Aalborg Universitet

Modeling of Reverberant Radio Channels Using Propagation Graphs

Pedersen, Troels; Steinböck, Gerhard; Fleury, Bernard Henri

Published in:I E E E Transactions on Antennas and Propagation

DOI (link to publication from Publisher):10.1109/TAP.2012.2214192

Publication date:2012

Document VersionEarly version, also known as pre-print

Link to publication from Aalborg University

Citation for published version (APA):Pedersen, T., Steinböck, G., & Fleury, B. H. (2012). Modeling of Reverberant Radio Channels UsingPropagation Graphs. I E E E Transactions on Antennas and Propagation, 60(12), 5978 - 5988 .https://doi.org/10.1109/TAP.2012.2214192

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1

Modeling of Reverberant Radio Channels UsingPropagation Graphs

Troels Pedersen, Gerhard Steinböck, and Bernard H. Fleury

Abstract— In measurements of in-room radio channel re-sponses an avalanche effect can be observed: earliest signalcomponents, which appear well separated in delay, are followedby an avalanche of components arriving with increasing rate ofoccurrence, gradually merging into a diffuse tail with exponen-tially decaying power. We model the channel as a propagationgraph in which vertices represent transmitters, receivers, andscatterers, while edges represent propagation conditions betweenvertices. The recursive structure of the graph accounts for theexponential power decay and the avalanche effect. We derivea closed form expression for the graph’s transfer matrix. Thisexpression is valid for any number of interactions and isstraightforward to use in numerical simulations. We discuss anexample where time dispersion occurs only due to propagation inbetween vertices. Numerical experiments reveal that the graph’srecursive structure yields both an exponential power decay andan avalanche effect.

Index Terms— Radio channels, channel impulse response, re-verberation, signal flow graphs, multiple scattering, indoor radiochannels.

I. INTRODUCTION

Engineering of modern indoor radio systems for commu-nications and geolocation relies heavily on models for thetime dispersion of the wideband and ultrawideband radiochannels [1]–[3]. From measurement data, as exemplified inFig. 1, it appears that the spatial average of the channelimpulse response’s squared magnitude (referred to as thespatially averaged delay-power spectrum) observed in in-roomscenarios exhibits an avalanche effect: The earliest signalcomponents, which appear well separated in time, are followedby an avalanche of components arriving with increasing rate ofoccurrence, gradually merging into a diffuse tail with exponen-tially decaying power. The diffuse tail is often referred to asa “dense multipath component” and is commonly attributedto diffuse scattering from rough surfaces and objects whichare small compared to the wavelength [5], [6]. A similaravalanche effect is well-known in room acoustics [7] whereit is attributed to recursive scattering of sound waves. Indoorradio propagation environments are particularly exposed torecursive scattering as electromagnetic waves may be reflectedback and forth in between walls, floor, and ceiling. Thus, theavalanche effect and the exponential power decay may occurdue to recursive scattering rather than diffuse scattering.

Recursive scattering phenomena have been previously takeninto account in a number of radio channel models. The works[8]–[11] use the analogy to acoustical reverberation theory to

Department of Electronic Systems, Aalborg University, Fredrik BajersVej 7, DK-9220 Aalborg East, Denmark. Email: {troels,gs,fleury}@es.aau.dk.

This work was co-funded by the European Union via the project ICT-248894 Wireless Hybrid Enhanced Mobile Radio Estimators 2 (WHERE2).

1

Modeling of Reverberant Radio Channels UsingPropagation Graphs

Troels Pedersen, Gerhard Steinböck, and Bernard H. Fleury

Abstract— In measurements of in-room radio channel re-sponses an avalanche effect can be observed: earliest signalcomponents, which appear well separated in delay, are followedby an avalanche of components arriving with increasing rateofoccurrence, gradually merging into a diffuse tail with exponen-tially decaying power. We model the channel as a propagationgraph in which vertices represent transmitters, receivers, andscatterers, while edges represent propagation conditionsbetweenvertices. The recursive structure of the graph accounts fortheexponential power decay and the avalanche effect. We derivea closed form expression for the graph’s transfer matrix. Thisexpression is valid for any number of interactions and isstraightforward to use in numerical simulations. We discuss anexample where time dispersion occurs only due to propagation inbetween vertices. Numerical experiments reveal that the graph’srecursive structure yields both an exponential power decayandan avalanche effect.

Index Terms— Radio channels, channel impulse response, re-verberation, signal flow graphs, multiple scattering, indoor radiochannels.

I. I NTRODUCTION

Engineering of modern indoor radio systems for commu-nications and geolocation relies heavily on models for thetime dispersion of the wideband and ultrawideband radiochannels [1]–[3]. From measurement data, as exemplified inFig. 1, it appears that the spatial average of the channelimpulse response’s squared magnitude (referred to as thespatially averaged delay-power spectrum) observed in in-roomscenarios exhibits an avalanche effect: The earliest signalcomponents, which appear well separated in time, are followedby an avalanche of components arriving with increasing rateofoccurrence, gradually merging into a diffuse tail with exponen-tially decaying power. The diffuse tail is often referred toasa “dense multipath component” and is commonly attributedto diffuse scattering from rough surfaces and objects whichare small compared to the wavelength [5], [6]. A similaravalanche effect is well-known in room acoustics [7] whereit is attributed to recursive scattering of sound waves. Indoorradio propagation environments are particularly exposed torecursive scattering as electromagnetic waves may be reflectedback and forth in between walls, floor, and ceiling. Thus, theavalanche effect and the exponential power decay may occurdue to recursive scattering rather than diffuse scattering.

Recursive scattering phenomena have been previously takeninto account in a number of radio channel models. The works[8]–[11] use the analogy to acoustical reverberation theory to

Department of Electronic Systems, Aalborg University, Fredrik Bajers Vej7, DK-9220 Aalborg East, Denmark. Email: {troels,gs,fleury}@es.aau.dk.

This work was co-funded by the European Union via the projectICT-248894 Wireless Hybrid Enhanced Mobile Radio Estimators 2 (WHERE2).

0 50 100 150

−110

−105

−100

−95

−90

−85

−80

−75

Delay [ns]

rms(

S21

) [d

B(1

0 G

Hz)

]

Office LOS

Direct comp.

Single dominantcomponents

Diffusetail

Fig. 1. Spatially averaged delay-power spectrum measured within an officeof 5×5×2.6m3obtained as the rms value of impulse response for 30×30receiver positions on a square horizontal grid with steps of1cm. Abscissaincludes cable delays; reference to signal bandwidth (10GHz) shifts ordinateby –100dB. Reprinted from [4] with permission (c© 2002 IEEE).

predict the exponential decay. As a matter of fact, there existsa well-developed theory of electromagnetic fields in cavities[12], [13], but in this context too the avalanche effect hasreceived little attention. Recursive scattering between particlesin a homogeneous medium is a well-known phenomenonstudied by Foldy [14] and Lax [15], [16]. The solution, givenby the so-called Foldy-Lax equation [17], has been appliedin the context of time-reversal imaging by Shi and Nehorai[18]. The solution is, however, intractable for heterogeneousindoor environments. In [19] the radio propagation mechanismis modeled as a “stream of photons” performing continuousrandom walks in a homogeneously cluttered environment. Themodel predicts a delay-power spectrum consisting of a singledirectly propagating “coherent component” followed by anincoherent tail. Time-dependent radiosity [20]–[23] accountingfor delay dispersion has been recently applied to design amodel for the received instantaneous power [24]. Thereby,the exponential power decay and the avalanche effect can bepredicted.

Simulation studies of communication and localization sys-tems commonly rely on synthetic realizations of the channelimpulse response. A multitude of impulse response modelsexist [2], [3], [25], but only few account for the avalancheeffect. The models [26]–[28] treat early components via a ge-ometric model whereas the diffuse tail is generated via anotherstochastic process; the connection between the propagationenvironment and the diffuse tail is, however, not considered.

Fig. 1. Spatially averaged delay-power spectrum measured within an officeof 5×5×2.6m3obtained as the rms value of impulse response for 30×30receiver positions on a square horizontal grid with steps of 1 cm. Abscissaincludes cable delays; reference to signal bandwidth (10GHz) shifts ordinateby –100dB. Reprinted from [4] with permission ( c© 2002 IEEE).

predict the exponential decay. As a matter of fact, there existsa well-developed theory of electromagnetic fields in cavities[12], [13], but in this context too the avalanche effect hasreceived little attention. Recursive scattering between particlesin a homogeneous medium is a well-known phenomenonstudied by Foldy [14] and Lax [15], [16]. The solution, givenby the so-called Foldy-Lax equation [17], has been appliedin the context of time-reversal imaging by Shi and Nehorai[18]. The solution is, however, intractable for heterogeneousindoor environments. In [19] the radio propagation mechanismis modeled as a “stream of photons” performing continuousrandom walks in a homogeneously cluttered environment. Themodel predicts a delay-power spectrum consisting of a singledirectly propagating “coherent component” followed by anincoherent tail. Time-dependent radiosity [20]–[23] accountingfor delay dispersion has been recently applied to design amodel for the received instantaneous power [24]. Thereby,the exponential power decay and the avalanche effect can bepredicted.

Simulation studies of communication and localization sys-tems commonly rely on synthetic realizations of the channelimpulse response. A multitude of impulse response modelsexist [2], [3], [25], but only few account for the avalancheeffect. The models [26]–[28] treat early components via a ge-ometric model whereas the diffuse tail is generated via anotherstochastic process; the connection between the propagationenvironment and the diffuse tail is, however, not considered.

2

Ray tracing methods may also be used for site-specific simu-lations [29]. However, ray tracing methods commonly predictthe signal energy to be concentrated into the single dominantcomponents whereas the diffuse tail is not well represented.

In this contribution, we model the channel impulse responsefor the purpose of studying the avalanche effect and, inparticular, its relation to recursive scattering. The objective is amodel for which analytical results can be obtained allowing forcomputer simulation of channel realization as well as averageentities, such as the delay-power spectrum, without the needfor truncating the recursive scattering process. Expanding ona previous work [30], [31], we propose a unified approach formodeling of the transition from single dominant componentsto the diffuse tail. The propagation environment is representedin terms of a propagation graph, with vertices representingtransmitters, receivers, and scatterers, while edges representpropagation conditions between vertices. The propagationgraph accounts inherently for recursive scattering and thussignal components arriving at the receiver may have undergoneany number of scatterer interactions. Furthermore, we are ableto express the channel transfer matrix in closed form foran unlimited number of interactions. With this closed formexpression it is possible in praxis to numerically compute thechannel recursive scattering with infinite number of bounces.The proposed formalism also enables a virtual “dissection” ofthe impulse response in so-called partial responses to inspecthow recursive scattering leads to a gradual buildup of thediffuse tail.

