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

1 A Geometric Interpret ation of Fading in W ireless Netw orks: Theory and Ap plicati ons Martin Haenggi, Senior Member , IEEE Abstract In w ireless networks with ran dom n ode distribution, th e un derlying p oint pro cess mod el and the channel fading process are usually consider ed separately . A unified framew ork is introduced that permits the g eometric ch aracterization of fading by incorporatin g the fading pr ocess into the point process m odel. Concretely , assuming n odes are distributed in a stationary Poisson point pro cess in R d , the proper ties of the po int processes that de scribe the path loss with fading are a nalyzed. The m ain application s are connectivity an d broadc asting. Index T erms W ireless n etworks, geometry , point process, fading, connectivity , br oadcasting. I . I N T RO D U C T I O N A N D S Y S T E M M O D E L A. Motivation The path loss over a wireless link is well mode led by the p roduct o f a distance co mponent (often c alled lar ge-scale path loss) a nd a f ad ing component ( called small-scale f a ding or shadowi ng). It is usually assume d that the distance part is deterministic while the fading part is modeled as a random process. This distinction, howe ver , does not apply to many types of wireless n etworks, where the distance itself is subjec t to uncertainty . In this ca se it may be beneficial to cons ider the distance an d fading uncertainty jointly , i.e. , to define a stocha stic point proce ss that incorporates both. E quiv alen tly , one may regard the distance u ncertainty as a large-scale fading compone nt and the multipath fading unc ertainty as small-scale fading c omponen t. This paper is an e xtension of preliminary w ork that has app eared at ISIT 2006, Seattle, W A, and IS IT 2007, Nice, France. M. Hae nggi is with the Department of E lectrical Engineering, Univ ersity of Notre D ame, Notre Dame, IN 46556, USA. E-mail : mhaenggi@nd.e du Nov ember 4, 2018 DRAFT 2 W e intr oduce a framework that of fers such a geometrical interpretation of fading and s ome new ins ight into its effect on the network. T o obtain c oncrete an alytical results, we will often us e the Nakaga mi- m fading model, which is fairl y gene ral and of fers the a dvantage of inc luding the special cases of Rayleigh fading a nd no fading for m = 1 and m → ∞ , respec ti vely . The two main applications of the theoretical foun dations laid in Section 2 are co nnectivity (Section 3) and broadcas ting (Section 4). Connectivity . W e ch aracterize the geometric properties of the set of nodes that are directly con nected to the o rigin for arbitrary fading mod els, g eneralizing the resu lts in [1], [2]. W e also show that if the path los s expone nt equals the n umber o f network d imension, a ny fading model (with unit me an) is distrib ution-preserving in a sens e made precise later . Br oa dcasting. W e are interested in the single-hop br oad cast transport capacity , i.e. , the cumulated distance-weigh ted rate summed ov er the set of nodes that can succe ssfully dec ode a messag e sent from a transmitter at the origin. In particular , we prove that if the path loss expo nent i s smaller than the number of network d imensions plus one, this transport capac ity can be made arbitrarily large by letti ng the rate of transmission approach 0. In Section 5, we disc uss se veral othe r applications, including the maximum transmission distance , probabilistic progress, the ef fect of retransmissions, and localization. B. Notation and symbols For con venient reference, we provide a list of the symbols and variables used in the paper . Most of them a re also explaine d in the text. Note that slanted sans -serif symbols such as x and f denote ran dom variables, in c ontrast to x and f that are standa rd real nu mbers or “dummy” variables. Since we mo del the distributi on of the ne twork n odes as a stocha stic point proc ess, we use the terms points an d nodes interchangea bly . Nov ember 4, 2018 DRAFT 3 Symbol Definition/explanation [ k ] the set { 1 , 2 , . . . , k } 1 A ( x ) indicator function u ( x ) , 1 { x > 0 } ( x ) (unit step function) d number of dimensions of the network o origin in R d B a Borel subs et of R or R d c d , π d/ 2 / Γ(1 + d/ 2) (vol ume of the d -dim. unit ball) α path loss exponen t δ , d/α ∆ , ( d + 1) /α s minimum pa th gain for connec tion F , f fading distrib u tion (cdf), fading r . v . F X distrib ution of random variable X (cdf) Φ = { x i } path loss proces s before fading (PLP) Ξ = { ξ i } path loss proces s with fading (PLPF) ˆ Φ = { ˆ x i } points in Φ connec ted to origin ˆ Ξ = { ˆ ξ i } points in Ξ connec ted to origin Λ , λ counting measure and density for Φ ˆ N = ˆ Ξ( R + ) number of nodes connec ted to o # A cardinality of A C. P oisso n p oint pr oc ess mo del A well accepted model for the no de distribution in wireless ne tworks 1 is the homogeneo us P oisson point pr o cess (PPP) o f intensity λ . W ithout loss of generality , we c an assume λ = 1 (sca le-in variance). Node distribution. Le t the set { y i } , i ∈ N consis t of the p oints of a stationary Poisson point process in R d of