1

EVM-Constrained and Mask-Compliant MIMO-OFDM Spectral Precoding Shashi Kant, Student Member, IEEE, Mats Bengtsson, Senior Member, IEEE, Gabor Fodor, Senior Member, IEEE, Bo Goransson,¨ Member, IEEE, and Carlo Fischione, Senior Member, IEEE

the sinc functions of the OFDM signal [4]. The OOBE must Abstract—Spectral precoding is a promising technique to sup- be adequately suppressed since high OOBE causes significant press out-of-band emissions and comply with leakage constraints interference to the neighbouring adjacent channels. In practice, over adjacent frequency channels and with mask requirements on the unwanted emissions. However, spectral precoding may all OFDM systems are designed to comply not only with distort the original data vector, which is formally expressed as OOBE requirements, in terms of adjacent channel leakage the error vector magnitude (EVM) between the precoded and ratio (ACLR) and spectral emission mask, but also in-band original data vectors. Notably, EVM has a deleterious impact requirements, in terms of error vector magnitude (EVM) and on the performance of multiple-input multiple-output orthogonal other signal demodulation/detection requirements [5]. Simply, frequency division multiplexing-based systems. In this paper we propose a novel spectral precoding approach which constrains the EVM describes the distortion/noise incurred to the useful the EVM while complying with the mask requirements. We first (transmit) symbols and ACLR represents the amount of un- formulate and solve the EVM-unconstrained mask-compliant desired power that exists in the neighbouring carriers relative spectral precoding problem, which serves as a springboard to to the desired carrier power—the detailed description of these the design of two EVM-constrained spectral precoding schemes. metrics are in the subsequent sections. The first scheme takes into account a wideband EVM-constraint which limits the average in-band distortion. The second scheme There are a plethora of techniques to suppress/reduce OOBE takes into account frequency-selective EVM-constraints, and for the cyclic-prefix based OFDM—see, e.g., [6] and its consequently, limits the signal distortion at the subcarrier level. references—which can be categorized into time and frequency Numerical examples illustrate that both proposed schemes out- domain methods. Amongst them, the methods are, namely, perform previously developed schemes in terms of important guard band inclusion, filtering [7], windowing [8], cancellation performance indicators such as block error rate and system- wide throughput while complying with spectral mask and EVM carriers [9]–[11], and spectral precoding [1], [12]–[18]. constraints. Spectral precoding is one of the promising bandwidth Index Terms—Sidelobe suppression, spectral precoding, efficient techniques for OOBE reduction. It spectrally precodes MIMO, OFDM, EVM, out-of-band emissions, ACLR, Consensus the data symbols before OFDM modulation [13], [17], [18], ADMM, Douglas-Rachford Splitting. which reduces the OOBE without using the extra spectral re- sources contrary to cancellation carriers and without increasing I.INTRODUCTION the delay/time dispersion or penalizing the cyclic prefix of the transmitted signal unlike filtering/windowing. In Section II we Modern wireless communication systems, including fifth- review related works in addition to our contribution. generation (5G) New Radio (NR), adopt orthogonal frequency In this paper, we develop spectral precoding schemes division multiplexing (OFDM) with cyclic prefix [2]. The that comply with EVM-constraints and simultaneously meet reasons are that OFDM has several attractive characteristics OOBE requirements in terms of mask and ACLR, without such as robustness to the negative effects of time dispersive affecting signal processing at the receiver. Our objective channels and multi-path fading, simplicity in terms of equal- is to design low complexity spectral precoding algorithms ization and flexibility in terms of supporting both low and that operate well in wide-band multiple-input multiple-output high symbol rates and thereby supporting a variety of quality (MIMO)-OFDM systems. In the first part of the paper, we of service requirements. Further, OFDM-based systems can consider the problem of designing mask-compliant spectral facilitate dynamic spectrum sharing [3]. precoding without EVM constraints. In the second part of the Unfortunately, OFDM suffers from high out-of-band emis- paper, we propose to incorporate wideband and frequency- sions (OOBE) due to the discontinuities at the boundaries of selective EVM constraints in addition to mask compliance. the rectangular window and the high sidelobes associated with Our goal is to improve key performance indicators, including S. Kant, B. Goransson,¨ and G. Fodor are with Ericsson AB and KTH out-of-band and in-band performance metrics, because highly Royal Institute of Technology, Stockholm, Sweden (e-mail: {shashi.v.kant, spectral efficient MIMO-OFDM-based systems require low bo.goransson, gabor.fodor}@ericsson.com) EVM. Accordingly, we introduce a wideband EVM constraint, M. Bengtsson and C. Fischione are with KTH Royal Institute of Technology, Stockholm, Sweden (e-mail: {[email protected], which restricts the wideband average in-band distortion due carlofi@kth.se}) to spectral precoding. Finally, the frequency-selective EVM The work of S. Kant was supported in part by the Swedish Foundation for constraint offers more flexibility—but at the expense of in- Strategic Research under grant ID17-0114. Part of this paper on EVM-unconstrained OFDM spectral precoding was creased complexity—than using wideband constraints, because presented at IEEE SPAWC 2019 [1]. the frequency-selective constraints limit the in-band distortion ©2020 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. DOI: 2 power at the subcarrier (or group of subcarriers) level. As Notching the spectrum at well-selected discrete frequencies we will show and discuss, these design approaches represent often suppresses the whole signal spectrum and consequently at least three distinct ways of addressing the complexity- leads to OOBE reduction [13]. However, such a notching performance trade-off in the design of MIMO-OFDM spectral approach has a deleterious impact on the in-band performance, precoders. in terms of increased EVM and block error rate, which The rest of the paper is structured as follows. The next thereby penalizes the system-wide throughput. Notably, the section, which can be skipped by the reader familiar with edge subcarriers have very high EVM—see Fig. 2(c) in the the subject, discusses related work including impact of EVM numerical results Section VI-C—which decreases the through- and states our contribution. Section III discusses preliminaries put. Recognizing these problems, in [16] a weighted-notching and introduce the system model. Next, Section IV formulates spectral precoder method is proposed to reduce the EVM at the EVM-unconstrained problem and develops an alternating the edge subcarriers by spreading the total distortion over all direction method of multipliers (ADMM)-based algorithm the allocated subcarriers. Another extension of the notching [19], [20] and a coordinate-descent-based algorithm to solve spectral precoder, proposed in [23], performs precoding jointly this unconstrained problem. Section V formulates the EVM- over several consecutive OFDM symbols which improves the constrained problem and develops an ADMM-based solution in-band performance slightly over the single OFDM symbol- and yet an alternative approach based on the Douglas-Rachford based precoding but penalizes the latency since a large number algorithm [19], [21]. Section VI presents simulation results of OFDM symbols have to be buffered. that compare the performance of the proposed schemes with that of benchmarking schemes, and finally Section VII con- cludes the paper. 2) Constrained Spectral Precoding: On the contrary, mask- compliant spectral precoder (MSP) improves the in-band per- II.RELATED WORKSAND CONTRIBUTIONS formance over notching spectral precoder while only meeting a target mask instead of creating nulls at the selected dis- In this section, we first give an overview of the related crete frequencies at the expense of increased complexity [1], frequency domain techniques, then the impact of EVM on [24]. Because MSP is posed as an inequality constrained the system performance, and finally highlight our original convex optimization formulation, it typically does not yield contribution. a closed-form solution. Furthermore, the authors of [15] sug- gest utilizing a generic optimization solver, which generally A. Cancellation Carrier Techniques employs interior-point methods [25]. Therefore, the authors The cancellation carrier techniques, such as active in- in [1], [24], propose computationally efficient schemes to terference cancellation [9], cancellation carrier with power obtain mask-compliant spectrally precoded data symbols for constraint [10], and extended active interference cancellation single-input single-output (SISO)-OFDM systems. The authors [11], utilize the non-data bearing subcarriers solely for OOBE in [26] propose three EVM-constrained precoders for SISO reduction. These methods generally offer good sidelobe sup- systems, whereby two precoders are developed heuristically, pression with relatively low complexity, at the expense of and one of them is posed as a convex optimization problem. transmit power wastage and spectral resources. Besides signal- The optimization-based precoder is obtained by minimizing to-noise ratio (SNR) degradation due to transmit power sharing the `2-norm of the OOBE at a chosen set of discrete frequency with the non-data bearing cancellation carriers, the extended points subject to the EVM constraint—yielding no closed- active interference cancellation [11] induces intersymbol and form solution. The former precoders are ad hoc, but can intercarrier interferences to the desired/useful signal due to be seen as a scaled-form of the notching spectral precoder the placement of the non-orthogonal carriers. Additionally, such that they meet the EVM constraint but penalize the there are several practical implementation challenges, e.g., OOBE in an unsystematic way. Furthermore, the authors many cancellation carriers are required to maintain appropriate show numerically that the ad hoc precoders render superior power spectral density (PSD) of the composite signals to performance compared with the optimization-based precoder avoid the negative impact of intermodulation due to nonlinear in suppressing OOBE under EVM constraint. components [22], e.g., power amplifier, and consequently 3) Spectral Precoding in MIMO Systems: In [27] several failing to meet other OOBE requirements—in terms of mask linear receivers are investigated when notching spectral pre- and ACLR defined in NR-like standards [5]. coders is employed in a MIMO-OFDM system. In [18, Pa- per F], the authors extend the SISO-OFDM spectral precoding B. Spectral Precoding Techniques to massive MIMO-OFDM for a joint spatial and notching There are many variants of spectral precoding methods in spectral precoding, which exploits full downlink channel state the literature to suppress OOBE. We briefly review some of information at the transmitter due to channel reciprocity in these techniques subsequently. time division duplexing, to improve the in-band performance In the notching spectral precoder [13], the spectrally pre- at the receiver. Recent works analyzed the impact of the EVM coded signal essentially nulls/notches the OOBE at given at the receiver or over-the-air specifically in massive MIMO discrete frequencies in the out-of-band spectrum of the OFDM context. These works reveal that in beamforming systems the signal. The precoder is obtained in the closed-form by solv- EVM is also beamformed along the same direction as the data, ing equality constrained least-squares optimization problem. see e.g., [28]–[31]. 3

