Resampling and requantization of band-limited Gaussian stochastic signals with flat power spectrum

Resampli n g and requan tizat ion of band-li mited Gaussian sto c hastic signals with flat p ow er sp ect rum Marco Lanucara a , Riccardo Borg hi b a Eur op e an Sp ac e Op er ation Centr e, Eur op e an Sp ac e A genc y, D-64293 Darmstadt, Germany b Dip artimento di Elettr onic a Applic ata, Universit ` a de gli Studi “R oma T r e”, I-0014 6 R ome, Italy Abstract A theore tica l a nalysis, aimed at characterizing the de g radation induced by the resampling a nd requantization pro cess es applied to ba nd-limited Gauss ian sig- nals with flat power spectrum, av ailable through their digitized samples, is pre- sented. The analysis provides an efficient algorithm for computing the complete joint biv ariate discrete probability distribution ass o ciated to the true quantized version o f the Gaussian signal a nd to the quantit y e s timated a fter re sampling and requan tiza tion of the input digitize d sequence. The use of F ourier transform techn iques allows deriving approximate analytical expre ssions for the quantities of interest, as well a s implementing their efficien t co mputation. Numerical ex- per iments are found to be in goo d ag reement with the theoretical results, a nd confirm the v alidity of the whole appro ach. Key wor ds: quantization, interpolatio n, statistical s ig nal pro ce s sing 1. INTR ODUCTION Mo dern signal pro cessing co nsists of algorithms a pplied to sequences of n um- ber s, obtained by analogue to digital (A/D) con version of analo gue signals. The A/D conv ers ion implies sampling in time do main a nd amplitude quantization, the seco nd step being mandatory due to the finite length of the regis ters used for storing the samples amplitude in the pro c e s sing machine. If the effect of quan- tization is disr egarded, the exact recons truction of the analo gue signa l fro m its samples is guara nt eed by the sampling theorem, under the a s sumption that the signal itself is band-limited. Conv ers ely , when quantization is applied, the exact reconstructio n of the signal from the quantized sa mples is no long er po ssible. An imp ortant signal pro cessing task is the rate co nversion a pplied to a se- quence of nu mbers r e pr esenting a digitized s ignal. This tas k consis ts in obtaining samples of a sig nal taken at a certain r ate, say 1 /T 2 , based o n the samples of ∗ Corresp onding author Email addr esses: marco.lan ucara@es a.int (Marco Lanucara), borghi@uni roma3.it (Riccardo Borghi) Pr eprint submitte d to Elsevi e r Novemb er 3, 2018 the same signal av aila ble at a differ e n t rate, say 1 /T 1 . This problem was exten- sively studied in the past years, for b oth ra tio nal and irrational v alues of the ratio T 2 /T 1 , assumed to b e either lar ger (interpo lation problem) or smaller (dec- imation problem) than unit y [1, 2 , 3 , 4]. The above cited pap ers derive p ow erful techn iques ensuring that the rate conversion is p erfor med without deg radation in all treated cases, under the ass umption that the s ignals a re no t quantized. In a no n ideal condition, the input sequenc e av ailable at rate 1 /T 1 and the output sequence obtained at ra te 1 /T 2 as result of the ra te conv ersio n pro- cess are b oth quantized, in gene r al (but no t necessar ily) accor ding to the s a me quantization scheme. Requantization asso ciated to rate con version is applied in different contexts, like for example to signals received from r adio sources in many applications of r adio a stronomy [5, 6], or to co ded video data in image pro cessing [7]. In such cases it is o f int erest to establish theo retical b ounds for the degradation o ccurr ing due to the qua nt ization pro cess, a ffecting b o th the input a nd the output sequences of num b ers. The inclusion of quantization effects within the context of rate conv er sion was studied by the authors, in the sp ecific case of e x treme clipping, when o nly the sign of the analogue signal is r ecorded, i.e. when o nly one bit of information is asso ciated to the amplitude o f ea ch sample [8 ]. Under this hypothesis , and assuming that the input analogue sig na l is a realiza tion x ( t ) of a band-limited Gaussian pr o cess X with flat p ower spectr al density within the supp orting band- width, results in closed for m could b e obtained a b out the degradation effect, in that context identified with the probability of error betw een the quantized version of x ( t ) at any instant of time estimated through the av aila ble quantized samples, and the true quantized v alue of x ( t ) (the “ target”). The pr esent pap er is devoted to extending the r e s ults of Ref. [8] to the case of arbitrar y q uantization scheme, including multiple output levels, with the unique constraints of antisymmetry of the non-linear qua nt ization function. The ab ov e men tioned pro bability o f er ror, whic h was the metric used for q uantif ying the degradatio n effect in the binary cas e, is replaced by a complete biv aria te discr ete probability distribution o r , in the case o f larg e n um be r of outputs, by the cross- correla tion co efficient betw een the estimated quan tized v alue o f x ( t ) and the target. 