Fourier Transform
Properties
„Signal Theory” Zdzisław Papir
•Linearity
•Conjugate operation
•Amplitude and phase spectra
•Scaling
•Symmetry
•Shifting in time
•Shifting in frequency
•Modulation
•Convolution in time
•Signal ‘area’
Fourier Transform
Properties
„Signal Theory” Zdzisław Papir
•Linearity
•Conjugate operation
•Amplitude and phase spectra
•Scaling
•Symmetry
•Shifting in time
•Shifting in frequency
•Modulation
•Convolution in time
•Signal ‘area’
•Differentiation in time domain
•Integration in time domain
•Real and imaginary parts of a signal
•Limiting properties of Fourier transform
•Parseval and Rayleigh theorems
•Energy spectrum; fractional energy
„Signal Theory” Zdzisław Papir
Fourier Transform
Properties
Fourier transform properties
Basic assumptions
Y
t
y
X
t
x
„Signal Theory” Zdzisław Papir
LINEARITY
Y
X
t
y
t
x
t
y
t
x
t
y
t
x
F
F
F
F
F
Fourier transform properties
„Signal Theory” Zdzisław Papir
CONJUGATE OPERATION
*
*
X
t
x
For a real signal the formula holds:
*
*
,
X
X
t
x
t
x
t
x
R
Fourier transform properties
„Signal Theory” Zdzisław Papir
AMPLITUDE and PHASE SPECTRA
spectrum
phase
-
spectrum
amplitude
-
e
A
A
X
t
x
j
For a real signal the formula holds:
function
odd
an
is
spectrum
phase
-
function
even
an
is
spectrum
amplitude
-
A
A
Fourier transform properties
„Signal Theory” Zdzisław Papir
2
2
2
1
1
1
1
1
1
1
j
j
j
e
t
t
1
arctg
1
1
2
A
Fourier transform properties
AMPLITUDE and PHASE SPECTRA
„Signal Theory” Zdzisław Papir
frequency
amplitude spectrum
phase spectrum
SCALING
0
,
1
1
e
1
j
j
t
t
1
X
t
x
1
• Squeezing signal in the time domain broadens its
spectrum; extending a signal narrows its spectrum.
• The shorter is the signal the broader is its spectrum.
„Signal Theory” Zdzisław Papir
Fourier transform properties
SYMMETRY
Fourier transform properties
„Signal Theory” Zdzisław Papir
x
t
X
X
t
x
2
W
T
T
W
Wt
tT
T
T
T
t
2
Sa
2
2
/
Sa
2
Sa
W
T
2
W
W
Wt
2
Sa
SYMMETRY
x
t
X
X
t
x
2
x
t
X
d
e
x
d
e
x
t
X
d
e
x
t
X
t
dt
e
t
x
X
t
j
t
j
t
j
t
j
2
2
2
1
Fourier transform properties
„Signal Theory” Zdzisław Papir
SHIFTING in TIME
Change of a phase spectrum:
j
j
j
j
A
A
X
e
e
e
e
j
e
X
t
x
Fourier transform properties
„Signal Theory” Zdzisław Papir
SHIFTING in FREQUENCY
o
o
e
X
t
x
t
j
Fourier transform properties
„Signal Theory” Zdzisław Papir
MODULATION
o
o
o
o
o
o
o
2
1
cos
exp
exp
X
X
t
t
x
X
t
j
t
x
X
t
j
t
x
X(
)
X(
-
o
)/2
X(
+
o
)/2
-
o
+
o
Fourier transform properties
„Signal Theory” Zdzisław Papir
CONVOLUTION in TIME
d
t
y
x
t
y
t
x
PROPERTIES
Commutation
t
x
t
y
t
y
t
x
Association
t
z
t
y
t
x
t
z
t
y
t
x
Distribution with
addition
t
z
t
x
t
y
t
x
t
z
t
y
t
x
Fourier transform properties
„Signal Theory” Zdzisław Papir
Commutation
d
t
y
x
t
y
t
x
t
x
t
y
t
y
t
x
)
(
)
(
t
y
t
x
d
t
y
x
)
(
)
(
dz
d
z
t
t
x
t
y
dz
z
y
z
t
x
)
(
)
(
„Signal Theory” Zdzisław Papir
CONVOLUTION in TIME
d
t
y
x
t
y
t
x
„Signal Theory” Zdzisław Papir
Fourier transform properties
„Signal Theory” Zdzisław Papir
CONVOLUTION in TIME
expresses the amount of overlap
of one function x(τ) as it is shifted over another function y(τ).
