Patent Description:
The present disclosure specifically presents a transceiver device, which provides a new waveform design based on a Lagrange-Vandermonde (LV) multicarrier modulation scheme or a Vandermonde- Lagrange (VL) multicarrier modulation scheme, which may allow for single-tap equalization with a low complex transceiver design. The present invention also presents a transmitter device, in particular for a multicarrier modulation scheme such as the LV or the VL multicarrier modulation scheme. The invention also presents a receiver device, in particular for a multicarrier modulation scheme such as the LV or the VL multicarrier modulation scheme.

The 3rd Generation Partnership Project (3GPP) agreed to use the Orthogonal Frequency Division Multiplexing (OFDM) (e.g., with some small modifications) for the Fifth Generation (<NUM>) mobile communications. Although, such decision may be useful in terms of backwards compatibility to the Fourth Generation (<NUM>) wireless systems, it is not the most efficient technique for all possible use cases. Moreover, the OFDM may be used as the modulation format (or the waveform) in both downlink (DL) and uplink (UL) transmissions. Furthermore, while the OFDM use is well-known for DL transmissions, it is new for the UL transmissions in the mobile communication networks. In addition, having the same waveform in both UL and DL transmissions may enable an easier communication for the device-to-device communication in future releases. However, the future mobile systems may be highly heterogeneous, and may also be characterized by a large range of possible use cases, for example, ranging from enhanced Mobile BroadBand (eMBB) over enhanced Machine Type Communications (eMTC) to Ultra-Reliable Low latency Communications (URLLC) in the vehicular communications. Therefore, a more flexible waveform design is of great significance to handle all the aforementioned use cases.

Two conventional OFDM-based schemes, referred to as the Cyclic Prefix (CP) OFDM scheme <NUM> and the Zero-Padding (ZP) OFDM scheme <NUM>, are schematically illustrated in <FIG> and <FIG>, respectively. The CP-OFDM scheme <NUM> and ZP-OFDM scheme <NUM> are known to guarantee Inter-Symbol Interference (ISI) cancellation. Moreover, assuming that K subcarriers have been used in both schemes, the frequency domain received signal (at the output of the demodulator) is given by Eq. <NUM>: <MAT> where <MAT> is the frequency response channel at the kth subcarrier, and F is a Discrete Fourier Transform (DFT) K × K matrix given by Eq. <NUM>: <MAT>.

However, it has the obvious drawback that the symbol sk(n) transmitted on the kth subcarrier cannot be recovered when it is hit by a channel zero (Hk = <NUM>). In this case, the Perfect Recovery (PR) condition has not been satisfied.

Generally, a Lagrange-Vandermonde scheme has also been proposed for Code-Division Multiple Access (CDMA) systems. <FIG> schematically illustrates a conventional scheme based on a Mutually-Orthogonal Usercode-Receiver (AMOUR) block diagram <NUM>. The AMOUR system is the most generalized framework for quasi-synchronous blind CDMA which has been proposed.

In the AMOUR system <NUM>, following operations may be performed:.

However, the conventional devices and methods have the following disadvantages:.

Although there exist techniques for providing a multicarrier modulation scheme (e.g., the conventional ZP-OFDM scheme and the conventional CP-OFDM scheme), it is generally desirable to provide improved devices and methods, e.g. for providing a multicarrier modulation scheme.

<NPL>, discloses developing a generalized frequency-hopping (GFH) orthogonal frequency-division multiple-access (OFDMA) system as a structured long code direct-sequence code-division multiple-access (DS-CDMA) system in order to bridge frequency-hopped multicarrier transmissions with long code DS-CDMA.

<NPL>, discloses a mutually-orthogonal usercode-receiver (AMOUR) system for quasi-synchronous blind code-division multiple-access (CDMA) that eliminates multiuser interference (MUI) deterministically and mitigates fading regardless of the unknown multipath and the adopted signal constellation.

In view of the above-mentioned problems and disadvantages, embodiments of the present invention aim to improve the conventional devices and methods. An objective is thereby to provide devices and methods for providing a new multicarrier modulation scheme.

The objective of the present invention is achieved by the embodiments provided in the enclosed independent claims. Advantageous implementations of the embodiments are further defined in the dependent claims. In the following, parts of the description and drawings referring to embodiments not covered by the claims, are not part of the invention, but are illustrative examples necessary for understanding the invention.

In particular the embodiments of the invention proposes devices and methods based on two multicarrier modulation schemes referred to as Lagrange-Vandermonde modulation scheme and Vandermonde-Lagrange modulation scheme that may generalize the conventional ZP-OFDM and CP-OFDM, respectively, while satisfying the PR condition.

The main advantages of the embodiments of the invention can be summarized as follows:.

A first aspect of the invention provides a transmitter device for a multicarrier modulation as claimed in claim <NUM>.

The transmitter device of the first aspect may provide, for example, a precoder or a modulation scheme with (perfect) inter-symbol interference cancellation.

In some embodiments, the transmitter device may obtain the plurality of signature roots, for example, the transmitter device may receive a feedback message from the receiver device. The feedback message may indicate the radius of a circle that the plurality of signature roots uniformly distributed on its circumference. Moreover, the transmitter device may obtain the signature roots based on the radius of the circle.

