Integer-sequence regularised orthogonal frequency division multiplexing

Abstract

A method of generating a modified OFDM signal comprising: (i) inputting an OFDM signal 614; (ii) defining a set of perturbation vector candidates 616; (iii) selecting one or more perturbation vectors

Description

Field The present specification relates to Integer-sequence regularised Orthogonal Frequency Division Multiplexing. In particular, the present invention relates to an iterative process for selecting a candidate perturbation vector to be superimposed on the original OFDM signal. Background The current OFDM waveform technologies used in cellular, Wi-Fi, and Satellite systems are not well-suited for highly spectral-efficient modulations like 4k-QAM or higher. This is due to their vulnerability to power amplifier nonlinearities, resulting in significant in-band distortions and out-of-band energy leakage. Wireless multi-carrier waveforms, such as orthogonal frequency-division multiplexing (OFDM) signals, typically possess high peak-to-average power ratio (PAPR). Multicarrier signals with high PAPR may suffer from nonlinear distortion which is caused by the clipping effect of the power amplifier (PA). This is because the amplified signal power will eventually saturate to a certain level for PAs. The power amplification range can be dynamically adjusted to preserve the power amplification linearity (e.g., large range of input backoff (IBO)). However this results in the decrease of PA efficiency because power supply is always ‘on’ to keep the PA functioning. Various techniques have been proposed to handle the PAPR reduction problem. Four such techniques are discussed below. 1. Signal pre-distortion, (e.g., ADMM) Pre-distortion techniques achieve desirable PAPR control performance by intentionally introducing both in-band and out-of-band distortion to the OFDM signal. This results in an increased in-band error vector magnitude (EVM) and out-of-band interference to the system. 2. Signalling-based approaches, (e.g., selective mapping and partial transmit sequence) Techniques within this category may create multiple versions of an OFDM signal through a certain devised mechanism; and the one that has the minimum PAPR is selected for the transmission. Extra signalling overhead is required in order to let the receiver successfully recover the signal; this results in a loss of spectrum efficiency. 3. Tone reservation / injection Tone reservation necessitates a certain number of unused subcarriers; the unused subcarriers are then used to generate the peak cancellation signal. Thus, some PAPR control is achieved with the decrease of spectral efficiency. The tone injection technique does not sacrifice the spectral efficiency; while in principle, the algorithms naturally require a searching process in order to find the appropriate candidates. Effort has been put in to achieve a better trade-off between the PAPR reduction performance and the computational complexity. However, the PAPR control performance is still suboptimal. 4. DFT-spread precoding (i.e., SC-FDMA in 3GPP) The sub-band DFT-spread technique was widely used in LTE systems. Prior to the inverse discrete-Fourier transform (IDFT) operation, a sub-band DFT operation is conducted to couple the data symbols. All of the above techniques have sub-optimal performance by some measure (e.g. data rate or spectral efficiency) and are not able to support the reliable signal transmission using 4k-QAM or even higher-order modulation schemes. Summary A first aspect of this disclosure provides a method of generating a modified Orthogonal Frequency Division Multiplexed, OFDM, signal, the method comprising: (i) inputting an OFDM signal; (ii) defining a set of perturbation vector candidates; (iii) selecting one or more perturbation vectors from the set of perturbation vector candidates, by calculating which of the one or more perturbation vectors results in the lowest distortion of the OFDM signal or the lowest peak-to-average power ratio of the OFDM signal; (iv) summing each of the selected one or more perturbation vectors with separate instances of the OFDM signal to create one or more modified OFDM signals in one or more calculation paths; (v) performing steps (ii)-(iv) iteratively on the one or more modified OFDM signals until a stopping criteria is satisfied; and (vi) when the stopping criteria is satisfied, outputting a final modified OFDM signal. The method may further comprise, after inputting the OFDM signal, initialising a perturbated vector using the OFDM signal. The method may further comprise, after inputting the OFDM signal, adding a plurality of initialising perturbation vectors to respective instances of the OFDM signal to create a plurality of starting vectors. Defining the set of perturbation vector candidates may comprise: determining the n largest instantaneous peak powers of the OFDM signal; determining to what extent each of a plurality of possible perturbation vectors reduces the peak powers of the n largest instantaneous peak powers of the OFDM signal; and selecting the perturbation vectors which minimise the peak powers of the n largest instantaneous peak powers of the OFDM signal. Calculating which of the one or more perturbation vectors results in the lowest distortion of the OFDM signal may comprise calculating i*(t) = arg min sk where i*(t) represents the one or more perturbation vectors resulting in the lowest distortion of the OFDM signal, ck is defined by ek := ||zfc(t) - #(zfc(t))|| and g(zk(t)) is an entry-wise non-linear function modelling a clipping effect of a power amplifier and takes the form: 5(^n,Vn) — ’ Ae^, if 7n > 7th else with A denoting a maximum output amplitude of the power amplifier, given the input sequence z, ^denoting the phase of zn and In defining the instantaneous peak-to-average power ratio of Zn. Calculating which of the one or more perturbation vectors