The Poenial Effeciveness of he Deecion of Pulsed Signals in he Non-Uniform Sampling Arhur Smirnov, Sanislav Vorobiev and Ajih Abraham 3, 4 Deparmen of Compuer Science, Universiy of Illinois a Chicago, Chicago, IL, USA asmirn3@uic.edu Insiue of Informaion Sysems and Informaion, Sain-Peersburg Sae Universiy (SUAI), S Peersburg, ussia k53@aane.ru 3 IT4Innovaions - Cener of excellence, VSB - Technical Universiy of Osrava, Czech epublic 4 Machine Inelligence esearch Labs (MI Labs), WA, USA ajih.abraham@ieee.org Absrac We consider a non-uniform ime quanizaion of he opimal form of he signal by shifing one of he samples in he neighborhood of he poin a which he minimum eigenvalue of he covariance marix of he noise is equal o zero. Shown, ha in esing simple hypoheses, arbirarily large value of he signal-o-noise- raio is achieved by his shif on he oupu of he discree mached filer a finie energy of he signal and noise power. We discuss some aspecs of he ill condiioning of he problem and he a priori of uncerainy. Keywords: Deecion, eigenvalue, SN, signal deecion I. INTODUCTION The ask of improving he efficiency of deecion of pulsed signals is relevan o many applicaions of saisical radio engineering. Deecion of a deerminisic signal S wih known arrival ime in he addiive saionary Gaussian noise X wih correlaion marix B is described by a discree mached filer [] BG = S. Saisic (Z = X + AS - signal on he inpu wih a hypohesis H, A ampliude) α = G T Z provides he signal-o-noise raio of he discree mached filer: d = G T BG = S T B - S. Is dependence on he shape of he signal allows opimizing he efficiency of deecion of he appoinmen of he opimal signal: S op = AU min, U min - Eigenvecor of he marix corresponding o he minimum eigenvalue α min >. This delivers he signal-o-noise raio: d = A U T minb - U min = A U T minuλ - U T U min = E / λ min,u, Λ - marices of eigenvecors and eigenvalues, E = A - he signal energy [,]. Inverse proporionaliy of he signal-onoise raio o he minimum eigenvalue means ha when λ min he probabiliy of deecion is very high. Obviously, his condiion is saisfied when here are wo samples wih inerval Δ: eigenvecors and eigenvalues of he correlaion marix of he sandard noise: ρ B = σ is equal o he following expression: ρ ρ U =, ; Λ = σ + ρ increasing he value of Δ, implies ρ and λ min. The need o expand bandwidh ΔF which is followed from Δ makes his example no very ineresing. Bu here is anoher case o be considered when here are many samplings available. I can be shown ha if one sample is shifed, which implemens a non-uniform signal sampling in ime, hen in his case he value of α min will be much less when compared o uniform sampling. II. SIMPLE HYPOTHESIS Le s assume ha vecor X T =[x,,x n ] consiss of sampling of saionary Gaussian process x().singular value decomposiion [3] of he correlaion marix B x will be as following: B x = UΔU T. Eigenvecors U of his marix are he direcions of he ellipsoid axes of dispersion and eigenvalues Δ are lenghs of is axes. Bu a he same ime eigenvalues are he roos λ i > of he characerisic equaion: de (B x -αi) = λ n a λ n- + a λ n- - +(-) n λ n =, in which free erm is described as following: a = de n B x i= α. Le s now assume ha vecor X can be modified in some way ha one of he eigenvalues λ k. Then a n, ellipsoid degeneraes dispersion by bringing one of he diameers o zero. A he same ime having a condiion ha λ k when he signal is defined as S op = U k which according o [] is equivalen o d = /λ k. A nonuniform sampling can implemen such a ransformaion. For example, le s ake six samples X T =[x,,x 5 ]wih a correlaion funcion: (τ) = exp(-λ τ )(cos βτ + λβ - sinβ τ ), λ =.5, β = π, sampling inerval Δ=, hen he characerisic equaion will be as following: λ 6-6λ 5 +.466 λ 4 -.66λ 3 + 5.3878λ - = n i
