On the d-dimensional Quasi-Equally Spaced Sampling

1 On the d -dimensional Quasi-Equally Spaced Sampling Alessandro Nordio ⋆ , Carla-Fa biana Chiasserini ⋆ , Emanuele V iterbo ‡ ⋆ Dipartimento di Elettronica, Politecnico di T orino C. Duca degli Abruzzi 24, I-10129 T orino, Italy Phone: +39 011 090 4226; Fax +39 0 11 09 04099 E-mail: { alessandro.nordio,chi asserini } @poli to.it ‡ DEIS, Univ ersit ` a della Calabria V ia P . Bucci, Cubo 42C, 87036 Rende (CS), Italy Phone: +39 0 984 4 94778; Fax +39 0984 494713 E-mail: vit erbo@deis.unical.it Abstract W e study a class o f rand om matrices that appear in several communicatio n and sign al processing applications, and whose asymp totic eigenv alue distribution is clo sely related to the reconstruction error of an irregularly sampled bandlimited s ignal. W e focus on the case where t he random variables characterizing these matrices are d -dimensional vectors, independent, and quasi-equally spaced, i.e., they have an arbitrary distribution a nd their av erages are vertices of a d -d imensional grid . Alth ough a clo sed f orm expression of the eigenv alue distribution is still unknown, under these c ondition s we are able ( i ) to d eriv e the distribution m oments as the matrix size grows to infinity , while its aspect ratio is kept constant, and ( ii ) to show that the eigenvalue distribution ten ds to the Mar ˇ cenko-Pastur law as d → ∞ . These r esults can find applicatio n in se veral fields, as an example we show how they can b e used fo r th e estimation of the mean squar e error p rovided by linear reconstruc tion techniqu es. EDICS: DSP -RECO Signal re construction, DSP-SAMP Samp ling, SPC-PERF Pe rformance ana lysis and b ounds. October 25, 2021 DRAFT 2 I . I N T R O D U C T IO N Consider the c lass of random matrices of size (2 M + 1) × r , with entries g i ven by G = 1 √ 2 M + 1            e − j2 π M x 1 · · · e − j2 π M x r . . . . . . 1 · · · 1 . . . . . . e +j2 π M x 1 · · · e +j2 π M x r            (1) The ge neric element o f G can be written as: G ℓ,q = 1 √ 2 M +1 e j2 π ℓx q , ℓ = − M , . . . , M , q = 0 , . . . , r − 1 , where x q are independ ent random vari ables characterized by a p robability density function (pdf) f x q ( z ) , with 0 ≤ z ≤ 1 . Th ese matrices are V ande rmonde matrices with complex expon ential en tries; they ap pear in many s ignal/image process ing a pplications and have been studied in a nu mber of recent works, (se e e.g., [ 1]–[8]). More specifica lly , in the field of signa l processing for sen sor ne tworks, [1], [2] stud ied the performance of linear rec onstruction techniqu es for phy sical fields irregularly sampled by senso rs. In such sce nario, the rand om variables x q in (1) rep resent the coordina tes of the s ensor no des. The work in [3] add ressed the cas e where these coordina tes are u niformly distributed and s ubject to an un known jitter . In the field of communications, the stud y in [8] presented a number of a pplications where thes e matrices app ear , which range from mu ltiuser MIMO sys tems to multifold sca ttering. In sp ite of their numerous applications , fe w results are kn own for the V an dermonde matrices in (1). In particular , a close d form expres sion for the e igen v alue distribution of the Hermitian T oeplitz matrix GG † , as well as its asymptotic behavior , would be of great interest. As an example, in [1], [2], [6], it has been observed that the performanc e of linear techniqu es for recons tructing a sign al from a se t of irregularly- space d samples with known coo rdinates is a function of the asy mptotic e igen va lue distribution o f GG † . The as ymptotic eigen value distributi on o f GG † is de fined as the distrib ution of its eige n v alues, in the limit o f M and r gro wing to infinity while the ir ratio is kep t constant. Unfortunately , such d istrib ution is s till unkn own. In this work, we conside r a general formulation whic h extends the model in ( 1 ) to the d -dimensiona l domain. W e stud y the properties of rand om matrices of size (2 M + 1) d × r and entries given by ( G d ) ν ( ℓ ) ,q = 1 p (2 M + 1) d e − j2 π ℓ T x q (2) where the vectors x q = [ x q 1 , . . . , x q d ] T have indepe ndent e ntries, cha racterized by the p df f x qm ( z ) , October 25, 2021 DRAFT 3 q = 0 , . . . , r − 1 , m = 1 , . . . , d , and d is the numbe r of d imensions. The inv ertible function ν ( ℓ ) = d X m =1 (2 M + 1) m − 1 ℓ m (3) maps the vec tor o f integers ℓ = [ ℓ 1 , . . . , ℓ d ] T , ℓ m = − M , . . . , M onto a s calar index, i.e., the row index of the matrix G d . Notice that, when d = 1 , G d reduces to (1). For the matrix mod el in (2), we study the interes ting case where x q are inde pende nt, qu asi-equally spaced random variables in the d -dimensional hy percube [0 , 1) d . In other words, we ass ume that the av erages of x q are the vertices of a d -dimensional grid in [0 , 1) d . Th is is often the c ase arising in measureme nt sys tems aff ected by jitt er , or in senso r network dep loyments whe re the sen sors sampling the phys ical field c an only be roughly placed at e qually sp aced positions, due to terrain conditions and deployment practicality [9]. Note that the distribution of the rand om variables x can be of any kind, the only assump tion we make is o n their averages b eing equally spaced. Since an an alytic expression of the eigen v alue distrib ution of G d G † d is unknown, we deriv e a closed form express ion for its moments. This enables us to s how that, a s d → ∞ , the eigen value distrib ution ten ds to the Mar ˇ cenko-Pastur law [12]. At the end of the pa per , we present some numerica l results