In an implementation form of the first aspect, the feedback message indicates a radius of a circle, wherein the plurality of signature roots are uniformly distributed on the circumference of the circle.

In particular, the transmitter device may obtain a feedback message from a receiver device. The receiver device may determine (and may further provide to the transmitter device) the signature roots that are uniformly distributed on the circumference of the circle, for example, the plurality of the signature roots may spread over a circle having a radius of "a". Moreover, for K subcarriers, the signature roots may be given by <MAT>.

Moreover, every user has its own channel environment for which the signature roots may further be modified (e.g., optimized). In addition, the K signature roots of every user m may be modified (e.g., optimized) according to a certain metric.

In a further implementation form of the first aspect, the transmitter device is further configured to allocate a determined transmit power to each subcarrier of the multicarrier modulated signal according to a tuning factor estimated based on the radius of the circle. In particular, the transmitter device may use a precoder that tunes the transmit power over the multicarrier (or transmitter device filter bank). Moreover, the tuning factor (κk) may depend on the radius of the circle "a", e.g., on the value of the radius according to Eq. <NUM>: <MAT>.

In a further implementation form of the first aspect, wherein the plurality of signature roots are obtained based on <MAT> where ρk corresponds to a signature root related to the kth subcarrier, where a corresponds to the radius of the circle, and where K is the number of the subcarriers.

In a further implementation form of the first aspect, the feedback message indicates at least one vector for the plurality of signature roots.

For example, the receiver device may modify the plurality of the signature roots. Furthermore, the receiver device may send at least one vector for the plurality of signature roots which may indicate the modified signature roots. The vector may be a vector of the complex points. The transmitter device may use the vector and may construct the Vandermonde matrix or the Lagrange matrix based on the modified signature roots.

In a further implementation form of the first aspect, the transmitter device is further configured to allocate a determined transmit power to each subcarrier of a multicarrier modulated signal according to a tuning factor estimated based on the plurality of signature roots.

For example, the tuning factor may be estimated based on the plurality of signature roots (ρk) and according to Eq. <NUM>: <MAT>.

In a further implementation form of the first aspect, the transmitter device is further configured to perform, when constructing a Lagrange matrix, a zero-padding procedure on the multicarrier modulated signal; or perform, when constructing a Vandermonde matrix, a cyclic-prefix procedure on the multicarrier modulated signal.

A second aspect of the application, not covered by the claims, provides a receiver device for a multicarrier modulation scheme, the receiver device being configured to determine a plurality of signature roots, wherein each signature root is a nonzero complex point; construct a Lagrange matrix or a Vandermonde matrix from the plurality of signature roots; and perform a demodulation of a multicarrier modulated signal based on the Lagrange matrix or the Vandermonde matrix. The receiver device of the second aspect may satisfy a Perfect Recovery (PR) condition. For example, in some embodiments, the transmitter device may use the precoder or the modulation scheme with perfect ISI cancellation. Moreover, a linear receiver device with single tap equalization may be provided. With the knowledge of channel sate information at this stage, a linear reduced-complexity receiver device may be provided which may satisfy the perfect recovery condition.

In an implementation form of the second aspect, the receiver device is further configured to determine a radius of a circle based on channel state information of a communication channel, wherein the determined plurality of signature roots are uniformly distributed on the circumference of the circle.

For example, for K subcarriers, the receiver device may determine (choose) the signature roots that are uniformly distributed on the circumference of the circle.

In a further implementation form of the second aspect, the receiver device is further configured to send a feedback message to a transmitter device indicating the radius of the circle.

As discussed, the receiver device may choose the plurality of signature roots which are uniformly distributed on the circumference of the circle. The circle may have the radius of "a". Moreover, the receiver device may send a feedback message to the transmitter device which may indicate the radius "a" of the circle.

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

The receiver device may further modify the radius "a" through an optimization block using a metric (such as the Mean Squared Error (MSE)) and may further obtain aopt. The aopt is the radius "a" which may be modified, optimized, etc. Moreover, the receiver device may send a feedback message to the transmitter device and may provide the "aopt" to the transmitter device.

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.

For example, the receiver device may include a p refinement algorithm or a p refinement unit which may be configured to modify (e.g., refine, optimize) the plurality of signature roots. In particular, the p refinement algorithm may be based on the machine learning algorithm such as the gradient descent algorithm.

In a further implementation form of the second aspect, the receiver device is further configured to determine at least one vector for the plurality of signature roots, based on the individual modification of each signature root; and send a feedback message to the transmitter device indicating the at least one vector for the plurality of signature roots.

For example, at least one signature root may be modified. Moreover, the receiver device may determine at least one vector for the modified signature root and may further provide the vector to the transmitter device.

The receiver may uses aopt or the refined signature points to construct the receive filters (e.g., the Vandermonde matrix) and it may further feedback the aopt or the refined signature points to the transmitter device in order to construct the transmit filters (e.g., the Vandermonde matrix).

In a further implementation form of the second aspect, the receiver device is further configured to perform a one-tap equalization on the demodulated signal, based on the plurality of signature roots.

