4th Euopea Sigal Pocessig Cofeece (EUSIPCO 6), Floece, Italy, Septembe 4-8, 6, copyight by EURASIP Extedig Laplace ad z Tasfom Domais Michael J Coithios Pofesso, Ecole Polytechique de Motéal Uivesité de Motéal 5 Chemi de Polytechique, Motéal, Qc, H3T J4 Caada michaelcoithios@polymtlca ABSTRACT A geealisatio of the Diac-delta fuctio ad its family of deivatives ecetly poposed as a meas of itoducig impulses o the complex plae i Laplace ad z tasfom domais is show to exted the applicatios of Bilateal Laplace ad z tasfoms Tasfoms of twosided sigals ad sequeces ae made possible by extedig the domai of distibutios to cove geealized fuctios of complex vaiables The domais of Bilateal Laplace ad z tasfoms ae show to exted to two-sided expoetials ad fast-isig fuctios, which, without such geealized impulses have o tasfom Applicatios iclude geealized foms of the samplig theoem, a ew type of spatial covolutio o the s ad z plaes ad solutios of diffeetial ad diffeece equatios with twosided ifiite duatio focig fuctios ad sequeces INTRODUCTION eealized fuctios have expaded cosideably the domai of existece of the Fouie tasfom []-[5] Weighted specta leadig to impulses o the complex Laplace ad z tasfom plaes have bee poposed fo the expoetial decompositio of fiite duatio sigals [6], [7] The decompositio of ifiite duatio complex expoetial cotiuous-time ad discete-time sigals leads i geeal to divegig itegals ad summatio eealizig the Diac-delta impulse has fo objective to defie tasfoms fo a class of fuctios which leads to itegals that ae ot absolutely coveget I this pape, the distibutio theoetic basis of the geealizatio is peseted, followed by popeties of the ew distibutios ad the esultig Bilateal tasfoms Applicatios of the bilateal tasfoms to the solutio of diffeetial ad diffeece equatios ae illustated though examples COMPLEX-DOMAIN DISTRIBUTIONS A geealised distibutio s (), associated with Laplace tasfom complex domai, as a geealised fuctio of a complex vaiable s + jω, may be defied as a itegal alog a staight lie cotou i the s plae extedig fom a poit s j to s + j of the poduct of s () with a test fuctio Φ Fo coveiece we efe to this itegal by the symbol I [ Φ ], o simply I [ ], ad use the Φ otatio [ ], I Φ s < s Φ s Φ ds The test fuctio Φ has deivatives of ay ode alog staight lies i the s plae goig though the oigi, ad teds to zeo moe apidly tha ay powe of s Fo example, if the geealised distibutio is the geealised impulse ξ () s [8], [9] we may wite [ Φ ] < ξ, Φ jφ (), σ ξ ds, σ I s s s Φ Basic Popeties I the followig a selectio of basic popeties of geealized distibutios i the cotext of the
4th Euopea Sigal Pocessig Cofeece (EUSIPCO 6), Floece, Italy, Septembe 4-8, 6, copyight by EURASIP cotiuous-time domai ad Laplace tasfom is icluded due to thei impotace i evaluatig tasfoms Shift i s Plae s + jω we may eite tig < s ( s), Φ s ( s) Φ( sds ) tig s s y, ds dy we obtai < s ( s ), Φ < y, Φ ( y+ s ) R [ y] σ Scalig γ be a eal costat We ca wite ( γ ) ( γ ) < s, Φ s s Φ s ds tig γ s y, γ ds dy we obtai < ( γ s), Φ < ( y), Φ ( y/ γ ) R [ y] γσ γ Poduct with a Odiay Fuctio s F s We ca wite Coside the poduct < F, Φ [ s], F if F Φ R < Φ C, the class of test fuctios Covolutio Deotig by () s the covolutio of two geealised distibutios, with y Σ+ jω, we may wite I < () s s, Φ s Σ+ j ( y) ( s y) dy, Σ j, R [ y] Σ < Φ I < y