inten sity 1 , ordered acc ording to their Euc lidean distance k y i − o k to the origin o . De fine a new 1 In particular, if nod es mo ve around randomly and independ ently , or if sensor no des are deployed fr om an airplane i n large quantities. Nov ember 4, 2018 DRAFT 4 one-dimension al (gene rally inhomo geneou s) PPP { r i , k y i − o k} such that 0 < r 1 < r 2 < . . . a.s. Let α > 0 be the path loss expon ent of the network and Φ = { x i , r α i } be the path loss pr ocess (before fading) (PLP). Let { f , f 1 , f 2 , . . . } be a n iid s tochastic proce ss with f drawn from a distrib ution F , F f with unit mea n, i.e. , E f = 1 , and sup p f ⊂ R + . Finally , let Ξ = { ξ i , x i / f i } be the path loss pr oc ess with fad ing (PLPF). In order to treat the cas e of n o fading in the same framew ork, we will allow the degenerate case F ( x ) = u ( x − 1) , resu lting in Φ = Ξ . Note that the fading is static (unles s mentioned otherwise), and that { ξ i } is no lon ger o rdered in general. W e will also interpret these point processe s as random counting me asures , e.g. , Φ( B ) = # { Φ ∩ B } for a ny Borel su bset B of R . Connectivity . W e are interested in co nnectivity to the origin. A no de i is c onnected if its p ath loss is smaller than 1 /s , i.e. , if ξ i < 1 /s . The process es of connected no des are denoted as ˆ Φ = { x i : ξ i < 1 /s } (PLP) and ˆ Ξ = { ξ i : ξ i < 1 /s } = Ξ ∩ [0 , 1 /s ) (PLPF). Counting measures. Let Λ b e the cou nting meas ure asso ciated with Φ , i.e. , Λ( B ) = E Φ( B ) for Bo rel B . For Λ([0 , a )) = E Φ([0 , a )) , we will also use the shortcut Λ( a ) . Similarl y , let ˆ Λ be the counting measure for ˆ Φ . All the po int p rocesses cons idered admit a dens ity . Let λ ( x ) = d Λ( x ) / d x and an d ˆ λ ( x ) = d ˆ Λ( x ) / d x be the densities of Φ and ˆ Φ , respec ti vely . F ading model. T o obtain c oncrete resu lts, we frequently use the Nak agami- m (power) f ading model. The distrib ution and density are F ( x ) = 1 − Γ ic ( m, mx ) Γ( m ) (1) f ( x ) = m m x m − 1 exp( − mx ) Γ( m ) , (2) where Γ ic denotes the uppe r incomplete gamma func tion. Th is distribution is a single-parameter version of the gamma distrib u tion where both parameters are the same such that the mean is 1 alw ays. D. The standard network For ease of exposition, we often consider a stan dard networ k 2 that ha s the following parameters: δ , d/α = 1 (path loss exponent equa ls the number of d imensions) a nd Ra yleigh fading, i.e. , F ( x ) = (1 − e − x ) u ( x ) . Fig. 1 sho ws a PPP of inten sity 1 in a 16 × 16 squ are, with the n odes marked that can be reached from the cente r , assuming a path gain threshold of s = 0 . 1 . The disk shows the maximum transmission distance in the non-fading case. 2 The term “standard” here refers to the fact that in this case the analytical ex pressions are particularly simple. W e do not claim that these parameters are the ones most frequen tly observ ed in r eality . Nov ember 4, 2018 DRAFT 5 −8 −6 −4 −2 0 2 4 6 8 −8 −6 −4 −2 0 2 4 6 8 Fig. 1. A Poisson point process of intensity 1 in a 16 × 16 square. The reachable nodes by the center node are indicated by a bold × for a path gain threshold of s = 0 . 1 , a path loss exponen t of α = 2 , and R ayleigh fading (standard network). The circle indicates the range of successful transmission in the no n-fading case. I ts radius is 1 / √ s ≈ 3 . 16 , and there are about π /s ≈ 31 nodes inside. I I . P R O P E RT I E S O F T H E P O I N T P RO C E S S E S Proposition 1 The pr o cesse s Φ , Ξ , and ˆ Ξ are P oiss on. Pr oo f: { y i } is Poisson by definition, s o { r i } a nd Φ = { x i } a re Poiss on by the mapping theorem [3]. Ξ is Poisson since f i is iid, and ˆ Ξ( R ) = Ξ ([0 , 1 /s )) . The Poisson property of ˆ Φ will be established in Prop. 6. Cor . 2 states some basic facts a bout these point process es that result from their Poisson property . Corollary 2 (Basic properties.) (a) Λ( x ) = E Φ ([0 , x )) = c d x δ and λ ( x ) = c d δ x δ − 1 . In particular , for δ = 1 , Φ is stationar y (on R + ). (b) r i is governed by the generalized gamma pdf f r i ( r ) = e − c d r d d ( c d r d ) i r Γ( i ) , (3) Nov ember 4, 2018 DRAFT 6 and x i is distributed ac cording to the cdf F x i ( x ) = 1 − Γ ic ( i, c d x δ ) Γ( i ) , . (4) The expected p ath loss without fading is E x i = c − 1 /δ d Γ( i + 1 / δ ) Γ( i ) . (5) In particular , for the standa r d ne twork, the x i ar e Erlang with E x i = i/c d . (c) The distribution function o f ξ i is F ξ i ( x ) = 1 − Z ∞ 0 F ( r /x )  c i d δ r δi − 1 exp( − c d r δ ) Γ( i )  d r . (6) F or δ = 1 an d Nakagami- m fad ing, the pdf of ξ i is f ξ i ( x ) = m m +1  m + i − 1 m  c i d x i − 1 ( m + c d x ) m + i . (7) In particular , F ξ 1 ( x ) = 1 −  m c d x + m  m (8) and E ξ i = mi c d ( m − 1) for m > 1 (9) V ar ξ i = m 2 i ( m + i − 1) c 2 d ( m − 1) 2 ( m − 2) for m > 2 . (10) F or the s tandard networks, F ξ i ( x ) =  c d x c d x + 1  i . (11) Pr oo f: (a) Since the original d -dimensional proces s { y i } is sta tionary , the expected number of points in a ball of radius x aroun d the origin is c d x d . The one -dimensional proce ss { r i } has the