The Topology of Wireless Communication Supported in part by a grant from the Israel Science Foundation.

In this paper we study the topological properties of wireless communication maps and their usability in algorithmic design. We consider the SINR model, which compares the received power of a signal at a receiver against the sum of strengths of other interfering signals plus background noise. To describe the behavior of a multi-station network, we use the convenient representation of a reception map. In the SINR model, the resulting SINR diagram partitions the plane into reception zones, one per station, and the complementary region of the plane where no station can be heard. SINR diagrams have been studied in [3] for the specific case where all stations use the same power. It is shown that the reception zones are convex (hence connected) and fat, and this is used to devise an efficient algorithm for the fundamental problem of point location. Here we consider the more general (and common) case where transmission energies are arbitrary (or non-uniform). Under that setting, the reception zones are not necessarily convex or even connected. This poses the algorithmic challenge of designing efficient point location techniques for the non-uniform setting, as well as the theoretical challenge of understanding the geometry of SINR diagrams (e.g., the maximal number of connected components they might have). We achieve several results in both directions. We establish a form of weaker convexity in the case where stations are aligned on a line and use this to derive a tight bound on the number of connected components in this case. In addition, one of our key results concerns the behavior of a -dimensional map, i.e., a map in one dimension higher than the dimension in which stations are embedded. Specifically, although the -dimensional map might be highly fractured, drawing the map in one dimension higher “heals” the zones, which become connected (in fact hyperbolically connected). In addition, as a step toward establishing a weaker form of convexity for the -dimensional map, we study the interference function and show that it satisfies the maximum principle. This is done through an analysis technique based on looking at the behavior of systems composed on lines of densely placed weak stations, as the number of stations tends to infinity, keeping their total transmission energy fixed. Finally, we turn to consider algorithmic applications, and propose a new variant of approximate point location.

Keywords:

Wireless communication, signal to interference plus noise ratio (SINR), point location, convexity

1 Introduction

Background and motivation:

The use of wireless technology in communication networks is rapidly growing. This trend imposes increasingly heavy loads on the resources required by wireless networks. One of the main resources required for such communication is radio spectrum, which is limited by nature. Hence careful design of all aspects of the network is crucial to efficient utilization of its resources. Good planning of radio communication networks must take advantage of all its features, including both physical properties of the channels and structural properties of the entire network. While the physical properties of channels have been thoroughly studied, see [8, 21]. Relatively little is known about the topology and geometry of the wireless network structure and their influence on performance issues.

There is a wide range of challenges in wireless communication for which better organization of the communication network may become useful. Specifically, understanding the topology of the underlying communication network may lead to more sophisticated algorithms for problems such as scheduling, topology control and connectivity. We study wireless communication in free space; this is simpler than the irregular environment of radio channels in a general setting, which involves reflection and shadowing. We use the Signal to Interference-plus-Noise Ratio (SINR) model which is widely used by the Electrical Engineering community, and is recently being explored by Computer Scientists as well. Let

In this model, a receiver at point successfully receives a message from the sender if and only if , where is the environmental noise, the constant denotes the minimum SINR required for a message to be successfully received, is the path-loss parameter and is the set of concurrently transmitting stations using power assignment . Within this context, we focus on one specific algorithmic challenge, namely, the point location problem, defined as follows. Given a query point , it is required to identify which of the transmitting stations is heard at , if any, under interference from all other transmitting stations and background noise . Obviously, one can directly compute for every in time and answer the above question accordingly. Yet, this computation may be too expensive, if the query is asked for many different points . Avin et al. [3] initiated the study of the topology and geometry of wireless communication in the SINR model, and its application to the point location problem, in the relatively simple setting of uniform powers, namely, under the assumption that all stations transmit with the same power level. They show that in this setting, the SINR diagram assumes a particularly convenient form: the reception zones of all senders are convex and “fat”. They later exploit these properties to devise an efficient data structure for point location queries, resulting in a logarithmic query time complexity.