Propagation graphs may be defined according to a spe-cific scenario for site-specific prediction, or be generatedas outcomes of a stochastic process for use in e.g. MonteCarlo simulations. In the present contribution we consideran example of a stochastically generated propagation graphsuitable for Monte Carlo simulations on a desktop computer.In this example model, delays occur only due to propagationin between vertices; the interactions with the scatterers causeno additional delay. The simulations reveal that the graph’srecursive structure yields both an exponential power decay andan avalanche effect in the generated impulse responses.

II. REPRESENTING RADIO CHANNELS AS GRAPHS

In a typical propagation scenario, the electromagnetic signalemitted by a transmitter propagates through the environmentwhile interacting with a number of objects called scatter-ers. The receiver, which is usually placed away from thetransmitter, senses the electromagnetic signal. If a line ofsight exists between the transmitter and the receiver, directpropagation occurs. Also, the signal may arrive at the receiverindirectly via one or more scatterers. In the following werepresent propagation mechanisms using graphs allowing forboth recursive and non-recursive scattering. We first introducethe necessary definitions of directed graphs.

A. Directed Graphs

Following [32], we define a directed graph G as a pair(V, E) of disjoint sets of vertices and edges. Edge e ∈ E withinitial vertex denoted by init(e) and terminal vertex denoted by

2

Ray tracing methods may also be used to do site-specificsimulations [29]. However, ray tracing methods commonlypredict the signal energy to be concentrated into the singledominant components whereas the diffuse tail is not wellrepresented.

In this contribution, we model the channel impulse responsefor the purpose of studying the avalanche effect and, inparticular, its relation to recursive scattering. The objectiveis a model for which analytical results can be obtainedallowing for computer simulation of both realizations of thechannel response and of average entities, such as the delay-power spectrum, without the need for truncating the recursivescattering process. Expanding on a previous work [30], [31],we propose a unified approach for modeling of the transitionfrom single dominant components to the diffuse tail.Thepropagation environment is represented in terms of a propaga-tion graph, with vertices representing transmitters, receivers,and scatterers, while edges represent propagation conditionsbetween vertices. The propagation graph accounts inherentlyfor recursive scattering and thus signal components arrivingat the receiver may have undergone any number of scattererinteractions.Furthermore, we are able to express the channeltransfer matrix in closed form for an unlimited number ofinteractions. With this closed form expression it is possible inpraxis to numerically compute the channel recursive scatteringwith infinite number of bounces.Furthermore, the formalismenables a virtual “dissection” of the impulse response in so-called partial responses to inspect how recursive scatteringleads to a gradual buildup of the diffuse tail.

Propagation graphs may be defined according to a spe-cific scenario for site-specific prediction, or be generatedas outcomes of a stochastic process for use in e.g. MonteCarlo simulations.In the present contribution we consideran example of a stochastically generated propagation graphsuitable for Monte Carlo simulations on a desktop computer.In this example model, delays occur only due to propagationin between vertices; the interactions with the scatterers causeno additional delay.The simulations reveal that the graph’srecursive structure yields both an exponential power decayandan avalanche effect in the generated impulse responses.

II. REPRESENTINGRADIO CHANNELS AS GRAPHS

In a typical propagation scenario, the electromagnetic signalemitted by a transmitter propagates through the environmentwhile interacting with a number of objects called scatter-ers. The receiver, which is usually placed away from thetransmitter, senses the electromagnetic signal. If a line ofsight exists between the transmitter and the receiver, directpropagation occurs. Also, the signal may arrive at the receiverindirectly via one or more scatterers. In the following werepresent propagation mechanisms using graphs allowing forboth recursive and non-recursive scattering. We first introducethe necessary definitions of directed graphs.

A. Directed Graphs

Following [32], we define a directed graphG as a pair(V , E) of disjoint sets of vertices and edges. Edgee ∈ E with

bTx4

bTx3

bTx2

bTx1

Vt

bRx1

bRx2

bRx3

Vr

b

S1b

S2

bS3

b

S4

bS5

b S6

Vs

Ed = {(Tx1,Rx1), (Tx2,Rx1), (Tx3,Rx1)}Et = {(Tx2, S3), (Tx3, S3), (Tx4, S6), (Tx4, S1)}Er = {(S1,Rx3), (S3,Rx3), (S6,Rx2)}Es = {(S1, S2), (S2, S1), (S3, S1), (S2, S4), (S4, S3), (S4, S5)}

Fig. 2. A propagation graph with four transmit vertices, three receive verticesand six scatterer vertices.

initial vertex denoted byinit(e) and terminal vertex denoted byterm(e) is said to be outgoing from vertexinit(e) and ingoingto term(e). We consider graphs without parallel edges. Thus,there exists no pair of edgese and e′ such thatinit(e) =init(e′) and term(e) = term(e′). In this case an edgee canbe identified by the vertex pair(init(e), term(e)) ∈ V2 andwith a slight abuse of notation we writee = (init(e), term(e))andE ⊆ V × V .

B. Propagation Graphs

We define a propagation graph as a directed graph wherevertices represent transmitters, receivers and scatterers. Edgesrepresent the propagation conditions between the vertices.Thus, the vertex setV of a propagation graph is a unionof three disjoint sets:V = Vt ∪ Vr ∪ Vs, the set oftransmittersVt = {Tx1, . . . ,TxNt}, the set of receiversVr = {Rx1, . . . ,RxNr}, and the set of scatterersVs ={S1, . . . , SNs}. The transmit vertices are considered as sourcesand have only outgoing edges. Likewise, the receivers areconsidered as sinks with only incoming edges. The edge setE can thus be partitioned into four disjunct sets asE =Ed ∪ Et ∪ Er ∪ Es, with direct edges inEd = E ∩ (Vt × Vr),transmitter-scatterer edges inEt = E ∩ (Vt × Vs), scatterer-receiver edges inEr = E ∩ (Vs×Vr), and inter-scatterer edgesin Es = E ∩ (Vs × Vs). Fig. 2 shows an example propagationgraph.

The signals propagate in the graph in the following way.Each transmitter emits a signal that propagates via its outgoingedges. The signal observed by a receiver vertex is the sumof the signals arriving via the ingoing edges. A scatterersums up the signals on its ingoing edges and re-emits thesum-signal on the outgoing edges. As a signal propagatesalong an edge, or interacts with a scatterer, it undergoesdelay and dispersion in time. The specific delay dispersionendured by a signal depends on the particular propagationmechanism along its edges. Assuming these mechanisms tobe linear and time-invariant, this effect can be representedas a convolution with an impulse response or, in the Fourier

Fig. 2. A propagation graph with four transmit vertices, three receive verticesand six scatterer vertices.

term(e) is said to be outgoing from vertex init(e) and ingoingto term(e). We consider graphs without parallel edges. Thus,there exists no pair of edges e and e′ such that init(e) =init(e′) and term(e) = term(e′). In this case an edge e canbe identified by the vertex pair (init(e), term(e)) ∈ V2 andwith a slight abuse of notation we write e = (init(e), term(e))and E ⊆ V × V .

B. Propagation Graphs

We define a propagation graph as a directed graph wherevertices represent transmitters, receivers and scatterers. Edgesrepresent the propagation conditions between the vertices.Thus, the vertex set V of a propagation graph is a unionof three disjoint sets: V = Vt ∪ Vr ∪ Vs, the set oftransmitters Vt = {Tx1, . . . ,TxNt}, the set of receiversVr = {Rx1, . . . ,RxNr}, and the set of scatterers Vs ={S1, . . . ,SNs}. The transmit vertices are considered as sourcesand have only outgoing edges. Likewise, the receivers areconsidered as sinks with only incoming edges. The edge setE can thus be partitioned into four disjunct sets as E =Ed ∪ Et ∪ Er ∪ Es, with direct edges in Ed = E ∩ (Vt × Vr),transmitter-scatterer edges in Et = E ∩ (Vt × Vs), scatterer-receiver edges in Er = E ∩ (Vs×Vr), and inter-scatterer edgesin Es = E ∩ (Vs × Vs). Fig. 2 shows an example propagationgraph.

The signals propagate in the graph in the following way.Each transmitter emits a signal that propagates via its outgoingedges. The signal observed by a receiver vertex is the sumof the signals arriving via the ingoing edges. A scatterersums up the signals on its ingoing edges and re-emits thesum-signal on the outgoing edges. As a signal propagatesalong an edge, or interacts with a scatterer, it undergoesdelay and dispersion in time. The specific delay dispersionendured by a signal depends on the particular propagationmechanism along its edges. Assuming these mechanisms tobe linear and time-invariant, this effect can be representedas a convolution with an impulse response or, in the Fourierdomain, as a multiplication with a transfer function. Hence,the signal arriving to vertex vn′ via edge e = (vn, vn′) reads

3

Ae(f)Cn(f), where Cn(f) is the signal emitted by vertex vnand Ae(f) denotes the transfer function associated to edgee. In other words, the transfer function Ae(f) describes theinteraction at the initial vertex vn and the propagation fromvn to vn′ .

C. Weighted Adjacency Matrix of a Propagation Graph

Propagation along the edges is described via a transfermatrix A(f) which can be viewed as an adjacency matrixwith each entry coinciding with the edge transfer function ofthe corresponding edge. Thus, the weighted adjacency matrixA(f) ∈ C(Nt+Nr+Ns)×(Nt+Nr+Ns) of the propagation graphG is defined as

[A(f)]nn′ =

{A(vn,vn′ )(f) if (vn, vn′) ∈ E ,0 otherwise,

(1)

i.e., entry n, n′ of A(f) is the transfer function from vertexvn to vertex vn′ of G. Selecting the indexing of the verticesaccording to

vn ∈

Vt, n = 1, . . . , Nt

Vr, n = Nt + 1, . . . , Nt +Nr

Vs, n = Nt +Nr + 1, . . . , Nt +Nr +Ns,

(2)

the weighted adjacency matrix takes the form

A(f) =

0 0 0D(f) 0 R(f)T(f) 0 B(f)

, (3)

where 0 denotes the all-zero matrix of the appropriate dimen-sion and the transfer matrices

D(f) ∈ CNr×Nt connecting Vt to Vr, (4)

R(f) ∈ CNr×Ns connecting Vs to Vr, (5)

T(f) ∈ CNs×Nt connecting Vt to Vs, and (6)

B(f) ∈ CNs×Ns interconnecting Vs. (7)

The special structure of A(f) reflects the structure of thepropagation graph. The first Nt rows are zero because wedo not accept incoming edges into the transmitters. Likewisecolumns Nt + 1, . . . , Nt + Nr are all zero since the receiververtices have no outgoing edges.

