2. Pursuit-evasion game

#### 2.1. Relative motion kinematics

The interceptor's movement consists of two orthogonal channels. The pursuit-evasion game is decomposed into two 2D channels.

Figure 1 shows the geometric diagram of the interceptor-target relative motion kinematics. A Cartesian reference system is denoted by "X-O-Y." The interceptor and target are denoted by "M" and "T". The line-of-sight angle is denoted by "q". The relative distance is denoted by "r". Flight path angles of the interceptor and target are denoted by "φm" and "φt". Velocities of the interceptor and target are denoted by "Vm" and "Vt".

Endgame relative motion kinematics are given by

$$\begin{cases} \dot{r} = V\_t \cos\left(\varphi\_t - q\right) - V\_m \cos\left(\varphi\_m - q\right), \\\ r\dot{q} = V\_t \sin\left(\varphi\_t - q\right) - V\_m \sin\left(\varphi\_m - q\right). \end{cases} \tag{1}$$

r0 ¼ r0 þ ~r0, ∣~r0∣ ≤ δr0 , v0 ¼ v0 þ v~0, ∣v~0∣ ≤ δv0 ,

Adaptive Robust Guidance Scheme Based on the Sliding Mode Control in an Aircraft Pursuit-Evasion Problem

where initial relative speed and distance are denoted by "v0" and "r0"; observations of v0 and r0 are denoted by "v0" and "r0"; observation deviations of v0 and r0 are denoted by "v~0" and

Remark 1. v0 and r0 are detected by a radar on ground or aircraft carrier and are sent to the interceptor via the data link only once. δv0 and δr0 are treated to be maximum observation

For successfully intercepting the target, the line-of-sight angular velocity should be constrained [5, 6]. In this chapter, seeker and autopilot loops are not considered. With this premise, relative equation of the line-of-sight angular velocity q\_ is obtained as Eq. (4). However, 1=r and 2v=r in Eq. (4) are obtained as Eq. (3), which indicates relative speed v alters as r, v, q\_, atr, and amr vary, and v0 and r0 are preset. In accordance with the characteristic of an interceptor's engine, the thrust along the line of sight almost does not change. Moreover, the target is usually escaping orthogonally to the line of sight to increase the line-ofsight angular velocity. Although acceleration component of the target along the line-of-sight is subsistent, the relative speed does not change too much with limited energy and time. Assume that acceleration components along the line of sight of the target and interceptor are

> r\_ ¼ v, v\_ ¼ rq\_2:

z\_1 ¼ z2, z\_2 ¼ z1q\_2:

According to Eq. (9), initial states for Eq. (11) to get r0 and v0 are obtained as

(9)

29

http://dx.doi.org/10.5772/intechopen.72177

(10)

(11)

"~r0"; upper boundaries of v~0 and ~r0 are denoted by "δv0 " and "δr0 ".

deviation of the detector.

zero. Simplify Eq. (3) into

Define z1 ¼ r and z2 ¼ v. Equation (10) becomes

2.2. Kinematics simplification

Figure 1. Relative motion kinematics.

Let the relative speed v ¼ r\_. Eq. (2) is obtained as

$$\begin{cases} \dot{v} = r\dot{q}^2 + \left[\dot{V}\_t \cos\left(\varphi\_t - q\right) - V\_t \dot{\varphi}\_t \sin\left(\varphi\_t - q\right)\right] \\ \quad - \left[\dot{V}\_m \cos\left(\varphi\_m - q\right) - V\_m \dot{\varphi}\_m \sin\left(\varphi\_m - q\right)\right], \\\ r\ddot{\eta} = -2\dot{r}\dot{\eta} + \left[\dot{V}\_t \sin\left(\varphi\_t - q\right) + V\_t \dot{\varphi}\_t \cos\left(\varphi\_t - q\right)\right] \\ \quad - \left[\dot{V}\_m \sin\left(\varphi\_m - q\right) + V\_m \dot{\varphi}\_m \cos\left(\varphi\_m - q\right)\right]. \end{cases} \tag{2}$$

