This presentation describes the Fourier Transform used in different mathematical and physical applications.
The presentation is at an Undergraduate in Science (math, physics, engineering) level.
Please send comments and suggestions to improvements to solo.hermelin@gmail.com.
More presentations can be found at my website http://www.solohermelin.com.
4. Fourier Transform
( ){ } ( ) ( ) ( )
=
−=−= ∫∫
+∞
∞−
=
+∞
∞−
a
F
aa
d
a
jfdttjtaftaf
ta
ωτ
τ
ω
τω
τ
1
expexp:F
( ) ( ){ } ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ){ }ωωωωω
ω
ωω FjdttjjtfF
d
d
dttjtftfF
nn
n
n
−=−−=→−== ∫∫
+∞
∞−
+∞
∞−
FF expexp:
SOLO
( )tf
-1
F
F
( )ωFProperties of Fourier Transform
Scaling4
Derivatives5
Proof:
( )taf
-1
F
F
a
F
a
ω1
Proof:
Corollary: for a = -1
( )tf −
-1
F
F
( )ω−F
( ) ( )tftj
n
−
-1
F
F ( )ω
ω
F
d
d
n
n
( )tf
td
d
n
n
-1
F
F
( ) ( )ωω Fj
n
( ) ( ){ } ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ){ }ωω
π
ω
ωωω
π
ω
ωωω Fj
d
tjjFtf
td
dd
tjFFtf
nn
n
n
1-1-
FF ==→== ∫∫
+∞
∞−
+∞
∞−
2
exp
2
exp
5. Fourier TransformSOLO
( )tf
-1
F
F
( )ωFProperties of Fourier Transform
Convolution6
Proof:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )ωωωττωττωτωτ
ττωττωττττωτττ
τ
212121
212121
expexpexp
expexpexp:
FFFdjfdduujufjf
ddttjtfjfdtdtfftjdtff
ut
=
−=
−−=
−−−−=
−−=
−
∫∫ ∫
∫ ∫∫ ∫∫
∞+
∞−
∞+
∞−
∞+
∞−
=−
+∞
∞−
+∞
∞−
+∞
∞−
+∞
∞−
+∞
∞−
F
( ) ( )tftf 21
-1
F
F ( ) ( )ωω 21
* FF( ) ( ) ( ) ( )∫
+∞
∞−
−= τττ dtfftftf 2121 :*
-1
F
F ( ) ( )ωω 21
FF
The animations above graphically illustrate the convolution of two rectangle functions (left) and two
Gaussians (right). In the plots, the green curve shows the convolution of the blue and red curves as a
function of t, the position indicated by the vertical green line.
The gray region indicates the product as a function of g (τ) f (t-τ) , so its area as a function of t is
precisely the convolution.
http://mathworld.wolfram.com/Convolution.html
7. Signal Duration and BandwidthSOLO
( )tf
-1
F
F
( )ωFRelationships from Parseval’s Formula
( ) ( ) ( ) ( )∫∫
+∞
∞−
+∞
∞−
= ωωω
π
dFFdttftf 2
*
12
*
1
2
1
Parseval’s Formula7
( ) ( ) ,2,1,0
2
1
2
22
== ∫∫
∞+
∞−
∞+
∞−
nd
d
Sd
dttst m
m
m
ω
ω
ω
π
Choose ( ) ( ) ( ) ( )tstjtftf
m
−== 21 ( ) ( )tftj
n
−
-1
F
F ( )ω
ω
F
d
d
n
n
and use 5a
Choose ( ) ( ) ( )
n
n
td
tsd
tftf == 21 and use 5b
( )tf
td
d
n
n
-1
F
F
( ) ( )ωω Fj
n
( ) ( ) ,2,1,0
2
1 22
2
== ∫∫
∞+
∞−
∞+
∞−
ndSdt
td
tsd m
n
n
ωωω
π
( ) ( ) ( ) ( ) ( ) ( ) ,2,1,0,,2,1,0
2
* ==
= ∫∫
+∞
∞−
+∞
∞−
mnd
d
Sd
S
j
dt
td
tsd
tstj m
m
n
n
n
n
mm
ω
ω
ω
ωω
π
Choosec ( ) ( )
n
n
td
tsd
tf =1
( ) ( ) ( )tstjtf
m
−=2
8. Fourier TransformSOLO
( )tf
-1
F
F
( )ωFProperties of Fourier Transform
Modulation9
Shifting: for any a real8
Proof:
( ) ttf 0cos ω -1
F
F
( ) ( )[ ]00
2
1
ωωωω −++ FF
Proof:
( ) ( )[ ]tjtjt 000 expexp
2
1
cos ωωω −+=
( )atf −
-1
F
F ( ) ( )ωω ajF −exp ( ) ( )tajtf exp
-1
F
F ( )aF −ω
( ){ } ( ) ( ) ( ) ( )( ) ( ) ( )ωωττωτω
τ
Fajdajfdttjatfatf
at
−=+−=−−=− ∫∫
+∞
∞−
=−
+∞
∞−
expexpexp:F
( ) ( ){ } ( ) ( ) ( ) ( ) ( )( ) ( )aFdttajtfdttjtajtftajtf −=−−=−= ∫∫
+∞
∞−
+∞
∞−
ωωω expexpexp:expF
use shifting property with a=±ω0
9. ( )atf −
-1
F
F ( ) ( )ωω ajF −exp
Fourier TransformSOLO
( )tf
-1
F
F
( )ωFProperties of Fourier Transform (Summary)
Linearity1
( ) ( ){ } ( ) ( )[ ] ( ) ( ) ( )ωαωαωαααα 221122112211 exp: FFdttjtftftftf +=−+=+ ∫
+∞
∞−
F
Symmetry2
( )tF
-1
F
F
( )ωπ −f2
Conjugate Functions3 ( )tf *
-1
F
F
( )ω−*
F
Scaling4 ( )taf
-1
F
F
a
F
a
ω1
Derivatives5 ( ) ( )tftj
n
−
-1
F
F ( )ω
ω
F
d
d
n
n
( )tf
td
d
n
n
-1
F
F
( ) ( )ωω Fj
n
Convolution6
( ) ( )tftf 21
-1
F
F ( ) ( )ωω 21
* FF( ) ( ) ( ) ( )∫
+∞
∞−
−= τττ dtfftftf 2121
:*
-1
F
F ( ) ( )ωω 21
FF
( ) ( ) ( ) ( )∫∫
+∞
∞−
+∞
∞−
= ωωω
π
dFFdttftf 2
*
12
*
1
2
1
Parseval’s Formula7
Shifting: for any a real8
( ) ( )tajtf exp
-1
F
F ( )aF −ω
Modulation9 ( ) ttf 0
cos ω -1
F
F
( ) ( )[ ]00
2
1
ωωωω −++ FF
( ) ( ) ( ) ( ) ( ) ( )∫∫∫
+∞
∞−
+∞
∞−
+∞
∞−
−=−= ωωω
π
ωωω
π
dFFdFFdttftf 212121
2
1
2
1
23. Fourier Transform
( )
>
<
=Π
2
1
0
2
1
1
t
t
t
2
1
2
1
−
( )tΠ
t
Rectangle
1
( ) ( )
−<−
<
>
=
τ
ττ
τ
τ
t
tt
t
t
2/1
2/
2/1
lim
Limiter τ
( )[ ] ( )tt sgnlimlim2 0
=→ ττ
0
( )tτlim
t
2/1
2/1−
τ
τ−
SOLO
Special Symbols
( )
>
<−
=Λ
10
11
t
tt
t
11−
( )tΛ
t
Triangle
1
( )
<
>
=
00
01
t
t
tH
0
( )tH
t
Heaviside
unit step
1
( )
<−
>
=
01
01
sgn
t
t
t
0
( )tsgn
t
Signum 1
1−
0
( )t
td
d
τlim
t
( )τ2/1
ττ−
Area = 1td
d
24. Fourier Transform
( ) ( ) ( )
( )
≠
=∞
=
>
≤
=
<−
≤
>
=
= →→→
00
0
0
2/1
lim
2/1
2/
2/1
limlimlim: 000
t
t
t
t
t
tt
t
td
d
t
td
d
t
τ
ττ
τ
ττ
τ
δ ττττ
SOLO
Special Symbols
0
( )t
td
d
τlim
t
( )τ2/1
ττ−
Area = 1td
d
δ (t) function
Since ( )( ) ( )tt sgn
2
1
limlim
0
=
→
ττ
we have also
δ (t) function is defined as:
( ) ( )t
td
d
t sgn
2
1
=δ
0
( )t
td
d
τlim
t
( )τ2/1
ττ−
Area = 1
0 t
( )tδ
Area = 1
0→τ
( ) ( )
−<−
<
>
=
τ
ττ
τ
τ
t
tt
t
t
2/1
2/
2/1
lim
Limiter τ
( )[ ] ( )tt sgnlimlim2
0
=
→ ττ
0
( )tτlim
t
2/1
2/1−
τ
τ−
25. Fourier TransformSOLO
Special Symbols
Properties of δ (t) function
0
( )t
td
d
τlim
t
( )τ2/1
ττ−
Area = 1
0 t
( )tδ
Area = 1
0→τ
( ) ( )tt −= δδδ (t) is a even function:2
( )
( )
≠
=∞
=
>
≤
=
→
00
0
0
2/1
lim
0
t
t
t
t
t
τ
ττ
δ τ
1
3 ( ) ( )
( ) ( )
( ) ( ) ( ) ( )[ ]00
2
1
++−=−=− ∫∫
+∞
∞−
−=+∞
∞−
τττδτδ
δδ
ffdtttfdtttf
uu
Proof:
( ) ( )
( ) ( )
( ) ( ) ( ) ( )[ ] ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]
( ) ( )[ ]00
2
1
00
2
1
lim
2
1
lim
sgn
2
1
limsgn
2
1
limsgnlim
2
1
lim
0
0
sgn
2
1
++−=
++−−−−+−+=
−+−+=
−−−=−=−
→∞
+
−
−
→∞
+
−
→∞
+
−→∞
+
−
→∞
=+
−
→∞
∫∫
∫∫∫
ττ
ττ
ττττδ
τ
τ
δ
ff
fTfTffTfTftfdtfdTfTf
tfdtttftdtfdtttf
T
T
T
T
T
T
T
T
TT
T
T
T
t
dt
d
tT
T
T
4 Fourier Transform ( ){ } ( ) ( ) ( ) ( ) 10exp
2
1
0exp
2
1
exp =++−=−= ∫
+∞
∞−
jjdttjtt ωδδF
26. Fourier Transform
( )
−=
+
=
=
=
−=
=
+
=
→
→
→
→
→
−
→
→
εεε
εε
εε
επ
εεπ
ε
ε
ε
π
δ
ε
ε
ε
ε
ε
ε
ε
ε
ε
x
Ln
x
x
J
x
Ai
x
x
x
x
x
x
2
exp
1
lim
11
lim
1
lim
sin
1
lim
4
exp
2
1
lim
lim
lim
1
2
0
/1
0
0
0
2
0
1
0
220
SOLO
Special Symbols
δ (t) function
The δ (t) function can be defined as the following limit as ε→0
Ai is the Airry function,
( ) ∫
∞
+=
0
3
3
cos
1
dttx
t
xAi
π ( ) ( )[ ]∫
+
−
−−=
π
π
τττ
π
dxnjxJn
sinexp
2
1
Friedrich Wilhelm
Bessel
1784 - 1846
Edmond Nicolas
Laguerre
1834 - 1886
Jn (x) is the Bessel function of the first kind,
and Ln (x) is the Laguerre polynomial of arbitrary positive order.