a nd applications whe re the moments and the asymptotic a pproximation to the e igen v alue distrib ution o f G d G † d can b e of great use. I I . P R E V I O U S R E S U L T S A N D P RO B L E M F O R M U L AT I O N As a first step, we briefly revie w previous results on the G d matrices. In a o ne-dimensiona l domain ( d = 1 ), the work in [1] considered a n irregularly sampled b andlimited s ignal, which is recons tructed using linear techniqu es a nd a ssuming the samples co ordinates to be kn own. The performance of the reconstruction s ystem was d eri ved as a func tion of the eigenv alue distribution f λ (1 , β , z ) o f the ma trix T 1 = β G 1 G † 1 , wh ere β is the aspec t ratio 1 of G 1 [1], [2]. An explicit expression of the momen ts E [ λ p 1 ,β ] = Z ∞ 0 z p f λ (1 , β , z ) d z was attained in [4], [5], for the specific case whe re x q are un iformly distributed in [0 , 1) . Also, in the case where x q are indepe ndent, quasi-equally space d ran dom vari ables, the ana lytic expression of the second moment of the eigen v alue distributi on of T , i.e., E [ λ 2 1 ,β ] , was obtained in [3]. Then, in [7] the moments f λ (1 , β , z ) were deriv ed for an arbitrary distrib ution f x q ( z ) . In [4], [5], the d -dimensiona l mo del (2) was also in vestigated. The re, the properties of the random matrices G d were studied in the ca se where the vectors x q = [ x q 1 , . . . , x q d ] T have ind epende nt entries, 1 The aspect ratio of G is the ratio between the number of ro ws and the number of columns of the matrix October 25, 2021 DRAFT 4 uniformly distribut ed in the h ypercube [0 , 1) d . Under s uch assump tions, and for gi ven d and aspect ratio β , an ana lytic expression of the mome nts o f f λ ( d, β , z ) was derived and it was shown that, as d → ∞ , f λ ( d, β , z ) tends to the Mar ˇ cenko-Pastur law [12 ], i.e., lim d →∞ f λ ( d, β , z ) = f MP ( β , z ) = p ( c 1 − z )( z − c 2 ) 2 π z β where c 1 , c 2 = (1 ± √ β ) 2 , 0 < β ≤ 1 , c 2 ≤ x ≤ c 1 . The following sections detail the problem ad dressed in this work and introduce s ome use ful notations. A. The quasi-equa lly spaced multidimension al model W e c onsider the matrix class in (2) and a ssume tha t the vectors x are inde penden t, q uasi-equa lly space d random variables in the d -dimensional hyp ercube [0 , 1) d , i.e., the averages of x are the vertices of a d -dimension al grid in [0 , 1) d . W e define ρ as the number of vertices per dimension, thus the total nu mber of vertices is r = ρ d . W e denote the coordinate of a ge neric vertex of the g rid by the vector q /ρ ∈ [0 , 1) d , where q = [ q 1 , . . . , q d ] T , is a n integer vector and q m = 0 , . . . , ρ − 1 . For notation simplicity and in ana logy with (3), we ide ntify the vertex with c oordinate q /ρ by the sc alar index µ ( q ) = d X m =1 ρ m − 1 q m (4) Notice that 0 ≤ µ ( q ) ≤ r − 1 is an in vertible function a nd allows us to write x µ ( q ) = q ρ + ˜ x µ ( q ) ρ where the average E [ x µ ( q ) ] = q ρ + 1 2 ρ is the coordina te of the sample iden tified by the sca lar label µ ( q ) an d 1 is the all on es vector . Furthermore, we ass ume that the entries of the vectors ˜ x µ ( q ) are i.i.d. with pdf f ˜ x ( z ) which does not depend on r , M , or q . By using this notation, the entries of G d are then given b y ( G d ) ν ( ℓ ) ,µ ( q ) = 1 p (2 M + 1) d e − j2 π ℓ T x µ ( q ) (5) while the aspe ct ratio is β = (2 M + 1) d r =  2 M + 1 ρ  d (6) The He rmitian T oeplitz matrix T d = β G d G † d is define d as ( T d ) ν ( ℓ ) ,ν ( ℓ ′ ) = 1 ρ d X q e − j2 π x T µ ( q ) ( ℓ − ℓ ′ ) (7) October 25, 2021 DRAFT 5 where P q represents a d -dimensiona l sum over all vec tors q such that q m = 0 , . . . , ρ − 1 , m = 1 , . . . , d . Our goals are (i) to deri ve the analytic expression of the mome nts of f λ ( d, β , z ) with q uasi-equally space d vec tors x µ ( q ) (Section III), and (ii) to show tha t a s d → ∞ , f λ ( d, β , z ) tends to the Mar ˇ cenko- Pastur law (Section IV). I I I . C L O S E D F O R M E X P R E S S I O N O F T H E M O M E N T S O F T H E A S Y M P T OT I C E I G E N V A L U E P D F Follo wing the app roach adopted in [13 ], [14], in the limi t for M an d r growi ng to infin ity with c onstant aspec t ratio β and d imension d , we comp ute the closed form express ion of E [ λ p d,β ] , which can be o btained from the powers of T d as [15], E [ λ p d,β ] = lim M ,r → + ∞ β T r  E X  T p d   (2 M + 1) d (8) In (8) the s ymbol T r identifies the matrix trace operator , and the average E X [ · ] is comp uted over the set of random variables X = { x 0 , . . . , x r − 1 } . An efficient method to c ompute (8) exploits s et partitioning . Indeed, note that the p ower T p d is the matrix product of p copies of T d . Th is o peration yields exponen tial terms, wh ose exponen ts a re given by a sum of p terms of the form x T µ ( q i ) ( ℓ i − ℓ [ i +1] ) (see also (22) in Appendix A ). The av erage of this sum depends on the numbe r of distinct vectors q i , an d all possible cases can be described as p artitions of the s et P = { 1 , . . . , p } . In p articular , the ca se wh ere in the set { q 1 , . . . , q p } there are 1 ≤ k ≤ p d istinct vectors, co rresponds to a partition of P in k subs ets. It follo ws that a funda mental s tep to calculate (8) is the co mputation o f a ll possible partitions of set P . Before p roceeding further in our analys is, we therefore introduce some u seful d efinitions related to se t partitioning. A. Definitions Let the integer p denote the moment order and let the vector µ = [ µ 1 , . . . , µ p ] be a poss ible co mbination of p integers. In our spec ific case, ea ch entry