Computing Radio Paths in an Urban EnvironmentBoaz Ben-Moshe∗, Nir Shvalb∗, Moti Shani∗, Paz Carmi† and Elhanan Shifman∗

∗Ariel University Center, Israel †Ben-Gurion University, Israel

Abstract—This work1 presents a new radio paths computationframe-work especially designed for complex indoor RF fieldprediction. We propose a new algorithm utilizing a geometricvisibility graph of a building to traverse all possible bounded ra-dio paths. These paths are needed to compute the signal strengthas received at given receiver location. We have implemented thesuggested algorithm and performed a set of experiments testingthe radio paths over complex buildings. The main conclusionis that the new algorithm is both (i) Accurate: predicts thesignal strength within complex buildings. (ii) Runtime efficient:can compute all relevant radio paths even on relatively complexstructures comprised of thousands of walls in a matter of seconds.

I. INTRODUCTION

In order to meet wireless communication demands, anincreasing number of antennas are being deployed in urbanareas. However, the cost of their installation accounts fora significant portion of wireless communication hardwaresystems. For this reason, the communication research com-munity is concerned with predicting the required throughputand quality of signals. Furthermore, WiMAX (802.16) [1],MIMO, (802.11.n) [2] and Beam Forming technologies areall characterized by greater bandwidth and range than thecommonly used WiFi, emphasizing the importance of theproblem hereby discussed. In light of the above, our algorithmis valuable in aspects of computation time reduction ([3]).

Most simulation systems are based on empirical propagationcharacteristics of the environment. Such designs usually im-plement statistical models for the geometric environment thatdescribes the expected performance of a propagation path (see[4] for an extensive survey on statistical propagation models).To enable an efficient interpolation at the field-estimationstage, a sophisticated database is required. The propagationprediction model, also named Radio Frequency (RF) model,provides two types of parameters: large-scale path loss andsmall-scale fading statistics. The former is used to determineand optimize the coverage of a base-station placement, whilethe latter provides tools to improve the receiver design. An-other approach is a ray-tracing-based radio wave propagationprediction model (e.g., [5]). Such systems consider each raypath as a sample, while the union of all ray paths betweenthe transmitter and receivers builds up to form the samplespace. Since each sample stands for different path components(like reflections and diffractions), each sample illuminatesdifferent sets of receivers, and contributes differently to thefinal prediction results. Thus, for an RF prediction system,which uses such a model, the termination conditions shouldcarefully be considered.

1Supported by the Israeli Short Range Consortium (ISRC).

The issue of approximating the strength of a signal receivedfrom a given transmitter in urban surroundings has beenextensively studied (e.g., [4], [6], [7], [8], [3]). Still, sincesuch tools should utilize RF-propagation model computation,simulating such complex networks requires significant com-puting power and time (e.g., [9]). Nevertheless, if we notethat one is usually interested in estimating the radio field ina sub-space rather than in the entire space (one would not beinterested in calculating the radio field near the ceiling, forexample), we may focus only on computing the radio fieldwithin a predetermined sub-space. This, of course, furthershortens computational time. Thus we may think of a pointto point ray tracing approach as a pure geometric spatialbilliard problem: Given a building structure, a billiard ball’sinitial position and a target hole’s fixed position, compute alldirections at which one can target the ball towards the hole,transversing a predetermined upper-bound path-length. Thisis actually a sub-problem of the one we consider here, sincewhen addressing a radio path one should consider not onlyreflections but penetration phenomenon as well (future workwill consider diffraction and absorption phenomena as well).

In this paper we suggest a new frame-work especiallydesigned for approximating the signal strength in complexurban Environment. The new In Door Radio Paths algorithm(namely IDRP) computes the set of significant radio pathsbetween the transmitter and the receiver using pure geometricproperties of the building and then this set is used by asophisticated RF model in order to allow accurate and efficientthe combine signal strength prediction.

The paper is organized as follows: In section II we presentan overview of the basic RF models applied for predicting asignal strength at a given point. In section III we introduce thegeometric properties of a radio path between a transmitter anda receiver, with respect to the walls between them. In sectionIV we introduce our IDRP algorithm for predicting all radiopaths and its asymptotic runtime. Finally in sections V and VIwe present an implementation of the suggested algorithm anddraw some conclusions.

II. RF PROPAGATION MODEL OUTLINE

Henceforth we will refer to our computational playground asthe building (denoted by B), that is to say, the relevant portionof space that should be taken into consideration regarding theRF signal power. Explicitly, a building is the set of all itsbuilding elements.