27. Fourier TransformSOLO
Special Symbols
δ (t) function
The δ (t) function can be defined also by the limit n→∞
( )
+
= →∞
x
xn
x n
2
1
sin
2
1
sin
2
1
lim
π
δ
( ) ( )
( )
( )tnsincn
tnn
xnnx
n
n
n
→∞
→∞
→∞
=
Π=
−=
lim
lim
explim 22
πδ( )
>
≤
=Π
2/10
2/11
x
x
x
( ) ( )
x
x
xsinc
π
πsin
=
30. Fourier Transform
( )
∆>−
∆≤−
∆=∆
2/0
2/
1
:
0
0
fff
fff
ffS f
SOLO
δ (f) function
Define:
In the time domain we obtain:
( ) ( ) ( )
( )
tfj
ff
ff
tfjff
ff
tfjtfj
ff e
tf
tf
tj
e
f
fde
f
fdefSts 0
0
0
0
0
2
2/
2/
22/
2/
22 sin
2
11 π
π
ππ
π
π
π ∆
∆
=
∆
=
∆
==
∆+
∆−
∆+
∆−
+∞
∞−
∆∆ ∫∫
For any function Φ (f), defined at f=f0- and f=f 0+ , we have
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )[ ]+−
−
+
+
−
Φ+Φ=Φ
∆
+Φ
∆
=
Φ
∆
+Φ
∆
=Φ
∆
=Φ
∆−
→∆
∆+
→∆
∆−
→∆
∆+
→∆
∆+
∆−
→∆
+∞
∞−
∆
→∆ ∫∫∫∫
00
2/
0
2/
0
2/
0
2/
0
2/
2/
00
2
11
lim
1
lim
1
lim
1
lim
1
limlim
0
0
0
0
0
0
0
0
0
0
ffff
f
ff
f
dff
f
dff
f
dff
f
dfffS
f
ff
f
ff
f
f
f
ff
f
ff
f
f
ff
ff
f
f
f
( ) ( )0
0
lim fffS f
f
−=∆
→∆
δ
( ) ( )0
0
:lim fffS f
f
−=∆
→∆
δ ( ) tfj
f
f
ets 02
0
lim π
=∆
→∆
( ) ( )
∫
+∞
∞−
−−
=− tdeff tffj 02
0
π
δ
31. Fourier Transform
( ) ( )∑−=
+=
N
Nn
N Tntftf :
SOLO
fN (f) N-Periodic Extension of a function f (t)
Define
N- extension of f (t)
32. Fourier Transform
( ) ( )∑−=
+=
N
Nn
N Tntt δδ :
SOLO
δN (f) function
Define
Let find the Fourier transform of δN (f)
( ) ( ) ( )
( )[ ]
[ ]
( )Tf
TNf
ee
j
j
ee
eee
eee
e
ee
etdenTttdetf
TfjTfj
TNfjTNfj
TfjTfjTfj
TNfjTNfj
Tfj
Tfj
NTfjTNfj
N
Nn
Tnfj
N
Nn
tfjtfj
NN
π
π
δδ
ππ
ππ
πππ
ππ
π
π
ππ
πππ
sin
2
1
2sin
2
2
1
1
2
1
2
2
1
2
2
1
2
2
1
2
2
1222
222
+
=
−
−
=
=
−
−
=
−
−
=
=+==∆
−
+−
+
−
+
+−
+−
−=−=
+∞
∞−
−
+∞
∞−
−
∑∑ ∫∫
We can see that
( ) ( )[ ]
( )
( )[ ]
( )
( ) ,2,1,0
sin
12sin
sin
1212sin
±±=∆=
+
=
+
+++
=
+∆ kf
Tf
TNf
kTf
NkTNf
T
k
f NN
π
π
ππ
ππ
N-extension of δ (t)
33. Fourier TransformSOLO
δN (f) function (continue – 1)
( ) ( )∑−=
+=
N
Nn
N Tntt δδ : ( ) ( )[ ]
( ) ∑−=
=
+
=∆
N
Nn
Tnfj
N e
Tf
TNf
f π
π
π 2
sin
12sin
δN (t) is a periodic function with a time period of T .
ΔN (f) is a periodic function with a frequency period of f0 = 1/T .
( )
( )
( )
( )
( )
( )
( )
[ ]
Tn
n
TTnj
e
dfedff
N
Nn
n
n
N
Nn
T
T
TnfjN
Nn
T
T
Tnfj
T
T
N
1sin1
2
00
01
2/1
2/1
22/1
2/1
2
2/1
2/1
====∆ ∑∑∑ ∫∫ −=
≠←
=←
−=
−
−−=
+
−
+
−
π
π
π
π
π
34. Fourier TransformSOLO
δN (f) function (continue – 2)
When N → ∞ the peak goes to infinity and the null-to-null bandwidth goes to zero.
This resembles to a delta function. To prove that this is the case let compute:
ΔN (f) is a periodic function with a frequency period of f0 = 1/T , with peak amplitude of
(2 N+1) and null-to-null bandwidth of 2/ [(2N+1) T].
( )
( )
( )
T
dff
T
T
N
1
2/1
2/1
=∆∫
+
−
( ) ( )
( )
( )
( )
( )
( )
( )0
1
limlim
2/1
2/1
2
2/1
2/1
Φ=Φ=Φ∆ ∑ ∫∫ −=
+
−
∞→
+
−
∞→ T
dffedfff
N
Nn
T
T
Tnfj
N
T
T
N
N
π
35. Fourier TransformSOLO
δN (f) function (continue – 3)
( )
( )
( )
T
dff
T
T
N
1
2/1
2/1
=∆∫
+
−
( ) ( )
( )
( )
( )
( )
( )
( )0
1
limlim
2/1
2/1
2
2/1
2/1
Φ=Φ=Φ∆ ∑ ∫∫ −=
+
−
∞→
+
−
∞→ T
dffedfff
N
Nn
T
T
Tnfj
N
T
T
N
N
π
Therefore ( ) ( )
( )
( )
∑∫
∞+
−∞=
+
−
∞→
∞
+=∆=∆
m
T
T
N
N T
m
f
T
dfff δ
1
lim:
2/1
2/1
36. Fourier TransformSOLO
δN (f) function (continue – 4)
Let compute the convolution between f (t) and δN (f)
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )tfTntfdTntfdTntfttf N
N
Nn
N
Nn
N
Nn
N =+=+−=+−=∗ ∑∑ ∫∫ ∑ −=−=
+∞
∞−
+∞
∞− −=
ττδτττδτδ :
Therefore ( ) ( ) ( )ttftf NN δ∗=
Using this relation the Fourier Transform of fN (t) is given by
( ) ( ) ( ) ( ) ( )[ ]
( )Tf
TfN
fFffFfF NN
π
π
sin
12sin +
=∆= ( ) ( ) ( ) ( )Tntfttftf
N
Nn
NN +=∗= ∑−=
δ
If N → ∞ then
( ) ( ) ( ) ( )
( ) ∑∑
∑
∞+
−∞=
∞+
−∞=
+∞
−∞=
∞∞
−
=
−=
−=∆=
mm
m
T
m
f
T
m
F
TT
m
ffF
T
T
m
f
T
fFffFfF
δδ
δ
11
1
( ) ( )
∑
∑
∑
∞+
−∞=
∞+
−∞=
−
+∞
−∞=
∞
=
−
=
+=
m
t
T
m
j
m
n
e
T
m
F
T
T
m
f
T
m
F
T
Tntftf
π
δ
2
1
1
1
F
37. Fourier TransformSOLO
δN (f) function (continue – 4)
( ) ∑
+∞
−∞=
∞
−
=
m T
m
f
T
m
F
T
fF δ
1
( ) ( ) ∑∑
+∞
−∞=
+∞
−∞=
∞
=+=
m
t
T
m
j
n
e
T
m
F
T
Tntftf
π21
f∞ (t) is a periodic function with a time period of T .
