Fourier
Transform
„Signal Theory” Zdzisław Papir
•Periodicity of Fourier series
•Limiting behaviour of Fourier series
•Limiting form of Fourier series
•Fourier transform pairs
•Existence of Fourier transform
Fourier
Transform
„Signal Theory” Zdzisław Papir
•Periodicity of Fourier series
•Limiting behaviour of Fourier series
•Limiting form of Fourier series
•Fourier transform pairs
•Existence of Fourier transform
Periodicity of Fourier
series
„Signal Theory” Zdzisław Papir
T
e
X
T
t
x
e
X
t
x
n
T
t
jn
n
n
t
jn
n
2
,
o
o
o
sawtooth signal
time t
period T = 1
Limiting behaviour
of Fourier series
„Signal Theory” Zdzisław
Papir
x(t)
time t
-T/2
x
T
(t)
Periodic extension of a signal window x
T
(t)
through Fourier series
+T/2
t
x
t
x
T
T
Limiting behaviour of Fourier
series
„Signal Theory” Zdzisław Papir
„Signal Theory” Zdzisław Papir
T
e
jn
T
T
X
T
n
2
,
1
1
1
1
o
o
t
e
t
t
x
1
Limiting behaviour of Fourier
series
„Signal Theory” Zdzisław Papir
T
e
jn
T
T
X
T
n
2
,
1
1
1
1
o
o
0
0
2
o
T
n
T
T
X
T
Limiting behaviour of Fourier
series
„Signal Theory” Zdzisław Papir
Limiting behaviour of Fourier
series
0
4
1
2
2
1
T
T
T
e
T
X
amplitude of the 1st spectrum line of an exponential puls
Fourier series window
0
4
1
2
2
1
T
T
T
e
T
X
„Signal Theory” Zdzisław Papir
Limiting behaviour of Fourier
series
2
2
2
4
1
n
T
e
T
X
T
n
Fourier series
window T
amplitude spectrum –
exponential pulse
„Signal Theory” Zdzisław Papir
Limiting behaviour of Fourier
series
2
2
2
4
1
n
T
e
T
X
T
n
Fourier series
window 3T
amplitude spectrum –
exponential pulse
„Signal Theory” Zdzisław Papir
Limiting behaviour of Fourier
series
2
2
2
4
1
n
T
e
T
X
T
n
amplitude spectrum –
exponential pulse
Fourier series
window 10T
„Signal Theory” Zdzisław Papir
Limiting behaviour of Fourier
series
2
2
2
4
1
n
T
e
T
X
T
n
amplitude spectrum – exponential pulse
Fourier series window
100T
„Signal Theory” Zdzisław Papir
j
X
j
e
T
TX
T
n
T
n
1
1
1
1
Squeezing Fourier series coefficients in FREQUENCY:
n
T
n
T
n
j
e
T
TX
jn
e
T
TX
n
n
1
1
1
1
o
o
Limiting behaviour of Fourier
series
Squeezing Fourier series coefficients in AMPLITUDE:
„Signal Theory” Zdzisław Papir
Riemann integral
a
b
x
f(x
)
n
x
n
x
f
b
a
n
x
n
n
n
dx
x
f
x
x
f
S
n
0
max
Limiting form of Fourier
series
„Signal Theory” Zdzisław Papir
Fourier series coefficients:
2
2
2
2
o
1
T
T
t
j
T
T
t
jn
n
dt
e
t
x
T
TX
dt
e
t
x
T
T
X
n
n
dt
e
t
x
X
T
TX
t
j
T
n
lim
FORWARD FOURIER TRANSFORM:
Limiting form of Fourier series
„Signal Theory” Zdzisław Papir
Fourier series:
d
e
X
t
x
t
x
t
j
T
T
1
lim
INVERSE FOURIER TRANSFORM:
n
n
n
t
j
T
n
t
jn
n
n
t
jn
n
T
e
T
TX
t
x
e
T
TX
e
T
X
t
x
2
1
2
1
o
o
o
„Signal Theory” Zdzisław Papir
n
T
t
t
t
j
n
t
jn
T
t
t
jn
n
t
jn
T
t
t
jn
d
e
x
t
x
e
d
e
x
t
x
e
d
e
x
T
t
x
n
0
0
0
0
0
0
0
0
0
0
2
1
2
1
1
0
n
t
jn
n
e
X
t
x
0
)
(
T
t
t
t
jn
n
dt
e
t
X
T
X
0
0
0
)
(
1
Fourier Integral Theorem
„Signal Theory” Zdzisław Papir
Fourier transform
n
T
t
t
t
j
d
e
x
t
x
n
0
0
2
1
d
d
e
x
t
x
t
j
2
1
dt
e
t
x
X
t
j
d
e
d
e
x
t
x
t
j
j
2
1
Fourier integral theorem
Forward
Fourier
transform
„Signal Theory” Zdzisław Papir
Inverse Fourier transform
d
d
e
x
e
t
x
j
t
j
2
1
dt
e
t
x
X
t
j
Inverse
Fourier
transform
d
e
X
t
x
t
j
2
1
Fourier transform pairs
„Signal Theory” Zdzisław Papir
TRANSFORM
dt
e
t
x
X
t
j
d
e
X
t
x
t
j
1
INVERSE
FORWARD
TRANSFORM
PAIRS
X
t
x
X
t
x
t
x
X
1
F
F
Fourier transform pairs
„Signal Theory” Zdzisław Papir
j
dt
e
dt
e
e
X
e
t
t
e
t
t
x
t
j
t
j
t
t
t
1
1
0
,
0
,
0
o
1
o
1
FORWARD FOURIER TRANSFORM:
j
e
t
t
1
1
1
Fourier transform pairs
„Signal Theory” Zdzisław Papir
j
e
t
t
1
1
1
FOURIER TRANSFORM:
t
e
t
t
x
1
time t
j
dt
e
dt
e
e
X
e
t
t
e
t
t
x
t
j
t
j
t
t
t
1
1
0
,
0
,
0
o
1
o
1
Fourier transform pairs
„Signal Theory” Zdzisław Papir
x
x
x
T
T
dt
e
X
t
T
t
T
t
t
x
T
T
t
j
T
sin
Sa
,
2
Sa
2
,
1
2
,
0
2
2
-
2
Sa
T
T
t
T
FOURIER TRANSFORM:
T/2
-T/2
1
t
T
fT
T
t
x
Sa
2
Sa
frequency f
„Signal Theory” Zdzisław Papir
x
x
x
T
T
dt
e
t
X
t
T
t
T
t
T
t
t
x
t
j
T
T
sin
Sa
,
4
Sa
2
1
2
,
2
1
2
,
0
2
-
4
Sa
2
1
2
T
T
t
T
FOURIER TRANSFORM:
T/2
-T/2
t
T
Fourier transform pairs
2
Sa
4
Sa
2
2
fT
T
t
x
frequency f
„Signal Theory” Zdzisław Papir
Existence of Fourier
transform
Dirichlet conditions are
necessary
for Fourier transform
existence.
• Signal x(t) must have only a finite number of maxima
and minima, as well as a finite number of discontinuities
over the entire range [–, + ].
• Signal x(t) is also allowed to be unbounded provided that
it is absolutely integrable:
dt
t
x
„Signal Theory” Zdzisław Papir
Summary
•
Fourier series is a spectral decomposition of periodic signal or
produces a periodic extension of signal window.
•
Fourier transform is a tool for spectral decomposition of
nonperiodic signals.
•
Fourier transform is a limiting case of Fourier series with signal
window being extended up to infinity.
•
Dirichlet conditions are necessary for Fourier transform
existence.
•
In engineering applications it is commonly assumed that signals
of limited energy are Fourier transformable.