WO2023057045A1

WO2023057045A1 - Apparatuses and methods for slepian-based waveform communications

Info

Publication number: WO2023057045A1
Application number: PCT/EP2021/077372
Authority: WO
Inventors: Kamel TOURKI; Merouane Debbah; Rostom ZAKARIA
Original assignee: Huawei Technologies Co., Ltd.
Priority date: 2021-10-05
Filing date: 2021-10-05
Publication date: 2023-04-13

Abstract

A wireless transmitter apparatus (110) comprises a processing circuitry (120) configured to encode a plurality of data bits into a plurality of data symbols and modulate, based on a plurality of discrete prolate spheroidal, DPS, sequences, the plurality of data symbols into a guard interval-free baseband waveform. Each DPS sequence has a duration T being an integer multiple J of a sampling time T s , wherein a number N p of DPS sequences of the plurality of DPS sequences is smaller than the integer multiple J. The wireless receiver apparatus (160) comprises a communication interface (180) configured to receive an RF signal, and a processing circuitry (170) configured to generate a baseband waveform based on the received RF signal, demodulate, based on the plurality of DPS sequences the received baseband waveform into a plurality of data symbols, and decode the plurality of data symbols.

Description

APPARATUSES AND METHODS FOR SLEPIAN-BASED WAVEFORM COMMUNICATIONS TECHNICAL FIELD The present disclosure relates to wireless communications in general. More specifically, the present disclosure relates to apparatuses and methods for communication using Slepian-based waveforms in a fifth generation (5G) network and beyond. BACKGROUND In the upcoming next generation wireless networks, in order to satisfy diversified requirements, the wireless networks, e.g.5G networks, need to access different frequency bands. A multi-layer spectrum layout has been defined accordingly where a super data layer relies on a frequency band above 6 GHz (e.g. frequencies in a range from 24.25 to 29.5 GHz and in a range from 37 to 43.5 GHz) to address specific use cases requiring extremely high data rates, such as enhanced mobile broadband (eMBB). The coverage and capacity layer relies on a frequency band in the 2 to 6 GHz range (e.g. C-band) to deliver the best compromise between capacity and coverage. The typical applications fall into ultra-reliable low-latency communications (URLLC), massive machine-type communications (mMTC), and eMBB. The coverage layer exploits frequencies below 2 GHz (e.g.1.8 GHz) providing wide-area and deep indoor coverage. The typical applications fall into URLLC, mMTC, and eMBB. The coverage and capacity layer is of great significance, since most of the 5G use cases rely on it. It has been stated in 3GPP Release 16 that a contiguous frequency band of 100 MHz has been assigned to this layer. However, a single-band solution to jointly increase the capacity and coverage range could raise several challenges, because (i) by increasing the size of each individual channel, the receiver complexity will increase accordingly, (ii) by adopting channel aggregation in one or more hardware units, the spectral efficiency (SE) will be challenging since 5G communication systems use a filtered orthogonal frequency division multiplexing (f-OFDM) based waveform that requires guard bands and (iii) using both polarizations (horizontal and vertical) for each channel is challenging. Consequently, multi-band solutions may be adopted by combining transmissions of one or more channels in more than one frequency band. Thereby, the capacity and coverage may be achieved separately, where ultra-high capacity is achieved due to the higher band, while higher reliability is achieved due to the lower band. Figure 1 shows an exemplary transmission and reception scheme 10 of a spectrally- localized waveform based on f-OFDM used by a transmitter apparatus 11 and a receiver apparatus 16. By allowing the filter length to exceed the cyclic prefix (CP) length of OFDM and designing a filter appropriately, the f-OFDM waveform can achieve a desirable frequency localization for bandwidths as narrow as a few tens of subcarriers, while keeping the inter-symbol interference/inter-carrier interference (ISI/ICI) within an acceptable limit. However, while a frequency localization may be achieved by means of filtering, the data bandwidth cannot be confined. This lack of design flexibility limits the potential SE gains. Furthermore, f-OFDM is not well localized in time and the filtering will not satisfy the key performance indicators (KPIs) that are being discussed for networks beyond 5G in terms of latency. SUMMARY Thus, it is an objective of the present disclosure to provide improved apparatuses and methods for efficient data transmission with low complexity in wireless networks. The foregoing and other objectives are achieved by the subject matter of the independent claims. Further implementation forms are apparent from the dependent claims, the description and the figures. Generally, embodiments disclosed herein provide a communication scheme based on guard interval-free (GIF) Slepian-based waveforms (SWF), i.e. Slepian-based waveforms not involving any guard interval such as cyclic prefix (CP) or zero padding (ZP). In other words, the use of CP and ZP is relaxed and the DPS generation is performed over ^^ samples instead of just N samples (for the case of using a guard interval). GIF SWF is well localized in the time-frequency domain, thus offering higher spectral efficiency (SE) compared to, for example, the 5G waveform f-OFDM. Furthermore, embodiments disclosed herein provide methods and apparatuses for signal domain equalization that are capable of dealing with doubly selective channels. This allows providing embodiments with low complexity for single-user single-input single-output (SISO) transceivers, i.e. transmitters and/or receivers, while the extension to multiple-input multiple-output (MIMO) transceivers remains straightforward. More specifically, according to a first aspect, a wireless transmitter apparatus is provided. The wireless transmitter apparatus comprises processing circuitry configured to encode a plurality of data bits into a plurality of data symbols. The processing circuitry is further configured to modulate, based on a plurality (namely Np with Np < J) of discrete prolate spheroidal, DPS, sequences, the plurality of data symbols into a baseband waveform. Each DPS sequence has a duration ^^ being an integer multiple ^^ of a sampling time ^^ _^^, and the number ^^ _^^ of DPS sequences of the plurality of DPS sequences is

