Continuous Fourier Transform

### 21.4.2 Continuous Fourier Transform

The Fourier transform of a function f is defined by

F(ω)= | ∫ |
+∞
−∞
| e−i ω t f(t) dt =F | ⎧ ⎨ ⎩ | f(t) | ⎫ ⎬ ⎭ | (ω),

where ω∈ℝ. The fourier command computes the Fourier transform by table lookup and application of transform properties.

- fourier takes one mandatory argument and several optional arguments:
  - expr, an expression which defines a function f(t)=expr.
  - Optionally, t, the variable for f (by default t).
  - Optionally, ω, the variable for the Fourier transform (by default omega).
  - Optionally, a sequence of arguments in form opt=bool, where bool is a boolean value and opt is one of the following:
    - int or integrate or Int or Integrate, for specifying whether symbolic integration should be attempted for unknown transform pairs (by default, this is disabled).
    - diff or derive, for specifying whether to attempt transforming the derivatives of unknown transform pairs (by default, this is disabled).
- fourier(expr ⟨,t,ω ⟩) returns the Fourier transform F(ω) of f(t).

The inverse Fourier transform, as its name implies, takes a Fourier transform F(ω) and returns the original function f(t). It is given by:

f(t)= |
1
2 π
|  | ∫ |
+∞
−∞
| ei ω t F(ω) dω=F−1 | ⎧ ⎨ ⎩ | F(ω) | ⎫ ⎬ ⎭ | (t)= |
1
2π
| F | ⎧ ⎨ ⎩ | F(ω) | ⎫ ⎬ ⎭ | (−t).

The ifourier command computes the inverse Fourier transform by using the above identity.

- ifourier takes one mandatory argument and several optional arguments:
  - expr, an expression which defines a function F(ω)=expr.
  - Optionally, ω, the variable for F (by default omega).
  - Optionally, t, the variable for the original function f (by default t).
  - Optionally, a sequence of arguments in form opt=bool, where bool is a boolean value and opt is one of the following:
    - int or integrate or Int or Integrate, for specifying whether symbolic integration should be attempted for unknown transform pairs (by default, this is disabled).
    - diff or derive, for specifying whether to attempt finding unknown transform pairs by transforming the derivative of the original (by default, this is disabled).
- ifourier(expr ⟨,ω,t ⟩) returns the inverse Fourier transform f(t) of F(ω).

The Fourier command is the inert form of fourier. It is used by fourier and ifourier when returning unknown transforms.

##### Transforming rational functions.

An arbitrary rational function can be transformed as long as its full partial fraction decomposition can be found.

#### Examples

Find the Fourier transform of f(t)=t/t3−19 t+30.

F:=fourier(t/(t^3-19t+30))

π | ⎛ ⎝ | 5 i e5 i ω−21 i e−3 i ω+16 i e−2 i ω | ⎞ ⎠ | sign | ⎛ ⎝ | ω | ⎞ ⎠
56

Indeed:

ifourier(F)

t
t3−19 t+30

Find the transform of f(t)=t2+1/t2−1.

F:=fourier((t^2+1)/(t^2-1))

2 π | ⎛ ⎝ | δ | ⎛ ⎝ | ω | ⎞ ⎠ | −sign | ⎛ ⎝ | ω | ⎞ ⎠ | sinω | ⎞ ⎠

Indeed:

ifourier(F)

t2+1
t2−1

Find the transform of f(t)=2/1+4t4.

F:=fourier(2/(1+4t^4))

π | ⎛ ⎜ ⎜ ⎝ | sin | ⎛ ⎜ ⎜ ⎝ |
⎪ ⎪ | ω | ⎪ ⎪
2
| ⎞ ⎟ ⎟ ⎠ | +cos | ⎛ ⎜ ⎜ ⎝ |
⎪ ⎪ | ω | ⎪ ⎪
2
| ⎞ ⎟ ⎟ ⎠ | ⎞ ⎟ ⎟ ⎠ | e |
− |
⎪ ⎪ | ω | ⎪ ⎪
2

Indeed:

ifourier(F)

2
4 t4+1

As a special case, you can transform all poylnomials. For example:

fourier(3x^2+2x+1,x,s)

2 π | ⎛ ⎝ | δ | ⎛ ⎝ | s | ⎞ ⎠ | +2 i δ | ⎛ ⎝ | s,1 | ⎞ ⎠ | −3 δ | ⎛ ⎝ | s,2 | ⎞ ⎠ | ⎞ ⎠

##### Transforming general functions.

A range of other (generalized) functions and distributions can be transformed, as demonstrated in the following examples.