In actual wireless communication systems, however, most wireless communication devices can modify their transmission power. Moreover, it has been demonstrated convincingly that allowing transmitters to use different power levels increases the efficiency of various communication patterns in terms of resource utilization (particularly, energy consumption and communication time). Hence it is important to develop both a deep understanding of the underlying structural properties and suitable algorithmic techniques for handling various communication-related problems in non-uniform wireless networks as well. In particular, it may be useful to develop algorithms for solving the problem of point location in such networks. Unfortunately, it turns out that once we turn to the more general case of non-uniform wireless networks, the picture becomes more involved, and the topological features of the SINR diagram are more complicated than in the uniform case. In particular, simple examples (with as few as five stations, as illustrated later on) show that the reception zone of a station is not necessarily connected, and therefore is not convex. Other “nice” features of the problem in the uniform setting, such as fatness, are no longer satisfied as well. Subsequently, algorithmic design problems become more difficult. In particular, the point location problem becomes harder, and cannot be solved directly via the techniques developed in [3] for the uniform case.

In this paper we aim to improve our understanding of the topological and geometric structure of the reception zones of SINR diagrams in the general (non-uniform) case. The difficulty in point location with variable power follows from several independent sources. First, one must overcome the fact that the number of connected cells is not always known (and there are generally several connected cells). A second problem is that the shape of each connected cell is no longer as simple as in the uniform case. Yet another problem is the possibility of singularity points on the boundaries of the reception zones. (Typically, those problems become harder in higher dimensions, but as seen later, this is not always the case for wireless networks.)

Nevertheless, we manage to establish several properties of SINR diagrams in non-uniform networks that are slightly weaker than convexity, but are still useful for tackling our algorithmic problems, such as satisfying the maximum principle of the interference function and enjoying hyperbolic convexity. To illustrate these properties, let us take a look at the simplest example where a problem already occurs. When we look at two stations in one dimension, the reception zones are not connected. Surprisingly, when we look at the same example in two dimensions (instead of one), the reception zones of both stations become connected. As shown later on, this is no coincidence. Moreover, when we examine closely the two-dimensional case, we see that the reception zones are no longer convex but actually hyperbolic convex (as opposed to non convex in the one dimensional case). We use this strategy of adding a dimension to the original problem and moving from Euclidean geometry to hyperbolic geometry to solve the point location problem.

Contributions:

In this paper we aimed toward gaining better understanding of SINR maps with non-uniform power. Better characterization of reception map has a theoretical as well as practical motivation. The starting point of our work is the following observation: in non-uniform setting, reception zones are neither convex nor fat. In addition, they are not connected. The loss of these “niceness” properties, previously established for the uniform power setting [3], appears even for the presumably simple case where all stations are aligned on a line.

This raises several immediate questions. The first is a simple “counting” question that has strong implications on our algorithmic question: What is the maximal number of reception cells that may occur in an SINR diagram of a wireless network on transmitters. The second question has a broader scope: Are there any “niceness” properties that can be established in non-uniform setting. Specifically, we aim toward finding other (weaker but still useful) forms of convexity that are satisfied by cells in non-uniform reception maps. Apart from their theoretical interest, these questions are also of considerable practical significance, as obviously, having reception zones with some form of convexity might ease the development of protocols for various design and communication tasks.

We establish two weaker forms of convexity and show their theoretical as well as algorithmic implications. Starting with the one-dimensional case, where stations are aligned on a line, we show that although the zones are not convex, they are convex in a region that is free from stations. We then use this “No-Free-Hole” (NFH) property to establish the fact that in one dimension, the number of reception cells generated by stations is bounded above by (and this can be realized). For the general setting where stations are embedded in , the problem of bounding the number of connected cells seems to be harder, even for . We are able to show that the number of reception cells is no more than and provide examples with reception cells for a single station. Do -dimensional zones enjoy the NFH property? Although this remains an open question, we make two major advances in this context.

