Optimal Sensor Configurations for Rectangular Target Dectection

Optimal Sensor Configurations f or Rectangular T arget Dectectio n Franc ¸ ois-Alex Bourque and Bao U. Nguyen Abstract — Optimal search strategies where targets are ob- serv ed at se veral different angles are f ound. T argets a re assumed to ex hib it recta ngular symmetry and ha ve a unif ormly- distributed orientation. By recta ngular symmetry , it i s meant that one side of a target is th e mirror image of its opposite side. Findin g an optimal solution is generally a hard problem. For tunately , symmetry prin ciples allo w analytical and intuitive solutions to be f ound . One such op timal search strategy consists of choosing n angles ev enly separated on the h alf-circle and leads to a lower bound of the probability of not detecting targets. As no prior knowledge of the targ et orientation i s required, such search strategies are also robust, a desirable feature in sear ch and d etection missions. I . I N T RO D U C T I O N In m ine hun ting op erations it is k nown th at the dete ction perfor mance imp roves when a target is o bserved many times at d ifferent aspect ang les [1]– [6]. Similarly , classification algorithm s [7]–[9 ] and sensor-arrays deployed for target localization and tracking [10]–[1 3] benefit from multi-aspect observations. This fact is, h owe ver , of ten overlooked. For example, the formu la fo r the prob ability of detecting a ta rget in a rand om search derived by K oopman is widely u sed yet it assumes no ang ular depend ence [1 4]. In this paper, search strategies that min imize the overall probab ility of not de tecting a target observed at se veral different angles are ide ntified. Find ing an o ptimal an gular configur ation is a p riori intractable as it is m ulti-dimen sional in the sense that each o bservation is ind epende nt o f one another and hence each observation angle must be considered as a separate dimensio n. Wha t is more, the explicit expr es- sion for the overall pro bability of detection can be ho pelessly complicated e ven when the probability o f d etection f or a single observation is simple an d the n umber of ob servations is small. The novelty of our approac h lies in the fact that this normally intractable problem is solved using an elegant symmetry argum ent. Specifically , ta rgets are assumed to exhibit re ctangular symmetry . That is, the left-han d side o f a target is the m irror image of its right-hand side, and its rear end is the mirror im age of its front end. M any targets can be approx imated with this class of symmetry including canoes, ships, sub marines, mines an d hum an bodies. Optimal an gles are then sh own to be ev enly distributed on multiples of the h alf-circle as in Ref. [13 ]. Howev er, th is constitutes a d eparture from the cur rent literature on senso r F-.A. Bourque and B. U. Nguyen are scientists with Defence Researc h and Dev elopment Canada Centre for Operati onal Research and Analysis, Nationa l Defence Headquart ers, 101 Colonel By driv e, Otta wa, Canada. Email enquiries should be sent to alex.bou rque@drdc-rddc .gc.ca. x Fig. 1. Recta ngular target observed at angle x . geometry [10]–[ 13] as our result is deriv ed fo r finite-extent targets rather than f or p oint targets. The simplicity of the solution implies that no complicate d calculations are required prior to a sear ch as long as th e target has the assumed approx imate symmetry . Th is