2. PRELIMINARIES Let X be a s to chastic pro cess with real rea lizations x ( t ), which is s tationary and er go dic, with ze r o mean v alue and Gaussian statistics. The pro cess is suppo sed to be band limited (BL for shor t), with flat p ower sp ectrum within the supp orting bandwidth [ − W, W ]. On denoting b y σ the standard devia tion of the pr o cess, it is well known that[9] h x ( t 1 ) x ( t 2 ) i = σ 2 sinc  t 2 − t 1 T  , (1) where T = 1 / 2 W is the inv er se of the Nyquist frequency , the sinc function is defined by sinc( ξ ) = sin( π ξ ) / ( π ξ ), and the sym bo l h·i represents the exp ected 2 v alue of its ar gument. Samples o f the signal x ( t ) are taken at k nown instants k T , so tha t x k = x ( k T ) denotes the k th s ample. It is known that the signal x ( t ) ca n b e expanded (in a mean-squa re sense) as [9] x ( t ) = + ∞ X k = −∞ x k sinc  t − k T T  . (2) After the sampling, a quantization of the contin uous v alue is p erformed, via a nonlinear function f ( x ), so that the final output, say the sequenc e { u m } , can be expressed as follows: u m = f ( x m ) , m = 0 , ± 1 , ± 2 , . . . (3) The function f is assumed to b e an antisymmetric, piecewis e function with an even num b er of outputs, i.e., none of the output levels equals zero . F or 2 M output levels we deno te the (p o sitive) disco nt inuit y points b y 0 = a 1 < a 2 < . . . < a M < a M +1 = ∞ , a s shown in Fig . 1 where f ( x ) is plotted for x > 0 . Note that the output levels y 1 < y 2 < . . . < y M are also rep orted. Of course, the case of quantization functions with o dd num b er of levels can be treated as well, by using the same metho dolog y we a re going to pre s ent. W e study the degradation as s o ciated to the reconstruction o f the digitized version of the signa l x ( t ) at a time no t belo ng ing to the sampling grid k T , based on the knowledge of its dig itized s amples u m given in Eq. (3). In view of the stationarity of the pr o cess X , such a problem consis ts in finding an estima te, say ˜ u ( λ ), of the target v alue f [ x ( λ T )], where λ ∈ [0 , 1] is a dimensionless parameter. T o this aim, we first as sume that a sinc interpolatio n is used to estimate x ( λ T ), and then the qua n tization function f is applied. The v alidity of this approa ch was demo ns trated in Ref. [8], for any antisymmetric non linea r function f . F ollowing the notations and results g iven in Ref. [8], the estimation of the digitized s ample ˜ u ( λ ) can be written as ˜ u ( λ ) = f [ w ( λ )] , (4) where w ( λ ) = A f + ∞ X i = −∞ u i φ i , (5) with A f = h x f ( x ) i h f 2 ( x ) i , (6) and φ i = sinc ( λ − i ) . (7) In general, ˜ u ( λ ) differs, even in a mean-squa re sense, from the target v alue f [ x ( λT )]. The effect of the degradation can b e accounted for by determining the biv ariate discrete probability distr ibutio n p i,j , equal to the proba bility tha t ˜ u = y j when the targe t is equal to y i , i.e., p i,j = Pr { f [ x ( λT )] = y i and ˜ u ( λ ) = y j } , (8) 3 for i, j = ± 1 , ± 2 , . . . , ± M . The task of our a nalysis is therefore to ev aluate the discr ete probability distribution o f Eq. (8), as a function of λ , for the most g eneral ca se of M ≥ 1. The cross-c o rrelatio n c o efficient b etw een the estimated and the target v alues will also be ev aluated, from which the resampling and requantization-induced degradatio n could b e easily inferred. 3. THEORETICAL ANAL YSI S The aim of the present section is to show the theoretical bas is of our appr o ach for solving the pro blem stated in the previo us section. W e retrieve the biv aria te probability distribution in Eq. (8 ) by fir st ev aluating its mixed moments up to the order 2 M − 1, which is sufficient for the distr ibutio n to b e fully reconstructed. The subsequent step concerns the ev aluation of the mixed moments, whic h is achiev ed by employing a p ow e rful and efficie nt metho d, mak ing use of F o urier transform (FT for s hort) techniques. Consider the mixed moments, say µ n,m , defined as µ n,m = h f ( x ) n f ( w ) m i , (9) with n, m = 0 , . . . , 2 M − 1. Note that, due the antisymmetry of f , the mo ment s v anish whenever n a nd m ha ve different pa rity . The biv a riate proba bilit y dis- tribution p i,j can b e ar ranged as a 2 M × 2 M matrix, say P , which is defined as P =                   p − M , − M . . . p − M , − 1 p − M , 1 . . . p − M ,M . . . . . . . . . . . . . . . . . . p − 1 , − M . . . p − 1 , − 1 p − 1 , 1 . . . p − 1 ,M p 1 , − M . . . p 1 , − 1 p 1 , 1 . . . p 1 ,M . . . . . . . . . . . . . . . . . . p M , − M . . . p M , − 1 p M , 1 . . . p M ,M                   . (10) By definition, the mixed momen ts are rela ted to the biv a riate distribution ac- cording to the relation µ n,m = X i,j p i,j y n i y m j , (11) which ca n b e ca st in a matrix form µ = Y PY † , (12) 4 where the da gger denotes the trans po se and the 2 D matric e s µ and Y are defined by µ =       µ 0 , 0 . . . µ 0 , 2 M − 1 . . . . . . . . . µ 2 M − 1 , 0 . . . µ 2 M − 1 , 2 M − 1       , (13) and Y =                   1 . . . 1 1 . . . 1 y − M . . . y − 1 y 1 . . . y M y 2 − M . . . y 2 − 1 y 2 1 . . . y 2 M y 3 − M . . . y 3 − 1 y 3 1 . . . y 3 M . . . . . . . . . . . . . . . . . . y 2 M − 1 − M . . . y 2 M − 1 − 1 y 2 M − 1 1 . . . y 2 M − 1 M                   , (14) resp ectively . Since Y is a V andermo nde matr ix , and since all y i ’s a re different, its inv er se is alwa ys defined, so that the whole biv ariate proba bility distribution P is tr ivially given by P = Y − 1 µ ( Y † ) − 1 . (15) Concerning the cor relation co efficient, this can also b e derived from the k nowl- edge of the mixe d moments defined in Eq. (9) in the fo llowing way: ρ = µ 1 , 1 √ µ 0 , 2 µ 2 , 0 . (16) The ev aluation o f the mixed moments µ n,m per tinent to a t y pical 2 M -levels quantization function is not a trivial task. Similarly to the approach used b y Banta fo r ev aluating a uto correla tion functions of quantized sig nals [10], w e ma ke use of a FT technique. W e start from the FT of the function [ f ( x )] n , s ay F n ( p ), which is defined by [ f ( x )] n = Z + ∞ −∞ F n ( p ) exp(2 π i xp ) d p, (17) where F n ( p ) de notes the F ourier transform of [ f ( x )] n , i . e., F n ( p ) = F { [ f ( x )] n } = Z + ∞ −∞ [ f ( x )] n exp( − 2 π i xp ) d x, (18) with F {·} denoting the F ourier transform op erator. In App endi x A 