0
1
x
1
0
2
y
1
0
-2
y
1
t
y
1
t
1
t
y
1
2
t
2
2
t
t
S
d
t
y
x
S
t - 2
t
0
S
Fourier transform properties
t
Convolution in time
„Signal Theory” Zdzisław Papir
dt
d
y
t
x
t
j
e
)
(
)
(
)
(
)
(
t
y
t
x
d
y
dt
e
t
x
t
j
)
(
)
(
dz
dt
z
t
d
y
dz
e
z
x
z
j
)
(
)
(
)
(
d
y
dz
e
z
x
z
j
)
(
)
(
)
(
d
y
dz
e
z
x
j
z
j
e
)
(
)
(
d
y
X
j
e
)
(
)
(
)
(
)
(
Y
X
)
(
)
(
t
y
t
x
Convolution in time
„Signal Theory” Zdzisław Papir
„Signal Theory” Zdzisław Papir
CONVOLUTION in FREQUENCY
d
Y
X
t
y
t
x
Y
X
t
y
t
x
2
1
2
1
Fourier transform properties
„Signal Theory” Zdzisław Papir
SIGNAL ‘AREA’ (dc component of the signal)
0
0
0
X
X
dt
e
t
x
dt
t
x
t
j
Fourier transform properties
W
W
dt
Wt
Wt
W
Wt
Wt
Wt
W
W
0
sin
sin
Sa
2
2
„Signal Theory” Zdzisław Papir
W
W
W
W
W
W
W
dt
Wt
Wt
sgn
0
,
0
,
sin
sgn
0
,
0
,
1
sin
sin
1
j
j
j
j
t
dt
t
t
j
dt
t
t
j
dt
t
e
t
t
j
F
F
j
t
j
t
2
sgn
sgn
1
„Signal Theory” Zdzisław Papir
DIFFERENTIATION in TIME
0
lim
,
t
x
X
j
dt
t
dx
X
t
x
t
X
j
dt
e
t
x
j
e
x
e
x
dt
e
t
x
j
e
t
x
dt
e
t
x
dt
t
dx
t
j
j
j
t
j
t
j
t
j
lim
lim
F
DIFFERENTIATION in TIME
emphasizes rapid changes
of signal level what is reflected in a relative „amplification”
of higher frequencies (responsible for these changes).
Fourier transform properties
„Signal Theory” Zdzisław Papir
INTEGRATION in TIME
j
X
X
d
x
X
t
x
t
0
If the signal does not contain a dc component,
X(
= 0) = 0, then:
j
X
d
x
X
t
x
t
Fourier transform properties
INTEGRATION in TIME
smooths rapid changes
of a signal level by a relative attenuation of high frequencies.
„Signal Theory” Zdzisław Papir
INTEGRATION in TIME
t
X
d
x
t
t
t
d
t
x
t
t
x
d
x
t
t
1
1
1
1
F
,
0
,
1
The proof of the property
INTEGRATION in TIME
is based on a convolution representation
of an integral.
0
,
1
0
,
0
t
t
t
1
Fourier transform properties
„Signal Theory” Zdzisław Papir
REAL and IMAGINARY PARTS of the SIGNAL
j
X
X
t
x
X
X
t
x
2
2
*
*
Im
Re
Fourier transform properties
„Signal Theory” Zdzisław Papir
EVEN and ODD SIGNALS
Fourier transform properties
R
R
dt
t
t
x
X
t
x
t
x
t
x
0
cos
2
,
even signal real Fourier transform
dt
t
t
x
j
X
t
x
t
x
t
x
0
sin
2
,
R
odd signal imaginary Fourier transform
„Signal Theory” Zdzisław
Papir
EVEN and ODD PARTS of the SIGNAL
Fourier transform properties
2
2
o
e
p
n
t
x
t
x
t
x
t
x
t
x
t
x
t
x
t
x
t
x
X
j
t
x
X
t
x
Im
Re
o
e
„Signal Theory” Zdzisław Papir
LIMITING PROPERTIES
of FOURIER TRANSFORM (Riemann)
0
lim
X
Decaying rate of Fourier transform (for T-pulses) is:
provided continuous derivatives exist
const
X
X
n
n
n
2
2
2
lim
1
lim
0
1
lim
Fourier transform properties
.