smaller than the integer multiple ^^. The processing circuitry is further configured to process the baseband waveform into a passband waveform. As will be described in more detail further below, this processing is done without using a guard interval, i.e. the passband waveform does not include a guard interval. As already mentioned above, these waveforms will be referred to herein as guard interval-free (GIF) Slepian-based waveforms (SWF). Thus, the wireless transmitter apparatus according to the first aspect is operable with improved performance in doubly selective channels while eliminating the need for inserting a time domain guard interval, can be implemented with low complexity, satisfies the 5G design criteria, satisfies the scalable numerology, is suitable to band and carrier aggregation requirements, and is well suited for MIMO. In a further possible implementation form, the processing circuitry is further configured to generate the plurality of DPS sequences using a Slepian matrix ^^[ ^^, ^^] based on the following equation:

wherein ^^ and ^^ denote an integer index in the range 1 to ^^, and ^^ _^^ denotes the bandwidth of the passband with a central frequency ^

In a further possible implementation form, the processing circuitry is configured to generate the plurality of DPS sequences using the Slepian matrix [ ] b determining

^^ _^^ eigenvectors of the Slepian matrix

In a further possible implementation form, the ^^ _^^ eigenvectors of the Slepian matrix are the eigenvectors with the largest eigenvalues.

In a further possible implementation form, the processing circuitry is further configured to generate a singular value decomposition, SVD, of the Slepian matrix ^^^[ ^^, ^^^] of the form ^^{^} and the processing circuitry is configured to determine the eigenvectors as the

first ^^ _^^ columns of the matrix ^^. In a further possible implementation form, the eigenvectors are ordered according to the values of their eigenvalues. Thus, the plurality of DPS sequences are well localized in the time-frequency domain. In a further possible implementation form, the processing circuitry is further configured, for a further passband, to: modulate, based on a further plurality (namely ^

) DPS sequences a plurality of further data symbols into a further baseband waveform, wherein each further DPS sequence has the duration ^^, i.e. ^^ = ^^ ∙ ^^ _^^, and wherein a number of further DPS sequences of the further plurality of DPS sequences is smaller than the integer multiple ^^; and process the further baseband waveform into a further passband waveform. As will be appreciated, in this way, the further passband waveform does not include a guard interval. In a further possible implementation form, the wireless transmitter apparatus further comprises a communication interface configured to generate a RF signal based on the passband waveform. In a further possible implementation form, the processing circuitry is configured to encode the plurality of data bits into the plurality of data symbols based on a low-density parity- check (LDPC) code. According to a second aspect, a method of operating a wireless transmitter apparatus is provided. The method comprises a step of encoding a plurality of data bits into a plurality of data symbols. The method comprises a further step of modulating, based on a plurality (namely ^^ _^^ with ^^ _^^ < ^^) of discrete prolate spheroidal, DPS, sequences the plurality of input symbols into a baseband waveform, wherein each DPS sequence has a duration ^^ being an integer multiple ^^ of a sampling time

and wherein the number ^^ _^^ of DPS sequences of the plurality of DPS sequences is smaller than the integer multiple ^^. The method comprises a further step of processing the baseband waveform into a passband waveform. As will be appreciated, in this way, the passband waveform does not include a guard interval. The method according to the second aspect of the present disclosure can be performed by the wireless transmitter apparatus according to the first aspect of the present disclosure. Thus, further features of the method according to the second aspect of the present disclosure result directly from the functionality of the wireless transmitter apparatus according to the first aspect of the present disclosure as well as its different implementation forms described above and below. According to a third aspect, a wireless receiver apparatus is provided. The wireless receiver apparatus comprises a communication interface configured to receive a RF signal from the wireless transmitter apparatus. The wireless receiver apparatus further comprises processing circuitry configured to generate a baseband waveform based on the received RF signal, demodulate, based on a plurality (namely ^^) of discrete prolate spheroidal, DPS, sequences the received baseband waveform into a plurality of data symbols. Each DPS sequence has a duration ^^ being an integer multiple ^^ of a sampling time and the number of DPS sequences of the plurality of DPS

sequences is equal to the integer multiple ^^. The processing circuitry is further configured to decode the plurality of data symbols. Thus, the wireless receiver apparatus is operable with improved performance in doubly selective channels while eliminating the need for inserting a time domain guard interval, implemented with low complexity, satisfies the 5G design criteria, satisfies the scalable numerology, is suitable to bands and carrier aggregation requirements, and is well suited for MIMO. In a further possible implementation form, the processing circuitry is further configured to generate a singular value decomposition, SVD, of a Slepian matrix ^^^[ ^^, ^^^]:

in the form of ^

and to generate the plurality of DPS sequences based on the matrix ^^ ^{^^} for demodulation. In a further possible implementation form, the processing circuitry is further configured to equalize the plurality of data symbols by processing an inter symbol interference (ISI) of the plurality of data symbols as a noise component. In a further possible implementation form, the processing circuitry is configured to equalize the plurality of data symbols by processing the inter symbol interference (ISI) of the plurality of data symbols as a noise component based on an equalization matrix ^^

where: enotes a first effective channel matrix defined as

, denotes a second effective channel matrix defined as

enotes a lower triangular portion of a channel matrix, denotes an upper triangular portion of the channel matrix representing the ISI,

denotes a covariance matrix, denotes the identity matrix, = , and

denotes the modulation matrix stacking the first ^^ _^^ of the plurality of DPS sequences.

Thus, the inter-symbol interference (ISI) due to the guard interval removal can be effectively prevented, the problem of DPS sequences being orthogonality broken when going through time-varying channels and the one tap equalization becoming consequently inadequate is solved, and an advanced yet low complex receiver is provided. In a further possible implementation form, the processing circuitry is further configured to equalize the plurality of data symbols by reducing an inter symbol interference (ISI) of the plurality of data symbols. In a further possible implementation form, the processing circuitry is configured to estimate the ISI of the plurality of output symbols using a plurality of equalized data symbols based on a previously received further baseband waveform or a previously received portion of the baseband waveform and reduce the ISI of the plurality of data symbols by compensating the estimated ISI. In a further possible implementation form, the processing circuitry is configured to equalize the plurality of data symbols by reducing the inter symbol interference (ISI) of the plurality of data symbols based on an equalization matrix ^^

where: denotes a first effective channel matrix defined as

, denotes a second effective channel matrix defined as

denotes a lower triangular portion of a channel matrix,

denotes an upper triangular portion of the channel matrix representing the ISI, ^^ ^{^^} ^^ denotes a covariance matrix, ^^ denotes the identity matrix,

^^ denotes the modulation matrix stacking the first ^^ _^^ of the plurality of DPS sequences, and ^^ denotes a diagonal matrix representing a residual interference covariance. Thus, the inter-symbol interference (ISI) due to the guard interval removal can be effectively prevented, the problem of DPS sequences being orthogonality broken when going through time-varying channels and the one tap equalization becoming consequently inadequate is solved, and an advanced yet low complex receiver is provided. In a further possible implementation form, the processing circuitry comprises a low-density parity-check (LDPC) decoder for decoding the plurality of data symbols. In an embodiment, the LPDC decoder may use log-likelihood ratio (LLR) expressions. According to a fourth aspect, a method of operating a wireless receiver apparatus is provided. The method comprises a step of receiving a RF signal. The method comprises a further step of generating a baseband waveform based on the received RF signal. The method comprises a further step of demodulating, based on a plurality (namely ^^) of discrete prolate spheroidal, DPS, sequences the received baseband waveform into a plurality of data symbols, wherein each DPS sequence has a duration ^^ being an integer multiple ^^ of a sampling time and wherein a number of DPS sequences of