F∞ (f) is a periodic function with a frequency period of f0 = 1/T .
We obtained the Fourier Series description of a periodic function
( ) ( )∫∑
+∞
∞−
+∞
−∞=
∞ =
== tdetf
TT
m
F
T
aeatf
t
T
m
j
m
m
t
T
m
j
m
ππ 22 11
If we define
( )
( )
>
≤
=
2/0
2/
0
Tt
Tttf
tf ( ) ( )∫
+
−
−
=
2/
2/
2
0
T
T
tfj
tdetffF π
then
( ) ( )∫∑
+
−
+∞
−∞=
∞ ==
2/
2/
22 1
T
T
t
T
m
j
m
m
t
T
m
j
m tdetf
T
aeatf
ππ
47. Laplace’s TransformSOLO
Laplace L-Transform (continue – 1)
The Inverse Laplace’s Transform (L -1
) is given by: ( ) ( )∫
∞+
∞−
=
j
j
ts
dsesF
j
tf
π2
1
Using Jordan’s Lemma (see “Complex Variables” presentation or the end of this one)
Jordan’s Lemma Generalization
If |F (z)| ≤ M/Rk
for z = R e iθ
where k > 0 and M are constants, then
for Γ a semicircle arc of radius R, and center at origin:
( ) 00lim <=∫Γ
→∞
mzdzFe zm
R
where Γ is the semicircle, in the left part of z plane.
x
yΓ
R
we can write
( ) ( ){ } ( ) ( )∫∫
∞+
∞−
+
+
===
j
j
tsts
f
f
dsesF
j
dsesF
j
sFtf
σ
σ
ππ 2
1
2
11-L
( ) ( ){ } ( ) ( ) ( )∫∫∫ =+==
∞+
∞−
dsesF
j
dsesF
j
dsesF
j
sFtf ts
C
ts
j
j
ts
πππ 2
1
2
1
2
1
0
1-L
If the F (s) has no poles for σ > σf+, according to Cauchy’s Theorem
we can use a closed infinite region to the left of σf+, to obtain
48. Laplace’s TransformSOLO
Properties of Laplace L-Transform
s - Domaint - Domain
( )tf ( ) ( ) { } +
>= ∫
∞
−
f
st
sdtetfsF σRe
0
1 ( ) { } if
M
i
ii zsFc σmaxRe
1
>∑=
Linearity ( )∑=
M
i
ii tfc
1
3 ( ) ( ) ( )
( ) ( )
( )+−+−+−
−−−− 000 1121 nnnn
ffsfssFs Differentiation
( )
n
n
td
tfd
4 ( ) ( )∫∞−
→ +
+
t
t
df
ss
sF
ξξ
0
lim
1Integration ( )∫∞−
t
df ξξ
5 ( )
s
sFReal Definite
Integration
( )∫
t
df
0
ξξ
( )∫∫
t
ddf
0 0
ξλλ
ξ ( )
2
s
sF
2
a
s
F
a
1Scaling ( )taf
51. ( )tf
( ) ( )∑
∞
=
−=
0n
T Tntt δδ
( ) ( ) ( ) ( ) ( )∑
∞
=
−==
0
*
n
T
TntTnfttftf δδ
( )tf *
( )tf
T t
( ) ( ){ } ( ) σσ <== +∫
∞
−
f
ts
dtetftfsF
0
L
SOLO
Sampling and z-Transform
( ) ( ){ } ( ) σδδ <
−
==
−== −
∞
=
−
∞
=
∑∑ 0
1
1
00
sT
n
sTn
n
T
e
eTnttsS LL
( ) ( ){ }
( ) ( ) ( )
( ) ( ){ } ( ) ( )
<<
−
=
=
−
==
−
∞+
∞−
−−
∞
=
−
∞
=
+∫
∑∑
0
00
**
1
1
2
1
σσσξξ
π
δ
δ
ξ
σ
σ
ξ f
j
j
tsT
n
sTn
n
d
e
F
j
ttf
eTnfTntTnf
tfsF
L
L
L
( )
( ) ( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
−
=
−
−
=
−
=
∑∫
∑∫
∑
−−
−
−−
Γ
−−
−−
Γ
−−
∞
=
−
ts
e
ofPoles
tsts
F
ofPoles
tsts
n
nsT
e
F
Resd
e
F
j
e
F
Resd
e
F
j
eTnf
sF
ξ
ξξ
ξ
ξξ
ξ
ξ
ξ
π
ξ
ξ
ξ
π
1
1
0
*
112
1
112
1
2
1
Poles of
( ) Ts
e ξ−−
−1
1
Poles of
( )ξF
planes
T
nsn
π
ξ
2
+=
ωj
ωσ j+
0=s
Laplace Transforms
The signal f (t) is sampled at a time period T.
1Γ
2
Γ
∞→R
∞→R
Poles of
( ) Ts
e ξ−−
−1
1
Poles of
( )ξF
planeξ
T
nsn
π
ξ
2
+=
ωj
ωσ j+
0=s
Z Transform
52. Fourier Transform
( )tf
( ) ( )∑
∞
=
−=
0n
T Tntt δδ
( ) ( ) ( ) ( ) ( )∑
∞
=
−==
0
*
n
T
TntTnfttftf δδ
( )tf *
( )tf
T t
SOLO
Sampling and z-Transform (continue – 1)
( ) ( )
( )
( )
( )
( ) ( ) ∑∑
∑∑
∞+
−∞=
∞+
−∞=
−−→
∞+
−∞=
−−
+→
+=
−
−−
+=
−
+
−=
+
−
−−
−=
−
−=
−−
−−
nn
Tse
n
ts
T
n
js
T
n
js
e
ofPoles
ts
T
n
jsF
TeT
T
n
jsF
T
n
jsF
e
T
n
js
e
F
RessF
ts
n
ts
π
π
π
π
ξ
ξ
ξ
ξπ
ξ
π
ξ
ξ
ξ
ξ
21
2
lim
2
1
2
lim
1
1
2
2
1
1
*
Poles of
( )ξF
ωj
σ
0=s
T
π2
T
π2
T
π2
Poles of
( )ξ*
F plane
js ωσ +=
The signal f (t) is sampled at a time period T.
The poles of are given by( )ts
e ξ−−
−1
1
( )
( )
T
n
jsnjTsee n
njTs π
ξπξπξ 2
21 2
+=⇒=−−⇒==−−
( ) ∑
+∞
−∞=
+=
n T
n
jsF
T
sF
π21*
53. Fourier TransformSOLO
F F-1
frequency-B/2 B/2
B
F F-1
-B/2 B/2
B
1/Ts-1/Ts frequency
Sample
Sampling a function at an interval Ts (in time domain)
Anti-aliasing filters is used to enforce band-limited assumption.
causes it to be replicated
at
1/ Ts intervals in the other (frequency) domain.
Sampling and z-Transform (continue – 2)
54. Fourier Transform
( )tf
( ) ( )∑
∞
=
−=
0n
T Tntt δδ
( ) ( ) ( ) ( ) ( )∑
∞
=
−==
0
*
n
T
TntTnfttftf δδ
( )tf *
( )tf
T t
SOLO
Sampling and z-Transform (continue – 3)
0=z
planez
Poles of
( )zF
C
The signal f (t) is sampled at a time period T.