5 it is sho wn that F n ( p ) =                y n m δ ( p ) + 1 π p M − 1 X j =1  y n j − y n j +1  sin(2 π a j +1 p ) , n even , − i y n 1 π p + i π p M − 1 X j =1  y n j − y n j +1  cos(2 π a j +1 p ) , n o dd , (19) where δ ( p ) denotes the Dira c distr ibutio n. As far as the momen t µ n,m is concerned, on susbstituting from Eq. (17) into Eq. (9) we ha v e µ n,m = h Z + ∞ −∞ Z + ∞ −∞ F n ( p ) F m ( p ′ ) exp[i2 π x ( λT ) p ] exp[i2 π w ( λ ) p ′ ] d p d p ′ i , (20) where the dep endence on λ has b een made expli cit. Finall y , on i n ter- c hanging the in teg rals with the av erages, w e obtain µ n,m = Z + ∞ −∞ Z + ∞ −∞ F n ( p ) F m ( p ′ ) h exp { 2 π i[ x ( λT ) p + w ( λ ) p ′ ] }i d p d p ′ . (21) The qua n tit y in the av era ge can b e written, star ting fro m Eq s. (2) and (5), as h exp { 2 π i[ x ( λT ) p + w ( λ ) p ′ ] }i = h exp ( 2 π i X k z k ) i , (22) where the zero- mea n, statistically indep endent , random v ariables z k = φ k [ x k p + A f f ( x k ) p ′ ] , (23) hav e b een defined. Moreov er, due to the above statis tica l indep endence, we hav e h exp(i2 π Z ) i = + ∞ Y k = −∞ h exp(i2 π z k ) i , (24) where Z = + ∞ X k = −∞ z k . (25) As we will see shortly , the average in the r.h.s. of Eq. (24) can be calculated, for any k , for the c o nsidered c la ss o f qua n tization functions. How ever, the pr es- ence of the infinite pro duct do e s not allow an exa ct closed form for the mixed moments to b e provided and makes their numerical estima tio n cumber some. In order to overcome such difficulties, we ar e going to implement suitable appr ox- imations which will simplify the deriv ation of the moment s. First of all, it should b e noted that the random v ariables z k are not no rmally distributed and, due to the prefactor φ k , are also not 6 iden tically distributed, b eing σ 2 z k → 0 for k → ±∞ . The ev aluation of the probabili t y di stribution function of infinite sums of not i den ti cally distributed indi p endent random v ariables repres en ts a task far from b eing trivi al, as witnesse d b y past and current lite ratures[ 1 1, 12, 1 3, 14, 15, 16, 17], and is b eyond the scop e of the prese n t work . In particular, it is clear that the us e of the central lim it theorem cannot b e in vok ed to find the probability densi t y function of Z . T o giv e a simple evidence of this, consider the case λ = 0 , for whic h φ k = δ 0 ,k , and th us Z = z 0 that, as said ab ov e, is not normally dis tributed, unless p ′ = 0 . In the general case λ 6 = 0 , w e use the approac h outl ined in Ref. [14], so that the random v ariable Z is first written as the s um of t w o, statistically indep endent, random v ariables, sa y Z C and Z I , defined as Z C = X k ∈N z k , (26) and Z I = X k / ∈N z k , ( 27) resp ectively , with N b eing a suitable finite set of N consecutive in- dices, N = { i 1 , i 2 , . . . , i N } . In particular, w e c ho ose N ( h ) =    {− h + 1 , − h + 2 , . . . , h − 1 , h } , h > 0 ∅ , h = 0 , (28) with N = 2 h b eing the num b er of consecutiv e i ndices form ing the set. 1 In the ab o ve decomp o sition, the v ariable Z C retains the co efficients φ k for which k is close to the interv al of interest o f λ , while the v ariable Z I corresp onds to the tails of the sum in Eq. (25), with co efficients φ k suc h that | φ k ( λ ) | is slowly v arying wi th res p ect to k , b eing | φ k ( λ ) | ≤ λ | λ − k | , (29) whic h, for large v alues of | k | , go es li ke 1 / | k | . F oll o w i ng Ref. [14], we assume that Z I retains appro xim ately a normal dis tribution, so that h exp(i2 π Z I ) i ≃ exp( − 2 π 2 σ 2 Z I ) , (30) where the v ariance σ 2 Z I = h Z 2 I i is σ 2 Z I = p 2 P I + Q 2 I ( p ′ 2 + 2 p p ′ ) , (31) 1 The explicit dependence of the set N on the v ariable h will not b e shown in the subsequen t formulas. 7 with P I = X k / ∈N φ 2 k h x 2 k i = σ 2 1 − X k ∈N φ 2 k ! , (32) Q 2 I = A 2 f X k / ∈N φ 2 k h f 2 ( x k ) i = A 2 f h f 2 i 1 − X k ∈N φ 2 k ! , (33) where the fact that X k φ 2 k = 1 has b een used. In Sec . 5, in order to minimize the computational effort, we hav e used the ab ov e decomp osition with the minimum po ssible v alue o f h , i.e., h = 1 a nd N = { 0 , 1 } . As w e will s ee, such a c hoice provides quite sa tisfactory results. As far as the Z C is c o ncerned, this v ariable is now defined as a sum o f a finite n umber o f terms, which makes not pr ohibitive the co mputation o f the quantit y h ex p(i2 π Z C ) i , as required b y Eq. (35). In particular, in App e ndix B it is shown that h exp(i2 π Z C ) i = exp − 2 π 2 σ 2 p 2 X k ∈N φ 2 k ! × Y k ∈N X j X s X q 1 2 ( − 1) q exp(i2 π s A f y j φ k p ′ ) × e r f  a j + q − i 2 π s φ k σ 2 p σ √ 2  , (34) where er f ( · ) denotes the error function [18], j = 1 , ..., M , s ∈ {− 1 , +1 } , and q ∈ { 0 , 1 } . h exp(2 π i Z ) i = h exp(2 π i Z C ) ih exp(2 π i Z I ) i , (35) where Accordingly , on reca lling Eq. (30), throug h Eqs. (31)-(33) we even tually found h exp(i2 π Z ) i ≈ exp  − 2 π 2 ( σ 2 p 2 + Q 2 I p ′ 2 + 2 Q 2 I p p ′ )  × Y k ∈N X j X s X q 1 2 ( − 1) q exp(i2 π s A f y j φ k p ′ ) × e r f  a j + q − i 2 π s φ k σ 2 p σ √ 2  . (36) It is not difficult to show that, once Eqs. (36) a nd (19) a re substituted into 8 Eq. (21), the mixed moments take the following for m (see App endix C): µ n,m = y n + m M + M − 1 X j =1 y m M ( y n j − y n j +1 ) I ( e, 1) (ˆ a j +1 ) + M − 1 X j =1 y n M ( y m j − y m j +1 ) I ( e, 2) (ˆ a ′ j +1 ) + M − 1 X j =1 M − 1 X j ′ =1 ( y n j − y n j +1 ) ( y m j ′ − y m j ′ +1 ) I ( e, 3) (ˆ a j +1 , ˆ a ′ j ′ +1 ) , (37) for even n a nd m a nd µ n,m = y n + m 1 I ( o ) (0 , 0 ) − M − 1 X j =1 y m 1 ( y n j − y n j +1 ) I ( o ) (ˆ a j +1 , 0) − M − 1 X j =1 y n 1 ( y m j − y m j +1 ) I ( o ) (0 , ˆ a ′ j +1 ) + M − 1 X j =1 M − 1 X j ′ =1 ( y n j − y n j +1 ) ( y m j ′ − y m