of the vector µ is given b y the expression in (4), i.e., µ i = µ ( q i ) an d, thus, can range betwee n 0 and r − 1 . W e de fine: • the sc alar integer 1 ≤ k ( µ ) ≤ p as the number of distinct e ntries o f the vector µ ; • γ ( µ ) as the vector o f integers, of length k ( µ ) , whose entries γ j ( µ ) , j = 1 , . . . , k ( µ ) , are the entries of µ without repetitions, in orde r of appe arance within µ ; • P j ( µ ) a s the set of indice s of the entries of µ w ith value γ j ( µ ) , j = 1 , . . . , k ( µ ) ; October 25, 2021 DRAFT 6 • the vector ω ( µ ) = [ ω 1 ( µ ) , . . . , ω p ( µ )] such tha t, for any given j = 1 , . . . , k ( µ ) , we hav e ω i ( µ ) = j if i ∈ P j ( µ ) , i = 1 , . . . , p . Example 1: Let µ = [1 , 5 , 2 , 8 , 5 , 3 , 2] , then k ( µ ) = 5 since the entries of µ take 5 distinct values (i.e., { 1 , 5 , 2 , 8 , 3 } ). Such values, taken in order of app earance in µ form the vector γ ( µ ) = [1 , 5 , 2 , 8 , 3] . Th e value γ 1 = 1 appears a t position 1 in µ , therefore P 1 ( µ ) = { 1 } . The value γ 2 = 5 appe ars a t p ositions 2 and 5 in µ , therefore P 2 ( µ ) = { 2 , 5 } . Similarly P 3 ( µ ) = { 3 , 7 } , P 4 ( µ ) = { 4 } , and P 5 ( µ ) = { 6 } . By using the se ts P j we build the vector , ω ( µ ) . For each j = 1 , . . . , k we assign the value j to every ω i such that i ∈ P j . For example, ω 2 = ω 5 = 2 since the integers 2 and 5 are in P 2 . In con clusion ω ( µ ) = [1 , 2 , 3 , 4 , 2 , 5 , 3] . Furthermore, we defi ne: • Ω p as the set of p artitions of P ; • Ω p,k as the set of p artitions of P in k s ubsets, 1 ≤ k ≤ p , with p ∪ k =1 Ω p,k = Ω p . Note tha t: (i) the cardinality of Ω p , denoted by B ( p ) = | Ω p | , is the p -th Bell numb er [16] an d (ii) the cardinality o f Ω p,k , d enoted by S ( p, k ) = | Ω p,k | , is a Stirling number o f the second k ind [17]. From the ab ove definitions, it follows that: 1) the vector µ induces a partiti on of the set P which is iden tified by the sub sets P j ( µ ) . Thes e subs ets have the following properties k ( µ ) ∪ j =1 P j ( µ ) = P , P j ( µ ) ∩ P j ′ ( µ ) = ∅ for j 6 = j ′ Even tho ugh the pa rtition identified by µ is often represe nted as {P 1 , . . . , P k ( µ ) } , by its de finition, an equ i valent rep resentation of such pa rtition is giv en by the vector ω ( µ ) . T herefore, from now on we will refer to ω ( µ ) as a partition of the p element set P induced by µ (for simplicity , howe ver , often we will not exp licit the de penden cy of ω on µ ); 2) k ( ω ) = k ( µ ) , since the e ntries of ω take all po ssible values in the se t { 1 , . . . , k ( µ ) } ; 3) P j ( ω ) = P j ( µ ) , for j = 1 , . . . , k ( µ ) . At last, we defin e M ( ω ) as the s et of µ inducing the same p artition ω of P . October 25, 2021 DRAFT 7 Example 2: L et r = 3 and p = 3 . Sinc e µ = [ µ 1 , . . . , µ p ] and µ i = 0 , . . . , r − 1 , i = 1 , . . . , p , we have r p = 27 possible vectors µ , name ly , { [0 , 0 , 0] , [0 , 0 , 1] , . . . , [2 , 2 , 1] , [2 , 2 , 2] } . Ea ch µ ide ntifies a partition ω ∈ Ω 3 ,k , with k = 1 , . . . , 3 , a s d escribed in Example 1. The sets of partitions Ω 3 ,k , are given by Ω 3 , 1 = { [1 , 1 , 1] } , Ω 3 , 2 = { [1 , 1 , 2] , [1 , 2 , 1] , [1 , 2 , 2] } , an d Ω 3 , 3 = { [1 , 2 , 3] } , and have cardinality S (3 , 1) = 1 , S (3 , 2) = 3 and S (3 , 3) = 1 , respectively . The s et of vectors µ identifying the pa rtition ω = [1 , 1 , 1] , i.e., M ([1 , 1 , 1]) , is g i ven by: M ([1 , 1 , 1]) = { [0 , 0 , 0] , [1 , 1 , 1] , [2 , 2 , 2] } . Similarly , M ([1 , 1 , 2]) = { [0 , 0 , 1] , [0 , 0 , 2] , [1 , 1 , 0] , [1 , 1 , 2] , [2 , 2 , 0] , [2 , 2 , 1] } M ([1 , 2 , 1]) = { [0 , 1 , 0] , [0 , 2 , 0] , [1 , 0 , 1] , [1 , 2 , 1] , [2 , 0 , 2] , [2 , 1 , 2] } M ([1 , 2 , 2]) = { [0 , 1 , 1] , [0 , 2 , 2] , [1 , 0 , 0] , [1 , 2 , 2] , [2 , 0 , 0] , [2 , 1 , 1] } M ([1 , 2 , 3]) = { [0 , 1 , 2] , [0 , 2 , 1] , [1 , 0 , 2] , [1 , 2 , 0] , [2 , 0 , 1] , [2 , 1 , 0] } B. Closed form expression of E [ λ p d,β ] By using the defin itions in S ection III- A a nd by applying set partitioning to (8), we ca n s tate the first main res ult of this work: Theorem 3 .1: Let T d be a (2 M + 1) d × (2 M + 1) d Hermitian random matrix a s defin ed in (7), where the properties o f the random vectors x µ ( q ) are des cribed in Section II-A. Then, for a ny given β and d , the p -th momen t of the a symptotic eigen value distribution of T d is given by: E [ λ p d,β ] = p X k =1 k X h =1 β p − h X ω ∈ Ω p,k X ω ′ ∈ Ω k,h u ( ω ′ ) v ( ω , ω ′ ) d (9) where u ( ω ′ ) = ( − 1) k − h h Y j ′ =1 ( |P j ′ ( ω ′ ) | − 1)! (10) v ( ω , ω ′ ) =                          Z H p k Y j =1 C  − j2 π β 1 /d w j ( ω )  d y h = 1 Z H p k Y j =1 C  − j2 π β 1 /d w j ( ω )  h Y j ′ =1 δ D   X i ′ ∈P j ′ ( ω ′ ) w i ′ ( ω )   d y 1 < h < k Z H p k Y j =1 δ D ( w j ( ω )) d y h = k (11) October 25, 2021 DRAFT 8 T ABLE I P A RT I T I O N S E T S Ω n,m F O R n = 1 , 2 , 3 , A N D 1 ≤ m ≤ n . E A C H PA RT I T I O N I S R E P R E S E N T E D T H R O U G H I T S A S S O C I AT E D V E C T O R ω A N D T H E V A L U E O F u ( ω ) ω , u ( ω ) m = 1 m = 2 m = 3 n = 1 [1], 1 n = 2 [1,1], -1 [1,2], 1 n = 3 [1,1,1], 2 [1,1,2], -1 [1,2,1], -1 [1,2,2], -1 [1,2,3], 1 and v ( ω , ω ′ ) = 1 for k = 1 . In (11), w e de fined H p as the p -dimensional hypercu be [ − 1 / 2 , 1 / 2) p , C ( s ) = E ˜ x [e sz ] as the c haracteristic func tion of ˜ x , δ D ( · ) a s the Dirac’ s delta, a nd w j ( ω ) = X i ∈P j ( ω ) y i − y [ i +1] y i ∈ R , i = 1 , . . . , p , and j = 1 , . . . , k ( ω ) . Pr oof: The proof can be foun d in Append ix A. W ith the aim to give a n intuitiv e explanation of the a bove express ions, no te