results in the lowest peak-to-average power ratio of the OFDM signal comprises calculating i*(t) = arg min pk (t), Vfc ikeX(t) where Pkty denotes the peak-to-average power ratio of zfc W. The stopping criteria may be that the selected one or more perturbation vectors do not reduce the distortion of the OFDM signal or the peak-to-average power ratio of the OFDM signal relative to the previous iteration. The stopping criteria may be that the selected one or more perturbation vectors do not reduce the distortion of the OFDM signal or the peak-to-average power ratio of the OFDM signal by more than a predetermined threshold relative to the previous iteration. The stopping criteria may be that the number of calculation paths exceeds a predetermined threshold. The method may further comprise: inputting the final modified OFDM signal into a power amplifier to create an amplified modified OFDM signal; and transmitting the amplified modified OFDM signal. A second aspect of this disclosure provides a computer-readable storage medium arranged to store computer program instructions which, when executed, performs the method according to the first aspect. A third aspect of this disclosure provides an apparatus comprising communication circuitry and processing circuitry configured to: (i) input an OFDM signal; (ii) define a set of perturbation vector candidates; (iii) select one or more perturbation vectors from the set of perturbation vector candidates, by calculating which of the one or more perturbation vectors results in the lowest distortion of the OFDM signal or the lowest peak-to-average power ratio of the OFDM signal; (iv) sum each of the selected one or more perturbation vectors with separate instances of the OFDM signal to create one or more modified OFDM signals in one or more calculation paths; (v) perform steps (ii)-(iv) iteratively on the one or more modified OFDM signals until a stopping criteria is satisfied; and (vi) when the stopping criteria is satisfied, output a final modified OFDM signal. The processing circuitry may comprise: one or more processors; and one or more memories coupled to the one or more processors, wherein the one or more memories include instructions executable by the one or more processors. The apparatus may be configured to transmit the final modified OFDM signal using the communication circuitry. The apparatus may be a network node. The apparatus may be a base station or base transceiver station. The apparatus may be a UE (User Equipment). The UE may be a mobile phone. Brief Description of the Figures So that the general concepts set out in the foregoing sections can be more fully understood, embodiments thereof will be described with reference to the accompanying drawings, in which: Figure 1 is a block diagram illustrating a conventional OFDM transceiver architecture; Figure 2 is a block diagram illustrating an l-OFDM transceiver architecture according to the present invention; Figure 3 is a block diagram showing a system architecture of a conventional SC-FDMA; Figure 4 is a block diagram 400 showing a system architecture of a conventional pre-distorted OFDM (with ADMM); Figure 5 is a block diagram 500 illustrating an l-OFDM transceiver architecture according to the present invention Figure 6a is a flowchart illustrating a process of l-OFDM waveform generation using a S-CMP procedure; Figure 6b is a flowchart illustrating a process of l-OFDM waveform generation using a S-CMP procedure; Figure 7 is a flowchart illustrating a process of l-OFDM waveform generation using a single-path list-encoding CMP (S-list-CMP) procedure; Figure 8 is a flowchart illustrating a process of l-OFDM waveform generation using a multi-path CMP (M-CMP) procedure; Figure 9 is a flowchart illustrating a process of l-OFDM waveform generation using a multi-path with list encoding CMP (M-list-CMP) procedure; Figure 10 shows the 4k-QAM symbol error rate (SER) as a function of bit-energy-to-noise ratio (EbNo) for prior art systems and for a system according to the present invention; Figure 11 shows the 4k-QAM symbol error rate (SER) as a function of bit-energy-to-noise ratio (EbNo) for prior art MIMO systems and for a MIMO system according to the present invention; Figure 12 illustrates the implementation of the l-OFDM technique for the conventional linearly-precoded MIMO-OFDM downlink scenario. Figure 13 is a flowchart illustrating the concept of waveform clustering and selection (online stage); Figure 14 is a flowchart illustrating the concept of waveform clustering and selection (offline stage). Figure 15 illustrates a network node configured to perform any of the techniques described herein; and Figure 16 illustrates a user equipment (UE) configured to perform any of the techniques described herein. Detailed description To address the limitations discussed above, a novel waveform technology, named l-OFDM, has been developed by the inventors. This technology remains backward-compatible with OFDM-based waveforms while being exceptionally resilient to power amplifier nonlinear distortions. Through extensive simulations, l-OFDM has shown its ability to reduce the in-band distortion by over 20 dB without causing additional out-of-band energy leakage. This innovation enables seamless integration of very high-order modulations, significantly enhancing spectral efficiency and overall system performance. The invention described herein tackles the problem of PAPR control, i.e., given an arbitrary OFDM signal, the technique will output a low-peak-power version of the signal which offers exceptional resilience to the clipping effect. Figure 1 is a block diagram 100 illustrating a conventional OFDM transceiver architecture. Figures 2 to 4 show how conventional techniques result in a clipping effect or distortion of the signal. Figure 2 is a block diagram 200 showing a system architecture of a conventional CP-OFDM highlighting the amplitude