Poularikas A. D. Distributions, Delta Function The Handbook of Formulas and Tables for Signal Processing. Ed. Alexander D. Poularikas Boca Raton: CRC
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1 Pulrik A. D. Diribui, Del Fuci The Hbk f Frmul Tble fr Sigl Prceig. E. Aleer D. Pulrik Bc R: CRC Pre LLC, 999
2 5 Diribui, Del Fuci 5. Te Fuci 5. Diribui 5.3 Oe-Dimeil Del Fuci 5.4 Emple 5.5 Tw-Dimeil Del Fuci Referece 5. Te Fuci 5.. A Te Fuci ϕ( i rel-vlue fuci f he rel iepee vrible h c be iffereie rbirr umber f ime, which i ieicl zer uie fiie iervl. Emple Prperie f Te Fuci. If f( c be iffereie rbirril fe, ψ( f(ϕ( e fuci.. If f( i zer uie fiie iervl, ψ( f( τϕ( τ τ, < <, i e fuci. 3. A equece f e fuci, {ϕ (} <, cverge zer if ll ϕ re ieicll zer uie me iervl iepee f ech ϕ, well ll f i erivive, e uifrml zer. Emple 5. [ ] < ep /( ϕ(, e fuci 0 ϕ( ϕ ϕ(. 4. Te fuci belg e D, where D i lier vecr pce uch h if ϕ D ϕ D, he ϕ ϕ D ϕ D fr umber. 999 b CRC Pre LLC
3 5. Diribui 5.. Defiii A iribui (r geerlize fuci g( i prce f igig ur rbirr e fuci ϕ( umber N g [ϕ(]. A iribui i l fucil. Emple 5.3 implie h u( i iribui h ig umber ech ϕ( equl i re. 5.. Prperie. Lieri-hmgeei: u( τϕ ( ϕ( N [ ϕ(] f g([ ϕ ( ϕ (] g( ϕ ( g( ϕ (. Shifig: 3. Sclig: g ( ϕ( g ( ϕ( g ( ϕ ( g ( ϕ 4. Eve iribui: 5. O iribui: g ( ϕ( 0, ϕ( g ( ϕ( 0, ϕ( eve 6. Derivive: 7. h erivive: g( ϕ( ϕ( g ( g ( ϕ( ϕ( ( g ( 8. Pruc wih rir fuci: prvie h f( ϕ( belg he e f e fuci. 9. Cvlui: [ g ( f(] ϕ( g ([ f( ϕ(] g ( τ g( τ τ ϕ( g( τ g( τ ϕ( τ 999 b CRC Pre LLC
4 Defiii A equece f iribui {g (} i i cverge he iribui g( if fr ll ϕ belgig he e f e fuci. 0. Ever iribui i he limi, i he ee f iribui, f equece f ifiiel iffereible fuci.. If g ( g( r ( r( (r beig iribui, he umber, he g ( g(, g ( r ( g( r(, g ( g(. A iribui g( m be iffereie m ime eire. The erivive f iribui lw ei, i i iribui. 5.3 Oe-Dimeil Del Fuci 5.3. Defiii lim g ( ϕ( g( ϕ( δ( 0 0 δ( ϕ( ϕ( 0, ϕ( i eig fuci 5.3. Prperie TABLE 5. Prperie f Del Fuci Del Fuci Prperie δ( δ( δ δ( δ( δ δ( δ( δ( δ( ; δ( eve fuci δ( f( f( 0 δ( f( f( f( δ( f( 0 δ( 999 b CRC Pre LLC
5 TABLE 5. Prperie f Del Fuci (ciue f( δ( f( δ( δ( 0 Aδ( Aδ( A Del Fuci Prperie f( δ( cvlui f( τ δ( τ τ f( δ( δ( δ( τ δ( τ τ δ[ ( ] N N N N N N δ( T δ( T ( N δ( T δ( f( f ( 0 δ( f ( f( δ( f( 0 f( ( δ( f ( 0 δ( f( δ( f( 0 δ( δ( (! δ (, m m m δ( m! δ( ( m m m, > m!, m < 0 δ( δ( 0, fuci δ( f ( f( δ( f( ( k 0 δ( δ( δ( u ( k k k! f( 0 δ( k k k!( k! δ( δ( δ( (, i eve if i eve, if i. δ( (i δ( 999 b CRC Pre LLC
6 TABLE 5. Prperie f Del Fuci (ciue δ( u( u( δ( u( δ( g( δ( Del Fuci Prperie δ( r( δ[(] r zer f r(, 0 r( δ( δ[(] r r( r( r zer f (, 0, 0 r( r( δ(i δ( π δ( δ( δ( δ( [ δ( δ( ] / ε e δ( lim ε επ ω δ( lim i ω π δ( lim ε π ε ε δ( ε lim ε 0 π ( ε δ( cωω π f ( [ u ( ( u ( u ( ] δ( u ( ( δ( u ( δ( cmb ( δ( T, f ( cmb ( f ( T δ( T T T COMB ω cmb ω δ ω ω π ω ( F { T ( } ( T jω lim e ω πδ( 999 b CRC Pre LLC
