Fourier Integral Examples
Fourier Integral Examples
In[8]:=
FourierTransform[UnitStep[t+1]-UnitStep[t-1],t,w]
Out[8]=
2 Sin[w]
--------
w
Recall that C(w) is 1/2 F(w).
In[9]:=
f[t_]:=1/Pi Integrate[Sin[w]/w Exp[I w t],{w,-Infinity,Infinity}]
In[10]:=
f[t]
Out[10]=
If[Im[1 - t] == 0 && Im[1 + t] == 0,
Pi (Sign[1 - t] + Sign[1 + t])
------------------------------, ComplexInfinity] / Pi
2
Do not be thrown off by the output. Look at the graph.
In[11]:=
Plot[Evaluate[f[t]], {t,-2,2}];
Does the fourier transform in Mathematica converge to the proper value? (Ignore the warning messages you see.)
In[12]:=
f[-1]
f[1]
Out[12]=
1 - 2
Out[13]=
1 - 2
In[14]:=
Let's look at the Fourier Integral representation of the function that is sin(t) from -3Pi to Pi and zero elsewhere.
In[15]:=
FourierTransform[(UnitStep[t+3 Pi]-UnitStep[t-Pi])Sin[t],t,w]
Out[15]=
-3 I Pi w
E 3/2 2
---------- - (Pi (--------------------------- +
2 2 2
-1 + w Sqrt[Pi] Sqrt[Pi (1 - w) ]
2 2
(2 Sqrt[Pi (1 - w) ]
2 2 1/4
(-1 + Sqrt[Pi] (Pi (1 - w) )
1
Sqrt[---------------------]
2 2
Pi Sqrt[Pi (1 - w) ]
2 2 5/2 2
Cos[Sqrt[Pi (1 - w) ]])) / (Pi (1 - w) )))\
3/2 2
/ 4 - (Pi (--------------------------- +
2 2
Sqrt[Pi] Sqrt[Pi (1 + w) ]
2 2
(2 Sqrt[Pi (1 + w) ]
2 2 1/4
(-1 + Sqrt[Pi] (Pi (1 + w) )
1
Sqrt[---------------------]
2 2
Pi Sqrt[Pi (1 + w) ]
2 2 5/2 2
Cos[Sqrt[Pi (1 + w) ]])) / (Pi (1 + w) )))\
I 3/2 1
/ 4 + (- Pi Sqrt[---------------------]
2 2 2
Pi Sqrt[Pi (1 - w) ]
2 2 2 2 1/4
Sin[Sqrt[Pi (1 - w) ]]) / (Pi (1 - w) ) -
I 3/2 1
(- Pi Sqrt[---------------------]
2 2 2
Pi Sqrt[Pi (1 + w) ]
2 2 2 2 1/4
Sin[Sqrt[Pi (1 + w) ]]) / (Pi (1 + w) )
Yuck! There is a valuable lesson to be learned here. It may be better to evaluate the Fourier transform from the definition instead of using the Mathematica command.
In[16]:=
F[w_]:=Integrate[Sin[t] Exp[- I w t],{t,-3 Pi, Pi}]
F[w]
Out[16]=
-I Pi w 3 I Pi w
E E
-(--------) + ---------
2 2
-1 + w -1 + w
In[17]:=
Simplify[ComplexExpand[F[w]]]
Out[17]=
I (2 Cos[Pi w] + 2 I Sin[Pi w]) Sin[2 Pi w]
-------------------------------------------
2
-1 + w
Recall that C(w) is 1/2 F(w).
In[18]:=
g[t_]:=1/Pi Integrate[1/2 F[w] Exp[I w t],{w,-Infinity,Infinity}]
In[19]:=
Plot[Evaluate[g[t]], {t,-6 Pi,3 Pi}];
Another valuable lesson. Look at the scale of your graph before you judge!
In[20]:=
Plot[Evaluate[g[t]], {t,-6 Pi,3 Pi},PlotRange->{-2,2}];
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