The z-Transform is defined as:
( ){ } ( ) ( )
( )
( ) ( )
( )
−
−===
∑
∑
=
−
→
∞
=
−
=
iF
iF
i
iF
Ts
FofPoles
T
F
n
n
ze
ze
F
zTnf
zFsFtf
ξξ
ξ
ξ
ξξ
ξξξ
1
0
*
1
lim:Z
( )
( )
<
>≥
= ∫
−
00
0
2
1 1
n
RzndzzzF
jTnf
fC
C
n
π
55. Fourier TransformSOLO
Sampling and z-Transform (continue – 4)
( ) ( ) ( )∑∑
∞
=
−
+∞
−∞=
=
+=
0
* 21
n
nsT
n
eTnf
T
n
jsF
T
sF
πWe found
The δ (t) function we have:
( ) 1=∫
+∞
∞−
dttδ ( ) ( ) ( )τδτ fdtttf =−∫
+∞
∞−
The following series is a periodic function: ( ) ( )∑ −=
n
Tnttd δ:
therefore it can be developed in a Fourier series:
( ) ( ) ∑∑
−=−=
n
n
n T
tn
jCTnttd πδ 2exp:
where: ( )
T
dt
T
tn
jt
T
C
T
T
n
1
2exp
1
2/
2/
=
= ∫
+
−
πδ
Therefore we obtain the following identity:
( )∑∑ −=
−
nn
TntT
T
tn
j δπ2exp
Second Way
56. Fourier Transform
( ) ( ){ } ( ) ( )∫
+∞
∞−
−== dttjtftfF νπνπ 2exp:2 F
( ) ( ) ( )∑∑
∞
=
−
+∞
−∞=
=
+=
0
* 21
n
nsT
n
eTnf
T
n
jsF
T
sF
π
( ) ( ){ } ( ) ( )∫
+∞
∞−
== ννπνπνπ dtjFFtf 2exp2:2-1
F
SOLO
Sampling and z-Transform (continue – 5)
We found
Using the definition of the Fourier Transform and it’s inverse:
we obtain ( ) ( ) ( )∫
+∞
∞−
= ννπνπ dTnjFTnf 2exp2
( ) ( ) ( ) ( ) ( ) ( )∑∫∑
∞
=
+∞
∞−
∞
=
−=−=
0
111
0
*
exp2exp2exp
nn
n
sTndTnjFsTTnfsF ννπνπ
( ) ( ) ( )[ ]∫ ∑
+∞
∞−
+∞
−∞=
−−== 111
*
2exp22 νννπνπνπ dTnjFjsF
n
( ) ( ) ∑∫ ∑
+∞
−∞=
+∞
∞−
+∞
−∞=
−=
−−==
nn T
n
F
T
d
T
n
T
FjsF νπνννδνπνπ 2
11
22 111
*
We recovered (with –n instead of n) ( ) ∑
+∞
−∞=
+=
n T
n
jsF
T
sF
π21*
Second Way (continue)
Making use of the identity: with 1/T instead of T
and ν - ν 1 instead of t we obtain: ( )[ ] ∑∑
−−=−−
nn T
n
T
Tnj 11
1
2exp ννδννπ
( )∑∑ −=
−
nn
TntT
T
tn
j δπ2exp
57. Claude Elwood Shannon
1916 – 2001
http://en.wikipedia.org/wiki/Claude_E._Shannon
Fourier TransformSOLO
Henry Nyquist
1889 - 1976
http://en.wikipedia.org/wiki/Harry_Nyquist
Nyquist-Shannon Sampling Theorem
The sampling theorem was implied by the work of Harry Nyquist in
1928 ("Certain topics in telegraph transmission theory"), in which
he showed that up to 2B independent pulse samples could be sent
through a system of bandwidth B; but he did not explicitly consider
the problem of sampling and reconstruction of continuous signals.
About the same time, Karl Küpfmüller showed a similar result, and
discussed the sinc-function impulse response of a band-limiting
filter, via its integral, the step response Integralsinus; this band-
limiting and reconstruction filter that is so central to the sampling
theorem is sometimes referred to as a Küpfmüller filter (but seldom
so in English).
The sampling theorem, essentially a dual of Nyquist's result,
was proved by Claude E. Shannon in 1949 ("Communication in
the presence of noise"). V. A. Kotelnikov published similar
results in 1933 ("On the transmission capacity of the 'ether' and
of cables in electrical communications", translation from the
Russian), as did the mathematician E. T. Whittaker in 1915
("Expansions of the Interpolation-Theory", "Theorie der
Kardinalfunktionen"), J. M. Whittaker in 1935 ("Interpolatory
function theory"), and Gabor in 1946 ("Theory of
communication").
http://en.wikipedia.org/wiki/Nyquist-Shannon_sampling_theorem
58. Fourier TransformSOLO
Nyquist-Shannon Sampling Theorem (continue – 1)
• Signal can be recovered if Fourier spectrum of the sampling signal do not overlap.
• Start with a band limited signal s (t) ( )
2
0
fB
fforfS >≡
• Sample s (t) at a time period Ts, replicates
spectrum every 1/Ts Hz.
( ) ∑
∞+
−∞=
−=
k sT
kfjSfS
1
2* π
fjs π2=
( ) ( ) ( )
−= ∑
+∞
−∞=n
sTnttsts δ* ( )
−= ∑
∞+
−∞=k sT
jksSsS
π2
*
L-1
L
F
F-1
59. Fourier Transform
2
1
2
B
T
B
s
−<
SOLO
Nyquist-Shannon Sampling Theorem (continue – 2)
• Signal can be recovered if Fourier spectrum of the
sampling signal do not overlap.
B
B
Ts
=
>
2
2
1
(Nyquist Sampling Rate)
• Complex signal band-limited to B/2 Hz requires B complex samples/second, or
2 B real samples/seconds (twice the highest frequency)
• Start with a band-limited signal f (t) ( )
2
0
fB
fforfF >≡ • Sample f (t) at a time period Ts,
replicates spectrum every 1/Ts Hz.
Nyquist-Shannon Sampling Theorem:
60. Fourier TransformSOLO
The Discrete Time Fourier Transform (DTFT)
• Start with a band limited signal s (t) ( )
2
0
fB
fforfS >≡
• Sample s (t) at a time period Ts, replicates
spectrum every 1/Ts Hz.
( )
−= ∑
∞+
−∞=k sT
kfSfS
1
*
( ) ( ) ( )
( ) ( )∑
∑
∞+
−∞=
+∞
−∞=
−=
−=
n
ss
n
s
TntTns
Tnttsts
δ
δ*
( ) ( )∫
+∞
∞−
−
= tdetsfS tfj π2
( ) ( )∫
+∞
∞−
= fdefSts tfj π2F
F-1
Continuous Fourier Transform
F
F-1
Discretization of a Continuous Signal ( ) ( )∫
+∞
∞−
== fdefSTnts sTnfj
s
π2
( ) ( ) ( )∑∑
∞+
−∞=
−
=
∞+
−∞=
−
==
n
n
f
f
j
s
T
f
n
Tnfj
sDTFT
s
s
s
s
eTnseTnsfS
π
π
2
1
2
:
DTFT provides an approximation of the continuous-time Fourier transform.
Discrete Time Fourier Transform
(DTFT)
Define
61. Fourier TransformSOLO
The Discrete Time Fourier Transform (DTFT) (continue-1)
• Signal can be recovered if Fourier spectrum of the sampling signal do not overlap.
Discretization of a Continuous Signal ( ) ( )∫
+∞
∞−
== fdefSTnts sTnfj
s
π2
DTFT-1
DTF
T
Discrete Time Fourier Transform
(DTFT)
( ) ( ) ( )∑∑
∞+
−∞=
−
=
∞+
−∞=
−
==
n
n
f
f
j
s
T
f
n
Tnfj
sDTFT
s
s
s
s
eTnseTnsfS
π
π
2
1
2
:
We can see that
( ) ( ) ( ) ( )∑∑
∞+
−∞=
−
−∞+
−∞=
+
−
===+
n
DTFT
nkj
n
f
f
j
s
n
n
f
fkf
j
ssDTFT fSeeTnseTnsfkfS ss
s
1
2
22
π
ππ
The Discrete Time Fourier Transform SDTFT (fs) is periodic with period fs.