The input signal vector X(f) is defined as

X(f) = [X1(f), . . . , XNt(f)]T, (8)

where Xm(f) is the spectrum of the signal emitted by trans-mitter Txm, and [ · ]T denotes the transposition operator. Theoutput signal vector Y(f) is defined as

Y(f) = [Y1(f), . . . , YNr(f)]T, (9)

where Ym(f) is the Fourier transform of the signal observedby receiver Rxm. Similar, to X(f) and Y(f) we let Z(f)denote the output signal vector of the scatterers:

Z(f) = [Z1(f), . . . , ZNs(f)]T (10)

with the nth entry denoting the Fourier transform of thesignal observed at scatterer vertex Sn. By the definition of the

3

domain, as a multiplication with a transfer function. Hence,the signal arriving to vertexvn′ via edgee = (vn, vn′) readsAe(f)Cn(f), whereCn(f) is the signal emitted by vertexvnand Ae(f) denotes the transfer function associated to edgee. In other words, the transfer functionAe(f) describes theinteraction at the initial vertexvn and the propagation fromvn to vn′ .

C. Weighted Adjacency Matrix of a Propagation Graph

Propagation along the edges is described via a transfermatrix A(f) which can be viewed as an adjacency matrixwith each entry coinciding with the edge transfer function ofthe corresponding edge. Thus, the weighted adjacency matrixA(f) ∈ C(Nt+Nr+Ns)×(Nt+Nr+Ns) of the propagation graphG is defined as

[A(f)]nn′ =

{A(vn,vn′ )(f) if (vn, vn′) ∈ E ,0 otherwise,

(1)

i.e., entryn, n′ of A(f) is the transfer function from vertexvn to vertexvn′ of G. Selecting the indexing of the verticesaccording to

vn ∈

Vt, n = 1, . . . , Nt

Vr, n = Nt + 1, . . . , Nt +Nr

Vs, n = Nt +Nr + 1, . . . , Nt +Nr +Ns,

(2)

the weighted adjacency matrix takes the form

A(f) =

0 0 0D(f) 0 R(f)T(f) 0 B(f)

, (3)

where0 denotes the all-zero matrix of the appropriate dimen-sion and the transfer matrices

D(f) ∈ CNr×Nt connectingVt to Vr, (4)

R(f) ∈ CNr×Ns connectingVs to Vr, (5)

T(f) ∈ CNs×Nt connectingVt to Vs, and (6)

B(f) ∈ CNs×Ns interconnectingVs. (7)

The special structure ofA(f) reflects the structure of thepropagation graph. The firstNt rows are zero because wedo not accept incoming edges into the transmitters. LikewisecolumnsNt + 1, . . . , Nt + Nr are all zero since the receiververtices have no outgoing edges.

The input signal vectorX(f) is defined as

X(f) = [X1(f), . . . , XNt(f)]T, (8)

whereXm(f) is the spectrum of the signal emitted by trans-mitter Txm, and[ · ]T denotes the transposition operator. Theoutput signal vectorY(f) is defined as

Y(f) = [Y1(f), . . . , YNr(f)]T, (9)

whereYm(f) is the Fourier transform of the signal observedby receiverRxm. Similar, to X(f) and Y(f) we let Z(f)denote the output signal vector of the scatterers:

Z(f) = [Z1(f), . . . , ZNs(f)]T (10)

X(f) Y(f)

Z(f)

Vt Vr

Vs

D(f)

T(f) R

(f)

B(f)

Fig. 3. Vector signal flow graph representation of a propagation graph.Vertices represent vertex sets of the propagation graph with associated vectorsignals. Signal transmission between the sets are represented by the edges andassociated transfer matrices.

with the nth entry denoting the Fourier transform of thesignal observed at scatterer vertexSn. By the definition of thepropagation graph, there are no other signal sources than thevertices inVt. Assuming linear and time-invariant propagationmechanisms, the input-output relation in the Fourier domainreads

Y(f) = H(f)X(f), (11)

whereH(f) is theNr×Nt transfer matrix of the propagationgraph.

The structure of the propagation graph unfolds in the vectorsignal flow graph depicted in Fig. 3. The vertices of the vectorsignal flow graph represent the three setsVt, Vr, and Vs

with the associated signalsX(f),Y(f), andZ(f). The edgetransfer matrices of the vector signal flow graph are the sub-matrices ofA(f) defined in (4)–(7).

III. T RANSFERMATRIX OF A PROPAGATION GRAPH

In the following we derive the transfer matrix of a prop-agation graph. In Subsection III-A we first discuss how theresponse of a graph is composed of signal contributions prop-agating via different propagation paths. This representationis, albeit intuitive, impractical for computation of the transfermatrix of graphs with cycles. Thus in Subsections III-B andIII-C we give the transfer matrix and partial transfer matricesof a general propagation graph in closed form. Subsection III-D treats the graphical interpretation of reciprocal channels.The section concludes with a discussion of related results inthe literature.

A. Propagation Paths and Walks

The concept of a propagation path is a corner stone in mod-eling multipath propagation. In the literature, this concept ismost often defined in terms of the resulting signal componentsarriving at the receiver. A shortcoming of this definition isthat it is often hard to relate to the propagation environment.The graph terminology offers a convenient alternative. A walkℓ (of length K) in a graphG = (V , E) is defined as asequenceℓ = (v(1), v(2), . . . , v(K+1)) of vertices inV suchthat (v(k), v(k+1)) ∈ E , k = 1, . . .K. We say thatℓ is a walkfrom v(1) to v(K+1) wherev(1) is the start vertex andv(K+1)

Fig. 3. Vector signal flow graph representation of a propagation graph.Vertices represent vertex sets of the propagation graph with associated vectorsignals. Signal transmission between the sets are represented by the edges andassociated transfer matrices.

propagation graph, there are no other signal sources than thevertices in Vt. Assuming linear and time-invariant propagationmechanisms, the input-output relation in the Fourier domainreads

Y(f) = H(f)X(f), (11)

where H(f) is the Nr×Nt transfer matrix of the propagationgraph.

The structure of the propagation graph unfolds in the vectorsignal flow graph depicted in Fig. 3. The vertices of the vectorsignal flow graph represent the three sets Vt, Vr, and Vswith the associated signals X(f),Y(f), and Z(f). The edgetransfer matrices of the vector signal flow graph are the sub-matrices of A(f) defined in (4)–(7).

III. TRANSFER MATRIX OF A PROPAGATION GRAPH

In the following we derive the transfer matrix of a prop-agation graph. In Subsection III-A we first discuss how theresponse of a graph is composed of signal contributions prop-agating via different propagation paths. This representationis, albeit intuitive, impractical for computation of the transfermatrix of graphs with cycles. Thus in Subsections III-B andIII-C we give the transfer matrix and partial transfer matricesof a general propagation graph in closed form. Subsection III-D treats the graphical interpretation of reciprocal channels.The section concludes with a discussion of related results inthe literature.

A. Propagation Paths and Walks

The concept of a propagation path is a corner stone in mod-eling multipath propagation. In the literature, this concept ismost often defined in terms of the resulting signal componentsarriving at the receiver. A shortcoming of this definition isthat it is often hard to relate to the propagation environment.The graph terminology offers a convenient alternative. A walk` (of length K) in a graph G = (V, E) is defined as asequence ` = (v(1), v(2), . . . , v(K+1)) of vertices in V suchthat (v(k), v(k+1)) ∈ E , k = 1, . . .K. We say that ` is a walk

4

from v(1) to v(K+1) where v(1) is the start vertex and v(K+1)

is the end vertex; v(2), . . . , v(K) are called the inner verticesof `. We define a propagation path as a walk in a propagationgraph from a transmitter to a receiver. Consequently, all (ifany) inner vertices of a propagation path are scatters. A signalthat propagates along propagation path ` traverses K + 1 (notnecessarily different) edges and undergoes K interactions. Werefer to such a propagation path as a K-bounce path. Thezero-bounce path ` = (v, v′) is called the direct path, fromtransmitter v to receiver v′. As an example, referring to thegraph depicted in Fig. 2, it is straightforward to verify that`1 = (Tx1,Rx1) is a direct path, `2 = (Tx4,S6,Rx2) isa single-bounce path, and `3 = (Tx4,S1,S2,S1,Rx3) is a3-bounce path.

We denote by Lvv′ the set of propagation paths in Gfrom transmitter v to receiver v′. The signal received at v′

originating from transmitter v is the superposition of signalcomponents each propagating via a propagation path in Lvv′ .Correspondingly, entry (v, v′) of H(f) reads

Hvv′(f) =∑

`∈Lvv′

H`(f), (12)

where H`(f) is the transfer function of propagation path `.The number of terms in (12) equals the cardinality of Lvv′ ,

which may, depending on the structure of the graph, be finiteor infinite. As an example, the number of propagation pathsis infinite if v and v′ are connected via a directed cycle in G,i.e. if a propagation path contains a walk from an inner vertexback to itself. The graph in Fig. 2 contains two directed cycleswhich are connected to both transmitters and receivers.

In the case of an infinite number of propagation paths,computing Hvv′(f) directly from (12) is infeasible. Thisproblem is commonly circumvented by truncating the sum in(12) to approximate Hvv′(f) as a finite sum. This approach,however, calls for a method for determining how many termsof the sum should be included in order to achieve reasonableapproximation.

In the frequently used “K-bounce channel models”, propa-gation paths with more than K interactions are ignored. Thisapproach is motivated by the rationale that at each interaction,the signal is attenuated, and thus terms in (12) resulting frompropagation paths with a large number of bounces are weakand can be left out as they do not affect the sum much.This reasoning, however, holds true only if the sum of thecomponents with more than K interactions is insignificant,which may or may not be the case. From this consideration,it is clear that the truncation criterion is non-trivial as itessentially necessitates computation of the whole sum beforedeciding whether a term can be ignored or not.

B. Transfer Matrix for Recursive and Non-Recursive Propa-gation Graphs

As an alternative to the approximation methods applied tothe sum (12) we now give an exact closed-form expressionfor the transfer matrix H(f). Provided that the spectral radiusof B(f) is less than unity, the expression holds true for anynumber of terms in the sum (12) and thus holds regardlesswhether the number of propagation paths is finite or infinite.

Theorem 1: If the spectral radius of B(f) is less than unity,then the transfer matrix of a propagation graph reads

H(f) = D(f) + R(f)[I−B(f)]−1T(f). (13)

According to Theorem 1 the transfer matrix H(f) consistsof the two following terms: D(f) representing direct propa-gation between the transmitters and receivers and R(f)[I −B(f)]−1T(f) describing indirect propagation. The conditionthat the spectral radius of B(f) be less than unity implies thatfor any vector norm ‖ · ‖, ‖Z(f)‖ > ‖B(f)Z(f)‖ for non-zero ‖Z(f)‖, cf. [33]. For the Euclidean norm in particularthis condition implies the sensible physical requirement thatthe signal power strictly decreases for each interaction.