A third aspect of the invention provides a transceiver device comprising a transmitter according to the first aspect or one of the implementation form of the first aspect and a receiver device according to second aspect or one of the implementation form of the second aspect, as claimed in claim <NUM>.

The transceiver device of the third aspect may comprise the transmitter device (according to the first aspect or one of the implementation form of the first aspect) which may provide the precoder or the modulation scheme with perfect ISI cancellation. Moreover, the transceiver device of the third aspect may further comprise the receiver device (according to second aspect or one of the implementation form of the second aspect) which may be based on a linear reduced-complexity receivers with single tap equalization that satisfies the perfect recovery condition.

A fourth aspect of the application, not covered by the claims, provides a transceiver device for a multicarrier modulation scheme, the transceiver device comprising a transmitter device configured to generate a multicarrier modulated signal based on constructing a Lagrange matrix or a Vandermonde matrix; and a receiver device configured to perform a demodulation of the multicarrier modulated signal based on constructing the other matrix from the Lagrange matrix or the Vandermonde matrix constructed by the transmitter device.

In particular, the transceiver device of the fourth aspect may be based on (e.g., it may provide) the two multicarrier modulation schemes referred to as Lagrange-Vandermonde modulation scheme and Vandermonde-Lagrange modulation scheme that may generalize the conventional ZP-OFDM and CP-OFDM, respectively.

A fifth aspect of the invention provides a method for being implemented at a transmitter device as claimed in claim <NUM>.

In an implementation form of the fifth aspect, the feedback message indicates a radius of a circle, wherein the plurality of signature roots are uniformly distributed on the circumference of the circle.

In a further implementation form of the fifth aspect, the method further comprises allocating a determined transmit power to each subcarrier of the multicarrier modulated signal according to a tuning factor estimated based on the radius of the circle.

In a further implementation form of the fifth aspect, the plurality of signature roots are obtained based on <MAT> where ρk corresponds to a signature root related to the kth subcarrier, where a corresponds to the radius of the circle, and where K is the number of the subcarriers.

In a further implementation form of the fifth aspect, the feedback message indicates at least one vector for the plurality of signature roots.

In a further implementation form of the fifth aspect, the method further comprises allocating a determined transmit power to each subcarrier of a multicarrier modulated signal according to a tuning factor estimated based on the plurality of signature roots.

In a further implementation form of the fifth aspect, the method further comprises performing, when constructing a Lagrange matrix, a zero-padding procedure on the multicarrier modulated signal; or performing, when constructing a Vandermonde matrix, a cyclic-prefix procedure on the multicarrier modulated signal.

A sixth aspect of the application, not covered by the claims, provides a method for being implemented at a receiver device, the method comprising determining a plurality of signature roots, wherein each signature root is a nonzero complex point; constructing a Lagrange matrix or a Vandermonde matrix from the plurality of signature roots; and performing a demodulation of a multicarrier modulated signal based on the Lagrange matrix or the Vandermonde matrix.

In an implementation form of the sixth aspect, the method further comprises determining a radius of a circle based on channel state information of a communication channel, wherein the determined plurality of signature roots are uniformly distributed on the circumference of the circle.

In a further implementation form of the sixth aspect, the method further comprises sending a feedback message to a transmitter device indicating the radius of the circle.

In a further implementation form of the sixth aspect, the method further comprises computing a metric for evaluating the radius of the circle and/or the plurality of signature roots, based on the channel state information of the communication channel.

In a further implementation form of the sixth aspect, the method further comprises modifying 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 sixth aspect, the method further comprises determining at least one vector for the plurality of signature roots, based on the individual modification of each signature root; and sending a feedback message to the transmitter device indicating the at least one vector for the plurality of signature roots.

In a further implementation form of the sixth aspect, the method further comprises performing a one-tap equalization on the demodulated signal, based on the plurality of signature roots.

A seventh aspect of the application, not covered by the claims, provides a method for being implemented at a transceiver device, the method comprising generating, at a transmitter device, a multicarrier modulated signal based on constructing a Lagrange matrix or a Vandermonde matrix; and performing, at a receiver device, a demodulation of the multicarrier modulated signal based on constructing the other matrix from the Lagrange matrix or the Vandermonde matrix constructed by the transmitter device.

In an implementation form of the seventh aspect, the method further comprises allocating, at the transmitter device, a determined transmit power to each subcarrier of the multicarrier modulated signal according to a tuning factor estimated based on the radius of the circle.

In a further implementation form of the seventh aspect, the method further comprises determining the plurality of signature roots ( ρk ) based on <MAT> where ρk corresponds to a signature root related to the kth subcarrier, where a corresponds to the radius of the circle, and where K is the number of the subcarriers.

In a further implementation form of the seventh aspect, the method further comprises sending a feedback message from the receiver device to the transmitter device, wherein the feedback message indicates at least one vector for the plurality of signature roots (ρk).

In a further implementation form of the seventh aspect, the method further comprises allocating at the transmitter device, a determined transmit power to each subcarrier of a multicarrier modulated signal according to a tuning factor estimated based on the plurality of signature roots (ρk).