For simplification, we get

$$
\dot{\upsilon} = r\dot{\eta}^2 + a\_{tr} - a\_{mr\prime} \tag{3}
$$

$$\ddot{q} = -2\frac{v}{r}\dot{q} + \frac{1}{r}a\_{t\eta} - \frac{1}{r}a\_{m\eta} \tag{4}$$

where acceleration components of the interceptor and target along the line-of-sight are denoted by "amr" and "atr"; acceleration components of the interceptor and target orthogonal to the line-of-sight are denoted by "amq"and "atq". The equations of them are formulated as

$$a\_{tr} = \dot{V}\_t \cos\left(\varphi\_t - q\right) - V\_t \dot{\varphi}\_t \sin\left(\varphi\_t - q\right),\tag{5}$$

$$a\_{mr} = \dot{V}\_m \cos\left(\varphi\_m - q\right) - V\_m \dot{\varphi}\_m \sin\left(\varphi\_m - q\right),\tag{6}$$

$$a\_{t\eta} = \dot{V}\_t \sin\left(\varphi\_t - \eta\right) + V\_t \dot{\varphi}\_t \cos\left(\varphi\_t - \eta\right),\tag{7}$$

$$a\_{m\eta} = \dot{V}\_m \sin\left(\varphi\_m - \eta\right) + V\_m \dot{\varphi}\_m \cos\left(\varphi\_m - \eta\right). \tag{8}$$

Assume the line-of-sight angular velocity is accurately observed at each instant. Initial relative speed and distance and error boundaries of them are obtained as

Adaptive Robust Guidance Scheme Based on the Sliding Mode Control in an Aircraft Pursuit-Evasion Problem http://dx.doi.org/10.5772/intechopen.72177 29

Figure 1. Relative motion kinematics.

2. Pursuit-evasion game

28 Adaptive Robust Control Systems

2.1. Relative motion kinematics

decomposed into two 2D channels.

interceptor and target are denoted by "Vm" and "Vt".

(

Endgame relative motion kinematics are given by

Let the relative speed v ¼ r\_. Eq. (2) is obtained as

8 >>>>>><

>>>>>>:

For simplification, we get

The interceptor's movement consists of two orthogonal channels. The pursuit-evasion game is

Figure 1 shows the geometric diagram of the interceptor-target relative motion kinematics. A Cartesian reference system is denoted by "X-O-Y." The interceptor and target are denoted by "M" and "T". The line-of-sight angle is denoted by "q". The relative distance is denoted by "r". Flight path angles of the interceptor and target are denoted by "φm" and "φt".  The equations of them are formulated as

Assume the line-of-sight angular velocity is accurately observed at each instant. Initial relative

q€ ¼ �2

speed and distance and error boundaries of them are obtained as

(1)

(2)

$$\begin{cases} \begin{aligned} r\_0 &= \overline{r}\_0 + \check{r}\_{0\prime} & |\check{r}\_0| \le \delta\_{r\_{0\prime}} \\ \upsilon\_0 &= \overline{\upsilon}\_0 + \check{\upsilon}\_{0\prime} & |\check{\upsilon}\_0| \le \delta\_{\upsilon\_{0\prime}} \end{aligned} \end{cases} \tag{9}$$

where initial relative speed and distance are denoted by "v0" and "r0"; observations of v0 and r0 are denoted by "v0" and "r0"; observation deviations of v0 and r0 are denoted by "v~0" and "~r0"; upper boundaries of v~0 and ~r0 are denoted by "δv0 " and "δr0 ".

Remark 1. v0 and r0 are detected by a radar on ground or aircraft carrier and are sent to the interceptor via the data link only once. δv0 and δr0 are treated to be maximum observation deviation of the detector.