C. Impact of EVM on the System Performance number x. The i-th element of a vector a∈Cm×1 is denoted In practical transceivers, the hardware imperfections stem by a[i] ∈ C, and element in the i-th row and j-th column m×n from both the digital and the analogue components. These of the matrix A ∈ C is denoted by A [i, j] ∈ C. The m×n may include clip noise, filter ripple or distortion, in-phase i-th row and j-th column vector of a matrix A ∈ C 1×n m×1 and quadrature (IQ) mismatch, non-linearity, local oscillator are represented as A [i, :] ∈ C and A [:, j] ∈ C , inducing phase noise, sampling clock offsets, timing and respectively. An i-th higher order vector and matrix are m×1 m×1 m×n frequency error, see, e.g., [32], [33]. These imperfections denoted as x[i] ∈ C or xi ∈ C and X[i] ∈ C . have detrimental impact on the MIMO-OFDM performance, We form a matrix by stacking the set of higher order  M×1 N  1×N M notably in high data rate achieving systems [34]–[39]. In vectors a[n]∈C n=1 and b[m]∈C m=1 column- practice, EVM metric evaluation provides insightful and useful wise and row-wise as A = [a[1],..., a[N]] ∈ CM×N and information on the link quality, in terms of SNR, seen at B = [b[1]; ... ; b[M]] ∈ CM×N , respectively. The dimensions the receiver due to the aggregated digital and analogue hard- of the vectors/matrices are equally applicable for both C and ware imperfections [40]–[42]. The NR-like standards mandate R. The transpose and conjugate transpose of a vector or T H (transmit) EVM requirements, see Table IV, on the composite matrix are denoted by (·) and (·) , respectively. The complex ∗ inband distortion emanating from the various sources but conjugate is represented by (·) . The K×K identity matrix is measured at the (ideal) receiver by using pilots or reference written as IK . The expectation operator is denoted by E{·}. signals as described in, e.g., [5, Appendix B]. An i-th iterative update is denoted by (·)(i). To minimize the impact of EVM onto the system throughput performance, in general, there are at least two approaches: III.PRELIMINARIES 1) constraining the EVM at the transmitter and 2) mitigating In this section, we introduce the downlink MIMO-OFDM or cancelling the (transmit) EVM seen at the receiver. We system model followed by performance metrics useful for the pursue the former approach because NR-like standards [5], spectral precoding design. [43] stipulate the minimum EVM requirements according to the considered modulation alphabet, which must be satisfied by the base station or user equipment devices. Hence, in A. System Model and Out-of-Band Emissions this paper, we constrain the EVM of spectral precoding in We consider the OFDM-based single-user MIMO downlink, addition to mask compliance (which implicitly may ensure where the base station is equipped with NT transmit (Tx) the fulfilment of minimum ACLR requirements). antennas, and the user equipment (UE) is equipped with NR receive (Rx) antennas as depicted in Fig. 1. Additionally, we D. Contribution of the Paper reckon spatial multiplexing transmission scheme with NL ≤ In this paper, we design computationally efficient al- min {NT,NR} spatial layers. gorithms for the mask-compliant spectral precoding with A spatially precoded symbol vector x[k] at k-th subcarrier (un)constrained EVM for the MIMO-OFDM-based systems for a given OFDM symbol can be formed by N ×1 that scale linearly with the number of supported transmit x[k] = W [k]s[k] ∈ C T , (1) antennas and do not require additional signal processing at where, the vector s[k] ∈ S NL×1 belongs to a complex- the receiver. More specifically, valued finite-alphabet set S , e.g., corresponding to a 2Q- • we propose and accomplish a wideband and frequency- quadrature amplitude modulation (QAM) constellation with selective EVM-constrained and mask-compliant spectral NT×NL Q ∈ {2, 4, 6, 8}. The given MIMO precoder W [k] ∈ C precoding formulation. is chosen, e.g., from a codebook in 3GPP NR [44]. • we develop two highly efficient algorithms by decom- Towards this end, we introduce a frequency-domain posing the large-scale spectral precoding optimization (spatially-precoded) data matrix X by stacking each (column) problems for both unconstrained and constrained EVM vector x [k] (cf. Eq. (1)) column-wise for all the subcarriers into subproblems, where each subproblem yields closed- within a given N-point inverse discrete Fourier transform such form or efficient solution, which has a very low com- that putational complexity compared to the general purpose h i X := [x [1] ,..., x [N]] := dT; ... ; dT ∈ NT×N , (2) solver. We propose solutions for (un)constrained EVM: 1) 1 NT C N×1 ADMM-based algorithms, referred to as ADMM/EVM- where we define a spatially precoded vector dj ∈ C for constrained ADMM (EADMM) and 2) specialized al- j-th transmit antenna as : T N×1 gorithms, dubbed as semi-analytical spectral precoding dj = (X [j, :]) ∈ C . (3) (SSP)/EVM-constrained SSP (ESSP). Note that part of Similarly, we define a spectrally precoded frequency-domain data matrix X as [1] is used for the EVM-unconstrained part of the present h T T i X := [x [1] ,..., x [N]] := d ; ... ; d ∈ NT×N , paper. 1 NT C (4) • we finally present exhaustive simulations using a 5G NR where the spectrally (and spatially) precoded data symbol at (Release-15 compliant) inhouse link-level simulator [1]. k-th subcarrier is represented as x [k]. Further, we define a N×1 E. Notation column vector dj ∈ C of a spectrally precoded vector corresponding to j-th transmit antenna branch as Let the set of complex and real numbers be denoted by C
T N×1 and R, respectively. <{x} denotes the real part of a complex dj := X [j, :] ∈ C . (5) 4

OFDM Digital and Analog Analog and Digital OFDM Mod Processing Processing DeMod

MIMO Spectral MIMO Precoder Precoding Receiver

Wireless Channel Wireless Processing Frequency-Selective

OFDM Digital and Analog Analog and Digital OFDM Mod Processing Processing DeMod

Fig. 1: Simplified block diagram of a single-user MIMO-OFDM transceiver with spectral precoding. R BW/2 B. Performance Metrics Sdesiredchannel(f) df ACLR:= −BW/2 ,  −BW 3BW  We utilize two figure-of-merits, i.e., OOBE and inband dis- 2 2  R R  tortions, for the spectral precoding design and its performance. max Sleftchannel(f) df, Srightchannel(f) df −3BW BW  The inband distortions are evaluated in terms of EVM and also 2 2 block error rate (BLER) (or equivalently throughput). More where Sdesiredchannel is the PSD in the desired carrier having specifically, we have employed EVM metric for the spec- BW bandwidth including the guard band; and similarly, tral precoding design. Nonetheless, there exists a nonlinear Sleftchannel and Srightchannel correspond to the PSD on the mapping between EVM and BLER or equivalently received left and the right side of the desired carrier having same SNR, see, e.g., [41], [45]. Therefore, EVM metric is sufficient bandwidth BW as the desired carrier, respectively. for the spectral precoding design and performance evaluation. The OOBE requirements are typically characterized by the We would like to accentuate that we do not directly use spectral mask and ACLR, among others [5], [43]. Based on our ACLR for the spectral precoding design, but rather we use the practical experience, if the base station fulfils the spectral mask OOBE power at the considered discrete frequency points. requirement with a suitable implementation margin then the 2) In-Band Distortion: The considered in-band distortion ACLR requirement is also achieved; and the converse is also for the spectral precoding design is EVM, which can be true. Therefore, we have solely considered an appropriately quantified as a loss in the demodulated signal quality. It can 1 discretized mask as a spectral precoding design parameter. be described mathematically per j-th transmit antenna as However, for the numerical performance evaluation, we also n 2o dj − dj evaluate ACLR for completeness. E 2 EVMj := j = n o . (7) 1) Out-of-Band Emissions: The OOBE is typically quanti- 2 E kdjk2 fied in terms of the operating band unwanted emissions and (conducted) ACLR, whose definitions are given below. For the spectral precoding design, we assume that the average transmit signal power per transmit antenna branch Definition 1 (Operating band unwanted emissions [5, Sec- n 2o E kdjk is fixed. tion 6.6.4], referred to as mask). Unwanted emissions that 2 are immediately outside the base station channel bandwidth resulting from the modulation process and non-linearity in the IV. EVM-UNCONSTRAINEDLARGE-SCALE MSP transmitter but excluding spurious emissions. (LS-MSP) IN MIMO-OFDM The unwanted OOBE due to the OFDM frequency-domain In this section, we first consider a previously proposed signal dj at the M considered discrete frequency points ν = mask-compliant spectral precoding without EVM constraint [ν1, . . . , νM ] can be described by p(ν) = Adj. that utilizes a generic convex optimization solver. It is known T 1×N We now define A[m, :] := a (νm) ∈ C , where that it suffers from high computational complexity, notably A [m, k] := a(νm, k) can be derived in discrete form as [13]: in large-scale systems. Afterwards, to mitigate the complex- ity of computing the LS-MSP, we propose a divide-and-     1 (νm − k) conquer approach that breaks the original problem into smaller a(νm, k) = √ exp jπ (NCP − N + 1) · N N rank 1 quadratic-constraint problems, where each small prob-   (νm−k) lem yields a closed-form solution. Lastly, based on these so- sin π N (N + NCP) · , (6) lutions, we develop two specialized first-order low-complexity  (νm−k)  sin π N algorithms. In particular, the first one is based on the consensus ADMM while the second one is derived by employing the where NCP corresponds to cyclic prefix length in samples. coordinate descent scheme of a dual variable and capitalizing Definition 2 (Adjacent channel leakage ratio (ACLR) [5, on closed-form of the rank 1 constraint. Section 6.6.3]). ACLR is the ratio of the filtered mean power 1EVM measurements in NR standard are stipulated with a zero-forcing centred on the assigned channel frequency to the filtered mean receiver, see, e.g., [5, Annex B], which can be seen as an equalized EVM measure. On contrast, the EVM for the spectral precoding design can be power centred on an adjacent channel frequency. The (worst- defined as an unequalized EVM since the unequalized EVM value would be case) ACLR can mathematically be expressed as conservative, i.e., it can not be less than the equalized EVM, see, e.g., [46]. 5

