Fourier Transform represents a function
s
(
t
)
{\displaystyle s\left(t\right)}
as a "linear combination" of complex sinusoids at different frequencies
ω
{\displaystyle \omega \,}
. Fourier proposed that a function may be written in terms of a sum of complex sine and cosine functions with weighted amplitudes.
In Euler notation the complex exponential may be represented as:
e
j
ω
t
=
cos
(
ω
t
)
+
j
sin
(
ω
t
)
{\displaystyle e^{j\omega t}\,=\cos(\omega t)+j\sin(\omega t)}
Thus, the definition of a Fourier transform is usually represented in complex exponential notation.
The Fourier transform of s (t ) is defined by
S
(
ω
)
=
∫
−
∞
∞
s
(
t
)
e
−
j
ω
t
d
t
.
{\displaystyle S\left(\omega \right)=\int \limits _{-\infty }^{\infty }s\left(t\right)e^{-j\omega t}\,dt.}
Under appropriate conditions original function can be recovered by:
s
(
t
)
=
1
2
π
∫
−
∞
∞
S
(
ω
)
e
j
ω
t
d
ω
.
{\displaystyle s\left(t\right)={\frac {1}{2\pi }}\int \limits _{-\infty }^{\infty }S\left(\omega \right)e^{j\omega t}\,d\omega .}
The function
S
(
ω
)
{\displaystyle S\left(\omega \right)}
is the Fourier transform of
s
(
t
)
{\displaystyle s\left(t\right)}
. This is often denoted with the operator
F
{\displaystyle {\mathcal {F}}}
, in the case above,
S
(
ω
)
=
F
(
s
(
t
)
)
{\displaystyle S\left(\omega \right)={\mathcal {F}}\left(s(t)\right)}
The function
s
(
t
)
{\displaystyle s\left(t\right)}
must satisfy the Dirichlet conditions in order for
s
(
t
)
{\displaystyle s\left(t\right)}
for the integral defining Fourier transform to converge.
Forward Fourier Transform(FT)/Anaysis Equation
S
(
ω
)
=
∫
−
∞
∞
s
(
t
)
e
−
j
ω
t
d
t
.
{\displaystyle S\left(\omega \right)=\int \limits _{-\infty }^{\infty }s\left(t\right)e^{-j\omega t}\,dt.}
Inverse Fourier Transform(IFT)/Synthesis Equation
s
(
t
)
=
1
2
π
∫
−
∞
∞
S
(
ω
)
e
j
ω
t
d
ω
.
{\displaystyle s\left(t\right)={\frac {1}{2\pi }}\int \limits _{-\infty }^{\infty }S\left(\omega \right)e^{j\omega t}\,d\omega .}
Relation to the Laplace Transform [ edit | edit source ]
In fact, the Fourier Transform can be viewed as a special case of the bilateral Laplace Transform . If the complex Laplace variable s were written as
s
=
σ
+
j
ω
{\displaystyle s=\sigma +j\omega \,}
, then the Fourier transform is just the bilateral Laplace transform evaluated at
σ
=
0
{\displaystyle \sigma =0\,}
. This justification is not mathematically rigorous, but for most applications in engineering the correspondence holds.
×
Time Function
Fourier Transform
Property
1
z
(
t
)
=
x
(
t
)
±
y
(
t
)
{\displaystyle z(t)=x(t)\pm \ y(t)}
Z
(
ω
)
=
X
(
ω
)
±
Y
(
ω
)
{\displaystyle Z(\omega )=X(\omega )\pm \ Y(\omega )}
Linearity
2
Z
(
t
)
{\displaystyle Z(t)}
2
π
z
(
−
ω
)
{\displaystyle 2\pi z(-\omega )}
Duality
3
c
x
(
t
)
{\displaystyle c\,x(t)}
, c = constant
c
X
(
ω
)
{\displaystyle c\,X(\omega )}
Scalar Multiplication
4
d
x
(
t
)
d
t
{\displaystyle {\frac {dx(t)}{dt}}}
j
ω
X
(
ω
)
{\displaystyle j\omega \,X(\omega )}
Differentiation in time domain
5
∫
−
x
t
x
(
τ
)
d
τ
{\displaystyle \int \limits _{-x}^{t}x(\tau )d\tau }
X
(
ω
)
j
ω
{\displaystyle {\frac {X(\omega )}{j\omega }}}
, if
∫
−
∞
∞
x
(
t
)
d
t
=
0
{\displaystyle \int \limits _{-\infty }^{\infty }x(t)\,dt=0}
Integration in Time domain
6
t
x
(
t
)
{\displaystyle t\,x(t)}
j
d
X
(
ω
)
d
ω
{\displaystyle j\,{\frac {dX(\omega )}{d\omega }}}
Differentiation in Frequency Domain
7
x
(
−
t
)
{\displaystyle x(-\,t)}
X
(
−
ω
)
{\displaystyle X(-\,\omega )}
Time reversal
8
x
(
a
t
)
{\displaystyle x(a\,t)}
1
|
a
|
X
(
ω
a
)
{\displaystyle {\frac {1}{\left|a\right|}}X\left({\frac {\omega }{a}}\right)}
Time Scaling
9
x
(
t
−
a
)
{\displaystyle x(t\,-\,a)}
e
−
j
ω
a
X
(
ω
)
{\displaystyle e^{-\,j\omega \,a}\,X(\omega )}
Time shifting
10
x
(
t
)
cos
ω
0
t
{\displaystyle x(t)\cos {\omega _{0}\,t}}
1
2
[
X
(
ω
+
ω
0
)
+
X
(
ω
−
ω
0
)
]
{\displaystyle {\frac {1}{2}}\left[X(\omega \,+\,\omega _{0})\,+\,X(\omega \,-\,\omega _{0})\right]}
Modulation
11
x
(
t
)
sin
ω
0
t
{\displaystyle x(t)\sin {\omega _{0}\,t}}
1
2
j
[
X
(
ω
−
ω
0
)
−
X
(
ω
+
ω
0
)
]
{\displaystyle {\frac {1}{2j}}\left[X(\omega \,-\,\omega _{0})\,-\,X(\omega \,+\,\omega _{0})\right]}
Modulation
12
e
−
a
t
x
(
t
)
{\displaystyle e^{-\,a\,t}x(t)}
X
(
ω
+
a
)
{\displaystyle X(\omega \,+\,a)}
Frequency shifting
13
x
1
(
t
)
×
x
2
(
t
)
{\displaystyle x_{1}(t)\times \,x_{2}(t)}
1
2
π
∫
−
π
π
X
1
(
λ
)
X
2
(
ω
−
λ
)
d
λ
{\displaystyle {\frac {1}{2\pi }}\int \limits _{-\pi }^{\pi }X_{1}(\lambda )\,X_{2}(\omega \,-\lambda )\,d\lambda }
Convolution