the plurality of DPS sequences is equal to the integer multiple J. The method comprises a further step of decoding the plurality of data symbols. In a further possible implementation form, the method according to the fourth aspect further comprises equalizing the plurality of data symbols by processing an inter symbol interference (ISI) of the plurality of data symbols as a noise component. In a further possible implementation form, the method according to the fourth aspect further comprises equalizing the plurality of data symbols by reducing an inter symbol interference (ISI) of the plurality of data symbols. The method according to the fourth aspect of the present disclosure can be performed by the wireless receiver apparatus according to the third aspect of the present disclosure. Thus, further features of the method according to the fourth aspect of the present disclosure result directly from the functionality of the wireless receiver apparatus according to the third aspect of the present disclosure as well as its different implementation forms described above and below. According to a fifth aspect, a computer program product is provided, comprising a computer-readable storage medium for storing program code which causes a computer or a processor to perform the method according to the second aspect or the method according to the fourth aspect when the program code is executed by the computer or the processor. Details of one or more embodiments are set forth in the accompanying drawings and the description below. Other features, objects, and advantages will be apparent from the description, drawings, and claims. BRIEF DESCRIPTION OF THE DRAWINGS In the following, embodiments of the present disclosure are described in more detail with reference to the attached figures and drawings, in which: Fig.1 shows a block diagram illustrating a filtered OFDM waveform; Fig.2 shows a schematic diagram illustrating a wireless transmitter apparatus and a wireless receiver apparatus according to an embodiment; Fig.3a shows a schematic diagram illustrating processing functionality implemented by a wireless transmitter apparatus according to an embodiment; Fig.3b shows a schematic diagram illustrating processing functionality implemented by a wireless receiver apparatus according to an embodiment; Fig.4 shows a schematic diagram illustrating a DPS sequence generation implemented by a wireless transmitter apparatus and a wireless receiver apparatus according to an embodiment; Fig.5 shows a schematic diagram illustrating MMSE with ISI mitigation implemented by a wireless receiver apparatus according to an embodiment; Fig.6 shows GIF-SWF and f-OFDM symbols, where J = N+L; Fig.7a shows a figure illustrating the power spectral densities of GIF-SWF and f-OFDM according to an embodiment; Fig.7b shows a figure illustrating a GIF-SWF and f-OFDM IAPR comparison, according to an embodiment; Fig.8a shows a figure illustrating the GIF-SWF and f-OFDM performance comparison considering ISI as noise according to an embodiment; Fig.8b shows a graph diagram illustrating the GIF-SWF and f-OFDM performance comparison considering ISI mitigation according to an embodiment; Fig.9 shows a graph diagram illustrating PSDs of GIF-SWF and f-OFDM for uplink transmission according to an embodiment; Fig.10a shows a graph diagram illustrating GIF-SWF and f-OFDM performance comparison in terms of BLER as a function of signal to noise ratio according to an embodiment; Fig.10b shows a graph diagram illustrating GIF-SWF and f-OFDM performance comparison in terms of SE as a function of signal to noise ratio according to an embodiment; Fig.11a shows a graph diagram illustrating DPS sequences in time domain according to an embodiment; Fig.11b shows a graph diagram illustrating DPS sequences in two consecutive SWF symbols according to an embodiment; Fig.12 shows a flow diagram illustrating a method of operating a wireless transmitter apparatus according to an embodiment; and Fig.13 shows a flow diagram illustrating a method of operating a wireless receiver apparatus according to an embodiment. In the following, identical reference signs refer to identical or at least functionally equivalent features. DETAILED DESCRIPTION OF THE EMBODIMENTS In the following description, reference is made to the accompanying figures, which form part of the disclosure, and which show, by way of illustration, specific aspects of embodiments of the present disclosure or specific aspects in which embodiments of the present disclosure may be used. It is understood that embodiments of the present disclosure may be used in other aspects and comprise structural or logical changes not depicted in the figures. The following detailed description, therefore, is not to be taken in a limiting sense, and the scope of the present disclosure is defined by the appended claims. For instance, it is to be understood that a disclosure in connection with a described method may also hold true for a corresponding device or system configured to perform the method and vice versa. For example, if one or a plurality of specific method steps are described, a corresponding device may include one or a plurality of units, e.g. functional units, to perform the described one or plurality of method steps (e.g. one unit performing the one or plurality of steps, or a plurality of units each performing one or more of the plurality of steps), even if such one or more units are not explicitly described or illustrated in the figures. On the other hand, for example, if a specific apparatus is described based on one or a plurality of units, e.g. functional units, a corresponding method may include one step to perform the functionality of the one or plurality of units (e.g. one step performing the functionality of the one or plurality of units, or a plurality of steps each performing the functionality of one or more of the plurality of units), even if such one or plurality of steps are not explicitly described or illustrated in the figures. Further, it is understood that the features of the various exemplary embodiments and/or aspects described herein may be combined with each other, unless specifically noted otherwise. Figure 2 shows a schematic diagram illustrating a communication network 100 comprising a wireless transmitter apparatus 110 according to an embodiment for exchanging one or more analog passband RF signals with a wireless receiver apparatus 160 according to an embodiment via a wireless RF communication channel 150. The wireless transmitter apparatus 110 comprises a processing circuitry 120, which may comprise or be configured to implement a modulator 125 (as will be described in more detail below in the context of figure 3a). The wireless transmitter apparatus 110 may further comprise a communication interface 130 for sending a passband waveform via the communication channel 150 to the wireless receiver apparatus 160. The wireless transmitter apparatus 110 may further comprise a memory 140 to store data and executable program code which, when executed by the processing circuitry 120 causes the wireless transmitter apparatus 110 to perform the functions, operations and methods described herein. As illustrated in figure 2, the wireless receiver apparatus 160 comprises a communication interface 180 configured to receive the passband signal from the wireless transmitter apparatus 110 via the communication channel 150. The wireless receiver apparatus 160 further comprises a processing circuitry 170 configured to generate 171 the baseband waveform based on the received passband signal. In an embodiment, the processing circuitry 170 may comprise or be configured to implement a demodulator 173 (as will be described in more detail below in the context of figure 3b). The wireless receiver apparatus 160 may further comprise a memory 190 to store data and executable program code which, when executed by the processing circuitry 170 causes the wireless receiver apparatus 160 to perform the functions, operations and methods described herein. The processing circuitries 120, 170 of the wireless transmitter apparatus 110 and/or the wireless receiver apparatus 160 may be implemented in hardware and/or software. The hardware may comprise digital circuitry, or both analog and digital circuitry. Digital circuitry may comprise components such as application-specific integrated circuits (ASICs), field- programmable arrays (FPGAs), digital signal processors (DSPs), or general-purpose processors. Figure 3a shows a schematic diagram illustrating a processing scheme implemented by the processing circuitry 120 of the wireless transmitter apparatus 110 based on SWF. The processing circuitry 120 is configured to encode a plurality of data bits into a plurality of data symbols. To this end, in an embodiment, the processing circuitry 120 may comprise or implement an encoder. The processing circuitry 120 (in an embodiment the modulator 125 implemented by the processing circuitry 120) is configured to modulate, based on a plurality, namely