A. EVM-Unconstrained Proposed Problem Formulations Algorithm 1 ADMM

 N×1 NT M The work in [13] was extended in [15] for single-antenna Inputs: dj ∈ C j=1, {γm; a (νm)}m=1, ρ ∈ R>0 (I) N×1 OFDM, referred to as MSP, such that only the mask constraint Output(s): dj ∈ C ∀j = 1,...,NT (0) (0) needs to be fulfilled, i.e., 1: Initialization: ym = 0N×1 and zm = 0N×1 2 2 2: for i = 1, 2,...,I do minimize dj − dj subject to A dj  γ, (8) " M # N×1 2 dj ∈C (i) 1 X  (i−1) (i−1) d = dj + ρ ym + zm (12a) j (1+ρM) where the inequality constraint is element-wise and the target m=1 M×1 3: parfor m = 1,...,M do . % run parallel mask γ ∈ R is given. The solution could not be expressed in (semi) closed-form and thereby the authors proposed to (i)  (i) (i−1) ym = projCm dj − zm (12b) solve this MSP problem via a generic quadratic programming (i) (i−1) (i) (i) solver [15], e.g., CVX [47]. zm = zm + ym − dj (12c) Based on our key observation, we rewrite the problem 4: end parfor 5: end for (8) as described below, without any loss of convexity. More (I) 6: return d specifically, the constraint in (8) can be decomposed into M j √ rank 1 constraints such that it becomes  H   b−|u x| 2 x+ u uHx
, if xHAx > b minimize d − d kuk2|uHx| e j j 2 = 2 (13) d j  H H x, if x Axe ≤ b. subject to dj Am dj ≤ γm, ∀m=1 ...,M, (9) Proof. See Appendix B.  where ∗ T N×N Am = a (νm) a (νm) ∈ C (10) If M > 1, then no closed-form is known yet. Hence, in the sequel, we propose low-complexity algorithms that essentially and rank{A } = 1. Problem (9) is referred to as LS-MSP m break down the LS-MSP problem into smaller subproblems, that facilitates large-scale optimization. where each subproblem admits closed-form solution capital- izing on Theorem 1. B. Efficient Algorithms for EVM-Unconstrained LS-MSP 1) EVM-Unconstrained ADMM LS-MSP (referred to as The proximal operator, prox, is used for the algorithm ADMM): In our first proposal, we utilize ADMM with con- design, whose definition is given below. sensus optimization to solve the LS-MSP problem by rewriting Problem (9) Definition 3 (proximal mapping [48], [49]). Given a proper M
X closed convex function f : domf 7→ (−∞ , +∞], then the minimize f dj + XCm (ym) N×1 proximal mapping of f is the operator given by dj ,ym∈C m=1   y = d ∀m=1, . . . , M, 1 2 subject to m j proxλf (x) = arg min f(z) + kx − zk2 (11) z∈dom
f 2λ where the (non-equalized) squared EVM f dj := 2 for any x ∈ domf , where domf corresponds to the domain of dj − dj is a convex and differentiable function. function f and λ > 0. 2 The non-differentiable indicator function XCm (ym) Definition 4 (proximal mapping of the indicator function [48], with the rank 1 constraint set is given by  H [49]). Let f : domf 7→ (−∞ , +∞] be an indicator function, Cm = ym : ym Am ym − γm ≤ 0 . f(x):=XC (x) where C is a nonempty set XC (x)=0 if x∈C Algorithm 1 summarizes the proposed recipe for the otherwise XC (x)=+∞, then the proximal mapping of a given ADMM-based spectral precoding, cf. [1] for details, where set C is an orthogonal projection operator projC onto the I denotes the total number of iterations. The convergence same set, i.e., analysis of the ADMM algorithm is given in Appendix A.  1  2) EVM-Unconstrained SSP LS-MSP (referred to as SSP): prox (x) = arg min X (x) + kx − zk2 λXC C 2 In our second proposal, we derive an optimal semi-analytical z∈domf 2λ   algorithm, dubbed as SSP, based on the Karush-Kuhn-Tucker 1 2 = arg min kx − zk = projC (x) . (KKT) conditions [25] for the constrained optimization (9). z∈C 2 2 We form the Lagrangian of (9) by introducing the Lagrange If M = 1, the orthogonal projection onto the rank 1 multipliers {µm} as follows: quadratic constraint is obtained in closed-form as described M in the following theorem:
2 X  H  L dj, {µm} = dj − dj 2 + µm dj Amdj − γm . Theorem 1 (projection onto the rank 1 quadratic con- m=1 N×1 (14) straint). Let C ⊆ C and C 6= ∅ given by C = n o Utilizing the KKT conditions, the stationarity condition yields x ∈ CN×1 : xHAxe − b ≤ 0 , where Ae = uuH ∈ CN×N (15c) and the Lagrange multipliers {µ } are obtained iter- is rank 1 matrix and b ∈ , then the proximal operator m R≥0 atively in a coordinate descent fashion [50] as outlined in prox (x) = proj (x) XC C Algorithm 2—see Appendix C for a detailed derivation. 6

TABLE I: Comparison of online complexity of various EVM-unconstrained and mask-compliant algorithms Method Complexity: Real Multiplications Complexity: Real Additions 4.5
MSP [15] O N NT [24] – ADMM I (9MN +M +N +1) NT I (14MN +2N +M +1) NT  2 2 2 2

 2 2 2 2 2

SSP 5MN +I 4M N +12MN +5M N +3MN +M NT 2MN +I 2M N +6MN +6M N −2MN +M NT

Algorithm 2 SSP main computational complexity of step (15a) is due to the  N×1 NT  m 2 M matrix inversion, but no online inversion is necessary due Inputs: dj ∈ C j=1, γm ∈R; λ1 = ka (νm) k2 m=1. (I) to the sum of rank 1 matrices—see Lemma 2, which are Output(s): d ∈ N×1 ∀j = 1,...,N j C T 2 1: Initialization: (M −1) N + NM complex multiplications, 2 (M −1) N r    λm   1 T 1 complex additions, (M −1) N real multiplications, and 2: µm = m a (νm) dj − 1 ∀m=1,...,M (cf. λ1 γm Lemma 1); φ = 0. (M −1) real additions for each frequency point, iteration, 3: for i = 1,...,I do and transmit antenna. The complexity of the computation m = 1,...,M 2
4: for do of both α1 and α2 are 2N for each frequency point, −1  M  iteration, and transmit antenna. The step (15b) need N −1 X (using Lemma 2) G\m = IN + µnAn complex multiplications, 1 real multiplication, and 1 real n=1;n6=m addition. Hence, the total online real multiplications and (15a) I N T −1 T −1 ∗ real additions for iterations and T transmit antennas are α1 = a (νm) G dj ; α2 = a (νm) G a (νm)  2 2 2 2


\m \m 5MN +I 4M N +12MN +5M N +3MN +M NT √ and2MN +I2M 2N 2 +6MN 2 +6M 2N −2MN +M 2

N
,  α1 exp (−ιφ) − γm  T µm ← < √ (15b) respectively. γm α2 5: end for 6: end for V. EVM-CONSTRAINED LS-MSP IN MIMO-OFDM 7: return M !−1 X dj = IN + µm Am dj (using Lemma 2) (15c) In this section, we firstly extend the LS-MSP Problem (9) m=1 such that the spectrally precoded symbol yearns to keep the EVM below the desired level by sacrificing OOBE perfor- mance in terms of ACLR, referred to as EVM-constrained C. Complexity Analysis MSP (EMSP). Unfortunately, adding EVM constraint in ad- We analyze the run-time complexity, in terms of required dition to the mask constraint poses challenges to develop a real-valued multiplications and real-valued additions but ig- computationally efficient algorithm. Thanks to ADMM, which nore the offline complexity. Notice, we convert all the com- offers a divide-and-conquer approach, incorporating EVM plex multiplications and additions into equivalent real-valued constraint becomes easy and the algorithm is referred to as multiplications and additions, i.e., 1 complex multiplication is EADMM. However, SSP becomes prohibitively complex to equivalent to 4 real multiplications and 2 real additions, and 1 support EVM constraint. Therefore, we have proposed an complex addition corresponds to 2 real additions. Furthermore, alternative operator splitting framework based on the Douglas- we assume that all the subcarriers are allocated, which will Rachford algorithm [19], [21], referred to as EVM-constrained give the worst-case complexity analysis. Table I summarizes SSP (ESSP), that employs the iterative SSP algorithm inter- the complexity of the mask-compliant precoding schemes. nally for the mask constraint and the outer loop of Douglas- ADMM: The initialization step requires no multiplica- Rachford supports the EVM constraint. tions/additions. In step (12a), there are (M +1) N complex additions, N + 1 real multiplications, and 1 real addition per iteration and transmit antenna. The dominating online A. EVM-Constrained Proposed Problem Formulations algebraic complexity is in the computation of the prox We pose wideband and frequency-selective EVM con- operator in step (12b), which is in the order of 2MN com- strained mask-compliant spectral precoding optimization prob- plex multiplications, 2MN complex additions, M (N +1) real lems and then develop low-complexity algorithms. We firstly multiplications, and M real additions per iteration and transmit perform the epigraph transformation [51] of Problem (9), antenna. However, due to distributed nature of consensus without losing convexity, such that the proposed wideband ADMM, the M subiterations can run in parallel per iteration EVM-constrained optimization problem reads as: cycle at the expense of increased memory requirements. In minimize ∆t (16a) step (12c), we need 2MN complex additions. Thus, ignoring N ×N X∈C T ,∆t∈R parallelization, the total run-time complexities for I iterations subject to X − X ≤ avg (16b) and NT transmit antennas are I (9MN +M +N +1) NT and F 2 I (14MN +2N +M +1) NT in terms of real multiplications T AX  Γ ∆t1M×N , (16c) and real additions, respectively. T SSP: The initialization step needs N complex and real where represents element-wise multiplication, and avg multiplications, respectively for each m-th frequency point denotes the desired wideband averaged EVM over all the m and antenna, where {λ1 } can be computed offline. The allocated subcarriers (frequency-domain) and also over all the 7 transmit antennas2. The target mask is Γ ∈ RM×NT , e.g., following feasibility problem: Γ [:, j] = [γ1; ... ; γm] and 1M×N is an all-ones M × NT N ×N T find X ∈ C T matrix. Note that the mask constraints can be the same for all the transmit antenna branches. Furthermore, the constraint subject to X[:, k]−X[:, k] 2 ≤  [k] ∀k ∈ T (19a) (16c) can be decomposed for each j-th transmit antenna and T 2 AX  Γ. each m-th discrete frequency point such that the constraint can H be read as dj Am dj ≤∆tγm. Now, the respective mask and wideband and frequency- To support more flexibility in terms of defining differ- selective EVM constraint sets are defined as following. The ent EVM constraints to different subcarriers, we extend the m-th mask constraint set corresponds to the rank 1 quadratic wideband EVM-constrained Problem (16) that offers different inequality, i.e., EVM constraint for each subcarrier. A frequency-selective n H o EVM-constrained optimization problem is Cm := X : dj Amdj − γm ≤ 0; ∀j = 1,...,NT , (20) minimize ∆t N ×N and the wideband EVM constraint set can be described by X∈C T ,∆t∈R  Ewb := X : X − X − avg ≤ 0 , (21) subject to X[:, k]−X[:, k] 2 ≤  [k] ∀k ∈T (17a) F 2 T whereas the frequency-selective EVM set can be expressed as AX  Γ ∆t1M×NT ,  Efs := X : X[:, k]−X[:, k] −  [k]≤0; ∀k ∈ T . (22) where  [k] is the desired EVM at k-th subcarrier and the set 2 T denotes all activated subcarriers. We will denote the EVM constraint set as E, which can be The main benefit of Problem (17) is that the EVM constraint wideband Ewb and/or frequency-selective Efs, unless stated per subcarrier or the group of subcarriers can be defined by the otherwise. upper layers depending on the channel link quality and/or the Now, we rewrite the feasibility problems (18) and (19) as allocation of the data/pilots appropriately. In other words, such the following unconstrained problem amenable to the latter frequency-selective EVM constraint may have a high threshold proposed efficient algorithms (or high allowable EVM) for the lower supported modulation ( M ) alphabet, e.g., for QPSK modulation—cf. Table IV. In contrast,

X
minimize F X := X E X + X Cm X , (23) NT×N the EVM constraint may have a low threshold for the higher X∈C m=1 modulation alphabet, e.g., for 64QAM. where the composite function F X
is a sum of non- An optimal solution to both problems (16) and (17) can differentiable indicator functions, i.e., X (·) and X (·), of be obtained via a general purpose optimization solver, e.g., E Cm constraint sets corresponding to the wideband or frequency- CVX [47]. However, as motivated in the previous section, selective EVM, and mask defined in (21) or (22) and (20), such general purpose algorithms employ interior-point-based respectively. Hence, we seek efficient methods to solve such methods whose complexity is prohibitively high. Hence, in a problem. order to develop efficient algorithms for the EVM-constrained and mask-compliant spectral precoding problems, we now in- Prior to developing the algorithms for the constrained EVM, stead transform the problems (16) and (17) into the feasibility we present the following theorems which are utilized for the problems by omitting the ∆t variable, i.e., problems (18) and subsequent algorithm development. (19), respectively. If ∆t ≤ 1 in problems (16) and (17), then Pn Theorem 2. If a function f (X) = k=1 fi (xk) the respective problems (18) and (19) are feasible. In other is separable across the variables column-wise words, we make an assumption that the problem is feasible X = [x1,..., xn] or row-wise X = [x1; ... ; xNT ], or has at least one solution, which implies ∆t ≤ 1, then the then the respective prox operators can be shown mask constraint (16c) can be expressed as following:   as proxf (X) = proxf1 (x1) ,..., proxfn (xn) or T 2 ∆t≤1   prox (X)= prox (x1); ... ; prox (xNT ) . AX  Γ ∆t1  Γ. f f1 fNT M×NT Consequently, we omit ∆t from the the mask constraint in Proof. Following the proximal operator Definition 3, the min- (16)/(17) for the feasibility problem. A wideband EVM con- imization of the separable function is equivalent to minimiza- straint Problem (16) can be posed as the following feasibility tion of respective functions {fk} independently [48], [49].  problem:

NT×N Theorem 3 (projection onto Frobenius norm ball [49, find X ∈ C p×q Lemma 6.26]). Let E ⊆ C and E 6= ∅ be given by subject to X − X ≤ avg (18a) p×q F E := Bk·k2 [C, r] = {X ∈C : kX −CkF ≤r}, then the 2 T proximal or orthogonal projection operator for the Frobenius AX  Γ. (18b) (or `2) norm ball, i.e., Bk·kF [C, r] with a given center C and Similarly, Problem (17), i.e., a frequency-selective EVM the radius r, is constraint with mask-compliant problem, can be posed as the proxλ (X) =proj [C,r] (X) X E Bk·kF   2It is straightforward to modify the problem to support wideband EVM- r constraint per transmit antenna branch. =C+ (X −C) . max {kX −CkF , r} 8

TABLE II: Complexity comparison of various EVM-constrained and mask-compliant algorithms Method Complexity: Real Multiplications Complexity: Real Additions 4.5
EMSP O N NT – EADMM (I (9MN +M +6N +2) NT) (I (14MN +10N +M −1)NT)  0 2 2 2 2

 0 2 2 2 2 2

ESSP I (5M +7) N +1+I 4M N +12MN +5M N +3MN +M NT I 2MN +14N −2+I 2M N +6MN +6M N −2MN +M NT

Algorithm 3 EADMM Algorithm 4 ESSP

M N M  NSC Inputs: X, {γm; a (νm)}m=1, and avg ∈R; {[k]∈R}k=1 Inputs: X, {γm; a (νm)}m=1, and  ∈ R or avg ∈ R (I) N ×N (I) N ×N Output(s): X ∈ C T Output(s): X ∈ C T (0) (0) (0) (0) 1: Initialization: Y m = 0NT×N and Zm = 0NT×N 1: Initialization: Xm = X and Zm = 0NT×N 2: for i = 1, 2,...,I do 2: for i = 1, 2,...,I do M 1 X  (i−1) (i−1) (i)  (i−1) (i−1) U = Y + Z Y =prox 2X −Z (solve SSP Alg. 2) M m m X C m=1 ( (i) projB X, (U) (26a) k·kF [ avg] X =projE (U) ≡ select (25a) (i) (i−1)  (i) (i−1) projB [X,] (U) k·k2 Z = Z + λi Y − X (26b)   (i) 3: parfor m = 1,...,M do . % run parallel (i)  (i) projB X, Z k·kF [ avg] X =projE Z ≡ select (i)  (i) (i−1)  (i) projB [X,] Z Y m = projCm X − Zm (25b) k·k2 (26c) (i) (i−1) (i) (i) Z = Z + Y − X (25c) m m m 3: end for (I) 4: end parfor 4: return X 5: end for (I) 6: return X Theorem 4 (Douglas-Rachford algorithm). Consider the fol- lowing problem B. Efficient Algorithms for EVM-Constrained LS-MSP minimize G X
+ H X
, (27) In this section, we develop two computationally efficient N ×N X∈C T algorithms to solve the aforementioned EVM-constrained where G and H are proper closed convex functions, and which problem (23). has at least one solution. Consider τ ∈ (0, ∞) and a sequence 1) EVM-Constrained ADMM LS-MSP solution (referred to of relaxation parameters λi ∈ (0, 2) ∀i ≥ 0 and satisfy as EADMM): We firstly express (23) amenable to ADMM: P i λi (2−λi) = +∞ with some arbitrary initial Z, then the M following iterative scheme
X
minimize X E X + X Cm Y m NT×N
X,Y m∈C m=1 X ← proxτG Z (28a)

subject to Y m =X ∀m=1, . . . , M, Z ← Z + λi proxτH 2X − Z − X (28b) The scaled-form consensus ADMM for the above problem can converges weakly to a solution to (27). be expressed as [19], [20] Proof. See, e.g., [21] [52, Corollary 5.2]. M 
X 2 Strikingly, if the problem (27) is infeasible, then for some X ←arg min X E X +ρ Y m −X +Zm F (24a) X m=1 cases one could still find an approximate solution through
2 Douglas-Rachford method—see, e.g., [53], [54]. Y m ←arg min X Cm Y m +ρ Y m −X +Zm F ∀m Y m The Douglas-Rachford algorithm is described for two func- (24b) tions. However, the problem (23) at hand has more than two 3 Zm ←Zm + Y m − X ∀m=1,...,M. (24c) functions . Thus, we reformulate (23) as

In the first step of our proposed EADMM LS-MSP algo- minimize X E X + X C X , (29) NT×N rithm, taking the derivative with respect to X and setting to X∈C  1 PM

PM
X =prox Y + Z i.e. where X C X := X Cm X such that the two- zero yields (1/M)X E M m=1 m m , , m=1 the orthogonal projection onto the wideband or frequency- operator Douglas-Rachford splitting can be employed. Since the proximal operator corresponding to the sum of M indicator selective EVM constraint (25a) —`2 norm ball (cf. Theo- functions X X
= PM X X
, i.e., prox doesn’t rem 3). The second step is an orthogonal projection onto the C m=1 Cm X C rank 1 quadratic constraint (cf. Theorem 1) yielding (25b). yield a closed-form, we approximate it by employing SSP. Algorithm 3 summarizes the proposed recipe for the EADMM We will show numerically that ESSP framework also requires based mask-compliant spectral precoding. The convergence relatively less number of iterations compared to EADMM to analysis of the EADMM algorithm is given in Appendix A. reach desired level of performance in terms of EVM and 2) EVM-Constrained SSP LS-MSP solution (referred to ACLR metrics at the cost of extra computational complexity as ESSP): We layout the definition of Douglas-Rachford 3One could reformulate Douglas-Rachford as consensus ADMM. However, algorithm and subsequently propose the modifications to in- we would like to employ SSP algorithm that offers a solution to the EVM- corporate SSP algorithm for the mask constraint. constrained LS-MSP. 9

TABLE III: Simulation Parameters for FDD NR (Rel-15) PDSCH Type-A TABLE IV: EVM Requirements [5] Parameters Test 1 Test 2 Test 3 Modulation Scheme EVM Threshold Subcarrier Spacing 15 kHz QPSK 17.5 % Carrier Bandwidth (PRB alloc.) 5 MHz (25 PRBs) 16QAM 12.5 % Carrier Spacing for ACLR 5 MHz upper and lower adjacent channels [5] 64QAM 8.0 % DL SU-MIMO NT,NR 2Tx, 2Rx 8Tx, 2Rx 2/8Tx, 2Rx Spatial Layers (rank) Fixed rank 1 adaptive (10% BLER) Spatial Precoding (codebook-based) adaptive (10% BLER) A. Performance Measures Modulation 64QAM adaptive (10% BLER) Code-rate 1/2 5/6 adaptive (10% BLER) We analyze the spectral precoding performance in terms of Channel Model TDL-A (300ns, 10Hz) & spatial correlation Low [43] two figure-of-merits, namely OOBE and in-band distortions, Channel & Noise power Practical LMMSE based HARQ max transmissions 4 (3 max retransmissions with rv {0, 2, 3, 1}) [55] in particular assuming base station supporting sub-6 GHz, e.g., Other Information LDPC; LMMSE-IRC receiver; no other impairments frequency ranges between 410 MHz and 7.125 GHz—referred to as FR1 in 5G NR [5, Section 5.1]. compared to EADMM, yet offering lower cost compared to 1) Out-of-Band Distortion: As mentioned in Section the generic interior-point based solvers. III-B1, mask and (conducted) ACLR are typically the per- formance metrics to quantify the operating band unwanted The proximal operator corresponding to the indicator func- emissions. tion for EVM constraint, either wideband X X
or Ewb In practical systems, the (digital) spectrum shaping is fol- frequency-selective constraint X X
, is an orthogonal pro- Efs lowed by other (non-linear) digital and analog processing as jection onto Euclidean norm ball (cf. Theorem 3). illustrated in Fig. 1. Consequently, there is some spectral The ESSP framework is summarized in Algorithm 4, where regrowth phenomenon due to such (non-linear) components we have performed a cyclic rotation of the Douglas-Rachford in the transmitter after spectrum shaping. Thus, an imple- algorithm steps such that the proximal operator corresponding mentation margin in terms of ACLR and mask requirements to the mask constraint occurs first in the given iteration cycle. are necessary to cope with spectral regrowth. Hence, OOBE performance must render better performance than the (overall) C. Complexity Analysis stipulated mask in the standard due to spectrum shaping to In this section, we analyze the worst-case run-time complex- account for the margin. ity, in terms of required real-valued multiplications and real- We have considered ACLR corresponding to the 1st ad- valued additions but ignore the offline complexity as described jacent carrier in both upper and lower frequencies, where in Section IV-C—see the summary in Table II. the minimum requirement is 45 dB—worst-case of measured EMSP: The computational complexity of solving the opti- ACLR in the upper and lower channels [5, Section 6.6.3]. It is mization problems (16) and (17)—almost similar to MSP and worth highlighting that these ACLR requirements are for the 4.5
using results in [24]—is O N NT . complete radio chain, i.e., measurements need to be performed EADMM: The step (25a) is additional compared to ADMM at the antenna connector. Thus, spectrum shaping may have Algorithm 1. So, the prox operator in the step (25a) some aggressive mask and ACLR requirements to meet the corresponding to the EVM constraint requires total com- minimum requirements at the antenna connector. plex multiplications (INNT), (I (3N −1) NT), and N + 1 2) In-Band Distortion: In these simulations, the in-band real multiplications. Hence, ignoring parallelization, the total distortion is not only quantified in terms of EVM but also in run-time complexities are (I (9MN +M +6N +2) NT) and terms of BLER [56] and throughput [57]. (I (14MN +10N +M −1)NT) in terms of real multiplica- For our link simulations, we present normalized through- tions and real additions, respectively. put, i.e., normalizing the throughput results by the maxi- ESSP: In step (26a), there are N complex multiplications mum achievable throughput without any spectral precoding and N real multiplications besides the computational com- or OOBE reduction and other hardware impairments or im- plexity of (iterative) prox operator (26a) approximated by perfections. the SSP algorithm. We need 2N complex additions and N real multiplications in the step (26b). Finally, in the prox B. Simulation Parameters and Assumptions operator (26c) corresponding to the EVM constraint same as The key simulation parameters for the physical downlink in EADMM need to be considered. Therefore, the total online shared channel (PDSCH) with type-A4 and the three investi- complexities in terms of real multiplications and additions are gated test scenarios are summarized in Table III, see, e.g., [5], given in Table II, where I0 corresponds to inner iterations using [58], for the detailed NR physical layer and performance SSP Algorithm 2. requirements. We have considered 15 kHz subcarrier spacing for the NR numerology unless otherwise mentioned. Further- VI.PERFORMANCE EVALUATION more, no supporting signals are transmitted besides PDSCH along with the demodulation reference signal (DMRS) for In this section, we evaluate the performance of the proposed the practical channel and noise variance estimation at the UE algorithms for mask-compliant spectral precoding that are both side. Note that for simulations purpose, we have considered EVM-unconstrained and EVM-constrained utilizing a 5G NR relatively narrow 5 MHz channel bandwidth with 25 physical (Rel-15) compliant inhouse link-level simulator. Moreover, we compare the performance of the proposed algorithms with the 4These data types refer to different PDSCH demodulation reference signals conventional spectral precoding algorithms, accordingly. allocation [44]. 10