Definition 1: A Building Element is either a rectangularor a triangular shaped planar element that is a part of a wall,floor or a ceiling, or the absence of such (like a window or adoor opening).

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Note that due to the point to point nature of our approachthe panel area of a building element may not be considered.

Definition 2: The Visibility Graph of a Building (V G(B))is the set of vertices V , together with the set of arcs E,where V is associated with the set of geometric buildingelements, and E is associated with the set of all possible directpaths between vertices. In other words, arcs correspond to theexistence of line(s) of sight (LOS) between building elements.When a signal travels through a building it bounces off(reflectance) and penetrates (transmittance) the building el-ements it crosses on its path. We will refer to both reflectanceand transference as bounces. A signal transmitted througha building element simply transverses its original incidencedirection vector k after penetration takes place. The signalreflection geometry, on the other hand, resembles a mirror-like light ray incidence geometry; that is: (1) The reflectedray and the normal to the reflection surface �n at the point ofthe incidence lie in the same plane, and (2) The angle betweenthe incident ray and the normal is equal to the angle betweenthe reflected ray and the same normal.

Obviously the signal power does not remain constantthroughout. Instead, just a portion of a signal bounces backwhile a complementary portion (these should sum to the in-cidence signal power) transmits through the building element.In order to explicitly formulate our model, we introduce thefollowing physical concept:

Definition 3: Polarization polarization vector �E0 is theproperty of a propagating signal that encloses the amplitude,phase and orientation of its oscillations. �E0 is situated on theplane perpendicular to the signal propagation direction k.Thus, for a transmitted signal one may write:

�Et

‖ �E0‖= tTE · cosβ · (k× n)+ tTM ·sinβ · ((k× n)× k). (1)

Similarly, for the reflected signal:

�Er

‖ �E0‖= rTE · cosβ · (k× n)+ rTM ·sinβ · ((k× n)× k) (2)

Where β is the angle between k × n and the incident po-larization vector �E0. tTE , rTE and tTM , rTM are well knownpenetration and reflection coefficients associated with differentTM and TE vector components of �E0 (these are referred to asthe Transverse Magnetic and Transverse Electric components- the polarization components which their associated field arenot in the propagation direction). These depend on the buildingelement material and the signal frequency:

t∗ =(1 − ρ2

∗)e−dkzi

1 − ρ2∗e−2dkzi, r∗ = ρ∗

1 − e−2dkzi

1 − ρ2∗e−2dkzi

where ε0 = 1, εr = 5.1 − 1.09i are the normalized airpermeability and concrete permeability respectively. θ0 is theangle of the signal wave vector �k with respect to the incidentsurface normal n. The angle given as θ = tan−1 sinθ0

�√εr−sin2θ0

extends between the wave vector of a penetrating signaland the incident surface normal −n. The surface normalcomponent of the wave vector kz is given by 2πf

c

√εrcosθ

and finally, ρ∗ here stand for:

ρTM = −εrcosθ0 −√

εr − sin2θ0

εrcosθ0 +√

εr − sin2θ0

,

ρTE =cosθ0 −

√εr − sin2θ0

cosθ0 +√

εr − sin2θ0

Since the physical RF model hereby described is straight-forward to calculate, from now on, we will discuss only theunderlying geometric problem. It should be noted that theaveraged values of equations 1,2 above agree with empiricalloss values for 2.4[GHz] known in the literature (e.g.,[7]).

III. THE GEOMETRIC PROPERTIES OF A RADIO PATH

In this section we introduce an algorithm for computingthe direction at which a directed signal from the transmitterT is received by the receiver R. When a signal hits a wallwe consider both the reflectance and transmittance phenomenadescribed above. Notice that, if a signal received by Rpenetrates a wall, the direction of the signal path is definedby the straight extension between T and R. Therefore, in thatcase, the path can be easily extracted.

Observe that if we have an oracle that informs us of agiven sequence of walls W (an ordered set), the followingobservation significantly simplifies computations regarding thereflectance phenomena:

Observation 4: Consider a signal bouncing off a wall w ∈W . Reflecting the image of B through the wall w will resultin a straight line presentation of the signal between T and thew-reflected R (within the w-reflected building).