fact should imp rove the task of plann ing the p ath of mo bile sen sors, such as u nmanne d vehicles, to search for fixed targets, as well as of dep loying a fixed sensor array to monitor traffic thr ough choke points. Assumptions and the minim ization prob lem are stated in Section II, while Section III presents a set of search strategies that minimizes the proba bility of no detection . Section IV provides a lower bound of n ot detec ting targets ac hiev able with these strategies, which is illu strated with a specific example in Section V. Conclusions includ ing fu ture work are d iscussed in Section V I. Su pplementar y lemmata used in Sections III and IV are foun d in the Appendix . I I . P RO B L E M S T A T E M E N T The depen dence of detection pro cess o n angle occurs often in sear ch and detection op erations. In gen eral, the effecti veness of such an operatio n also depen ds o n th e distance between the sensor and the target. Howev er, here, the p robability of d etection is assumed co nstant as a function of range and, hence, the focu s is o nly on the a ngular depend ence. For mo re d etails o n th e ra nge dep endenc e, refer to Ref. [6]. 1 1 Note that the probabil ity of de tection as a function of ra nge is primari ly a characte ristic of the sensor, while the probability of detection as a function of angle is primarily a characteri stic of the target. x  x x +  Fig. 2. Symmetries of the targ et: Reflecti on through the short axis of the target (dashed line) and reflec tion through the center of the target (dot- dashed line). As shown in Fig. 1, the problem is modeled on a two- dimensiona l plan and the observation angle, x , is defined as the counte r-clockwise angle measured in radian between the sensor beam and the short axis of a r ectangular (p ositiv e horizon tal axis) target. An observation angle of zero d egree correspo nds to the observation of the lo ng side of the target, while an observation angle of π / 2 degrees correspon ds to the observation of th e sho rt side of the target. T argets considered will have ap proxima te rectang ular symm etries as shown in Fig. 2. That is, they p ossess a reflection axis throug h their short axis ( left-righ t m irror symm etry) and a p oint reflection throug h th eir cen ter . 2 In what fo llows, th e probab ility o f no detec tion rather than the p robability of d etection is considered; one b eing the complement o f the oth er . Define the sing le probability of no detection as the pro bability o f not detecting the target at angle x a nd denote this sing le-value real function as g ( x ) . Note that the sing le proba bility of no detectio n is even due to the reflection symmetry throu gh the shor t axis o f th e target and period ic due to the reflection th rough the center of the target. Specifically , g ( x ) = g ( − x ) , g ( x ) = g ( x + π ) . Next, define the multip le pr obability of no detection as the pro bability of not detec ting a target after n observations. Let x be the orientation of the target. Let u i be the i - th ang le at whic h the target is observed relative to x and ~ µ = ( µ 0 , . . . , µ n − 1 ) b e the vector o f the n relative o b- servation a ngles. Assume the multiple ob servation detectio n process is a Bernoulli pro cess, i.e., all ob servations ar e indepen dent. Then , the multi-o bservation probab ility of no detection is modeled as the product of sin gle probabilities 2 Note t hat t he co mposition