j ′ +1 ) I ( o ) (ˆ a j +1 , ˆ a ′ j ′ +1 ) , (38) for odd n a nd m , where the functions I ( o ) ( · , · ), I ( e, 1) ( · ), I ( e, 2) ( · ), and I ( e, 3) ( · , · ), together with the symbols ˆ a j and ˆ a ′ j +1 are defined in Appendix C. 4. THEORETICAL RESUL TS The ev alua tion o f the momen ts could b e carried out, through Eqs. (37) and (38), for arbitrary antisymmetric qua nt ization functions of the form of Fig. 1. How ever, in the e x amples w e are going to show, w e used the quantization schemes identified b y Max as the results of an optimiza tion pro ces s aimed at minimizing the distortion r esulting from quantization [1 9]. F or conv e nie nc e , the geometries of the a b ov e quantization schemes a re r ep orted in T ab. 1 for v alues of M up to 4. Presenting the res ults rela ted to the whole biv aria te probability distribution is a non tr ivial task due to the discrete and 2D c ha r acter of the distribution itself. W e decided to pres e n t tw o examples of the p i,j distribution for M = 4 and for tw o fixed v alues of λ , namely λ = 0 . 05 and λ = 0 . 5, which are rep orted in T abs. 2 and 3, re sp e ctively . A meaningful pa rameter quantifying the amount of degradatio n induced by the quantization a nd resampling pro cesses is the cross-cor r elation co efficient, 9 defined in Eq . (16), who se b ehavior, a s a function of λ , is plotted in Fig . 2 for M = 1 , . . . , 4 . Note that, due to sy mmetry r easons, only the interv al [0 , 1 / 2] of λ is shown. As a general rema rk, it sho uld b e noted that, on increasing the n um be r of quantization levels, the corr elation co efficient increa s es appro aching 1 and dis- plays a plate au who se e x tension a pproaches the whole λ interv a l. Both b ehav- iors are exp ected, and they account for the fact that, for dense a nd non-clipping quantization s chemes, f ( x ) → x . Although the methodolo g y presen ted so fa r pro vides a complete so lutio n to the problem under inv es tigation, the inv olved computational effort increases prop ortiona lly to M 4 . As a matter of fact, its use for lar ge v alues o f M , i.e., fo r dense quantization schemes, is made difficult by pra ctical constraints related to the co mputation time required for ev aluating all inv olved int egrals and to the nu merical stability of the final results. In fact, a p oss ible drawbac k o ccurs when the V andermonde matr ix in E q. (14) has to b e inv er ted for la rge v alues of M , to der ive the full biv a riate probability distribution. In this cas e , how ever, it is preferable to deal with the pro blem in terms of the cros s-corr elation co efficient, which pr ovides a n adequate descr iption of the degr adation effect. 5. COMP ARISON W ITH N UME RICAL SIMULA TIONS W e p erfor med n umerical simulations a imed at quantitativ ely v er ifying the theoretical r esults presented in Sec. 4. In Fig. 3 a schematic blo ck diag ram expla ining the metho do logy a dopted for the simulations is sketched. A sequence o f r andom n um ber s nor mally dis - tributed with unit v ar ia nce and z ero mean is ge nerated, re pr esenting the samples of a re a lization of a Gaussia n pro cess X ta ken at the sa mpling p erio d which is ident ified as 1 sec. By constructio n, the pro cess X is BL b etw een -1/ 2 and 1 /2 Hz. The sa mples { x k } are used, along tw o parallel signal paths, to gener ate the v alues of f [ x ( λ )] a nd f [ w ( λ )] according to the reconstruction formulas (2) and (5). Only a finite num b er of terms is used for reco nstruction; the adopted selection of 200 terms is justified in Appendix D. The v a lues f [ x ( λ )] and f [ w ( λ )] are then used to es timate the mixed moments µ n,m and the discr ete pro babil- it y p i,j , indep endently . In particular, p i,j has been ev alua ted by coun ting the even ts f [ x ( λ )] = y i and f [ w ( λ )] = y j for a larg e num b er of realiza tions (in the order on 1 0 5 ). The mixed moments hav e also b een estimated by av er aging the pro duct f n ( x ) f m ( x ) ov er the same n um ber of rea lizations, in order to v e r ify the theoretical predictions ab out the co rrelation co efficient. T ables 4 and 5 give the biv aria te dis crete pro ba bilit y distribution e s timated from numerical s im ulations corr e sp o nding to the cas e M = 4, for λ = 0 . 05 , and λ = 0 . 5, resp ectively . They hav e to b e compared to tables 2 and 3, resp ec- tively . As w e can see , the ag reement b et ween the theo retical and exp erimental probability distributions is very go o d. As far as the corre la tion co efficient is concerned, Fig. 4 shows the extre mely go o d ag reement b etw een the theoretical v alues of ρ plotted in Fig. 2 (so lid curves) and the exp erimental r esults obtained by numerical s im ulations (circles). 10 Before concluding the present section, it is worth pro viding so me details ab out the c ho ice of the num b er of samples used in the reco nstruction formula of E q. (5). T o this a im, Appendix D contains a deta iled a nalysis concer ning the way the truncation o f the series in Eq. (5) a ffects the degradation of the reconstructed signal. In par ticular it is co nfirmed, by suitable numerical exper i- men ts, that a num b er of samples of ab out 200 is enough to v alida te the excellent agreement betw een theo r y a nd exp eriment previously displayed. 6. AN APPLICA TION TO DIGIT AL SIGNAL PROCESSING In the present section we illustrate a practical application o f our theor etical results. W e review the problem of the sampling rate increase (interpolatio n), making reference to the classical theory by Schafer and Rabiner [1]. In par- ticular we show how the degra dation originated from the re-sampling and re- quantization pr o cesses can be a ccounted for by using the theor etical expressions presented in Sec. 3. W e s ta rt from the top par t of the blo ck diagr a m of Fig. 6. The s ignal x ( t ), defined in Sec. 2, is first sampled at the r ate 1 /T . The obtained sequence { x k } is then interpolated and qua ntized, pro ducing the output sequence { u m } . The sampling rate ass o ciated to the sequence { u m } is 1 /T ′ , where T ′ /T = D /L , with D and L being integer n um ber s greater than 1, with D < L . 