#### 2.2. Kinematics simplification

For successfully intercepting the target, the line-of-sight angular velocity should be constrained [5, 6]. In this chapter, seeker and autopilot loops are not considered. With this premise, relative equation of the line-of-sight angular velocity q\_ is obtained as Eq. (4). However, 1=r and 2v=r in Eq. (4) are obtained as Eq. (3), which indicates relative speed v alters as r, v, q\_, atr, and amr vary, and v0 and r0 are preset. In accordance with the characteristic of an interceptor's engine, the thrust along the line of sight almost does not change. Moreover, the target is usually escaping orthogonally to the line of sight to increase the line-ofsight angular velocity. Although acceleration component of the target along the line-of-sight is subsistent, the relative speed does not change too much with limited energy and time. Assume that acceleration components along the line of sight of the target and interceptor are zero. Simplify Eq. (3) into

$$\begin{cases} \dot{r} = v, \\ \dot{v} = r\dot{q}^2. \end{cases} \tag{10}$$

Define z1 ¼ r and z2 ¼ v. Equation (10) becomes

$$\begin{cases} \dot{z}\_1 = z\_2, \\ \dot{z}\_2 = z\_1 \dot{q}^2. \end{cases} \tag{11}$$

According to Eq. (9), initial states for Eq. (11) to get r0 and v0 are obtained as

$$z\_1^{(1)}(t\_0) = \overline{r}\_{0\prime} \ z\_2^{(1)}(t\_0) = \overline{v}\_{0\prime} \tag{12}$$

s\_ ¼ r r

V\_ 1 ¼ � r

Next, the following is obtained:

where Δ is a tiny positive constant.

dynamically estimate ε [23, 24].

Equation (11) is solved with Eq. (12) or (14).  (9) or (13).  0. Finally s ! 0. Since s ¼ z

amq ¼ N � z

3.2. Improved guidance scheme based on the SMCG

Proposition 1. An unchanged constant k > 0 exists, so that

� r r

z ð Þ2 1 z ð Þ3 1

Because εsgnð Þ� q\_ atq > 0, j j Δv � ~v > 0, r > 0, and r > 0, we get V\_ 1 < 0. Using Lyapunov

Remark 2. The "sgn" function in Eq. (17) is replaced by the following function to suppress the jitter:

ε in Eq. (27) or (17) is unchanged, which indicates that an unchanged upper boundary of atq is employed to ensure the sliding mode's subsistence. By this means, the guidance command's jitter might exist in the vicinity of the sliding mode although "sgn" is already replaced in Eq. (27). For smoothing the command, the better way is to use the adaptive approach to

z ð Þ1 2 � � � � � � þ 2j j Δv

" #

� r r

r ðj j Δv � ~vÞq\_

Adaptive Robust Guidance Scheme Based on the Sliding Mode Control in an Aircraft Pursuit-Evasion Problem

� r r

j j v q\_ � 2 j j Δv � ~vÞq\_ � εsgnð Þ� q\_ atq � �}: � (21)

http://dx.doi.org/10.5772/intechopen.72177

31

1 ≤ r: (23)

1 ≥ r > 0: (24)

≥ 0: (25)

> 0: (26)

ð Þ1

q\_ þ ε q\_

1 q\_, that is, q\_ ! 0.

∣q\_∣ þ Δ, (27)

εsgnð Þ� q\_ atq � �q\_: (22)

2 � r 2 r

$$z\_1^{(2)}(t\_0) = \overline{r}\_0 + \delta\_{r0\prime} \ z\_2^{(2)}(t\_0) = \overline{v}\_0 + \delta\_{r0\prime} \tag{13}$$

$$z\_1^{(3)}(t\_0) = \overline{r}\_0 - \delta\_{r0} \cdot z\_2^{(3)}(t\_0) = \overline{v}\_0 - \delta\_{v0} \cdot \tag{14}$$

Equations (12)–(14) are employed to calculate Eq. (11). Boundaries of v and rare computed as

$$\begin{cases} |\Delta r| = \max \left\{ z\_1^{(2)} - z\_1^{(1)}, z\_1^{(1)} - z\_1^{(3)} \right\}, \\ |\Delta v| = \max \left\{ z\_2^{(2)} - z\_2^{(1)}, z\_2^{(1)} - z\_2^{(3)} \right\}. \end{cases} \tag{15}$$


