the plurality of signature roots, according to Eq. (3), as described above. Moreover, the receiver device 200 may construct the second Vandermonde matrix M according to Eq. (6), as described above. For example, the receiver device 200 may optionally comprise a demodulator block 214. Moreover, the receiver device 200, e.g., its demodulator block 214, may determine the plurality of signature roots ρ _k that are uniformly distributed on the circumference of the circle as, for instance, determined by the determination unit 213. Furthermore, the demodulator block 214 may construct the first Vandermonde matrix and the second Vandermonde matrix based on the plurality of signature roots ρ _k .

The receiver device 200 is further configured to perform a demodulation of a single-carrier modulated signal 120 received from the transmitter device 110, based on the first Vandermonde matrix, to obtain a demodulated signal 221. The modulation operation may be performed according to Eq. (4), as described above.

For example, the receiver device 200 may apply the first Vandermonde matrix E to a received signal in time domain r to obtain a demodulated signal y, according to Eq. (4), as described above.

For example, the receiver device 200 may receive the single-carrier modulated signal 120 from the transmitter device 110. Moreover, the demodulator block 214 of the receiver device 200 may perform the demodulation on the single-carrier modulated signal 120 using the constructed first Vandermonde matrix, and may further obtain the demodulated signal 221.

The receiver device 200 is further configured to obtain symbols 230 based on performing an iterative procedure on estimated symbols obtained by a frequency domain equalization, the iterative procedure comprising a set of ICR iterations.

For example, the receiver device 200 may optionally comprise an ICR block 215. Moreover, the ICR block 215 of receiver device 200 may perform the iterative procedure comprising the set of ICR iterations on the estimated symbols obtained by the frequency-domain equalization, and the receiver device 200 may further obtain the symbols 230.

The receiver device 200 may comprise processing circuitry (not shown in FIG. 2) configured to perform, conduct or initiate the various operations of the receiver device 200 described herein The processing circuitry may comprise hardware and software. The hardware may comprise analog circuitry or digital circuitry, or both analog and digital circuitry. The digital circuitry may comprise components such as application-specific integrated circuits (ASICs), field- programmable arrays (FPGAs), digital signal processors (DSPs), or multi-purpose processors. In one embodiment, the processing circuitry comprises one or more processors and a non- transitory memory connected to the one or more processors. The non-transitory memory may carry executable program code which, when executed by the one or more processors, causes the receiver device 100 to perform, conduct or initiate the operations or methods described herein.

FIG. 3 depicts a schematic view of a system 300 for single-carrier modulation, according to an embodiment of the present disclosure.

The system 300 comprises a transmitter device 110 that is configured to generate a single- carrier modulated signal based on constructing a Lagrange matrix or a Vandermonde matrix. The system 300 further comprises a receiver device, which may be the receiver device 100 or the receiver device 200 illustrated in FIG. 1 and FIG. 2, respectively.

FIG. 4 depicts a schematic view of the system 300 for a SC-LVDM for a doubly selective channel.

The system 300 comprises the transmitter device 110 that includes a precoder 401, a modulator 402, and a ZP block 403. Moreover, the communication channel 111 of the system 300 comprises the receiver filter 405. The system 300 further comprises the receiver device (e.g., it may be the receiver device 100 of FIG.1 or the receiver device 200 of FIG. 2), which includes the demodulator 114, 214. The demodulator 114, 214 may perform a demodulation of the single-carrier modulated signal based on the first Vandermonde matrix E and may further obtain a demodulated signal. The first iterative procedure is exemplarily shown as a tentative decision block comprising the PIC block 115, the second iterative procedure is exemplarily shown as the ICR block 215, the channel estimation block 112, 212, the determination unit 113, 213 (e.g., an optimization block), and the decision block 408.

Moreover, the receiver device 100 in the system 300 may perform the first iterative procedure and may further obtain symbols 130. Furthermore, the receiver device 200 in the system 300 may perform the second iterative procedure and may further obtain symbols 230. Further, the receiver device 100 may perform both the first iterative procedure and the second iterative procedure and may further obtain symbols 230. In the following, the system 300 is discussed as an example for time-frequency domain equalizations for the SC-LVDM, without limiting the present disclosure.