of a refle ction through the shor t axi s follo wed by a reflection through the center of the target is equi val ent to a reflection through the long axis of the target (forward/ba ckward m irror symmetry). of no de tection. In gener al, howe ver , the exact value of x , i.e., the o rientation of the target is u nknown. T o cir cumvent this problem, assume that the target’ s orientation is uniformly distributed and evaluate the average mu ltiple pro bability of no detectio n by G ( ~ µ ) . Then, G ( ~ µ ) = 1 π Z π 2 − π 2 dx n − 1 Y i =0 g ( x + µ i ) . (1) Therefo re, the problem amounts to find ing search strate- gies that minimize G ( ~ µ ) . For simplicity , the prob ability of no detection is taken to me an the average mu ltiple probability of no detection in what follows. I I I . A S E T O F O P T I M A L S E A R C H S T R A T E G I E S In this section, a co ndition ensuring that all par tial deriv a- ti ves o f the prob ability of no detection are eq ual to zero is d eriv ed. From this co ndition, a set of o ptimal search strategies is then identified and the probability of no detection is recast into a form used in the subsequent section. Let u s fir st in troduce some useful notation and definition s. Let i ∈ { 0 , . . . , n − 1 } , N i = { 0 , . . . , n − 1 } \{ i } and j ∈ N i . Define ∂ i = ∂ / ∂ i . Denote ~ µ ∗ as an optim al poin t of G ( ~ µ ) . Let a be an integer and b be a positi ve in teger . Define the m odulo operation as a mo d b = a − j a b k b. Let m be a non-negative integer and µ = π /n . Define ˜ G ( m, n ) = 1 π Z π 2 − π 2 dx n − 1 Y i =0 g ( x + miµ ) . Then the following holds. Lemma 3.1: The partial derivati ve can b e written as ∂ i G ( ~ µ ) = 1 2 π Z π 2 − π 2 dx g ′ ( x )  Y j ∈ N i g ( x + µ j − µ i ) − Y j ∈ N i g ( x − µ j + µ i )  . Pr o of: Apply the par tial deriv ativ e to the expression for G ( ~ µ ) g iv en by ( 1). The n ∂ i G ( ~ µ ) = 1 π Z π 2 − π 2 dx g ′ ( x + µ j ) Y j ∈ N i g ( x + µ j ) . Let x → x − µ i and note that Lemma A.1 applies and dictates that the integral is inv ariant un der this change of variable. Thus, ∂ i G ( ~ µ ) = 1 π Z π 2 − π 2 dx g ′ ( x ) Y j ∈ N i g ( x + µ j − µ i ) . (2) Because g ( x ) is ev en, ∂ i G ( ~ µ ) = 1 π Z π 2 − π 2 dx g ′ ( x ) Y j ∈ N i g ( − x − µ j + µ i ) . Let x → − x and remar k that g ′ ( x ) is odd. Th en ∂ i G ( ~ µ ) = − 1 π Z π 2 − π 2 dx g ′ ( x ) Y j ∈ N i g ( x − µ j + µ i ) . (3) And the result follows from the av erag e of (2) an d (3) . A con dition for optim izing G ( ~ µ ) is th us for th e integrands of all par tial der iv ati ves to be e qual to ze ro. The next Lemma identifies a set o f search strategies for which this co ndition holds, i.e. , optimizes th e prob ably of no detection. Lemma 3.2: Define ( m, n ) to be the search strategy such that the sep aration between two consecutive observations is a co nstant and equal to mπ /n with m a positive integer . Then the sear ch strategy ( m, n ) is an optimum of G ( ~ µ ) and defines a subset of all possible o ptimal search strategies. Pr o of: Evaluate the pr oduct in (2 ) at p oint ~ µ ∗ giving Y j ∈ N i g ( x + µ ∗ j − µ ∗ i ) . Assume that ( m, n ) is an op timal search strategy . Then, the definition of the strategy ( m, n ) imp lies that µ ∗ j − µ ∗ i = m [ − (2 i − j ) + i ] µ. From defin ition of the modulo n ote that 2 i − j =  2 i − j n  n + (2 i − j ) mod n. Define σ i ( j ) = (2 i − j ) mo d n and re call th at g ( x ) is