frequency
with
0
ansform
Fourier tr
2
2
T
x
T
x
n
,
1
4
4
sin
2
1
4
Sa
2
1
2
2
2
2
T
T
T
T
T
t
T
T/2
-T/2
t
T
T/2
-T/2
t
T
1
t
T
2
4
Sa
2
1
2
T
T
2
1
„Signal Theory” Zdzisław Papir
LIMITING PROPERTIES
of FOURIER TRANSFORM
„Signal Theory” Zdzisław Papir
LIMITING PROPERTIES
of FOURIER TRANSFORM
-5
-4
-3
-2
-1
0
1
2
3
4
5
0
+T/2
-T/2
Pulse „raised cosine”
3
2
2
2
1
~
2
2
sin
2
,
2
,
2
cos
1
2
1
T
T
X
T
T
t
t
T
t
x
T
T
PARSEVAL THEOREM
0
2
2
2
1
2
1
d
X
d
X
dt
t
x
E
t
x
R
i(t) = x(t)
u(t) = x(t)
E
dt
t
x
dt
t
i
t
u
E
2
R = 1
„Signal Theory” Zdzisław Papir
Fourier transform properties
PARSEVAL THEOREM
dt
t
x
2
)
(
dt
t
x
t
x
)
(
)
(
*
dt
d
e
X
t
x
t
j
)
(
2
1
)
(
*
d
dt
e
t
x
X
t
j
)
(
)
(
2
1
*
d
X
X
)
(
)
(
2
1
*
d
X
2
)
(
2
1
„Signal Theory” Zdzisław Papir
Marc-Antoine PARSEVAL (1755 - †1836)
Very little is known of Antoine Parseval's life.
Parseval had only
five publications, all presented to the Académie des
Sciences.
The second was Mémoire sur les séries et sur
l'intégration complète
d'une équation aux differences partielle linéaires du
second ordre,
à coefficiens constans dated 5 April 1799, contains the
result known
today as Parseval's theorem.
Parseval's result was not published until his five
papers were all
published by the Académie des Sciences in 1806.
Before that it was
known by members of the Academy and appeared in
works by Lacroix
and Poisson before Parseval's papers were printed.
Parseval was never honoured with election to
the Académie
des Sciences. He remains a somewhat shadowy figure
and it is hoped
that research will one day provide a better
understanding of his life
and achievements.
d
X
dt
t
x
2
2
2
1
(no picture available)
„Signal Theory” Zdzisław Papir
RAYLEIGH THEOREM
d
Y
X
dt
t
y
t
x
t
y
t
x
*
*
2
1
,
C
Rayleigh theorem is the Parseval theorem
stated for two different signals.
„Signal Theory” Zdzisław Papir
Fourier transform properties
FRACTIONAL ENERGY
ENERGY SPECTRAL DENSITY
2
2
2
1
1
,
X
S
X
dv
v
X
E
E
E
E
E
dv
v
X
E
dv
v
X
E
E
u
o
2
o
2
1
0
,
1
,
0
„Signal Theory” Zdzisław Papir
Fourier transform properties
2
Sa
T
T
t
T
FRACTIONAL ENERGY
„Signal Theory” Zdzisław Papir
Fourier transform properties
frequency f
rectangular pulse – fractional energy
amplitude spectrum
fractional energy
0
,
Sa
2
o
2
u
f
dv
v
f
E
fT
Summary
• In most cases we determine Fourier transforms using
previously proved properties and known Fourier
transform pairs.
It is not recommended to use the definition formulas
of the Fourier transform.
• Convolution and Parseval theorems are of most
considerable significance.
• Convolution concept is used for signal filtering description.
• Parseval theorem is a starting point for a spectral analysis
of random processes.