First, we consider the -dimensional SINR map of a wireless network whose stations are embedded in -dimensional space, and establish a much stronger property. It turns out, that while in the -dimensional space the network’s SINR map might be highly fractured, going one dimension higher miraculously “heals” the reception zones, which become connected (in fact, hyperbolically connected or hyperbolically convex). This may have practical ramifications. For instance, considering stations located in the 2-dimensional plane, one realizes that their reception zones in 3-dimensional space are connected, which aids in answering point location queries in this realistic setting.

Turning back to the -dimensional map, we consider a well known property of harmonic functions, namely, the maximum principle. Generally speaking, the maximum principle refers to the case where the maximum value of the function in a given domain, is attained at the circumference of that domain. Does the SINR function follow the maximum principle? This is yet another open question. If so, NFH property is followed. As a step toward achieving this goal, we then examine the properties of the interference function (appearing in the denominator of the SINR function), and establish the fact that this function satisfies the maximum principle. This is done through an analysis technique based on looking at the behavior of systems composed on lines of densely placed weak stations, as the number of stations tends to infinity, keeping their total transmission energy fixed.

Finally, we consider the point location task, defined as follows. Given a set of broadcasting stations and a point , we are interested in knowing whether the transmission of station is correctly received at . We present a construction scheme of a data structure (per station) that maintains a partition of the plane into three zones: a zone of all points that correctly receive the transmissions of , i.e., points with ; a zone where the transmission of cannot be correctly received, i.e., points with ; and a zone of uncertainty corresponding to points that might receive the transmission in a somewhat lower quality, i.e., points with , where is predefined performance parameter. Using this data structure, a point location query can be answered in logarithmic time.

Related work:

Our starting point is the work of Avin at. el. [3], where it is proven that if all transmitters use the same power then the reception zones are convex and fat. Several papers have shown that the capacity of wireless networks increases when transmitters can adapt their transmission power. In their seminal paper [1], Gupta and Kumar analyzed the capacity of wireless networks in the physical and protocol models. Moscibroda [13] analyzed the worst-case capacity of wireless networks, without any assumption on the deployment of nodes in the plane, as opposed to almost all previous works on this problem. Non-uniform power assignments can clearly outperform a uniform assignment [16, 15] and increase the capacity of a network. Therefore the majority of the literature on capacity and scheduling addresses non-uniform power. In the engineering community, the physical interference (SINR) model has been scrutinized for almost four decades. Assuming that the power of all transmitters is uniform, we know from [3] that the reception zones are convex and fat. Therefore the singularity points of a zone can be easily handled. Yet when power is not uniform, handling the singularity points becomes a major challenge. We remark that recently, Gabrielov, Novikov, and Shapiro have shown that the number of singular points of functions similar to the interference function is finite, see [7]. Maxwell conjectured that the number of singularity points in the interference function is bound by where is the number of transmitters; see [11] for more details. For illustration see Figure 1a.

Another challenge that one has to deal with in non-uniform networks is the possible existence of regions with very small gradient in the function, as exemplified in Figure 1b, which reflects the fact that the area containing all points such that cannot be bounded even for small .

Figure 1: Topographic view of (with no noise). Heights indicate SINR thresholds. (a) SINR map of 4-station non-uniform power network: singular points ( and ) and contour lines of . (b) Low gradient regions in SINR map in 2-station uniform power network. is unbounded when but finite for , illustrating the impossibility of getting uniformly bounds of the area between two SINR curves corresponding to two different threshold levels.

It is hoped that a better understanding of the topology of the SINR diagram will improve our understanding of the joint problem of scheduling and power control. The complexity of this problem in the physical model, taking into account the geometry of the problem, is unknown. Nevertheless, many algorithms and heuristics have been suggested, e.g., [5, 6, 10, 15, 22, 24]. See [14] for a more detailed discussion of these approaches. Recently, Kesselheim [9] has shown how to achieve a constant approximation for the capacity problem with power control, for doubling metric spaces. His algorithm yields approximation for general metrics. Halldórsson and Mitra [17] show tight characterizations of capacity maximization under power control, using oblivious power assignments in general metrics.

