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This example show how to do FFT on a continouse time function, then do an IFFT to recover the original function. This is done in Matlab and Maple

1 Matlab

clear all; 
t = linspace(-pi,pi,100); 
dt = t(2)-t(1); 
fs = 1/dt; 
w = linspace(0,fs,100); 
 
y = sin(t) + 2*sin(3*t); 
g = exp(-w) .* fft(y); 
h = ifft(g); 
plot(real(h));

2 Maple

#showing how to do FFT and IFT in maple 10. Nasser M. Abbasi. 
restart; 
with(inttrans): 
with(LinearAlgebra): 
with(plots): 
N:=100: 
 
#the function to fft 
y:= t->sin(t) + 2*sin(3*t): 
 
linspace:=proc(fromP,toP,n) 
          local incr,data,i,T,L; 
          incr:=(toP-fromP)/(n-1); 
          L:=Vector(1..n,[]); 
          T:=fromP; 
          for i from 1 to n do 
              L[i]:=T; 
              T:=T+incr; 
          end do: 
          return(L); 
end proc: 
 
data:=Vector(1..N,[]): 
g:=Vector(1..N,[]): 
T:=linspace(-Pi,Pi,N): 
 
for i from 1 to N do 
    data[i]:=evalf(y(T[i])); 
end do: 
 
Y:=DiscreteTransforms:-FourierTransform( data,algorithm=DFT,padding=0 ): 
adj:=evalf(sqrt(1/N)): #adjust as maple has normalization factor 
Y:=Y/adj: 
 
dt:=T[2]-T[1]: 
fs:=1/dt: 
w:=evalf(linspace(0,fs,N)): 
s:=[seq(exp(-w[i]),i=1..N)]: 
g:=Vector(N,zip((x,y)->x*y,s,convert(Y,list))): 
h:=DiscreteTransforms:-InverseFourierTransform( g ): 
h:=h*sqrt(1/N):#adjust the IFFT due to maple normalization again 
 
listplot(map(Re,h),color=red,labels=["t","y(t)"],title="result of IFFT");

3 Source code download

1. Matlab fft_example.m
2. Maple maple.txt