A Geometric Interpretation of Fading in Wireless Networks: Theory and Applications

A Geometric Interpretation of Fading in Wireless Networks: Theory and Applications

Abstract

In wireless networks with random node distribution, the underlying point process model and the channel fading process are usually considered separately. A unified framework is introduced that permits the geometric characterization of fading by incorporating the fading process into the point process model. Concretely, assuming nodes are distributed in a stationary Poisson point process in , the properties of the point processes that describe the path loss with fading are analyzed. The main applications are connectivity and broadcasting.

W

ireless networks, geometry, point process, fading, connectivity, broadcasting.

1 Introduction and System Model

1.1 Motivation

The path loss over a wireless link is well modeled by the product of a distance component (often called large-scale path loss) and a fading component (called small-scale fading or shadowing). It is usually assumed that the distance part is deterministic while the fading part is modeled as a random process. This distinction, however, does not apply to many types of wireless networks, where the distance itself is subject to uncertainty. In this case it may be beneficial to consider the distance and fading uncertainty jointly, i.e., to define a stochastic point process that incorporates both. Equivalently, one may regard the distance uncertainty as a large-scale fading component and the multipath fading uncertainty as small-scale fading component.

We introduce a framework that offers such a geometrical interpretation of fading and some new insight into its effect on the network. To obtain concrete analytical results, we will often use the Nakagami- fading model, which is fairly general and offers the advantage of including the special cases of Rayleigh fading and no fading for and , respectively.

The two main applications of the theoretical foundations laid in Section 2 are connectivity (Section 3) and broadcasting (Section 4).

Connectivity. We characterize the geometric properties of the set of nodes that are directly connected to the origin for arbitrary fading models, generalizing the results in [1, 2]. We also show that if the path loss exponent equals the number of network dimension, any fading model (with unit mean) is distribution-preserving in a sense made precise later.

Broadcasting. We are interested in the single-hop broadcast transport capacity, i.e., the cumulated distance-weighted rate summed over the set of nodes that can successfully decode a message sent from a transmitter at the origin. In particular, we prove that if the path loss exponent is smaller than the number of network dimensions plus one, this transport capacity can be made arbitrarily large by letting the rate of transmission approach 0.

In Section 5, we discuss several other applications, including the maximum transmission distance, probabilistic progress, the effect of retransmissions, and localization.

1.2 Notation and symbols

For convenient reference, we provide a list of the symbols and variables used in the paper. Most of them are also explained in the text. Note that slanted sans-serif symbols such as and denote random variables, in contrast to and that are standard real numbers or “dummy” variables. Since we model the distribution of the network nodes as a stochastic point process, we use the terms points and nodes interchangeably.

 SymbolDefinition/explanation[k]the set {1,2,…,k}1A(x)indicator functionu(x)≜1{x⩾0}(x) (unit step function)dnumber of dimensions of the networkoorigin in RdBa Borel subset of R or Rdcd≜πd/2/Γ(1+d/2) (volume~{}of the d-dim.~{}unit ball)αpath loss exponentδ≜d/αΔ≜(d+1)/αsminimum path gain for connectionF,\mathsfslffading distribution (cdf), fading r.v.FXdistribution of random variable X (cdf)Φ={\mathsfslxi}path loss process before fading (PLP)Ξ={ξi}path loss process with fading (PLPF)^Φ={^\mathsfslxi}points in Φ connected to % origin^Ξ={^ξi}points in Ξ connected to originΛ,λcounting measure and density for Φ^N=^Ξ(R+)number of nodes connected to o#Acardinality of A

1.3 Poisson point process model

A well accepted model for the node distribution in wireless networks1 is the homogeneous Poisson point process (PPP) of intensity . Without loss of generality, we can assume (scale-invariance).

Node distribution. Let the set , consist of the points of a stationary Poisson point process in of intensity , ordered according to their Euclidean distance to the origin . Define a new one-dimensional (generally inhomogeneous) PPP such that a.s. Let be the path loss exponent of the network and be the path loss process (before fading) (PLP). Let be an iid stochastic process with drawn from a distribution with unit mean, i.e., , and . Finally, let be the path loss process with fading (PLPF). In order to treat the case of no fading in the same framework, we will allow the degenerate case , resulting in . Note that the fading is static (unless mentioned otherwise), and that is no longer ordered in general. We will also interpret these point processes as random counting measures, e.g., for any Borel subset of .