Let compute
( ) ( )
( )
( )
( )
( )
( )
( ) ( ) ( )[ ]
( )
( )∑ ∑
∑ ∫∫ ∑∫
∞+
−∞=
∞+
−∞=
=←
≠←
+
−
−
∞+
−∞=
+
−
−
+
−
∞+
−∞=
−
+
−
=
−
−
=
−
=
==
n
s
sn
nm
nm
ss
f
fs
nm
f
f
j
s
n
f
f
nm
f
f
j
s
f
f n
nm
f
f
j
s
f
f
m
f
f
j
DTFT
Tms
Tnm
nm
fTns
f
nm
j
e
Tns
fdeTnsdfeTnsdfefS
s
s
s
s
s
s
s
s
s
s
s
s
1sin
2
1
0
2/
2/
2
2/
2/
22/
2/
22/
2/
2
π
π
π
π
πππ
( ) ( )∑
+∞
−∞=
−
=
n
Tnfj
sDTFT
s
eTnsfS π2
: ( ) ( )
( )
( )
∫
+
−
=
s
s
s
T
T
nTfj
DTFTss dfefSTTns
2/1
2/1
2π
62. Fourier TransformSOLO
The Discrete Time Fourier Transform (DTFT) (continue-2)
Normalization of the frequency
DTFT-1
DTFT
( ) ( )∑
+∞
−∞=
−
=
n
Tnfj
sDTFT
s
eTnsfS π2
: ( ) ( )
( )
( )
∫
+
−
=
s
s
s
T
T
nTfj
DTFTss dfefSTTns
2/1
2/1
2π
( ) ( )[ ]
[ ]2/1,2/1
2/1,2/1
:
*
*
+−∈
+−∈
=
f
TTf
Tff
ss
s
( ) ( )∑
+∞
−∞=
−
=
n
nfj
DTFT ensfS *2*
: π
DTFT-1
DTFT
( ) ( )∫
+
−
=
2/1
2/1
*2
** dfefSns nfj
DTFT
π
Example ( ) 1,,1,002
−== −
NneAns nfj
π
( ) ( )
( )
( )
( ) ( )
( ) ( )
( )
( )
( )[ ]
( )[ ]
( )( )1*
0
0
*
*
**
**
*2
*21
0
*2*
0
0
0
00
00
0
0
0
*sin
*sin
1
1
−−−
−−
−−
−−−
−−−
−−
−−−
=
−−
−
−
=
−
−
=
−
−
== ∑
Nffj
ffj
Nffj
ffjffj
NffjNffj
ffj
NffjN
n
nffj
DTFT
e
ff
Nff
A
e
e
ee
ee
A
e
e
AeAfS
π
π
π
ππ
ππ
π
π
π
π
π
|SDTFT(f*)|
Normalized Frequency
63. Fourier TransformSOLO
The Discrete Time Fourier Transform (DTFT) (continue-3)
( ) ( )∑
+∞
−∞=
−
=
n
nfj
DTFT ensfS *2*
: π
DTFT-1
DTFT
( ) ( )∫
+
−
=
2/1
2/1
*2
** dfefSns nfj
DTFT
π
Example ( )
≥=
=
=
−
22&8,,00
21,,10,902
nn
ne
ns
nfj
π
( )
≥=
=
=
−
27&4,,00
26,,10,302
nn
ne
ns
nfj
π
Frequency Resolution Increases with Observation Time N Ts
DTFT
DTFT
64. Fourier Transform
( ) ( )∑
−
=
−
=
1
0
2
:
N
n
nk
N
j
sDFT eTnskS
π
SOLO
The Discrete Fourier Transform (DFT)
Assume a periodic sequence, sampled at a time period Ts, such that s (n Ts) = s [(n+kN) Ts]
The Discrete Fourier Transform (DFT) requires an input function that is discrete
and whose non-zero values have a limited (finite) duration.
Unlike the Discrete-time Fourier transform (DTFT), it only evaluates enough frequency
components to reconstruct the finite segment that was analyzed. Its inverse transform
cannot reproduce the entire time domain, unless the input happens to be periodic (forever).
Therefore it is often said that the DFT is a transform for Fourier analysis of finite-domain
discrete-time functions
For the sequence s (0), s (Ts),…,s [(N-1) Ts] we define the Discrete Fourier Transform:
65. Fourier Transform
( ) ( ) ( )∑∑
−
=
−
=
−
==
1
0
1
0
2
:
N
n
nk
s
N
n
nk
N
j
sDFT WTnseTnskS
π
SOLO
The Discrete Fourier Transform (DFT) (continue – 1)
For the sequence s (0), s (Ts),…,s [(N-1) Ts] we define the Discrete Fourier Transform:
where is a primitive N'th root of unity
and is periodic
N
j
eW
π2
:
−
=
n
Nm
N
j
n
N
j
Nmn
N
j
Nmn
WeeeW =
=
=
−−
+
−
+
1
222 πππ
( )
( )
( )
( )
( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
[ ]
( )
( )
( )
( )[ ]
( )[ ]
N
N
N s
s
s
s
s
s
W
NNNNNNN
NNNNNNN
NN
NN
NN
S
DFT
DFT
DFT
DFT
DFT
TNs
TNs
Ts
Ts
Ts
WWWWW
WWWWW
WWWWW
WWWWW
WWWWW
NS
NS
S
S
S
⋅−
⋅−
⋅
⋅
⋅
=
−
−
−−−−−−−
−−−−−−−
−−
−−
−−
1
2
2
1
0
1
2
2
1
0
1121211101
1222221202
1222221202
1121211101
1020201000
[ ] NNN sWS = [ ]NW is a Vandermonde type of Matrix
66. Fourier TransformSOLO
The Discrete Fourier Transform (DFT) (continue – 2)
nNmn
WW =+
[ ] [ ] N
H
NN I
N
WW
1
=
N
j
eW
π2
−
= 1
2
* −
== WeW N
j
π
[ ]
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
=
−−−−−−−
−−−−−−−
−−
−−
−−
1121211101
1222221202
1222221202
1121211101
1020201000
NNNNNNN
NNNNNNN
NN
NN
NN
N
WWWWW
WWWWW
WWWWW
WWWWW
WWWWW
W
[ ] [ ]
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
==
−+−−+−−−−−−
−+−−+−−−−−−
+−+−−−
+−+−−−
+−+−−−
1112121110
2122222120
2122222120
1112121110
0102020100
*
NNNNNNN
NNNNNNN
NN
NN
NN
T
N
H
N
WWWWW
WWWWW
WWWWW
WWWWW
WWWWW
WW
Let multiply those two matrices
[ ] [ ]( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( )
( )
( )
( )
( )
=
≠=
−
−
=
−
−
==
+++++=
−
−
−
−−
=
−
+−−−−
∑
mkN
mk
W
W
W
W
W
WWWWWWWWWW
mk
mk
N
mk
NmkN
j
jmk
mNNkmjjkmkmk
mk
H
NN
0
1
1
1
1
1
1
0
111100
,
Where IN is the NxN identity matrix
67. Fourier Transform
( ) ( ) ( )∑∑
−
=
−
=
−
==
1
0
1
0
2
:
N
n
nk
s
N
n
nk
N
j
sDFT WTnseTnskS
π
SOLO
The Discrete Fourier Transform (DFT) (continue – 3)
For the sequence s (0), s (Ts),…,s [(N-1) Ts] we defined the Discrete Fourier Transform:
[ ] NNN sWS = [ ]NW is a Vandermonde type of Matrix
We found that
[ ] [ ] N
H
NN I
N
WW
1
= Where IN is the NxN identity matrix
Therefore the Inverse Discrete Fourier Transform (IDFT) is
[ ] N
H
NN SW
N
s
1
=
( ) ( ) ( )∑∑
−
=
−
=
−
==
1
0
21
0
11 N
n
nk
N
j
DFT
N
k
nk
DFTs ekS
N
WkS
N
Tns
π
D.F.T.
I.D.F.T.
68. Fourier TransformSOLO
The Discrete Fourier Transform (DFT) (continue – 4)
Second way to find the Inverse Discrete Fourier Transform (IDFT). Let compute:
( ) ( )
( )
( )
( )
∑ ∑∑∑∑
−
=
−
=
−−−
=
−
=
−−−
=
+
==
1
0
1
0
21
0
1
0
21
0
2 N
n
N
k
rnk
N
j
s
N
k
N
n
rnk
N
j
s
N
k
rk
N
j
DFT eTnseTnsekS
πππ
( )
( )
( )
( )
( )
( )[ ] ( )[ ]
( ) ( )
( )[ ]
( )
( )[ ] ( )[ ]
( ) ( )
( )[ ]
( )
( )
( )
( )[ ] ( )[ ]
( ) ( )
≠−
=−
=
−+
−
−+−
−
−
−
−
=
−+
−
−+−
−
−
=
−+
−−
−+−−
=
−
−
=
−
−
=
−−
−−
−−
−−
−
=
−−
∑
Nmrn
NmrnN
rn
N
jrn
N
rnjrn
rn
N
rn
N
rn
rn
N
rn
N
jrn
N
rnjrn
rn
N
rn
rn
N
jrn
N
rnjrn
e
e
e
e
e
rn
N
j
rnj
rn
N
j
N
rn
N
j
N
k
rnk
N
j
0
cossin
cossin
sin
sin
cossin
cossin
sin
sin
2
sin
2
cos1
2sin2cos1
1
1
1
1
2
2
2
2
1
0
2
ππ
ππ
π
π
π
π
ππ
ππ
π
π
ππ
ππ
π
π
π
π
π
( ) ( )[ ] ,2,1,0
1
0
2
±±=+=∑
−
=
+
mTmNrsNekS s
N
k
rk
N
j
DFT
π
69. Fourier Transform
( ) ( ) ( )∑∑
−
=
−
=
−
==
1
0
1
0
2
:
N
n
nk
s
N
n
nk
N
j
sDFT WTnseTnskS
π
SOLO
The Discrete Fourier Transform (DFT) (continue – 1)
For the sequence s (0), s (Ts),…,s [(N-1) Ts] we define the Discrete Fourier Transform:
where is a primitive N'th root of unity
and is periodic
N
j
eW
π2
:
−
=
n
Nm
N
j
n
N
j
Nmn
N
j
Nmn
WeeeW =
=
=
−−
+
−
+
1
222 πππ
( )
( )
( )
( )
( )
( )
( )
( )
( )[ ]
( )[ ]
⋅−
⋅−
⋅
⋅
⋅
=
−
−
−−
−−
−−
−−
s
s
s
s
s
NN
NN
NN
NN
DFT
DFT
DFT
DFT
DFT
TNs
TNs
Ts
Ts
Ts
WWWWW
WWWWW
WWWWW
WWWWW
WWWWW
NS
NS
S
S
S
1
2
2
1
0
1
2
2
1
0
12210
23320
23420
12210
00000
70. Fourier TransformSOLO
The Discrete Fourier Transform (DFT) (continue – 5)
The DFT ant Inverse DFT (IDFT) are given by
( ) ( )∑
−
=
+
=
1
0
2
1 N
k
nk
N
j
DFTs ekS
N
Tns
π
( ) ( )∑
−
=
−
=
1
0
2
:
N
n
nk
N
j
sDFT eTnskS
π
IDFT
DFT
with the periodic properties
( )[ ] ( )
,2,1,0 ±±=
=+
m
TnsTmNns ss
( ) ( )
,2,1,0 ±±=
=+
m
kSNmkS DFTDFT
The sequence s (0), s (Ts),…,s [(N-1) Ts] can be interpreted to be a sequence of finite
length, given for r = 0, 1,…,N-1, and zero otherwise or a periodic sequence, defined
for all r.