of orthonormal discrete prolate spheroidal (DPS) sequences of order 1 to the plurality of data symbols into a baseband

waveform. Each DPS sequence has a duration T being an integer multiple ^^ of a sampling time

and the number

sequences of the plurality of DPS sequences is smaller than the integer multiple ^^. The processing circuitry 120 of the wireless transmitter apparatus 110 is further configured to process the baseband waveform into a passband waveform, as described in more detail further below. As will be described in more detail further below, this processing is done without using a guard interval, i.e. the passband waveform signal does not include a guard interval. As already mentioned above, these waveforms are referred to herein as guard interval-free (GIF) Slepian-based waveforms (SWF). The processing circuitry 120 may further comprise or implement a parallel to serial converter 123. The communication channel 150 may be used for transmitting a signal using the passband, i.e. frequency band ^^ _^^ with a central frequency ^

. According to an embodiment, the processing circuitry 120 may choose

DPS sequences, which may be orthonormal, of length

with a confined energy in

given by the first ^

eigenvectors of the Slepian matrix whose elements may eb calculated by

( ) . An eigenvalue decomposition may be calculated by

, where the eigenvectors of ^^ are given by the columns of the matrix ^^, and the eigenvalues by the diagonal matrix ^^. The columns of the matrix ^^

representing the DPS sequences may be ordered according to their eigenvalues

As illustrated in Figure 4, the processing circuitry 120 of the wireless transmitter apparatus 110 may be further configured to generate a modulation matrix S of size stacking

the first DPS sequences. The modulation matrix S may be used by a GIF-SWF modulator 125 implemented by the processing circuitry 120, where the DPS sequences’ duration is

_^ and ^^ _^^ is the sampling time. The Slepian matrix C of size ^^ × ^^ may be generated 125a by the processing circuitry 120 for generating 125b the matrix S of size as defined above.

Figure 3b shows a schematic diagram illustrating a processing scheme implemented by the processing circuitry 170 of the wireless receiver apparatus 160 based on SWF. The processing circuitry 170 may be configured for signal domain equalization by implementing or comprising an equalizer 175 and may be configured to generate the plurality of DPS sequences based on the matrix ^^ ^{^^} of size ^^ × ^^ for demodulating in sub- band ^^ _^^. The demodulator 173 may be configured for processing the received passband signal to demodulate, based on a plurality, namely ^^ of orthonormal DPS sequences of order 1 to ^^, the received passband waveform into a plurality of output symbols. Each DPS sequence has the duration ^^ and the number of DPS sequences of the plurality of DPS sequences is equal to the integer multiple ^^. The processing circuitry 170 is further configured to decode the plurality of data symbols into a plurality of data bits. In an embodiment, the processing circuitry 170 may implement a decoder 179 configured to decode the plurality of data symbols into a plurality of data bits. The processing circuitry 170 of the of the wireless receiver apparatus 160 may be further configured to implement a module for channel estimation 177, which will be described further below. According to an embodiment, the processing circuitry 120, e.g. the encoder may be configured to encode the plurality of data bits into the plurality of input symbols based on a LDPC code. The encoded data bits may be mapped into ^^ _^^ complex symbols following a quadrature amplitude modulation of order and stacked into a vector ^

Therefore, the ^^-th GIF-SWF transmitted s

ymbol may be calculated by the processing

The signal received by the wireless receiver apparatus 160 and processed by the processing circuitry 170, may be given by:

where H is the

lower triangular time-varying channel matrix,

is the

upper triangular time-varying channel matrix that represents the ISI, is the previous data

vector, and ^^ is the

additive white Gaussian noise (AWGN) vector. The demodulator 173 may be configured to generate the demodulated signal based on the matrix U^H given by:

where ^ (as defined above) is a unitary matrix stacking all DPS in its columns. Consequently,

is still AWGN with covariance matrix

identity matrix. In the following, the effective channels are denoted as and

In time-varying channels, the channel impulse response may vary even within one GIF- SWF symbol ^^ _^^. Furthermore, the time-domain guard interval removal may induce ISI as stated above, making the detection processing challenging. Subsequently, to address this issue, the processing circuitry 170 of the wireless receiver apparatus 160 may implement a first scheme and/or a second scheme, wherein the minimum mean squared error (MMSE) may be used for equalization by the equalizer 175. The first scheme may not rely on ISI cancellation techniques and the second scheme may take into account ISI. According to an embodiment with respect to the first scheme, the MMSE equalization by the equalizer 175 considers ISI as noise and may be calculated by

which is applied to the demodulated signal ^^ _^^ as

, where the

equalization output samples are given by ^̃ , where

are the diagonal entries of

, and is an AWGN with variance

According to an embodiment with respect to the second scheme, the MMSE equalization by the equalizer 175 performs the ISI mitigation, i.e. reducing the ISI of the plurality of data symbols, wherein the previous equalized SWF symbol may have been buffered

in a buffer 175c, as illustrated in Fig.5. The estimation of ISI may be obtained from the expectation of the previous detected vector