Connectivity. We are interested in connectivity to the origin. A node is connected if its path loss is smaller than , i.e., if . The processes of connected nodes are denoted as (PLP) and (PLPF).

Counting measures. Let be the counting measure associated with , i.e., for Borel . For , we will also use the shortcut . Similarly, let be the counting measure for . All the point processes considered admit a density. Let and and be the densities of and , respectively.

Fading model. To obtain concrete results, we frequently use the Nakagami- (power) fading model. The distribution and density are

 F(x) =1−Γic(m,mx)Γ(m) (1) f(x) =mmxm−1exp(−mx)Γ(m), (2)

where denotes the upper incomplete gamma function. This distribution is a single-parameter version of the gamma distribution where both parameters are the same such that the mean is always.

1.4 The standard network

For ease of exposition, we often consider a standard network2 that has the following parameters: (path loss exponent equals the number of dimensions) and Rayleigh fading, i.e., .

Fig. 1 shows a PPP of intensity 1 in a square, with the nodes marked that can be reached from the center, assuming a path gain threshold of . The disk shows the maximum transmission distance in the non-fading case.

2 Properties of the Point Processes

Proposition 1

The processes , , and are Poisson.

{IEEEproof}

is Poisson by definition, so and are Poisson by the mapping theorem [3]. is Poisson since is iid, and . The Poisson property of will be established in Prop. 6.

Cor. 2 states some basic facts about these point processes that result from their Poisson property.

Corollary 2 (Basic properties.)

1. and . In particular, for , is stationary (on ).

2. is governed by the generalized gamma pdf

 f\mathsfslri(r)=e−cdrdd(cdrd)irΓ(i), (3)

and is distributed according to the cdf

 F\mathsfslxi(x)=1−Γic(i,cdxδ)Γ(i),. (4)

The expected path loss without fading is

 E\mathsfslxi=c−1/δdΓ(i+1/δ)Γ(i). (5)

In particular, for the standard network, the are Erlang with .

3. The distribution function of is

 Fξi(x)=1−∫∞0F(r/x)(cidδrδi−1exp(−cdrδ)Γ(i))dr. (6)

For and Nakagami- fading, the pdf of is

 fξi(x)=mm+1(m+i−1m)cidxi−1(m+cdx)m+i. (7)

In particular,

 Fξ1(x)=1−(mcdx+m)m (8)

and

 Eξi =micd(m−1)for m>1 (9) Varξi =m2i(m+i−1)c2d(m−1)2(m−2)for m>2. (10)

For the standard networks,

 Fξi(x)=(cdxcdx+1)i. (11)
{IEEEproof}
1. Since the original -dimensional process is stationary, the expected number of points in a ball of radius around the origin is . The one-dimensional process has the same number of points in , and , so . For , is constant.

2. Follows directly from the fact that is stationary Poisson. ((3) has been established in [4].)

3. The cdf is with distributed according to (4). (7) is obtained by straightforward (but tedious) calculation.

Remarks:

• For general (rational) values of , , and , can be expressed using hypergeometric functions.

• (8) approaches as , which is the distribution of . Similarly, and .

• Alternatively we could consider the path gain process . Since , the distribution functions look similar.

• In the standard network, the expected path loss does not exist for any , and for , the expected path gain is infinite, too, since both and are exponentially distributed. For , , and for , .

• For the standard network, the differential entropy is for and grows logarithmically with . For Nakagami- fading . For the path gain process in the standard network, the entropy has the simple expression

 h(ξ−1i)=i+1i+log(πi), (12)

which is monotonically decreasing, reflecting the fact that the variance is decreasing with .

• The are not independent since the are ordered. For example, in the case of the standard network, the difference is exponentially distributed with mean , thus the joint pdf is

 f\mathsfslx1…\mathsfslxn(x1,…,xn)=cnde−cdxn10

where denotes the (positive) order cone (or hyperoctant) in dimensions.

Proposition 3

For and any fading distribution with mean ,

 Ξ(B)\lx@stackreld=Φ(B)∀B⊂R+,

i.e., fading is distribution-preserving.

{IEEEproof}

Since is Poisson, independence of and for is guaranteed. So it remains to be shown that the intensities (or, equivalently, the counting measures on Borel sets) are the same. This is the case if for all ,

 E(#{\mathsfslxi:\mathsfslxi>a,ξia}),

i.e., the expected numbers of nodes crossing from the left (leaving the interval ) and the right (entering the same interval) are equal. This condition can be expressed as

 ∫a0λ(x)F(x/a)dx=∫∞aλ(x)(1−F(x/a))dx∀a>0.