same nu mber of points in [0 , x ) , a nd x i = r α i , so E Φ([0 , x )) = c d x δ . For δ = 1 , λ ( x ) = c d is constant. (b) Follo ws directly from the fact tha t { y i } is stationary Poisson . ((3) ha s b een established in [4].) (c) The cdf P [ ξ i < x ] is 1 − E x i ( F ( x i /x )) with x i distrib uted a ccording to (4). (7) is obtained by straightforward (b ut tediou s) calculation. Remarks : - For gen eral (rational) values of m , d , and α , F ξ i can be express ed using hypergeometric functions. Nov ember 4, 2018 DRAFT 7 - (8) ap proaches 1 − e xp( − c d x ) as m → ∞ , which is the d istrib ution of x 1 . Similarly , lim m →∞ E ξ i = i/c d = E x i and lim m →∞ V ar ξ i = i/c 2 d = V ar x i . - Alternati vely we could consider the path gain pr oce ss ξ − 1 i . Since F ξ − 1 i ( x ) = 1 − F ξ i (1 /x ) , the distrib ution functions look similar . - In the s tandard network, the expected path loss E ξ i does not exist for any i , a nd for i = 1 , t he expected path gain is infinite, too, s ince bo th x 1 and f are expone ntially dis trib u ted. For i > 1 , E ( ξ − 1 i ) = c d / ( i − 1) , and for i > 2 , V ar( ξ − 1 i ) = 2 c 2 d / (( i − 1)( i − 2) ) . - For the stand ard network, the differential entropy h ( ξ i ) , E [ − ln f ξ i ( ξ i )] is 2 − log c d for i = 1 and grows logarithmically with i . For Nakag ami- m fading h ( ξ 1 ) = 1 + 1 /m − log c d . For the path gain process in the standard network, the entropy ha s the simple expression h ( ξ − 1 i ) = i + 1 i + log  π i  , (12) which is monotonically decreasing , reflecting the fact that the vari ance V ar ξ − 1 i is dec reasing with i − 2 . - The ξ i are no t indepe ndent since the x i are ordered. For example, in the cas e of the stan dard network, the dif ference x i +1 − x i is exponentially distrib uted with mea n 1 /c d , thus the joint pdf is f x 1 ... x n ( x 1 , . . . , x n ) = c n d e − c d x n 1 0 0 , E ( # { x i : x i > a, ξ i < a } ) = E (# { x i : x i < a, ξ i > a } ) , i.e. , the expe cted numbe rs of node s c rossing a from the left (leaving the interval [0 , a ) ) and the right (entering the same interv al) are equal. This condition can be expressed as Z a 0 λ ( x ) F ( x/a ) d x = Z ∞ a λ ( x )(1 − F ( x/a )) d x ∀ a > 0 . Nov ember 4, 2018 DRAFT 8 If δ = 1 , λ ( x ) = c d , and the condition reduces to Z 1 0 F ( x ) d x = Z ∞ 1 (1 − F ( x )) d x , which holds since Z 1 0 (1 − F ( x )) d x | {z } 1 − R 1 0 F ( x ) d x + Z ∞ 1 (1 − F ( x )) d x = E f = 1 . An immediate cons equenc e is t hat a receiv er c annot decide on the amount o f fading p resent in the network if δ = 1 and g eographica l distanc es are n ot known. Corollary 4 F or Nakagami- m fading, δ = 1 , and any a > 0 , the expected number o f node s with x i < a and ξ i > a , i.e. , nodes that leave the interval [0 , a ) du e to fading, is E (# { x i : x i < a, ξ i > a } ) = c d a m m − 1 Γ( m ) e − m . (14) The same numb er of nodes is expected to enter this interval. F or Rayleigh fading ( m = 1 ), the fraction of nodes leaving any interval [0 , a ) is 1 /e . Pr oo f: E (# { x i : x i < a } ) = Λ( a ) = c d a , and for Nakagami- m , the fraction o f no des leaving the interval is Z 1 0 F ( x ) d x = m m − 1 Γ( m ) e − m . Clearly , fading ca n be interpreted a s a stochastic mapping from x i to ξ i . So, { x i } are the points in the geogra phical do main (they ind icate distance ), wh ereas { ξ i } are the p oints in the pa th loss domain, since ξ i is the actual path loss including fading. This mapp ing results in a pa rtial reordering of the nodes, as visualized in Fig. 2. In the path loss domain, the connected nod es are s imply gi ven by { ˆ ξ i } = { ξ i } ∩ [0 , 1 /s ] . Fig. 3 i llustrates the situation for 200 node s rand omly ch osen from [0 , 5] with a thresho ld s = 1 . Before fading, we expe ct 40 nod es inside. From these , a fraction e − 1 is moving out (right triangles), the res t stays in (marked by × ). From the one s outside, a fraction (1 − e − 4 )( ae ) ≈ 9% moves in (left triangles), the rest stays out (circles). For the standard n etwork, the probability o f p oint reordering due to fading ca n b e ca lculated explicitly . Let P i,j , P [ ξ i > ξ i + j ] . By this definition, P i,j = P [ x i / f i > x i + j / f i + j ] = P  x i x i + y j > f i f i + j  . (15) Nov ember 4, 2018 DRAFT 9 1/s x ξ Fig. 2. The points of a P oisson point process x i are mapped and reordered according to ξ i := x i / f i , where f i is iid expon ential with unit mean. In the lower axis, the nodes to the left of the thresho ld 1 /s are conn ected to the origin (path loss smaller t han 1 /s ). 0 1 2 3 4 5 0 1 2 3 4 5 good fading bad fading Geographical domain Path loss domain Fig. 3. Illustration of the Rayleigh mapping. 