Furthermore, the soft symbol used for the soft interference cancellation may be calculated 175d by the processing circuitry 170 with:

where are the diagonal entries of with ̅ and ̅ being the

previous effective channel matrix and its corresponding MMSE equalization, respectively. Consequently, following a calculation 175e of the effective channel

as defined above, the soft interfering vector

may be canceled out 175a from the current demodulated vector as:

and the resulting MMSE equalizer is calculated 175b by

where s a diagonal matrix representing the residual interference

covariance, whose diagonal entries are calculated 175f by:

In the particular case of QPSK mapping, the processing circuitry 170 may compute the diagonal matrix

According to an embodiment, subsequently, either by using the first scheme or the second scheme for equalization, the log-likelihood ratio (LLR) of the

bit of the

complex sample in the transmit vector

may be approximated by the decoder 179 implemented by the processing circuitry 170 for the channel estimation 177 as:

where is a binary vector, is the symbol mapping

e is the set of all vectors with the element {-1} respectively {+1}

in their entry. In the particular case of quadratic phase-shift keying (QPSK) mapping, the LLRs may be calculated by the processing circuitry 160 with:

Thus, embodiments disclosed herein provide a Slepian-based waveform using DPS sequences that eliminates the need for inserting a time-domain guard interval, which is well localized in the time-frequency domain allowing it to get rid of ISI mitigation techniques. As will be appreciated, the embodiments disclosed above and below for a single-band transmission are still valid for multi-band transmission. Embodiments disclosed herein provide a low-complexity transceiver implementation for a system designed to deal with doubly selective channels and high mobility. Figure 6 illustrates an equalization of GIF-SWF vector symbol duration 603 to the f-OFDM symbol duration 601 for assessing the performance of an embodiment. The GIF-SWF performance may be assessed in third-generation partnership project (3GPP) channels. As a benchmark, the f-OFDM system transmitting over 5 MHz may be used using

300 data (active) subcarriers with a FFT size = 512 and subcarrier spacing = 15

KHz. Hence, the sampling frequency may be set to

= 7.68 MHz and L = 36, which corresponds to a maximum delay spread = 4.69 µs. The f-OFDM filter may be given

by:

where = 129 and ^^ = 1.8715 is the normalized cut-off frequency. Furthermore, QPSK

symbols for transmission using a carrier frequency

= 3.5 GHz and the LDPC code rate of ½ may be chosen. In the following, a comparison will be carried out where the performance results will be discussed assuming perfect CSI knowledge at the receiver. For assessing performance, the GIF-SWF vector symbol duration 603 may be set equal to the f-OFDM symbol duration 601

Considering = 350 leads to a SE gain of 16.7% compared to f-OFDM. The power

spectral densities (PSDs) of GIF-SWF (curve 701) and f-OFDM (curve 703) are illustrated in Figure 7a and instantaneous-to-average power ratio (IAPR) of GIF-SWF and f-OFDM are illustrated in Figure 7b. As can be seen, GIF-SWF provided by embodiments disclosed herein exhibits lower out-of-band emission and lower IAPR while providing a higher spectral efficiency than f-OFDM. In Figure 7b

= 300 is chosen for a fair comparison. Figure 8a illustrates the block error rate of GIF-SWF without an ISI mitigation technique according to embodiments of the first scheme described above and Figure 8b illustrates the block error rate of GIF-SWF with the ISI mitigation technique according to embodiments of the second scheme described above. In order to focus on the ISI impact, frequency selective channels were used to show that the SWF used by embodiments disclosed herein can get rid of an ISI mitigation technique, thus keeping the receiver complexity lower, while eliminating the use of any guard interval. Indeed, the same BLER performances are achieved whether or not the ISI mitigation technique is considered. Figure 9 illustrates the PSD of GIF-SWF (curve 901) and f-OFDM (curve 903) for an uplink transmission according to an embodiment, where the f-OFDM transmitter will send 4 resource blocks (RBs) among 106 that could be sent over 20 MHz. Therefore

the FFT size = 2048 and L = 144, thus J = 2192. It is pointed out, that both

waveforms have data bandwidth

of 720 KHz (=12*4*15 KHz). However, the needed guard band is defined referring to the 3GPP mask requirement, which is at -18 dBm/30 KHz for uplink, giving 130 KHz and 2 KHz for f-OFDM and GIF-SWF, respectively. Since 4 RBs out of 106 were used, the guard sub-band

is scaled by 4/106. The SE expression may be given by where

is (TBS

size for 4 PRB) × (1-simulated BLER) and TBS is the transport block size. Considering the Tapped Delay Line model C (TDL-C 300ns) channels, the performance have been provided in terms of BLER as shown in Figure 10a and SE as shown in Figure 10b as function of signal to noise ratios (SNRs). A very high velocity has been considered where v = 500 Km/h leads to a Doppler frequency spread of = 1.85 KHz, assumed that

f-OFDM is using full MMSE, as GIF-SWF. As will be appreciated, GIF-SWF MMSE complexity order is O(J³) while f-OFDM MMSE complexity order is O( . However, a SNR gain of 0.5 dB is obtained for GIF-SWF over f-