71. Fourier Transform
( ) ( )∑
−
=
−
=
1
0
2
:
N
n
nk
N
j
sDFT eTnskS
π
SOLO
The Discrete Fourier Transform (DFT) (continue – 6)
The DFT ant Inverse DFT (IDFT) are given by
( ) ( )∑
−
=
+
=
1
0
2
1 N
k
nk
N
j
DFTs ekS
N
Tns
π
IDFT
DFT
( ) ( )∑
+∞
−∞=
−
=
n
nfj
DTFT ensfS *2*
: π
( ) ( )∫
+
−
=
2/1
2/1
*2
** dfefSns nfj
DTFT
π
IDTFT
DTFT
The DTFT ant Inverse DTFT (IDTFT) where given by
We can see that DFT is a sampled version of DTFT by tacking:
( ) ( )[ ]
[ ]2/1,2/1
2/1,2/1
1,,1,0
*
*
+−∈
+−∈
−==⇒==
f
TTf
Nk
TN
k
f
N
k
fTf
ss
s
s
( ) ( ) ( ) 1,,1,0:
1
0
2
−=== =
−
=
−
∑ NkfSeTnskS
sTN
k
fDTFT
N
n
nk
N
j
sDFT
π
72. Fourier TransformSOLO
The Discrete Fourier Transform (DFT) (continue –7)
We can see that DFT is a sampled version of DTFT :
( ) ( ) ( ) 1,,1,0:
1
0
2
−=== =
−
=
−
∑ NkfSeTnskS
sTN
k
fDTFT
N
n
nk
N
j
sDFT
π
By changing f0 from 0.25 to 0.275 we move |SDTFT (f)| to the right, and since the sampling
points didn’t change, we obtain different |SDFT (k)| values.
73. Fourier TransformSOLO
The Discrete Fourier Transform (DFT) (continue – 8)
We can see that DFT is a sampled version of DTFT :
( ) ( ) ( ) 1,,1,0:
1
0
2
−=== =
−
=
−
∑ NkfSeTnskS
sTN
k
fDTFT
N
n
nk
N
j
sDFT
π
Increase sampling density from N=20 to N=60.
74. SOLO
Properties of The Discrete Fourier Transform (DFT) (continue – 9)
( )mns − ( )
mk
N
j
DFT ekS
π2
−
Linearity1 ( ) ( )nsns 2211 αα +
Shift of a Sequence2
3
4
5
Periodic Convolution
6
7
Conjugate
8
9
IDFT
DFT ( ) ( )∑
−
=
−
=
1
0
2
:
N
n
nk
N
j
DFT enskS
π
( ) ( )∑
−
=
+
=
1
0
2
1 N
k
nk
N
j
DFT ekS
N
ns
π
( ) ( )kSkS DFTDFT 2211 αα +
( ) ( )nsns 21 , Periodic Sequence
(Period N)
( ) ( )kSkS DFTDFT 21 , DFT
(Period N)
( )
nl
N
j
ens
π2
−
( )lkSDFT −
( ) ( )∑
−
=
−⋅
1
0
21
N
m
mnsms
( ) ( )kSkS DFTDFT 21 ⋅
( ) ( )nsns 21 ⋅
( ) ( )∑
−
=
−⋅
1
0
21
1 N
l
DFTDFT lkSlS
N
( )ns∗
( )kSDFT −
∗
( )ns −∗
( )kSDFT
∗
Real & Imaginary ( )[ ]nsRe
( )[ ]nsImj
( ) ( ) ( )[ ] 2/kSkSkS DFTDFTeven −+=
∗
( ) ( ) ( )[ ] 2/kSkSkS DFTDFTodd −−=
∗
75. SOLO
Properties of The Discrete Fourier Transform (DFT) (continue – 10)
( ) ( ) ( )[ ] 2/: nsnsnseven −+= ∗
( )kSDFTReEven Part10
11
12 Symmetric Proprties
(only when s (n) is real)
Parseval’s Formula
IDFT
DFT ( ) ( )∑
−
=
−
=
1
0
2
:
N
n
nk
N
j
DFT enskS
π
( ) ( )∑
−
=
+
=
1
0
2
1 N
k
nk
N
j
DFT ekS
N
ns
π
( ) ( )nsns 21 , Periodic Sequence
(Period N)
( ) ( )kSkS DFTDFT 21 , DFT
(Period N)
( )lkSDFT −
( ) ( )
( )[ ] ( )[ ]
( )[ ] ( )[ ]
( ) ( )
( ) ( )
−−∠=∠
−=
−−=
−=
−=
∗
kSkS
kSkS
kSmkSm
kSkS
kSkS
DFTDFT
DFTDFT
DFTDFT
DFTDFT
DFTDFT
II
ReRe
Odd Part ( ) ( ) ( )[ ] 2/: nsnsnsodd −−= ∗
76. Fourier TransformSOLO
Fast Fourier Transform (FFT)
John Wilder Tukey
1915 – 2000
http://en.wikipedia.org/wiki/John_Tukey
James W. Cooley
1926 -
http://www.ieee.org/portal/pages/about/awards/bios/2002kilby.html
The Cooley-Tukey algorithm, is the most common fast
Fourier transform (FFT) algorithm. It re-expresses the
discrete Fourier transform (DFT) of an arbitrary composite
size N = N1N2 in terms of smaller DFTs of sizes N1 and N2,
recursively, in order to reduce the computation time to O(N
log N) for highly-composite N (smooth numbers).
FFTs became popular after J. W. Cooley of IBM and
John W. Tukey of Princeton published a paper in 1965
reinventing the algorithm (first invented by Gauss) and
describing how to perform it conveniently on a
computer
77. Fourier TransformSOLO
Fast Fourier Transform (FFT)
The radix-2 DIT Algorithm
The radix-2 decimation-in-time (DIT) FFT is the simplest and most common form of the
Cooley-Tukey algorithm, although highly optimized Cooley-Tukey implementations
typically use other forms of the algorithm as described below. Radix-2 DIT divides a DFT
of size N into two interleaved DFTs (hence the name "radix-2") of size N/2 with each
recursive stage.
( ) ( ) ( )∑∑
−
=
−
=
−
==
1
0
1
0
2
:
N
n
nk
s
N
n
nk
N
j
sDFT WTnseTnskS
π
For the sequence s (0), s (Ts),…,s [(N-1) Ts] we define the Discrete Fourier Transform:
1,1, 22/1
2
*
2
+==−====→= −−−
−
ππ
ππ
jNj
evenN
NN
j
N
j
eWeWWeWeW
Suppose N is a power of 2; i.e. N=2L
(L is integer). Since N is a even integer, let compute
SDFT (k) by separate s (nTs) into two (N/2)-point sequences consisting of the even-numbered
points (n=2r) and odd numbered points (n=2r+1).
( ) ( )
( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
( )
∑∑
∑∑
−
=
−
=
−
=
+
−
=
++=
++=
12/
0
2
12/
0
2
12/
0
12
12/
0
2
122
122
N
n
kr
N
k
N
N
n
kr
N
N
n
kr
N
N
n
kr
NDFT
WrsWWrs
WrsWrskS
78. Fourier TransformSOLO
Fast Fourier Transform (FFT)
The radix-2 DIT Algorithm (continue – 1)
2/
2/
222
2
N
N
j
N
j
N WeeW ==
=
−−
ππ
We divided the N-point DFT into two N/2-points DFTs.
( ) ( ) ( )
( )
( ) ( )
( )
( )
( )
( )
( )
( )
( )
kH
N
n
kr
N
k
N
kG
N
n
kr
N
N
n
kr
N
k
N
N
n
kr
NDFT
WrsWWrs
WrsWWrskS
∑∑
∑∑
−
=
−
=
−
=
−
=
++=
++=
12/
0
2/
12/
0
2/
12/
0
2
12/
0
2
122
122
Since
79. Fourier TransformSOLO
Fast Fourier Transform (FFT)
The radix-2 DIT Algorithm (continue – 2)
We divided the N-point DFT into two N/2-points DFTs.
Reduction of an 8-points FFT to two
4-points FFTs
A 2-points FFT
Reduction of an 4-points FFT to two
2-points FFTs
81. Fourier TransformSOLO
Fast Fourier Transform (FFT)
The radix-2 DIT Algorithm (continue – 2)
( ) ( )kkj
kN
N
j
Nk
N eeW 1
2
2/
−==
= −
−
π
π
We divided the N-point DFT into two N/2-points DFTs.