of the waveform before and after being fed in the power amplifier. The amplitude of an exemplary conventional OFDM signal measured at the output of different processing blocks is illustrated. It can be observed that PA clipping imposes significant distortion to the OFDM signal. Figure 3 is a block diagram 300 showing a system architecture of a conventional SC-FDMA highlighting the amplitude of the waveform before and after being fed in the power amplifier. The amplitude of an exemplary conventional OFDM signal measured at the output of different processing blocks is illustrated. Although the clipping effect is less than that illustrated in Figure 2, there is still a distortion to the OFDM signal due to PA clipping. Figure 4 is a block diagram 400 showing a system architecture of a conventional pre-distorted OFDM (with ADMM) highlighting the amplitude of the waveform before and after being fed in the power amplifier. The amplitude of an exemplary conventional OFDM signal measured at the output of different processing blocks is illustrated. Although the clipping effect is reduced by pre-distortion, the corresponding in-band signal is also distorted. I-OFDM transceiver model Figure 5 is a block diagram 500 illustrating an l-OFDM transceiver architecture according to the present invention. Compared with the conventional OFDM system of Figure 1, the l-OFDM transceiver architecture has two additional processing blocks, an l-OFDM operation 502 at the transmitter side, after the modulator, but before the PA Linearisation and a modulo operation 504 at the receiver side. The time-domain l-OFDM signal takes the following form: (1) where x := FHs denotes the original time-domain OFDM signal and i = aFHwis a time-domain superimposed sequence that is orthogonal to x in the modulo domain; w e %Ndenotes the frequency-domain complex integer perturbation vector, s € C^denotes the frequency-domain data symbol block and 8n,Vne[i,JV] e denotes the n-th entry of the symbol block which is randomly drawn from a finite-alphabet set l<S| = In a more practical sense, J denotes the number of states incorporated in the constellation, e.g., J = 4096 for 4k-QAM modulation. FH denotes the normalised inverse discrete-Fourier transform (IDFT) matrix. In order to accurately capture the amplitude of the continuous-time domain signal, FH is usually configured as an oversampled IDFT matrix (typically with the oversampling factor of 4). Note that l-OFDM can easily swich to OFDM by setting i = O This makes l-OFDM backward-compatible to OFDM systems. A key aspect of this invention is the procedure (following the principle of correlation matching pursuit (CMP)) specifically designed to find the time-domain superimposed sequence iwhich effectively helps minimise the PAPR and the impact of nonlinearclipping distortion. This procedure is discussed in greater detail below. Assuming that the perturbation vector has been properly found, given an arbitrary OFDM signal, the following discussion will now focus on describing the signal transmission and reception procedures. Given the mathematical expressions above, the signal before being fed into the PA takes the following equivalent form: z = FH (s + aw) (2) Since the invention deals with the nonlinear clipping effect incurred by the PA, it is assumed that the PA’s memory effect has been removed by digital pre-distorter which corresponds to the ‘PA Linearisation’ block in Figure 2. Clipping effects induced by the PA can be modelled as an entry-wise nonlinear function. A simple example of PA clipping (called ‘soft limiter’) takes the following expression: r if 7n >7th Zn’w) = t *n, else with ^denoting the maximum output amplitude of the PA, given the input sequence z. V’ndenotes the phase of zn. Tn defines the instantaneous peak-to-average power ratio of zn, i.e., and 7th denotes the clipping threshold which is usually referred to the input backoff (IBO). Apart from the ‘soft limiter’ model, there exists other models of PA, e.g., SSPA. We denote the frequency-domain response of the transmitted signal after PA clipping as z = g(s + .where ^(') stands for the frequency-domain nonlinear clipping function. For the sake of simplicity, the CP insertion is omitted in this discussion. The received signal in the frequency domain (obtained after OFDM demodulation) takes the form as Y = DZ + v (5) where D — diag{Hi... ■ ,¾} denotes the channel frequency response (CFR) matrix and v denotes the additive white Gaussian noise (AWGN). After the one-tap channel equalisation, we yield S = D-1Y; then, a modulo operation is conducted to remove the impact of u?; mathematically, this is written as s = S — a round(—) a Note that ckdenotes the constant scaling factor which is used to facilitate the modulo operation. And round(-) denotes the rounding operation. The value of a is a configurable parameter which may affect both PAPR control performance as well as the signal detection performance (at the receiver). I-OFDM principle Overview The following discussion focuses on 1) principle of the iterative process (called CMP process) devised for finding the superimposed sequence i; 2) a list encoding scheme which is designed to preserve more diversity during the CMP process and 3) a multi-path initialisation scheme which is designed to exploit more diversity. The configurations of the techniques can be categorised into a) single-path CMP (S-CMP) scheme, b) multi-path CMP (M-CMP) scheme and c) their list-encoding based counterparts. Generally, the M-CMP scheme can be seen as a parallel of the S-CMP processes, as each individual path proceeds with their own process separately. Thus, for the sake of simplicity, we firstly introduce the CMP principle for PAPR control by taking the S-CMP process as an example. The same principle is then applicable to the M-CMP process. S-CMP procedure is an iterative process. The original time-domain OFDM signal is fed as