**1. Introduction**

Tumour morphology might be heterogeneous with alternating regions of relatively vital tumour tissue, mild hypoxia, severe hypoxia and necrosis [1]. Tumour hypoxia manifests itself predominantly in solid tumours [2]. Central necrosis of tumours occurs as a result of hypoxia and is caused by uncontrolled oncogene-driven proliferation without efficient vasculature, inducing a nutrient and oxygen shortage [1]. As a morphological

**Citation:** Noortman, W.A.; Vriens, D.; Mooij, C.D.Y.; Slump, C.H.; Aarntzen, E.H.; van Berkel, A.; Timmers, H.J.L.M.; Bussink, J.; Meijer, T.W.H.; de Geus-Oei, L.-F.; et al. The Influence of the Exclusion of Central Necrosis on [18F]FDG PET Radiomic Analysis. *Diagnostics* **2021**, *11*, 1296. https://doi.org/10.3390/ diagnostics11071296

Academic Editor: Giorgio Treglia

Received: 1 June 2021 Accepted: 15 July 2021 Published: 19 July 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

marker, central necrosis is associated with poor prognosis in a variety of cancers [3–5], including non-small cell lung carcinomas (NSCLC) [6,7]. Larger regions of necrosis can be detected on 2-[18F]fluoro-2-deoxy-D-glucose positron emission tomography ([18F]FDG PET/CT) as an often centrally located region with little to no tracer uptake.

Radiomics aims to quantify the geometry and tracer uptake, including uptake heterogeneity, of tumours by using first order, shape and texture features and hypothesizing that these features can be used for tumour characterisation, prognostic stratification and response prediction in precision medicine [8]. It is uncertain whether regions of central necrosis should be added to the delineation of the tumour, since the effect of delineation methods on the predictive value of the radiomic signature remains unknown [9]. Semiautomatic tumour delineation methods used in radiomic analysis apply isocontours by using fixed or adaptive thresholds [10,11] or more advanced algorithms, such as the fuzzy locally adaptive Bayesian (FLAB) algorithm [12]. Although these methods have shown to be highly reproducible [10,13], they often underestimate the true (anatomical) tumour volume by excluding (up to a certain degree) regions of low tracer uptake. Some studies manually add the excluded regions of low tracer uptake to the volume of interest (VOI), but this is not always clearly reported. It is hypothesised that the addition of a region of central necrosis to the VOI may influence all three radiomic feature classes numerically. First order features might be affected by the addition of voxels with low grey levels and this skews the intensity histogram. Shape features might be influenced by the different 3D morphology of the VOI when central necrosis is included. Texture features, representing spatial relationships between voxels in terms of run lengths or size zones of the same voxel values or combinations of neighbouring voxel values, might change as well. The introduction of an area of central necrosis might, for instance, result in long runs with low values that will change the run length matrix and, as a result, the feature values. The Image Biomarker Standardisation Initiative (IBSI), which is an independent international collaboration working towards standardising the extraction of image biomarkers, provides reporting guidelines for radiomic studies but, up to this point, does not specify the need to report on the inclusion/exclusion of necrosis while describing the used segmentation method [14]. Moreover, the effect of the delineation method, including whether or not to include central necrosis, on the performance of the radiomic signature for predicting underlying tumour biology remains unknown [9].

This study explores how central necrosis influences PET radiomic analysis by assessing the differences in radiomic features and the predictive performance of features extracted from VOIs delineated using an isocontour method with and without the manual addition of the region of central necrosis for two datasets of NSCLC and pheochromocytomas or paragangliomas (PPGL), catecholamine-producing neuroendocrine tumours that arise from the chromaffin cells of the adrenal medulla and extra-adrenal sympathetic paraganglia [15].