#### Examples

fourier(Dirac(x-1)+Dirac(x+1),x)

2 cosω |  |  |  |  |  |  |  |  |  |

fourier(exp(-2*abs(x-1)),x,s)

4 e−i s
s2+4

fourier(BesselJ(3,x),x,s)

− |
s | ⎛ ⎝ | 4 s2−3 | ⎞ ⎠ | ⎛ ⎝ | −i sign | ⎛ ⎝ | s+1 | ⎞ ⎠ | +i sign | ⎛ ⎝ | s−1 | ⎞ ⎠ | ⎞ ⎠
√ |
−s2+1

F:=fourier(sin(x)*sign(x),x,s)

− |
2
s2−1

ifourier(F,s,x)

sign | ⎛ ⎝ | x | ⎞ ⎠ | sinx

fourier(log(abs(x)),x,s)

− |
π
⎪ ⎪ | s | ⎪ ⎪
| −2 π γ δ | ⎛ ⎝ | s | ⎞ ⎠

F:=fourier(abs(2t-3)^(-3/4))

Γ | ⎛ ⎜ ⎜ ⎝ |
1
4
| ⎞ ⎟ ⎟ ⎠ |
√
√ |
2
| +2
|  | ⎛ ⎜ ⎝ | ⎪ ⎪ | ω | ⎪ ⎪ |
1
4
| ⎞ ⎟ ⎠ |
3
| e |
− |
3 i ω
2
⎛ ⎜ ⎝ | 2 |
1
4
| ⎞ ⎟ ⎠ |
3
|  | ⎪ ⎪ | ω | ⎪ ⎪

ifourier(F)

1
⎛ ⎜ ⎝ | ⎪ ⎪ | −2 t+3 | ⎪ ⎪ |
1
4
| ⎞ ⎟ ⎠ |
3

fourier(rect(x),x,s)

2 sin | ⎛ ⎜ ⎜ ⎝ |
s
2
| ⎞ ⎟ ⎟ ⎠
s

fourier(exp(-abs(x))*sinc(x),x,s)

arctan | ⎛ ⎝ | s+1 | ⎞ ⎠ | −arctan | ⎛ ⎝ | s−1 | ⎞ ⎠

fourier(1/sqrt(abs(x)),x,s)

√ |
2
|  | √ |
π
√ |
⎪ ⎪ | s | ⎪ ⎪

F:=fourier(1/cosh(2x),x,s)

π
e |
− |
1
4
| π s
| +e |
1
4
| π s

ifourier(F,s,x)

2
e−2 x+e2 x

F:=fourier(Gamma(1+i*x/3),x,s)

6 π e |
− | ⎛ ⎝ | 3 s+e−3 s | ⎞ ⎠

ifourier(F,s,x)

Γ | ⎛ ⎜ ⎜ ⎝ |
1
3
| i x+1 | ⎞ ⎟ ⎟ ⎠

fourier(atan(x/4)/x,x,s)

π ugamma | ⎛ ⎝ | 0,4 | ⎪ ⎪ | s | ⎪ ⎪ | ⎞ ⎠

assume(a>0):; fourier(exp(-a*x^2+b),x,s)

√ |
a
|  | √ |
π
| e |
− |
s2
4 a
| +b
a

Piecewise functions can be transformed if defined as

piecewise(t<a1,f1,x<a2,f2,…,t<an,fn,f0)

where f0,…,fn are expressions and a1,a2,…,an are real numbers such that a1<a2<⋯<an . Inequalities may be strict or non-strict. For example:

f:=piecewise(t<-1,exp(t+1),t<1,1,exp(2-2t)):; F:=fourier(f)

−i ω sinω+3 ω cosω+4 sinω
ω | ⎛ ⎝ | ω−2 i | ⎞ ⎠ | ⎛ ⎝ | ω+i | ⎞ ⎠

ifourier(F)

θ | ⎛ ⎝ | −t+1 | ⎞ ⎠ | −θ | ⎛ ⎝ | −t−1 | ⎞ ⎠ | +θ | ⎛ ⎝ | t−1 | ⎞ ⎠ | e−2 t+2+θ | ⎛ ⎝ | −t−1 | ⎞ ⎠ | et+1

You can verify that the above expression coincides with f(t) by plotting them together with the plot command. Alternatively, you can apply piecewise to the result (see Section 5.1.3):

piecewise(ans())

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ |
et+1, | t<−1
1, | t<1
e |
− | ⎛ ⎝ | 2 t−2 | ⎞ ⎠
| ,
| otherwise

##### Convolution Theorem and applications.

The Fourier transform behaves nicely when applied to convolutions. Recall that the convolution f∗ g (see Section 21.2.5) of two functions f and g is defined by

(f∗ g)(t)= | ∫ |
+∞
−∞
| f(u) g(t−u) du.

The Convolution Theorem states that F(f∗ g)=F(f)·F(g).

As an example, we use this result for computing the convolution of f(t)=e−|t| with itself. Note that Xcas is unable to compute f∗ f from the definition:

f(t):=exp(-abs(t)):; int(f(u)*f(t-u),u=-inf..inf)

undef |  |  |  |  |  |  |  |  |  |

Using Fourier transform:

F:=fourier(f(t))

2
ω2+1

collect(ifourier(F^2))

| ⎛ ⎝ | ⎪ ⎪ | t | ⎪ ⎪ | +1 | ⎞ ⎠ | e |
− | ⎪ ⎪ | t | ⎪ ⎪

The above result is the desired convolution (f∗ f)(t)=∫−∞+∞f(u) f(t−u) du.