that the right hand side of (9) c ounts all possible partitions of the set P = { 1 , . . . , p } , C ( s ) in (11) ac counts for the gene ric distrib ution of the variables ˜ x , a nd the qua ntity w j ( ω ) represen ts the indices pa iring that a ppears in the exponent of the gen eric entry o f the power T p d . T o further c larify the mo ments c omputation, T able I rep orts an exa mple of partition s ets Ω n,m for n = 1 , . . . , 3 and 1 ≤ m ≤ n , w hile Exa mple 3 s hows the computation o f the second moment of the eigen v alue distributi on. October 25, 2021 DRAFT 9 Example 3: W e co mpute the analytic expres sion of E [ λ 2 d,β ] . Us ing (9), we get: E [ λ 2 d,β ] = 2 X k =1 k X h =1 β 2 − h X ω ∈ Ω 2 ,k X ω ′ ∈ Ω k,h u ( ω ′ ) v ( ω , ω ′ ) d By exp anding this expres sion and us ing T a ble I, we obtain E [ λ 2 d,β ] = β v ([1 , 1] , [1]) d − β v ([1 , 2] , [1 , 1]) d + v ([1 , 2] , [1 , 2]) d W e no tice that, for k = 1 , v ([1 , 1] , [1]) = 1 . The term v ([1 , 2] , [1 , 2]) refers instead to the case k = h = 2 , an d it is giv en by v ([1 , 2] , [1 , 2]) = Z H 2 2 Y j =1 δ D ( w j ([1 , 2])) d y with w 1 ([1 , 2]) = y 1 − y 2 and w 2 ([1 , 2]) = y 2 − y 1 . It follo ws that v ([1 , 2] , [1 , 2]) = Z H 2 δ D ( y 1 − y 2 ) δ D ( y 2 − y 1 ) d y = 1 Finally , v ([1 , 2] , [1 , 1]) = Z H 2 2 Y j =1 C  − j2 π β 1 /d w j ([1 , 2])  d y = Z H 2    C  − j2 π β 1 /d ( y 1 − y 2 )     2 d y Thus, we write E [ λ 2 d,β ] = 1 + β − β  Z H 2    C  − j2 π β 1 /d ( y 1 − y 2 )     2 d y  d I V . C O N V E R G E N C E T O T H E M A R ˇ C E N K O - P A S T U R D I S T R I B U T I O N In this se ction we s how that the asymptotic e igen v alue distributi on of the ma trix T d tends to the Mar ˇ cenko-Pastur law [12], a s d → ∞ . This is e quiv alent to prove that, a s d → ∞ , the p -th moment of λ d,β tends to the p -th moment of the Mar ˇ cenko-P astur distrib ution with p arameter β , for ev ery p ≥ 1 . Theorem 4 .1: Let T d be a (2 M + 1) d × (2 M + 1) d Hermitian random matrix a s defin ed in (7), where the properties of the rando m vectors x µ ( q ) are described in Section II-A. Le t E [ λ p d,β ] be the p -th mome nt of the as ymptotic eigen v alue distributi on of T d , given by Theo rem 3.1. The n, for any given β , lim d →∞ E [ λ p d,β ] = E [ λ p ∞ ,β ] = p X k =1 β p − k N ( p, k ) (12) October 25, 2021 DRAFT 10 where N ( p, k ) are the Narayan a numbe rs [18], [19] and E [ λ p ∞ ,β ] are the N arayana polynomials , i.e., the moments o f the Mar ˇ cenko-Pastur distribution [12]. Pr oof: W e first loo k a t the expression of the p -th a symptotic moment and obs erve tha t, for h = k , the contributi on of the term in the right hand s ide of (9) reduces to p X k =1 β p − k X ω ∈ Ω p,k X ω ′ ∈ Ω k,k u ( ω ′ ) v ( ω , ω ′ ) d (13) The cardinality of Ω k ,k is S ( k , k ) = 1 and Ω k ,k = { [1 , . . . , k ] } . Thus, we only cons ider ω ′ = [1 , . . . , k ] . Moreover , u sing (10) we h av e u ([1 , . . . , k ]) = 1 sinc e e ach subset P j ′ ([1 , . . . , k ]) has cardinality 1 , j ′ = 1 , . . . , k . T herefore, the term in (13) become s p X k =1 X ω ∈ Ω p,k β p − k v ( ω , [1 , . . . , k ]) d Using (11) with h = k , we h ave: v ( ω , [1 , . . . , k ]) = Z H p k Y j =1 δ D ( w j ( ω )) d y ∆ = v ( ω ) (14) Hence, the co ntrib ution to the p -th momen t reduc es to p X k =1 β p − k X ω ∈ Ω p,k v ( ω ) d (15) In [4], [5] it is s hown that, as d → ∞ , (15) tends to the Naraya na polynomial o f order p . It follows tha t, in order to p rove the theorem, it is en ough to show that for h < k the c ontrib ution of the term in the right hand s ide of (9), to the expres sion of the p -th asymptotic mo ment, vanishes as d → ∞ . In practice we have to s how that, for each ω ∈ Ω p,k and ω ′ ∈ Ω k ,h , with h < k , lim d →∞ v ( ω , ω ′ ) d = 0 or , equ i valently , that | v ( ω , ω ′ ) | < 1 . W e first notice that for 1 < h < k | v ( ω , ω ′ ) | =       Z H p k Y j =1 C  − j2 π β 1 /d w j ( ω )  h Y j ′ =1 δ D   X i ′ ∈P j ′ ( ω ′ ) w i ′ ( ω )   d y       ≤ Z H p       k Y j =1 C  − j2 π β 1 /d w j ( ω )  h Y j ′ =1 δ D   X i ′ ∈P j ′ ( ω ′ ) w i ′ ( ω )         d y = Z H p k Y j =1    C  − j2 π β 1 /d w j ( ω )     h Y j ′ =1 δ D   X i ′ ∈P j ′ ( ω ′ ) w i ′ ( ω )   d y (16) October 25, 2021 DRAFT 11 Moreover , we have:    C  − j2 π β 1 /d w j ( ω )     =     Z + ∞ −∞ exp  − j2 π β 1 /d w j ( ω ) z  f ˜ x ( z ) d z     ( a ) ≤ Z + ∞ −∞    exp  − j2 π β 1 /d w j ( ω ) z  f ˜ x ( z )    d x = Z + ∞ −∞ f ˜ x ( z ) d z = 1 (17) The equa lity ( a ) arises if the co ndition w j ( ω ) = 0 is always verified, othe rwise, if w j ( ω ) 6 = 0 ,   C  − j2 π β 1 /d w j ( ω )    < 1 . Next, we make the following obse rv ations: ( i ) sinc e we co nsider partitions ω ′ of the form { 1 , . . . , k } in h subsets with h < k , then at least one o f the sets P j ′ ( ω ′ ) has c ardinality |P j ′ ( ω ′ ) | > 1 ; ( ii ) the term h Y j ′ =1 δ D   X i ′ ∈P j ′ ( ω ′ ) w i ′ ( ω )   giv es a non-zero co ntrib ution to the integral in (16) only when P i ′ ∈P j ′ ( ω ′ ) w i ′ ( ω ) = 0 . Hence , if |P j ′ ( ω ′ ) | > 1 for some j ′ , then some w i ′ ( ω ′ ) 6 = 0 will provide a n on-zero contribution to the integral in (16). In this c ase, we can write | v ( ω , ω ′ ) | ≤ Z H p k Y j =1    C  − j2 π β 1 /d w j ( ω )     h Y j ′ =1 δ D   X i ′ ∈P j ′ ( ω ′ ) w i ′ ( ω )   d y < Z H p h Y j ′ =1 δ   X i ′ ∈P j ′ ( ω ′ ) w i ′ ( ω )   d y ≤ 1 (18) which proves the c laim. When h = 1 , a gain, there is a meas urable subs et of H p for which w j ( ω ) 6 = 0 , h ence, | v ( ω , ω ′ ) | ≤ Z H p k Y j =1    C  − j2 π β 1 /d w j ( ω )     d y < 1 i.e., the strict inequa lity holds. In Figure 1, we show the empirical e igen v alue distribution of the matrix T d for β = 0 . 