In order to deal with doubly selective channels, a channel estimation (CE) is proposed. For example, the channel estimation unit 112 of the receiver device 100 or the channel estimation unit 212 of the receiver device 200 may run a channel estimation algorithm for obtaining the CSI or estimated channel matrix, in order to deal with doubly selective channels.

The system 300 may show a low-complexity implementation and its performance may outperform the performance of the conventional SC-FDE.

For example, the receiver device 100 or the receiver device 200 may estimate (e.g., based on the MSE), the modified, optimized radius of the circle a _opt , which may be used for the blocks of the receiver device and performing the equalization processing.

The receiver device 100 or the receiver device 200 may further perform a time-frequency domain equalization procedure, for example, in order to overcome one or more issues related to time- varying channel.

For instance, the receiver device 100 or the receiver device 200 may overcome an issue related to outdated CSI that breaks the optimization of the plurality of signature roots. Moreover, the receiver device 100 or the receiver device 200 may overcome an issue related to the ISI that makes the one-tap equalization inadequate.

The above issues may be solved by a time-frequency domain equalization discussed in the following. The received time-domain signal is given by Eq. (10) as follows: Eq. (10) where Ht is the time-varying channel matrix, where h(n, Z) = h _c (nT _s , IT _S ) and h _c (t, τ) is the time-varying impulse response of the channel that includes transmitter-receiver filters as well as the doubly selective propagation effects.

In the received signal expression, the channel impulse response may vary even within one SC- LVDM symbol, which may make the detection processing challenging.

For example, at a first step, the receiver device 100 or the receiver device 200 may run a channel estimation algorithm. Moreover, the receiver device 100 or the receiver device 200 may estimate some values of optimized radius a _opt .

Furthermore, at a second step, the receiver device 100 or the receiver device 200may use the estimated channel to detect the SC-LVDM symbols, and may further performs operations for removing a contribution of ISI

In particular, during the first step, a channel estimation (e.g., a channel estimation algorithm) may be performed. For example, by assuming that τ _max and f _D are the delay spread and the Doppler spread, respectively, and T _s is the sampling period at the receiver device, both τ _max and f _D may be measured. Moreover, the NT _S may be the frame duration.

Furthermore, a Slepian basis expansion (SBE) may be used (generally known to the skilled person), where h _c (t, τ) will be presented for t G [kNT _s , (k + 1)N T _S ) by using: Q + 1 coefficients that remain invariant per block, but are allowed to change with k, and

2. Q + 1 Slepian sequences that capture the time variation, but are common for all k,

Further, each time-varying delay tap of the channel impulse response may be approximated as follows: where and where are the integer floor and the the integer ceiling, respectively.

FIG. 5 depicts a diagram illustrating the transmitted SC-LVDM frames, wherein NP training sequence vectors are inserted every D transmitted symbol vectors (one pilot vector followed by (D-1) SC-LVDM symbols), wherein is the size of the transmitted SC-LVDM symbols.

In the diagram illustrated in FIG. 5, NP = 6 and every pilot symbol vector is followed by (D - 1) SC-LVDM symbols (vectors).

In the following, an exemplary procedure is presented for estimating the (Q + 1) × (L + 1) coefficients, discussed above, which may lead to the channel estimation over the coherence time period (NT _S ), and using the SBE approximation, without limiting the present disclosure to this specific procedure.

The receiver device 100 or the receiver device 200 may stack the received NP pilot vectors (where one is buffered) in y _b (see FIG. 6). Moreover, y _b may have a size of ((NP × P) X 1) and may be obtained according to Eq. (12) as follows: Eq. (12) where

Eq. (13) where Eq. (14) is the vector stacking the q ^th coefficient of every l ^th delay tap to be determined, and Eq. (15) has a size ((NP x P) x ((Q + 1) x (L + 1))), where, in above Eq. (15),

Eq (16) is a Toeplitz matrix formed by the transmitted pilot vectors. For example, the Toeplitz matrix may be formed by a number of (K + 1) modulated pilot symbols followed by a number of L- zeros padded symbols, i.e., L zeros are added at the end of the modulated pilot symbols, as shown in FIG. 6, and is a P X P diagonal matrix and can be obtained according to Eq. (17) as follows:

Eq. (17) where

Eq. (18)

FIG. 6 depicts a diagram illustrating NP stacked pilot vectors comprising (NP — 1) new pilot vectors shown along with one buffered pilot vector.