-20 NO OOBER NO OOBER NSP -50 NSP [13] 35 NSP NSP [13] 70 MSP(CVX); 2 MSP (CVX); 2 -52 ADMM; 2 MSP (CVX); 2 65 -40 30 ADMM; 2 -54 SSP: 2 60 SSP; 2 NO OOBER -3 -2.5 MSP(CVX); 1 25 SSP; 2 ADMM; 1 MSP (CVX); 1 55 ADMM; 2 NSP [13] -60 MSP (CVX); 2 SSP; 1 20 ADMM; 1 MSP (CVX); 1 NO OOBER 3GPP MASK SSP; 1 50 ADMM; 2 15 -80 ACLR [dB] ACLR SSP; 2 45 1: all overlap 2: all overlap SSP; 1 ADMM; 1 MSP (CVX); 1 10 1 40 ADMM; 1 NO OOBER -100 2

PSD [dBm/100 KHz] [dBm/100 PSD 5

SSP; 1 NSP [%] EVM (Average) mask compliant schemes: all overlap 1 1 2 3 0 10 10 10 -20 -10 0 10 20 5 10 15 20 25 Iterations (i) Frequency [MHz] PRBs (Frequency-Domain) (a) ACLR-vs.-Iterations. (b) PSD at the final iteration. (c) EVM [%] at the final iteration. Fig. 2: (a) Convergence behaviour of EVM-unconstrained algorithms considering two different target masks, while (b) and (c) are achieved PSD and EVM at the final iteration. resource block (PRB)s allocation, even though the proposed (6). For NSP α = 1, and ENSP α ∈ R is a tunable parameter methods can be employed for arbitrary bandwidths. Further- to meet the desired EVM. more, we have employed a low spatial correlation model, but the complexities of the proposed algorithms are oblivious of C. Simulation Results spatial correlation—cf. Table I and II. However, the inband In this section, we present simulation results for both un- performance may degrade with increasing correlation, notably constrained and constrained EVM spectral precoding methods. for high spatial rank setup. In Table IV, we have provided In Fig. 2, we present the convergence behaviour of the 3GPP NR wideband average EVM5 requirements [5] as a proposed EVM-unconstrained algorithms in terms of OOB and reference according to the considered modulation alphabet. in-band performance considering Test 3 (cf. Table III) and two In addition to the parameters given in Table III, the discrete different target masks, i.e., mask-1 and mask-2. frequencies for mask compliant precoders are selected as Figure 2(a) shows the ACLR performance against itera- ν ∈{∓5010, ∓4995, ∓2565, ∓2550} kHz, where the negative tions. Evidently, SSP converges in 2-3 iterations for both and positive frequencies correspond to the left and right side target masks to achieve the same ACLR results as rendered of the out-of-band (OOB) of the occupied signal spectrum, by MSP(CVX). Under relaxed mask-1 constraint, ADMM respectively. Notice that these discrete frequencies can be requires nearly 80 iterations, while for aggressive mask-2 it asymmetrically selected for the OOBE suppression. The con- requires approximately 800 iterations. On the other hand, we sidered mask, referred to as mask-1 is γmask−1 := γ1 = have observed that CVX achieves the MSP solution in nearly [−75, −75, −65, −65] dBm/100 kHz, corresponding to left/right 25 iterations, but we could not manage to extract the result for side of the signal spectrum. Furthermore, we have considered every single iteration; hence, we show the final result rendered another aggressive target mask-2, i.e., γmask−2 := γ2 = by the CVX solver corresponding to the last iteration in the [−85, −85, −75, −75] dBm/100 kHz for ACLR and EVM per- figures. We have also observed similar behaviour for EVM formance, particularly. For this work, we have not optimized against iterations. For the subsequent EVM-unconstrained the selection of the discrete frequencies’ set for the precoding. results, we consider mask-1 and fix the iterations of ADMM Based on our numerical grid-search utilizing Test 3 with 2 and SSP as 80, and 2, respectively. Furthermore, we have transmit antennas (cf. Table III), we found a suitable ρ = 10 numerically observed that these spectral precoders have a for consensus ADMM algorithm (cf. Algorithm 1). For the negligible impact on the peak to average power ratio. MSP (8) solution, we have employed CVX wrapper with Figure 2(b) exhibits the average PSD versus frequency for SDPT3 solver [47]. Note that we do not have any additional the proposed algorithms and both masks considering final radio hardware impairments at the transmitter and/or receiver iterations of the respective algorithms. The NR mask corre- besides the distortion generated at the transmitter by the sponding to a medium range BS with the maximum output of considered spectral precoders, if enabled. 38 dBm [5, Section 6.6.4] is also shown for the completeness, Furthermore, for benchmarking purpose, we have also but the mask is normalized according to the normalized evaluated notching spectral precoder (NSP) [13] and EVM- transmit signal power of 0 dBm, i.e., approximately −21.5 constrained NSP (ENSP) [26], where for each transmit an- dBm/100 kHz, in the link simulations. All the proposed methods tenna branch NSP and ENSP are performed independently. in addition to the prior art fulfils the 3GPP NR mask. More specifically, for each j-th transmit antenna, we perform Figure 2(c) depicts the average EVM distribution per PRBs −1 H H [58]. The edge PRBs have relatively high distortion power dj =Gdj, where G=IN −αA AA A and A[m, k] in compared to the central PRBs. Notice that the NR bandwidth is slightly asymmetric with respect to the direct-current carrier. 5The NR standard puts a requirement on the equalized EVM, cf. [5, Moreover, the discrete selected frequency points (ν) are not Section 6.5.2.2], e.g., for 64QAM, the transmitter is allowed to induce EVM symmetric with respect to the direct-current carrier. Therefore, up to 8%. As mentioned in footnote 1 (Page 4), the unequalized EVM is an upper bound for the equalized EVM, i.e., unequalized EVM is tougher one could consequently observe that the EVM distribution in requirement than equalized, see, e.g., [46]. the frequency-domain is asymmetric. 11

Violation of the 10 MSP 2 50 MSP 10 MASK-1 Constraint ESSP: STOPPING CRITERION Constrained EVM schemes: all overlap

ESSP (I'=2) ESSP (I'=1) 8 8.04 0 8 10 45 7.96 Meeting the EMSP 6 288 290 292 294 296 ESSP (I'=3) -2 MASK-1 Constraint MSP (CVX); 1 MSP (CVX); 1 ENSP [26] 10 MSP(CVX); 1 EVM ENSP [26] EMSP (CVX) 4 0 ENSP [26] 40 CONSTRAINED EMSP (CVX) 10 ENSP EADMM [%] EVM ACLR [dB] ACLR & ENSP -4 EMSP(CVX) ESSP (I'=1) EADMM 10 EMSP: EADMM 2 ESSP (I'=1) t=344.17 EADMM ESSP (I'=2)

ESSP (I'=1) ESSP (I'=3) ESSP (I'=2) (Wideband Average) Average) (Wideband -6 -5 ESSP (I'=2) 35 NR MIN ACLR 0 ESSP (I'=3) OOBE Power to MASK-1 to Power OOBE 10 ESSP (I'=3) 0 1 2 3 -5 0 5 10 10 10 10 100 101 102 103 Frequency [MHz] (Outer) Iterations (i) (Outer) Iterations (i) (a) Ratio of OOBE power to mask-1. (b) ACLR-vs.-Iterations of ESSP. (c) EVM-vs.-Iterations. Fig. 3: Convergence behaviour of wideband 8% EVM-constrained spectral precoding. Subsequently, we present the performance of EVM- coding algorithms. In the case of ESSP, we illustrate the per- constrained while considering mask-1 constraint. In Fig. 3, we formance with three different internal iterations corresponding exhibit the convergence behaviour of the proposed wideband to the prox operator for the sum of the indicator functions EVM-constrained algorithms in terms of the in-band, namely, to the mask constraints, i.e., calling SSP algorithm. Strikingly, EVM, and OOB performance, namely, ACLR and PSD. For the ACLR performance accomplished by ESSP without stop- these simulations, we have considered 8% wideband unequal- ping criterion diverges after typically 2 iterations, although ized EVM. the wideband EVM performance convergence behaviour is Figure 3(a) illustrates that the problem (23) is infeasible, consistent over the iterations, i.e., meeting the wideband EVM i.e., the intersection of the two convex sets corresponding to constraint—see Fig. 3(c). Remarkably, the result rendered by mask and EVM constraints are empty, by considering a single EADMM do not digress or diverge over iterations, unlike OFDM symbol and by randomly selecting any of the transmit observed in Douglas-Rachford-based ESSP algorithm without antenna branch corresponding to Test 3. More specifically, we early stopping criterion. Thus, ESSP requires a stopping crite-  2 rion to render an approximate solution. We choose to terminate show OOBE power to the target mask, i.e., Adj (∆tγ), the outer iterations within ESSP when the ACLR performance where denotes elementwise division. If this ratio is larger degrades with the increasing iterations. Hence, we employ an than 1, then it implies that the target mask constraint is early stopping criterion for the results presented in the sequel, violated at the considered ν-th discrete frequency point. On which typically terminates the algorithm at around 2 outer the contrast, if this ratio is less than and equal to 1, the iterations (for all the considered test cases). mask constraint is met by the spectrally precoded OFDM symbol. Notice that, in the case of EMSP, ∆t = 344.17 Figure 4 demonstrates the convergence behaviour of was obtained from CVX [47] for the given OFDM symbol frequency-selective EVM algorithms. The frequency-selective and transmit antenna branch, which implies that the original EVM constraints are arbitrarily chosen to exhibit the efficacy problem is infeasible. As it can be observed from the figure, of the proposed frequency-selective algorithms. The first lower the solution rendered by the MSP naturally meets the mask and upper edge PRBs were set to have EVM requirement  constraint since it is an inequality constrained minimization 20%, 20%, 19%, 19%, 16%, 15%, 14%, 13%, 12.5%, 12.5%,  problem, i.e., this ratio will always be less than or equal 12.5%, 12.5% such that the edge most subcarrier of the edge to 1 as it is a feasible problem. The spectrally precoded PRB corresponds to 20% EVM and the inner most subcarrier symbol generated by ENSP algorithm does not meet any of corresponds to 12.5% EVM. Furthermore, all the subcarriers the 8 mask constraints while the EVM constraint is kept to of the second, third, and fourth edge PRBs are set to have 8%. The symbol generated by EMSP meets the target mask the flat target of 12%, 9.5%, and 8% EVM, respectively. The constraints but after considering ∆t = 344.17, which evidently remaining central subcarriers are set to 7%. This represents penalizes the achieved ACLR compared to the ACLR achieved a wideband average EVM of ∼ 8.9% across the subcarriers. through MSP algorithm in order to meet the wideband 8% The EVM distribution over subcarriers are shown in Fig. 4(a), EVM constraint. Furthermore, the precoded symbol rendered where all the proposed methods meet the frequency-selective by EADMM and ESSP apparently show that for some discrete EVM requirement. In Fig. 4(c), all the frequency-selective frequency points mask constraint is met while for some points EVM algorithms have similar wideband EVM performance. it fails. Therefore, long-term average of PSDs or ACLRs over The imposed mask and frequency-selective EVM constraints several slots of OFDM signals generated by the proposed make the problem infeasible which is evident from the algorithms outperform significantly heuristic ENSP method in divergence of ACLR metric rendered by ESSP algorithms terms of out-of-band performance. In other words, we can without any stopping criterion as depicted in Fig. 4(b). We construe from the figure that the EVM constraint penalizes observe that the ACLR performance delivered by EADMM the achievable ACLR compared to the ACLR achieved by are quite stable over iterations even though the problem is unconstrained EVM problem. infeasible. Reiterating that the ENSP can not be employed or Figure 3(b) demonstrate the ACLR performance against extended to support frequency-selective EVM requirement. iterations for all the proposed EVM-constrained spectral pre- In Fig. 5, we exhibit in-band and OOBE performance 12