If , , and the condition reduces to

 ∫10F(x)dx=∫∞1(1−F(x))dx,

which holds since

 ∫10(1−F(x))dx1−∫10F(x)dx+∫∞1(1−F(x))dx=E\mathsfslf=1.

An immediate consequence is that a receiver cannot decide on the amount of fading present in the network if and geographical distances are not known.

Corollary 4

For Nakagami- fading, , and any , the expected number of nodes with and , i.e., nodes that leave the interval due to fading, is

 E(#{\mathsfslxi:\mathsfslxia})=cdamm−1Γ(m)e−m. (14)

The same number of nodes is expected to enter this interval. For Rayleigh fading (), the fraction of nodes leaving any interval is .

{IEEEproof}

, and for Nakagami-, the fraction of nodes leaving the interval is

 ∫10F(x)dx=mm−1Γ(m)e−m.

Clearly, fading can be interpreted as a stochastic mapping from to . So, are the points in the geographical domain (they indicate distance), whereas are the points in the path loss domain, since is the actual path loss including fading. This mapping results in a partial reordering of the nodes, as visualized in Fig. 2. In the path loss domain, the connected nodes are simply given by .

Fig. 3 illustrates the situation for 200 nodes randomly chosen from with a threshold . Before fading, we expect 40 nodes inside. From these, a fraction is moving out (right triangles), the rest stays in (marked by ). From the ones outside, a fraction moves in (left triangles), the rest stays out (circles).

For the standard network, the probability of point reordering due to fading can be calculated explicitly. Let . By this definition,

 Pi,j=P[\mathsfslxi/\mathsfslfi>\mathsfslxi+j/\mathsfslfi+j]=P[\mathsfslxi\mathsfslxi+\mathsfslyj>\mathsfslfi\mathsfslfi+j]. (15)

is Erlang with parameters and , is the distance from to and thus Erlang with parameters and , and the cdf of is . Hence

 Pi,j= E\mathsfslx,\mathsfsly(\mathsfslxi2\mathsfslxi+\mathsfslyj) = ∫∞0∫∞0x2x+yci+jdxi−1yj−1Γ(i)Γ(j)e−cd(x+y)dxdy.

does not depend on . Closed-form expressions include , and . Generally can be determined analytically. For , we obtain . Further, , which is the probability that an exponential random variable is larger than another one that has twice the mean.

In the limit, as , , which is the probability that a node has the largest fading coefficient among nodes that are at the same distance. Indeed, as , a.s. for any and finite .

While the are dependent, it is often useful to consider a set of independent random variables, obtained by conditioning the process on having a certain number of nodes in an interval (or, equivalently, conditioning on ) and randomly permuting the nodes. In doing so, the points and , are iid distributed as follows.

Corollary 5

Conditioned on :

1. The nodes are iid distributed with

 fa\mathsfslxi(x)=λ(x)Λ(x)=δ(xa)δ1x,0⩽x

and cdf .

2. The path loss with fading is distributed as

 Faξi(x)=1−∫a0F(y/x)δ(ya)δ1ydy. (17)
3. For the standard network,

 Faξi(x)=xa(1−e−a/x) (18)
4. For Rayleigh fading and ,

 Faξi(x)=√π2√xaerf(√ax). (19)
{IEEEproof}

As in (6), the cdf is given by with distributed as (16).

3 Connectivity

Here we investigate the processes and of connected nodes.

3.1 Single-transmission connectivity and fading gain

Proposition 6 (Connectivity)

Let a transmitter situated at the origin transmit a single message, and assume that nodes with path loss smaller than can decode, i.e., are connected. We have:

1. is Poisson with .

2. With Nakagami- fading, the number of connected nodes is Poisson with mean

 E^Nm=cd(ms)δΓ(δ+m)Γ(m) (20)

and the connectivity fading gain, defined as the ratio of the expected numbers of connected nodes with and without fading, is

 E^NmE^N∞=1mδΓ(δ+m)Γ(m)=E(\mathsfslfδ). (21)
Proof.
1. The effect of fading on the connectivity is independent (non-homogeneous) thinning by .

2. Using (a), the expected number of connected nodes is

 ∫∞0^λ(x)dx=∫∞0cdδxδ−1Γic(m,msx)Γ(m)dx

which equals in the assertion. Without fading, , which results in the ratio (21).