As discussed, the receiver device 100 or the receiver device 200 may determine the radius of the circle based on the determined CSI. For example, the receiver device 100 or the receiver device 200, particularly in the CE block 112, 212, may apply the linear MMSE channel estimator and may further obtain the estimated channel as follows:

Eq. (19) where

Eq. (20)

Furthermore, it may be assumed that R _h is known at the receiver device 100 or the receiver device 200. For example, since R _h depends only on the channel delay profile, it may be determined by the receiver device 100 or the receiver device 200.

The aforementioned details are summarized in the following algorithm, wherein the (estimated) channel matrix H _Det , which is the output of the algorithm, may be used to detect the actual (D- V)(NP-V) received SC-LVDM symbols (see FIG. 6).

Therefore, the receiver device 100 or the receiver device 200 may perform algorithm 1 to output and obtain (e.g., take) the m-th ((L + 1) x k) channel matrix as follows:

H _a,m = H _det (: , (1: K) + (m - 1)K), where m = 1 : (D - 1)NP.

Therefore, Eq. (21) may be obtained as follows: Eq. (21) Moreover, the determined radius a may be modified (optimized) at the receiver device 100 or the receiver device 200 through the corresponding determination unit 113, 213 (e.g., the optimization block) and by using a metric (such as the MSE).

Moreover, the optimized radius of the circle may depend on the channel realization, and may be different for different LVDM symbols. Therefore, the receiver device 100 or the receiver device 200 may obtain, for each LVDM symbol, a respective optimized radius of the circle a _opt . The optimized radius(s) a _{opt m} obtained for different LVDM symbols may further be used, to build different blocks of the receiver device 100 or the receiver device 200. For example, for the m-th LVDM symbol, the receiver device 100 or the receiver device 200 uses m-th radius of the circle a _{opt m} , estimated according to Eq. (22) as follows:

Eq. (22) where Eq. (23)

Moreover, each frame is structured such that, one pilot symbol comprising K + L samples is followed by (D - 1) LVDM symbols, where each LVDM symbol comprises K + L samples, too.

The above-mentioned algorithm for joint estimation may further be summarized according to Eq. (24).

The inputs of the algorithm are R _h and y _b , and the output of the algorithm is H _Det . Moreover, the algorithm includes five computing steps, wherein each computing step of the algorithm is indicated with a bullet point as follows:

Eq. (24)

The next step is the detection phase. During the detection phase, the receiver device 100 or the receiver device 200 may use the m-th ((L + 1) × K) channel matrix H _{D, m} = H _Det (: , (1: K) + (m — 1)K) to detect the m-th SC-LVDM received symbol based on performing two iterative procedures using time-frequency processing (see FIG. 8).

In the following, the first iterative procedure is also referred to as the tentative decision step, and the second iterative procedure is also referred to as the interference cancellation refinement step.

Reference is now made to FIG. 7, which depicts a diagram 700 illustrating the receiver device 100 obtaining symbols based on performing the first iterative procedure comprising a set of PIC iterations.

As can be derived from FIG. 7, the first iterative procedure (i.e., the tentative decision) is made based on the parallel interference cancellation technique that is applied to the demodulated received signal 121 in the frequency domain y, given by: y = E[m] r, Eq. (25) where r is the received signal 120 (in the time domain), and the Vandermonde matrix can be obtained as follows: Eq. (26) where In the sequel, the index m is omitted.

The receiver device 100 may further obtain the second matrix T 702. For example, the receiver device 100 may obtain that may represent the ISI in the frequency domain.

Moreover, the receiver device 100 may have a one-tap equalizer unit that may perform the one- tap equalization by apply the diagonal matrix 703 given on the demodulated signal 121 or on a remainder of the demodulated signal obtained after removing a contribution of the first ISI from estimated symbols in the previous PIC iteration, and obtain the symbols based on removing the contribution of the estimated first ISI in the current PIC iteration, from the estimated symbols.

