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
TeXTtxeXtxn
Ttjnn
n
tjnn 2, o
oo
sawtooth signal
time t
period T = 1
Limiting behaviourof Fourier series
„Signal Theory” Zdzisław Papir
x(t)
time t-T/2
xT(t)
Periodic extension of a signal window xT(t)through Fourier series
+T/2
txtx TT
Limiting behaviour of Fourier series
„Signal Theory” Zdzisław Papir
„Signal Theory” Zdzisław Papir
TejnT
TX Tn
2,1
1
11o
o
tettx 1
Limiting behaviour of Fourier series
„Signal Theory” Zdzisław Papir
TejnT
TX Tn
2,1
1
11o
o
0
02o
Tn
T
TX
T
Limiting behaviour of Fourier series
„Signal Theory” Zdzisław Papir
Limiting behaviour of Fourier series
04
1221
T
T
T
eTX
amplitude of the 1st spectrum line of an exponential puls
Fourier series window
04
1221
T
T
T
eTX
„Signal Theory” Zdzisław Papir
Limiting behaviour of Fourier series
222 4
1
nT
eTX
T
n
Fourier series window T
amplitude spectrum – exponential pulse
„Signal Theory” Zdzisław Papir
Limiting behaviour of Fourier series
222 4
1
nT
eTX
T
n
Fourier series window 3T
amplitude spectrum – exponential pulse
„Signal Theory” Zdzisław Papir
Limiting behaviour of Fourier series
222 4
1
nT
eTX
T
n
amplitude spectrum – exponential pulse
Fourier series window 10T
„Signal Theory” Zdzisław Papir
Limiting behaviour of Fourier series
222 4
1
nT
eTX
T
n
amplitude spectrum – exponential pulse
Fourier series window 100T
„Signal Theory” Zdzisław Papir
j
Xj
eTTX T
n
T
n
1
1
1
1
Squeezing Fourier series coefficients in FREQUENCY:
n
Tn
T
n j
eTTX
jn
eTTX
n
n
1
1
1
1o
o
Limiting behaviour of Fourier series
Squeezing Fourier series coefficients in AMPLITUDE:
„Signal Theory” Zdzisław Papir
Riemann integral
a b
x
f(x)
nx
nxf
b
an
x
n
nn dxxfxxfS n 0max
Limiting form of Fourier series
„Signal Theory” Zdzisław Papir
Fourier series coefficients:
2
2
2
2
o1
T
T
tj
T
T
tjnn
dtetxTTX
dtetxT
TX
n
n
dtetxXTTX tj
T n
lim
FORWARD FOURIER TRANSFORM:
Limiting form of Fourier series
„Signal Theory” Zdzisław Papir
Fourier series:
deXtxtx tjT
T
1lim
INVERSE FOURIER TRANSFORM:
n
n
n
tjT
n
tjnn
n
tjnnT
eTTXtx
eTTXeTXtx
2
1
2
1o
oo
„Signal Theory” Zdzisław Papir
n
Tt
t
tj
n
tjnTt
t
jn
n
tjnTt
t
jn
dextx
edextx
edexTtx
n0
0
00
0
0
00
0
0
2
1
2
1
1
0
n
tjnneXtx 0)(
Tt
t
tjnn dtetXTX
0
0
0)(1
Fourier Integral Theorem
„Signal Theory” Zdzisław Papir
Fourier transform
n
Tt
t
tj dextx n0
02
1
ddextx tj
2
1
dtetxX tj
dedextx tjj
2
1
Fourier integral theorem
ForwardFouriertransform
„Signal Theory” Zdzisław Papir
Inverse Fourier transform
ddexetx jtj
2
1
dtetxX tj
InverseFouriertransform
deXtx tj
2
1
Fourier transform pairs
„Signal Theory” Zdzisław Papir
TRANSFORM
dtetxX tj
deXtx tj1
INVERSEFORWARD
TRANSFORMPAIRS
Xtx
Xtx
txX1
FF
Fourier transform pairs
„Signal Theory” Zdzisław Papir
jdtedteeX
ette
ttx
tjtjt
tt
1
1
0,
0,0
o
1
o
1
FORWARD FOURIER TRANSFORM:
j
et t
1
11
Fourier transform pairs
„Signal Theory” Zdzisław Papir
j
et t
1
11
FOURIER TRANSFORM:
tettx 1
time t
jdtedteeX
ette
ttx
tjtjt
tt
1
1
0,
0,0
o
1
o
1
Fourier transform pairs
„Signal Theory” Zdzisław Papir
xxxT
TdteX
tTt
Tttx
T
T
tj
T
sinSa,2
Sa
2,1
2,0
2
2-
2
SaT
TtT
FOURIER TRANSFORM:
T/2-T/2
1 tT
fTT
tx Sa
2Sa
frequency f
„Signal Theory” Zdzisław Papir
xxxT
TdtetX
tTtTt
Tttx
tjT
T
sinSa,4
Sa2
1
2,21
2,0
2
-
4
Sa2
1 2 TTtT
FOURIER TRANSFORM:
T/2-T/2
tT
Fourier transform pairs
2
Sa4
Sa 22 fTTtx
frequency f
„Signal Theory” Zdzisław Papir
Existence of Fourier transformDirichlet conditions are necessaryfor 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:
dttx
„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.