.4. -. -.4 -.6 -.93λ +.9 = has a minimum soluion λ =.6 wih corresponding signal: S T op = U T = [.747;.4;.5475;.5475;.4;.747] which carries ou a signalo-noise raio d = 3.85. This value deermines a poenial effeciveness of deecion [4] wih uniform sampling. Bu if 3 = 3 will be changed o 3 = 3.75, hen he characerisic equaion, by having λ =.3 will be as following: λ 6-6λ 5 +.55λ 4 -.88λ 3 + 3.8685λ -.347λ+ +4.3634= which carries ou a signal-o-noise raio for he following signal: S op = [.33;.5646;.84; -.84;.5646; -.33] Then a marix of eigenvalues for a non-uniform sampling will be as following:.33.367.3969.567.4558.5646.3.56.98.4775.884.636.67.48.534 U =.884.636.67.48.534.5646.3.56.98.477.33.367.3969.567.4558.939.443.499.499.443.939 Figure shows a dependency λ = φ( ), which was obained on he inerval < < for he correlaion funcion: (τ) = exp(-λ τ )(cos βτ + λβ - sinβ τ ), λ =.5, β = π, eigenvalue a poin changes a sign, hen λ ( )=, and herefore d = /λ k a poin undergoes a disconinuiy of he second kind. λ T.8.6.4. -. -.4 -.6 3 4 funcion d ( ) is no defined. A seep growh of he signalo-noise raio when defines a poenial effeciveness of he deecion of he signal of opimal form when having a non-uniform sampling. Poenial propery d wih finie energy of he signal can be called he non-uniform sampling mehod "super deecion". Figure. Signal-o-noise raio By modifying he vecors of he correlaion marix B x wih operaor A = B / B -/ x [], we find ou ha due o he srucure of he marix B here is a sequence of complex numbers generaed. When you generae he "semi-infinie" sequence wih a given correlaion vecor paining of a discree whie noise linear sysem wih he weigh vecor, which is he discree soluion of he inegral equaion [,5] below: 4 3 - - -3-4 ( ) ( ) h ()h+τ d = τ Figure 3 shows he samplings of posiive definie funcion and is reproducion wih solving ĥ() wih uniform sampling wih he inerval Δ =..8.6.4. D а б T.5.5 = 6 4 - -4-6 D.363.363.363.363.363.8.6.4 =.3.5.5 3 4 5 τ. -. Figure. Minimal Eigen value, sampling nodes -.4 -.6 -. τ The signal-o-noise raio d around poin changes very seeply. Bu in he Figure (d = D) he value of he 4 6 8 Figure 3. Correlaion funcion 4 6 8
The gap of he funcion d () causes poor condiionaliy of mehod of non-uniform sampling. For example, signals: S T =[.334; -.5596;.687;.867; -.5596;.334], S T =[.334; -.56;.873;.873; -.56;.334] correspond o he signal-o-noise raio d = -65.4 and d = 65.4 In his case, super deecion can be obained near poins, ; 3, 3; 4 when and ohers. Figure 4 shows he minimal eigenvalues and examples of he opimal non-uniform sampling wih he following samplings received wih sampling inerval Δ =.. Sampling nodes T, 3 T 3, 4 T 4. (τ) = exp(-λ τ )(cos βτ + λβ - sinβ τ ), (τ) = exp(-λ τ )+ +cos βτ Table corresponds o he resuls of modeling. Table. Modeling funcion using differen parameers # funcion 3 3 4 (3).364.68.36.45 3.748 3.49 (6).7.574.69.356 3.635 3. λ,.8 (7).36.69.33.43 3.695 3..6 -, -,4 -,6 T.4. -. 3 4 The minimal eigenvalue changes is sign wih he firs sampling node is shifed near he poin =.57, which is shown in Figure 6. -,8 - =.68 =.36 -.4 -.6 λ.5.5 3 3 4 5.4.9.8 Figure 4. Minimal Eigenvalues sampling nodes..7.6.5 -..4 λ.. 3 3 3 -. T3 -. 3 =.45 -.3 3 = 3.748.5 3 3.5 4.8.6.4. 3 4 -. -.4 -.6 3 4 5 -.4 -.6 =.57.5.5 3 4 5 Figure 6. Minimal Eigenvalues, sampling nodes The shifing of oher sampling nodes leaves λ > and a he same ime d. If he coefficien α changes he value of o zero, hen he signal-o-noise raio d =. I is very imporan o noe ha he mehod of nonuniform sampling requires an expended, bu finie frequency bandwidh of he deecor..3.. Figure 5. Minimal Eigenvalues, sampling nodes The same ype of modeling (Δ =., n =6) was done for he noise wih he following corresponding correlaion funcions: III. COMPLEX HYPOTHESIS Pracical implemenaion of he "super deecion" requires a special sudy relaed o he specific problems. In radio engineering, as a rule, he noise is saionary a limied inervals of ime; his is why adapive sysems wih channel measuring noise characerisics are used. Error of