55 , d = 1 , 2 , 3 , and ˜ x un iformly d istrib uted in [0 , 1] . The emp irical distribution is c ompared to the Mar ˇ cenko-Pastur distrib ution (solid line). W e obse rve that as, d increas es, the Mar ˇ cenko-Pastur distributi on law bec omes a good a pproximation of f λ ( d, β , z ) . In p articular , the two cu rves are relatively close for sma ll z , a lready for d = 3 . October 25, 2021 DRAFT 12 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 f λ (d, β ,z) z d=1 M=52 d=2 M=11 d=3 M=4 Marcenko-Pastur Fig. 1. Comparison between the Mar ˇ cenko -Pastur distribution and the empirical distribu tion obtained for β = 0 . 55 and d = 1 , 2 , 3 in t he quasi equally space case, and uniform f ˜ x ( z ) V . A P P L I C A T I O N S Here we prese nt some applications whe re the results d eri ved in this work c an be used . The clos ed form expression of the mome nts of f λ ( d, β , z ) , g i ven by (34), can be a use ful bas is for pe rforming d econ voluti on ope rations, a s propos ed in [8]. As for the asymptotic ap proximation, we show below how to exp loit ou r results for the e stimation of the MSE provided by linear reconstruction techniques o f irregularly sa mpled signals. Let us as sume a g eneral linear sys tem model affected by a dditi ve noise. For simplicity , cons ider a one-dimension al sign al, s ( x ) . When observed over a finite interv al, it admits a n infinite Fourier series expansion [1], [2]. W e can think of the lar gest index M of the non-negligible Fourier coefficients of the expansion as the app roximate one-side d b andwidth of the signal. W e therefore rep resent s ( x ) by using 2 M + 1 complex harmonics as s ( x ) = 1 √ 2 M + 1 M X k = − M a ℓ e j2 π ℓx (19) October 25, 2021 DRAFT 13 Now , consider that the signal is observed within o ne period interval [0 , 1) and sa mpled in r points placed at pos itions x = [ x 0 , . . . , x r − 1 ] T , x q ∈ [0 , 1) , q = 0 , . . . , r − 1 . The complex nu mbers a ℓ represent amplitudes and p hases of the harmonics in s ( x ) . The signal samp les s = [ s ( x 0 ) , . . . , s ( x r − 1 )] T can be written as s = G † a , whe re the matrix G is gi ven in (1). The signal discrete sp ectrum is gi ven by the 2 M + 1 complex vector a = [ a − M , . . . , a 0 , . . . , a M ] T . W e can now write the linea r mod el for a measureme nt sample vector p = [ p ( x 0 ) , . . . , p ( x r − 1 )] T taken at the sa mpling points x q p = s + n = G † a + n (20) where n is a random vector rep resenting measuremen t noise . Th e g eneral problem is to reconstruct s or a given the n oisy meas urements p [4], [5]. A commo nly used parameter to meas ure the quality of the estimate of the reconstructed signal is the me an square error (MSE). In [1]–[3] it ha s been shown that, when linear reconstruction techn iques are use d a nd the sa mple c oordinates are known, the asymp totic MSE (i.e., a s the number of h armonics and the number of samples tend to infin ity while their ratio is kept constan t) is a function of the asy mptotic eigen value distribution of the matrix T = β GG † , i.e. , MSE = E λ  β λ S NR m + β  (21) where the random variable λ has distribution f λ ( d, β , z ) and SNR m is the signal-to-noise ratio on the measure. W e therefore exploit our asymp totic approximation to f λ ( d, β , z ) to compute (21). Figure 2 shows the MSE obtained as a function of the signal-to-noise ratio SNR m . The curves with markers labeled b y “ d = 1 , 2 , 3 ” refer to the ca ses where the s ignal ha s dimens ion d a nd the sampling points are quasi-equally spac ed with jitter ˜ x , un iformly distributed over [0 , 1) , and β = 0 . 729 . The cu rve labeled by “MP” (thick line) repo rts the results deri ved through o ur asymptotic ( d → ∞ ) ap proximation to the eigenv alue distribution, w hile the curve labeled by “Equally spa ced” (dashed line) repres ents the MSE achieved under a perfec t e qually spa ced s ample placement, i.e., wh en the eigen value distributi on is giv en b y f λ ( d, β , z ) = δ D ( z − 1) . Notice tha t the MSE grows as d increases a nd tends to the MSE obtained b y a Ma r ˇ cenko-P astur eigenv alue distrib ution. Instead, a s expected, the “Equa lly spa ced” curve represents a lower bound to the system performance. Figure 3 p resents similar results but obtained for d = 2 a nd different values of β . W e obs erve that the MSE obtained through our asymp totic approximation (the curve labe led by “MP”) giv es excellent results for values of β as small as 0. 2, even whe n co mpared against the n umerical res ults derived by fixing d = 2 . For β = 0 . 6 (i.e., when the ratio of the number of sign al ha rmonics to the nu mber o f sample s increases ), the approximation b ecomes slightly loos er , an d the MSE compu ted by using the Mar ˇ cenko- Pastur distrib ution giv es an up per limit to the qua lity of the recons tructed signal. Note that the s maller October 25, 2021 DRAFT 14 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 -10 0 10 20 30 40 50 MSE SNR m [dB] d=1 (M=52, r=144, β =0.729) d=2 (M=11, r=729, β =0.726) d=3 (M=4, r=1000, β =0.729) MP, β =0.729 Equally spaced, β =0.729 Fig. 2. MSE as a function of the signal-to-noise ratio for d = 1 , 2 , 3 . The curves are compared with the results obtained through our asymptotic analysis (MP) and with the equally spaced case the β , the higher the oversampling rate relative to the equally s paced minimal s ampling rate β = 1 . W e thus obs erve how o ur bound beco mes tighter a s the oversampling rate increases . T o c onclude, we de scribe s ome area s in sign al p rocessing whe re the a bove s