Proof: Let Hk(f) denote the transfer matrix for all k-bounce propagation paths, then H(f) can be decomposed as

H(f) =

∞∑

k=0

Hk(f), (14)

where

Hk(f) =

{D(f), k = 0

R(f)Bk−1(f)T(f), k > 0.(15)

Insertion of (15) into (14) yields

H(f) = D(f) + R(f)

( ∞∑

k=1

Bk−1(f)

)T(f). (16)

The infinite sum in (16) is a Neumann series converging to[I−B(f)]−1 if the spectral radius of B(f)) is less than unity.Inserting this in (16) completes the proof.

The decomposition introduced in (14) makes the effect ofthe recursive scattering directly visible. The received signalvector is a sum of infinitely many components resulting fromany number of interactions. The structure of the propagationmechanism is further exposed by (16) where the emitted vectorsignal is re-scattered successively in the propagation environ-ment leading to the observed Neumann series. This allowsfor modeling of channels with infinite impulse responses byexpression (13).

It is possible to arrive at (13) in an alternative, but lessexplicit, manner:

Proof: It is readily observed from the vector signal flowgraph in Fig. 3 that Z(f) can be expressed as

Z(f) = T(f)X(f) + B(f)Z(f). (17)

Since the spectral radius of B(f) is less than unity we obtainfor Z(f) the solution

Z(f) = [I−B(f)]−1T(f)X(f). (18)

Furthermore, according to Fig. 3 the received signal is of theform

Y(f) = D(f)X(f) + R(f)Z(f). (19)

Insertion of (18) in this expression yields (13).We remark that the above two proofs allow for propagation

paths with any number of bounces. This is highly preferable, asthe derived expression (13) is not impaired by approximation

5

errors due to the truncation of the series into a finite numberof terms as it occurs when using K-bounce models.

A significant virtue of the expression (13) is that prop-agation effects related to the transmitters and receivers areaccounted for in the matrices D(f), T(f) and R(f), but donot affect B(f). Consequently, the matrix [I−B(f)]−1 onlyneeds to be computed once even though the configuration oftransmitters and receivers changes. This is especially advanta-geous for simulation studies of e.g. spatial correlation as thisleads to a significant reduction in computational complexity.

C. Partial Transfer Matrices

The closed form expression (13) for the transfer matrix ofa propagation graph accounts for propagation via an arbitrarynumber of scatterer interactions. For some applications it is,however, relevant to study only some part of the impulseresponse associated with a particular number of interactions.One case is where a propagation graph is used to generateonly a part of the response and other techniques are used forthe remaining parts. Another case is when one must assessthe approximation error when the infinite series is truncated.In the following we derive a few useful expressions for suchpartial transfer matrices.

We define the K : L partial transfer matrix as

HK:L(f) =

L∑

k=K

Hk(f), 0 ≤ K ≤ L, (20)

i.e., we include only contributions from propagation paths withat least K, but no more than L bounces. It is straightforwardto evaluate (20) for K = 0, and L = 0, 1, 2:

H0:0(f) = D(f) (21)H0:1(f) = D(f) + R(f)T(f) (22)H0:2(f) = D(f) + R(f)T(f) + R(f)B(f)T(f). (23)

This expansion of the truncated series is quite intuitive butthe obtained expressions are increasingly complex for largeL. Theorem 2 gives a closed form expression of the partialtransfer function HK,L(f) for arbitrary K and L:

Theorem 2: The partial response HK:L(f) is given by

HK:L(f) =

D(f) + R(f)[I−BL(f)][I−B(f)]−1T(f),

K = 0, L ≥ 0

R(f)[BK−1(f)−BL(f)][I−B(f)]−1T(f),

0 < K ≤ L,

provided that the spectral radius of B(f) is less than unity.Proof: The partial transfer function for 0 ≤ K ≤ L reads

HK:L(f) =

∞∑

k=K

Hk(f)−∞∑

k′=L+1

Hk′(f)

= HK:∞(f)−HL+1:∞(f). (24)

For K = 0 we have H0:∞(f) = H(f) by definition; forK ≥ 1 we have

HK:∞(f) = R(f)

∞∑

k=K−1Bk(f)T(f)

= R(f)BK−1(f)

∞∑

k=0

Bk(f)T(f)

= R(f)BK−1(f)[I−B(f)]−1T(f). (25)

Inserting (25) into (24) completes the proof.Theorem 2 enables closed-form computation of HK:L(f)

for any K ≥ L. We have already listed a few partial transfermatrices in (21), (22), and (23). By definition the partialresponse HK:K(f) equals HK(f) for which an expressionis provided in (15). The transfer function of the K-bounceapproximation is equal to H0:K(f). Another special caseworth mentioning is HK+1:∞(f) = H(f)−H0:K(f) availablefrom (25), which gives the error due to the K-bounce approx-imation. Thus the validity of the K-bounce approximationcan be assessed by evaluating some appropriate norm ofHK+1:∞(f).

D. Reciprocity and Reverse Graphs

In most cases, the radio channel is considered reciprocal. Aswe shall see shortly, the graph terminology accommodates aninteresting interpretation of the concept of reciprocity. For anypropagation graph we can define the reverse graph in whichthe roles of transmitter and receiver vertices are swapped.The principle of reciprocity states that the transfer matrixof the reverse channel is equal to the transposed transfermatrix of the forward channel, i.e., a forward channel withtransfer matrix H(f) has a reverse channel with transfer matrixH(f) = HT(f). In the sequel we mark all entities related tothe reverse channel with a tilde.

We seek the relation between the forward graph G = (V, E)and its reverse G = (V, E) under the assumption of reciprocity.More specifically, we are interested in the relation betweenthe weighted adjacency matrix A(f) of G and the weightedadjacency matrix A(f) of G. We shall prove the followingtheorem:

Theorem 3: For a propagation graph G = (V, E) withweighted adjacency matrix A(f) and transfer matrix H(f)there exists a reverse graph G = (V, E) such that V = V , E ={(v, v′) : (v′, v) ∈ E}, A(f) = AT(f), and H(f) = HT(f).

Proof: We prove the Theorem by constructing a suit-able propagation graph such that the reciprocity condition isfulfilled. We first note that the set of transmitters, receivers,and scatterers is maintained for the reverse channel, thus thevertex set of G is V with Vt = Vr, Vr = Vt, and Vs = Vs. Itis immediately clear from the structure of G that the reversegraph G has no ingoing edges to Vt and no outgoing edgesfrom Vr and G is thus a propagation graph. Assuming thevertex indexing as in (2), the weighted adjacency matrix of Gis of the form

A(f) =

0 D(f) T(f)0 0 0

0 R(f) B(f)

, (26)

6

6

Y(f) X(f)

Z(f)

Vr = Vt Vt = Vr

Vs = Vs

D(f)

T(f) R

(f)

B(f)

Fig. 4. Vector signal flow graph representation of a reverse propagation graph.Compared to the forward graph depicted in Fig. 3 all edges arereversed.

where the transfer matricesD(f), R(f), T(f), andB(f) aredefined according to Fig. 4. EquatingA(f) and A(f) weobtain the identitiesD(f) = DT(f), B(f) = BT(f), T(f) =RT(f), and R(f) = TT(f). The relation betweenE and Enow follows from the definition of the weighted adjacencymatrix. The input-output relation of the reverse channel readsY(f) = H(f)X(f) whereX(f) is the signal vector emittedby the vertices inVt and Y(f) is the signal vector receivedby the vertices inVt. Considering Fig. 4 and arguing as inSection III-B yields for the reverse channel

H(f) = D(f) + R(f)[I− B(f)

]−1

T(f) (27)

Inserting forD(f), R(f), T(f), and B(f) leads to

H(f) = DT(f) +TT(f)[I−BT(f)

]−1RT(f)

= HT(f), (28)

where the last equality results from (13).In words Theorem 3 says that for a propagation graph there

exists a reverse graph which fulfills the reciprocity condition.Furthermore, an example of a propagation graph fulfillingthe reciprocity condition is the reverse graphG obtained byreversing all edges ofG while maintaining the edge transferfunctions.

E. Related Recursive Scattering Models

We provide a few examples of recursive models to assistthe reader in recognizing models which can be represented bythe graphical structure.

In [18] Shi and Nehorai consider a model for recursive scat-tering between point scatterers in a homogeneous background.The propagation between any point in space is described bya scalar Green’s function. The transfer function obtained byapplying the Foldy-Lax equation can also be obtained froma propagation graph by defining the sub-matrices ofA(f) asfollows. The model does not include a directed term and thusD(f) = 0. The entry of[T(f)]m1n is the Green’s functionfrom transmit vertexm1 to scatterern times the scatteringcoefficient of scatterern′. Similarly, the entry[R(f)]nm2

isthe Green’s function from the position of scatterern to receiverm2. The entry[B(f)]nn′ , n 6= n′ is the Green’s function fromthe position of scatterern to the position of scatterern′ times

the scattering coefficient of scatterern. Since a point scattererdoes not scatter back on itself, the diagonal entries ofB(f)are all zero. As can be observed from the above definitions,the assumption of homogeneous background medium leads tothe special case withEd = ∅, Et = Vt×Vs, Er = Vs×Vr, andEs = Vs × Vs.

Another modeling method that can be conveniently de-scribed using propagation graphs is (time-dependent) radiosity[20]–[24]. In these methods, the surfaces of the objects in thepropagation environment are first divided in smaller patches,then the power transfer between pairs of patches is definedin terms of the so-called “delay-dependent form factor”, andfinally the power transfer from the transmitter to the receiveris estimated. The delay-dependent form factor is essentiallya power impulse response between patches. It appears thatfor these algorithms no closed-form solution feasible fornumerical evaluation is available in the literature. Thus [20]–[24] resort to iterative solutions which can be achieved afterdiscretizing the inter-patch propagation delays. The time-dependent radiosity problem can be expressed in the Fourierdomain in terms of a propagation graph where each patchis represented by a scatterer, while the entries ofA(f) aredefined according to delay dependent form factor. Using thisformulation, a closed form expression of the channel transfermatrix appears immediately by Theorem 1 with no need forquantization of propagation delays.

F. Revisiting Existing Stochastic Radio Channel Models

It is interesting to revisit existing radio channel models bymeans of the just defined framework of propagation graphs.Such an effort may reveal some structural differences betweenmodels, which are not apparent merely from the mathematicalformulation. It is, however, a fact that the interpretationofa transfer function as a propagation graph is not unique—it is straightforward to construct different propagation graphswhich yield the same transfer matrix. Thus different propa-gation graphs may be associated to the same radio channelmodel. In the sequel we construct graphs for two well-knownstochastic channel models, i.e., the model by Turinet al. [34]and the model by Saleh and Valenzuela [35]. We include hereonly enough detail to allow for the discussion of the structureof the associated graphs and refer the reader to the originalpapers for the full description.