periodical. Th en, g ( x + µ ∗ j − µ ∗ i ) = g ( x − mσ i ( j ) µ + mi µ ) . Lemma A.2 implies that the ma p σ i ( j ) is a bijection fr om the set of j to itself. Therefor e, Y j ∈ N i g  x + µ ∗ j − µ ∗ i  = Y j ∈ N i g ( x − mσ i ( j ) µ + miµ ) = Y j ∈ N i g ( x − mj µ + miµ ) = Y j ∈ N i g  x − µ ∗ j + µ ∗ i  . And Lemma 3.1 then e ntails that ∂ i G ( ~ µ ∗ ) = 0 f or all i . For an optim al search strategy ( m , n ) , it will b e usef ul in what follows to cast the in tegral in the fo llowing f orm. Lemma 3.3: Let the search strategy be ( m, n ) . Then the probab ility o f n o detection is equal to ˜ G ( m, n ) . Pr o of: First, remar k that on ly th e difference betwee n two consecutive observations is specified in Lem ma 3. 2. Thus, liberty exists in the choice of the absolute referen ce angle. Without loss o f g enerality , take µ ∗ 0 to b e this absolute referenc e angle. Then, µ ∗ i = µ ∗ 0 + miµ. Substituting this expression into (1) y ields G ( ~ µ ∗ ) = 1 π Z π 2 − π 2 dx n − 1 Y i =0 g ( x + µ ∗ 0 + miµ ) . Next, let x → x − µ ∗ 0 . Then Lemma A.1 en tails that the domain of integration is in variant under such a sh ift of the integration variable. I V . A L OW E R B O U N D O F T H E P R O B A B I L I T Y O F N O D E T E C T I O N In th e previous section, a set of optima l search strategies was iden tified, the ( m, n ) search strategies. In this section, a lower bo und of the probability of no d etection ach iev able with the se strategies is pr oven. Let u s fir st in troduce some useful notation and definition s. Let g cd ( · , · ) be the greatest c ommon divider . Let r , q and p be strictly po siti ve integers such that m = pq , n = pr and p = gcd( m, n ) . Let i ∈ { 0 , . . . , n − 1 } , j ∈ { 0 , . . . , r − 1 } and k ∈ { 1 , . . . , p } . Define h ( x ) = g ( x ) g ( x + m µ ) . . . g ( x + ( r − 1) mµ ) . Lemma 4.1: The following iden tities hold : p Y k =1 h ( x + ( k − 1) µ ) = n − 1 Y i =0 g ( x + iµ ) , (4) n − 1 Y i =0 g ( x + mi µ ) = h ( x ) p . (5) Pr o of: Consider the first identity . Using the defin ition of h ( x ) , write th e left-han d side of (4) as p Y k =1 r − 1 Y j =0 g ( x + mj µ + ( k − 1) µ ) . The definition s of m and n and of th e modu lo operatio n imply that mj = n  q j r  + p ( q j mo d r ) . Using this expression for mj and the periodicity of g ( x ) , the r ight-han d side further beco mes equal to p Y k =1 r − 1 Y j =0 g ( x + p ( q j mo d r ) µ + ( k − 1 ) µ ) . Next, from Lemm a A.3, the map σ ( j ) = q j mo d r is known to be a b ijection from the set of j to itself. Th erefore , the produ ct can also b e written as p Y k =1 r − 1 Y j =0 g ( x + pj µ + ( k − 1) µ ) . Finally , fr om Lemma A.4, the set of ( j, k ) pairs can be mapped to the set of i using the b ijection σ ( k − 1 , j ) = ( k − 1) + pj . Th erefore, p Y k =1 r − 1 Y j =0 g ( x + pj µ + ( k − 1) µ ) = n − 1 Y i =0 g ( x + iµ ) . Now , con sider th e second identity . From Lemma A.4, the set of i can be mapp ed to the set of ( j, k ) p airs using th e bijection σ ( j, k − 1) = j + r ( k − 1) . Therefor e, the left-hand side of (5) beco mes p Y k =1 r − 1 Y j =0 g ( x + mj µ + m ( k − 1) r µ ) . Next, note that the d efinitions of m an d n im ply mr = nq and that th e definitio n of µ imp lies nµ = π , fr om which follows that m ( k − 1 ) rµ = ( k − 1) q π . This equa lity and the p eriodicity of g ( x ) , then imply that g ( x + mj µ + m ( k − 1) r µ ) = g ( x + mj µ ) . Finally , f rom the definition of h ( x ) , p Y k =1 r − 1 Y j =0 g ( x + mj µ ) = p Y k =1 h ( x ) = h ( x ) p . Theor em 4.2 (Lower Bo und Estimate): For any ( m, n ) search strategies, ˜ G ( m, n ) ≥ ˜ G (1 , n ) . Pr o of: Consider a given ( m, n ) pair . Then, Lemm a 4. 