ystem mod el a nd results find ap plication. i) Spectral e stimation with nois e. Spec tral estimation from high p recision sa mpling an d q uantization of ba ndlimited s ignals uses me asurement sys tems which a re us ually affected by jitter [20]. In s uch applications the quantization noise c orresponds to the measureme nt noise and the jitter is caused by the limit ed acc uracy o f the timing circuits. In this ca se the sampling points a re mismatched with respect to the nominal values, thus for d = 1 we have: x q = q r + ˜ x q r with some sampling rate 1 /r . Note that the exact positions o f the samp les are not k nown an d the cas e studied in this pa per (i.e., MSE with exact p ositions) gives a lower bound to the recons truction error . ii) Signal reconstruction in sen sor network s. Sensor ne tworks, whose no des sa mple a phys ical field, like air temperature, light intensity , po llution le vels or rain fall s, typically represent an example of quas i- October 25, 2021 DRAFT 15 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 -10 0 10 20 30 40 50 MSE SNR m [dB] β =0.2, d=2 β =0.2, MP β =0.2, Equally spaced β =0.6, d=2 β =0.6, MP β =0.6, Equally spaced Fig. 3. MS E as a function of the si gnal-to-noise ratio for β = 0 . 2 , 0 . 6 . The curves are obtained for d = 2 and compared against both the equally spaced case and the results deri ved through our asymptotic analysis ( MP) equally spac ed sa mpling [3], [9], [21], [22]. Indeed, often sensors are not regularly deployed in the area of interest due to terrain co nditions and d eployment p racticality and, thus, the phy sical field is not regularly sampled in the s pace domain. Sens ors report the data to a common proces sing un it (or sink nod e), which is in charge of recons tructing the sens ed field, ba sed on the receiv ed samples an d on the knowledge of their coordinates. If the field ca n b e approximated as bandlimited in the spa ce domain, then an estimate of the discrete sp ectrum c an b e o btained by using linear reconstruction techniques [3], [23], ev en in p resence of additive noise. In this c ase, our ap proximation a llo ws to compute the MSE on the rec onstructed field. iii) S tochastic sa mpling in co mputer graphics and imag e pr ocessing. J ittered sampling was first examined by Balak rishnan in [24], who ana lyzed it a s an und esirable ef fect in sampling continuo us time functions. More than twenty years later , Cook [25] realized that the effect of stochastic sa mpling can be advantageo us in compu ter graphics to redu ce aliasing artifacts, and con sidered jittering a regular grid as an effecti ve sampling technique. Anothe r example of s ampling with jitter was rec ently October 25, 2021 DRAFT 16 proposed in [26], for robust authentication of images. V I . C O N C L U S I O N S W e studied the behavior of the eigen v alue distribution of a c lass of rando m matrices, which find large application in signa l a nd imag e process ing. In particular , by u sing asy mptotic ana lysis, we deriv ed a closed-form expression for the moments of the eigen v alue distribution. Using these momen ts, we showed that, as the signal dimension goes to infinity , the as ymptotic eigen value d istrib ution tends to the Mar ˇ cenko- Pastur law . T his result a llo wed us to ob tain a s imple and ac curate bound to the signa l reconstruction error , which can fin d a pplication in several fields, such as jittered sampling, sens or networks, computer graphics and image processing . A P P E N D I X A P R O O F O F T H E O R E M 3 . 1 Using (7 ), the term T r E X  T p d  in (8) ca n be written as : T r E X  T p d  = E X   X ℓ 1 ( T p d ) ν ( ℓ 1 ) ,ν ( ℓ 1 )   = E X   X ℓ 1 · · · X ℓ p ( T d ) ν ( ℓ 1 ) ,ν ( ℓ 2 ) · · · ( T d ) ν ( ℓ p ) ,ν ( ℓ 1 )   = 1 r p X ℓ 1 · · · X ℓ p X q 1 · · · X q p E X " exp − j2 π p X i =1 x T µ ( q i ) ( ℓ i − ℓ [ i +1] ) !# = 1 r p X L ∈L d X Q ∈Q d E X " exp − j2 π p X i =1 x T µ ( q i ) ( ℓ i − ℓ [ i +1] ) !# (22) where Q d and L d are sets of integer matrices such that Q d =  Q | Q = [ q 1 , . . . , q p ] , q i = [ q i, 1 , . . . , q i,d ] T , q i,m = 0 , . . . , ρ − 1  L d =  L | L = [ ℓ 1 , . . . , ℓ p ] , ℓ i = [ ℓ i, 1 , . . . , ℓ i,d ] T , ℓ i,m = − M , . . . M  and [ i + 1] =    i + 1 1 ≤ i < p 1 i = p October 25, 2021 DRAFT 17 A. Set partitioning W e now apply the definitions in Se ction III-A in order to re write (22) u sing set partitioning. In particular by considering the vector µ = µ ( Q ) ∆ = [ µ 1 , . . . , µ p ] T where µ i = µ ( q i ) and q i is the i -th co lumn of Q , we obs erve that: • the vector µ is uniquely de fined by Q , and a gi ven µ unique ly define s a matrix Q ∈ Q d since µ ( · ) is a n in verti ble function; • a given µ induce s a partition ω ( µ ) ; • since r is the numb er of values tha t the entries µ i can take, there exist r ! / ( r − k ( µ ))! matrices Q ∈ Q d generating a gi ven p artition of P ma de o f k ( µ ) subs ets. In other words r ! / ( r − k ( µ ))! distinct µ ’ s yield the sa me partition ω ( µ ) . Since the random vec tors x µ ( q ′ ) and x µ ( q ′′ ) are indepe ndent for q ′ 6 = q ′′ , for any giv en Q the average operator in (22) factorizes into k ( µ ) terms, i.e. , E X " exp − j2 π p X i =1 x T µ ( q i ) ( ℓ i − ℓ [ i +1] ) !# = E X " exp − j2 π p X i =1 x T µ i ( ℓ i − ℓ [ i +1] ) !