The one-tap equalization is given by: Eq. (27) where the diagonal matrix 703 and the suffix is for the Z-th PIC iteration with y ⁽⁰⁾ = y.

Furthermore, the receiver device 100 may perform a post-equalization using a Vandermonde matrix 704 M and may further obtain a time-domain signal as follows: Eq. (28) where the matrix representing the ISI, and reduces to the identity in the absence of time selectivity (time-invariant channels).

Moreover, as discussed, the first iterative procedure comprises the set of PIC iterations. Each PIC iteration may remove (in the frequency domain) the contribution of the ISI from y given by: Eq. (29) where the first matrix 701 is V = E(: ,1: A) given by the first K columns of the matrix E 706 (e.g., determined by the demodulator unit 114). For example, the receiver device 100 may further comprise a post-equalization block 707 that may perform a post-equalization by using the second Vandermonde matrix M 704, and may further obtain . The output of the post- equalization goes through a hard decision unit 705 in order to obtain a hard decision (e.g., the obtained symbol 130) according to . The output value of the hard decision unit 705, should be the same among the symbols of the used constellation (for example, the QPSK symbols).

The receiver device 100 performs the first iterative procedure until a converging of the set of PIC iterations meets a predefined criterion. For example, the tentative detector may continue reiterating until converging.

Reference is made to FIG. 8, which depicts a diagram 800 illustrating performing a first iterative procedure comprising a set of PIC iterations followed by a second iterative procedure comprising a set of ICR iterations.

For example, the receiver device 100 may obtain symbols 130 based on performing the first iterative procedure comprising a set of PIC iterations, and further performing a second iterative procedure comprising a set of ICR iterations. As can be derived from FIG. 8, the obtained symbols 130, i.e. the output of the first iterative procedure indicated by in FIG. 8, are fed to the second iterative procedure comprising the set of ICR iterations. Moreover, the receiver device 200 may obtain symbols 230 based on performing the iterative procedure comprising the set of ICR iterations, e.g., as discussed above with respect to FIG. 2.

During the second iterative procedure (the interference cancellation refinement), the receiver device 100 or the receiver device 200 obtains a cancelation filter 801.

For example, the cancelation filter 801 may be a canceller filter G that may be given by: Eq. (30)

Furthermore, the iterative procedure comprises the set of ICR iterations in which, during each ICR iteration, is input to a one-tap time-domain equalizer block 813 as follows: Eq. (31) where is the output of the first iterative procedure (i.e., the tentative decision). In other words, a refinement of the equalization made by PIC during the first iterative procedure will be carried out. Furthermore, in one-tap time-domain equalizer block 813, the inversion of the diagonal matrix 803 may be applied on the filtered received signal The output of the one-tap time domain equalizer block 813 is a soft decision indicated by and may be obtained based on

Next, the soft decision that is output by the one-tap time-domain equalizer block 813 goes to the hard decision block 804, providing a hard decision (i.e., the obtained symbol 130 for the device 100 or the obtained symbol 230 for the device 200), according to: Eq. (32)

In the diagram illustrated in FIG. 8, the H ^H 802 is the matched filter.

The receiver device 100 or the receiver device 200 may perform the second iterative procedure until a converging of the set of ICR iterations meets a predefined criterion. For example, the ICR may continue reiterating until converging and the and the resulting SC-LVDM symbol 230 is given by Reference is now made to FIG. 9, FIG. 10, FIG. 11, FIG. 12, FIG. 13, FIG. 14A, and FIG. 14B, which depict respective diagrams illustrating performance results of the proposed receiver devices 100, 200 in a doubly selective channel. A comparison will be carried out, and the performance and/or the complexity tradeoff will be discussed. Moreover, the sensitivity of the performance and the optimized radius a _opt to the channel estimation errors are presented, in which the complexities are generally known to the skilled person.

1. Doubly Selective Fading Channels

For example, K = 64 and L = 16 may be set. Moreover, the maximum Doppler spread is set to f _D = 1 KHz while the subcarrier spacing is set to Δf = 30 KHz and 15 KHz in FIG. 9, FIG. 10 and FIG. 11, respectively. Further, the quadratic phase-shift keying (QPSK) modulation is considered.