( ) ( ) ( ) ( )
[ ]
( )
( ) ( )
( )
( )
∑∑
−
=
−
−
=
+
++=++=
12/
0
1
2/
12/
0
2/
2/2/
N
n
kn
N
Nk
N
N
n
Nnk
N
kn
NDFT WWNnsnsWNnsWnskS
k
Since N/2 is an even integer (N=2L
)
( ) ( ) ( )[ ]
( )
( )
( )
( )
( )
tgofFFTN
N
n
nl
N
WW
N
N
n
nl
N
ng
DFT WngWNnsnslkS
NN
L
2/
12/
0
2/
2
12/
0
2
2/
2
2/2 ∑∑
−
=
=
=
−
=
=++==
( ) ( ) ( )[ ]
( )
( )
( )
( )
( )
thofFFTN
N
n
nl
N
WW
N
N
n
nl
N
nh
n
NDFT WnhWWNnsnslkS
NN
L
2/
12/
0
2/
2
12/
0
2
2/
2
2/12 ∑∑
−
=
=
=
−
=
=+−=+=
82. Fourier TransformSOLO
Fast Fourier Transform (FFT)
The radix-2 DIT Algorithm (continue – 3)
We divided the N-point DFT into two N/2-points DFTs.
Reduction of an 8-points FFT to two
4-points FFTs
Reduction of an 4-points FFT to two
2-points FFTs
A 2-points FFT
(Butterfly)
84. Fourier Transform
( ) ( ) 1,,1,0:
1
0
2
−== ∑
−
=
−
NkeTnskS
N
n
nk
N
j
sDFT
π
8 64 24 64 8
16 256 64 256 24
32 1024 160 1024 64
64 4096 384 4096 160
128 16384 896 16384 384
SOLO
Fast Fourier Transform (FFT)
Arithmetic Operations for a Radix FFT versus DFT
For N = 2L
we have L stages of Radix FFT and:
For N-point DFT we have:
For each row we have N complex additions and N complex multiplications, therefore for
the N rows we have
Number of complex additions DFT = Number of complex multiplications DFT = NxN=N2
Number of complex additions FFT =N L=N log2 N
Number of complex additions FFT =N/2 (multiplications per stage) x L -1 =N/2 log2 (N/2)
Operation
Complex additions Complex multiplications
DFT DFTFFT FFT
N=2L
Approximate number of Complex Arithmetic Operations Required for 2L-point DFT and FFT computations
85. SOLO Complex Variables
Laurent’s Series (1843)
Power Series
If f (z) is analytic inside and on the boundary of the ring
shaped region R bounded by two concentric circles C1 and
C2 with center at z0 and respective radii r1 and r2 (r1 > r2),
then for all z in R:
Pierre Alphonse Laurent
1813 - 1854
C1
x
y
R
C2R2
R1
z0
z
z'
r
P1
P0
z'( ) ( )
( )∑∑
∞
=
−
∞
= −
+−=
1 00
0
n
n
n
n
n
n
zz
a
zzazf
( )
( )
,2,1,0'
'
'
2
1
2
1
0
=
−
= ∫ +−−
nzd
zz
zf
i
a
C
nn
π
( )
( )
,2,1,0'
'
'
2
1
1
1
0
=
−
= ∫ +
nzd
zz
zf
i
a
C
nn
π
Proof:
Since z is inside R we have R1 <|z-z0|=r < R2 , and |z’-z0|= R1 on C1 and R2 on C2.
Start with the Cauchy’s Integral Formula:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
∫∫∫∫∫∫ −
−
−
=→
−
+
−
+
−
+
−
=
212
0
1
1
01
'
'
'
'
'
'
'
'
'
'
'
'
'
'
'
'
'
'
0
CCC
P
P
P
PC
dz
zz
zf
dz
zz
zf
zfdzdz
zz
zf
dz
zz
zf
dz
zz
zf
dz
zz
zf
zf
86. SOLO Complex Variables
Laurent’s Series (continue - 1)
Power Series
Pierre Alphonse Laurent
1813 - 1854
C1
x
y
R
C2R2
R1
z0
z z'
r
Proof (continue – 1):
Since z and z’ are inside R we have R1 >|z-z0|=r >R2, |z’-z0|=R1.
From Cauchy’s Integral Formula: ( ) ( ) ( )
∫∫ −
−
−
=
21
'
'
'
'
'
'
CC
dz
zz
zf
dz
zz
zf
zf
Use the identity:
α
α
ααα
α −
+++++≡
−
−
1
1
1
1 12
n
n
For I integral:
−
−
−
−
−
+
−
−
++
−
−
+
−
=
−
−
−
−
=
−
− nn
zz
zz
zz
zzzz
zz
zz
zz
zz
zz
zzzzzz 0
0
0
0
1
0
0
0
0
0
0
00
'
'
1
1
''
1
'
1
'
1
1
'
1
'
1
( )
( )
( )
( )
( )
( ) ( )
( )
( )
( )
( ) ( )
( ) ( )
( ) ( ) ( ) ( ) ( ) n
n
n
R
C
n
n
n
zs
C
n
za
C
za
C
Rzzzazzzaza
zzzz
zdzf
i
zz
zz
zz
zdzf
i
zz
zz
zdzf
izz
zdzf
i
n
n
+−⋅++−⋅+=
−−
−
+
−
−
++−
−
+
−
=
∫
∫∫∫
−
0000100
0
0
1
0
0
02
00
2
0
2
01
2
00
2
''
''
2
'
''
2
1
'
''
2
1
'
''
2
1
π
πππ
( )
∫ −1
'
'
'
2
1
C
zd
zz
zf
iπ
We have:
( )
( )
n
n
n
C
n
n
n
R
r
rR
MR
dR
rRR
Mr
zzzz
zdzfzz
R
−
=
−
≤
−−
−
≤ ∫∫ 11
1
2
0
1
110
0
2''
''
2 0
π
θ
ππ
where |f (z)|<M in R and r/R1< 1, therefore: 0
∞→
→
n
nR
87. SOLO
Complex Variables
Laurent’s Series (continue - 2)
Power Series
Pierre Alphonse Laurent
1813 - 1854
C1
x
y
R
C2R2
R1
z0
z z'
r
Proof (continue – 1):
Since z and z’ are inside R we have R1 >|z-z0|=r > R2, |z’-z0|=R2.