s y Φ s ds the itegal o the ight, beig i the fom of a covolutio with a test fuctio, belogs to the class of test fuctios Diffeetiatio < s, Φ s s Φ s ds Itegatig by pats we obtai,, < s Φ s < s Φ s ad, by epeated diffeetiatio, ( ), < s Φ s ( ) <, Φ Multiplicatio of the Deivative Times a Odiay Fuctio Coside the poduct F() s We ca wite < s F(), s Φ s s F Φ s ds Itegatig by pats we obtai < s F(), s Φ s <, F Φ [ s] F s 3 DISCRETE TIME DOMAIN R <, Φ A distibutio z may be defied as the value of the itegal, deoted I Φ, of its poduct with a test fuctio Φ Symbolically, we wite, I Φ z < z Φ z z z z Φ z dz whee the cotou of itegatio is a cicle of adius ceted at the oigi i the z plae Simila popeties to those of the cotiuous time domai ae ecouteed i the discete-time domai 4 THE ENERALISED DIRAC_DELTA IMPULSE IN THE s DOMAIN The geealised Diac-delta impulse deoted ξ s [8], [9] may be defied by the elatio, < ξ Φ j ξ Φ ds jφ (), σ j, σ If F() s is aalytic at s the
4th Euopea Sigal Pocessig Cofeece (EUSIPCO 6), Floece, Italy, Septembe 4-8, 6, copyight by EURASIP, F < ξ j ξ F ds jf(), σ j, σ Some impotat popeties ae summaized i the followig Diffeetiatio s σ jω tig + we may eite ( s s ), < s ( s), Φ s ( s) Φ( sds ) ( ) < ξ Φ R [ s] σ ( ) ( s ) j Φ, σ, σ σ Covolutio ξ( s a) ξ( s b) jξ s ( a+ b) [ ] Covolutio with a Odiay Fuctio ξ( s s ) F jf( s s ) Multiplicatio of a Impulse Times a Odiay Fuctio ξ s a F s F a ξ s a Multiplicatio by the th deivative of the Impulse Applyig the popety of the deivative times a odiay fuctio we obtai ξ () s Fs () F() ξ () s F () ξ() s Moe geeally we obtai ( ) k ( k) ( k) F() s ξ () s () F () ξ () s k k 5 APPLICATION TO DISCRETE-TIME ENERALISED IMPULSES The discete-time domai geealised impulse will be deoted by the symbol ψ ad is equivalet to the symbol ζ ( z ) used ealie [8], that is, ψ ζ ( z ) jφ (), < ψ, Φ z, If X ( z ) is aalytic at z the jf(), ψ ( zfzdz ), z Diffeetiatio < z, Φ z ψ z ( j( ) ) (), Φ, 6 DIFFERENTIAL AND DIFFERENCE EQUATIONS WITH TWO-SIDED FORCIN FUNCTIONS I what follows the steady state solutio of diffeetial ad diffeece equatios ae evaluated with o-causal ifiite duatio two sided fuctios, ie extedig fom to Example To fid the steady state solutio of the diffeetial equatio y + 3y + 3y + y t + 8 + Applyig the Laplace tasfom we have s 3 + 3s + 3s+ Y s 4πξ s + 6πξ s ie Y s πξ 4πξ + 6 3 ( s + 3s + 3s+ ) 4πξ + 6πξ s 3 ( s + 3s + 3s+ ) F 3 ( s + 3s + 3s+ ) ( s + ) 3 4π ξ + 6πξ 4π{ F( ) ξ F ( ) ξ } + 6πξ 4πξ + 4π ( 3) ξ + 6πξ ( s ) Y s F s s s + y t t Example To evaluate the steady state solutio of the diffeece equatio y[ ] by[ ] x[ ] with x[ ] a Applyig the z tasfom ( ) πψ ( πψ ( a z) Y ( Y z bz a z ) 3
4th Euopea Sigal Pocessig Cofeece (EUSIPCO 6), Floece, Italy, Septembe 4-8, 6, copyight by EURASIP F z π ( ψ Y z F z a z π F a a z ψ a z ψ, ( ba ) Hece + a y [ ] a b Example 3 To obtai the steady state solutio of the same diffeece equatio with x [ ] Applyig the z tasfom to both sides of the equatio we have Witig we have ( ) πψ Y z bz z z Y z F z πψ z ( π z ( ψ Y z F z z Now F z ψ z F ψ z F ψ z, π ( ( F Hece π π Y ψ + ψ, ( b) ( b) ( b) ( b) b b y [ ] + b b b 7 ENERALIZED SAMPLIN IMPULSE TRAIN eealized samplig impulse tais which gow o decay expoetially o as a special case ae the commo uifom costat level tai ca be tasfomed by the exteded Bilateal Laplace ad z tasfoms Fig