1 allows to write ˜ G (1 , n ) = 1 π Z π 2 − π 2 dx p Y k =1 h ( x + ( k − 1) µ ) , ˜ G ( m, n ) = 1 π Z π 2 − π 2 dx h ( x ) p . Next, let that l ∈ { 0 , . . . , p − 1 } and define λ l ( x ) = p − l Y k =1 h ( x + ( k − 1) µ ) . Assume th at Z π 2 − π 2 dx λ l ( x ) h ( x ) l ≤ Z π 2 − π 2 dx λ l +1 ( x ) h ( x ) l +1 . Then ˜ G (1 , n ) = 1 π Z π 2 − π 2 dx λ 0 ( x ) ≤ 1 π Z π 2 − π 2 dx λ 1 ( x ) h ( x ) ≤ . . . ≤ 1 π Z π 2 − π 2 dx λ l ( x ) h ( x ) l ≤ . . . ≤ 1 π Z π 2 − π 2 dx λ p − 1 ( x ) h ( x ) p − 1 = 1 π Z π 2 − π 2 dx h ( x ) p = ˜ G ( m, n ) . T o show that the assumption holds true, proc eed by generating a partially ordere d set. L et δ l = ( p − l − 1) µ an d note that λ l ( x ) = λ l +1 ( x ) h ( x + δ ) . Then, the first elemen t of the partially orde red set is Z π 2 − π 2 dx λ l +1 ( x ) h ( x + δ ) h ( x ) l . And the last elemen t is Z π 2 − π 2 dx λ l +1 ( x ) h ( x ) l +1 . For greater clarity , the indice of λ l +1 ( x ) and δ l are suppressed in what follows. Next, ap ply recu rsiv ely t times h ( x ) a h ( x + δ ) b ≤ 1 2 h ( x ) a − b h h ( x ) 2 a + h ( x + δ ) 2 a i from the first element. After summing the resulting geometric series, the element gene rated is Z π 2 − π 2 dx λ ( x )  " 1 −  1 2  t # h ( x ) l +1 +  1 2  t h ( x ) l +1 − 2 t h ( x + δ ) 2 l  . Remark howe ver that b eyond a r ecursion step defined such that 2 t +1 > l + 1 ≥ 2 t , the power of h ( x ) becomes negati ve in the second term. This is problematic be cause the last element of th e partially or dered set has only p ositiv e powers of h ( x ) . T o circumvent this prob lem, let x → x − δ in the the second ter m of the elem ent. Since the domain of integration remains unchan ged ( Lemma A.1), h ( x ) is even (Lemma A.5), and λ ( x ) = λ ( x − δ ) (Corollary A.6), then the seco nd term is also eq ual to Z π 2 − π 2 dx λ ( − x ) h ( − x ) l +1 − 2 t h ( − x + δ ) 2 l Finally , lettin g x → − x , the t -elemen t becomes Z π 2 − π 2 dx λ ( x )  " 1 −  1 2  t # h ( x ) l +1 +  1 2  t h ( x + δ ) l +1 − 2 t h ( x ) 2 l  And the power of h ( x ) is now greater than or eq ual to that of h ( x + δ ) in the second term allowing again the recursiv e application of the ineq uality . The algor ithm th us loops throug h succe ssi ve r ecursion steps an d chan ges o f v ariable. Note that for l + 1 = 2 t , the last element is generated after th e first iteration. Lemma A.7 states th at this is the on ly case for which the a lgorithm terminates in a finite number of iterations. For all other cases, the last element arises by letting the numbe r of iteration s go to infin ity . V . A N A NA L Y T I C A L E X A M P L E In the previous section, a lower bound of the probability of no detec tion was fou nd for a n ar bitrary g ( x ) . In this sectio n, the pro bability of no d etection is ev aluated and th e inequa lity of Theorem 4.2 is explicitly verified when g ( x ) = sin ( x ) 2 Lemma 5.1: Let g ( x ) = sin ( x ) 2 . Th en ˜ G ( m, n ) = 2 p (2 p − 1 )!! 