2 Preliminaries

2.1 Geometric notions

Throughout, we consider the -dimensional Euclidean space (for ). The distance between points and point is denoted by . A ball of radius centered at point is the set of all points at distance at most from , denoted by . Unless stated otherwise, we assume the 2-dimensional Euclidean plane, and omit . The basic notions of open, closed, bounded, compact and connected sets of points are defined in the standard manner. A point set is said to be open if all points are internal points, and closed if its complement is open. If there exists some real such that for every two points , then is said to be bounded. A compact set is a set that is both closed and bounded. The closure of , denoted , is the smallest closed set containing . The boundary of a point set , denoted by , is the intersection of the closure of and the closure of its complement, i.e., . Let denote the length of . A connected set is a point set that cannot be partitioned to two non-empty subsets such that each of the subsets has no point in common with the closure of the other (i.e., is connected if for every such that and , either or .). A maximal connected subset is a connected point set such that is no longer connected for every .

We use the term zone to describe a point set with some “niceness” properties. Unless stated otherwise, a zone refers to the union of an open connected set and some subset of its boundary. It may also refer to a single point or to the finite union of zones.

Let and let . Then is the characteristic polynomial of a zone if .

Denote the area of a bounded zone (assuming that it is well-defined) by . For a non-empty bounded zone and an internal , denote the maximal and minimal radii of w.r.t. by

and define the fatness parameter of with respect to to be . The zone is said to be fat with respect to if is bounded by some constant.

2.2 Wireless networks

We consider a wireless network , where is the dimension, is a set of transmitting radio stations embedded in the -dimensional space, is an assignment of a positive real transmitting power to each station , is the background noise, is a constant that serves as the reception threshold (to be explained soon), and is the path-loss parameter. We sometimes wish to consider a network obtained from by modifying one of the parameters while keeping all other parameters unchanged. To this end we employ the following notation. Let be a network identical to except its dimension is . and are defined in the same manner. For notational simplicity, also refers to the point in the -dimensional space where the station resides, and moreover, when , the point in the Euclidean plane is denoted . The network is assumed to contain at least two stations, i.e., . The energy of station at point is defined to be . The energy of a set of stations at a point is defined to be . Fix some station and consider some point . We define the interference of to be the energy of at , denoted . The interference of a set of stations at a point is defined to be . The signal to interference & noise ratio (SINR) of at point is defined as

(1)

Observe that is always positive since the transmitting powers and the distances of the stations from are always positive and the background noise is non-negative.

In certain contexts, it is convenient to consider the reciprocal of the SINR function, namely, defined as

(2)

When the network is clear from the context, we may omit it and write simply , , and .

The fundamental rule of the SINR model is that the transmission of station is received correctly at point if and only if its SINR at is not smaller than the reception threshold of the network, i.e., . If this is the case, then we say that is heard at . We refer to the set of points that hear station as the reception zone of , defined as

This definition is necessary since is undefined at points in and in particular at itself. In the same manner we refer to the set of points that hear no station (due to the background noise and interference) defined as

An SINR diagram is a “reception map” characterizing the reception zones of the stations. This map partitions the plane into zones; a zone for each station , , and a zone where no successful reception exists to any of the stations.

It is important to note that a reception zone, , is not necessarily connected. A maximal connected component within a zone is referred to as a cell. Let be the cell in .

Hereafter, the set of points where the transmissions of a given station are successfully received is referred to as its reception zone, and a cell is a maximal connected set or component in a given reception zone. Hence the reception zone is a set of cells, given by , where is the number of cells in . Analogously, is composed of connected cells, . Overall, the topology of a wireless network is arranged in three levels: The reception map is at the top of the hierarchy. It is composed of reception zones, , . Each zone is composed of reception cells. For a pictorial description see Figure 2.