With respect to FIG. 9, for example, when the receiver device 100 performs the first iterative procedure (only the set of PIC iterations). As can be derived from FIG. 9, the SC-LVDM obtained by the receiver device 100 outperforms SC-FDE by 10 dB at BER = 10 ^-4 .

Furthermore, it can be derived that the PIC converges within 2 iterations with SC-LVDM, while it needs 8 iterations to converge for SC-FDE. Moreover, the overall complexity of PIC is O(2IK ² ), where I is the number of iterations.

FIG. 10 shows the performance results when only the iterative procedure comprising the set of ICR iterations is performed.

In particular, a received signal 120 is demodulated by the first Vandermonde matrix E 706, afterwards, a one-tap equalization is performed at one-tap equalizer block 703. At next, a post- equalization is made at block 705. Furthermore, a first hard decision is made and the obtained symbol 130 went directly through the ICR processing, i.e., without undergoing a set of PIC iterations. As can be derived from FIG. 10, when using the second iterative procedure only with the conventional SC-LVDM and SC-FDE schemes (no PIC iterations in Stage 1), the performance is saturated as it is depicted in FIG. 10. Furthermore, the second iterative procedure converges within 3 iterations for both schemes, SC-LVDM and SC-FDE, while it brings performance enhancement in low signal to noise ratios (SNRs). Here, the complexity order is O((1 + 4L)IK ), where I is the number of iterations in the second iterative procedure and L is the channel delay spread.

With respect to FIG. 11, the performance results are provided for the SC-LVDM and the SC- FDE. In this case, at first, the iterative procedure comprising the set of PIC iterations is performed on the demodulated signal 121. Moreover, the first result of hard decision is the obtained symbol 130 (i.e., the output of PIC processing). At next, the obtained symbol 130 undergone the second iterative procedure comprising the set of ICR iterations, for refinement and the symbols 230 are obtained.

As can be derived from the diagram illustrated in FIG. 11, the SC-LVDM outperforms SC-FDE when using the proposed scheme which converges faster than when using PIC only. Furthermore, the second iterative procedure (the set of ICR iterations) provides performance enhancement in low SNR range while the first iterative procedure (the set of PIC iterations) relaxes the performance saturation that the second iterative procedure (the set of ICR iterations) would bring (see FIG. 10), if it would be applied alone. Thus, performing the first iterative procedure and the second iterative procedure together may enable the proposed scheme stronger than performing either the first iterative procedure alone or the second iterative procedure alone.

Next, simulation results in the third-generation partnership project (3GPP) channels are provided.

2. 3GPP Channels

For example, K = 64 is set. Further, the carrier frequency f _c = 3.5 GHz, the velocity is v = 200 Km/h, Doppler spread is f _D = 648 Hz, and subcarriers spacing Δf = 30 KHz. Further, the QPSK modulation is considered.

