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TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
1
Transformasi Fourier
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
2
What do we hope to achieve with theFourier Transform?We desire a measure of the frequencies present in a wave. the spectrum
It will be nice if our measure also tells us when each frequency occurs.
Ligh
t el
ectr
ic f
ield
Time
Plane waves have only one frequency, w.
This light wave has many frequencies. And the frequency increases in time (from red to blue).
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
3
Fourier Cosine Series
f( t) 1
Fm cos(mt)
m0
•Because cos(mt) is an even function (for all m), we can write an even function, f(t), as:
• where the set {Fm; m = 0, 1, … } is a set of coefficients that define the series. • And where we’ll only worry about the function f(t) over the interval (–π,π).
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
4
The Kronecker delta function
,
1 if
0 if m n
m n
m n
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
5
Finding the coefficients, Fm, in a Fourier Cosine Series
Fourier Cosine Series: To find Fm, multiply each side by cos(m’t), where m’ is another integer, and integrate:
But since:
0
1( ) cos( )m
m
f t F mt
f (t) cos(m' t) dt
1
m0
Fm cos(mt) cos(m' t) dt
, '
'cos( ) cos( ' )
0 ' m m
if m mmt m t dt
if m m
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
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Finding the coefficients, Fm, in a Fourier Cosine Series_2
So: only the m’ = m term contributes
Dropping the ‘ from the m: yields the
coefficients for any f(t)!
, '
0
1( ) cos( ' ) m m m
m
f t m t dt F
( ) cos( )mF f t mt dt
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
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Fourier Sine Series
•Because sin(mt) is an odd function (for all m), we can write any odd function, f(t), as:
•where the set {F’m; m = 0, 1, … } is a set of coefficients that define the series. •where we’ll only worry about the function f(t) over the interval (– π,π).
f (t) 1
F m sin(mt)
m0
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
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Finding the coefficients, F’m, in a Fourier Sine Series
•Fourier Sine Series:
•To find F’m, multiply each side by sin(m’t), where m’ is another integer, and integrate:
•But since:
f (t) 1
F m sin(mt)
m0
0
1( ) sin( ' ) sin( ) sin( ' )m
m
f t m t dt F mt m t dt
, '
'sin( ) sin( ' )
0 ' m m
if m mmt m t dt
if m m
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
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Finding the coefficients, F’m, in a Fourier Sine Series_2
• So: only the m’
= m term contributes
• Dropping the ‘ from the m:
yields the coefficients
for any f(t)!
, '
0
1( ) sin( ' ) m m m
m
f t m t dt F
( ) sin( )mF f t mt dt
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
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Fourier Series
even component odd component
where
and
0 0
1 1( ) cos( ) sin( )m m
m m
f t F mt F mt
Fm f (t) cos(mt) dt F m f (t) sin(mt) dt
So if f(t) is a general function, neither even nor odd, it can be written:
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
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Fourier Transform
f (t) cos(mt) dt i f (t) sin(mt) dtF(m) º Fm – i F’m =
Let’s now allow f(t) to range from –∞ to ∞, so we’ll have to integrate from –∞ to ∞, and let’s redefine m to be the “frequency,” which we’ll now call ω :
( ) ( ) exp( )F f t i t dt
The FourierTransform
F(ω) is called the Fourier Transform of f(t). It contains equivalent information to that in f(t). We say that f(t) lives in the “time domain,” and F(ω) lives in the “frequency domain.” F(ω) is just another way of looking at a function or wave.
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
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The Inverse Fourier Transform
The Fourier Transform takes us from f(t) to F(w). How about going back?
Recall our formula for the Fourier Series of f(t) :
Now transform the sums to integrals from –∞ to ∞, and again replace Fm with F(ω). Remembering the fact that we introduced a factor of i (and including a factor of 2 that just crops up), we have:
'
0 0
1 1( ) cos( ) sin( )m m
m m
f t F mt F mt
1( ) ( ) exp( )
2f t F i t d
Inverse Fourier
Transform
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
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The Fourier Transform and its Inverse
So we can transform to the frequency domain and back. Interestingly, these functions are very similar.
There are different definitions of these transforms. The 2π
can occur in several places, but the idea is generally the same.
Inverse Fourier Transform
FourierTransform ( ) ( ) exp( )F f t i t dt
1
( ) ( ) exp( )2
f t F i t d
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
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Fourier Transform Notation
There are several ways to denote the Fourier transform of a function. If the function is labeled by a lower-case letter, such as f, we can write:
f(t) ® F(w) If the function is labeled by an upper-case letter, such as E, we can write:
or:
( ) ( )E t E ( ) { ( )}E t E t F
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
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The Spectrum
We define the spectrum of a wave E(t) to be:
2{ ( )}E tF
This is our measure of the frequencies present in a light
wave.
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
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Example: the Fourier Transform of arectangle function: rect(t)
( sinc(F Imaginary Component = 0
F(w)
w
1/ 21/ 2
1/ 2
1/ 2
1( ) exp( ) [exp( )]
1[exp( / 2) exp(
exp( / 2) exp(
2
sin(
F i t dt i ti
i ii
i i
i
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
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Sync(x) and why it's important
• Sinc(x/2) is the Fourier transform of a rectangle function.
• Sinc2(x/2) is the Fourier transform of a triangle function.
• Sinc2(ax) is the diffraction pattern from a slit.
It just appears everywhere...
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
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The Fourier Transform of the trianglefunction, D(t), is sinc2( /w )
w0
2sinc ( / 2)1
t0
( )t1
1/2-1/2∩
We’ll prove this when we learn about convolution.
Sometimes people use L(t), too, for the triangle
function.
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
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Example: the Fourier Transform of adecaying exponential: exp(-at) (t > 0)
0
0 0
0
( exp( )exp( )
exp( ) exp( [ )
1 1exp( [ ) [exp( ) exp(0)]
1[0 1]
1
F at i t dt
at i t dt a i t dt
a i ta i a i
a i
a i
1(F i
ia
A complex Lorentzian!
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
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Example: the Fourier Transform of aGaussian, exp(-at2) is itself!
2 2
2
{exp( )} exp( )exp( )
exp( / 4 )
at at i t dt
a
F
t0
2exp( )at
w0
2exp( / 4 )a∩
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
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Existence of Fourier Transform
Some functions don’t have Fourier transforms.The condition for the existence of a given F(w) is:
Functions that do not asymptote to zero in both the +¥
and –¥ directions generally do not have Fourier transforms.
So we’ll assume that all functions of interest go to zero at ±∞.
( )f t dt
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution
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Fourier Transform Symmetry Properties
Expanding the Fourier transform of a function, f(t):
Expanding further:
( ) [Re{ ( )} Im{ ( )}] [cos( ) sin( )]F f t i f t t i t dt
( ) Re{ ( )} cos( ) Im{ ( )} sin( )F f t t dt f t t dt
¬Re{F(w)}
¬Im{F(w)}
¯ ¯
= 0 if Re or Im{f(t)} is odd = 0 if Re or Im{f(t)} is even
Even functions of w Odd functions of w
Im{ ( )} cos( ) Re{ ( )} sin( )i f t t dt i f t t dt