OFDM. Furthermore, SE gain is obtained for GIF-SWF over f-OFDM due to guard band saving and BLER outperformance. Figure 11a illustrates DPS sequences in the time domain and Figure 11b illustrates DPS sequences within two consecutive SWF symbols according to an embodiment. As will be appreciated, the DPS sequences are well localized in the time domain so that the adverse ISI impact is limited. Figure 12 shows a flow diagram illustrating a method 1200 implemented by the wireless transmitter apparatus 110 for exchanging one or more analog passband RF signals with the wireless receiver apparatus 160 via the wireless RF communication channel 150 according to an embodiment. The method 1200 comprises a step 1201 of encoding the plurality of data bits into the plurality of data symbols. The method 1200 further comprises a step 1203 of modulating, based on the plurality of discrete prolate spheroidal, DPS, sequences the plurality of data symbols into the baseband waveform, wherein each DPS sequence has a duration T being an integer multiple of a sampling time and wherein the number

of DPS sequences of the

plurality of DPS sequences is smaller than the integer multiple ^^. The method 1200 further comprises a step 1205 of processing the baseband waveform into the passband waveform. As the method 1200 can be implemented by the wireless transmitter apparatus 110, further features of the method 1200 result directly from the functionality of the wireless transmitter apparatus 110 and its different embodiments described above and below. Figure 13 shows a flow diagram illustrating a method 1300 implemented by the wireless receiver apparatus 160 for exchanging one or more analog passband RF signals with the wireless transmitter apparatus 110 via the wireless RF communication channel 150 according to an embodiment. The method 1300 comprises a step 1301 of receiving the RF signal. The method 1300 further comprises a step 1303 of generating the baseband waveform based on the received RF signal. The method 1300 further comprises a step 1305 of demodulating, based on the plurality of discrete prolate spheroidal, DPS, sequences the received baseband waveform into the plurality of data symbols, wherein each DPS sequence has the duration ^^ being an integer multiple ^^ of the sampling time and wherein the number of DPS sequences of the

plurality of DPS sequences is equal to the integer multiple ^^. The method 1300 further comprises a step 1307 of decoding the plurality of data symbols. As the method 1300 can be implemented by the wireless receiver apparatus 160, further features of the method 1300 result directly from the functionality of the wireless receiver apparatus 160 and its different embodiments described above and below. The person skilled in the art will understand that the "blocks" ("units") of the various figures (method and apparatus) represent or describe functionalities of embodiments of the present disclosure (rather than necessarily individual "units" in hardware or software) and thus describe equally functions or features of apparatus embodiments as well as method embodiments (unit = step). In the several embodiments provided in the present application, it should be understood that the disclosed system, apparatus, and method may be implemented in other manners. For example, the described embodiment of an apparatus is merely exemplary. For example, the unit division is merely logical function division and may be another division in an actual implementation. For example, a plurality of units or components may be combined or integrated into another system, or some features may be ignored or not performed. In addition, the displayed or discussed mutual couplings or direct couplings or communication connections may be implemented by using some interfaces. The indirect couplings or communication connections between the apparatuses or units may be implemented in electronic, mechanical, or other forms. The units described as separate parts may or may not be physically separate, and parts displayed as units may or may not be physical units, may be located in one position, or may be distributed on a plurality of network units. Some or all of the units may be selected according to actual needs to achieve the objectives of the solutions of the embodiments. In addition, functional units in the embodiments of the invention may be integrated into one processing unit, or each of the units may exist alone physically, or two or more units are integrated into one unit.

Claims

CLAIMS 1. A wireless transmitter apparatus (110), comprising processing circuitry (120) configured to: encode a plurality of data bits into a plurality of data symbols; modulate, based on a plurality of discrete prolate spheroidal, DPS, sequences, the plurality of data symbols into a baseband waveform, wherein each DPS sequence has a duration T being an integer multiple J of a sampling time wherein a number

of DPS

sequences of the plurality of DPS sequences is smaller than the integer multiple J; and process the baseband waveform into a passband waveform.

2. The wireless transmitter apparatus (110) of claim 1, wherein the processing circuitry is further configured to generate the plurality of DPS sequences using a Slepian matrix based on the following equation:

where p and q denote an integer index in the range 1 to J, and denotes the bandwidth

of the passband with a central frequency

3. The wireless transmitter apparatus (110) of claim 2, wherein the processing circuitry is configured to generate the plurality of DPS sequences using the Slepian matrix by determining eigenvectors of the Slepian matrix

4. The wireless transmitter apparatus (110) of claim 3, wherein the eigenvectors

of the Slepian matrix are the eigenvectors with the largest eigenvalues.

5. The wireless transmitter apparatus (110) of claim 3 or 4, wherein the processing circuitry (120) is further configured to generate a singular value decomposition of the Slepian matrix

of the form

and wherein the processing circuitry is configured to determine the eigenvectors as the first

columns of the matrix ^^.

6. The wireless transmitter apparatus (110) of claim 4 or 5, wherein the eigenvectors are ordered according to the values of their eigenvalues.