#### **2. Materials and Methods**

#### *2.1. Patient Population, Data Acquisition and Image Reconstruction*

Subjects from two cohorts of patients who underwent an [18F]FDG PET/CT in a single academic centre were retrospectively included to study the effect of different aspects of central necrosis on the radiomic analysis. A cohort of patients with non-small cell lungcarcinomas (NSCLC, *n* = 35), generally presenting a high tumour-to-background ratio, and a cohort of patients with pheochromocytomas or paragangliomas (PPGL, *n* = 77), generally presenting a low tumour-to-background ratio, were included.

The NSCLC cohort is a previously published prospective cohort [16]. Patients underwent a dynamic [18F]FDG PET/CT scan with the primary tumour located centrally in the field of view using the Biograph Duo or Biograph 40 mCT (Siemens Healthineers, Erlangen, Germany) at the Radboud University Medical Center between 2009 and 2014. Only tumours with a diameter larger than 30 mm were included to minimise the influence of partial volume effects and to be able to reliably quantify uptake heterogeneity [17]. Imaging was in accordance with European Association of Nuclear Medicine (EANM)

guidelines for tumour PET imaging [18]. Patients fasted for at least 6 h before imaging and serum glucose levels were below 8 mmol/L. Directly after the start of the acquisition, a standardised infusion of 3.45 MBq of [18F]FDG per kilogram of body weight started. The final time frame (50–60 min p.i.) of the dynamic series was used in the current study. Voxel sizes were 2.56 × 2.56 × 3.38 and 1.59 × 1.59 × 2.03 mm3 for the Biograph Duo PET/CT and Biograph 40 mCT PET/CT, respectively. This study has been reviewed and approved by the Commission on Medical Research Involving Human Subjects Region Arnhem-Nijmegen, the Netherlands. All patients signed an informed consent form.

The PPGL patients who underwent a [18F]FDG PET/CT scan in the Radboud University Medical Center between 2011 and 2018 were retrospectively included. A selection of these patients has previously been described [15,19]. Static PET/CT images were acquired using the Biograph 40 mCT (Siemens Healthineers, Erlangen, Germany), in accordance with aforementioned EANM guidelines [18]. Patients fasted for at least 6 h and serum glucose levels were below 8 mmol/L. Image acquisition (3 or 4 min per bed position) started 60 (55–75) minutes after intravenous administration of [18F]FDG (dosage according to a non-linear dosage regimen based on body weight; details can be found in Supplementary File S1). The reconstructed voxel size was 3.18 × 3.18 × 3.00 mm3. This retrospective database study has been reviewed and approved by the Commission on Medical Research Involving Human Subjects Region Arnhem-Nijmegen, the Netherlands. Informed consent was waived due to the retrospective nature of the study. Patients that objected to the use of their anonymised data were excluded.

Additional details on patient preparation, data acquisition, image reconstruction, image processing and radiomic analysis can be found in Supplementary File S1: the IBSI reporting guidelines [14].


**5. Final considerations**

The health safety and environmental management system model shown in this chapter resulted from the application of a methodology called Ergonomic Design Thinking. It is based in two equivalent project management approaches easily found in design and ergonomics projects. Preliminary experiences had already shown that the Design Thinking approach proved itself effective not only for design project actions [20]. Using the general concept derived by Tim Brown's original approach, EDT adds the basic principle of ergonomics: No one knows better the work performed than workers themselves. And this is particularly true for ergonomics projects, since ergonomists and human factors professionals know exactly what is necessary to allow a perfect synchronicity between their logic and knowledge and the correct listening and interpretation of what some have been calling "the voices of the shop floor" [21].

Naturally, much more real-world experiences are needed to master the technique and to affirm without shadow of a doubt that this model is more effective and others. It is definitely better than the ones the authors have previously utilized. Hopefully, the publication of this text will provide an opportunity to disseminate the method and its tools to a more broad and global audience. To wrap up the discussion, a list of important issues that should be addressed before employing any participatory action method is presented down below.