In a further implementation form of the seventh aspect, the method further comprises performing, at the transmitter device, when constructing a Lagrange matrix, a zero-padding procedure on the multicarrier modulated signal; or performing, at the transmitter device, when constructing a Vandermonde matrix, a cyclic-prefix procedure on the multicarrier modulated signal.

In a further implementation form of the seventh aspect, the method further comprises determining, at the receiver device, a radius of a circle based on channel state information of a communication channel, wherein the determined plurality of signature roots (ρk) are uniformly distributed on the circumference of the circle.

In a further implementation form of the seventh aspect, the method further comprises computing a metric for evaluating the radius of the circle and/or the plurality of signature roots (ρk), based on the channel state information of the communication channel.

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

In a further implementation form of the seventh aspect, the method further comprises determining, at the receiver device, at least one vector for the plurality of signature roots (ρk), based on the individual modification of each signature root; and sending a feedback message to the transmitter device indicating the at least one vector for the plurality of signature roots (ρk).

In a further implementation form of the seventh aspect, the method further comprises performing, at the receiver device, a one-tap equalization on the demodulated signal, based on the plurality of signature roots (ρk).

<FIG> is a schematic view of a transmitter device <NUM> for a multicarrier modulation scheme, according to an embodiment of the present invention.

The transmitter device <NUM> for the multicarrier modulation scheme is configured to obtain a plurality of signature roots ρk based on receiving a feedback message <NUM> from a receiver device <NUM>, wherein each signature root is a nonzero complex point.

The transmitter device <NUM> is further configured to construct a Lagrange matrix <NUM>-L or a Vandermonde matrix <NUM>-V from the plurality of signature roots ρk.

The transmitter device <NUM> is further configured to generate a multicarrier modulated signal <NUM>-L, <NUM>-V based on the Lagrange matrix <NUM>-L or the Vandermonde matrix <NUM>-V.

The transmitter device <NUM> may comprise processing circuitry (not shown) configured to perform, conduct or initiate the various operations of the transmitter device <NUM> 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 multipurpose 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 transmitter device <NUM> to perform, conduct or initiate the operations or methods described herein.

Moreover, in some embodiments, the transmitter device <NUM> may further be incorporated in a transceiver device.

<FIG> is a schematic view of a receiver device <NUM> for a multicarrier modulation scheme.

The receiver device <NUM> for the multicarrier modulation scheme is configured to determine a plurality of signature roots ρk, wherein each signature root is a nonzero complex point. The receiver device <NUM> is further configured to construct a Lagrange matrix <NUM>-L or a Vandermonde matrix <NUM>-V from the plurality of signature roots ρk.

The receiver device <NUM> is further configured to perform a demodulation <NUM>-V, <NUM>-L of a multicarrier modulated signal <NUM>-L, <NUM>-V based on the Lagrange matrix <NUM>-L or the Vandermonde matrix <NUM>-V.

The receiver device <NUM> may comprise processing circuitry (not shown) configured to perform, conduct or initiate the various operations of the receiver device <NUM> 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 multipurpose 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 <NUM> to perform, conduct or initiate the operations or methods described herein.

Moreover, in some embodiments, the receiver device <NUM> may further be incorporated in a transceiver device.

<FIG> is a schematic view of a transceiver device <NUM> for a multicarrier modulation scheme, according to an embodiment of the present invention.

The transceiver device <NUM> comprises a transmitter device <NUM> configured to generate a multicarrier modulated signal <NUM>-L, <NUM>-V based on constructing a Lagrange matrix <NUM>-L or a Vandermonde matrix <NUM>-V.

The transceiver device <NUM> further comprises a receiver device <NUM> configured to perform a demodulation <NUM>-V, <NUM>-L of the multicarrier modulated signal <NUM>-L, <NUM>-V based on constructing the other matrix <NUM>-V, <NUM>-L from the Lagrange matrix or the Vandermonde matrix constructed by the transmitter device <NUM>.

For example, the transceiver device <NUM> may be based on a LV multicarrier modulation scheme. For instance, the transmitter device <NUM> of the transceiver device <NUM> may generate the multicarrier modulated signal <NUM>-L based on constructing a Lagrange matrix <NUM>-L. Moreover, the receiver device <NUM> may obtain the multicarrier modulated signal <NUM>-L and may further construct the Vandermonde matrix <NUM>-V from the plurality of signature roots ρk. Furthermore, the receiver device <NUM> may perform the demodulation <NUM>-V of the multicarrier modulated signal <NUM>-L based on the Vandermonde matrix <NUM>-V.

Similarly, the transceiver device <NUM> may be based on a VL multicarrier modulation scheme. For instance, the transmitter device <NUM> of the transceiver device <NUM> may generate the multicarrier modulated signal <NUM>-V based on constructing a Vandermonde matrix <NUM>-V. Moreover, the receiver device <NUM> may obtain the multicarrier modulated signal <NUM>-V and may further construct the Lagrange matrix <NUM>-L from the plurality of signature roots ρk. Furthermore, the receiver device <NUM> may perform the demodulation <NUM>-L of the multicarrier modulated signal <NUM>-V based on the Lagrange matrix <NUM>-L.

The transceiver device <NUM> may comprise processing circuitry (not shown) configured to perform, conduct or initiate the various operations of the transceiver device <NUM> 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 multipurpose 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 transceiver device <NUM> to perform, conduct or initiate the operations or methods described herein.