Proof: The proof is by induction on the number ofsignal bounces b. Since we use the same paradigm for theinduction hypothesis we start by demonstrating our claim onthe somewhat trivial base case. For the base case where b = 1:denote by wi the wall hit by the signal path and further denotepi to be the hitting point. The signal paths (T , pi) and (pi,R)are both straight lines. Thus we can ignore all other walls (b = 1). Denote the plane defined by the triangle �(T , pi,R)by P . Notice that P would intersect the reflected R denotedby R′ as well. Furthermore, notice that the segment extendingfrom T and pi contains R′. Assume b > 1 and let wb bethe last wall the signal bounces off of, with point pb the lastbouncing point. Prior to the last bounce, induction hypothesisimplies that the signal path from T to the reflected pb (withtheir respective reflections) is a straight line. Therefore, wecan consider the first b − 1 bounces as penetrations. Finally,base case 2 implies that reflecting through wb concludes theproof.Since the number of walls that a signal may hit is physicallybounded we can examine all possible sequences, dismissingthe need for an oracle.

IV. BOUNDED LENGTH INDOOR RADIO PATHS

ALGORITHM

We will now introduce our algorithm for computing abuilding Visibility Graph (V G(B)). Next, given a desired pathlength (i.e., the number of signal bounces) we extract fromthe V G(B) all bounded length paths between the transmitter

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and receiver2. Then, for each such path we compute its virtualreflected structure and conclude whether such a path can berealized. Finally we compute the total contribution (power,phase of �E) of all realizable paths.

A. Preprocessing: computing the visibility graph

We now describe the preprocessing stage in which a Vis-ibility Graph (V G(B)) of the building B is computed. Theset of vertices V is associated with all building elements (i.e.,facets of walls, ceilings and floors). To be more accurate, eachbuilding element is associated with two vertices in V G(B)(i.e., face-facet and back-facet). However, n-polygonal (withn > 4) elements, as well as non-simply-connected elements(such as doors or windows) are not included in V . Instead,these are divided into several simply shaped elements; eachis associated with a vertex in V G(B). Each pair v1, v2 ∈ Vis connected by an arc e12 ∈ E if and only if there exists aspatial LOS between their associated building elements (thatis, without intersecting any other building element along theway). In the planar case there are few efficient algorithmsfor computing V G(B) (e.g., [10]). Yet, computing V G(B)of a building is a rather complex task, both theoreticallyand practically. S. Teller and M. Hohmeyer [11] showed thatfour lines with general Plucker coordinates determine a singleline incident on the four input lines. In practice, however,input degeneracies may result in 0,1,2, or various infinitiesof incident lines. We use this property in order to design anO(n5) algorithm for computing the exact V G(B) in a generalposition.

Observation 5: General position assumption is not applica-ble for real-life buildings. Indeed, most buildings have manyparallel walls and grid based coordinate values.

In light of that, the algorithm runtime is hardly practical forlarge buildings (with hundreds of walls) and it requires specialcare for degenerate input lines. Yet, a careful examinationyields the following:

Observation 6: Any path in the V G(B) corresponds to asingle (or non) spatial geometric radio path in the actualbuilding.

Proof: Recall that we only consider planar building ele-ments. Further recall that each building element correspondsto a pair of vertices in V G(B). Therefore, since V G(B) isconstructed via successive building reflections (which reflectthe receiver as well), one may think of the geometric radiopath connecting T and the reflected receiver R as a straightline (i.e., all wall bounces may be thought of as penetrations).

Observation 7: We may compute a relaxed approximationof V G(B) which represents ‘over-visibility’. Yet, we mustensure that if there exists a direct LOS between two buildingelements, there will be an edge between their correspondingvertices in V G(B).

2In the actual implementation each building element is associated with apath loss value, and we bound the signal power loss, rather then the pathlength.

Motivated by the room-structure of most buildings andthe observations we have made, we define a more practicalalgorithm for computing the V G(B)

Algorithm 1: The Visibility Graph PseudocodeData: Geometric structure of B; T and R.Result: A relaxed visibility graph of B with T , R.Transform each simple building element into a pair of1

connected vertices;Divide the building into rooms;2

Divide each floor and ceiling into patches according to3

each room projection;Define each door or window in the room as a virtual wall4

(of zero width);Connect all the inner elements of each room (walls, floor,5

ceiling, doors and windows);

The actual algorithm is somewhat more complex. Forexample, consider staircase visibility computation. Dividinga building into rooms requires several technical approaches.Given the transmitter T and the receiver R loci, we addthem to V G(B) by connecting each with the room verticesthey are in. In other words, any geometric element in Bwhich can directly ”see” T will be arcwise connected to itin V G(B). We then, denote the extended visibility graph byV G(B, T ,R). Notice that V G(B) should only be computedonce, and, furthermore, if T and R loci are to be altered onemay add them to V G(B) and simply reconnect both to thebuilding elements in the room they are in, thus making ourstrategy time practical.