2 The change o f the sampling rate fro m 1 /T to 1 /T ′ is op erated acco rding to the pr escriptions given in Ref. [1]. More pr ecisely , after the fir st block the sampling ra te is increased by the integer fa c tor L , by inserting a sequence of L − 1 zero-v alued samples betw een an y t wo consecutive elements of the or iginal sequence. The seq uence so obtained is filtered through an idea l low-pass filter having a no rmalized cutoff frequency π /L and gain L . The output of the filter, is decimated by selecting a sample every D and even tually qua n tized by the function f . F ollowing Ref. [1], it is p os sible to show that the r e lation betw e e n the input sequence { x k } and the output sequence { u m } is given by u m = f " X k x k φ k ( ˜ λ m ) # = f [ x ( mT ′ )] , (39) where ˜ λ m = D L m . In the bo ttom part of Fig. 6, the sa me pro ces s ing scheme is ado pted as suming that the signal x ( t ) is only av aila ble thr ough its digitized samples { f ( x k ) } . The input sequence { f ( x k ) } is suitably scaled by the normaliza tion fa c to r A f , which will b e set to one, a ssuming the use of an idea l quantizer f [19]. The outco me of this pr o cessing is now repres ent ed by the sequence { ˜ u m } , whe r e ˜ u m = f " X k f ( x k ) φ k ( ˜ λ m ) # . (40) 2 W e li mit ourselves to the case of inte rp olation, for which T ′ < T . 11 W e note that Eq. (40) c oincides with Eqs. (4) a nd (5), so that we can apply our theo retical results to the pr esent situation. E ach e le men t o f the sequence { ˜ u m } is a degraded version of the cor resp onding element of the sequence { u m } (the targe t). Such a degra dation is not stationar y with respe ct to the “time”, represented by the index m . In fact, when mD /L is an integer num ber there is indeed no deg radation, whe r eas when the same quantit y has fractional part equal to 1 / 2, we know that the degradatio n is maximum (see Fig. 2). The ov erall deg radation b etw een the tw o sequences c an be quantit atively accounted for b y the 0-delay tempora l degree of coher ence, sa y γ , which is defined by γ = u m ˜ u m q u 2 m q ˜ u 2 m , (41) where the bar denotes the tempo ral (i.e., m ) av er age. The ab ov e definition can be used, provided that it is shown to b e indep endent of the par ticular realiza- tion x ( t ). After straightforward a lgebra, it is p o ssible to prov e the fo llowing theoretical expre ssion for γ , in terms of the mixed momen ts µ n,m , defined in Eq. (9): γ = 1 L L − 1 X i =0 µ 1 , 1 ( λ i ) √ µ 2 , 0 v u u t 1 L L − 1 X i =0 µ 0 , 2 ( λ i ) , (4 2) where λ i = i /L , ( i = 0 , . . . , L − 1), a nd the fa ct that µ 2 , 0 do es not dep end o n λ has b een made explicit. It sho uld b e noted that, for the tr eated in terp olation problem, only the v alue of L is re lev ant fo r the degradatio n. The quantit y in Eq. (41) is easily measur a ble b y implemen ting the blo ck diagram of Fig. 6, whereas expres sion in Eq . (42) is purely theor etical, bas ed on the expr ession of the moments obtained in Sec. 3. T o provide the exp erimental verification of the iden tity b etw e e n the t w o e q uations, the v a rious DSP blo cks of Fig. 6 hav e b een implemented. In particular, a FIR filter has been used to implemen t the filter blo ck, based on windowing the ideal impulse resp onse cor - resp onding to a rec tangular tr ansfer function by a Hamming window, similar ly as w e did in Ref. [8]. V alues of γ obta ine d for v alues of L from 2 up to 30, and for v a rious v alues o f D , hav e b een experimentally ev aluated and rep orted is Fig. 7 for M = 1 , 2 , 3 , 4. Soli d curv es represe n t the theoretica l v a lues provided by Eq. (42). The a g reement is excellent. 7. CONCLUSIONS The purp ose o f this pap er was to extend the studies ab out the r ate co nv er- sion applied to signal s equences, by taking into account the degradatio n effect asso ciated to the quantization pro cess. 12 In particular , we addressed the problem of computing the de g radation in- duced b y the r esampling a nd re-quantization of a B L s tationary and er go dic signal with Gaussia n statistics and flat p ower sp ectrum within the supp orting bandwidth, av a ila ble through its qua nt ized sa mples. The analysis pr ovides the algorithm for quant itatively characterizing the degradatio n effect induced by the resampling and re-quantization pro cesses in terms o f the knowledge of the complete biv ar iate discr ete pr obability distribu- tion ass o ciated to the ta rget and the estimated quantized signals , or in terms of the cor r elation co efficie nt betw een the tw o quan tities. The analysis makes use of FT representation of the quan tization function and of its p ow er s, to allow the application of linea r a nalysis techniques. Numerical exp e riments hav e also bee n implemented in or der to v alidate the theor etical metho dology and r esults. The compar is on show ed an exce llent ag reement b etw een theor y and sim ula tions. Finally , w e provided an ex ample of application o f our theoretical res ults to a n impo rtant area of Digital Signal Pro ce ssing, the sa mpling r ate c o nv ersio n. The class o f stochastic pro cesses c o nsidered in the present pap er represents a fundamen tal mo del with imp ortant applications in radio astro nomy , where the received noise-like signal or iginated fro m radio sources can b e modeled, after filtering, with go o d accura cy by the ideal pro cess her