In the following, some mathematical basics and notation are briefly discussed, that may be used by the transmitter device <NUM> and/or the receiver device <NUM> and/or the transceiver device <NUM>, without limiting the present invention.

For example, from a set of K distinct nonzero complex points <MAT>, that are refered to as signature roots, a Vandermonde matrix may be constructed. The Vandermonde matrix, is a K × P matrix, given by Eq. <NUM>: <MAT>.

Moreover, note that, if <MAT>, therefore, VK×K = FK×K which is the Discerete Fouriuer Transofrm (DFT) matrix given above.

Furthermore, the Lagrange basis polynomials (e.g., a K polynomials) may be obtained according to Eq. <NUM> <MAT> where, κk is a tuning factor that normalizes the transmitter device filter (Fk) energy. Moroever, a Lagrange matrix may be constructed, given by Eq. <NUM>: <MAT>.

Note that, Fk(ρl) = κk δ(k - l) where k,l ∈ [<NUM>, K - <NUM>]. Furthermore, the following identity may be verified: <MAT> where κk are the tuning factors defined above.

Reference is made to <FIG> which is an exemplarily scheme of the transceiver device <NUM> comprising the transmitter device <NUM> using a Lagrange matrix for modulation and the receiver device <NUM> using a Vandermonde matrix for demodulation.

In the block diagram of the LV modulator of <FIG>, the transceiver device <NUM> (i.e., being based on a LV modulator) is exemplarily shown for K signature roots. The transceiver device <NUM> comprises the transmitter device <NUM> which includes a precoder <NUM>, a modulator <NUM> and a ZP block <NUM>.

The precoder <NUM> may apply the tuning factors κk, for example, for allocating the determined transmit power, which may be KxK diagonal matrix (Ω) in <FIG>. Moreover, the modulator <NUM> uses the Lagrange matrix (R in <FIG>) which has a size of KxK (for example, it may construct a Lagrange matrix <NUM>-L and may further generate a multicarrier modulated signal <NUM>-L based on the Lagrange matrix <NUM>-L).

Furthermore, the ZP block <NUM> may be used for the zero-padding procedure, where every input block of K symbols will be trailed by L zeros. Therefore, it may provide and may further output block symbols with the length of P, where P =K + L.

Moreover, the communication channel of the transceiver device <NUM> comprises the transmitter filter (Tx filter) <NUM> and the receiver filter (Rx filter) <NUM> (for example, they may be raised cosine filters). In addition, the parameter C <NUM> which is a propagation channel of order L may be obtained according to Eq. <NUM>: <MAT>.

Furthermore, the convolution of the Tx filter <NUM>, the C <NUM> and the Rx filter may be given by a channel matrix H.

The transceiver device <NUM> further comprises the receiver device (Rx) <NUM> which includes the demodulator <NUM>, the one-tap Equalizer unit <NUM> and the decision block <NUM>.

The demodulator <NUM>, perform a demodulation based on constructing a matrix E which is a Vandermonde matrix having a size of KxP. The one-tap equalizer <NUM> uses a KxK diagonal matrix (for example, it may construct a Vandermonde matrix <NUM>-V and may further perform a demodulation <NUM>-V of a multicarrier modulated signal <NUM>-L based on the Vandermonde matrix <NUM>-V).

Furthermore, a convolution of the modulation, channel, and demodulation, is given by Eq. <NUM>: <MAT>.

Note that, the following operations or conditions may be performed or satisfied.

The proposed multicarrier modulation scheme (e.g., the Lagrange-Vandermonde multicarrier modulation scheme presented in <FIG>) may generalize the conventional ZP-OFDM modulation, and may further satisfy the PR condition. At next, an exemplarily procedure is provided which discusses that this generalization may be achieved while satisfying the transmit power constraint.

As discussed above, in some embodiments, the plurality of signature roots may be modified (e.g., they may migrate, refined, optimized, or the like). However, if the transceiver device send using K signature roots, the optimization should be carried out over <MAT>where the complexity increases with the K.

This problem may be solved based on operations performed in the following two steps including step <NUM> and step <NUM>:.

In the embodiment of <FIG> in which the tranceiver device is based on an LV Modulator, all of the Tx Filters ( Fk ) may have the same energy and may be normalized by <MAT>, ∀k and Eq. <NUM> may further be obtained: <MAT>.

Furthermore , The Lagrange matrix R reduces to a Vandermonde, given by Eq. <NUM>: <MAT>.

Note that, when R reduces to a Vandermonde matrix, a low-complex transceiver may be implemented (for example, based on a simple one-tap equalization and no matrix inversion is required as the AMOUR system <NUM> in <FIG>).

Moreover, if a = <NUM>, therefore, the following operation is satisfied:
<MAT>.

From the above operations (e.g., the Eq. <NUM>) it may be determined that the LV modulator (i.e., the Lagnrange-Vandemonde multicarrier modulation scheme of the invention) generalizes the conventional ZP-OFDM multicarrier modulation scheme.

Furthermore, if a = <NUM> is considered, therefore, D may be the Channel frequency response while satisfying the PR condition.