The seminal model [34] by Turinet al. can be expressed inthe frequency domain as

H(f) =∞∑

ℓ=0

αℓ exp(−j2πfτℓ), (29)

whereαℓ is the complex gain andτℓ denotes the delay ofcomponentℓ. Apart from the direct component, to which weassign indexℓ = 0, the indexℓ is merely a dummy indexwhich does not indicate any ordering of terms in (29). The set,{(τℓ, αℓ) : ℓ = 1, 2, . . . } is a marked Poisson point process on[τ0,∞) with complex marks{αℓ : ℓ = 1, 2, . . . }.

The propagation graph we seek should contain a propaga-tion path for each term in the sum (29). We notice that mod-ifying the termαℓ exp(−j2πfτℓ) has no effect on the other

Fig. 4. Vector signal flow graph representation of a reverse propagation graph.Compared to the forward graph depicted in Fig. 3 all edges are reversed.

where the transfer matrices D(f), R(f), T(f), and B(f) aredefined according to Fig. 4. Equating A(f) and A(f) weobtain the identities D(f) = DT(f), B(f) = BT(f), T(f) =RT(f), and R(f) = TT(f). The relation between E and Enow follows from the definition of the weighted adjacencymatrix. The input-output relation of the reverse channel readsY(f) = H(f)X(f) where X(f) is the signal vector emittedby the vertices in Vt and Y(f) is the signal vector receivedby the vertices in Vt. Considering Fig. 4 and arguing as inSection III-B yields for the reverse channel

H(f) = D(f) + R(f)[I− B(f)

]−1T(f) (27)

Inserting for D(f), R(f), T(f), and B(f) leads to

H(f) = DT(f) + TT(f)[I−BT(f)

]−1RT(f)

= HT(f), (28)

where the last equality results from (13).In words Theorem 3 says that for a propagation graph there

exists a reverse graph which fulfills the reciprocity condition.Furthermore, an example of a propagation graph fulfillingthe reciprocity condition is the reverse graph G obtained byreversing all edges of G while maintaining the edge transferfunctions.

E. Related Recursive Scattering Models

We provide a few examples of recursive models to assistthe reader in recognizing models which can be represented bythe graphical structure.

In [18] Shi and Nehorai consider a model for recursive scat-tering between point scatterers in a homogeneous background.The propagation between any point in space is described bya scalar Green’s function. The transfer function obtained byapplying the Foldy-Lax equation can also be obtained froma propagation graph by defining the sub-matrices of A(f) asfollows. The model does not include a directed term and thusD(f) = 0. The entry of [T(f)]m1n is the Green’s functionfrom transmit vertex m1 to scatterer n times the scatteringcoefficient of scatterer n′. Similarly, the entry [R(f)]nm2

is

the Green’s function from the position of scatterer n to receiverm2. The entry [B(f)]nn′ , n 6= n′ is the Green’s function fromthe position of scatterer n to the position of scatterer n′ timesthe scattering coefficient of scatterer n. Since a point scattererdoes not scatter back on itself, the diagonal entries of B(f)are all zero. As can be observed from the above definitions,the assumption of homogeneous background medium leads tothe special case with Ed = ∅, Et = Vt×Vs, Er = Vs×Vr, andEs = Vs × Vs.

Another modeling method that can be conveniently de-scribed using propagation graphs is (time-dependent) radiosity[20]–[24]. In these methods, the surfaces of the objects in thepropagation environment are first divided in smaller patches,then the power transfer between pairs of patches is definedin terms of the so-called “delay-dependent form factor”, andfinally the power transfer from the transmitter to the receiveris estimated. The delay-dependent form factor is essentiallya power impulse response between patches. It appears thatfor these algorithms no closed-form solution feasible fornumerical evaluation is available in the literature. Thus [20]–[24] resort to iterative solutions which can be achieved afterdiscretizing the inter-patch propagation delays. The time-dependent radiosity problem can be expressed in the Fourierdomain in terms of a propagation graph where each patchis represented by a scatterer, while the entries of A(f) aredefined according to delay dependent form factor. Using thisformulation, a closed form expression of the channel transfermatrix appears immediately by Theorem 1 with no need forquantization of propagation delays.

F. Revisiting Existing Stochastic Radio Channel Models

It is interesting to revisit existing radio channel models bymeans of the just defined framework of propagation graphs.Such an effort may reveal some structural differences betweenmodels, which are not apparent merely from the mathematicalformulation. It is, however, a fact that the interpretation ofa transfer function as a propagation graph is not unique—it is straightforward to construct different propagation graphswhich yield the same transfer matrix. Thus different propa-gation graphs may be associated to the same radio channelmodel. In the sequel we construct graphs for two well-knownstochastic channel models, i.e., the model by Turin et al. [34]and the model by Saleh and Valenzuela [35]. We include hereonly enough detail to allow for the discussion of the structureof the associated graphs and refer the reader to the originalpapers for the full description.

The seminal model [34] by Turin et al. can be expressed inthe frequency domain as

H(f) =

∞∑

`=0

α` exp(−j2πfτ`), (29)

where α` is the complex gain and τ` denotes the delay ofcomponent `. Apart from the direct component, to which weassign index ` = 0, the index ` is merely a dummy indexwhich does not indicate any ordering of terms in (29). The set,{(τ`, α`) : ` = 1, 2, . . . } is a marked Poisson point process on[τ0,∞) with complex marks {α` : ` = 1, 2, . . . }.

7 7

a) Turin et al.

bTx b Rx

b

b

bb

b

b

b) Saleh & Valenzuela

bTx b Rx

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

Fig. 5. Propagation graph representations of: a) a realization of the modelby Turin et al. [34]; and b) a realization of the Saleh-Valenzuela model [35].

terms in (29). This structure can be captured by constructinga propagation graph in which modifying vertices or edges in apropagation path leaves others unchanged. Graph theory offersthe convenient notion of “independent walks” [32]: Two ormore walks are called independent if none of them containsan inner vertex of another. In particular, propagation paths areindependent if they do not traverse the same scatterers. Thusfor independent propagation pathsℓ, ℓ′, changing pathℓ bychanging its edge transfer functions or by deleting edges fromit, has no effect on pathℓ′. We therefore associate to eachterm in (29) an independent propagation path. Assuming forsimplicity all paths to be single bounce paths this results inthe graph shown in Fig. 5(a).

The model by Saleh and Valenzuela [35] can be formulatedas a second-order Turin model:

H(f) =

∞∑

ℓ=0

exp(−j2πfτℓ)

∞∑

ℓ′=0

αℓℓ′ exp(−j2πfτℓℓ′), (30)

whereℓ is a cluster index andℓ′ is a index for the componentswithin a cluster. The set{τℓ : ℓ = 1, 2, . . . } is a Poissonprocess on[τ0,∞) and conditioned on the cluster delays{τℓ; ℓ = 0, 1, . . . }, the family of sets{(αℓℓ′ , τℓℓ′) : ℓ′ =1, 2, . . . }, ℓ = 0, 1, . . . , are independent marked Poissonprocesses. Using a similar argument as for the Turin model,leads to the graph depicted in Fig. 5(b). The transmitter isconnected to a set of super-ordinate scatterers, each scatterercorresponding to a “cluster”. These cluster-scatters are thenconnected to the receiver via independent single-bounce pathspassing through sub-ordinate scatterers. In this graph, deletingcluster scattererℓ makes a whole signal clusterℓ disappearin (30). Similarly, removing the sub-ordinate scattererℓ′ thecomponent with indexℓℓ′ in (30) vanishes.

We end the discussion of the Saleh-Valenzuela model bymentioning that many equivalent graphical interpretations maybe given. As an example, one may consider a graph structureas that of the reverse graph of Fig. 5(b). In such a structure,thetransmitter is directly connected to the sub-ordinate scattererswhereas the receiver is connected to the clusters. Indeed by

reversing any number of clusters we obtain an equivalentpropagation graph. These graphs share the common propertythat they contain no cycles.

IV. EXAMPLE : STOCHASTIC MODEL FOR IN-ROOM

CHANNEL

The concept of propagation graph introduced until nowcan be used for describing a broad range of channel models.In this section we apply these general results to a specificexample scenario where scatterer interactions are consideredto be non-dispersive in delay. This resembles the case whereall wave interactions can be considered as specular reflections,or considering points scatterers. The intent is to exemplify thatthe experimentally observed avalanche effect and diffuse tailcan be explained using only scatterer interactions that by them-selves yield no delay dispersion. We specify a method feasiblefor generating such a graph in Monte Carlo simulations. Themodel discussed in this example is a variant of the modelproposed in [30], [31].

A. Weighted Adjacency Matrix

We define the weighted adjacency matrix according to ageometric model of the environment. We consider a scenariowith a single transmitter, a single receiver, andNs scatterers,i.e., the vertex set readsV = Vt∪Vr∪Vs with Vt = {Tx},Vr ={Rx}, and Vs = {S1, . . . , SNs}. To each vertexv ∈ Vwe assign a displacement vectorrv ∈ R3 with respect to acoordinate system with arbitrary origin. To edgee = (v, v′)we associate the Euclidean distancede = ‖rv − rv′‖, the gainge, the phaseφe, and the propagation delayτe = de/c, wherec is the speed of light. The edge transfer functions are definedas

Ae(f) =

{ge(f) exp j(φe − 2πτef); e ∈ E0; e 6∈ E . (31)

The edge gains{ge(f)} are defined according to

g2e(f) =

1(4πfτe)2

; e ∈ Ed1

4πfµ(Et)· τ−2

e

S(Et); e ∈ Et

14πfµ(Er)

· τ−2e

S(Er); e ∈ Er

g2

odi(e)2 ; e ∈ Es

(32)

where odi(e) denotes the number of edges frominit(e) toother scatterers and for anyE ′ ⊆ E

µ(E ′) =1

|E ′|∑

e∈E′

τe and S(E ′) =∑

e∈E′

τ−2e , (33)

with | · | denoting cardinality. The weight of the direct edgeis selected according to the Friis equation [36] assumingisotropic antennas at both ends. The weights of edges inEtandEr also account for the antenna characteristics. They arecomputed at the average distance to avoid signal amplificationwhen scatterers are close to a transmitter or receiver, namelywhen the far-field assumption is invalid. The value of inter-scatterer gain is for simplicity assumed fixed, i.e.,ge(f) =g, e ∈ Es.

Fig. 5. Propagation graph representations of: a) a realization of the modelby Turin et al. [34]; and b) a realization of the Saleh-Valenzuela model [35].