One last issue is important to address. The best thing about both EDT methodology and our proposed HSE management system model is that they do not employ intricate, complex tools or instruments. It is completely "open source", so there is no proprietary software to buy, copyrighted materials or any similar exclusivity resource. Even some known and common tools used during ergonomic assessments, like the Corlett Diagram and the NIOSH equation for horizontal load handling, for example, are not under any trademark imposition. It was designed as a collective experiment in which all free men is more than welcome to join in.


*2.2. Predictive Response to Immunotherapy Score (PRIS)*

We tried to develop a specific disease index for predicting SLIT efficacy so that patients could choose whether to undergo SLIT based on their chance of success. We identified eight parameters that might contribute to SLIT responsiveness. Each parameter was assigned a score range, and three to five groups were established (Table 2). The parameters were chosen based on clinical practice, literature review, and previous work evaluating AIT responsiveness and possible predictive factors. The parameters included age, clinical features, disease onset, number of allergen sensitizations, presence of symptoms following exposure to the allergen(s) to which the patient is sensitized, specific IgE/total IgE ratio, IgE level for CRD, and allergen dominance (Table 2).

**Table 2.** Predictive Response to Immunotherapy Score (PRIS).

a Assessed with Skin Prick Test and/or specific IgE; ImmunoCAP 250, Phadia, Sweden. b The number of dominant allergens was assessed as described in the Methods Section 2.5 (Immunotherapy).

Total PRIS was calculated for each patient when he/she completed the diagnostic evaluation. Each patient was informed about his/her PRIS value and spontaneously decided to undergo SLIT treatment and be enrolled in this study. PRIS value could potentially range from 20 to 100. Therefore, PRIS stratification in quartiles would be as follows: first quartile = PRIS ≥ 80; second quartile = 80 > PRIS ≥ 60; third quartile = 60 > PRIS ≥ 40; fourth quartile = PRIS < 40. However, the PRIS values of patients enrolled in this study ranged from 41 to 93. Therefore, we had no patients in the fourth quartile.


**6. User selection algorithms**

**5.5. Condition number**

38 Contemporary Issues in Wireless Communications

of **w***i* is not matched to **h***i*

defined as [33]:

expression [34]:

where tr(**H**¯(*S*))= **H**(*S*) *F*

the following combinatorial problem:

The ZF-based beamforming methods are in general power inefficient since the spatial direction

power penalty and a strong SINR degradation at the receivers. In numerical analysis a metric to measure the invertibility of a matrix is given by the condition number. In MIMO system this metric is used to measure how the eigenvalues of a channel matrix spread out and to indicate multipath richness for a given channel realization. The less spread of the eigenvalues, the larger the achievable capacity in the high SNR regime. For the matrix **H**(*S*) the condition number is

> max min | ( ( ))| ( ( )) | ( ( ))| l

and *κ*(**H**(*S*)) is an indication of the multiplexing gain of a MIMO system. Another definition of the condition number is given by the product **A A**-1 for a given non-singular square

If the condition number is small, the matrix **H**(*S*) is said to be well-conditioned which implies that as *κ*(**H**(*S*))→1 the total achievable sum rate in the MISO systems under ZF-based beamforming can achieve a large portion of the sum rate of the inter-user interference free scenario. Problem (4) can be sub-optimally solved by a set of users with the minimum condition

:| | arg min ( ( )) *Nt*

Another important metric to estimate matrix condition is given by the Demmel condition number. For such metric several applications in MIMO systems have been proposed in recent years, e.g., link adaptation, coding, and beamforming [34]. The Demmel condition number can be seen as the ratio between the total energy of the channels of **H**(*S*) over the magnitude of the smallest eigenvalue of **H**¯(*S*) in the current channel realization and is given by the following

> min tr( ( )) ( ( )) ( ( )) *D*

channel. By using (30) the set of users that sub-optimally solves (4) is given by the solution of

S

2 , i.e., the Frobenius norm is related with the overall energy of the

l= S S

**H H**

ÌW = = S S

k

S S **H** (29)

S **H**

l= S S

k

**H**

matrix **A** [20] and (28) generalizes that metric for any matrix **H**(*S*) ∈ ℂ

k

k

number and such set is formally described as:

. Inverting a ill-conditioned channel matrix **H**(*S*) yields a significant

**H** (28)

**H** (30)

where |*S*|≤ *Nt*.