Moreover, a procedure for modifying the radius of the circle may be provided. For example, the transceiver device <NUM> (e.g., its receiver device <NUM>) may modify (e.g., optimize) the radius of the circle.

Without loss of generalities, it may be derived that both LV and VL modulators end up with the same optimization metric's expression. In the following, the LV modulator scheme is discussed, while VL modulator may be deduced accordingly.

Referring to <FIG>, the received signal (at the input of the demodulator E) may be given by: <MAT>.

Therefore, the demodulated signal is given by: <MAT>.

Moreover, the one tap-equalization is given by: <MAT> here, it may be determined that, a perfect recovery of s is satisfied.

In addition, a method, among other, for optimizing the radius "a" is to minimize the mean squared error (MSE) given by Eq. <NUM> as follow: <MAT>.

Moreover, in some embodiments, a uniform power allocation over subcarriers (defined by signature roots) may be used, and by using the same tuning factor <MAT>, the MSE expression is given by the MSE = K-<NUM>E{uHu} and according Eq. <NUM>: <MAT>.

Therefore, the aopt may be determined as <MAT>.

Additionally, in some embodiments, the power allocation may be optimized, for example, by using different κk that minimize the MSE given by Eq. <NUM> as follow: <MAT>.

The xk =|κk|-<NUM>|C(ρk)|-<NUM> may be set, and the problem formulation may be according to Eq. <NUM> as follow: <MAT>.

Furthermore, the optimal κk and the MSEmin may be given by Eq. <NUM> and Eq. <NUM> as: <MAT> and <MAT>.

Consequently, the aopt may be determined as <MAT>.

For example, the signature roots that uniformly spread over a circle of radius aopt may be used, and an algorithm may further be applied that may optimize the signature roots individually following a specific optimization metric. In particular, a machine learning techniques may be used in this step.

In the following this step is exemplarily referred to as the "signature roots refinement". A detailed description of this step is provided, for example, in <FIG> and <FIG>. Reference is made to <FIG> which is an exemplarily scheme of the transceiver device <NUM> comprising the transmitter device <NUM> using a Vandermonde matrix for modulation and the receiver device <NUM> using a Lagrange matrix for demodulation.

In the block diagram of the VL modulator of <FIG>, the transceiver device <NUM> (i.e., being based on a VL modulator) is exemplarily shown for K signature roots. The transceiver device <NUM> comprises the transmitter device <NUM> which includes a precoder <NUM> and a modulator <NUM>.

The precoder <NUM> of the transmitter device <NUM> may apply the tuning factors κk, for example, for allocating the determined transmit power, which may be KxK diagonal matrix (Ω).

Moreover, the modulator <NUM> of the transmitter device <NUM> uses the Vandermonde matrix V (in <FIG>) of size PxK, where P =K + L. For example, it may construct a Vandermonde matrix <NUM>-V and may further generate a multicarrier modulated signal <NUM>-V based on the Vandermonde matrix <NUM>-V.

The transceiver device <NUM> further comprises the receiver device (Rx) <NUM> which includes CP removal block <NUM>, the demodulator <NUM>, the one-tap Equalizer unit <NUM> and the decision block <NUM>.

The CP removal block <NUM> may be given by [<NUM>K×L IK×K] where IK×K is the identity matrix.

The demodulator <NUM>, perform a demodulation based on constructing a matrix L which is a Lagrange matrix of size KxK. For example, it may construct a Lagrange matrix <NUM>-L and may further perform a demodulation <NUM>-L of a multicarrier modulated signal <NUM>-V based on the a Lagrange matrix <NUM>-L.

The one-tap equalizer <NUM> uses a KxK diagonal matrix, and its output is provided to the decision block <NUM>.

The proposed multicarrier modulation scheme (e.g., the Vandermonde-Lagrange multicarrier modulation scheme presented in <FIG>) may generalize the conventional CP-OFDM modulation scheme, and may further satisfy the PR condition.

As discussed above, the plurality of signature roots may be modified. However, if sending using K signature roots, the modification (e.g., optimization) may be carried out over <MAT>where the complexity increases with K.

For example, the plurality of signature roots (ρk) may be uniformly distributed on the circumference of the circle, e.g., uniformly spread over a circle of radius a, such that ρk = <MAT>.

In the embodiment of <FIG> in which the transceiver device is based on a VL Modulator, a Lagrange basis polynomials may be used at the receiver device, given by Eq. <NUM>: <MAT>.

Moreover, the Lagrange matrix L reduces to a Vandermonde, given by Eq. <NUM>: <MAT>.

From the above operations it may be determined that the VL modulator (i.e., the Vandemonde- Lagnrange multicarrier modulation scheme of the invention) generalizes the conventional CP-OFDM multicarrier modulation scheme.

Similar to the embodiment of <FIG> (i.e., being based on the LV modulator), it may be derived that both LV and VL modulators end up with the same optimization metric's expression. A repeated derivation of equations for the VL modulator is omitted, as it can be derived by the skilled person.

In the following this step is exemplarily referred to as the "signature roots refinement". A detailed description of this step is provided, e.g., in <FIG> and <FIG>.