e a na lysed. In s uch a ppli- cations c o arse quantization (1 or 2 bits) is commonly applied, and the obtained sequences are often s ub ject to res a mpling and requantization. Of course, the results o btained in the present pap er m us t b e in terpreted a s a first step to - ward the extension of the metho dology origina lly de velop ed in Ref. [8] to other impo rtant classes o f sto chastic signals. Within the same p er sp ective another impo rtant topic, which will be the sub ject of forthco ming studies, concerns the developmen t of an asymptotic analysis dealing with the ca se of dens e and no- clipping quantization schemes, aimed at de r iving close d- form limit expr essions describing the degradation effect. Finally , w e wish to suggest a pos sible, future extension of our w ork whic h also coul d b e of in terest for the radio- astrono m y com- m un i t y . In particular, we refer at the classi cal problem in very large baseline interferometry (VLBI), of different an tennas observing the same ob ject and pro du ci ng digi tized representations of the receiv ed signal which are then cross-correlated to extract observ ables of inter- est. The decorrelation induced by the quan tization pro cess has b een studied in v arious work s in the past, for example in Ref. [2 0]. W e now think to the case, pres en tl y of practical in terest, whe re the receiv ed do wnl ink sig n al is di gitized at the different antennas according to di f- feren t sampli ng and quan tization sc hemes. In suc h a case the data streams ha ve to b e brough t to a comm on quantizat ion sc heme and sampling rate prior to cross-correlation. W e b eli e v e that the addi- tional de-correlation induced by the re-sampling and re-quantiza tion can b e studie d by use of the metho ds exp osed in our pape r. 13 Ac knowledgmen t W e wish to thank T uri Mar ia Spinozzi for his inv aluable help during a ll the phases of the prepara tion of the manuscript. A. Deri v ation of Eq. (19) W e start from Eq. (18) whic h, by FT in version, gives F n ( p ) = Z + ∞ −∞ [ f ( x )] n exp( − i2 π px ) d x, (43) where, due to the piecewi se c haracter of the quan tization function, [ f ( x )] n =                y n M − M − 1 X j =1  y n j +1 − y n j  rect  x 2 a j +1  , n even , y n M sign( x ) − M − 1 X j =1  y n j +1 − y n j  sign( x ) rect  x 2 a j +1  , n o dd , (44) where the function rect ( x ) is defined as rect( x ) =    1 , | x | ≤ 1 / 2 , 0 , | x | > 1 / 2 , (45) and s i gn ( x ) de no tes the signum function, i. e., sign( x ) =            1 , x > 0 , − 1 , x < 0 , 0 , x = 0 . (46) F urthermo re, on taking in to accoun t that F n rect  x 2 a o = sin(2 π ap ) π p , (47) and F n sign( x ) rect  x 2 a o = = F  rect  x − a/ 2 a  + F  rect  x + a/ 2 a  = = − 2i s in( π ap ) F n rect  x a o = − 2i π p sin 2 ( π ap ) = = − i π p [1 − cos(2 π ap )] , (48) after som e alge bra Eqs. (43) and (44) lead to Eq. (19). 14 B. Deriv ation of Eq. (34) W e start from h exp(2 π i Z C ) i = Y k ∈N ψ k ( p, p ′ ) , (49) where ψ k ( p, p ′ ) = h exp { 2 π i φ k [ x k p + A f f ( x k ) p ′ ] }i = = Z + ∞ −∞ p x ( x ) exp { 2 π i φ k [ x p + A f f ( x ) p ′ ] } d x, (50) where p x ( x ) is the pdf of the Ga us sian pro cess X . Due to the piecewise character of the f ( x ), Eq. (50) can b e written as ψ k ( p, p ′ ) = M X j =1 Z a j +1 a j p x ( x ) exp { 2 π i φ k [ x p + A f y j p ′ ] } d x + M X j =1 Z − a j − a j +1 p x ( x ) exp { 2 π i φ k [ x p − A f y j p ′ ] } d x. (51) F urthermor e , on changing the int egration v ariable x in − x in the s econd integral, after trivial algebra we obtain ψ k ( p, p ′ ) = = M X j =1 exp(2 π i φ k A f y j p ′ ) Z a j +1 a j p x ( x ) exp(2 π i φ k x p ) d x + c . c ., (52) where c.c. stands for c omplex c onjugate . The la st integral can be analytically expressed in terms of err o r function, and precis ely Z a j +1 a j p x ( x ) exp(2 π i φ k x p ) d x = 1 2 exp( − 2 π 2 φ 2 k σ 2 p 2 ) ×  erf  a j +1 − 2i π φ k σ 2 p √ 2 σ  − erf  a j − 2i π φ k σ 2 p √ 2 σ  . (53) On substituting from Eq. (53) into Eq. (52), and o n intro ducing tw o binary indices, s ay q ∈ { 0 , 1 } a nd s ∈ { − 1 , 1 } , we hav e ψ k ( p, p ′ ) = − e xp( − 2 π 2 φ 2 k σ 2 p 2 ) × X j,q,s ( − 1) q 2 exp(2 π i s φ k A f y j p ′ ) erf  a j + q − 2i s π φ k σ 2 p √ 2 σ  . (54) Finally , on substituting E q . (54) int o Eq. (49), after trivia l algebra Eq. (34) naturally follows. 15 C. Some com putational remarks First of a ll, we note that E q. (36) can b e forma lly r ewritten in the following wa y: h exp(i2 π Z ) i = 1 2 M exp  − 2 π 2 ( σ 2 p 2 + Q 2 I p ′ 2 + 2 Q 2 I p p ′ )  × X j X s X q Y k ∈N ( − 1) qk exp(i2 π s k A f y j k p ′ ) × e r f  a j k + q k − i 2 π s k φ k σ 2 p σ √ 2  , (55) where the vectorial indices j , s , a nd q ar e defined by j = [ j i 1 , j i 2 , . . . , j i N − 1 , j i N ] , s = [ s i 1 , s i 2 , . . . , s i N − 1 , s i N ] , q = [ q i 1 , q i 2 , . . . , q i N − 1 , q i N ] , (56) with j l ∈ { 1 , . . . , M } , s l ∈ {− 1 , 1 } , and q l ∈ { 0 , 1 } . The vectorial indices have bee n introduced to make the for m ulas more co mpact. The use of the indices is esemplificated by the following statement: X j ( · ) = X j i 1 X j i 2 . . . X j i N − 1 X j i N ( · ) . (57) F urthermor e , we introduce t wo dimensio nless v aria bles, say ξ and η , in pla ce of p and p ′ , resp ectively , which ar e defined by ξ = √ 2 π σ p, η = √ 2 π Q I p ′ , (58) so that Eq. (55) b ecomes h exp(i2 π Z ) i = 1 2 M exp  − ( ξ 2 + η 2 + 2 αξ η )  × X j X s X q Y k ∈N ( − 1) q k exp(i β j k s k φ k η ) × e r f (ˆ a j k + q k − i s k φ k ξ ) , (59) where ˆ a j = a j σ √ 2 , α = Q I σ , β j = y j √ 2 A f Q I . (60) 16 As far as the pro duct F n ( p ) F m ( p ′ ) is concer ned, from Eq. (1 9) we have F n ( p ) F m ( p ′ ) = y n + m M δ ( ξ ) δ ( η ) + M − 1 X j =1 y m M ( y n j − y n j +1 ) sin(2ˆ a j +1 ξ ) π ξ δ ( η ) + M − 1 X j =1 y n M ( y m j − y m j +1 ) sin(2ˆ a ′ j +1 η ) π η δ ( ξ ) + M − 1 X j =1 M − 1 X j ′ =1 ( y n j − y n j +1 ) ( y m j ′ − y m j ′ +1 ) sin(2ˆ a j +1 ξ ) π ξ sin(2ˆ a ′ j ′ +1 η ) π η , (61) for even v alues of b oth n and m , and F n ( p ) F m ( p ′ ) = − y n + m 1 1 π ξ 1 π η + M − 1 X j =1 y m 1 ( y n j − y n j +1 ) cos(2ˆ a j +1 ξ ) π ξ 1 π η + M − 1 X j =1 y n 1 ( y m j − y m j +1 ) cos(2ˆ a ′ j +1 η ) π η 1 π ξ − M − 1 X j =1 M − 1 X j ′ =1 ( y n j − y n j +1 ) ( y m j ′ − y m j ′ +1 ) cos(2ˆ a j +1 ξ ) π ξ cos(2ˆ a ′ j ′ +1 η ) π η , (62) for o dd v alues of b o th n and m , where ˆ a ′ j = ˆ a/α . Finally , on substituting from Eqs. (59), (61) and (62) into Eq. (21) w e obtain 17 Eqs. (38) and (37), where I ( e, 1) (ˆ a ) = X j , s , q ( − 1) q × Z d ξ ex p( − ξ 2 ) sin(2ˆ a ξ ) π ξ Y k ∈N erf(ˆ a j k + q k − i s k ξ φ k ) , I ( e, 2) (ˆ a ′ ) = X j , s , q ( − 1) q erf  ˆ a ′ + Γ s , j 2  Y k ∈N erf ( ˆ a j k + q k ) , I ( e, 3) (ˆ a, ˆ a ′ ) = X j , s , q ( − 1) q Re Z d ξ ex p( − ξ 2 ) sin(2ˆ a ξ ) π ξ × erf  ˆ a ′ + Γ s , j 2 + i αξ  Y k ∈N erf(ˆ a j k + q k − i s k ξ φ k ) , I ( o ) (ˆ a , ˆ a ′ ) = X j , s , q ( − 1) q Im Z d ξ ex p( − ξ 2 ) cos(2ˆ a ξ ) π ξ × erf  ˆ a ′ + Γ s , j 2 + i αξ  Y k ∈N erf(ˆ a j k + q k − i s k ξ φ k ) , (63) and ( − 1) q = Y k ∈N ( − 1) q k , Γ s , j = X k ∈N s k β j k φ k . (64) D. Analysi s of degradation effect in case a fini te num b er of samples is us ed for sig nal reconstruction The reco nstruction formula in Eq. (5) assumes that an infinite num ber of samples ca n b e used for estimating w ( λ ). In prac tice , the sum will b e made ov er a finite num ber of samples, a pplying some type o f windowing function. F o r instance, in the case of a rectangula r window one has simply that 3 ˜ w ( λ ) = A f X i ∈G u i φ i , (65) The set G is made by consecutive indices distributed around 0, which we supp ose including the set N defined in E q . (28). In particular, it is stra ightf orward to 3 In the present annex we will indicate with til de all terms which cha nge due to the trun- cation. 18 show that the constant A f , whic h ensures minimum po s sible degr adation, is still defined as in the case of infinite sum, i.e., by Eq. (6). It is not difficult to show that the theoretical analysis de velop ed still remains v alid, provided tha t the ra ndom v a riable Z I defined in Eq. (27) be replaced by a new rando m v ar iable, sa y ˜ Z I , defined as the sum of tw o statistica lly indep endent terms, a s fo llows: ˜ Z I = X k / ∈N k ∈G z k + X k / ∈G p x k φ k . (66) Then, on applying a similar methodolo gy a s done for the case of Z I , one can assume that h exp(i2 π ˜ Z I ) i c an b e a pproximated by e xp( − 2 π 2 σ 2 ˜ Z I ), where σ 2 ˜ Z I = p 2 P I + ˜ Q 2 I ( p ′ 2 + 2 p p ′ ) , (67) and ˜ Q 2 I = A f h f 2 i X k ∈G φ 2 k − X k ∈N φ 2 k ! , (68) provided tha t 1. a s uitable set N ha s bee n selected accor ding to the prescr iptions of Sec. 4 (e.g. { 0 , 1 } ); 2. G is m uch la rger than N . The whole theor etical analysis develope d in the pap er is now en tirely a pplicable having c a re to r eplace Q I with ˜ Q I . T o giv e a numerical evidence abo ut the effect of the finite n um ber of sa m- ples, Fig. 5 s hows the b ehavior, a s a function of the total num b er of samples, of the corre la tion co efficient, ev a lua ted for λ = 1 / 2 and for the 1 -bit quantization function. The dots are representativ e of the outcomes of numerical s im ulations, 4 while the so lid curve represents the r e sults obtained by applying the theoretical analysis, together with the prescr iption given by Eq. (68). It is eviden t that the agree ment b etw een the v alues o f ρ obtained from n umerical simulations and those derived thro ugh the theoretical ana ly sis in the prese n t a nnex is quite satisfactory even w hen small num b er of terms is us e d for re construction. Addi- tionally , it is also clear that selecting 20 0 sa mples in the r econstruction formula is well r epresentativ e of the ideal condition, s ince the v alues of ρ have reached their a symptotic r egime. 4 Of course, the num b er of terms used for reconstructing x ( λ ), according to Eq. (2), was k ept constan t during al l simulations to the relativ ely large of num ber of 500. 19 References [1] R. W. Schafer, L. R. Rabiner, “A digital signal pr o cessing approach to int erp olation,” P ro ceedings of the IEE E, V ol. 61, Nr. 6, pp. 69 2 - 7 02 (1973). [2] R.E . Cro chiere and L. R. Rabiner, “Interpo lation and Decimation of Digital Signals: A T utor ial Re v iew,” Pro c. IEEE , V ol. 6 9, Nr. 3 , 30 0-331 (1 981). [3] R. E. Cro chiere a nd L. R. Rabiner, Multir ate Digital Signal Pr o c essing, Pr entic e-Hal l, Englewo o d Cliffs, NJ, 1983 . [4] T. A. Ramstad, “Digital Me tho ds for Conv e r sion B e tw een Arbitrar y Sam- pling F r equencies,” IE EE T rans. Acous t. Sp eech and Sig. Pro cess., V ol. ASSP-32, Nr. 3, pp. 5 7 7-591 , (198 4). [5] B . R. Carls on, P . E. Dewdney , T. A. Bur gess, R. V. Casor so, W. T. Petra- chenk o, and W. H. Cannon, “The S2 VLBI Co r relator : A Correla tor fo r Space VLBI and Geodetic Signal Pro ces sing,” Publ. Astron. Soc . Pacific V ol. 11 1 , 102 5-104 7 (19 9 9). [6] S. Iguchi, T. Kuray ama, N. Kaw aguchi, and K. Kaw ak ami, “Gig abit Digital Filter Bank: Digital B ack end Subsystem in the VERA Data-Acquisition System,” Publ. Astron. So c. J a pan V ol. 57 , 25 9-27 1 (200 5). [7] Y. Nak a jima, H. Hori, T. K anoh, “Rate conversion of MPEG co ded video by re-quantization pro cess,” P ro ceedings o f the IEEE 199 5 In ter national Conference on Imag e Pro ce s sing, V ol. 3, 4 08 - 41 1 (1995). [8] M. Lanucara a nd R. Bo r ghi, “Res ampling of band-limited Gaus sian ran- dom signals with fla t p ow er sp ectrum, av aila ble throug h 1-bit quantized samples,” IEEE