200 points x i are cho sen uniformly randomly in [0 , 5] . Plotted are the points ( x i , x i /f i ) , where the f i are dra wn iid exponential with mean 1. Consider the i nterv al [0 , 1] ( i.e. , assume a threshold s = 1 ). Points mark ed by × are points that remain inside [0 , 1] , those marked by o remain outside, the ones marked with left- and right-pointing triangles are the ones that mov ed in and out, respecti vely . The node marked with a double triangle i s the furthest reachable node. On average the same number of nodes mov e in and out. Note that not all points are shown , since a fraction e − 1 is mapped outside of [0 , 5] . Nov ember 4, 2018 DRAFT 10 x i is Erlang with pa rameters i and c d , y j is the distance from x i to x i + j and thus Erlang with pa rameters j and c d , and the cdf of z := f n / f n + m is F z ( x ) = x/ ( x + 1) . Hence P i,j = E x , y  x i 2 x i + y j  = Z ∞ 0 Z ∞ 0 x 2 x + y c i + j d x i − 1 y j − 1 Γ( i )Γ( j ) e − c d ( x + y ) d x d y . P i,j does not depe nd on c d . Closed-form expressions include P 1 , 1 = 1 − ln 2 ≈ 0 . 30 7 , and P 1 , 2 = 3 − 4 ln 2 ≈ 0 . 227 . Generally P k ,k can be determined analytically . For k = 1 , 2 , 3 , 4 , we obtain 1 − ln 2 , 12 ln 2 − 8 , 16 7 / 2 − 120 ln 2 , 1120 ln 2 − 776 . Further , lim k →∞ P k ,k = 1 / 3 , which is the probability that an exponential ran dom vari able is lar ger than another one that has twice the mean. In the limit, as i → ∞ , P i,j = 1 / ( j + 1) , which is the probability tha t a node has the largest fading coefficient among j + 1 nodes that are at the same distance. Inde ed, as i → ∞ , x i + j < x i (1 + ǫ ) a .s. for any ǫ > 0 an d finite j . While the ξ i are depende nt, it is often useful to cons ider a s et of independen t random variables, ob tained by conditioning the proc ess on having a certain number of nodes n in a n interv al [0 , a ) (or , equ i valently , conditioning on x n +1 = a ) and randomly permuting the n no des. In doing s o, the n points { x i } and { ξ i } , i = 1 , 2 , . . . , n are iid distributed as follo ws . Corollary 5 Conditioned on x n +1 = a : (a) The nodes { x i } n i =1 ar e iid distrib uted with f a x i ( x ) = λ ( x ) Λ( x ) = δ  x a  δ 1 x , 0 6 x < a (16) and cdf F a x i ( x ) = ( x/a ) δ . (b) The path loss with fading { ξ i } n i =1 is distributed as F a ξ i ( x ) = 1 − Z a 0 F ( y /x ) δ  y a  δ 1 y d y . (17) (c) F or the standar d network , F a ξ i ( x ) = x a  1 − e − a/x  (18) (d) F or Rayleigh fad ing and δ = 1 / 2 , F a ξ i ( x ) = √ π 2 r x a erf  r a x  . (19) Pr oo f: As in (6), the cdf is given by 1 − E ( F ( y /x )) with y distributed as (16). Nov ember 4, 2018 DRAFT 11 I I I . C O N N E C T I V I T Y Here we in vestigate the processes ˆ Φ and ˆ Ξ = Ξ ∩ [0 , 1 /s ) of connected nodes. A. Single-transmission c onnectivity 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 1 /s can decode, i.e . , ar e conne cted. W e h ave: (a) ˆ Φ is P ois son with ˆ λ ( x ) = λ ( x )(1 − F ( sx )) . (b) W ith Nakagami- m fading, the numb er ˆ N = ˆ Φ( R + ) of connected nodes is P oisson with mean E ˆ N m = c d ( ms ) δ Γ( δ + m ) Γ( m ) (20) and the c onnectivity fading g ain , defined as the ratio of the expected numbe rs of connected nodes with and without fading, is E ˆ N m E ˆ N ∞ = 1 m δ Γ( δ + m ) Γ( m ) = E ( f δ ) . (21) Pr oo f: (a) The ef fect of fading on the conne cti vity is inde penden t (non-homog eneous ) thinning by 1 − F ( sx ) = P [ x/ f < 1 /s ] . (b) Using (a), the expected n umber of connected nodes is Z ∞ 0 ˆ λ ( x ) d x = Z ∞ 0 c d δ x δ − 1 Γ ic ( m, msx ) Γ( m ) d x which e quals E ˆ N m in the assertion. W ithout fading, E ˆ N ∞ = lim m →∞ = Λ(1 /s ) = c d s − δ , which results in the ratio (21). Remarks : 1) (20) is a ge neralization of a result in [1] where the connecti vity of a node in a t wo-dimensional network with Ra yleigh fading was studied. 2) E ˆ N can also be expressed as E ˆ N = ∞ X i =1 P [ ξ i < 1 /s ] . (22) The relationship with part (b) ca n be v iewed as a simple instan ce of Ca mpbell’ s theorem [5]. Since ˆ N is Poisson, the probability of isolation is P ( ˆ N = 0) = exp( − E ˆ N ) . 3) E ˆ N 1 = c d s − δ Γ( δ + 1) , and E ˆ N ∞ = c d s − δ . For δ = 1 , ˆ N does not depend on the type (or presen ce) of fading. Nov ember 4, 2018 DRAFT 12 1 2 3 4 5 0 0.5 1 1.5 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 m δ fading gain Fig. 4. Connecti vit y fa ding gain for Naka gami- m f ading as a fun ction of δ ∈ [0 , 3 / 2] and m ∈ [1 , 5] . For δ = 1 , the gain is 1 i ndepend ent of m (thick line). 4) The connectivity fading gain equa ls the δ -th moment of the fading distributi on, which, by definition, approach es one as the fading vanishes, i.e. , as m → ∞ . For a fixed δ , it is d ecreasing in m if δ > 1 , increasing if δ < 1 , and equal to 1 for all m if δ = 1 . It also equals 1 if δ = 0 . For a fixed m , it is not monoton ic with δ , but exhibits a minimum at some δ min ∈ (0 , 1) . The fading g ain a s a function of δ and m is p lotted in Fig. 4. For Rayleigh fading and δ = 1 / 2 , the fading gain is π / 2 , and the minimum is assu med at δ min ≈ 0 . 462 , corresponding to α ≈ 4 . 33 for d = 2 . So, depending on the type of fading and the ratio of the number o f network dimensions to the path los s exponent α , fading can increase or d ecrease the number of connected nodes. 5) For the standard n etwork, E ˆ N = c d /s and the probability of isolation is e − c d /s . 