Reference is made to <FIG> which is a schematic view for signaling exchange indicating a radius aopt of a circle.

The present invention may provide (e.g., identify and propose) a new waveform that may satisfy the perfect recovery condition while keeping a low complex transceiver implementation. Without limiting the present invention, the signaling exchange indicating the radius of the circle is exemplarily discussed for a transceiver device <NUM> being based on a transceiver device <NUM> comprising a transmitter device <NUM> using a Lagrange matrix <NUM>-L for modulation <NUM>-L and a receiver device <NUM> using a Vandermonde matrix <NUM>-V for demodulation <NUM>-V. However, such a signaling exchange for a transceiver device <NUM> being based on a VL modulator can also be deduced accordingly and a repeated description (i.e., for a transceiver being based on a VL modulator) is omitted, since the VL modulator will follow same steps.

In the signalling exchange the following three operations may be performed.

In some embodiments of the invention, the above mentioned step <NUM> (i.e., Step I: choosing the plurality of signature roots) may only be performed (i.e., the above step may be enough).

Moreover, in some embodiments, (e.g., depending on the use case), the above mentioned step <NUM> (i.e., Step <NUM>: modifying the plurality of the signature roots) may further be performed, which is exemplarily discussed, e.g., in <FIG> and <FIG>.

Reference is made to <FIG> which is a schematic view for signaling exchange indicating the signature root refinement.

Without the limiting the present invention, the signaling exchange indicating the signature root refinement is exemplarily discussed for a transceiver device <NUM> being based on a transceiver device <NUM> comprising a transmitter device <NUM> using a Lagrange matrix <NUM>-L for modulation <NUM>-L and a receiver device <NUM> using a Vandermonde matrix <NUM>-V for demodulation <NUM>-V. However, such a signaling exchange for a transceiver device <NUM> being based on a VL modulator can also be deduced accordingly and a repeated description (i.e., for a transceiver being based on a VL modulator) is omitted, since the VL modulator will follow same steps.

For example, the signature roots that uniformly spread over a circle of radius aopt may be used, and an algorithm may further be applied that may optimize the signature roots individually following a specific optimization metric. In particular, a machine learning techniques may be used in this step. <FIG> illustrates the signalling exchange corresponding to the Step <NUM>.

For the signalling exchange of the signature roots refinement, the following operations may be performed.

References are made from <FIG> which illustrate two exemplarily channel realization.

At a first step, the signature roots may be obtained (e.g., determined, generated) such that they are uniformly spread over a circle of radius a, for example, according to <MAT>.

The significance of aopt and its impact on the overall system performance is exemplarily described.

For example, for a system of K = <NUM> subcarriers, and the channel spread L = <NUM>, two channel realization including channel realization <NUM> and channel realization <NUM> may be determined as follow, where the C(z) is the channel response:.

Furthermore, considering the optimization metric, the MSE (by using a uniform power allocation, therefore, same κ over the subcarriers may be applied).

In the example of channel realization <NUM> which is illustrated in <FIG>, the optimum radius is <NUM> (i.e., aopt = <NUM>). Note that, if using the ZP-OFDM (a = <NUM>), the signal cannot be efficiently recovered since <MAT> is almost <NUM> (see <FIG>).

However, in the example of channel realization <NUM> which is illustrated in <FIG>, the best choice is when the radius is equal to <NUM>, then the LV scheme reduces to the ZP-OFDM.

In the following, the performance results are presented, in terms of BER as a function of the signal-to-noise ratio (SNR).

References are made from <FIG> which illustrate the performance results for a uniform and an optimized power allocation at the transmitter device based on a frequency selective channels with uniform (<FIG>) and exponential power delay profile (<FIG>.

When using K = <NUM> subcarriers, the channel spread L of <NUM> (i.e., L = <NUM>), and further carrying out the performance where the transmitter device uses the uniform and the optimized power allocation (for example, a precoder with different tuning factors) and assuming the frequency selective channels with uniform (e.g., <FIG>) and exponential power delay profile (pdp) (e.g., <FIG>.

With reference to <FIG> (the uniform pdp) and <FIG> (Exponential pdp with factor α = <NUM>), it can be derived that, both of the LV scheme (represented by the dashed curves) including for the LV (uniform power allocation) and the LV (optimized power allocation) (e.g., always) outperforms the ZP-OFDM schemes (represented by the solid lines).

Moreover, the performance of both schemes increases when using the optimal power allocation.

<FIG>, <FIG> have been depicted using perfect channel state information (CSI) at the receiver device. In the following, the performance results are shown using imperfect CSI at the receiver device (i.e., channel estimation errors). Without limiting the present invention, the performance results (i.e., <FIG>) are presented for the frequency selective channels using uniform power delay profile.

References are made from <FIG> which illustrate comparison of performance results under perfect and imperfect CSI, when the transmitter device is using uniform power allocation (<FIG>), and when the transmitter device is using optimal power allocation (<FIG>).

As can be derived from <FIG>, the LV modulation scheme outperforms the ZP-OFDM under the imperfect CSI conditions. This result also illustrates the robustness of the present invention to the channel conditions.

As discussed, in some embodiments, the signature roots may be modified (e.g., refined, migrated, optimized, etc.). For example, the "Step <NUM>: modifying the plurality of the signature roots may be performed".