EMSP(CVX); WB EVM 8% 25 EMSP(CVX); FS EVM 47.5 ESSP: STOPPING CRITERION 9 EADMM; FS EVM ESSP(I'=2) FS EVM ESSP(I'=2) ESSP (I'=1); FS EVM 45 8 20 ESSP (I'=2); FS EVM 8.85 EMSP 8.8 ESSP (I'=3); FS EVM 42.5EMSP 7 (FS) 8.75 15 FREQUENCY-SELECTIVE (FS) EVM-CONSTRAINT (WB) ESSP(I'=1) ESSP(I'=3) WB EVM 7.1 40 6 112 114 116 118 120 7 EMSP(CVX); WB EVM 8% ESSP(I'=1) ESSP(I'=3) EMSP(CVX); FS EVM 10 6.9 37.5 EMSP(CVX); WB EVM 8% EADMM; FS EVM [%] EVM 5 90 95 100 [dB] ACLR EMSP(CVX); FS EVM ESSP (I'=1); FS EVM 35 EADMM; FS EVM ESSP (I'=2); FS EVM 4 ESSP (I'=1); FS EVM 5 EADMM ESSP (I'=3); FS EVM 32.5 Average) (Wideband EADMM ESSP (I'=2); FS EVM (Average) EVM [%] EVM (Average) 3GPP NR MIN ACLR ESSP (I'=3); FS EVM WIDEBAND (WB) EVM-CONSTRAINT 0 1 2 3 3 0 10 10 10 10 100 101 102 103 0 100 200 300 (Outer) Iterations (i) (Outer) Iterations (i) Subcarriers (b) ACLR-vs.-Iterations of ESSP. (c) EVM-vs.-iterations. (a) EVM [%] over subcarriers. Fig. 4: Convergence behaviour of frequency-selective EVM-constrained spectral precoding.

50 MSP (CVX); 1 MSP (CVX) NO OOBER MSP (CVX) -20 ENSP [26] 47.5 -52.5 MSP(CVX); 1 ENSP [26] 30 EMSP (CVX) EMSP (CVX) EMSP (CVX) EADMM ENSP 45 -53 EADMM -40 ESSP (I'=2) NO OOBER ESSP 18 ESSP EADMM -53.5 3GPP NR MASK 20 16 42.5 MSP (CVX); 1 14 -2.6 -2.4 EMSP/EADMM/ESSP 12 ENSP [26] ENSP 40 EMSP (CVX) 4 6 8 10 12 14 ACLR [dB] ACLR ENSP -60 EADMM 10 37.5 NO OOBER ESSP (I'=2) NR MIN ACLR MEETING MASK [%] EVM (Average)

PSD [dBm/100 KHz] [dBm/100 PSD EVM CONSTRAINED 35 REQUIREMENT MSP 0 0 1 2 3 -80 10 10 10 10 -20 -10 0 10 20 0 100 200 300 (Outer) Iterations (i) Frequency [MHz] Subcarriers (a) ACLR-vs.-Iterations for Test 3 (b) PSD for Test 3 (8Tx2Rx). (c) EVM distribution for Test 3 (8Tx2Rx). (8Tx2Rx).

0 MSP 1 NO OOBER 100 10 MSP (0.50) NO OOBER NO OOBER (0.50) NO OOBER MSP (CVX) NO OOBER (0.83) NO OOBER (0.83) NO OOBER 0.8 ENSP [26] (0.50) MSP (0.50) MSP [13] EMSP (CVX) (0.83) MSP (0.83) MSP [13] EADMM (0.50) ENSP[26] (0.50) ENSP [26] 0.6 ESSP (0.83) ENSP[26] (0.83) ENSP [26]

64QAM; (0.50) EMSP 64QAM; (0.50) EMSP BLER coderate 0.83 (0.83) EMSP BLER coderate 0.83 (0.83) EMSP 0.4 8% WB EVM (0.50) EADMM (0.50) EADMM CONSTRAINED 64QAM (0.83) EADMM (0.83) EADMM 0.2 -1 (0.50) ESSP -1 64QAM 10 coderate 0.5 10 coderate 0.5 (0.50) ESSP EVM UNCONSTRAINED (0.83) ESSP (0.83) ESSP Normalized Throughput Normalized 0 0 20 40 0 20 40 -10 0 10 20 30 40 SNR [dB] SNR [dB] SNR [dB] (d) BLER-vs.-SNR for Test 1 (2Tx2Rx, (e) BLER-vs.-SNR for Test 2 (8Tx2Rx, (f) Throughput-vs.-SNR for Test 3 Rank 1). Rank 1). (8Tx2Rx). Fig. 5: Performance of wideband 8% EVM-constrained spectral precoding including ESSP with stopping criterion. of the proposed 8% wideband EVM-constrained algorithms, be corroborated by the EMSP performance comparing with where ESSP employs 2 inner and 2 outer iterations (with the performance achieved by the MSP (CVX). Moreover, it stopping criterion) for all the 3 test cases. Further, EADMM can be noticed that all the algorithms have different ACLR employs 40 iterations for the performance evaluation. More performance and unfortunately do not meet the minimum specifically, we show the in-band performance, in terms of 45 dB ACLR requirement for the 1st channel. Still, all the pro- BLER and (normalized) throughput against received SNR posed algorithms render the ACLR between 44 dB and 45 dB. at the user-equipment and EVM distribution for the final Conspicuously, ESSP achieves 44.95 dB ACLR, which is 0.05 chosen iterations of the respective algorithms. We also depict dB below the minimum 45 dB requirement. Moreover, it can the OOBE performance, in terms of ACLR versus (outer) be observed that all the proposed algorithms meet the 3GPP iterations and average PSDs of the the final chosen iterations mask requirement as illustrated in the PSD—see Fig. 5(b). of the respective algorithms. Therefore, one could employ a relaxed additional spectrum shaping method, for instance, a transmit windowing or filtering Particularly, Figure 5(a) depicts ACLR performance against that would use less than 9% of the cyclic prefix length to meet iterations utilizing Test 3 (similar results are observed for other the ACLR requirement with some implementation margin. test scenarios). Since the problem was infeasible, the EVM In other words, one can combine spectral precoding with constraint penalizes the achievable ACLR compared to the other (time-domain) spectrum shaping methods, see, e.g., [18] ACLR achieved by unconstrained EVM problem, which can 13 and thereby improve OOBE performance. For completeness, unconstrained spectral precoding method, namely 1) ADMM Fig. 5(c) depicts the distribution of EVM over subcarriers (in and 2) SSP—which is derived by utilizing KKT conditions, frequency-domain) after the final iterations of the respective capitalizing on the closed-form solution of rank 1 quadratic algorithms. form, and a coordinate descent scheme. In the second part—in In Fig. 5(a), the ACLR performance rendered by the heuris- stark contrast to the unsystematic prior art on wideband EVM- tic ENSP method is, inadequately low, 41.24 dB compared constrained spectral precoding—we formulated the optimal to the proposed principled methods that are between 44 dB wideband and frequency-selective constrained EVM problems and 45 dB, i.e., ENSP has 3-4 dB loss in ACLR but sim- in conjunction with mask-compliant spectral precoding. Sub- ilar wideband EVM. Furthermore, ENSP fails to meet the sequently, employing the divide-and-conquer approach, we mask requirement, cf. Fig. 5(b). Thus, ENSP would need an extended and developed “hardware-friendly” algorithms for aggressive additional spectrum shaping method, for instance the EVM-constrained spectral precoding method, referred to as transmit windowing with nearly more than 30%-40% cyclic 1) EADMM and 2) ESSP—which uses the Douglas-Rachford prefix length, to meet both mask and ACLR requirement. operator splitting technique to meet an EVM constraint while Hence, the gain from such spectral precoding vanishes and internally utilizes SSP for mask constraint. The proposed may be futile to employ in a realistic system. algorithms should be treated as vendor-specific transmitter In Fig. 5, we also manifest the in-band performance of module like filtering, which implies that they are 3GPP NR the proposed algorithms considering wideband 8% EVM con- standard transparent to the transmitter and the receiver. Nu- straint. In particular, we show BLER against received SNR merical results corroborate that the proposed low-complexity at the user-equipment for the Test 1 and Test 2 since there algorithms can meet the target EVM constraints and the 3GPP is no link adaptation in these tests except for the spatial NR mask by suppressing OOBE. precoding matrix indicator (PMI) adaptation, where all the This is arguably the first work that proposes computationally proposed EVM-constrained algorithms have similar perfor- affordable EVM-constrained and yet mask-compliant spectral mance. Furthermore, we present normalized throughput versus precoding. received SNR for the Test 3 with 8 transmit antennas since this test employs fast link adaptation, i.e., spatial precoder (PMI), APPENDIX A rank/layers, and modulation and code rate adaptation (referred CONVERGENCE ANALYSIS:ADMM—ALGORITHM 1 AND to as channel quality indicator in the 3GPP standard) with 10% EADMM—ALGORITHM 3 target BLER on long-term average. From these figures, firstly, We present the convergence of Algorithm 1 and Algorithm 3 we observe that the distribution of the EVM in the frequency in a consolidated manner. domain matters since meeting minimum 8% wideband EVM Theorem 5 (Global convergence of consensus ADMM or requirement for 64QAM may not be sufficient to meet the EADMM). Consider an either EVM unconstrained (9) or receiver demodulation performance. Secondly, we observe that constrained problem (23) that can be unified as, i.e., increasing number of the transmit antennas from 2 to 8 for the M low-rank scenario does not improve the in-band performance X

minimize fm Y m + g X gain. We have observed even with 8 transmit antennas that the N ×N N ×N {Y m∈C T },X∈C T m=1 (30) EVM emanating from spectral precoding is beamformed in the same direction as the signal, similar observations for (large subject to Y m = X, scale) MIMO for the EVM stemming from other sources, see, where non-differentiable indicator function fm := X Cm to e.g., [28]–[31]. The EVM-constrained algorithms improve the the mask constraint set (see (20)) for all m = 1,...,M is receiver demodulation performance compared to the EVM- closed convex proper (common to both ADMM and EADMM). unconstrained spectral precoding. However, there is a potential The closed convex proper function is either differentiable : to improve receiver performance while meeting the OOBE g = X −X F (EVM unconstrained) or non-differentiable requirements when the transmitter can be aware of the full indicator function g := X E (for either wideband (21) or channel state information of the receiver(s), which will be frequency-selective (22) EVM constraint). Suppose (30) has addressed in future work. at least one solution. Now, assume subproblems of ADMM and EADMM have solutions, and so-called dual residual  (i+1) (i) VII.CONCLUSION limi→+∞ X −X =0 and primal residual limi→+∞ In this paper we investigated the problem of spectral precod-  (i+1) (i+1) Y m −X = 0, ∀m = 1,...,M, and ρ ∈ R>0 with ing with mask compliant properties. The problem is formally n (0) (0) (0)o posed as some optimizations that offer an explicit trade-off some arbitrary initial X , Y m , Zm . Then, Algorithm 1 n (i)o between EVM and OOBE suppression for large-scale MIMO- and Algorithm 3, at any limit point, X converge to a OFDM systems, but have the disadvantage of not having KKT point of (30). closed-form solutions. To mitigate the complexity of such problems, we proposed a divide-and-conquer approach that Proof. Towards the convergence analysis goal of the pro- decomposes the spectral precoding problem into smaller prob- posed ADMM and EADMM algorithms, we follow the proof lems having each either a closed-form or an efficient solution. given in, e.g., [20]. Now, we form the (unaugmented) La- More specifically, in the first part of the paper, we devel- grangian of the unified problem (30) proposed to be solved oped two computationally efficient algorithms for the EVM- by ADMM/EADMM such that 14

 M  M  2  H 2  L Y m m=1 ,X, Zm m=1 can be formed as L (z, µ) = kx − zk2 + µ u z − b . M M According to the KKT conditions [25], in particular due to X

X n  H
o complementary slackness condition, if the given x is feasible, := fm Y m +g X + 2< Tr Zm Y m −X , m=1 m=1 that is, fulfils the constraint then z = x and the Lagrange (31) multiplier would correspond to µ = 0. However, if the given where Tr is a trace operator. Using (31), then according x does not fulfil the constraint, then µ > 0 and the inequality to KKT optimality conditions—see, e.g., [25], in particular, constraint can be converted to the equality√ constraint such H 2 H n ? oM that constraint u z = b ⇐⇒ u z = b exp (ιθ) for stationarity condition at the optimal primal values Y m m=1 some unknown angle θ. Now, we assume that the angle ? n ? oM θ is known, then the Lagrangian can be expressed as and X , and dual variable Zm satisfy  √  m=1 L (z, µ0) = kx − zk2 + µ0 uHz − b exp (ιθ) .   2 ∂ n ? oM ? n ? oM Now, according to the KKT conditions, we set gradient of 0 ∈ ∗ L Y m , X , Zm  ? m=1 m=1 0 0 L (z, µ ) with respect to z to 0, such that z = √x − µ u. ∂ X H 0 u x− b exp(ιθ) Plugging this in the constraint yields µ = 2 . M √ kuk2 H  ? X ? 0  u x− b exp(ιθ)  ⇐⇒0 ∈ ∂g X − Zm (32) Thus, z = x − µ u = x − 2 u. Following kuk2 m=1 2 [59], the optimal θ that minimizes the objective kx − zk2 = ∂ n ? oM ? n ? oM  √ 2  uHx− b exp(ιθ)  0 ∈ ∗ L Y m , X , Zm u is an angle of xHu. Thus, exp (ιθ) =  ?  m=1 m=1 kuk2 ∂ Y 2 2 m uHx H . Hence, the projection result follows and given in (13).  ?  ? |u x| ⇐⇒0 ∈ ∂fm Y m + Zm, (33) APPENDIX C  ? n ? o where, for EVM unconstrained, ∂g X = X −X DERIVATION OF SSPALGORITHM  ? We present the derivations of the SSP algorithm, whose and, for EVM constrained, ∂g X =∂X E ≡N E , where N E pseudo-code is outlined in Algorithm 2. In the SSP algorithm, corresponds to a normal cone (see, e.g., [49, Chapter 3]); and M  ?  n (0)o ∂f Y =∂X ≡N . The primal feasibility satisfies the set of Lagrange multipliers µm are initialized m m Cm Cm m=1 ? ? assuming that the considered m-th multiplier is present while Y m − X = 0 ∀m = 1,...,M. (34) others are absent, i.e., boiling down to rank 1 case where the Towards this end, we analyze the proposed Algorithm 1 and multiplier is computed in closed-form by following [59]. Algorithm 3, which for sufficiently large iterations satisfy Lemma 1. Let M = 1, then a closed-form solution to the the abovementioned optimality conditions using the stated Lagrange multiplier µm is (i+1) assumptions. In the first step of (E)ADMM, X mini- s !  (i+1) 1 λm  mizes the update step-1 (cf. (24a)), i.e., 0 ∈ ∂g X − µ = a (ν )T d 1 − 1 , (35) m λm m j γ  (i) (i+1) (i) 1 m PM ρ Y −X +Z . Using dual variable update m=1 m m m 2  (i+1) where µ ≥ 0 and λ1 = ka (νm) k2. (24c) and rearranging the terms yields 0 ∈ ∂g X − (i+1)  (i+1) (i) Proof. Following [59, Section IIIB] and performing algebraic PM ρZ + PM ρ Y −Y . Now, we state m=1 m m=1 m m manipulations, we derive the closed-form solution.   (i+1) (i) Y m −Y m → 0 when i → ∞ because of the assumption After initialization, at every given iteration, we apply  (i+1) (i) that the dual residual X −X → 0 and the primal coordinate descent iterative scheme for the SSP algorithm  (i+1) (i+1) (i+1) that essentially utilizes the closed-form solution for rank 1 residual Y m −X →0. Thus, asymptotically, X scenario.
update satisfies the stationarity condition (32). Similarly, Using Lagrangian L dj, {µm} (14), the stationarity con-  (i+1)  (i+1) (i+1) (i) we have 0 ∈ ∂fm Y m + ρ Y m −X +Zm = dition of the KKT conditions, i.e., setting the gradient of L (·)  (i+1) (i+1) with respect to dj to 0, yields ∂fm Y m +ρZm in the step-2 update (cf. (24b)), where −1 in the last equality have used the dual variable update (24c). M ! X −1 Thus, step-2 always satisfies the stationarity condition (33) dj = IN + µmAm dj := G dj. (36) for sufficiently large i. m=1 Finally, primal feasibility (34) is satisfied by the assumption For brevity, we define  (i+1) (i+1) Y −X =0, when i → +∞. M ! m  X
G≡G (µm):= IN + µmAm = G\m +µmAm , APPENDIX B m=1 PROOFOF (13): PROJECTIONONTORANK 1 ELLIPSOID (37) 2 The projection problem reads minimize kx − zk M z 2 P H H where G\m := IN + µnAn. subject to z Aze −b ≤ 0 where Ae = uu . So, the Lagrangian n=1\m 15