# = k ( µ ) Y j =1 E x γ j   exp   − j2 π x T γ j X i ∈P j ( µ ) ℓ i − ℓ [ i +1]     = k ( µ ) Y j =1 E x γ j h ζ ρ x T γ j ˆ w j ( µ ) i (23) indeed, for every i ∈ P j ( µ ) , we have µ i = γ j . In the last line of (23), we exploited the following two definitions ζ = exp ( − j2 π /ρ ) and ˆ w j ( µ ) = X i ∈P j ( µ ) ℓ i − ℓ [ i +1] (24) Also, note tha t, in the product in (23), ea ch factor depe nds on a single random vector , x γ j . Since x µ ( q ) = q /ρ + ˜ x µ ( q ) /ρ and µ ( · ) is in vertible then, b y defin ing ¯ x γ j = µ − 1 ( γ j ) we have x γ j = ¯ x γ j /ρ + ˜ x γ j /ρ and E x γ j h ζ ρ x T γ j ˆ w j ( µ ) i = ζ ¯ x T γ j ˆ w j ( µ ) E ˜ x γ j h ζ ˜ x T γ j ˆ w j ( µ ) i = ζ ¯ x T γ j ˆ w j ( µ ) E ˜ x h ζ ˜ x T ˆ w j ( µ ) i (25) October 25, 2021 DRAFT 18 In the last term of (25) we removed the su bscript γ j from the argument o f the average opera tor , since the d istrib ution of ˜ x γ j does n ot de pend o n γ j . Summarizing, the term T r E X  T p d  in (8) ca n b e written as T r E X  T p d  = 1 r p X Q ∈Q d X L ∈L d k ( µ ) Y j =1 ζ ¯ x T γ j ˆ w j ( µ ) E ˜ x h ζ ˜ x T ˆ w j ( µ ) i (26) Since e ach Q is uniquely identified by a vector µ , we can observe that X Q ∈Q d f ( µ ) = X ω ∈ Ω p X µ ∈M ( ω ) f ( µ ) = p X k =1 X ω ∈ Ω p,k X µ ∈M ( ω ) f ( µ ) (27) for every function f ( µ ) . Rec all that, in (27), M ( ω ) represents the se t of µ inducing a gi ven pa rtition ω . From the defin itions in Section III-A, it follo ws tha t, if µ induces ω , then k ( µ ) = k ( ω ) , P j ( µ ) = P j ( ω ) , and ˆ w j ( µ ) = ˆ w j ( ω ) , j = 1 , . . . , k ( ω ) . T herefore, T r E X  T p d  = 1 r p p X k =1 X ω ∈ Ω p,k X µ ∈M ( ω ) X L ∈L d k Y j =1 ζ ¯ x T γ j ˆ w j ( µ ) E ˜ x h ζ ˜ x T ˆ w j ( µ ) i = 1 r p p X k =1 X ω ∈ Ω p,k X µ ∈M ( ω ) X L ∈L d k Y j =1 ζ ¯ x T γ j ˆ w j ( ω ) E ˜ x h ζ ˜ x T ˆ w j ( ω ) i = 1 r p p X k =1 X ω ∈ Ω p,k X L ∈L d X µ ∈M ( ω )   k Y j =1 ζ ¯ x T γ j ˆ w j ( ω )     k Y j =1 E ˜ x h ζ ˜ x T ˆ w j ( ω ) i   ( a ) = 1 r p p X k =1 X ω ∈ Ω p,k X L ∈L d η ( ω , L ) X µ ∈M ( ω ) k Y j =1 ζ ¯ x T γ j ˆ w j ( ω ) (28) In (28) we define d η ( ω , L ) = k Y j =1 E ˜ x h ζ ˜ x T ˆ w j ( ω ) i = k Y j =1 d Y m =1 E ˜ x m h ζ ˜ x m ˆ w jm ( ω ) i (29) where ˜ x m and ˆ w j m are the m -th entries of ˜ x and ˆ w j , respec ti vely . In the equality “(a)” we exploited the fact that the term ζ ˜ x T ˆ w j ( ω ) does not depend on µ and c an be factored from the su m over µ . As for the term P µ ∈M ( ω ) Q k j =1 ζ ¯ x T γ j ˆ w j , we have the following lemma. Lemma A. 1: Let ω ∈ Ω p,k , let ˆ w 1 , . . . , ˆ w k be vectors of size d with integer entries, de fined a s in (24). Let M ( ω ) be the se t of vec tors µ ind ucing ω . T hen X µ ∈M ( ω ) k Y j =1 ζ ¯ x T γ j ˆ w j = k X h =1 r h X ω ′ ∈ Ω k,h u ( ω ′ ) h Y j ′ =1 δ   X i ′ ∈P j ′ ( ω ′ ) ˆ w i ′ ( ω )   (30) where u ( ω ′ ) = ( − 1) k − h Q h j ′ =1 ( |P j ′ ( ω ′ ) | − 1)! , γ j = γ j ( µ ) , and whe re Ω k ,h is the set of vectors ω ′ of size k , rep resenting the partitions of the set P ′ = { 1 , . . . , k } in h subse ts, namely , P ′ 1 ( ω ′ ) , . . . , P ′ h ( ω ′ ) . October 25, 2021 DRAFT 19 Pr oof: The proof can be foun d in Append ix B. By a pplying the result of Le mma A.1 to (28), we get T r E X  T p d  = p X k =1 k X h =1 X ω ∈ Ω p,k X ω ′ ∈ Ω k,h r h u ( ω ′ ) r p X L ∈L d η ( ω , L ) h Y j ′ =1 δ   X i ′ ∈P j ′ ( ω ′ ) ˆ w i ′ ( ω )   (31) Considering that h Y j ′ =1 δ   X i ′ ∈P j ′ ( ω ′ ) ˆ w i ′ ( ω )   = h Y j ′ =1 d Y m =1 δ   X i ′ ∈P j ′ ( ω ′ ) ˆ w i ′ m ( ω )   and b y using (29) a nd (31), we h av e X L ∈L d η ( ω , L ) h Y j ′ =1 δ   X i ′ ∈P j ′ ( ω ′ ) ˆ w i ′ ( ω )   = X ℓ 1 ∈L 1 · · · X ℓ d ∈L 1 k Y j =1 d Y m =1 E ˜ x m h ζ ˜ x m ˆ w jm ( ω ) i h Y j ′ =1 d Y m =1 δ   X i ′ ∈P j ′ ( ω ′ ) ˆ w i ′ m ( ω )   =   X ℓ ∈L 1 k Y j =1 E ˜ x h ζ ˜ x ˆ w j ( ω ) i h Y j ′ =1 δ   X i ′ ∈P j ′ ( ω ′ ) ˆ w i ′ ( ω )     d = ψ M ( ω , ω ′ ) d (32) where the su bscript M highlights the dep endency of ℓ on M . In co nclusion, T r E X  T p d  = p X k =1 k X h =1 X ω ∈ Ω p,k X ω ′ ∈ Ω k,h r h u ( ω ′ ) r p ψ M ( ω , ω ′ ) d (33) T o compu te E [ λ p d,β ] , we cons ider the limit in (8). B y using the definition (6), we first notice that r h r p (2 M + 1) d = β p − h (2 M + 1) d ( p − h +1) Then, by using (33) in (8), we ob tain E [ λ p d,β ] = lim M ,r → + ∞ β p X k =1 k X h =1 β p − h (2 M + 1) d ( p − h +1) X ω ∈ Ω p,k X ω ′ ∈ Ω k,h u ( ω ′ ) ψ M ( ω , ω ′ ) d = p X k =1 k X h =1 β p − h X ω ∈ Ω p,k X ω ′ ∈ Ω k,h u ( ω ′ )  lim M →∞ ψ M ( ω , ω ′ ) (2 M + 1) p − h +1  d = p X k =1 k X h =1 β p − h X ω ∈ Ω p,k X ω ′ ∈ Ω k,h u ( ω ′ ) v ( ω , ω ′ ) d (34) The secon d equa lity in (34) holds sinc e, for a ny gi ven p , the s ums P ω ∈ Ω p,k and P ω ′ ∈ Ω k,h are over a finite number of terms, and the coefficients u ( ω ′ ) are finite and do not depend on M . Th erefore, the October 25, 2021 DRAFT 20 limit o perator can be swapped with the summa tions. The c oefficient v ( ω , ω ′ ) is defin ed as v ( ω , ω ′ ) = lim M →∞ ψ M ( ω , ω ′ ) (2 M + 1) p − h +1 = lim M →∞ 1 (2 M + 1) p − h +1 X ℓ ∈L 1 k Y j =1 E ˜ x h ζ ˜ x ˆ w j ( ω ) i h Y j ′ =1 δ   X i ′ ∈P j ′ ( ω ′ ) ˆ w i ′ ( ω )   ( a ) = lim M →∞ 1 (2 M + 1) p − h +1 X ℓ ∈L 1 k Y j =1 C ( − j2 π ˆ w j ( ω ) /ρ ) h Y j ′ =1 δ   X i ′ ∈P j ′ ( ω ′ ) ˆ w i ′ ( ω )   (35) where, in the equality ( a ) , we introduced the charac teristic fun ction of ˜ x , de fined as C ( s ) = E ˜ x [e sz ] . W e now consider three possible c ases: • if h = 1 , then Ω k , 1 = { [1 , . . . , 1 | {z } k ] } , thus we only conside r ω ′ = [1 , . . . , 1 | {z } k ] . Then, P 1 ( ω ′ ) = { 1 , . . . , k } and X i ′ ∈P 1 ( ω ′ ) ˆ w i ′ ( ω ) = X i ′ ∈{ 1 ,...,k } ˆ w i ′ ( ω ) = k X i ′ =1 ˆ w i ′ ( ω ) = k X i ′ =1 X i ∈P i ′ ( ω ) ℓ i − ℓ [ i +1] = p X i =1 ℓ i − ℓ [ i +1] = 0 (36) and b y c onsequ ence δ  P i ′ ∈P j ′ ( ω ′ ) ˆ w i ′ ( ω )  = 1 . Hence, v ( ω , ω ′ ) = Z H p k Y j =1 C  − j2 π β 1 /d w j ( ω )  d y (37) where, in ana logy with (24), we defin ed w j = X i ∈P j ( ω ) y i − y [ i +1] y i ∈ R , i = 1 , . . . , p . W e d enote by y the vector y = [ y 1 , . . . , y p ] T ; • if 1 < h < k , the a r gument of the δ ( · ) function in (35) is always a function o f the indice s ℓ i . Thus Z H p k Y j =1 C  − j2 π β 1 /d w j ( ω )  h Y j ′ =1 δ D   X i ′ ∈P j ′ ( ω ′ ) w i ′ ( ω )   d y where δ D ( · ) d enotes the Dirac’ s delta; October 25, 2021 DRAFT 21 • if h = k , the cardinality of Ω k ,h = Ω k ,k is S ( k , k ) = 1 and Ω k ,k = { [1 , . . . , k ] } . Thus, we only consider ω ′ = [1 , . . . , k ] . It follo ws that: v ( ω , ω ′ ) = Z H p k Y j =1 C  − j2 π β 1 /d w j ( ω )  k Y j ′ =1 δ D   X i ′ ∈P j ′ ([1 ,...,k ] ) w i ′ ( ω )   d y = Z H p k Y j =1 C  − j2 π β 1 /d w j ( ω )  δ D   X i ′ ∈P j ([1 ,...,k ]) w i ′ ( ω )   d y (38) Since P j ([1 , . . . , k ]) = { j } an d C (0) = 1 , we have v ( ω , [1 , . . . , k ]) = Z H p k Y j =1 C  − j2 π β 1 /d w j ( ω )  δ D ( w j ( ω )) d y = Z H p k Y j =1 C (0) δ D ( w j ( ω )) d y = Z H p k Y j =1 δ D ( w j ( ω )) d y (39) As a las t remark, if k = 1 , we have h = 1 and Ω p,k = Ω p, 1 = { [1 , . . . , 1 | {z } p ] } . Then w j ( ω ) = P p i =1 w i = 0 . Using (37), we obtain v ( ω , ω ′ ) = Z H p k Y j =1 C  − j2 π β 1 /d w j ( ω )  d y = Z H p k Y j =1 C (0) d y = 1 A P P E N D I X B P R O O F O F L E M M A A . 1 Recall that M ( ω ) deno tes the set of vectors µ = [ µ 1 , . . . , µ p ] induc ing the same partit ion ω . As define d in Section III-A, if ω ∈ Ω p,k , then eac h µ ∈ M ( ω ) con tains k distinct values, namely , γ = [ γ 1 , . . . , γ k ] where 0 ≤ γ j < r , j = 1 , . . . , k a nd γ j 6 = γ j ′ for each j, j ′ = 1 , . . . , k a nd j 6 = j ′ . Therefore, from (A.1) we can write X µ ∈M ( ω ) k Y j =1 ζ ¯ x T γ j ˆ w j = X γ 1 ,...,γ k 6 = k Y j =1 ζ ¯ x T γ j ˆ w j where the symbol P γ 1 ,...,γ k 6 = indicates a s um over the v ariables γ 1 , . . . , γ k with the constraint that γ j 6 = γ j ′ for every j, j ′ = 1 , . . . , k and j 6 = j ′ . Notice that the values γ j ( j = 1 , . . . , k ) are the sc alar counterparts of the integer v ectors v 1 , . . . , v k , v j = [ v j 1 , . . . , v j d ] T , 0 ≤ v j m < ρ , m = 1 , . . . , d , through the inv ertible function µ ( · ) , i.e., γ j = µ ( v j ) , j = 1 , . . . , k . Henc e, by de finition of ¯ x , we have ¯ x γ j = ¯ x µ ( v j ) = v j and October 25, 2021 DRAFT 22 in co nclusion X µ ∈M ( ω ) k Y j =1 ζ ¯ x T γ j ˆ w j = X v 1 ,..., v k 6 = k Y j =1 ζ v T j ˆ w j = X v 1 ,..., v k 6 = ζ v T 1 ˆ w 1 + ··· + v T k ˆ w k (40) W e now c ompute the last term of (40) b y summing over on e vari able at a time. W e first no tice that, for every set v 1 , . . . , v n of distinct vectors X v 6 = v 1 ,..., v n ζ v T ˆ w =    r − n ˆ w = 0 − P n j =1 ζ v T j ˆ w ˆ w 6 = 0 In pa rticular when w 6 = 0 , P v ζ v T ˆ w = 0 . Let us arbitrarily choos e the variable v k . If by hypo thesis w k 6 = 0 , then by summing (40) over v k we get X v 1 ,..., v k 6 = ζ v T 1 ˆ w 1 + ··· + v T k ˆ w k = − k − 1 X j =1 X v 1 ,..., v k − 1 6 = ζ v T 1 ˆ w 1 + ··· + v T k − 1 ˆ w k − 1 ζ v T j ˆ w k (41) W e c ompute separately each of the k − 1 contributions in (41). In particular , the ge neric j ′ -th term ( j = j ′ ) is given by − X v 1 ,..., v k − 1 6 = ζ v T 1 ˆ w 1 + ··· + v T k − 1 ˆ w k − 1 ζ v T j ′ ˆ w k = − X v 1 ,..., v k − 1 6 = ζ v T 1 ˆ w 1 + ··· + v T j ′ ( ˆ w j ′ + ˆ w k )+ v T k − 1 ˆ w k − 1 W e now procee d b y summing over the variable v j ′ . If by hypothes is ˆ w j ′ + ˆ w k 6 = 0 , this su mmation produces k − 2 terms. Ag ain, we con sider each term s eparately . This procedure repeats un til a su bset S of { 1 , . . . , k } is found, such that s = P i ∈S ˆ w i = 0 . In this case, the contributi on of the n -th sum is gi ven by r − ( k − n ) whe re n = |S | is the cardinality of S . Overall, a fter n sums the total contribution is ( − 1) n − 1 ( n − 1)!( r − ( k − n )) X v j ,j ∈{ 1 ,...,k }−S 6 = Y j ∈{ 1 ,...,k }−S ζ v T j ˆ w j The factor ( n − 1)! accoun ts for the n umber o f pe rmutations of the e lements in S , once the first eleme nt is fixed (remember that we arbitrarily chos e the fi rst v ariable o f the summa tion). The factor ( − 1) n − 1 takes into acco unt that we su mmed n − 1 times with the co ndition ˆ w 6 = 0 , which implies n − 1 sign chan ges. Eventually , the term P v j ,j ∈{ 1 ,...,k }−S 6 = Q j ∈{ 1 ,...,k }−S ζ v T j ˆ w j is s imilar to the last term in (40) where only k − n variables v a re in v olved. This proce dure repeats until we sum over all variables v . T his is equiv alent to che ck if for all pos sible partitions of { 1 , . . . , k } in h su bsets P 1 , . . . , P h , h = 1 , . . . , k the condition s 1 = s 2 = · · · = s h = 0 October 25, 2021 DRAFT 23 holds, with s j = P i ∈P j ˆ w i , n j = |P j | , and P j n j = k . In this c ase, the con trib ution is gi ven by h Y j =1 ( − 1) n j − 1 ( n j − 1)! p r ( n 1 , . . . , n h ) and it is 0 otherwise. Here p r ( n 1 , . . . , n h ) = ( r − ( k − n 1 ))( r − ( k − n 1 − n 2)) · · · ( r − ( k − n 1 − n 2 − · · · − n h − 1 )) . In co nclusion, we can write X v 1 ,..., v k 6 = ζ v T 1 ˆ w 1 + ··· + v T k ˆ w k = k X h =1 X ω ′ ∈ Ω k,h u ( ω ′ ) p r ( ω ′ ) h Y j ′ =1 δ   X i ′ ∈P j ′ ( ω ′ ) ˆ w i ′ ( ω )   where u ( ω ′ ) = ( − 1) k − h Q h j ′ =1 ( |P j ′ ( ω ′ ) | − 1)! and p r ( ω ′ ) is a polynomial in r of degree h . For lar ge r , p r ( ω ′ ) ≃ r h , thu s proving the lemma. R E F E R E N C E S [1] A. Nordio, C.-F . Chiasserini, and E. V iterbo “Quality of field reconstruction in sensor networks , ” IEEE INFOC OM Mini- Symposium , Anchorage, AK, May 2007. [2] A. Nordio, C. -F . Chiasserini, and E. 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