References are made from <FIG> which illustrate determining the radius of the circle (<FIG>) and further determining the signature roots using the radius of the circle (<FIG>).

For example, the Gradient descent algorithm may be used in order to perform the individual signature roots optimization (i.e., modifying the signature root). For instance, at first, the radius of the circle aopt may be used (i.e., which has been provided by Step <NUM>) and considering the K = <NUM> and the L = <NUM> (e.g., the results given by Step <NUM>). The determined radius of the circle in <FIG> may be used and the plurality of the signature roots may further be obtained (e.g., determined, generated, etc.), as it is illustrated in <FIG>.

Moreover, the plurality of the signature roots represented in <FIG> may further be modified (e.g., refined) using Gradient Descent algorithm (GDA). The results of signature roots refinement (using Step <NUM>) are depicted in <FIG>, for the same channel realization.

<FIG> shows the plurality of signature roots migrating toward new positions, and <FIG> shows the MSE decreasing with the GDA iterations.

AS it can be derived from <FIG>, the MSE degrades while the GDA algorithm is optimizing the plurality of the signature roots positions from an iteration to another.

<FIG> shows the overall performance of the LV modulator of the invention compared to the conventional ZP-OFDM performance.

The comparison of the performance is performed based on considering K = <NUM>, L = <NUM>, and using frequency selective channel following a uniform pdp (the results can be derived for a more general channel). Moreover, the comparison of performance results is performed using Step <NUM> only, and step <NUM> along with the Step <NUM> (which uses Step <NUM> as an intermediate results).

Note that, the Step <NUM> brings a significant improvement compared to LV and conventional ZP-OFDM using Step <NUM> only. For example,.

<FIG> shows a method <NUM> according to an embodiment of the invention for being implemented at a transmitter device <NUM>. The method <NUM> may be carried out by the transmitter device <NUM>, as it described above.

The method <NUM> comprises a step <NUM> of obtaining a plurality of signature roots ρk based on receiving a feedback message <NUM> from a receiver device <NUM>, wherein each signature root is a nonzero complex point.

The method <NUM> further comprises a step <NUM> of constructing a Lagrange matrix <NUM>-L or a Vandermonde matrix <NUM>-V from the plurality of signature roots ρk.

The method <NUM> further comprises a step <NUM> of generating a multicarrier modulated signal <NUM>-L, <NUM>-V based on the Lagrange matrix <NUM>-L or the Vandermonde matrix <NUM>-V.

<FIG> shows a method <NUM> for being implemented at a receiver device <NUM>. The method <NUM> may be carried out by the receiver device <NUM>, as it described above.

The method <NUM> comprises a step <NUM> of determining a plurality of signature roots ρk, wherein each signature root is a nonzero complex point;.

The method <NUM> further comprises a step <NUM> of performing a demodulation <NUM>-L, <NUM>-V of a multicarrier modulated signal <NUM>-V, <NUM>-L based on the Lagrange matrix <NUM>-L or the Vandermonde matrix <NUM>-V.

<FIG> shows a method <NUM> for being implemented at a transceiver device <NUM>. The method <NUM> may be carried out by the transceiver device <NUM>, as it described above.

The method <NUM> comprises a step <NUM> of generating, at a transmitter device <NUM>, a multicarrier modulated signal <NUM>-L, <NUM>-V based on constructing a Lagrange matrix <NUM>-L or a Vandermonde matrix <NUM>-V.

The method <NUM> further comprises a step <NUM> of performing, at a receiver device <NUM>, a demodulation <NUM>-V, <NUM>, L of the multicarrier modulated signal <NUM>-L, <NUM>-V based on constructing the other matrix <NUM>-V, <NUM>-L from the Lagrange matrix or the Vandermonde matrix constructed by the transmitter device <NUM>.

Claim 1:
A transmitter device (<NUM>) for a multicarrier modulation scheme, the transmitter device (<NUM>) being configured to:
obtain a plurality of signature roots (ρk) based on receiving a feedback message (<NUM>) from a receiver device (<NUM>), wherein each signature root is a nonzero complex point, wherein the feedback message (<NUM>) indicates a radius (a) of a circle,
wherein the transmitter device (<NUM>) is configured to obtain the plurality of signature roots (ρk) based on <MAT>
where ρk corresponds to a signature root related to the kth subcarrier, where a corresponds to the radius of the circle, and where K is the number of the subcarriers of a user, and wherein the feedback message (<NUM>) further indicates at least one vector for the plurality of signature roots (ρk) that indicates a plurality of individually modified signature roots;
allocate a determined transmit power to each subcarrier of the multicarrier modulated signal (<NUM>-L, <NUM>-V) according to a tuning factor (κk) estimated based on the radius (a) of the circle, wherein the tuning factor is defined by <MAT>
construct a Lagrange matrix (<NUM>-L) or a Vandermonde matrix (<NUM>-V) from the plurality of individually modified signature roots (ρk); and
generate a multicarrier modulated signal (<NUM>-L, <NUM>-V) based on the Lagrange matrix (<NUM>-L) or the Vandermonde matrix (<NUM>-V).