6) The expected number of connected n odes ˆ N a with x i < a is E ˆ N a = c d a δ F a ξ i (1 /s ) . (23) where F a ξ i is gi ven in (17). Corollary 7 Under Nagak ami- m fading, a uniformly randomly chosen connected node ˆ x ∈ ˆ Φ has mea n E ˆ x = δ ( δ + m ) ms ( δ + 1) , (24) which is 1 + δ /m times the va lue without fading. Nov ember 4, 2018 DRAFT 13 Pr oo f: A rand om connected node ˆ x is distrib u ted according to f ˆ x ( x ) = ˆ λ ( x ) E ˆ N . (25) W ithout fading, the distributi on is s δ δ x δ − 1 , 0 6 x 6 1 /s , resulting in a n expectation of δ/ ( s ( δ + 1)) . For Rayleigh fading, for example, the density f ˆ x is a gamma d ensity with mean δ /s , so the av erage connec ted node is 1 + δ times further away tha n without fading. B. Connectivity with r etransmissions Assuming a bloc k fading network and n tr ansmissions of the same pac ket, what is the process of n odes that receiv e the pa cket at least on ce? Corollary 8 In a network with iid block fading, the d ensity of the pr oc ess of node s ˆ λ n that r eceive at least one of n transmissions is ˆ λ n ( x ) = (1 − F ( sx ) n ) c d δ x δ − 1 . (26) Pr oo f: This is a straightforward gene ralization of Prop. 6(a). So, in a standard network, the number of connected nodes with n transmissions E ˆ N n = Z ∞ 0 ˆ λ n ( x ) d x = c d s (Ψ( n + 1) + γ ) , (27) where Ψ is the d igamma function (the logarithmic de ri vativ e of the ga mma function), which grows with log n . Alternati vely if the thresho ld s k for the k -th transmission is chose n as s k , s 1 /k , k ∈ [ n ] , the expected n umber of nodes reached increases linearly with the number of transmissions. I V . B RO A D C A S T I N G A. Br oa dcasting reliability Proposition 9 F or δ = 1 and Na kagami- m fading, m ∈ N , the probabilit y that a randomly chosen node x ∈ [0 , a ) can be reached is p m ( ˜ s ) = 1 ˜ s 1 − exp( − m ˜ s ) m − 1 X k =0 m k (1 − k/m ) k ! ˜ s k ! , (28) where ˜ s , as . p m is increasing in m for all ˜ s > 0 and con verg es uniformly to lim m →∞ p m ( ˜ s ) = m in { 1 , ˜ s − 1 } . (29) Nov ember 4, 2018 DRAFT 14 Pr oo f: p m ( ˜ s ) is giv en by p m ( ˜ s ) = Z 1 0 (1 − F ( ˜ sx )) d x = Z 1 0 Γ( m, m ˜ sx ) Γ( m ) d x . (30) For m ∈ N , this is p m ( ˜ s ) = m − 1 X k =0 Z 1 0 exp( − m ˜ sx ) ( m ˜ sx ) k k ! d x , (31) which, after some manipulations, yields p m ( ˜ s ) = 1 ˜ s   1 − 1 m exp( − m ˜ s ) m − 1 X k =0 k X j =0 ( m ˜ s ) j j !   (32) = 1 ˜ s        1 − exp( − m ˜ s ) m − 1 X k =0 m k (1 − k/m ) k ! ˜ s k | {z } P m − 1 ( ˜ s )        . (33) The polyno mial P m − 1 is the T aylor expansion of order m of (1 − ˜ s ) exp( m ˜ s ) at ˜ s = 0 (the c oefficient for ˜ s m is zero). So exp( − m ˜ s ) P m − 1 ( ˜ s ) = 1 − s + O ( s m +1 ) from which the limit 1 for ˜ s < 1 follows. For ˜ s > 1 , the exponential d ominates the polynomial so that their prod uct tends to zero and 1 / ˜ s remains as the limit. The co n vergence to m in { 1 , ˜ s − 1 } is the expec ted behavior , since without fading a node is c onnec ted if it is positioned within [0 , 1 /s ] ( ˜ s < 1 ) a nd for a rand omly chose n no de in [0 , a ] for a > 1 /s or ˜ s > 1 , this has probability 1 /as . So with increas ing m , deri vati ves of h igher and higher order be come 0 at ˜ s = 0 . From the previous d iscussion we know that p m ( ˜ s ) = 1 + O ( ˜ s m ) . Calcula ting the coef ficient for ˜ s m yields p m ( ˜ s ) = 1 − m m Γ( m + 2) ˜ s m + O ( ˜ s m +1 ) . (34) The m -th order T aylor expansion a t ˜ s = 0 is a lower bound. Upper bounds a re o btained by truncating the polynomial; a natural choice is the first-order version 1 + ( m − 1) ˜ s to obtain  1 − m m Γ( m + 2) ˜ s m  + < p m ( ˜ s ) 6 min  1 , 1 ˜ s (1 − exp ( − m ˜ s )(1 + ( m − 1) ˜ s ))  . (35) Using the lower bound , we can establish the follo wing Co rollary . Corollary 10 ( ǫ -reachability .) If as < (Γ( m + 2) · ǫ ) 1 /m m . (36) at least a fr action 1 − ǫ of the nodes x i ∈ [0 , a ) a r e co nnected. In the s tandard network (specializing to m = 1 ), the sufficient condition is as < 2 ǫ , (37) Nov ember 4, 2018 DRAFT 15 This follo ws directly from the lo wer bound in (35). Remarks : - For m → ∞ , the b ound (36) is no t tight s ince the RHS conv er ges to 1 /e f or all positi ve ǫ (by Stirling’ s approximation), while the exact condition is as < 1 / (1 − ǫ ) . - The s uf ficient con dition (37) is tight (within 7 %) for ǫ < 0 . 1 . W ith p 1 ( as ) = (1 − e − as ) /as , the condition p 1 ( as ) > 1 − ǫ can be solved exactly using the Lambert W function: as < W ( − q e − q ) + q , where q , 1 1 − ǫ . (38) A li near approximation yields the same bo und as before, while a quadratic exp ansion yields t he sufficient co ndition as < 2 ǫ + 4 / 3 ǫ 2 which is within 3 . 9% for ǫ < 0 . 