the input and used as initialisation of the perturbated vector. S-CMP process proceeds at each iteration by summing the perturbated vector up with a perturbation vector which is selected from a pre-defined finite candidate set, according to an established criterion. In such a way, the perturbated vector is iteratively updated. The entire CMP process converges once the stopping criterion is satisfied and the perturbated vector is output as the final l-OFDM waveform. The criterion used at each iteration for selecting perturbation vector candidates is the key determinant factor of the PAPR control performance. Intuitively, the criterion is designed to primarily satisfy the following requirements: 1. At a certain iteration, a subset of perturbation vector candidates is formed; the perturbation vector candidates therein can reduce multiple entry-wise instantaneous powers of the perturbated vector, e.g., the ten largest entry-wise peak powers. Note that the number of peak powers is also configurable. 2. Among those collected perturbation vector candidates, the candidate that offers the minimum clipping distortion (or alternatively, the minimum PAPR) is selected to update the perturbated vector. The criterion designed for the list-encoding scheme satisfies the first requirement; however, listencoding scheme may select, at each iteration, multiple perturbation vector candidates instead of a single candidate; and then use each selected candidate vector to individually update their corresponding perturbated vector. CMP Principle and S-CMP Process Recalling that the l-OFDM waveform is formed by z = x + i and i = O!FHw. S-CMP process is designed to iteratively find i At each iteration, the perturbated vector is updated as z(t) = z(t - 1) + i*(t) (7) where denotes the desired perturbation vector; it is selected from a pre-defined candidate vector set, i.e., € 4 ZW denotes the perturbated vector updated at the t-th iteration; in the single-path scheme, the perturbated vector is initialised as the original OFDM signal, i.e., z(0) = x. The candidate set is defined as = {wln £ [1, N],£ G [1, L]}, where = a^nxt (8) denotes the perturbation vector candidates. Respectively, f^and e X denote the n-th column of Fh and the complex-valued integer drawn from an integer set X, e.g., ~ {±1> ±Qwith L = 4 It is worth noting that only corresponds to a single perturbation integer By sorting lzn,Vn(t)|2 in the descending order, we can collect the first ^peak largest instantaneous powers’ index and form an index set as ni e I^G)I = M>eak Accordingly, denotes the index of the largest instantaneous power, i.e., the largest peak power. Knowing the instantaneous powers I^.VniWPthat are of interest, we can determine whether the perturbation vector candidates, f, can contribute to the multi-peak power reduction. We defined the entry-wise correlation between the perturbated vector and the candidate perturbation vector as (n4) = zgX :           14? p with conj{'} denoting the conjugate operation and ^{’} denotes the real part; the superscript denoting the entry’s index. We define the peak power variation as A _ I ~(n*) (f — 1 )|2 _ \z^n^ (l — 1 ) I I2 th 1 ' 'I 1 ' ' T W 1 ; an equivalent interpretation of the ni peak power being th. minimised by the perturbation is A >0. This leads to a sufficient and necessary condition for the ni peak power to be minimised, i.e., . Collecting those candidate perturbation vectors that can minimise the selected instantaneous powers allows to form a subset, i.e., := {ifc|p£ Vnt 6 $(i)},^4(t) c A (recalling that the subscript ndenotes the index of columns ofIDFT matrix and £denotes the index of perturbation integer, it could be more convenient to merge them into a unified subscript, i.e., fc G |n practice, the indices nand ^in the equations above can be replaced with k. Selected perturbation vector candidates shall satisfy the condition of llz(^ _ 1) + M2 ^llxll2 which means that the corresponding energy increase (induced by the superimposed sequences) is controlled within an acceptable level. £ >^indicates the allowed maximum energy increase (e.g., 1 dB); such a threshold is pre-configured considering that significant power increase leads to reduced PAPR but also energy efficiency loss. One of the criterions that can be used at each iteration, for selecting the best candidate, i.e., from can be expressed by i*(<) = arg min Ek (10) with Sfcis defined by (please see equation (3) for the definition of Ek := ||zfc(t) -s(zfc(t))|| (11) i.e., the Euclidean distance between the perturbated sequence (zkW = ZG - 1) + ifc) updated by ik at the t-th iteration, and its counterpart output by the PA, which undergoes the clipping effect. Equations (10) and (11) imply that the candidate that offers the ‘minimum distortion’ is selected for the update. Note: • varies iteration by iteration with the process proceeding. This is because at each iteration, the considered peak powers’ location (collected by $W) vary and thus incurs the perturbation vector candidates demonstrate different peak power reduction behaviour (please see equation (9)). • The iterative process converges if the Euclidean distance defined in (11) cannot be further reduced through two consecutive iterations. • An alternative criterion for selecting the candidate perturbation vector is the ‘minimum-PAPR’ criterion, i.e., selecting the candidate that offers the minimum PAPR instead of the distortion. Although this will lead to suboptimal performance, to some extent the performance is acceptable, e.g., for 4k-QAM. Mathematically, the criteria for selecting the best candidate to minimize PAPR can take the form i*(t) — arg min pfc(f),Vfe (lib) Where Pk W denotes the PAPR of zk(Q • In (11), the setup of 9(‘) necessitates a known IBO value which can be pre-configured close to the worst-case scenario, e.g., 4 dB or even lower. This will ensure the waveform output by l-OFDM schemes to be