7. The wireless transmitter apparatus (110) of any one of the preceding claims, wherein the processing circuitry (120) is further configured, for a further passband, to: modulate, based on a further plurality of DPS, sequences a plurality of further data symbols into a further baseband waveform, wherein each further DPS sequence has the duration ^^, wherein a number of further DPS sequences of the further plurality of DPS sequences is smaller than the integer multiple ^^; and process the further baseband waveform into a further passband waveform.

8. The wireless transmitter apparatus (110) of any one of the preceding claims, further comprising a communication interface (130) configured to generate an radio frequency, RF, signal based on the passband waveform.

9. The wireless transmitter apparatus (110) of any one of the preceding claims, wherein the processing circuitry is configured to encode the plurality of data bits into the plurality of data symbols based on a low-density parity-check code.

10. A method (1200) of operating a wireless transmitter apparatus (110), wherein the method (1200) comprises: encoding (1201) a plurality of data bits into a plurality of data symbols; modulating (1203), based on a plurality of discrete prolate spheroidal, DPS, sequences the plurality of data symbols into a baseband waveform, wherein each DPS sequence has a duration ^^ being an integer multiple ^^ of a sampling time wherein a number of

DPS sequences of the plurality of DPS sequences is smaller than the integer multiple ^^; and processing (1205) the baseband waveform into a passband waveform.

11. A wireless receiver apparatus (160), comprising: a communication interface (180) configured to receive a radio frequency, RF, signal; and processing circuitry (170) configured to: generate a baseband waveform based on the received RF signal, demodulate, based on a plurality of discrete prolate spheroidal, DPS, sequences the received baseband waveform into a plurality of data symbols, wherein each DPS sequence has a duration ^^ being an integer multiple ^^ of a sampling time

and wherein a number of DPS sequences of the plurality of DPS sequences is equal to the integer multiple ^^, and decode the plurality of data symbols.

12. The wireless receiver apparatus (160) of claim 11, wherein the processing circuitry (170) is further configured to: generate a singular value decomposition of a Slepian matrix

in the form of

and generate the plurality of DPS sequences based on the matrix

for demodulation.

13. The wireless receiver apparatus (160) of claim 12, wherein the processing circuitry (170) is further configured to equalize the plurality of data symbols by processing an inter- symbol interference of the plurality of data symbols as a noise component.

14. The wireless receiver apparatus (160) of claim 13, wherein the processing circuitry is configured to equalize the plurality of data symbols by processing the inter-symbol interference of the plurality of data symbols as a noise component based on an equalization matrix ^^:

where: denotes a first effective channel matrix defined as

, denotes a second effective channel matrix defined as

denotes a lower triangular portion of a channel matrix, ^_^ denotes an upper triangular portion of the channel matrix, ^^{^} ^^ denotes a covariance matrix,

denotes the identity matrix,

^^ denotes the modulation matrix stacking the first

of the plurality of DPS sequences.

15. The wireless receiver apparatus (160) of claim 12, wherein the processing circuitry (170) is configured to equalize the plurality of data symbols by reducing an inter-symbol interference, ISI, of the plurality of data symbols.

16. The wireless receiver apparatus (160) of claim 15, wherein the processing circuitry (170) is configured to: estimate the ISI of the plurality of output symbols using a plurality of equalized data symbols based on a previously received further baseband waveform or a previously received portion of the baseband waveform; and reduce the ISI of the plurality of data symbols by compensating the estimated ISI.

17. The wireless receiver apparatus (160) of claim 15 or 16, wherein the processing circuitry (170) is configured to equalize the plurality of data symbols by reducing the ISI of the plurality of data symbols based on an equalization matrix ^^: ^^

where: denotes a first effective channel matrix defined as ^

, ^ denotes a second effective channel matrix defined as

denotes a lower triangular portion of a channel matrix, ^ denotes an upper triangular portion of the channel matrix, ^ ^^ denotes a covariance matrix,

denotes the identity matrix,

denotes the modulation matrix stacking the first

of the plurality of DPS sequences, and denotes a diagonal matrix representing a residual interference covariance.

18. The wireless receiver apparatus (160) of any one of claims 11 to 17, wherein the processing circuitry comprises a low-density parity-check decoder (179) for decoding the plurality of data symbols.

19. A method (1300) of operating a wireless receiver apparatus (160), comprising: receiving (1301) a radio frequency, RF, signal; generating (1303) a baseband waveform based on the received RF signal; demodulating (1305), based on a plurality of discrete prolate spheroidal, DPS, sequences the received baseband waveform into a plurality of data symbols, wherein each DPS sequence has a duration ^^ being an integer multiple ^^ of a sampling time

and wherein a number of DPS sequences of the plurality of DPS sequences is equal to the integer multiple and decoding (1307) the plurality of data symbols.

20. The method (1300) of claim 19, further comprising equalizing the plurality of data symbols by processing an inter-symbol interference of the plurality of data symbols as a noise component.

21. The method (1300) of claim 19, further comprising equalizing the plurality of data symbols by reducing an inter-symbol interference of the plurality of data symbols.

22. A computer program product comprising a computer-readable storage medium for storing program code which causes a computer or a processor to perform the method (1200) of claim 10 or the method (1300) of any one of claims 19 to 21 when the program code is executed by the computer or the processor.