25 . B. Br oa dcast transport sum-distanc e an d capacity Assuming the origin o transmits, the set of nodes that receiv e the mes sage is { ˆ x i } . W e sha ll determine the br o adcast trans port sum-distance D , i.e. , the expected sum over the all the distanc es ˆ x 1 /α i from the origin: D , E   X x ∈ ˆ Φ x 1 /α   (39) Proposition 11 The br oadcast trans port sum-distance for Nak agami- m fad ing is D m = c d δ ∆ 1 ( ms ) ∆ Γ( m + ∆) Γ( m ) , (40) and the (br oa dcast) fading gain D m /D ∞ is D m D ∞ = 1 m ∆ Γ( m + ∆) Γ( m ) = E ( f ∆ ) . (41 ) Pr oo f: From Ca mpbell’ s theorem E   X x ∈ ˆ Φ x 1 /α   = Z ∞ 0 x 1 /α ˆ λ ( x ) d x = c d δ Z ∞ 0 x 1 /α + δ − 1 (1 − F ( sx )) d x , which equals (40) for Nakagami- m f ading. Nov ember 4, 2018 DRAFT 16 W ithout fading, a node x i is connec ted if x i < 1 /s , therefore D ∞ = Z 1 /s 0 x 1 /α λ ( x ) d x (42) = c d δ ∆ s − ∆ = c d d d + 1 s − ∆ . (43) So the fading g ain D m /D ∞ is the ∆ -th moment of f as gi ven in (41). Remarks : 1) The fading gain is ind epende nt of the threshold s . D m ∝ s − ∆ for all m . It strongly rese mbles the connec ti vity gain (Prop. 6 ), the on ly dif ference being the parameter ∆ instead of δ . In pa rticular , D m is independent of m if ∆ = 1 . See Remark 3 to Prop. 6 and Fig. 4 for a discussion and visualization of the behavior of the ga in as a func tion of m and ∆ . 2) For Ra yleigh fading ( m = 1 ), D 1 = c d δ s − ∆ , and the fading gain is Γ(1 + ∆) . For d = α = 2 , D ∞ = 2 π 3 s 3 / 2 . 3) The formula for the broa dcast transport su m-distance reminds of an interference expression. Indeed, by simply replac ing x 1 /α by x − 1 , a well- known result on the me an interference is reproduced: Assuming each node transmits at unit power , the total interferenc e at the origin is E X x ∈ Φ x − 1 ! = Z ∞ 0 x − 1 λ ( x ) d x = c d δ δ − 1 x δ − 1    ∞ 0 which for δ < 1 d i ver g es due to the l ower bo und integration b ound ( i.e. , the one or two close st nodes) and for δ > 1 diver ges due to the uppe r bound ( i.e. , the large number of nodes that are far away). So far , we have ignored the ac tual rate of transmiss ion R and just used the threshold s for the sum- distance. T o ge t to the single-hop broad cast transport capacity C (in b it-meters/s/Hz), we relate the (bandwidth-normalized) rate of transmission R and the threshold s by R = log 2 (1 + s ) and define C , m ax R> 0 { R · D (2 R − 1) } = max s> 0 { log 2 (1 + s ) D ( s ) } . (44) Let D 1 m be the broadcas t transp ort sum-distance for s = 1 (see Prop. 11) s uch that D m = D 1 m s − ∆ . Proposition 12 F or Nakagami- m fading: (a) F or ∆ ∈ (0 , 1) , the br oadcas t transport ca pacity is achieved for R opt = W  − e − 1 / ∆ ∆  + ∆ − 1 log 2 , ∆ ∈ (0 , 1) . (45) Nov ember 4, 2018 DRAFT 17 The r esulting b r oad cast tr anspor t capac ity is tightly (within at mos t 0.13%) lowe r bounded by C m > D 1 m (∆) log 2 (∆ − 1 − ∆)  e ∆ − 1 − ∆ − 1  − ∆ . (46) (b) F or ∆ = 1 , C m = c d δ log 2 (47) independ ent of m , and R opt = 0 . (c) F or ∆ > 1 , the br oadc ast transpor t capa city increases without boun ds a s R → 0 , indepe ndent of the transmit p ower . Pr oo f: (a) D m ∝ s − ∆ , so C m ∝ R (2 R − 1) − ∆ which, for ∆ 6 1 , has a maximum a t R opt giv en in (45). The lower bo und stems from a n approximation o f R opt using W ( − exp( − 1 / ∆) / ∆) ' − ∆ which holds since for ∆ = 1 , the two expressions are identical, and the deri vati ve of the Lambe rt W expression is smaller than -1 for ∆ < 1 . (b) For ∆ = 1 , C m increases as the rate is lowered but remains bound ed as R → 0 . Th e limit is c d δ / log 2 . (c) For ∆ > 1 , R (2 R − 1) − ∆ is decreas ing with R , and lim R → 0 R (2 R − 1) − ∆ = lim R → 0 (log 2) − ∆ R 1 − ∆ = ∞ . Remarks : - The optima for R , s are indepen dent of the type of fading (parameter m ). - For ∆ < 1 , the o ptimum s is tightly lo we r bounded by s opt > exp(∆ − 1 − ∆) − 1 . (48) This is the expression a ppearing in the bound (46). - (c) is a lso appa rent from the express ion D ( s ) log 2 (1 + s ) , which, for s → 0 , is approximately D 1 m s 1 − ∆ / log 2 . So, t he intuition i s that in this re gime, the ga in from reaching additional nod es more than off sets the loss in rate. - For ∆ = 1 / (2 log 2) , s opt = R opt = 1 a nd C m = D 1 m . This is, h owe ver , no t the minimum. The capac ity is minimum arou nd ∆ ≈ 0 . 85 , depend ing slightly on m . Fig. 5 d epicts the op timum rate a s a function of ∆ , together with the lo wer bound (∆ − 1 − ∆) / log 2 , and Fig. 6 plots the broadca st transport capacity for Rayleigh fading and no f ading for a two-dimensional Nov ember 4, 2018 DRAFT 18 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 ∆ R opt Optimum rate vs. ∆ exact lower bound Fig. 5. Op timum t ransmission rates for ∆ ∈ [0 . 5 , 1 . 0] The optimum r ate i s 1 for ∆ = 1 / (2 log 2) ≈ 0 . 72 . network. The range ∆ ∈ [0 . 5 , 1 . 0] correspond s to a p ath loss expon ent range α ∈ [3 , 6] . It can be seen that Nakagami fading is harmful. For small values of ∆ , the cap acity for Ray leigh fading is about 10% smaller . C. Optimum br oa dcasting (su perpos ition cod ing) Assuming that nod es can decode at a rate c orresponding to their SNR, the broadca st transport capacity (without fading) is ˜ C = E " X x ∈ Φ x 1 /α log 2 (1 + x − 1 ) # (49) T o avoid problems with the singu larity of the path loss law at the origin, we replace the log by 1 for x < 1 . For x > 1 , we us e the lo wer bound log 2 (1 + x − 1 ) > 1 /x . Proce eding as in the proof of Prop. 11, we obtain ˜ C > c d δ  1 ∆ + Z ∞ 1 x ∆ − 2 d x  , (50) which is significantly lar ger than in the case with single-rate decod ing. For ∆ < 1 , ˜ C > c d δ ∆(1 − ∆) . (51) Nov ember 4, 2018 DRAFT 19 0.5 0.6 0.7 0.8 0.9 1 1.8 2 2.2 2.4 2.6 2.8 3 ∆ C Broadcast transport capacity vs. ∆ m=1 m= ∞ Fig. 6. Bro adcast transport capacity for d = 2 , ∆ ∈ [0 . 