resilient to the clipping distortion induced by PAs, which possesses IBO equal or greater than the pre-configured IBO value. Please note, this also means that the IBO knowledge of the practical PA is not required. In the previous discussions, l$W is defined as a configurable parameter which remains as constant during the iterations. While more practically, the number of reduceable instantaneous powers may not remain as a fixed number from iterations to iterations. Thus, l$W can be set as the maximum number of reduceable interested instantaneous powers for each iteration. Figure 6a is a flow chart 600 illustrating a process of l-OFDM waveform generation using the above described S-CMP procedure. At step 602, an OFDM signal is input. At step 604, a subset of candidate perturbation vectors is calculated. The subset of candidate perturbation vectors is selected from the whole set of candidate perturbation vectors by choosing those candidate perturbation vectors that minimise the selected instantaneous power. At step 606, a single perturbation vector, i*(t) is selected, i.e. chosen from the subset of candidates. This perturbation vector is the “best candidate” vector and is determined by calculating which of the subset of candidate perturbation vectors minimizes the distortion of the signal, or minimizes the peak-to-average power ratio of the signal. At step 608, the best candidate vector is summed with the OFDM signal and it is checked whether the distortion of the signal is reduced or not. If the distortion of the signal is reduced (Yes at step 608), the method proceeds to step 610, in which the perturbation vector is updated and the next iteration begins. The method then returns to step 604. Steps 604-610 are repeated iteratively until the distortion of the signal is not reduced (No at step 608). In this event, the method ends at step 612 with the final l-OFDM signal being output. Figure 6b is an alternative flowchart, illustrating the same process of l-OFDM waveform generation as in Figure 6a, using the above described S-CMP procedure. Step 614 comprises inputting an OFDM signal. Step 616 comprises defining a set of perturbation vector candidates. The set of perturbation vector candidates may referred to as a subset, since it may be selected from a wider set of candidate perturbation vectors by choosing those candidate perturbation vectors that minimise the selected instantaneous power. Step 618 comprises selecting one or more perturbation vectors from the set of perturbation vector candidates. The selection is performed by calculating which of the one or more perturbation vectors results in the lowest distortion of the OFDM signal or the lowest peak-to-average power ratio of the OFDM signal. Step 620 comprises summing each of the selected one or more perturbation vectors with separate instances of the OFDM signal to create one or more modified OFDM signals in one or more calculation paths. Step 622 comprises determining whether a stopping criteria has been satisfied. For example, the stopping criteria may be whether the distortion of the signal is reduced or reduced by more than a predetermined amount. If the stopping criteria is not met (No at step 622), the process returns to step 614 and the modified ODM signal is used as the input for a further iteration of steps 614-620. If the stopping criteria is met (Yes at step 622), the process proceeds to step 624 comprising outputting a final modified OFDM signal. List-Encoding Scheme Throughout the S-CMP process, I^WI 1 generally holds, while the ‘minimum-distortion’ criterion enables a single candidate perturbation vector to be selected. This would incur the loss of the path diversity and thus renders the PAPR control performance suboptimal. The list-encoding scheme is devised to preserve more path diversity. Compared with the ‘minimum distortion’ criterion, the difference arises from the final candidate vector selection. The list-encoding scheme allows the perturbation vector to be updated by different candidate perturbation vectors, separately and in parallel; such yields multiple paths, with each represented by a uniquely updated perturbation vector. Note that each path developed at current iteration may develop multiple paths at the next iterations; this results in the computational complexity exponentially increasing w.r.t the number of developed paths. We introduce a list along with a tailored stopping criterion to effectively control the computational complexity. The list is denoted by, £(*)> ~ where Qdenotes the total number of paths (i.e., updated perturbation vectors) that are developed before the stopping criterion is met. More specifically, as the iterative process proceeds, the number of paths, will increase cumulatively; and at a certain iteration, this number becomes larger than fl This means that the stopping criterion has been met and thus the best perturbation vector will be selected from the collected candidates. Note that the selection criterion can be either ‘minimum-distortion’ or ‘minimum-PAPR’. Note that the stopping criterion that applied to S-CMP is also compatible with S-list-CMP. In the case where the total number of collected candidates is always smaller than the list size, the list-encoding scheme may converge in a way that the process will stop if the list remains the same for arbitrary two consecutive iterations. Figure 7 is a flow chart 700 illustrating a process of l-OFDM waveform generation using the above described single-path list-encoding CMP (S-list-CMP) procedure. At step 702, an OFDM signal is input. At step 704, a subset of candidate perturbation vectors is calculated for each of a plurality of paths and at step 706 the paths are developed in parallel. Steps 704 and 706 together encompasses all of steps 604,606, 608 and 610 described with reference to Figure 6, but performed multiple times, i.e. once for each of a the plurality of paths. The