Figure 2: Schematic representation of a reception map, consisting of reception zones, each composed of several connected components referred to as cells.

The following definition is useful in our later arguments. Let 111When is clear from context we may omit it and simply write and . When refereing to reception zones or we may omit and simply write and . , be the characteristic polynomial of given by

(3)

Then iff .

Avin et al. [3] discuss the relationships between SINR diagram on a set of stations with uniform powers and the corresponding Voronoi diagram on . Specifically, it is shown that the reception zones are strictly contained in the corresponding Voronoi cells . SINR diagrams with non-uniform powers are related to the weighted Voronoi diagram of the stations instead of to the Voronoi diagram.

In the weighted version of Voronoi diagram [2], we consider a weighted system , where represents a set of points in d-dimensional Euclidean space and is an assignment of weights to each point . The weighted voronoi diagram of partitions the planes into zones, where

denotes the zones (of influence) of a point in , for every . The weighted Voronoi map denoted by , is composed of cells, edges and vertices. A cell corresponds to a maximal connected component in , . An edge is the relative interior of the intersection of two closed cells. Finally, a vertex is an endpoint of an edge. In the unweighted Voronoi diagram each zone corresponds to one connected cell. On the contrary, a weighted Voronoi map is composed of cells as was shown at [2]. For a given wireless network , we define the corresponding weighted Voronoi system in the following manner. The set of points corresponds to positions and , for every . In what follows we formally express the relation between and .

Lemma 2.1

, for every and .

Proof: Let . Let be such that . We prove that . Since , by (1)

where , and hence

The choice of implies that and the claim holds.    

Consider the way the “reception map” of a given network changes as goes to infinity while the other parameters (e.g., the set of stations, , the noise etc.) are fixed. The map converges to is denoted by

Lemma 2.2

, for every .

Proof: By Lemma 2.1, . It follows that for . But is simply . This can also be seen by considering the SINR function: as gets larger, the power of the station becomes negligible compared to distance between the station and the point . In other words, it gets closer to the uniform Voronoi diagram.    

We conclude this section by stating an important technical lemma from [3] that will be useful in our later arguments.

Lemma 2.3

[3] Let be a mapping consisting of rotation, translation, and scaling by a factor of . Consider some network and let , where . Then for every station and for all points , we have .

3 SINR diagrams of nonuniform networks: Basics

3.1 Disconnectivity of nonuniform power SINR maps

The SINR diagram is a central concept to this paper. We are interested in gaining some basic understanding of its topology. Specifically, we aim toward finding some “niceness” properties of reception zones and studying their usability in algorithmic applications. In previous work [3], Avin et al. consider the simplified case where all stations transmit with the same power. For a uniform power network, the reception zone of each station is known to be connected and to exhibit some desirable properties such as fatness and convexity. In the current work we study the general (and common) case of non-uniform transmission powers.

3.2 2-Station networks

This section provides a detailed characterization of the possible SINR diagrams in a system with two stations. Let be a network consisting of two stations embedded on the -axis with transmitting powers respectively, with a threshold parameter and path-loss parameter . For clarity of presentation, we first assume the simplified case where there is no background noise (i.e., ). This is represented by the network . The case of , corresponding to , is discussed at the end of this section.

Assume without loss of generality that is located at the origin and is located at , where . Recall that for a 2-station network with no background noise, the SINR formula takes the form

Assuming that , the formula takes the form

(4)

As may be expected, the parameter controlling the behavior of the system is the ratio . When , define and .

Lemma 3.1

The zone assumes one of the following three possible configurations.

(C1) If , then is a -dimensional disk, .

(C2) If , then is a complement of a -dimensional disk, .

(C3) If , then is a halfplane, .

See Figure 3 for illustration assuming .

Figure 3: Possible configurations of a 2-stations network, where is located at the origin and is located at (0,2), with the following energies and path loss parameter: (a) , and : and assumes configuration (C1), (b) , and : and assumes configuration (C2), (c) , and : and assumes configuration (C3).