5 , 1 . 0] and m = 1 and m = ∞ . For ∆ = 1 , the capacity is 2 π / (3 log 2) ≈ 3 . 02 irrespecti ve of m . F or the no fading case, the minimum occurs at ∆ = 1 / (2 log 2) , w here C = 2 π / 3 . For ∆ > 1 , this lo wer bound and thus ˜ C is unbounde d, in agreement w ith the previous result. The only dif ference is that for ∆ = 1 , ˜ C div er ges whereas C is finite. Note that sinc e log 2 (1 + x − 1 ) < 1 / ( x log 2) for x > 1 , the lower b ound is within a f actor log 2 of the correct value. If the actual Shann on capacity were cons idered for nodes that are very close, ˜ C w ould div er ge more quickly as ∆ → 0 ( α → ∞ ) sinc e the contrib u tion from the nodes within distance one would b e: ˜ C [0 , 1] > c d δ Z 1 0 − x ∆ − 1 log 2 x d x = 1 log(2)∆ 2 . (52) V . O T H E R A P P L I C A T I O N S A. Maximum transmission distance How far c an we expect to trans mit, i.e. , what is the (av erage) ma ximum transmission distance M , E  max x ∈ ˆ Φ { x 1 /α }  ? Let ˆ x be a uniformly randomly c hosen connected nod e. The pdf f ˆ x is gi ven by (25 ). The distri bution Nov ember 4, 2018 DRAFT 20 of the maximum x M of a Poisson number of R Vs is gi ven by the Gumbel distrib ution 3 F ˆ x M ( x ) = exp  − E ˆ N (1 − F ˆ x ( x )  . (53) So, in principle, M = E ( ˆ x 1 /α M ) can be c alculated. Howe ver , ev en for the standard network, where F ˆ x M ( x ) = exp( − c d s exp( − sx )) , the re do es not seem to exist a c losed-form expres sion. If the number o f connec ted no des was fixed to c d /s (instead of b eing Poisson distributed with this mean), we would have F ˆ x M ( x ) = (1 − e − xs ) c d /s with mean E ˆ x M = 1 s  Ψ  c d s + 1  + γ  . (54) Since Ψ is concave, this u pperbound s the true mean by Jens en’ s ineq uality . Finally , we in voke Jens en again by replacing E ( ˆ x 1 /α M ) by E ( ˆ x M ) 1 /α to obtain M <  1 s  Ψ  c d s + 1  + γ   1 /α . (55) W ithout much ha rm, Ψ( x ) could be replaced by (the slightly larger) log( x ) . Even replacing Ψ( x + 1) b y log( x ) s till appears to be an upper bound. The bound is quite tight, see Fig. 7. Also compa re wit h Fig. 1, where the most distant node is quite exactly 6 u nits away ( s = 0 . 1 ). The factor s − 1 /α is the boun d in the no n-fading case, s o the Rayleigh fading (diversity) gain for the maximum transmission distan ce is roughly log(1 /s ) 1 /α which grows without boun ds as s → 0 . B. Pr ob abilistic p r ogress In a ddition to the ma ximum transmission distance or the distance-rate produ ct, the product distance s times probability of su ccess ma y be of interest. W ithout considering the a ctual node positions, on e ma y want to maximize the c ontinuous pr ob abilistic p r ogress G ( x ) , max { x 1 /α P [ f > sx ] } . For the standard network with α = 2 , this is maximized at x = 1 / 2 s . If there was no fading, the op timum would be x = p 1 /s . Of course there is no guarantee that there is a node very clos e to this o ptimum location. Alternati vely , define the (discrete) probabilisti c p r ogress when transmitting to node i by G i , E  x 1 /α i · P [ f > s x i | x i ]  (56) W e would like to find i opt = arg max i G i . For the s tandard network, G i = E  x 1 /α i exp( − s x i )  = c i d ( s + c d ) i +1 /α Γ( i + 1 /α ) Γ( i ) . (57) 3 Note that the Gumbel cdf is not zero at 0 + . This reflects the fact that the number of connected nodes may be zero, in which case the maximu m t ransmission distance w ould be zero. Accordingly the pd f includ es a pulse at 0 , the term exp( − E ˆ N ) δ ( x ) . Nov ember 4, 2018 DRAFT 21 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 9 10 s M Expected max. transmission distance vs. threshold s simulation upper bound no fading Fig. 7. Expected maximum transmission distances for the standard two-dimension al network and s ∈ [0 . 05 , 1 . 0 0] . For comparison, the curv e s − 1 / 2 for the non-fading case is also displayed. The maximum o f G i cannot be foun d directly , but sinc e Γ( i + 1 / α ) / Γ( i ) is very tightly lower bounde d by i 1 /α we hav e G i / c i d i 1 /α ( s + c d ) i +1 /α (58) which, assuming a continuous parameter ˜ i , is maximized at ˜ i opt = 1 α log(1 + s c d ) . (59) Note that the same expression for i opt would be obtaine d if G i was approximated by the f actorization G ′ i = E ( x 1 /α i ) P [ ξ i < 1 /s ] . For the stand ard ne twork, E ( x 1 /α i ) = Γ( i +1 /α ) Γ( i ) c 1 /α d , and P [ ξ i < 1 /s ] = ( π / ( π + s )) i . So G ′ i dif fers from G i only by the factor (1 + s /c d ) 1 /α which is indep endent of i and quite s mall for typical s . Now , the question is how to round ˜ i opt to i opt . For large s , i opt = 1 . For small s , ˜ i opt ≈ c d / ( αs ) so i opt = ⌈ c d αs ⌉ (60) is a good choice. It can be verified that this is indeed the optimum. The expe cted distance to this i opt -th node is qu ite exac tly 1 / ( αs ) 1 /α . So in this non-oppo rtunistic setting wh en reliability matters, Rayleigh fading is harmful; it reduces the range of transmissions by a f actor α − 1 /α . Nov ember 4, 2018 DRAFT 22 C. Retransmissions and localization Proposition 13 (Retransmissions ) Consider a ne twork with bloc k Rayleigh fading. The expected number of nodes that r e ceive k o ut of n transmitted pa ck ets E N n k is E N n k = c d Γ(1 + δ ) ( k s ) δ , k ∈ { 0 , 1 , . . . , n } . (61) Pr oo f: Let p ( x ) , 1 − F ( sx ) . The de nsity of node s that receive k packets out of n transmissions is giv en by λ n k ( x ) = λ ( x )  n k  p ( x ) k (1 − p ( x )) n − k . (62) Plugging in p ( x ) = exp( − sx ) for Ra yleigh fading a nd integrating (62) yields E N n k = Λ n k ( R + ) . Remarks : - Interestingly , (61 ) is indep endent o f n . So, the mean numbe r of nodes that recei ve k pa ckets do es no t depend on how often the packet was transmitted. - Summing λ n k over k ∈ [ n ] reproduc es Cor . 