Remarks:

1. (20) is a generalization of a result in [1] where the connectivity of a node in a two-dimensional network with Rayleigh fading was studied.

2. can also be expressed as

 E^N=∞∑i=1P[ξi<1/s]. (22)

The relationship with part (b) can be viewed as a simple instance of Campbell’s theorem [5]. Since is Poisson, the probability of isolation is .

3. , and . For , does not depend on the type (or presence) of fading.

4. The connectivity fading gain equals the -th moment of the fading distribution, which, by definition, approaches one as the fading vanishes, i.e., as . For a fixed , it is decreasing in if , increasing if , and equal to for all if . It also equals if . For a fixed , it is not monotonic with , but exhibits a minimum at some . The fading gain as a function of and is plotted in Fig. 4. For Rayleigh fading and , the fading gain is , and the minimum is assumed at , corresponding to for . So, depending on the type of fading and the ratio of the number of network dimensions to the path loss exponent , fading can increase or decrease the number of connected nodes.

5. For the standard network, and the probability of isolation is .

6. The expected number of connected nodes with is

 E^Na=cdaδFaξi(1/s). (23)

where is given in (17).

Corollary 7

Under Nagakami- fading, a uniformly randomly chosen connected node has mean

 E^\mathsfslx=δ(δ+m)ms(δ+1), (24)

which is times the value without fading.

Proof.

A random connected node is distributed according to

 f^\mathsfslx(x)=^λ(x)E^N. (25)

Without fading, the distribution is , , resulting in an expectation of . ∎

For Rayleigh fading, for example, the density is a gamma density with mean , so the average connected node is times further away than without fading.

3.2 Connectivity with retransmissions

Assuming a block fading network and transmissions of the same packet, what is the process of nodes that receive the packet at least once?

Corollary 8

In a network with iid block fading, the density of the process of nodes that receive at least one of transmissions is

 ^λn(x)=(1−F(sx)n)cdδxδ−1. (26)
{IEEEproof}

This is a straightforward generalization of Prop. 6(a). So, in a standard network, the number of connected nodes with transmissions

 E^Nn=∫∞0^λn(x)dx=cds(Ψ(n+1)+γ), (27)

where is the digamma function (the logarithmic derivative of the gamma function), which grows with . Alternatively if the threshold for the -th transmission is chosen as , , the expected number of nodes reached increases linearly with the number of transmissions.

4 Broadcasting

4.1 Broadcasting reliability

Proposition 9

For and Nakagami- fading, , the probability that a randomly chosen node can be reached is

 pm(~s)=1~s(1−exp(−m~s)m−1∑k=0mk(1−k/m)k!~sk), (28)

where . is increasing in for all and converges uniformly to

 limm→∞pm(~s)=min{1,~s−1}. (29)
{IEEEproof}

is given by

 pm(~s)=∫10(1−F(~sx))dx=∫10Γ(m,m~sx)Γ(m)dx. (30)

For , this is

 pm(~s)=m−1∑k=0∫10exp(−m~sx)(m~sx)kk!dx, (31)

which, after some manipulations, yields

 pm(~s) =1~s(1−1mexp(−m~s)m−1∑k=0k∑j=0(m~s)jj!) (32) =1~s⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝1−exp(−m~s)m−1∑k=0mk(1−k/m)k!~skPm−1(~s)⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (33)

The polynomial is the Taylor expansion of order of at (the coefficient for is zero). So from which the limit for follows. For , the exponential dominates the polynomial so that their product tends to zero and remains as the limit.

The convergence to is the expected behavior, since without fading a node is connected if it is positioned within () and for a randomly chosen node in for or , this has probability . So with increasing , derivatives of higher and higher order become 0 at . From the previous discussion we know that . Calculating the coefficient for yields

 pm(~s)=1−mmΓ(m+2)~sm+O(~sm+1). (34)

The -th order Taylor expansion at is a lower bound. Upper bounds are obtained by truncating the polynomial; a natural choice is the first-order version to obtain

 (1−mmΓ(m+2)~sm)+

Using the lower bound, we can establish the following Corollary.

Corollary 10 (ϵ-reachability.)

If

 as<(Γ(m+2)⋅ϵ)1/mm. (36)

at least a fraction of the nodes are connected. In the standard network (specializing to ), the sufficient condition is

 as<2ϵ, (37)

This follows directly from the lower bound in (35).