The propagation graph we seek should contain a propaga-tion path for each term in the sum (29). We notice that mod-ifying the term α` exp(−j2πfτ`) has no effect on the otherterms in (29). This structure can be captured by constructinga propagation graph in which modifying vertices or edges in apropagation path leaves others unchanged. Graph theory offersthe convenient notion of “independent walks” [32]: Two ormore walks are called independent if none of them containsan inner vertex of another. In particular, propagation paths areindependent if they do not traverse the same scatterers. Thusfor independent propagation paths `, `′, changing path ` bychanging its edge transfer functions or by deleting edges fromit, has no effect on path `′. We therefore associate to eachterm in (29) an independent propagation path. Assuming forsimplicity all paths to be single bounce paths this results inthe graph shown in Fig. 5(a).

The model by Saleh and Valenzuela [35] can be formulatedas a second-order Turin model:

H(f) =

∞∑

`=0

exp(−j2πfτ`)∞∑

`′=0

α``′ exp(−j2πfτ``′), (30)

where ` is a cluster index and `′ is a index for the componentswithin a cluster. The set {τ` : ` = 1, 2, . . . } is a Poissonprocess on [τ0,∞) and conditioned on the cluster delays{τ`; ` = 0, 1, . . . }, the family of sets {(α``′ , τ``′) : `′ =1, 2, . . . }, ` = 0, 1, . . . , are independent marked Poissonprocesses. Using a similar argument as for the Turin model,leads to the graph depicted in Fig. 5(b). The transmitter isconnected to a set of super-ordinate scatterers, each scatterercorresponding to a “cluster”. These cluster-scatters are thenconnected to the receiver via independent single-bounce paths

passing through sub-ordinate scatterers. In this graph, deletingcluster scatterer ` makes a whole signal cluster ` disappearin (30). Similarly, removing the sub-ordinate scatterer `′ thecomponent with index ``′ in (30) vanishes.

We end the discussion of the Saleh-Valenzuela model bymentioning that many equivalent graphical interpretations maybe given. As an example, one may consider a graph structureas that of the reverse graph of Fig. 5(b). In such a structure, thetransmitter is directly connected to the sub-ordinate scattererswhereas the receiver is connected to the clusters. Indeed byreversing any number of clusters we obtain an equivalentpropagation graph. These graphs share the common propertythat they contain no cycles.

IV. EXAMPLE: STOCHASTIC MODEL FOR IN-ROOMCHANNEL

The concept of propagation graph introduced until nowcan be used for describing a broad range of channel models.In this section we apply these general results to a specificexample scenario where scatterer interactions are consideredto be non-dispersive in delay. This resembles the case whereall wave interactions can be considered as specular reflections,or considering points scatterers. The intent is to exemplify thatthe experimentally observed avalanche effect and diffuse tailcan be explained using only scatterer interactions that by them-selves yield no delay dispersion. We specify a method feasiblefor generating such a graph in Monte Carlo simulations. Themodel discussed in this example is a variant of the modelproposed in [30], [31].

A. Weighted Adjacency Matrix

We define the weighted adjacency matrix according to ageometric model of the environment. We consider a scenariowith a single transmitter, a single receiver, and Ns scatterers,i.e., the vertex set reads V = Vt∪Vr∪Vs with Vt = {Tx},Vr ={Rx}, and Vs = {S1, . . . ,SNs}. To each vertex v ∈ Vwe assign a displacement vector rv ∈ R3 with respect to acoordinate system with arbitrary origin. To edge e = (v, v′)we associate the Euclidean distance de = ‖rv− rv′‖, the gainge, the phase φe, and the propagation delay τe = de/c, wherec is the speed of light. The edge transfer functions are definedas

Ae(f) =

{ge(f) exp j(φe − 2πτef); e ∈ E0; e 6∈ E . (31)

The edge gains {ge(f)} are defined according to

g2e(f) =

1(4πfτe)2

; e ∈ Ed1

4πfµ(Et) ·τ−2e

S(Et) ; e ∈ Et1

4πfµ(Er) ·τ−2e

S(Er) ; e ∈ Erg2

odi(e)2 ; e ∈ Es

(32)

where odi(e) denotes the number of edges from init(e) toother scatterers and for any E ′ ⊆ E

µ(E ′) =1

|E ′|∑

e∈E′τe and S(E ′) =

∑

e∈E′τ−2e , (33)

8

with | · | denoting cardinality. The weight of the direct edgeis selected according to the Friis equation [36] assumingisotropic antennas at both ends. The weights of edges in Etand Er also account for the antenna characteristics. They arecomputed at the average distance to avoid signal amplificationwhen scatterers are close to a transmitter or receiver, namelywhen the far-field assumption is invalid. The value of inter-scatterer gain is for simplicity assumed fixed, i.e., ge(f) =g, e ∈ Es.

B. Stochastic Generation of Propagation Graphs

We now define a stochastic model of the sets {rv}, E , and{φe} as well as a procedure to compute the correspondingtransfer function and impulse response. The vertex positionsare assumed to reside in a bounded region R ⊂ R3 corre-sponding to the region of interest. The transmitter and receiverpositions are assumed to be fixed, while the positions of theNs scatterers {rv : v ∈ Vs} are drawn independently from auniform distribution on R.

Edges are drawn independently such that a vertex pair e ∈V × V is in the edge set E with probability Pe = Pr[e ∈ E ]defined as

Pe =

Pdir, e = (Tx,Rx)

0, term(e) = Tx

0, init(e) = Rx

0, init(e) = term(e)

Pvis otherwise

. (34)

The first case of (34) controls the occurrence of a directcomponent. If Pdir is zero, D(f) is zero with probabilityone. If Pdir is unity, the direct term D(f) is non-zero withprobability one. The second and third cases of (34) excludeingoing edges to the transmitter and outgoing edges from thereceiver. Thus the generated graphs will have the structuredefined in Section II-B. The fourth case of (34) excludes theoccurrence of loops in the graphs. This is sensible as a specularscatterer cannot scatter a signal back to itself. A consequenceof this choice is that any realization of the graph is looplessand therefore A(f) has zeros along its main diagonal. Thelast case of (34) assigns a constant probability Pvis of theoccurrence of edges from Vt to Vs, from Vs to Vs and fromVs to Vr.

Finally, the phases {φe : e ∈ E} are drawn independentlyfrom a uniform distribution on [0; 2π).

Given the deterministic values of parameters R, rTx, rRx,Ns, Pdir, Pvis and g, realizations of the (partial) transfer func-tion HK:L(f) and corresponding (partial) impulse responsehK:L(τ) can now be generated for a preselected frequencyrange [fmin, fmax], using the algorithm stated in Fig. 6 andAppendix.

The algorithm in Appendix invokes (13) once at eachfrequency point considered. The computational complexity isthus of order M . Evaluation of (13) at one frequency callsfor computation of the term R(f)(I −B(f))−1T(f). If thisis done by inverting the matrix I − B(f) the complexity isof order N3

s . If there are fewer transmitters or receivers thanscatterers the complexity of computing (13) can be reduced

1) Draw rv, v ∈ Vs uniformly on R2) Generate E according to (34)

3) Draw independent phases {φe : e ∈ E} uniformly on[0, 2π)

4) Compute A(f) within the frequency bandwidth using (31)

5) IF spectral radius of B(f) is larger than unity for somefrequency within the bandwidth THEN GOTO step 1

6) Estimate HK:L(f) and hK:L(τ) as described in Appendix

Fig. 6. Algorithm for generating full or partial transfer functions and impulseresponses for a preselected bandwidth.

TABLE IPARAMETER SETTINGS FOR THE NUMERICAL EXAMPLES

Parameters Symbol Values

Room size R [0, 5]× [0, 5]× [0, 2.6]m3

Transmitter position rTx [1.78, 1.0, 1.5]T m

Receiver position rRx [4.18, 4.0, 1.5]T mNumber of scatterers Ns 10Tail slope ρ –0.4dB/nsProb. of visibility Pvis 0.8Prob. of direct propagation Pdir 1Speed of light c 3 · 108 m/sTransmit signal X[m] Unit power Hann pulseNumber of frequency samples M 8192

somewhat depending on the particular numerical implementa-tion. Thus the complexity of the algorithm in Appendix is atmost cubic in Ns. In practice, e.g. with the settings in Table I,the complexity of this algorithm is low enough to be run on astandard desktop computer in less than a second per channelrealization.

C. Numerical Experiments

The effect of the recursive scattering phenomenon cannow be illustrated by numerical experiments. The parametersettings given in Table I are selected to mimic the experimentalsetup of [27] used to acquire the measurements reportedin Fig. 1. The room size and positions of the transmitterand receiver are chosen as in [27]. We consider the casewhere direct propagation occurs and set Pdir to unity. Theprobability of visibility Pvis and the number of scatterersNs are chosen to mimic the observed avalanche effect. Thevalue of g is set to match the tail slope ρ ≈ −0.4 dB/nsof the delay-power spectrum depicted in Fig. 1. The valueof g can be related to the slope ρ of the log delay-powerspectrum via the approximation ρ ≈ 20 log10(g)/µ(Es). Thisapproximation arises by considering the power balance for ascatterer: assuming the signal components arriving at the scat-terer to be statistically independent, neglecting the probabilityof scatterers with outdegree zero, and approximating the delaysof edges in Es by the average µ(Es) defined in (33). We remarkthat since µ(Es) depends on the realization of the graph, thegain coefficient g is a random variable.

Fig. 7 shows the amplitude of a single realization of thetransfer function. Overall, the squared amplitude of the transferfunction decays as f−2 due to the definition of {ge(f)}.Furthermore, the transfer function exhibits fast fading over theconsidered frequency band. The lower panel of Fig. 7 reports

9

10

[fmin, fmax] = [2, 3] [GHz]

|h(τ)|

[dB

]

τ [ns]

|H(f)|

[dB

]

f [GHz]

|H(f)|

[dB

]

f [GHz]

[fmin, fmax] = [1, 11] [GHz]

0 τTx,Rx20 40 60 80 100 120

1 2 3 4 5 6 7 8 9 10

2 2.2 2.4 2.6 2.8 3

−40

−30

−20

−10

0

10

20

30

40

−80

−70

−60

−50

−40

−70

−60

−50

−40

Fig. 7. Channel response for a specific realization of the propagation graph.Top: Transfer function in dB(20 log10 |H(f)|). Bottom: Impulse responsesin dB (20 log10 |h(τ)|) computed for two frequency bandwidths.