4 n p ! . (6) Pr o of: Recall that ˜ G ( m, n ) = 1 π Z π 2 − π 2 dx p Y k =1 r − 1 Y j =0 g ( x + j mµ ) . Then Lem ma A.8 im plies that ˜ G ( m, n ) = 1 π Z π 2 − π 2 dx p Y k =1 r − 1 Y j =0 g  x + j π r  . Let g ( x ) = sin ( x ) 2 and use the identity [15]: sin ( r x ) = 2 r − 1 r − 1 Y j =0 sin  x + j π r  . Then, ˜ G ( m, n ) = 1 π Z π 2 − π 2 dx p Y k =1 1 4 r − 1 sin ( r x ) 2 = 1 π Z π 2 − π 2 dx 1 4 p ( r − 1) sin ( r x ) 2 p Finally , carrying out the integral [16 ] an d r ecalling that n = pr , ˜ G ( m, n ) = 1 4 n − p (2 p − 1 )!! 2 p !! . Since 2 p !! = 2 p p ! , (6) follows. Cor olla ry 5.2 : For p = gc d (1 , n ) = 1 , ˜ G (1 , n ) = 2 4 n . And Theorem 4.2 follows since (2 p − 1 )!! = 1 × 3 × · · · × (2 p − 1 ) ≥ 1 × 2 × · · · × p = p ! . V I . C O N C L U S I O N S A N D F U T U R E WO R K In th is paper, the angu lar dependen ce of the detection process wh ich is often overlooked fo r search and detection mission is explicitly accoun ted for by a ssuming that the tar- get p ossesses r ectangular symmetry . One major co nsequen ce of this ap proxim ate symmetry is that the long side of a target is en dowed with the largest cr oss section, which results in the hig hest proba bility of detectio n given that the target is observed only once. Howev er, since the o rientation of th e target is in gen eral unk nown, there is likelihoo d that it will be im aged on the short side , i.e., the smallest cross-section. Therefo re, the p robability of not d etecting the target m ay n ot be zero even if the search area is en tirely covered. M aking se veral observations o f the target in order to increase the change of observing its lo ng side is o ne way to ad dress th is problem . Assuming that the observations are indepe ndent, an op ti- mal search stra tegy is then to observe the ta rget such that the separation between two co nsecutive observations is a constant and equal to a multiple of 180 degrees divided by the nu mber of observations. The resulting tactic is simp le, intuitive and robust (as no prior knowledge of the target orientation is requ ired). For example, two ob servations sep - arated b y 90 degree s or thr ee observations separated by 60 degrees will m inimize the p robab ility of no d etection. Having shown th at one of these search strategies leads to a lower bou nd o f the probability o f no detection, work is cu rrently un derway to prove th at it is also a globally minimal sear ch strategy . An other log ical extensio n of this work is to relax the assump tion that informatio n pr ovided by subseq uent o bservations is un correlated (i.e., follows a Bernoulli pro cess) as it en tails tha t the pro bability of not d etecting a ta rget decreases with in creasing nu mber of observations ev en if the observations are co-linear . Th is is questionab le as no ad ditional infor mation is gain ed. V I I . A C K N OW L E D G M E N T S One of the au thors (A. Bourq ue) would like to a cknowl- edge A. Perciv al f rom Defence R&D Canada - Atlantic for bringin g to his attentio n the issue of correlation effects in the d etection process. A P P E N D I X S U P P L E M E N TA RY L E M M A TA Lemma A.1 (Rotatio nal In variance): L et ω ∈ R and h ( x ) = h ( x + π ) . Then Z π 2 − π 2 dx h ( x − ω ) = Z π 2 − π 2 dx h ( x ) . Pr o of: Let x → x + ω on the RHS and break the integration interval o f the r esulting integral into [ − π 2 − ω , − π 2 [ and [ − π 2 , π 2 − ω ] . Then let x → x − π in the integral over the first interval and recall that by assumption h ( x ) is period ic. Lemma A.2: Let i ∈ { 0 , . . . , n − 1 } , N i = { 0 , . . . , n − 1 } \{ i } and j ∈ N i . Then σ i ( j ) = (2 