The performance results of SC-LVDM and SC-FDE in 3GPP ETU channels, using L = 10, are shown in FIG. 12. Moreover, it can be derived that the SC-LVDM outperforms the SC-FDE when using the proposed receiver devices in Extended Typical Urban (ETU) channels. Therein, the first iterative procedure uses 3 and 6 iterations for SC-LVDM and SC-FDE, respectively, while the second iterative procedure uses 3 iterations for both. Moreover, the second iterative procedure provides a performance enhancement of 3 dB for SC-LVDM, and a gain of 5 dB for SC-FDE. The performance results of SC-LVDM and SC-FDE in 3GPP EVB channels, using L = 39, are shown in FIG.13. Moreover, it can be derived that the SC-LVDM performs well in Extended Vehicular B (EVB) channels where the second iterative procedure of the proposed receiver devices brings 5 dB. Here, the SC-LVDM outperforms the SC-FDE. In FIG.9, FIG.10, FIG.11, FIG.12 and FIG.13, the performance result are obtained by using perfect CSI at the receiver devices 100 or 200. The perfect CSI may represent a realistic CSI, as it is generally known. For example, considering that the receiver devices need to know the channel, in order to perform the equalization, and consequently, detect the transmitted symbols. In these cases, it is assumed that the receiver devices know the channel, perfectly. Next, the performance results are shown using the CE algorithm, for obtaining the CSI, according to Eq. (11) to Eq. (24), as discussed above. Moreover, the robustness is shown for the channel estimation algorithm. In FIG. 14B, the Normalized Mean Square Error (NMSE) is shown for the assessment of channel estimation. The direct impact is two-fold, for example, the estimated channel should be used to determine optimized radius of the circle, and, moreover, the optimized radius of the circle should be used for building the blocks of the receiver device (for detection of symbols), as described above. Without limiting the present disclosure, in FIG.14A and FIG.14B, the performances are shown for the SC-LVDM and the SC-FDE in EVB channels. In FIG.14A and 14B, the sensitivity to the channel and a _opt in 3GPP EVB channels is obtained using L = 39. The pilot vectors have been arranged as it was discussed with respect to FIG.6, where NP = 6 and D = 5. FIG. 14A shows that the proposed channel estimation algorithm performs well. Further, the normalized MSE (NMSE) of a _opt estimation is depicted in FIG.14B. Therefore, it may be derived that the proposed advanced receiver devices remain robust and the SC-LVDM outperforms the SC-FDE in 3GPP EVB channels, when using either perfect or imperfect CSI. FIG. 15 shows a method 1500 according to an embodiment of the present disclosure for a receiver device 100 for single-carrier modulation. The method 1500 may be carried out by the receiver device 100 or the receiver device 200 as described above. Without limiting the present disclosure, in the following, the method 1500 is exemplarily discussed as a method being performed by the receiver device 100. The method 1500 comprises a step S1501 of determining CSI 101 of a communication channel 111 between the receiver device 100 and a transmitter device 110. The method 1500 further comprises a step S1502 of determining a plurality of signature roots based on the CSI 101, wherein each signature root of the plurality of signature roots is a nonzero complex point, and wherein the plurality of signature roots are uniformly distributed on a circumference of a circle in a complex plane. The method 1500 further comprises a step S1503 of constructing a first Vandermonde matrix based on the plurality of signature roots. The method 1500 further comprises a step S1504 of performing a demodulation of a single- carrier modulated signal 120 received from the transmitter device 110, based on the first Vandermonde matrix, to obtain a demodulated signal 121. The method 1500 further comprises a step S1505 of obtaining symbols 130 based on performing a first iterative procedure on the demodulated signal 121, the first iterative procedure comprising a set of PIC iterations. FIG. 16 shows a method 1600 according to an embodiment of the present disclosure for a receiver device 200 for single-carrier modulation. The method 1600 may be carried out by the receiver device 100 or the receiver device 200 as described above. Without limiting the present disclosure, in the following, the method 1600 is exemplarily discussed as a method being performed by the receiver device 200. The method 1600 comprises a step S1601 of determining CSI 201 of a communication channel 111 between the receiver device 200 and a transmitter device 110. The method 1600 further comprises a step S1602 of determining a plurality of signature roots based on the CSI 201, wherein each signature root of the plurality of signature roots is a nonzero complex point, and wherein the plurality of signature roots are uniformly distributed on a circumference of a circle in a complex plane. The method 1600 further comprises a step S1603 of constructing a first Vandermonde matrix and a second Vandermonde matrix based on the plurality of signature roots. The method 1600 further comprises a step S1604 of performing a demodulation of a single- carrier modulated signal 220 received from the transmitter device 110, based on the first Vandermonde matrix, to obtain a demodulated signal 221. The method 1600 further comprises a step S1605 of obtaining symbols 230 based on performing an iterative procedure on estimated symbols obtained by a frequency-domain equalization comprising a one-tap equalization followed by a post-equalization performed based on the second Vandermonde matrix, the iterative procedure comprising a set of interference cancellation refinement, ICR, iterations. The present disclosure has been described in conjunction with various embodiments as examples as well as implementations. However, other variations can be understood and effected by those persons skilled in the art and practicing the claimed disclosure, from the studies of the drawings, this disclosure and the independent claims. In the claims as well as in the description the word “comprising” does not exclude other elements or steps and the indefinite article “a” or “an” does not exclude a plurality. A single element or other unit may fulfill the functions of several entities or items recited in the claims. The mere fact that certain measures are recited in the mutual different dependent claims does not indicate that a combination of these measures cannot be used in an advantageous implementation.