T ra ns. Sig nal P ro cessing V o l. 55, 398 7-399 4 (2007 ). [9] W. B. Dav enpo rt a nd W. L. Ro ot, An Int r o duction to the The ory of R an- dom Signals and N oise (Wiley , New Y o rk, 1987 ). [10] E. D. Ba n ta, “ On the auto corr elation function of qua nt ized signal plus noise,” IEEE T ra ns. Infor mation Theory , V ol. 1 1 , 1 14-11 7 (19 6 5). [11] V. V. Petrov, Sums of indep endent r andom varia bles , (Spr inger-V er lag, Berlin, 19 75). [12] M. A. Lifshits, “On the low er tail pro babilities of so me random se r ies,” Ann. Pro bab. 25, 424- 442 (199 7). [13] R. E. Crandall, “Theory of R OOF w alks,” preprint, ht tp://p eople.reed.edu/ ∼ crandall/ pap ers/ROOF11.p df 20 [14] B. Sc hm uland, “Random Har monic Series ,” Amer. Ma th. Month., 110, 40 7- 416 (200 3 ). [15] L. V. Rozovsky , “Small devia tio n probabilities for a class of dis tributions with a p olynomia l decay at zer o,” J. Math. Sci. 139, 6 603-6 607 (2006). [16] F. Aurza da, “O n the Low er T ail Pr obabilities of Some Random Sequences in l p ,” J. Theor. Pr obab. 20, 843 858 (200 7). [17] A. A. Borovk ov and P .S. Ruzankin, “O n Small Deviations o f Ser ies ofW eighted Random V ar iables,” J. Theor. Proba b. 21, 628 649 (200 8). [18] M. Abramowitz, I. Stegun, Handb o ok of Mathematic al functions , (Dov er, New Y ork, 19 72). [19] J. Max, “Q ua nt izing for minimum disto rsion,” IEEE T r ansactions on In- formation Theory , V ol. 6, 7-12 (1960). [20] S. Iguchi a nd N. Kawaguc hi, “High-order sampling, ov er-sampling and dig- ital filtering techniques”, IEICE T rans. Commun., 8 5, 1806- 1816 (2 002). 21 List of T abl e s M a j ( j = 1 , . . . , M ) y j ( j = 1 , . . . , M ) 1 a 1 = 0 . y 1 = . 798 2 a 1 = 0 . y 1 = . 4528 a 2 = 0 . 981 6 y 2 = 1 . 510 3 a 1 = 0 . y 1 = 0 . 3177 a 2 = 0 . 658 9 y 2 = 1 . a 3 = 1 . 447 y 3 = 1 . 894 4 a 1 = 0 . y 1 = 0 . 2451 a 2 = 0 . 500 6 y 2 = 0 . 7560 a 3 = 1 . 050 y 3 = 1 . 344 a 4 = 1 . 748 y 4 = 2 . 152 . . . . . . . . . T able 1: Quan tization schemes, tak en from Ref. [19], s elected f or testing of the theoretical approac h. F or meaning of symbols r efer to Fig. 1. 0.04 0 0 0 0 0 0 0 0 0.10 0.01 0 0 0 0 0 0 0.01 0.14 0.0 1 0 0 0 0 0 0 0.01 0.16 0.0 1 0 0 0 0 0 0 0.01 0.16 0.01 0 0 0 0 0 0 0.01 0.14 0.01 0 0 0 0 0 0 0.0 1 0 .10 0 0 0 0 0 0 0 0 0.04 T able 2: Biv ariate discrete probability distribution p i,j , defined as in Eq. (8), calculated for M = 4 and λ = 0 . 05. i ndices i (rows) and j (columns) equal − M , . . . , − 1 , 1 , . . . , M . 22 0.03 0.01 0 0 0 0 0 0 0.01 0.08 0.0 2 0 0 0 0 0 0 0.02 0.12 0.0 3 0 0 0 0 0 0 0.03 0.13 0.0 3 0 0 0 0 0 0 0.03 0.13 0.03 0 0 0 0 0 0 0.03 0.12 0.02 0 0 0 0 0 0 0.0 2 0 .08 0.01 0 0 0 0 0 0 0 .01 0.03 T able 3: The same as in T able 2 but for λ = 0 . 5. 0.04 0 0 0 0 0 0 0 0 0.10 0.01 0 0 0 0 0 0 0.01 0.14 0.0 1 0 0 0 0 0 0 0.01 0.16 0.0 1 0 0 0 0 0 0 0.02 0.16 0.01 0 0 0 0 0 0 0.01 0.14 0.01 0 0 0 0 0 0 0.0 1 0 .09 0 0 0 0 0 0 0 0 0.04 T able 4: Biv ar i ate dis crete probability distri bution p i,j , estimated fr om numerical si mulat ions for M = 4 and λ = 0 . 05, to b e compared to T ab. 2. 0.03 0.01 0 0 0 0 0 0 0.01 0.08 0.0 2 0 0 0 0 0 0 0.02 0.12 0.0 2 0 0 0 0 0 0 0.02 0.14 0.0 3 0 0 0 0 0 0 0.03 0.14 0.02 0 0 0 0 0 0 0.02 0.12 0.02 0 0 0 0 0 0 0.0 2 0 .08 0.01 0 0 0 0 0 0 0 .01 0.03 T able 5: The same as in T able 4 but for λ = 0 . 5. Thi s table has to b e compared to T ab. 3. 23 List of Figure Captions Figure 1: Geome try of the quan tization f unction. Figure 2: Theoretical b ehavior of the cross-corr elation co efficient, defined in Eq. (16), as a function of λ , for M = 1 , . . . , 4. Figure 3: Block di agram showing the methodology adopted for making the n umerical simula- tions. The a v erage in the h . . . i -blo ck was made ov er 10 5 realizations, w hi le the sinc int erpo- lation was performed by using 200 terms. Figure 4: Compari son of the theoretical results shown in Fig. 2 (solid curves) to the experi- men tal results obtained through numerical simulations (circles). Figure 5: Beha vior of the correlation coefficient ρ as a function of the total n umber of samples used for reconstructing the v alues of w ( λ ) for λ = 1 / 2. Dots are r epresen tative of the nume rical simulations; the soli d curve represents the results provided by the theoretical analysis with the use of Eq. (68). The quan tization function corresponds to the 1-bit case. Figure 6: A DSP application. Sampling rate conv ersion applied to a signal x ( t ) a v ailable through it quan tized samples. Figure 7: Behav ior of the degradation betw een the sequences { u m } and { ˜ u m } in Fig. 6, experimentally ev al uated f or v alues of L fr om 2 up to 30, and for several v alues of D . The theoretical prediction given in Eq. (42) is also shown (the theoretical p oints ar e joined b y a solid curve which is used as a gui de for the eye). T he quantization functions corresp ond to M = 1 (op en circles), M = 2 (black dots), M = 3 (open squares), and M = 4 (black squares). 24 Figures a 1 =0 a 2 a 3 a M -1 a M x f (x) ... ... y 1 y 2 y M -1 y M ... Fig. 1 - Lanucara and Bor ghi 25 Fig. 2 - Lanucara and Bor ghi 26 f λ f f A f P i,j µ n,m x k f(x k ) w ( λ ) f [ w ( λ ) ] f [ x ( λ ) ] x ( λ ) Zero-mean random (Gaussian) number generator σ =1 Sinc interpolation T =1 Sinc interpolation T =1 Fig. 3 - Lanucara and Bor ghi 27 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 Normalized co rrelation coe fficient λ M=1 M=2 M=3 M=4 Fig. 4 - Lanucara and Bor ghi 28 0.5 0.6 0.7 50 100 150 200 250 300 350 400 Correlation coefficient Number of samples Fig. 5 - Lanucara and Bor ghi 29 D L H f f D L H f x( t ) x k = x( kT ) rate =1/ T rate = L / T rate =( L /D)/ T =1/ T ' u m = f [ x( mT' )] u m ~ A f 1/ T Fig. 6 - Lanucara and Bor ghi 30 0 5 10 15 20 25 30 L 0.75 0.80 0.85 0.90 0.95 1.00 Γ Fig. 7 - Lanucara and Bor ghi 31