[fmin, fmax] = [2, 3] [GHz]

[fmin, fmax] = [2, 3] [GHz]

Pow

er[d

B]

τ [ns]

Pow

er[d

B]

[fmin, fmax] = [1, 11] [GHz]

[fmin, fmax] = [1, 11] [GHz]

0 τTx,Rx 20 40 60 80 100 120

0 τTx,Rx 20 40 60 80 100 120

−20

−10

0

10

20

30

40

−20

−10

0

10

20

30

40

Fig. 9. Simulated delay-power spectra. Top panel: Ensembleaverage of 1000Monte Carlo runs. Bottom panel: Spatial average of a single graph realizationassuming the same grid as in Fig. 1, i.e., 900 receiver positions on a 30×30horizontal square grid with1 × 1 cm2 mesh centered at positionrRx givenin Table I.

effect. This numerical experiment lead to the observationthat the diffuse tail can be generated even when scattererinteractions are non-dispersive in delay. Furthermore, the ex-ponential decay of the delay-power spectra can be explainedby the presence of recursive scattering. As illustrated by thesimulation results the proposed model, in contrast to existingmodels which treat dominant and diffuse components sepa-rately, provides a unified account by reproducing the avalancheeffect.

APPENDIX

The transfer functionH(f) and impulse responseh(τ ) canbe estimated as follows:

1) ComputeM samples of the transfer matrix within thebandwidth[fmin, fmax]

H[m] = H(fmin+m∆f ), m = 0, 1, . . .M−1, (35)

where ∆f = (fmax − fmin)/(M − 1) and H( · ) isobtained using Theorem 1.

2) Estimate the received signaly(τ ) via the inverse discreteFourier transform

y(i∆τ ) = ∆f

M−1∑

m=0

H[m]X[m] exp(j2πim/M),

i = 0, . . .M − 1,

whereX[m] = X(fmin + m∆f ),m = 0, 1, . . .M − 1and∆τ = 1/(fmax − fmin).

The impulse response can be estimated by lettingX[f ] be awindow function of unit power

∫ fmax

fmin

|X(f)|2df ≈M∑

m=0

|X[m]]|2∆f = 1, (36)

whereX[m] = X(fmin +m∆f ). The window function mustbe chosen such that its inverse Fourier transform exhibits anarrow main-lobe and sufficiently low side-lobes;y(τ ) is thenregarded as a good approximation of the impulse responseof the channel and by abuse of notation denoted byh(τ ).Samples of the partial transfer matrix are obtained by replacingH( · ) by HK:L( · ) in (35). The corresponding received partialimpulse response is denoted byhK:L(τ ).

ACKNOWLEDGMENT

We wish to thank Dr. -Ing. Jürgen Kunisch and Dipl.-Ing.Jörg Pamp for their kind permission to use Fig. 1.

REFERENCES

[1] H. Hashemi, “Simulation of the urban radio propagation,” IEEE Trans.Veh. Technol., vol. 28, pp. 213–225, Aug. 1979.

[2] D. Molkdar, “Review on radio propagation into and withinbuildings,”Microwaves, Antennas and Propagation, IEE Proceedings H, vol. 138,no. 1, pp. 61–73, 1991.

[3] N. Alsindi, B. Alavi, and K. Pahlavan, “Measurement and modeling ofultrawideband TOA-based ranging in indoor multipath environments,”IEEE Trans. Veh. Technol., vol. 58, no. 3, pp. 1046–1058, Mar. 2009.

[4] J. Kunisch and J. Pamp, “Measurement results and modeling aspectsfor the UWB radio channel,” inIEEE Conf. on Ultra Wideband Systemsand Technologies, 2002. Digest of Papers, May 2002, pp. 19–24.

Fig. 7. Channel response for a specific realization of the propagation graph.Top: Transfer function in dB (20 log10 |H(f)|). Bottom: Impulse responsesin dB (20 log10 |h(τ)|) computed for two frequency bandwidths.

the corresponding impulse response for two different signalbandwidths. Both impulse responses exhibit an avalancheeffect as well as a diffuse tail of which the power decaysnearly exponentially with ρ ≈ −0.4 dB/ns. As anticipated,the transition to the diffuse tail is most visible in the responseobtained with the larger bandwidth.

The build up of the impulse response can be examined viathe partial impulse responses given in Fig. 8. Inspection of thepartial responses when K = L reveals that the early part ofthe tail is due to signal components with a low K while thelate part is dominated by higher-order signal components. Itcan also be noticed that as K increases, the delay at whichthe maximum of the K-bounce partial response occurs andthe spread of this response are increasing.

Fig. 9 shows two types of delay-power spectra. The upperpanel shows the ensemble average of squared amplitudes of1000 independently drawn propagation graphs for the twosignal bandwidths also considered in Fig. 7. Both spectraexhibit the same trend: A clearly visible peak due to the directsignal is followed by a tail with nearly exponential powerdecay. As expected, the first peak is wider for the case with1 GHz bandwidth than for the case with 10 GHz bandwidth.The tails differ by approximately 7 dB. This shift arises dueto the f−2 trend of the transfer function resulting in a higherreceived power for the lower frequencies considered in the1 GHz bandwidth case. For the 1000 generated propagationgraphs, the gain coefficient appears approximately Gaussiandistributed with mean 0.65 and standard deviation 0.029. Thebottom panel shows spatially averaged delay-power spectraobtained for one particular realization of the propagation

10

[fmin, fmax] = [2, 3] [GHz]

|h(τ)|

[dB

]

τ [ns]

|H(f)|

[dB

]

f [GHz]

|H(f)|

[dB

]

f [GHz]

[fmin, fmax] = [1, 11] [GHz]

0 τTx,Rx20 40 60 80 100 120

1 2 3 4 5 6 7 8 9 10

2 2.2 2.4 2.6 2.8 3

−40

−30

−20

−10

0

10

20

30

40

−80

−70

−60

−50

−40

−70

−60

−50

−40

Fig. 7. Channel response for a specific realization of the propagation graph.Top: Transfer function in dB(20 log10 |H(f)|). Bottom: Impulse responsesin dB (20 log10 |h(τ)|) computed for two frequency bandwidths.

[fmin, fmax] = [2, 3] [GHz]

[fmin, fmax] = [2, 3] [GHz]

Pow

er[d

B]

τ [ns]

Pow

er[d

B]

[fmin, fmax] = [1, 11] [GHz]

[fmin, fmax] = [1, 11] [GHz]

0 τTx,Rx 20 40 60 80 100 120

0 τTx,Rx 20 40 60 80 100 120

−20

−10

0

10

20

30

40

−20

−10

0

10

20

30

40

Fig. 9. Simulated delay-power spectra. Top panel: Ensembleaverage of 1000Monte Carlo runs. Bottom panel: Spatial average of a single graph realizationassuming the same grid as in Fig. 1, i.e., 900 receiver positions on a 30×30horizontal square grid with1 × 1 cm2 mesh centered at positionrRx givenin Table I.

effect. This numerical experiment lead to the observationthat the diffuse tail can be generated even when scattererinteractions are non-dispersive in delay. Furthermore, the ex-ponential decay of the delay-power spectra can be explainedby the presence of recursive scattering. As illustrated by thesimulation results the proposed model, in contrast to existingmodels which treat dominant and diffuse components sepa-rately, provides a unified account by reproducing the avalancheeffect.

APPENDIX

The transfer functionH(f) and impulse responseh(τ ) canbe estimated as follows:

1) ComputeM samples of the transfer matrix within thebandwidth[fmin, fmax]

H[m] = H(fmin+m∆f ), m = 0, 1, . . .M−1, (35)

where ∆f = (fmax − fmin)/(M − 1) and H( · ) isobtained using Theorem 1.

2) Estimate the received signaly(τ ) via the inverse discreteFourier transform

y(i∆τ ) = ∆f

M−1∑

m=0

H[m]X[m] exp(j2πim/M),

i = 0, . . .M − 1,

whereX[m] = X(fmin + m∆f ),m = 0, 1, . . .M − 1and∆τ = 1/(fmax − fmin).

The impulse response can be estimated by lettingX[f ] be awindow function of unit power

∫ fmax

fmin

|X(f)|2df ≈M∑

m=0

|X[m]]|2∆f = 1, (36)

whereX[m] = X(fmin +m∆f ). The window function mustbe chosen such that its inverse Fourier transform exhibits anarrow main-lobe and sufficiently low side-lobes;y(τ ) is thenregarded as a good approximation of the impulse responseof the channel and by abuse of notation denoted byh(τ ).Samples of the partial transfer matrix are obtained by replacingH( · ) by HK:L( · ) in (35). The corresponding received partialimpulse response is denoted byhK:L(τ ).

ACKNOWLEDGMENT

We wish to thank Dr. -Ing. Jürgen Kunisch and Dipl.-Ing.Jörg Pamp for their kind permission to use Fig. 1.

REFERENCES

[1] H. Hashemi, “Simulation of the urban radio propagation,” IEEE Trans.Veh. Technol., vol. 28, pp. 213–225, Aug. 1979.

[2] D. Molkdar, “Review on radio propagation into and withinbuildings,”Microwaves, Antennas and Propagation, IEE Proceedings H, vol. 138,no. 1, pp. 61–73, 1991.

[3] N. Alsindi, B. Alavi, and K. Pahlavan, “Measurement and modeling ofultrawideband TOA-based ranging in indoor multipath environments,”IEEE Trans. Veh. Technol., vol. 58, no. 3, pp. 1046–1058, Mar. 2009.

[4] J. Kunisch and J. Pamp, “Measurement results and modeling aspectsfor the UWB radio channel,” inIEEE Conf. on Ultra Wideband Systemsand Technologies, 2002. Digest of Papers, May 2002, pp. 19–24.

Fig. 9. Simulated delay-power spectra. Top panel: Ensemble average of 1000Monte Carlo runs. Bottom panel: Spatial average of a single graph realizationassuming the same grid as in Fig. 1, i.e., 900 receiver positions on a 30×30horizontal square grid with 1 × 1 cm2 mesh centered at position rRx givenin Table I.

graph. The simulated spatial averaged delay-power spectraexhibit the avalanche effect similar to the one observed inFig. 1. For the 10 GHz bandwidth case the power level of thediffuse tails agrees remarkably well with the measurement inFig. 1. The modest deviation of about 3 dB can be attributedto antenna losses in the measurement.

V. CONCLUSIONS

The outset for this work was the observation that the mea-sured impulse responses and delay-power spectra of in-roomchannels exhibit an avalanche effect: Multipath componentsappear at increasing rate and gradually merge into a diffuse tailwith an exponential decay of power. We considered the ques-tion whether the avalanche effect is due to recursive scattering.To this end we modelled the propagation environment as agraph in which vertices represent transmitters, receivers, andscatterers and edges represent propagation conditions betweenvertices. From this general structure the graph’s full and partialfrequency transfer matrices were derived in closed form. Theseexpressions can, by specifying the edge transfer functions, bedirectly used to perform numerical simulations.