Proof: Eq. (4) implies that

Letting , the condition on can be rewritten as

(5)

We begin with Claim (C3). If , then condition (5) can be written as , implying and Claim (C3) follows.

Next, we prove (C1) and (C2). Assume that . We first rewrite condition (5) in a circle form, by rearranging it as , or

We consider two cases.

Case 1: If , then

Hence the zone is composed of one cell defined by a circle centered at of radius . Claim (C1) follows.

Case 2: If , then

Hence the zone is composed of one cell defined by the complement of a circle centered at of radius , establishing (C2).    

Finally, we turn to the case where . It can be shown that assumes three configurations as well (see Lemma 3.1). The key difference between and is that the presence of noise induces only bounded zones. Consequently, (C2) and (C3) where is unbounded, are no longer feasible. These configurations are replaced by an equivalent ones where attains a bounded shape (i.e., a large enclosing disc for (C2) and an elliptic shape for (C3)).

By considering a 2-station network with non-uniform power it is apparent that the reception zones of non-uniform power networks are not convex, however connectivity is maintained. Unfortunately, although this is true for 2-stations systems, it does not hold in general. Connectivity might be broken even in networks with small number of participants, as illustrated by the 5-station system of Figure 4, where the reception zone of is composed of two connected cells. This raises the immediate question of bounding the maximal number of cells a given SINR diagram might have.

Figure 4: An instances of 5-station system with two connected cells of .

A seemingly promising approach to studying this question is considering the corresponding weighted Voronoi diagrams. Recall that by Lemma 2.1, . It therefore seems plausible that the number of weighted Voronoi cells (bounded by [2]) might upper bound the number of connected cells in the corresponding SINR diagram. Unfortunately, this does not hold in general, since it might be the case that a single weighted Voronoi cell corresponds to several connected SINR cells. This phenomenon is formally stated in the following lemma.

Lemma 3.2

There exists a wireless network such that a given cell of the corresponding weighted Voronoi diagram contains more than one cell of .

Proof: Let be a wireless network, where and is not connected, i.e., is composed of more than one cell. Let
, where , , where and , for every and . To avoid cumbersome notation, let and let be the corresponding weighted Voronoi diagram of . In what follows, we show that for sufficiently large , the network satisfies the conditions of the desired network . Specifically, it is easy to verify that for large enough , the weighted zone is connected. We next show that contains more than one connected cell of . First, observe that , and therefore is not connected as well. This follows by noting that and . Next, by the connectivity of and Lemma 2.1, it follows that . Since is not connected, the lemma follows.    

This lemma illustrates that the structural complexity of the SINR diagram cannot be fully captured by the weighted Voronoi diagram. Specifically, it implies that the number of connected cells in a non-uniform SINR diagram cannot be bounded by the number of weighted Voronoi cells, hence a different approach is needed. This challenge is extensively discussed in this paper, where we obtain bounds and provide extreme constructions with respect to the the number of connected cells for a given station. We conjecture that the obtained upper bounds are not tight, and our constructions are close to the limit. Yet so far, no formal proof is available.