7 TABLE 5. Prperie f Del Fuci (ciue Del Fuci Prperie lim ( j j ( e ω ω ω π δ ( lim ( j j e ω δ ω ω π The fllwig emple will elucie me f he el prperie he ue f he el fuci. 5.4 Emple Emple 5.4 Equivlece f eprei ivlvig he el fuci: (c i δ( δ( b c i δ( c c e δ( e δ( Emple 5.5 The vlue f he fllwig iegrl re: ( 4 5 δ( , ( c δ( e δ( k k [ ( ( ] k k Emple 5.6 The fir erivive f he fuci i: 6 ( u ( u( ( u ( u[ ( ] δ( δ( ([ u ( ]c ( c u( c i δ( c u( i ( u( i δ( u π u u u π π ( π i δ δ( π i ( π c π u u π δ [ ( π]c 999 b CRC Pre LLC
8 Emple 5.7 The vlue f he fllwig iegrl re: e δ( ( [ e i ] i ( ( 3 δ δ 3 δ( ( 3 ( 3 δ( 3 ( 3 ( ( 3 ( ( Emple 5.8 The vlue f he fllwig iegrl re: e 3 e δ( δ e δ e e e δ( 3 e δ[ ( 3] e δ( 3 e 5.5 Tw-Dimeil Del Fuci 5.5. Defiii δ(, δ( δ( δ(, δ( δ( 4 f(, ξηδ ( ξδ ( η ξη f(, A δ( δ( b p (, b, b he bur f A 5.5. Lie Me, A pa(, 0 herwie The fuci ϕ(δ( c be ierpree lie m he lie f ei ϕ(. Emple 5.9 p (δ( i lie m he -i wih ei e he -i frm. 999 b CRC Pre LLC
9 Emple 5.0 f(, δ( which i he prfile f f(, Lie M Curve α(, δ[α(,] i lie m he curve α(, 0 wih ei λ(, where α, α α α(, α Lie Me Alg - -Ae m α(, The lie me hve eiie lg he - -ireci give b α α m α(, m m α repecivel., α α α m m hece δ[α(,] δ( α(, 0 i he curve f α(,. α α Slui f α(, If we lve α(, 0 fr ee i h r wih i he we m regr δ[α(,] he lie m imilrl fr he lui Emple 5. If δ[ r ] he α(, r, α /, α /,, ± r,, ± r f(, ξη δ( ξ ξη f(, η η δα α δ α (, [ (, ] ( i, α i δα α δ α (, [ (, ] ( i, α. i δα [ (, ] δ( r δ( r r r δ( r δ( r, r. Al r δα [ (, ] δ( r δ( r r [ ] [ ] < [ ] Sice α he δ[α(,] δ(r r i rig el fuci wih ui ei α r lg r r b CRC Pre LLC
10 Emple 5. b If δ(α b c, he α(, α b c, α, α b, hece c, b c b b δ(α b c c b b c δ δ b Trfrmi f Crie fr δ( b c (ee Figure 5. cθ i θ, iθ c θ, b θ, k b b, c θ, i θ, ( b k k k /, ( b / k. b b b δ( b c δ c δ( k c δ( where c/ k. k k k FIGURE 5. Emple 5.3 f(,δ( b c k f b b, δ( k k where k b m b b The ei lg hi lie i f k,. k k c/ k The Fuci δ( b c, b c : Frm (5.5.5 b c b c δ( b c, b c δ δ b c b δ c b b c δ b c b b b bc δ b bc c b δ b c b δ( D, 999 b CRC Pre LLC
11 5.5.8 The fuci f(,δ( b c, b c f(,δ( b c, b c f(, δ(,. See (5.5.7 fr he vlue f D,,. D cmb( b c, b c cmb( b c, b c δ( b c δ( b c m m See (5.5.7 fr he vlue f D,, cmb( b, b 5.5. f(, cmb( b c, b c b bm m δ δ D D D D D m b bm m cmb( b, b D D D D δ δ D m b bm f cmb b c b c f D D D m (, (,, D D m b bm m δ δ D D D D 5.5. δ[α (,] δ[α (,] δα [ (, ] δα [ (, ] i δ( δ( α α i α α i where i, i re he crie f he iereci f he curve α (, α (,, Emple 5.4 (See Figure 5. α α(, α(, α (, α (,, α, α, α. δ[α (,] δ[α (,] δ( δ(. Ierec (, (,. α (,, α (,, α / /, α / /, α /, α / b CRC Pre LLC
12 Hece frm (5.5. δα [ (, ] δα [ (, ] [ δ( δ( δ( δ( ] δα [ (, ] δα [ (, ] δ( δ( δ( δ( fr <. FIGURE 5. Referece Gelf, I. M., e l., Geerlize Fuci, Vl. -6, Acemic Pre, New Yrk, NY Hki, R. F., Geerlize Fuci, Chicheer, Egl, 979. Lighhill, M. J., Iruci Furier Ali Geerlize Fuci, Cmbrige Uiveri Pre, New Yrk, NY, 959. Pulrik, A. D., Sigl em, i The Trfrm Applici Hbk, Eie b A. D. Pulrik, CRC Pre Ic., Bc R, Flri, b CRC Pre LLC
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