We considered as an example a graph-based stochasticmodel where all interactions are non-dispersive in delay ina scenario similar to an experimentally investigated scenariowhere the avalanche effect was observed. The impulse re-sponses generated from the model also exhibit an avalancheeffect. This numerical experiment lead to the observationthat the diffuse tail can be generated even when scattererinteractions are non-dispersive in delay. Furthermore, the ex-ponential decay of the delay-power spectra can be explainedby the presence of recursive scattering. As illustrated by thesimulation results the proposed model, in contrast to existingmodels which treat dominant and diffuse components sepa-rately, provides a unified account by reproducing the avalancheeffect.

10

9

4:∞4:4

3:∞3:43:3

2:∞2:42:3

h2:2(τ )

2:2

1:∞1:41:31:21:1

0:∞0:40:30:20:10:0

|h(τ)|

[dB

]

τ [ns]

2:2

h0:∞(τ )

0 25 50 75 100

0 50 100

−40

−30

−20

−10

0

10

20

30

40

−40

−20

0

20

40

Fig. 8. Partial responses obtained for one graph realization for the bandwidth[2, 3]GHz. The K : L settings are indicated in each miniature. Top row:K-bounce approximations. Right-most column: error terms resulting from(K−1)-bounce approximations. Main diagonal(K = L): K-bounce contributions.

the tail is due to signal components with a lowK while thelate part is dominated by higher-order signal components. Itcan also be noticed that asK increases, the delay at whichthe maximum of theK-bounce partial response occurs andthe spread of this response are increasing.

Fig. 9 shows two types of delay-power spectra. The upperpanel shows the ensemble average of squared amplitudes of1000 independently drawn propagation graphs for the twosignal bandwidths also considered in Fig. 7. Both spectraexhibit the same trend: A clearly visible peak due to the directsignal is followed by a tail with nearly exponential powerdecay. As expected, the first peak is wider for the case with1GHz bandwidth than for the case with 10GHz bandwidth.The tails differ by approximately 7 dB. This shift arises dueto thef−2 trend of the transfer function resulting in a higherreceived power for the lower frequencies considered in the1GHz bandwidth case.For the 1000 generated propagationgraphs, the gain coefficient appears approximately Gaussiandistributed with mean0.65 and standard deviation0.029. Thebottom panel shows spatially averaged delay-power spectraobtained for one particular realization of the propagationgraph. The simulated spatial averaged delay-power spectraexhibit the avalanche effect similar to the one observed in

Fig. 1. For the 10GHz bandwidth case the power level of thediffuse tails agrees remarkably well with the measurement inFig. 1. The modest deviation of about 3dB can be attributedto antenna losses in the measurement.

V. CONCLUSIONS

The outset for this work was the observation that the mea-sured impulse responses and delay-power spectra of in-roomchannels exhibit an avalanche effect: Multipath componentsappear at increasing rate and gradually merge into a diffusetailwith an exponential decay of power. We considered the ques-tion whether the avalanche effect is due to recursive scattering.To this end we modelled the propagation environment as agraph in which vertices represent transmitters, receivers, andscatterers and edges represent propagation conditions betweenvertices. From this general structure the graph’s full and partialfrequency transfer matrices were derived in closed form. Theseexpressions can, by specifying the edge transfer functions, bedirectly used to perform numerical simulations.

We considered as an example a graph-based stochasticmodel where all interactions are non-dispersive in delay ina scenario similar to an experimentally investigated scenariowhere the avalanche effect was observed. The impulse re-sponses generated from the model also exhibit an avalanche

Fig. 8. Partial responses obtained for one graph realization for the bandwidth [2, 3]GHz. The K : L settings are indicated in each miniature. Top row:K-bounce approximations. Right-most column: error terms resulting from (K−1)-bounce approximations. Main diagonal (K = L): K-bounce contributions.

APPENDIX

The transfer function H(f) and impulse response h(τ) canbe estimated as follows:

1) Compute M samples of the transfer matrix within thebandwidth [fmin, fmax]

H[m] = H(fmin+m∆f ), m = 0, 1, . . .M−1, (35)

where ∆f = (fmax − fmin)/(M − 1) and H( · ) isobtained using Theorem 1.

2) Estimate the received signal y(τ) via the inverse discreteFourier transform

y(i∆τ ) = ∆f

M−1∑

m=0

H[m]X[m] exp(j2πim/M),

i = 0, . . .M − 1, with X[m] = X(fmin + m∆f ) and∆τ = 1/(fmax − fmin).

The impulse response can be estimated by letting X[f ] be awindow function of unit power

∫ fmax

fmin

|X(f)|2df ≈M∑

m=0

|X[m]]|2∆f = 1, (36)

where X[m] = X(fmin +m∆f ). The window function mustbe chosen such that its inverse Fourier transform exhibits anarrow main-lobe and sufficiently low side-lobes; y(τ) is then

regarded as a good approximation of the impulse responseof the channel and by abuse of notation denoted by h(τ).Samples of the partial transfer matrix are obtained by replacingH( · ) by HK:L( · ) in (35). The corresponding received partialimpulse response is denoted by hK:L(τ).

ACKNOWLEDGMENT

We wish to thank Dr. -Ing. Jürgen Kunisch and Dipl.-Ing.Jörg Pamp for their kind permission to use Fig. 1.

REFERENCES

[1] H. Hashemi, “Simulation of the urban radio propagation,” IEEE Trans.Veh. Technol., vol. 28, pp. 213–225, Aug. 1979.

[2] D. Molkdar, “Review on radio propagation into and within buildings,”Microwaves, Antennas and Propagation, IEE Proceedings H, vol. 138,no. 1, pp. 61–73, 1991.

[3] N. Alsindi, B. Alavi, and K. Pahlavan, “Measurement and modeling ofultrawideband TOA-based ranging in indoor multipath environments,”IEEE Trans. Veh. Technol., vol. 58, no. 3, pp. 1046–1058, Mar. 2009.

[4] J. Kunisch and J. Pamp, “Measurement results and modeling aspectsfor the UWB radio channel,” in IEEE Conf. on Ultra Wideband Systemsand Technologies, 2002. Digest of Papers, May 2002, pp. 19–24.

[5] J. Poutanen, J. Salmi, K. Haneda, V. Kolmonen, F. Tufvesson, andP. Vainikainen, “Propagation characteristics of dense multipath compo-nents,” Antennas and Wireless Propagation Letters, IEEE, vol. 9, pp.791–794, 2010.

[6] J. Poutanen, J. Salmi, K. Haneda, V. Kolmonen, and P. Vainikainen,“Angular and shadowing characteristics of dense multipath componentsin indoor radio channels,” IEEE Trans. Antennas Propagat., vol. 59,no. 1, pp. 245–253, Jan. 2011.

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[10] R. F. Rudd, “The prediction of indoor radio channel impulse response,”in The Second European Conf. on Antennas and Propagation, 2007.EuCAP 2007., Nov. 2007, pp. 1–4.

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[12] T. H. Lehman, “A statistical theory of electromagnetic fields in complexcavities,” Otto von Guericke University of Magdeburg, Tech. Rep., May1993.

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[15] M. Lax, “Multiple scattering of waves,” Reviews of Modern Physics,vol. 23, no. 4, pp. 287–310, Oct. 1951.

[16] ——, “Multiple scattering of waves. II. The effective field in densesystems,” Physical Review, vol. 85, no. 4, pp. 621–629, Feb. 1952.

[17] M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scatteringof Light by Particles: Radiative Transfer and Coherent Backscattering.Cambridge University Press, May 2006.

[18] G. Shi and A. Nehorai, “Cramér-Rao bound analysis on multiplescattering in multistatic point-scatterer estimation,” IEEE Trans. SignalProcessing, vol. 55, no. 6, pp. 2840–2850, June 2007.

[19] M. Franceschetti, “Stochastic rays pulse propagation,” IEEE Trans.Antennas Propagat., vol. 52, no. 10, pp. 2742–2752, Oct. 2004.

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Troels Pedersen received the M.Sc.E. degree in dig-ital communications and the Ph.D. degree in wirelesscommunications from Aalborg University, Denmark,in 2004 and 2009, respectively. Since 2009 he hasbeen with Section Navigation and Communications,at Department of Electronic Systems, Aalborg Uni-versity, first as assistant professor and since 2012 asassociate professor. He received the ’Teacher of theYear 2011’ award by the Study Board for Electronicsand IT, Aalborg University. In Spring 2012 he wasa visiting professor at IETR, University Rennes 1,

France. His research interests lie in the area of statistical signal processingand communication theory, including sensor array signal processing, radiogeolocation techniques, radio channel modelling, and radio channel sounding.

Gerhard Steinböck received the DI (FH) degree intelecommunications from Technikum Wien, Austriain 1999. From 2000 to 2006, he worked as a R&Dengineer at the Austrian Institute of Technology(AIT), Vienna, Austria, contributing among otherthings in the hard- and software development of areal-time radio channel emulator. Gerhard Steinböckreceived the M.Sc.E. (cum laude) degree in wirelesscommunications from Aalborg University, Denmark,in 2008, where he is currently pursuing a Ph.D.degree in wireless communications. His research

interests lie in the area of wireless communications, radio channel modeling,radio channel estimation and sounding, and radio geolocation techniques.

Bernard H. Fleury (M’97–SM’99) received theDiploma in electrical engineering and mathematicsin 1978 and 1990, respectively, and the Ph.D. degreein electrical engineering from the Swiss FederalInstitute of Technology Zurich (ETHZ), Switzerland,in 1990.

Since 1997, he has been with the Department ofElectronic Systems, Aalborg University, Denmark,as a Professor of communication theory. He is theHead of the Section Navigation and Communica-tions, which is one of the eleven laboratories of

this Department. From 2006 to 2009, he was a Key Researcher with theTelecommunications Research Center Vienna (FTW), Austria. During 1978–1985 and 1992–1996, he was a Teaching Assistant and a Senior ResearchAssociate, respectively, with the Communication Technology Laboratory,ETHZ. Between 1988 and 1992, he was a Research Assistant with theStatistical Seminar at ETHZ.

Prof. Fleury’s research interests cover numerous aspects within communi-cation theory, signal processing, and machine learning – mainly for wirelesscommunications. His current scientific activities include stochastic modellingand estimation of the radio channel especially for MIMO systems operatingin harsh conditions, iterative message-passing processing with focus on thedesign of efficient feasible architectures for wireless receivers, localizationtechniques in wireless terrestrial systems, and radar signal processing. He hasauthored and co-authored more than 120 publications in these areas. He hasdeveloped with his staff a high-resolution method for the estimation of radiochannel parameters that has found a wide application and has inspired similarestimation techniques both in academia and in industry.