For MU-MIMO BC systems with ZF-based beamforming the optimum solution to the sum rate maximization problem implies the optimization over the power allocation and user selection as well as the correct dimension and orientation of the signal subspace used by each selected user [9]. Since this optimization is performed over several dimensions (time, space, signal space, power, users) the optimum solution for the MU-MIMO scenario has not been found. Even when the signal space spans over one dimension, i.e. single-antenna users, the optimum solution to the sum rate maximization problem (4) in MU-MISO BC systems is given by an exhaustive search whose computational complexity increases approximately with (*K Nt*). In order to find a fair trade-off between complexity and performance, a large number of works have addressed the sum rate maximization problem for ZFBF (e.g., [9], [21], [35] and ZFDP (e.g., [18], [19]) by designing low-complexity algorithms that find a sub-optimal yet acceptable solution to (4). The goal of such algorithms is to construct the solution set *S* of quasi-orthogonal users implementing different iterative approaches. Some works in the literature proposed greedy opportunistic algorithms that exploit the instantaneous channel information. In a greedy selection each new selected user finds a locally maximum for a given global objective function so that the final set of users attempt to converge to a close-to-optimal solution. Another approach to solve (4) is to use channel metrics and reformulate the original problem as an integer program. In this section we present the principles and characteristics of different user selection algorithms proposed in the literature. Our aim is to introduce generic structures of the user selection process and to illustrate under which conditions they can be used to maximize the sum rate and to reduce computational complexity for ZF-based beamforming.

#### **6.1. Metric-based selection**

The objective of the metric-based user selection is to find a set *S* of users with spatially quasiorthogonal channels such that the profit from beamforming is maximized. User selection algorithms can be design to relax the original optimization problem (4) by optimizing a particular channel metric. This kind of optimization may face two main problems: 1) the large search space *L* for the optimum solution, and 2) how to discriminate between metrics of sets with different cardinality without computing the objective function. Both problems are partially solved by imposing a constraint in the optimizations problem, i.e., |*S*|= *Nt* ∀*S* ⊆Ω in problem (4) or by operating in the high SNR regime. This yields a search space of size *L Nt* and metrics that can be easily compared since they are applied over user sets of the same size. In MU-MISO BC systems with ZF-based precoding and optimal power allocation, authors in [5] showed that there exist a critical SNR that depends on **H**(*S*) for which ∀ *P* ≤*P*0 the maximum sum rate is achieved by a subset *S* with cardinality |*S*|<*rank*(**H**(*S*)). For an operation point ∀ *P* >*P*0 the system achieves full multiplexing diversity, i.e., |*S*|=*rank*(**H**(*S*)). The metricbased user selection algorithms solve (4) in two phases. In the first phase *S* is selected in order to optimize a given metric and in the second phase the sum rate is evaluated for the previously defined set. This means that the user selection and the resource allocation problems are decoupled in this approach. The cardinality of *S* is fixed in the first phase and it may be modified during the second phase when the sum rate is evaluated. If power allocation is performed based on water-filling, this might yield a zero power allocation (no data transmis‐ sion) for some users due to the instantaneous value of *P*0.