i − j ) mo d n is a bijectio n fr om N i to itself and its own in verse. Pr o of: Composition gives σ i ( σ i ( j )) = (2 i − (2 i − j ) mo d n ) mo d n. Use the definition of the mo dulo twice to giv e σ i ( σ i ( j )) = j +  (2 i − j ) n  n −     j + j (2 i − j ) n k n n     n. Recall that j ∈ N i and note that j n < n . Then j j n + j (2 i − j ) n k n k = j (2 i − j ) n k n and σ i ( σ i ( j )) = j . Lemma A.3: Let r , q be positive integers such th at gcd ( r, q ) = 1 , i.e., r and q are co-pr imes. L et i ∈ { 0 , . . . , r − 1 } . Then the map σ ( i ) = q i mod r is a bijection o f the set of i to itself. Pr o of: Proceed with a p roof by contradiction . Assume this map is n ot a bijection. Then there exists a pair u, v ∈ { 0 , . . . , r − 1 } su ch that u 6 = v and q u mo d r = q v mo d r . Next, assume th at u > v then q ( u − v ) mo d r = q w mo d r = 0 where w = u − v . Th is implies th at q w = ry with y a positive integer , as w, q > 0 . Because q and r are co-pr imes, i.e., gcd ( r, q ) = 1 then w = rz with z > 0 , which leads to a con traction as w = u − v < r − 1 . Lemma A.4: Let u ∈ { 0 , . . . a − 1 } , v ∈ { 0 , . . . , b − 1 } , and w ∈ { 0 , . . . , n − 1 } wh ere n = ab . T hen σ ( u, v ) = u + av and σ − 1 ( w ) =  w mo d a,  w a  are b ijections. Pr o of: Composition gives σ  σ − 1 ( w )  = w mod a + a j w a k = w , where the last equality fo llows fro m th e definition of the modulo op eration. Similarly , σ − 1 ( σ ( u, v )) =  ( u + av ) mo d a,  u + av a  = ( u , v ) , where the last equality fo llows fro m th e definition of the modulo operatio n and since u < a . Therefor e, σ ( u, v ) and σ − 1 ( w ) are bo th one- to-one, onto, and inverse of each oth er . Lemma A.5: h ( x ) = h ( − x ) . Pr o of: Becau se of th e perio dicity of g ( x ) and nq = mr , g ( x + mj µ ) = g ( x + m ( j − r ) µ ) . Let j → − j + r . Then h ( x ) = g ( x ) . . . g ( x − ( r − 1 ) j µ ) and the proof fo llows since b y definition g ( x ) is even. Cor olla ry A.6 : Con sider λ l ( x ) and δ l = ( p − k − 1) µ . Then λ l ( x − δ l ) = h ( x − ( p − l − 1 ) µ ) . . . h ( x ) and λ l ( x − δ l ) = λ l ( − x ) sinc e h ( x ) is even b y Lemma A.5. Lemma A.7: Let i and c be po siti ve integers. Let { t i } be the set o f n on-negative integers such that 2 t i +1 > c ≥ 2 t i . And let { u i } be the set o f no n-negative in teger such that u i = c − 2 t i u i − 1 and u 0 = 1 . Th en, a fix point exists if and only if c = 2 l 1 . Pr o of: After i iterations, u i = c − 2 t i u i − 1 = c − 2 t i  c − 2 t i − 1 u i − 2  =  c − 2 t i  . . .  c − 2 1  . . .  = c  1 − 2 t i  . . .  1 − 2 t 2  . . .  + ( − 1) i 2 t i + ··· + t 1 . Then u i = 0 implies that the prime factorization of ( k + 1) must be 2 a where a is a n on-negative integer . Because 2 t 1 +1 > ( l + 1 ) = 2 a ≥ 2 t 1 , th en a = t 1 and u i = 0 for i > 0 . Lemma A.8: r − 1 Y j =0 g  x + mj π n  = r − 1 Y j =0 g  x + j π r  . Pr o of: Reca ll that r , q a nd p are positive integers such that m = pq , n = pr , an d p = gcd( m, n ) . The n g  x + mj π n  = g  x + q j π r  . Use the period icity o f g ( x ) an d the definition of the modu lo to fu rther write the right-h and side of the equality a s g  x + q j π r −  q j r  π  = g  x + ( q j mo d r ) π r  . Lemma A.3 implies that map σ ( j ) = q j mo d r is a bijectio n from the set of j to itself. R E F E R E N C E S [1] B. Zerr , E. Bovio, and B. 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