(* Ft.m copyright 1989 by Victor W. Sparrow *) (* Permission is granted to copy the file provided that the copyright notice is left intact and the copies are not distributed commercially. *) (* The author wishes to thank Roman Maeder, George Swenson, and Mike White, who helped in the development of this package. Dr. Maeder for his help in the treatment of complex numbers in pattern matching, and Drs. Swenson and White for their encouragement and suggestions. *) BeginPackage["Ft`"] (* Version 9: added Fta[], Ftb[], iFta[], iFtb[], conversion functions for alternate transform definitions. 04/10/89 *) (* Version 8: fixed delay with optional scaling for forward transforms. Also some other rearrangements. 04/04/89 *) (* Version 7: fixed differentiation transforms and some other things on 03/27/89 *) (* Version 6: more inverse transforms and many improvements to forward ones 03/23/89 *) (* Version 5: started adding inverse transforms on 03/22/89 *) (* Version 4: more usage statements, and additional improvements to many functions. 03/20/89 *) (* Version 3: more forward transforms 03/03/89 *) (* In version 2 Vic Sparrow, the author of this package, had a little help on complex numbers from Roman Maeder. 02/28/89 *) Ft::usage = "Ft[expr,t,omega] gives the symbolic Fourier transform of expr in t space to omega space." iFt::usage = "iFt[expr,t,omega] gives the symbolic inverse Fourier transform of expr in omega space to t space." Fta::usage = "Fta[expr,t,omega] gives the symbolic Fourier transform of expr in t space to omega space. In this transform the 2 Pi appears in the exponent of the exponential function." iFta::usage = "iFta[expr,t,omega] gives the symbolic inverse Fourier transform of expr in omega space to t space. In this transform the 2 Pi appears in the exponent of the exponential function." Ftb::usage = "Ftb[expr,t,omega] gives the symbolic Fourier transform of expr in t space to omega space. This transform includes Sqrt[2 Pi] as a scaling factor" iFtb::usage = "iFtb[expr,t,omega] gives the symbolic inverse Fourier transform of expr in omega space to t space. This transform includes Sqrt[2 Pi] as a scaling factor." convolution::usage = "convolution[a,b] is the convolution integral of a and b." delta::usage = "delta[t] is the dirac delta function of t." u::usage = "u[t] is the unit step function of t, u[t]=0 for t<0 and u[t]=1 for t>0." rect::usage = "rect[t] is the rectagular function of t, rect[t]=1 for Abs[t]<1/2 and rect[t]=0 otherwise." sinc::usage = "sinc[t] is the function Sin[t]/t." triangle::usage = "triangle[t] is the triangular function of t, triangle[t]=1-Abs[t] for Abs[t]<1 and triangle[t]=0 otherwise." Begin["`private`"] (* Fourier Transform Section: *) (* constants *) Ft[c_,t_,om_]:= 2 Pi c delta[om] /; FreeQ[c,t] (* linearity *) Ft[a_ + b_,t_,om_]:= Ft[a,t,om] + Ft[b,t,om] (* pick off constants *) Ft[c_ a_,t_,om_]:= c Ft[a,t,om] /; FreeQ[c,t] (* frequency translation *) Ft[a_ Exp[c_Complex omn_. t_],t_,om_]:= Ft[a,t,om-Im[c]omn] /; (FreeQ[omn,t] && Re[c]==0) (* amplitude modulation *) Ft[a_ Cos[omn_. t_],t_,om_]:= Ft[a,t,om+omn]/2 + Ft[a,t,om-omn]/2 /; FreeQ[omn,t] (* time differentiation *) Unprotect[Derivative] Derivative/: Ft[Derivative[n_Integer?Positive][a_][t_],t_,om_]:= (I om)^n Ft[a[t],t,om] Protect[Derivative] (* time convolution *) Ft[convolution[a_,b_],t_,om_]:= Ft[a,t,om] Ft[b,t,om] (* complex conjugate *) Ft[Conjugate[a_],t_,om_]:= Conjugate[Ft[a,t,-om]] (* sign function *) Ft[Sign[t_],t_,om_]:= 2 / (I om) (* delta function *) Ft[delta[t_],t_,om_]:= 1 (* step function *) Ft[u[t_],t_,om_]:= Pi delta[om] - I/om (* complex exponential *) Ft[Exp[c_Complex omn_. t_],t_,om_]:= 2 Pi delta[om - Im[c]omn] /; (FreeQ[omn,t] && Re[c]==0) (* cosine *) Ft[Cos[omn_. t_],t_,om_]:= Pi(delta[om -omn] + delta[om+omn]) /; FreeQ[omn,t] (* sine *) Ft[Sin[omn_. t_],t_,om_]:= -I Pi(delta[om - omn] - delta[om+omn]) /; FreeQ[omn,t] (* step exponential *) Ft[Exp[c_?NumberQ con_. t_] u[t_],t_,om_]:= 1/(-c con + I om) /; (FreeQ[con,t] && c<0 && Im[c]==0) (* step exponential 2 *) Ft[t_ Exp[c_?NumberQ con_. t_] u[t_],t_,om_]:= 1/(-c con + I om)^2 /; (FreeQ[con,t] && c<0 && Im[c]==0) (* double sided exponential *) Ft[Exp[c_?NumberQ con_. Abs[t_]],t_,om_]:= -2 c con/((- c con)^2 + om^2) /; (FreeQ[con,t] && c<0 && Im[c]==0) (* Gaussian: c=-1/(2 sigma^2) sigma=Sqrt[-1/2c] assume c<0 and sigma>0 *) Ft[Exp[c_?NumberQ ccon_. t_^2],t_,om_]:= Sqrt[- Pi/(c ccon)] Exp[om^2/( 4 c ccon)] /; (FreeQ[ccon,t] && c<0 && Im[c]==0) (* Sign in freq. domain. *) Ft[Power[t_,-1],t_,om_]:= -I Pi Sign[om] /; FreeQ[c,t] (* Rectangular fn. *) Ft[rect[t_],t_,om_]:= sinc[om/2] (* sinc fn. *) Ft[sinc[t_],t_,om_]:= Pi rect[om/2] (* Triangle fn. *) Ft[triangle[t_],t_,om_]:=(sinc[om/2])^2 (* Very general rules should be last. *) (* frequency differentiation *) Ft[t_^n_. a_,t_,om_]:= I^n Derivative[n][ Ft[a,t,om] ][om] /; FreeQ[n,t] && n>0 (* multiplication by delta function *) Ft[a_ delta[t_ + tnot_.],t_,om_]:= Limit[a Exp[-I om t],t->-tnot] /; FreeQ[tnot,t] (* scaling *) Ft[a_[alph_ t_],t_,om_]:= (1/Abs[alph] Ft[a[t],t,om/alph] /; FreeQ[alph,t] ) (* delay with optional scaling *) Ft[a_[alph_. t_ + tnot_],t_,om_]:= (1/Abs[alph] Ft[a[t],t,om/alph] Exp[I tnot om / alph] /; FreeQ[alph,t] && FreeQ[tnot,t]) (* frequency convolution for 2 *) Ft[a_[t_] b_[t_],t_,om_]:= convolution[Ft[a[t],t,om],Ft[b[t],t,om]]/(2 Pi) (* Inverse Fourier Transform Section: *) (* constants *) iFt[c_,t_,om_]:= c delta[t] /; FreeQ[c,om] (* linearity *) iFt[a_ + b_,t_,om_]:= iFt[a,t,om] + iFt[b,t,om] (* pick off constants *) iFt[c_ a_,t_,om_]:= c iFt[a,t,om] /; FreeQ[c,om] (* one sided exponential *) iFt[1/(c_ + I om_),t_,om_]:= Exp[-c t] u[t] /; FreeQ[c,om] (* t and one sided exponential *) iFt[1/(c_ + I om_)^2,t_,om_]:= t Exp[-c t] u[t] /; FreeQ[c,om] (* double sided exponential *) iFt[1/(c_ + om_^2),t_,om_]:= ( Exp[-Sqrt[c] Abs[t]] /(2 Sqrt[c]) /; FreeQ[c,om] ) (* Gaussian *) (* c=-sigma^2/2) sigma=Sqrt[-2c] assume c<0 and sigma>0 *) iFt[Exp[c_?NumberQ ccon_. om_^2],t_,om_]:= Exp[t^2/(4 c ccon)]/(2 Sqrt[-Pi c ccon]) /; (FreeQ[ccon,om] && c<0 && Im[c]==0) (* Sign[] in time *) iFt[1/om_,t_,om_]:= I Sign[t]/2 (* Sign[] for frequency *) iFt[Sign[om_],t_,om_]:= I/(Pi t) (* delta function *) iFt[delta[om_],t_,om_]:= 1/( 2 Pi) (* sine: this could be improved *) iFt[delta[om_-omnot_]-delta[om_+omnot_],t_,om_]:= I Sin[omnot t]/Pi /; FreeQ[omnot,om] (* cosine: this could be improved *) iFt[delta[om_-omnot_]+delta[om_+omnot_],t_,om_]:= Cos[omnot t]/Pi /; FreeQ[omnot,om] (* sinc function *) iFt[sinc[om_],t_,om_]:= rect[t/2]/2 (* rect function *) iFt[rect[om_],t_,om_]:= sinc[t/2]/(2 Pi) (* sinc squared *) iFt[sinc[om_ tau_.]^2,t_,om_]:= triangle[t/(2 tau)]/(2 tau) /; FreeQ[tau,om] (* time differentiation *) iFt[om_^n_. a_,t_,om_]:= (-I)^n Derivative[n][ iFt[a,t,om] ][t] /; FreeQ[n,om] && n>0 (* frequency differentiation *) Unprotect[Derivative] Derivative/: iFt[Derivative[n_Integer?Positive][a_][om_],t_,om_]:= (t/I)^n iFt[a[om],t,om] Protect[Derivative] (* general rules last *) (* multiplication by delta function *) iFt[a_ delta[om_ + omnot_.],t_,om_]:= Limit[a Exp[I om t]/(2 Pi),om->-omnot] /; FreeQ[omnot,om] (* scaling *) iFt[a_[alph_ om_],t_,om_]:= ( iFt[a[om],t/alph,om]/Abs[alph] /; FreeQ[alph,om] ) (* frequency translation *) iFt[a_[om_ + omnot_],t_,om_]:= iFt[a[om],t,om] Exp[-I omnot t]/; FreeQ[omnot,om] (* time translation *) iFt[a_[om_] Exp[c_Complex tnot_. om_],t_,om_]:= iFt[a[om],t + Im[c] tnot, om] /; FreeQ[tnot,om] && Re[c]==0 (* frequency multiplication *) iFt[a_[om_] b_[om_],t_,om_]:= convolution[iFt[a[om],t,om],iFt[b[om],t,om]] (* Alternative definition transform section. *) (* Alternative a: 2Pi only in exponent of expoentials. *) Fta[a_,t_,om_]:=Ft[a,t,2 Pi om] iFta[a_,t_,om_]:=iFt[2 Pi a, 2 Pi t, om] (* Alternative b: Sqrt[2 Pi] appears in both forward and inverse trasforms. *) Ftb[a_,t_,om_]:=Ft[a,t,om]/Sqrt[2 Pi] iFtb[a_,t_,om_]:=Sqrt[2 Pi] iFt[a,t,om] End[] EndPackage[]