In the literature of user selection several algorithms solve problem (4) optimizing the NSP by iteratively computing the sum of the effective channel gains each iteration. The optimum set that maximizes the NSP is given by the solution of (9) which is a combinatorial problem that requires the evaluation of *L Nt* matrix product and inversion operations. Therefore, the set *S* is found in a greedy fashion by selecting the user that locally maximizes the total sum of effective channel gains which roughly requires *L K* =∑*k* =1 *Nt*-1 (*K* - *k*) evaluations of the metric *f* (∙ ). Given *S*(*n* - 1) in the *n*th iteration, the optimum new user *i* ∈Ω achieves the largest effective channel gain when its channel is projected onto the orthogonal subspace spanned by the previously selected users =*Sp*(**H**(*S*(*n* - 1))). In [10], [21] the authors evaluate iteratively at the *n*th iteration the intersection of the null spaces of the previously selected users *S*(*n* - 1) with each new candidate user by means of GSO. Using this technique, it is not necessary to extract the basis of the subspace spanned by the channel matrix (SVD) or to compute the orthogonal projector matrix **Q***i* at each iteration [19]. Other user selection algorithms based on the NSP further reduce the required computational complexity by approximating any of the its alternative forms described in Section 5.1.1. The algorithm proposed in [26] approximates (16)

by using only the correlation coefficient between the users in *S*(*n*). The authors in [36] select users based on metric (17) where each user computes its own projector matrix and the algorithm approximates the NSP. The optimization of an approximation of the NSP yields a suboptimal set of users and performance degradation regarding to (8) yet some gains in terms of complexity are attained. The general structure of this kind of algorithms is presented in Alg. 1 and *f* (∙ ) is given by any metric defined in Section 5. Notice that the first selected user is the one with the strongest channel which not necessarily belongs to the optimum set. However, this criterion simplifies the initialization of *S* and does not yield a significant performance degradation, specially in the large *K* regime [21].

Another relevant feature of this generic structure is given by the adjustment of Ω in each iteration. This is relevant in the large *K* regime where the number of operations required to find the next user is roughly *K* which can be computationally demanding. In order to reduce the total number of metric evaluations, an optimization over Ω can be performed. For instance, by performing an optimization similar to (25) per iteration for a given *ε*-orthogonality target whose optimal value depends on *K* and *Nt* [37]. The principle behind this optimization is that the candidate users must satisfy the hyperslab condition Ω(n)={**h***i* ∀ *i* ∈ Ω(n-1): cos *θ***h***i* **g** ≤*ε*} for a given vector **g** [21]. The metric-based user selection does not guarantee full multiplexing gain or sum rate maximization and authors in [10], [25] have proposed a second optimization once that *S* have been found. This is referred in Alg. 1 as *Removal Optimization* and its objective is twofold: to discard inactive users that do not receive data and to maximize the achievable sum rate by optimizing the beamforming weights and powers of the active users.

**Alg 1.** Generic structure of the Metric-Based User Selection


*3.1. General*

All chemicals were purchased from commercial suppliers and used without further purification unless otherwise specified. Anhydrous THF was purchased from Acros, anhydrous Ba(ClO4)2 was purchased from Alfa Aesar, and deuterated solvents were purchased from deutero GmbH. Compounds **8b** [57], **9c** [63], **11a**, **12a** [59], **11b**, **12b** [64], **14a**, **15a**, **16a**, and **20** [26] were prepared according to the literature. NMR spectra of all compounds were recorded on an Agilent DD2-400 MHz NMR or an Agilent DD2-600 MHz NMR spectrometer with ProbeOne. Chemical shifts of the 1H, 19F, and 13C spectra were reported in parts per million (ppm) using TMS as an internal standard for 1H/13C and CFCl3 for 19F spectra. Mass spectrometric (MS) data were obtained on a Xevo TQ-S mass spectrometer (Waters) by electron spray ionization (ESI). The melting points were determined on a Galen III melting point apparatus (Cambridge Instruments & Leica) and are uncorrected. TLC detections were performed using Silica Gel 60 F254 sheets (Merck, Darmstadt, Germany). TLCs were developed by visualization under UV light (λ = 254 nm). Chromatographic separations were accomplished by using an automated silica gel column chromatography system Biotage Isolera Four and appropriate Biotage KP-SIL SNAP columns. UV/Vis- measurements were realized at a Specord 50 by Analytik Jena. The calculation of the stability constants was accomplished using HypSpec 1.1.18.