4 The No-free-hole property

Convexity was shown in [3] to play a significant role in showing that the reception zones of uniform SINR diagrams are connected. Unfortunately, as discussed in the previous section, reception zones of non-uniform SINR diagrams might be non-convex, even when the network is composed of only two stations. Is there any form of weaker convexity that can still be established? Are there excluded configurations in non-uniform diagrams? To address these questions, let’s examine several examples of non-convex shapes illustrated in Figure 5. Non-convex shapes can be classified into two types: (a) shapes with non-convex contour (Fig. 5a), (b) shapes with a convex contour but with a hole. Type (b) is further classified into two types; (b1) the hole contains at least one interfering station (Fig. 5b1) and (b2) the hole is free of stations (Fig. 5b2). Interestingly, though type (a) and (b1) are fairly common feasible configurations of cells in non-uniform SINR diagrams, all our attempts to generate a configuration of type (b2) have failed so far. We conjecture that type (b2) is an excluded configuration of cells in non-uniform SINR diagrams. In other words, we believe that every hole in a reception cell must contain at least one interfering station. This property (namely, that type (b2) is an excluded state) is hereafter termed “no-free-hole” or NFH for short, and is defined as follows. A collection of closed shapes in obeys the NFH property with respect to a set of stations if for every that is free of stations, if all its border points are reception points of , then all points of are reception points as well. Formally, if and , then also . This property turns out to be relevant for bounding the number of connected cells in . The next subsection is dedicated to proving the conjecture in the 1-dimensional case.

Figure 5: Classification of non-convex cells. (a) Non-convex contour; (b1) Convex contour with a hole that occupied by some interfering station; (b2) Convex contour with a hole that is free of stations.

4.1 The one-dimensional case

The purpose of this subsection is to show that the NFH property holds in the one dimensional case. This fact is later used in Subsection 5.1 to bound the number of connected cells in a one dimensional map. The analysis is organized as follows. In Subsection 4.1.1, we introduce the framework and establish some basic properties. In Subsection 4.1.2, we establish the NFH property for 3-station network and no background noise. In Subsection 4.1.3, the NFH property for -station network either with or without background noise is established.

4.1.1 Framework

We consider a network of the form . Let be the position of . In the following we may abuse notion by confusing between a station and its geometric location. Without loss of generality, we focus on . By adopting Eq. (1) to the current setting we get that

(6)

and is heard at iff . To establish NFH, we consider a segment such that where and , and show that under these conditions . Continuity follows simply by the fact that no station is located on . We need to show that for every .

First, we provide some notation useful for this section. Let and let . Then using Equation (6), the fundamental rule of the model of can be expressed by the 1-variate polynomial

(7)

such that is heard iff . The first derivative of is given by

(8)

In the same manner, the second derivative of is given by

(9)

Without loss of generality, let . Our analysis relies on the observation that a network may assume one of the following three configurations:
(C1) , for every .
(C2) .
(C3) for some and .
The NFH property follows easily for networks in configuration (C1) and (C2). In case the network assumes configuration (C3), the proof is more involved. We begin with configurations (C1) and (C2).

Claim 4.1

If the network assumes configuration (C1) or (C2), then has no local maximum in the interval .

Proof: Without loss of generality, let . Suppose first that is in configuration (C1). Since for every , it follows by Eq. (9) that for every and specifically for every , which establishes the claim. Next, assume that is in configuration (C2). In this case, the set of stations can be partitioned into two sets, namely, and . Since for every and , we have that for every and . In addition, one can verify that for every and . In summary, we get that for every , and the claim follows.    

Corollary 4.2

If assumes configuration (C1) or (C2), then .

Proof: By Eq. (7) it holds that . By Claim 4.1, we have that has no local maximum in the interval , hence for every we have , implying , and the claim follows.    

4.1.2 3-stations (no noise)

We now establish NFH for the special case of a non-uniform power network with three stations and no background noise, . Assume without loss of generality (by Lemma 2.3) that , and . Let , and .

Lemma 4.3

Let be a segment such that and . Then .

Proof: Due to Corollary 4.2, it remains to consider the case where assumes configuration (C3), i.e., and , see Figure 6. In this context, it is convenient to consider for the following characterizing polynomial

It follows that if and only if . Since , the polynomial has at most 4 roots. The claim follows by applying a counting argument on the number of roots of . Clearly, a root of is consumed whenever the polynomial changes its sign. Since , and , it follows that has roots in each of the intervals , and . Consequently, we are left with one undecided root.

We claim that every point is a reception point of . This is proven by contradiction. Assume to the contrary, that there is a non-reception point such that or . This would imply the existence of two roots that correspond to the intervals