result is that a subset of candidate perturbation vectors is calculated for each of the plurality of paths in parallel. The number of paths increases with each iteration. After each iteration of the plurality of paths, at step 708 it is checked whether the total number of paths surpasses a threshold value (Q). If the threshold is not exceeded (No at step 708), the iterative process of steps 704 and 706 continues. When the threshold number of paths is exceeded (Yes at step 708), the best path is selected at step 710. Each of the final perturbation vectors produced by the plurality of paths is assessed and the “best candidate” perturbation vector is selected, using the criteria discussed previously. The process ends at step 612 with the final I-OFDM signal being output. Multi-Path Initialisation Scheme The multi-path initialisation aims to exploit more path diversity. This is achieved by adding an initialising perturbation vector to the original OFDM signal which thus form an initialised path, i.e., MOHx + m,- (12) with J denoting the path index (-^path paths in total), [1> ^path]> i 7^ j Each initialised path proceeds with their individual S-CMP procedure. Note that multiple paths can proceed in parallel, and they may demonstrate different converging behaviour. Accordingly, there will be a final l-OFDM signal being obtained by each path, the one that offers the minimum distortion / PAPR is selected. Figure 8 is a flow chart 800 illustrating a process of l-OFDM waveform generation using the above described multi-path CMP (M-CMP) procedure. At step 802, an OFDM signal is input. At step 804, a multi-path initialization is performed by applying a different initialising perturbation vector to different instances of the input OFDM signal. At step 806, each of the multiple paths proceeds in parallel according to the S-CMP procedure described above with reference to Figure 6. In step 808, once each of the parallel paths has completed i.e. has output a final l-OFDM signal, a selection of the best path is made. The best path may be selected by selecting the l-OFDM signal with the lowest distortion relative to the original signal or lowest PAPR. At step 810, the final l-OFDM signal resulting from the selected path is output. Figure 9 is a flow chart 900 illustrating a process of l-OFDM waveform generation using a multi-path with list encoding CMP (M-list-CMP) procedure. At step 902, an OFDM signal is input. At step 904, a multi-path initialization is performed by applying a different initialising perturbation vector to different instances of the input OFDM signal. At step 906, each of the multiple paths proceeds in parallel according to the S-list-CMP procedure described above with reference to Figure 7. In step 908, once each of the parallel paths has completed i.e. has output a final l-OFDM signal or the total number of paths has exceeded a threshold number, a selection of the best path is made. The best path may be selected by selecting the l-OFDM signal with the lowest distortion relative to the original signal or lowest PAPR. At step 910, the final l-OFDM signal resulting from the selected path is output. The graphs shown in Figures 10 and 11 compares the performance of a system using the l-OFDM process according to embodiments of the present invention with prior art techniques. Figure 10 shows the 4k-QAM symbol error rate (SER) as a function of bit-energy-to-noise ratio (EbNo). In this example, the multi-path CMP (M-CMP) procedure was used and minimum PAPR candidate selection criteria was used. The power amplifier uses a soft limiter model and it is a single-input single-output (SISO) system. As can be seen, the error rate when the technique of the present invention is used tracks the ideal rate closely. In contrast, the prior art techniques suffer high error rates. Figure 11 shows the 4k-QAM symbol error rate (SER) as a function of bit-energy-to-noise ratio (EbNo). In this example, the multi-path CMP (M-CMP) procedure was used and minimum PAPR candidate selection criteria was used. The power amplifier uses a soft limiter model and it is a multi-input multi-output (MIMO) system. As can be seen, the l-OFDM waveform generation is compatible with both single-input singleoutput (SISO) and multiple-input multiple-output (MIMO) systems. For both uplink and downlink MIMO systems, precoding technologies may apply, e.g., downlink precoding and uplink precoding schemes. The present invention is also compatible with these precoding technologies. As an example, the figure 12 illustrates the implementation of the l-OFDM technique for the conventional linearly-precoded MIMO-OFDM downlink scenario. Note that the linear precoder is conducted by the base station to pre-compensate the fading effect, e.g., zero-forcing precoder, minimum mean squared error precoder and codebook based precoders. Although it is not shown in figure 12, finding the appropriate integer sequence in such a case can be considered jointly with the vector perturbation procedure in MIMO downlink transmissions. As an extension to the process already described herein, it is possible to cluster multiple OFDM signals based on their peak power patterns, and to find a perturbation vector which offers good peak power reduction for this cluster of OFDM signals. The peak power patterns, e.g., peak power indices, time-domain entry’s magnitude / angle, may be shared by multiple time-domain OFDM signals in the same cluster. By carefully defining the clusters and the corresponding integer sequence, a clustering-and-selection mechanism can be derived to establish the mapping between the OFDM signal and the appropriate integer sequence candidate(s). Through this procedure, the I-OFDM waveform for a number of initial OFDM signals can be found with significantly reduced computational complexity (compared with CMP procedures discussed above) and at the acceptable cost of peak power reduction