8. - (61) is valid even for k = 0 since E N n 0 = ∞ . - For the stan dard networks, the expres sion simplifies to E N n k = c d k s , which, when summed o ver k ∈ [ n ] , yields (27). Let x n k be the position of a randomly chose n node from the nodes that received k out of n packets. From Prop. 13, the pdf (normalized density) is f x n k ( x ) = λ n k ( x ) ( k s ) δ c d Γ(1 + δ ) , k ∈ [ n ] . (63) For the standard network, we have E x n n = ( ns ) − 1 , V ar x n n = ( ns ) − 2 , and E x n 1 = 1 s (Ψ( n + 1) + γ ) , which is again related to (27) (di vision by the con stant density c d ). The densities of the node s recei ving exactly k of 6 me ssage s is plotted in Fig. 8 for the stand ard network with α = 2 . This expres sion permits the ev a luation of the co ntrib ution that eac h additional transmission makes to the broadca st transport sum-distanc e and capacity . These results can also be ap plied in localiza tion. If a no de receives k o ut of n transmissions, E x n k is an obvious e stimate for its p osition, and V ar x n k for the uncertainty . Alternatively , if the path loss x ca n be measured, then the correspond ing node index ˆ i ( x ) can be determined by the ML estimate ˆ i ( x ) = arg max i f ξ i ( x ) , (64) Nov ember 4, 2018 DRAFT 23 0 1 2 3 4 0 0.5 1 1.5 2 2.5 3 x Density λ k 6 (x) k=6 k=1 k=0 sum λ k 6 (x), k ∈ [6] Fig. 8. Densities λ 6 k ( x ) for the standard network with α = 2 ( c d = π ) and s = 1 . The maximum of the density for k = n = 6 is λ 6 6 (0) = π . The dash ed curv e is the den sity of the nodes that receiv e at least 1 packe t. Normalized by E N 6 k these densities are the pdfs of x 6 k . with the pdf f ξ i giv en in Cor . 2. For the standard networks, for example, t he ML dec ision is ˆ i ( x ) = ⌈ c d /x ⌉ since ˆ i ( x ) = i ⇐ ⇒ c d i 6 x < c d i − 1 . (65) This is of course related to the fact E x i = i/c d . V I . C O N C L U D I N G R E M A R K S W e have offered a geome tric interpreta tion of fading in wireless networks which is b ased on a point proces s model that incorporates both ge ometry and fading. The frame work enables a nalytical in vestigations of the properties of wireless n etworks and the impact of fading, leading to clos ed-form results that are obtained in a rather con venien t manner . For Nak agami- m fading, it turns out that the connectivity fad ing gain is the δ -th moment of the fading distrib ution, while the fading g ain in the br oadcas t transport sum-distanc e is its ∆ -th moment. A path loss exponent larger than the n umber of dimensions d ( d + 1 for broadcasting) leads to a nega ti ve impa ct of fading. Interestingly , the broadcast tr anspor t capacity turns out to be unb ounded if ∆ > 1 , i.e. , if the path loss exponent is smaller than d + 1 . While this resu lt may be of interest for the design of ef ficient broadcas ting protoco ls, it also raises do ubts o n the v alidity of transport ca pacity as a performance metric. Nov ember 4, 2018 DRAFT 24 Generally , it can be observed that the parameters δ and/or ∆ app ear ubiqu itously in the expression s. So the network b ehavior critically dep ends on the rati o of the numbe r of dimen sions to the path loss exponent. Other applications con sidered includ e the maximum transmission distan ce, prob abilistic progres s, and the effect of retransmissions. W e are con vinced that there are many more that will benefit from the theoretical foundations laid in this paper . A C K N O W L E D G M E N T S The suppo rt of NSF (Grants CNS 04-4786 9, DMS 505624) and the D ARP A IT -MANET program (Grant W911NF-07-1-0028) is gratefully acknowledged. R E F E R E N C E S [1] D. Miorandi and E. Altman, “Coverage and Connectivity of Ad Hoc Networks in Presence of Channel Randomness, ” in IEEE INFOC OM’05 , (Miami, F L), Mar . 2005 . [2] M. Haenggi, “A Geometry-Inclusiv e Fading Model for Random W ireless Netw orks, ” in 200 6 IEEE International Symposium on Information T heory (ISIT’06) , (S eattle, W A), pp. 13 29–133 3, July 2006. A v ailable at http://www .nd.edu/ ∼ mhaenggi/pu bs/ isit06.pdf. [3] J. F . C. Kingman, P oisson Pr ocesses . Oxford Science Publications, 1993. [4] M. Haenggi, “On Distances in Uniformly Random Networks, ” IEEE T rans. on Information Theory , vol. 51, pp. 3584–358 6, Oct. 2005. A v ail able at http://www .nd.edu/ ∼ mhaenggi/pu bs/tit05.pdf. [5] D. S toyan, W . S . Kend all, and J. Mecke, Stochastic Geometry and its Application s . John Wiley & Sons, 1995. 2nd Ed. Nov ember 4, 2018 DRAFT