B. Computing all bounded length radio paths over the visi-bility graph

For practical signal power computation the number ofsignal bounces can be upper-bounded (i.e., we only need toconsider paths in V G(B)) which are at most c-hop length).Given a bounded length c, we traverse V G(B, T ,R) andcompute all possible paths between T and R of length ≤ c.For each such path γ ⊆ V G(B, T ,R) we compute its virtualreflected structure V RS(B, γ)

For each computed radio path γ (a valid geometric pathbetween T and R) one can compute the signal received at R(considering only γ), taking into consideration the properties(reflection, penetration, thickness, density, angle) of eachwall, which γ passes through.

C. Complexity and practical runtime

In this section we present the runtime complexity of thealgorithms followed by the practical runtime over commoninputs. Assuming |B| = n, |E(V (B))|

n = k. Computing V G(B)can be done in O(kn2), note: this preliminary stage (prepro-cessing) is only computed once per building. adding T andR to V G(B)) can be done in linear time O(n) (the algorithmsimply find the rooms which contains T and R). Computing

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T, a, k, d, h,R

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Fig. 1. The IDRP algorithm: at the first step (A) a simple example ofa 2D building is given; for simplicity we present each building elementusing a single node and a connected pair of vertices. (B) The buildingvisibility graph: V G(B). (C) Adding T and R to the recomputed visibilitygraph. (D) Consider the graph path between T and R in the visibilitygraph {T, a, k, d, h, R}. (E) The virtual building which is correlated withthe path and the validation algorithm: connecting T and R with a straightline, making sure the line intersects all the graph path nodes (the walls alongthe path are dashed). (F ) The actual radio-path of the visibility graph path{T, a, k, d, h, R} .

all relevant c bounded length radio paths over V G(B, T ,R)is O(2max(c,k) +nk). Note: for all practical scenarios one canassume k, c < 20 and k << n. For common buildings whereB has 20 rooms, n = 2000, c = 20, k = 7, The algorithmcomputed an average amount of 550, 000 possible radio pathwith in less than 10 seconds, where only about 100 radio pathswhere considered after the pruning stage.

V. EXPERIMENTAL RESULTS - VALIDITY TESTING

We have implemented a software package for computing thevisibility graph and computing all bounded length radio paths.As discussed, our implementation also includes a sophisticatedRF model in order to allow realistic propagation models suchas WiFi. The software was implemented in Java 1.6, whilethe 3D models are generated using Google sketch-up. Wehave tested the application on several models including largebuilding structures with thousands of walls and several floors.As an example see Figure 2 which is an implementation of

Algorithm 2: The IDRP strategy Pseudocode

Data: Geometric structure of B; The visibility graphV G(B).

Result: All possible paths γ between T and R.Given T and R, add two corresponding nodes to V G(B)1

;Compute all bounded-length paths γ between T and R2

over V G(B) ;for each path such γ do3

Construct the virtual reflected structure V RS(B, γ) ;4

if the direct spatial line between T and R in5

V RS(B, γ) hits all (and only) the building elementsthat correspond to vertices in γ then

Compute γ∗: the real radio path induced by γ;6

Add γ∗ to the set of radio paths.;7

IDRP on a campus building. In order to empirically validateour model results, we have used Agilent PNA Microwave Net-work Analyzer and executed a series of physical experiments.Measurements where carried out in a single sixteen roomsfloor in the campus.

A transmitter and a receiver where placed three rooms apart(15 meters from each other). Both antennas which we usedwhere Z-polarized. RF indoor propagation properties such aspermeability for WiFi frequency range had been extensivelystudied (theoretically and experimentally) and therefore areavailable in the literature. Accordingly, measurements werepreformed in frequencies range of 2.4[GHz] which corre-sponds to standard WiFi wireless LAN. Narrow band mea-surements (1.0625[MHz]) were converted to the time domainusing FFT (with a raised cosine window) yielding an impulseresponse. In order to test consistency, the receiver’s positionwas altered along a 1[m] straight line and measurements weretaken every 1[cm]. We followed [12] for the exact permeabilityparameters (for frequencies 2.412[GHz] − 2.462[GHz]) andachieved consistent and relatively accurate prediction3 to thesignal strength as shown in 3. It should be noted that wedid not use any kind of normalization factors, i.e., the resultsshown here were computed directly without any kind of fittingfunction.