Remarks:

• For , the bound (36) is not tight since the RHS converges to for all positive (by Stirling’s approximation), while the exact condition is .

• The sufficient condition (37) is tight (within 7%) for . With , the condition can be solved exactly using the Lambert W function:

 as

A linear approximation yields the same bound as before, while a quadratic expansion yields the sufficient condition which is within for .

4.2 Broadcast transport sum-distance and capacity

Assuming the origin transmits, the set of nodes that receive the message is . We shall determine the broadcast transport sum-distance , i.e., the expected sum over the all the distances from the origin:

 D≜E⎛⎜⎝∑\mathsfslx∈^Φ\mathsfslx1/α⎞⎟⎠ (39)
Proposition 11

The broadcast transport sum-distance for Nakagami- fading is

 Dm=cdδΔ1(ms)ΔΓ(m+Δ)Γ(m), (40)

and the (broadcast) fading gain is

 DmD∞=1mΔΓ(m+Δ)Γ(m)=E(\mathsfslfΔ). (41)
{IEEEproof}

From Campbell’s theorem

 E⎛⎜⎝∑\mathsfslx∈^Φ\mathsfslx1/α⎞⎟⎠ =∫∞0x1/α^λ(x)dx =cdδ∫∞0x1/α+δ−1(1−F(sx))% dx,

which equals (40) for Nakagami- fading.

Without fading, a node is connected if , therefore

 D∞ =∫1/s0x1/αλ(x)dx (42) =cdδΔs−Δ=cddd+1s−Δ. (43)

So the fading gain is the -th moment of as given in (41).

Remarks:

1. The fading gain is independent of the threshold . for all . It strongly resembles the connectivity gain (Prop. 6), the only difference being the parameter instead of . In particular, is independent of if . See Remark 3 to Prop. 6 and Fig. 4 for a discussion and visualization of the behavior of the gain as a function of and .

2. For Rayleigh fading (), , and the fading gain is . For , .

3. The formula for the broadcast transport sum-distance reminds of an interference expression. Indeed, by simply replacing by , a well-known result on the mean interference is reproduced: Assuming each node transmits at unit power, the total interference at the origin is

 Missing or unrecognized delimiter for \Big

which for diverges due to the lower bound integration bound (i.e., the one or two closest nodes) and for diverges due to the upper bound (i.e., the large number of nodes that are far away).

So far, we have ignored the actual rate of transmission and just used the threshold for the sum-distance. To get to the single-hop broadcast transport capacity (in bit-meters/s/Hz), we relate the (bandwidth-normalized) rate of transmission and the threshold by and define

 C≜maxR>0{R⋅D(2R−1)}=maxs>0{log2(1+s)D(s)}. (44)

Let be the broadcast transport sum-distance for (see Prop. 11) such that .

Proposition 12

For Nakagami- fading:

1. For , the broadcast transport capacity is achieved for

 Ropt=W(−e−1/ΔΔ)+Δ−1log2,Δ∈(0,1). (45)

The resulting broadcast transport capacity is tightly (within at most 0.13%) lower bounded by

 Cm⩾D1m(Δ)log2(Δ−1−Δ)(eΔ−1−Δ−1)−Δ. (46)
2. For ,

 Cm=cdδlog2 (47)

independent of , and .

3. For , the broadcast transport capacity increases without bounds as , independent of the transmit power.

{IEEEproof}
1. , so which, for , has a maximum at given in (45). The lower bound stems from an approximation of using which holds since for , the two expressions are identical, and the derivative of the Lambert W expression is smaller than -1 for .

2. For , increases as the rate is lowered but remains bounded as . The limit is .

3. For , is decreasing with , and .

Remarks:

• The optima for , are independent of the type of fading (parameter ).

• For , the optimum is tightly lower bounded by

 sopt⩾exp(Δ−1−Δ)−1. (48)

This is the expression appearing in the bound (46).

• (c) is also apparent from the expression , which, for , is approximately . So, the intuition is that in this regime, the gain from reaching additional nodes more than offsets the loss in rate.

• For , and . This is, however, not the minimum. The capacity is minimum around , depending slightly on .

Fig. 5 depicts the optimum rate as a function of , together with the lower bound , and Fig. 6 plots the broadcast transport capacity for Rayleigh fading and no fading for a two-dimensional network. The range corresponds to a path loss exponent range . It can be seen that Nakagami fading is harmful. For small values of , the capacity for Rayleigh fading is about 10% smaller.