measuremen of he correlaion funcion or he correlaion marix will lead o errors in seing he waveform (eigenvecor of he correlaion marix). You need o research he effeciveness of he deecion signal wih disored shapes o define he requiremens for he measuring channel. The repored resuls of he saisical analysis of he eigenvecors and eigenvalues associaed wih he mehod of principal componen analysis [5], which allocaes he maximum value. Saisical sudies of he minimal eigenvalue and he corresponding eigenvecor in he problem of deecion of independen ineres. The following example describes his kind of issues ha are in his kind of research should be allowed. Le us now assume ha he correlaion funcion has he following form and he following coefficiens are defined as following: (τ) = exp(-α τ )(cos βτ + αβ - sinβ τ ), α =.5, β = π, α * N(.5,.), β * N(π,.6), he arrival ime and sampling nodes are available. Le s define a sampling inerval Δ =, he second sample will be moved by poin (shown in Fig. 4) and is equal o =.7. If he coefficiens were known, hen he signal-o-noise raio would be equal o d =.3. The resuls of modeling such signals S wih random parameers chosen above are shown in he Figure 7. S h There could be a differen limiaion in he asynchronous sysems which is defined by disconinuiy of a funcion d = /λ () need of synchronizing of he deecor for maximizing he signal-o-noise raio by approaching a sample node o he poin of disconinuiy. If we assume ha he hird sample node of he correlaion funcion is opimized by moving around poin 3 =3.748 (Figure 5). Le us now assign 3 = 3.7, hen he corresponding signal will be as following: S T op = A [.337;,5645;.88; -.88; -.5645; -.337], which carries ou he signal-o-noise raio d = 54.4. If he synchronizer has an error δ N(,σ), σ =. (normal densiy is muliplied by 5 in Figure 8), he average signal-o-noise raio will be d = 58.8. Whine uniform discreizaion is applied, hen d r =3.85, hen nonuniform discreizaion is.8 db more efficien. In his case modeling a deecor is provided for NN = 3 for random values of 3. The value d = 6 saes for he signal ampliude A =.3333. Figure 9 shows he characerisics obained for signal wih 3 = 3.7: insabiliy of he signal generaor does no affec he signal efficiency deecion. Thus, he a priori uncerainy of he effeciveness of he mehod of nonuniform discreizaion can be significanly reduced when compared wih he poenially aainable efficiency, however, i is expeced ha i will overrun he mehod of uniform discreizaion..4 8 D, f. N = 6 45 4 -. 4 35 3 -.4 5 3 4 5 -.5.5..5. λ 5 Figure 7. Signals, hisogram of Eigenvalue The hisogram shows ha wih he random parameers chosen above; deecing a leas 4% of signal wih α < is impossible. The average value of posiive values of signal-o-noise raio is equal o d 5 is no a sign ha he deecion is more efficien because among all he values ha were obained when d > 4.5, some values are in he range d > 5. This example clearly shows ha an aemp o ge close o a super deecion, for example, in synchronous sysem can be accompanied by a se of heoreical and engineering asks in he wide range including bu no limied o he following: correlaion properies, organizing sysem ec. 5 3,66 3,67 3,68 3,69 3,7 3,7 3,7 3,73 3,74 Figure 8. Averaging signal-o-noise raio
D [4] V. Tikhonov, Saisical radio engineering, Sovie adio, 966, p. 678 [5] M. Kendall and A. Sewar, Mulideminsial saisical analysis and ses, Nauka, 976, p. 736 d = /3,9,8 F,, Figure 9. Signal characerisics IV. CONCLUSIONS In erms of addiive saionary Gaussian noise, opimum signal is in he form of he eigenvecor corresponding o is minimum eigenvalue. When esing simple hypoheses and iniial uniform sampling signal, one of he samples can be shifed o he poin where he minimum eigenvalue of he correlaion marix of he noise is zero. This achieves he poenial effeciveness of deecion - he oupu of a discree mached filer can be obained by arbirarily high signal-o - noise raio of he signal wih uni energy and he noise from he uni variance. A poin, he signal-o-noise raio undergoes a disconinuiy of he second kind which defines a bad condiionaliy of deecion wih non-uniform discreizaion. This condiionaliy complicaes esing he hypohesis. ACKNOWLEDGMENTS This work was suppored in he framework of he IT4 Innovaions Cenre of Excellence projec, reg. no. CZ..5/../.7 by operaional programme esearch and Developmen for Innovaions funded by he Srucural Funds of he European Union and sae budge of he Czech epublic, EU. EFEENCES [] S. Vorobiev, Effecive deermined signal deecion, GUAP,, p. 39 [] A. Smirnov, Deecion of signals wih normal correlaion inerference, [in press] [3]. Khorn and C. Johnson, Marix analysis, MI, 989, p. 656