shows the Laplace tasfom of a expoetially isig impulse tai The samplig theoem ca be eadily witte as a covolutio i the s plae ρ T () t δ ( t T ) be a samplig tai chose fo simplicity as a the usual uifom tai A sigal f ( t ), ideally sampled, is give by ρ fs t f t T t The sampled sigal spectum may be diectly witte i the fom π π f () t ρt () t F ξ s j π T T π f () t ρt () t F s j T T which is possible sice ow the Laplace tasfom of the impulse tai exists Similaly, samplig by a gowig o decayig impulse tai may be effectig i the s plae Fig The Laplace tasfom of a expoetially isig impulse tai 8 NEW EXTENDED TRANSFORMS Tables ad i the Appedix list basic ew Laplace ad z tasfoms As ca be see fom these tables, thaks to the geealised ew distibutios, the domais of existece of Laplace ad z tasfom ae exteded to fuctios that had to date o tasfom The Fouie tasfom, if it exists, ca be diectly obtaied as a special case fom the Laplace ad z tasfom, eve fo fuctios that lead to impulses o the imagiay axis 9 CONCLUSION Bilateal tasfoms domais ae exteded by the geealizatio of distibutios to complex vaiables I paticula the geealizatio of the Diac delta impulse i both Laplace ad z domais is show to exted these existece of these tasfoms ad thei applicatios to a lage class of two-sided sigals 4
4th Euopea Sigal Pocessig Cofeece (EUSIPCO 6), Floece, Italy, Septembe 4-8, 6, copyight by EURASIP ACKNOWLEDEMENT The autho wishes to ackowledge the geeous ivitatios to oe-yea visits of Jacob Beesty, éad Alegi, A Costatiidis, ad S Tzafestas to INRS-EMT Moteal, Uivesité de Nice, Impeial College, Lodo, Natioal Techical Uivesity of Athes, espectively, duig which pats of the peset wok wee caied out REFERENCES [] A Papoulis, The Fouie Itegal ad its Applicatios, Mcaw Hill, New Yok, 96 [] R N Bacewell, The Fouie Tasfom Ad Its Applicatios, Mcaw Hill, New Yok 986 [3] B Va de Pol, B ad H Bemme, Opeatioal Calculus, Cambidge Uivesity Pess, Cambidge, UK, 964 [4] A D Poulaikas, Edito-i Chief, The Tasfoms ad Applicatios Hadbook, CRC ad IEEE Pess,, Boca Rato, Floida [5] A V Qppeheim,, ad R W Schafe, Discete- Time Sigal Pocessig, Eglewood Cliffs, NJ, Petice Hall, 989 [6] M J Coithios,: A Weighted Z Spectum fo Mathematical Model Estimatio, IEEE Tas Comput Vol 45 No 5, pp53-58, May, 996 [7] M J Coithios, Laplace spectum fo expoetial decompositio ad pole zeo estimatio, IEE Poceedigs, Visio, Image ad Sigal Pocessig, pp 35-34, Octobe [8] M J Coithios, eealizatio of the Diac- Delta Impulse Extedig Laplace ad z Tasfom Domais, IEE Poc Visio, Image ad Sigal Pocessig, Vol 5 No, Apil 3, pp 69-8 [9] M J Coithios Complex-vaiable Distibutio theoy fo Laplace ad z tasfoms, IEE Poc Visio, Image ad Sigal Pocessig, Vol 5, No, Febuay 5, pp-97-6 ut () / s + πξ at e u() t /( s a) + πξ ( s a) αt e e αt cos( β t) π{[ ξ s ( α + jβ)] + ξ[ s ( α jβ)]} s α cos βt u( t) + πξ {[ s ( α+ jβ)] ( s α) + β + ξ[ s ( α jβ)]} t π dξ / ds t tut () ( ) ( ) πξ (!/ s + ) + ( ) πξ Table Exteded z Tasfoms of Basic Sequeces x[ ] Exteded z Tasfom X ( z ) πψ (z) a πψ ( z/ a) u [ ] z + πψ au [ ] ( z/ a) az + πψ Appedix Table Exteded Laplace Tasfom of Basic xc () t Fuctios Exteded Laplace Tasfom πξ at e π ξ ( s a) cosh( at ) πξ {[ s a] + ξ[ s+ a]} cosh( jβ t) π{ δω [ β] + ξω [ + β]} u [ ] ( ) i! ( ) Si (, ) z ( ) i+ z i + i i+ ( i ) + π ( ) S ( +, i) ψ + i+ ( i π ( ) S ( + ), i) ψ i i i 5