VI. CONCLUSIONS

Results show the suggested IDRP frame-work is accurateand relatively fast and can compute all significant radio pathsof complicated buildings in a matter of seconds. Travers-ing this set of significant radio paths enables us to predictthe signal strength at R. Moreover, appreciating the actualgeometric paths allow us to predict the signal phase. Thesuggested algorithm was able to sufficiently evaluate an indoorradio impulse. Thus, computing the superposition signal sum ismade possible. This property is essential for simulating MIMOtechnology (such as [2], [1]).

3The full description of the experiments can be found at:http:\\www.ariel.ac.il\projects\kcg.

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Fig. 2. The Radio Paths implementation: 16 different radio paths betweenT and R (located at the lower and upper floor).

Fig. 3. Two arbitrary positions simulation versus experimental results, in2.412GHz - WiFi channel-1. Notice that the experimental and theoreticalpeaks are concurrent and agree by value. Discrepancies may be a result ofmodeling and measuring errors.

We plan to generalize this work to support larger urbanregions (as shown in figure 4) and advance wireless technologyas WiMax and LTE .

Fig. 4. An example of large scale urban environment.

ACKNOWLEDGMENT

The authors wish to thank Yosi Pinhasi, Asher Yahalom,and Tsahi Birk for introducing us to the real-world of wireless

indoor communication and propagation. We thank Dana Porratand Yehuda Agiv for their assistance with the experiments.This work was supported by the Israeli Short Range Consor-tium (ISRC).

REFERENCES

[1] “802.16e-2005: IEEE standard for local and metropolitan area networks(wimax),” 2006.

[2] “802.11n-2009: IEEE standard for local and metropolitan area networks(mimo-wifi),” 2009.

[3] S. Topcu, H. Kymen, A. Altnta, and M. Aksun, “Propagation predictionand planning tools for digital and analog terrestrial broadcasting andland mobile services,” in Proc. of 50th Annual Broadcast Symposium.,vol. 32, 2000.

[4] N. Blaunstein, Radio Propagation in Cellular Networks. Artech HousePublishers, 1999. [Online]. Available: http://www.cse.bgu.ac.il/Articles/article.asp?ArticleID=29

[5] Z. Chen, A. Delis, and H. L. Bertoni, “Radio-wave propagation predic-tion using ray-tracing techniques on a network of workstations (now),”J. Parallel Distrib. Comput., vol. 64, no. 10, pp. 1127–1156, 2004.

[6] F. Ikegami, S. Yoshida, T. Takeuchi, and M. Umehira, “Propagationfactors controlling mean field strength on urban streets,” IEEE Trans.Antennas and Propagation, vol. 32, pp. 822–829, 1984.

[7] D. I. Laurenson, “Indoor radio channel propagation modelling by raytracing techniques,” 1994. [Online]. Available: http://www.see.ed.ac.uk/∼dil/thesis.pdf

[8] A. Schmitz and M. Wenig, “The effect of the radio wave propagationmodel in mobile ad hoc networks,” in MSWiM ’06: Proceedings of the9th ACM international symposium on Modeling analysis and simulationof wireless and mobile systems. New York, NY, USA: ACM, 2006, pp.61–67.

[9] A. Shlivinski, E. Heyman, A. Boag, and C. Letrou, “A phase-spacebeam summation formulation for ultrawide-band radiation,” Antennasand Propagation, IEEE Transactions on, vol. 52, no. 8, pp. 2042 –2056, 2004.

[10] E. Welzl, “Constructing the visibility graph for n line segments inO(n2) time,” Inform. Process. Lett., vol. 20, pp. 167–171, 1985.

[11] S. Teller and M. Hohmeyer, “Determining the lines through four lines,”journal of graphics tools, vol. 4, no. 3, pp. 11–22, 1999.

[12] Y. Pinhasi, A. Yahalom, and S. Petnev, “Propagation of ultra wide-bandsignals in lossy dispersive media,” in International IEEE Conference onMicrowaves, Communications, Antennas and Electronic Systems (IEEECOMCAS 2008), 2008.

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