performance. Figure 13 is a flowchart 1300 illustrating the concept of waveform clustering and selection. In this embodiment, it is assumed that there exists a single integer sequence for each class. If multiple integer sequences are configured for a specific class, a final decision process can be integrated for the selection of the candidate that offers the best peak power reduction performance. In step 1302 an OFDM signal is input. A corresponding classified index is output at step 1304. The classification is based on the signal’s time-domain features. The feature used for classification could be peak power, time-domain entry amplitude and / or phase, but is not limited to these examples. The classification outputs a cluster index which guides the selection of the perturbation vector incorporated in the perturbation vector set. The classification procedure can be realized through deep neural network with devised learning algorithm or, other explicit algorithms. Given the classified index as the input, at step 1306 a corresponding perturbation vector is output which is optimised for the corresponding class for clipping distortion minimization (or PAPR minimization). The calculated perturbation vector for each cluster is referred to the vector that has multiple modulo-domain integer entries, rather than having a single integer entry. At step 1308, the output perturbation vector is summed with the original OFDM signal to form the I-OFDM signal. At step 1310, the l-OFDM signal is output. In practice, the waveform clustering and selection procedure may be conducted in two stages, offline training stage and online stage. The process described above with reference to Figure 13 is the online stage. Prior to this, an offline training stage may be performed as illustrated in the flowchart 1400 of Figure 14. The offline training stage begins at step 1402, where a set of OFDM signals are input. Each OFDM signal in the set Is fed as a single input sequence into a pattern analysis module. At step 1404, the pattern analysis module assigns each input OFDM signal to a certain cluster. This assignment may be performed according to the peak-power-related patterns of the signal, e.g., amplitude, phase, peak power, etc. Thus, the OFDM signals in the same cluster may demonstrate similar peak-power-related patterns. By performing this pattern analysis on all of the input OFDM signals, a certain number ofclusters can be formed (step 1406). The cardinality of the OFDM signal set needs to be sufficiently large; such that each cluster has enough OFDM signals assigned to it; this is particularly helpful for finding the appropriate perturbation vector. For each cluster, at step 1408 an appropriate perturbation vector is calculated. This perturbation vector is used in step 1306 of the online stage, as described above, in order to minimizing the clipping distortion / PAPR for all the OFDM signals belonging to that cluster. At step 1410, the calculated perturbation vectors for all the clusters are collected by a set which will be used in the online stage. Figure 15 illustrates a network node 1500 configured to perform any of the techniques described herein. The network node 1500 may be a base station. Network node 1500 includes processing circuitry 1502 and communication circuitry 1504. The communication circuitry 1504 is configured to transmit and / or receive information to and / or from one or more other nodes. This may be performed via any communication technology. The communication circuitry 1504 may comprise a receiver configured to receive signals from a user equipment (UE) and a transmitter configured to transmit signals to a UE. The processing circuitry 1502 comprises a processor 1506 (which may be any combination of microprocessors, microcontrollers, digital signal processors, and digital logic) and a memory 1508. The processor 1506 and memory 1508 are operatively coupled together and the processing circuitry 1502 is operatively coupled to the communication circuitry 1504. Memory 1508 may comprise program code for execution by processor 1506, where the program code is configured to cause the processor 1506 (and thereby the processing circuitry 1502) to carry out any one or more of the techniques described herein. Figure 16 illustrates a user equipment (UE) 1600 as implemented in accordance with one or more embodiments. The UE 1600 may be configured to perform any of the techniques described herein. As shown, the UE 1600 includes processing circuitry 1602 and communication circuitry 1604. The communication circuitry 1604 (e.g., radio circuitry) is configured to transmit and / or receive information to and / or from one or more other nodes, e.g., via any communication technology. Such communication may occur via one or more antennas that are either internal or external to the UE 1600. The processing circuitry 1602 comprises one or more processors 1606 and is configured to perform processing described above, such as by executing instructions stored in memory 1608. It should be noted that any or several of the components illustrated in Figure 15 and Figure 16 could be implemented as a single unit or be divided into several units. Processing circuitry 1502 / 1602 may use software functioning in conjunction with a programmed microprocessor or a general purpose computer, and / or using an application specific integrated circuit (ASIC). Whilst certain embodiments of the invention have been described herein with reference to the drawings, it will be understood that many variations and modifications will be possible without departing from the scope of the invention as defined in the accompanying claims. For example, although some embodiments relate to mobile cellular communications, the techniques described herein can also be applied to Satellite Communications, Broadcasting Networks, WiFi, or any other suitable system that uses OFDM at any frequency.