The matrix inversion of G(µm) utilizing Sherman-Morrison REFERENCES formula [60] is −1 −1 ! [1] S. Kant, G. Fodor, M. Bengtsson, B. Goransson,¨ and C. Fischione, “Low- µmG AmG complexity OFDM spectral precoding,” in Proc. IEEE Int. Workshop G(µ )−1 = G−1 − \m \m , (38) m \m T −1 ∗ Signal Process. Adv. Wireless Commun. (SPAWC), Cannes, France, 2019. 1+µma (νm) G\ma (νm) [2] E. Dahlman, S. Parkvall, and J. Skold,¨ 5G NR: The next generation wireless access technology. Academic Press, 2018. and noting the fact that the matrix G(µm) is a sum of rank one [3] P. Setoodeh and S. Haykin, Fundamentals of cognitive radio. John matrices then the matrix inversion can be performed iteratively, Wiley & Sons, Inc., 2017. [4] R. Prasad, OFDM for wireless communications systems. Artech House, cf. Lemma 2. In order to obtain the set of {µm} Lagrange 2004. multipliers, we employ coordinate descent scheme [61]. Let [5] 3GPP TS 38.104, “NR; Base station (BS) radio transmission and us say we compute µm in a given cycle, then we fix other reception,” V15.9.0, 2020. multipliers µn ∀n but excluding µm. We propose to compute [6] X. Huang, J. A. Zhang, and Y. J. Guo, “Out-of-band emission reduction and update all the M Lagrange multipliers cyclically. In a and a unified framework for precoded OFDM,” IEEE Commun. Mag., vol. 53, no. 6, pp. 151–159, June 2015. given iteration cycle, we compute µm Lagrange multiplier [7] M. Faulkner, “The effect of filtering on the performance of OFDM and fixing other multipliers by utilizing the complementary systems,” IEEE Trans. Veh. Technol., vol. 49, no. 5, pp. 1877–1884, slackness condition of the KKT conditions and the dual Sep. 2000. [8] E. Bala, J. Li, and R. Yang, “Shaping spectral leakage: A novel low- feasibility condition: 1) If µm = 0, then dj = G\mdj (cf. complexity transceiver architecture for cognitive radio,” IEEE Veh. (37)), and 2) If µm > 0, then the inequality constraint should Technol. Mag., vol. 8, no. 3, pp. 38–46, July 2013. be an equality constraint. By plugging (36) and (37) in the [9] H. Yamaguchi, “Active interference cancellation technique for MB- OFDM cognitive radio,” in Proc. IEEE Eur. Microw. Conf., 2004, pp. constraint such that 1105–1108. H H ∗ T [10] S. Brandes, I. Cosovic, and M. Schnell, “Sidelobe suppression in OFDM dj Amdj −γm =dj a (νm) a (νm) dj − γm =0 systems by insertion of cancellation carriers,” in Proc. IEEE Veh. T −1 √ Technol. Conf., 2005, pp. 152–156. ⇐⇒a (νm) G (µm) dj = γm exp (ιφ) , (39) [11] D. Qu, Z. Wang, and T. Jiang, “Extended active interference cancellation where φ is a free parameter. We now obtain the Lagrange for sidelobe suppression in cognitive radio OFDM systems with cyclic prefix,” IEEE Trans. Veh. Technol., vol. 59, no. 4, pp. 1689–1695, May multiplier µm from (39) utilizing (38) after some algebraic 2010. manipulation [12] I. Cosovic, S. Brandes, and M. Schnell, “Subcarrier weighting: a method for sidelobe suppression in OFDM systems,” IEEE Commun. Lett., T
−1 √ a (νm) G\m + µmAm dj = γm exp (ιφ) (40) vol. 10, no. 6, pp. 444–446, June 2006. T −1 −1 ! [13] J. van de Beek, “Sculpting the multicarrier spectrum: A novel projection a (νm) G AmG dj precoder,” IEEE Commun. Lett., vol. 13, no. 12, pp. 881–883, 2009. ⇐⇒ a (ν )TG−1 d −µ \m \m m \m j m T −1 ∗ [14] H. Chen, W. Chen, and C. Chung, “Spectrally precoded OFDM and 1+µma (νm) G\ma (νm) OFDMA with cyclic prefix and unconstrained guard ratios,” IEEE Trans. √ = γ exp (ιφ) . (41) Wireless Commun., vol. 10, no. 5, pp. 1416–1427, May 2011. m [15] A. Tom, A. Sahin, and H. Arslan, “Mask compliant precoder for OFDM T −1 spectrum shaping,” IEEE Commun. Lett., vol. 17, no. 3, pp. 447–450, For brevity, let α1 := a (νm) G\mdj and α2 := 2013. T −1 ∗ a (νm) G a (νm) such that (41) can be rewritten as [16] R. Kumar and A. Tyagi, “Weighted least squares based spectral precoder \m for OFDM cognitive radio,” IEEE Wireless Commun. Lett., vol. 4, no. 6, µmα1α2 α1 √ pp. 641–644, Dec. 2015. α1 − = = γm exp (ιφ) [17] M. Mohamad, R. Nilsson, and J. van de Beek, “A novel transmitter 1 + µmα2 1 + µmα2  √  architecture for spectrally-precoded OFDM,” IEEE Trans. Circuits Syst. α1 exp (−ιφ) − γm I: Regular Papers, vol. 65, no. 8, pp. 2592–2605, Aug. 2018. ⇒µm = < √ . [18] M. Mohamad, “Spectrally precoded OFDM design and analysis,” PhD γm α2 dissertation, Department of Computer Science, Electrical and Space Engineering, Lulea˚ University, Lulea,˚ Sweden, 2019. APPENDIX D [19] P. L. Combettes and J.-C. Pesquet, “Proximal splitting methods in signal processing,” in Fixed-Point algorithms for inverse problems in science USEFUL LEMMA and engineering. Springer, New York, NY, 2011, pp. 185–212. Lemma 2 (Matrix inversion with sum of rank one matrices). [20] S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed Pr optimization and statistical learning via the alternating direction method The matrix inversion of Ar+1 = G + i=1 Hi with sum of of multipliers,” Foundations and Trends® in Machine Learning, vol. 3, −1 r rank one matrices Hi can be obtained iteratively Ak+1 = no. 1, pp. 1–122, 2011. −1 −1 −1 [21] J. Eckstein and D. P. Bertsekas, “On the Douglas-Rachford splitting A − gkA HkA ∀k = 1, . . . , r, where A1 = G and k k k method and the proximal point algorithm for maximal monotone opera- 1 1+Tr A−1H gk = /( ( k k)). tors,” Mathematical Programming, vol. 55, no. 1–3, pp. 293–318, Apr. 1992. Proof. Let G and G + H be invertible matrices, and H = [22] D. M. Pozar, Microwave Engineering, 4th Edition. Wiley, 2011. Pr Pk i=1 Hi with rank(Hi) = 1. Let Ak+1 = G+ i=1 Hi be [23] M. Mohamad, R. Nilsson, and J. van de Beek, “Minimum-EVM N- invertible. By initializing A1 = G, then utilizing [62] result, continuous OFDM,” in Proc. 2016 IEEE Int. Conf. on Commun. (ICC), we achieve A−1 = A−1 −g A−1H A−1 , for k = 1, . . . , r May 2016, pp. 1–5. k+1 k k k k k [24] R. Kumar and A. Tyagi, “Computationally efficient mask-compliant 1 1+Tr A−1H where gk = /( ( k k)).  spectral precoder for OFDM cognitive radio,” IEEE Trans. on Cogn. Commun. Netw., vol. 2, no. 1, pp. 15–23, 2016. [25] S. P. Boyd and L. Vandenberghe, Convex optimization. Cambridge ACKNOWLEDGEMENT University Press, 2004. [26] J. van de Beek and F. Berggren, “EVM-constrained OFDM precoding This paper is dedicated to the memory of Prof. Peter Handel¨ for reduction of out-of-band emission,” in Proc. IEEE Veh. Tech. Conf. who prematurely passed away. Fall, Sep. 2009, pp. 1–5. 16

[27] R. Pitaval and B. M. Popovi, “Linear receivers for spectrally-precoded [52] P. L. Combettes, “Solving monotone inclusions via compositions of MIMO-OFDM,” IEEE Commun. Lett., vol. 21, no. 6, pp. 1269–1272, nonexpansive averaged operators,” Optimization, vol. 53, no. 5–6, pp. June 2017. 475–504, Oct. 2004. [28] N. N. Moghadam, P. Zetterberg, P. Handel,¨ and H. Hjalmarsson, “Cor- [53] Y. Liu, E. K. Ryu, and W. Yin, “A new use of Douglas-Rachford splitting relation of distortion noise between the branches of MIMO transmit an- for identifying infeasible, unbounded, and pathological conic programs,” tennas,” in Proc. IEEE Int. Symp. Pers., Indoor, Mobile Radio Commun. Computational Optimization and Applications, vol. 177, p. 225–253, (PIMRC), Sep. 2012, pp. 2079–2084. Sep. 2019. [29] E. Sienkiewicz, N. McGowan, B. Goransson,¨ T. Chapman, and T. Elf- [54] E. K. Ryu, Y. Liu, and W. Yin, “Douglas-Rachford splitting and ADMM strom,¨ “Spatially dependent ACLR modelling,” in Proc. IEEE Conf. for pathological convex optimization,” Computational Optimization and Antenna Meas. Appl. (CAMA), Nov. 2014, pp. 1–4. Applications, vol. 74, p. 747–778, Sep. 2019. [30] E. G. Larsson and L. Van Der Perre, “Out-of-band radiation from [55] 3GPP TS 38.212, “NR; Multiplexing and channel coding,” V15.8.0, antenna arrays clarified,” IEEE Wireless Commun. Lett., vol. 7, no. 4, 2020. pp. 610–613, Aug. 2018. [56] 3GPP TS 34.121, “Terminal conformance specification, radio transmis- [31] N. N. Moghadam, G. Fodor, M. Bengtsson, and D. J. Love, “On sion and reception (FDD),” V6.4.0, 2006. the energy efficiency of MIMO hybrid beamforming for millimeter- [57] 3GPP TS 38.306, “NR; User equipment (UE) radio access capabilities,” wave systems with nonlinear power amplifiers,” IEEE Trans. Wireless V15.9.0, 2019. Commun., vol. 17, no. 11, pp. 7208–7221, Nov. 2018. [58] 3GPP TS 38.211, “NR; Physical channels and modulation,” V15.8.0, [32] G. Fettweis, M. Lohning, D. Petrovic, M. Windisch, P. Zillmann, and 2020. W. Rave, “Dirty RF: a new paradigm,” in Proc. IEEE Int. Symp. Pers., [59] K. Huang and N. D. Sidiropoulos, “Consensus-ADMM for general Indoor, Mobile Radio Commun. (PIMRC), vol. 4, 2005, pp. 2347–2355 quadratically constrained quadratic programming,” IEEE Trans. Signal Vol. 4. Process., vol. 64, no. 20, pp. 5297–5310, Oct. 2016. [60] C. D. Meyer, Matrix analysis and applied linear algebra. Society for [33] T. Schenk, RF imperfections in high-rate wireless systems. The Industrial and Applied Mathematics, 2000. Netherlands: Springer, 2008. [61] P. Tseng, “Convergence of a block coordinate descent method for [34]B.G oransson,¨ S. Grant, E. Larsson, and Z. Feng, “Effect of transmitter nondifferentiable minimization,” Journal of Optimization Theory & and receiver impairments on the performance of MIMO in HSDPA,” in Applications, vol. 109, no. 3, pp. 475–494, June 2001. Proc. IEEE Works. on Sign. Proc. Adv. in Wireless Commun., 2008, pp. [62] K. S. Miller, “On the inverse of the sum of matrices,” Mathematics 496–500. Magazine, vol. 54, no. 2, pp. 67–72, Mar. 1981. [35] C. Studer, M. Wenk, and A. Burg, “MIMO transmission with residual transmit-RF impairments,” in Proc. ITG Workshop Smart Antennas, Feb. 2010, pp. 189–196. [36] ——, “System-level implications of residual transmit-RF impairments in MIMO systems,” in Proc. European Conf. on Antennas and Propagation, Apr. 2011, pp. 2686–2689. [37] E. Bjornson,¨ J. Hoydis, and L. Sanguinetti, “Massive MIMO networks: Spectral, energy, and hardware efficiency,” Foundations and Trends®in Signal Processing, vol. 11, no. 3-4, pp. 154–655, 2017. [38] E. Bjornson,¨ J. Hoydis, M. Kountouris, and M. Debbah, “Massive MIMO systems with non-ideal hardware: Energy efficiency, estimation, and capacity limits,” IEEE Trans. Inf. Theory, vol. 60, no. 11, pp. 7112– 7139, Sep. 2014. [39] A. A. A. Boulogeorgos and A. Alexiou, “Analytical performance eval- uation of beamforming under transceivers hardware imperfections,” in Proc. IEEE Wireless Commun. and Networking Conf., 2019, pp. 1–7. [40] R. Hassun, M. Flaherty, R. Matreci, and M. Taylor, “Effective evaluation of link quality using error vector magnitude techniques,” in Proc. IEEE Wireless Commun. Conf., 1997, pp. 89–94. [41] H. A. Mahmoud and H. Arslan, “Error vector magnitude to SNR conversion for nondata-aided receivers,” IEEE Trans. Wireless Commun., vol. 8, no. 5, pp. 2694–2704, May 2009. [42] T. L. Jensen and T. Larsen, “Robust computation of error vector magnitude for wireless standards,” IEEE Trans. Commun., vol. 61, no. 2, pp. 648–657, Mar. 2013. [43] 3GPP TS 38.101-4, “User equipment (UE) radio transmission and reception; part 4: Performance requirements,” V15.5.0, 2020. [44] 3GPP TS 38.214, “NR; Physical layer procedures for data,” V15.9.0, 2020. [45] R. A. Shafik, M. S. Rahman, and A. R. Islam, “On the extended relationships among EVM, BER and SNR as performance metrics,” in Proc. Int. Conf. Elect. Comput. Eng., 2006, pp. 408–411. [46] K. Freiberger, “Measurement methods for estimating the error vector magnitude in OFDM transceivers,” PhD dissertation, Doctoral School of Information and Communication Engineering, Graz University of Technology, Austria, 2017. [47] M. Grant and S. Boyd, “CVX: Matlab software for disciplined convex programming, version 2.1,” http://cvxr.com/cvx, Mar. 2014. [48] N. Parikh, S. Boyd, N. Parikh, and S. Boyd, “Proximal algorithms,” Foundations and Trends® in Optimization, vol. 1, no. 3, pp. 123–231, 2013. [49] A. Beck, First-order methods in optimization. SIAM, 2017. [50] Z. Q. Luo and P. Tseng, “On the convergence of the coordinate descent method for convex differentiable minimization,” Journal of Optimization Theory and Applications, vol. 72, no. 1, pp. 7–35, Jan. 1992. [51] C.-Y. Chi, W.-C. Li, and C.-H. Lin, Convex optimization for signal processing and communications: from fundamentals to applications, 1st ed. CRC Press, 2017.