Claims

1. A method of generating a modified Orthogonal Frequency Division Multiplexed, OFDM, signal, the method comprising:(i) inputting an OFDM signal;(ii) defining a set of perturbation vector candidates;(iii) selecting one or more perturbation vectors from the set of perturbation vector candidates, by calculating which of the one or more perturbation vectors results in the lowest distortion of the OFDM signal or the lowest peak-to-average power ratio of the OFDM signal;(iv) summing each of the selected one or more perturbation vectors with separate instances of the OFDM signal to create one or more modified OFDM signals in one or more calculation paths;(v) performing steps (ii)-(iv) iteratively on the one or more modified OFDM signals until a stopping criteria is satisfied; and(vi) when the stopping criteria is satisfied, outputting a final modified OFDM signal.

2. The method of claiml, further comprising, after inputting the OFDM signal, initialising a perturbated vector using the OFDM signal.

3. The method of claim 1, further comprising, after inputting the OFDM signal, adding a plurality of initialising perturbation vectors to respective instances of the OFDM signal to create a plurality of starting vectors.

4. The method of any of claims 1 to 3, wherein defining the set of perturbation vector candidates comprises:determining the n largest instantaneous peak powers of the OFDM signal;determining to what extent each of a plurality of possible perturbation vectors reduces the peak powers of the n largest instantaneous peak powers of the OFDM signal; andselecting the perturbation vectors which minimise the peak powers of the n largest instantaneous peak powers of the OFDM signal.

5. The method of any of claims 1 to 4, wherein calculating which of the one or more perturbation vectors results in the lowest distortion of the OFDM signal comprises calculatingi*(t) = arg min skwhere i*(t) represents the one or more perturbation vectors resulting in the lowest distortion of the OFDM signal, sk is defined byet ■= IM^)-9(2^))11and g(zk(t)) is an entry-wise non-linear function modelling a clipping effect of a power amplifier and takes the form:with A denoting a maximum output amplitude of the power amplifier, given the input sequence z, ^denoting the phase of and 7« defining the instantaneous peak-to-average power ratio of ^n.

6. The method of any of claims 1 to 4, wherein calculating which of the one or more perturbation vectors results in the lowest peak-to-average power ratio of the OFDM signal comprises calculatingi*(i) = arg min pkify^kwhere denotes the peak-to-average power ratio of7. The method of any of claims 1 to 6, wherein the stopping criteria is that the selected one or more perturbation vectors do not reduce the distortion of the OFDM signal or the peak-to-average power ratio of the OFDM signal relative to the previous iteration.

8. The method of any of claims 1 to 6, wherein the stopping criteria is that the selected one or more perturbation vectors do not reduce the distortion ofthe OFDM signal or the peak-to-average power ratio of the OFDM signal by more than a predetermined threshold relative to the previous iteration.

9. The method of any of claims 1 to 8, wherein the stopping criteria is that the number of calculation paths exceeds a predetermined threshold.

10. The method of claims 1 to 9, further comprising:inputting the final modified OFDM signal into a power amplifier to create an amplified modified OFDM signal; andtransmitting the amplified modified OFDM signal.

11. A computer-readable storage medium arranged to store computer program instructions which, when executed, perform a method according to any of claims 1 to 10.

12. Apparatus comprising communication circuitry and processing circuitry configured to:(i) input an OFDM signal;(ii) define a set of perturbation vector candidates;(iii) select one or more perturbation vectors from the set of perturbation vector candidates, by calculating which of the one or more perturbation vectors results in the lowest distortion of the OFDM signal or the lowest peak-to-average power ratio of the OFDM signal;(iv) sum each of the selected one or more perturbation vectors with separate instances of the OFDM signal to create one or more modified OFDM signals in one or more calculation paths;(v) perform steps (ii)-(iv) iteratively on the one or more modified OFDM signals until a stopping criteria is satisfied; and(vi) when the stopping criteria is satisfied, output a final modified OFDM signal.

13. The apparatus of claim 12, wherein the processing circuitry comprises: one or more processors; andone or more memories coupled to the one or more processors, wherein the one or more memories include instructions executable by the one or more processors.5 14. The apparatus of claim 12 or claim 13, wherein the apparatus isconfigured to transmit the final modified OFDM signal using the communication circuitry.

15. The apparatus of any of claims 12 to 14, wherein the apparatus is a10 network node.

16. The apparatus of any of claims 12 to 14, wherein the apparatus is a UE.15

Images