From Cauchy’s Integral Formula: ( ) ( ) ( )
∫∫ −
−
−
=
21
'
'
'
'
'
'
CC
dz
zz
zf
dz
zz
zf
zf
Use the identity:
α
α
ααα
α −
+++++≡
−
−
1
1
1
1 12
n
n
For II integral:
−
−
−
−
−
+
−
−
++
−
−
+
−
=
−
−
−
−
=
−
−
− nn
zz
zz
zz
zzzz
zz
zz
zz
zz
zz
zzzzzz 0
0
0
0
1
0
0
0
0
0
0
00
'
'
1
1''
1
1
'
1
11
'
1
( ) ( )
( )
( )
( )
( )
( )
( )
( )
( ) ( )
( ) ( )
( ) ( ) ( ) ( ) n
n
n
R
C
n
n
n
za
C
n
za
CC
Rzzzazzza
zzzz
zdzfzz
i
zzzz
zdzf
izzzz
zdzf
i
zdzf
i
n
n
−
+−
+−
−
−
+−+−−
+−++−=
−−
−
+
−
−
++
−
−
+=
−
+−−
∫
∫∫∫
1
001
1
001
0
0
1
0
1
00
2
0
0
0
01
0
01
00
'
'''
2
1
1
'
''
2
11
'
''
2
1
''
2
1
π
πππ
( )
∫ −C
zd
zz
zf
i
'
'
'
2
1
π
We have:
( )
( )
n
n
n
C
n
n
n
r
R
rR
RM
dR
rRr
MR
zzzz
zdzfzz
R
−
=
−
≤
−−
−
≤ ∫∫−
2
2
2
2
0
2
2
2
0
0
2'
'''
2
1
0
π
θ
ππ
where |f (z)|<M in R and R2/r< 1, therefore: 0
∞→
− →
n
nR Return to Table of Contents
88. Z2 Transform
C1
x
y
R
C2r2 z0
z
r
z'
C
r1
SOLO
Z-Transform Two Sided
( ) ( )∑
∞
−∞=
−
=
n
n
zTnfzF
Example 1
( ) Tn
aTnf =
( ) ( )∫
−
=
C
n
dzzzF
j
Tnf 1
2
1
π
( )
<<
−
=
=
><
−
=
=
==
∑ ∑
∑
∑∑
−∞=
∞+
=
+∞
=∞+
−∞=
∞+
−∞=
−
1
0
0
0
/1
/
0
1
1
n k
T
T
Tk
TT
n
T
n
T
T
n
T
n
n
T
n
nTn
naz
az
az
a
z
a
z
z
a
nza
z
az
a
z
a
zazF
89. Z2 TransformSOLO
Z-Transform Two Sided
Example 2
−+
−+
<<
<<
gg
ff
r
z
r
rr
ξ
ξ
ξξ −+
<< gg
rzr
−−++ << gfgf rrzrr
( ) ( ){ } ( )∫
= −
C
d
z
GF
j
TngTnf ξ
ξ
ξξ
π
1
2
1
Z
−−++
−−
++
<<
<<<
><<
gfgf
fg
gf
rrzrr
nrrz
nrzr
0&/
0&/
ξ
ξ
( ) ( ){ } ( ) ( ) ( ) ( )
( ) ( ) ( )∫∫ ∑
∑ ∫∑
=
=
==
−
<<∞+
−∞=
−
−
+∞=
−∞=
−
<<
−
+∞=
−∞=
−
−+
<
−
>
+
C
r
z
r
C n
n
n
n
n
rr
C
n
n
n
n
d
z
GF
j
d
z
TngF
j
zTngdzF
j
zTngTnfTngTnf
gg
n
f
n
f
ξ
ξ
ξξ
π
ξ
ξ
ξξ
π
ξξ
π
ξ
ξ
11
1
2
1
2
1
2
1
00
Z
90. Z2 TransformSOLO
Z-Transform Two Sided
Example 2 (continue – 1)
{ }
><
−
=
−−
=
−−
<>
−
=
−
−
=
−−
=
∫
∫
→
<<
→
<<
zban
ba
z
ba
z
b
z
b
z
a
aResd
b
z
b
z
a
a
j
zban
z
ba
z
ba
Resd
z
baj
ba
TT
TT
TT
C
T
T
T
T
b
z
b
b
z
T
T
T
T
C
TT
TTTTa
b
z
a
TT
TnTn
T
T
T
T
T
T
&0
1111
1
2
1
&0
1
1
1
11
1
1
1
11
2
1
ξ
ξ
ξ
ξ
ξ
ξ
ξ
ξ
ξ
ξπ
ξξ
ξ
ξ
ξ
ξπ
ξ
ξ
ξ
ξ
Z
( ) ( ){ } ( )∫
= −
C
d
z
GF
j
TngTnf ξ
ξ
ξξ
π
1
2
1
Z
−−++
−−
++
<<
<<<
><<
gfgf
fg
gf
rrzrr
nrrz
nrzr
0/
0/
ξ
ξ
91. Z TransformSOLO
Properties of Z-Transform Functions
Z - Domaink - Domain
( )kf ( ) ( ) −+
∞
=
−
<<= ∑ ff
k
k
rzrzkfzF
0
1 ( ) −+
<<∑=
ii ff
M
i
ii rzrzFc minmax
1
Linearity ( )∑=
M
i
ii kfc
1
2 ( ) ( ) ,2,10 ==−− kkfmkf ( )zFz m−
Shifting
( )mkf − ( ) ( ) ( )
∑=
−−−
−+
m
k
kmm
zkfzFz
1
( )mkf + ( ) ( ) ( )
∑=
−
−
m
k
kmm
zkfzFz
1
( )1+kf ( ) ( )0fzFz −
3 Scaling ( )kfak ( ) ( ) ( ) −+
∞
=
−−−
<<= ∑ ff
k
k
razrazakfzaF
0
11
92. Z TransformSOLO
Properties of Z-Transform Functions (continue – 1)
4 Periodic Sequence ( )kf
( ) ( ) −+ <<
−
111
1
ffN
N
rzrzF
z
z
N = number of units in a period
Rf1- ,+ = radiuses of convergence in F(1) (z)
F(1) (z) = Z -Transform of the first period
5 Multiplication by k ( )kfk
( )
−+ <<− ff rzr
zd
zFd
z
6 Convolution ( ) ( ) ( ) ( )∑
∞
=
−=∗
0
:
m
mkhmfkhkf ( ) ( ) ( ) ( )−−++ <<⋅ hfhf rrzrrzHzF ,min,max
7 Initial Value ( ) ( )zFf
z ∞→
= lim0
8 Final Value ( ) ( ) ( ) ( ) existsfifzFzkf
zk
∞−=
→∞→
1limlim
1
Z - Domaink - Domain
( )kf ( ) ( ) −+
∞
=
−
<<= ∑ ff
k
k
rzrzkfzF
0
96. L2 Transform
( ) 0>=
−
aetf
ta
SOLO
Laplace Transform Two Sided
Example 2
{ }
ab
d
bsa
ee
fg
j
j
tbta
=<<−=−
−−
−
−
−=
−−
∞+
∞−
−−
∫
σσσσσ
ξ
ξξ
ξ
σ
σ
ξ
ξ
11
2L
{ }
( )
( )
>=<=−
+
+
<=<=
−
−
=
+
−
−
0&
1
0&
1
2
tsreala
as
tasreal
as
e
f
f
ta
σσ
σσ
L
( ) 0>=
−
betg
tb
Find the two sided Laplace transform of f (t) g (t)
{ }
( )
( )
>=<=−
+
+
<=<=
−
−
=
+
−
−
0&
1
0&
1
2
tsrealb
bs
tbsreal
bs
e
f
f
tb
σσ
σσ
L
{ }
ba
d
bsa
ee
gf
j
j
tbta
+=−<<<−
+−
−
+
=
++
∞+
∞−
−−
∫
σσσσσ
ξ
ξξ
ξ
σ
σ
ξ
ξ
11
2L
( )basbsa
Res
a
+−
−=
−−−
−=
=
111
ξξξ
( )basbsa
Res
a
++
−=
+−+
=
−=
111
ξξξ
C1
σ
ω
b−σ a
0<t
0
0
=∫<t
C2
σ
ω
b+σa−
0>t
0
0
=∫>t
97. SOLO
References
A. Papoulis, “The Fourier Integral and its Applications”, McGraw Hill, 1962
R.N. Bracewell, “The Fourier Transform and its Applications”, McGraw Hill, 1965, 1978
J.W. Goodman,“Introduction to Fourier Optics”, McGraw Hill, 1968
H. Stark, Ed. “Applications of Optical Fourier Transform”, Academic Press, 1982
A. Papoulis, “Systems and Transforms with Applications in Optics”, McGraw Hill, 1968
Fourier Transform
Athanasios Papoulis
1921-2002 Ronald N. Bracewell
1921 -
Joseph W. Goodman
William Ayer Professor, Emeritus
Packard 352
Department of Electrical Engineering
Stanford University
Stanford, CA 94305
Email: goodman@ee.stanford.edu
98. January 6, 2015 98
SOLO
Technion
Israeli Institute of Technology
1964 – 1968 BSc EE
1968 – 1971 MSc EE
Israeli Air Force
1970 – 1974
RAFAEL
Israeli Armament Development Authority
1974 – 2013
Stanford University
1983 – 1986 PhD AA
99. Raymond Paley
1907 - 1933
Norbert Wiener
1894 - 1964
Paley – Wiener Condition
A necessary and Sufficient condition for a square-integrable
function A (ω) ≥ 0 to be the Fourier spectrum of a causal function
is the convergence of the integral:
( )
∞<
+∫
+∞
∞−
ω
ω
ω
d
A
2
1
ln
SOLO
100. The Mellin Transform
( ) ( )∫
∞
−
=
0
1
: s
M exfsF
SOLO
Hjalmar Mellin
1854 - 1933
Putting: tdexdex tt −−
−=→=
( )11 −−−
= sts
ex
( ) ( )∫
+∞
∞−
−−
= tdeefsF tst
M
We can see that the Mellin Transform of the function f (t) is identical to the
Bilateral Laplace Transform of f (e-t
).
101. SOLO
Example
( )
∫
∞
0
sin
dk
k
kr
Let compute:
x
y
R
ε
A
B
C
D
E
F
G
H
Rx =Rx −=
For this use the integral: 0=∫ABCDEFGHA
zi
dz
z
e
Since z = 0 is outside the region of integration
0=+++= ∫∫∫∫∫
−
− BCDEF
ziR xi
GHA
zi
R
xi
ABCDEFGHA
zi
dz
z
e
dx
x
e
dz
z
e
dx
x
e
dz
z
e
ε
ε
∫∫∫∫∫∫
∞∞
∞→
→
−
∞→
→
∞→
→
−
−∞→
→
===
−
=+
00
0000
sin
2
sin
2
sin
lim2limlimlim dk
k
rk
idx
x
x
idx
x
x
idx
x
ee
dx
x
e
dx
x
e
R
R
R xixi
R
R xi
RR
xi
R ε
ε
ε
ε
ε
ε
ε
ε
πθθθε
ε ππ
ε
ε
π
θ
θ
ε
ε
ε
ε
θ
θθ
idideidei
e
e
dz
z
e i
ii
eii
i
eiez
GHA
zi
−==== ∫∫∫∫ →→
=
→
00
1
0
0
00
limlimlim
( ) 01
2
2
0
/2
/2sin
0
sin
00
∞→
−−
≥
−
=
→−=≤=≤= ∫∫∫∫∫
R
RRReRii
i
eRieRz
BCDEF
zi
e
R
dedededeRi
eR
e
dz
z
e i
ii
π
θθθθ
π
πθ
πθθ
π
θ
ππ
θ
θ
θ
θθ
Therefore: 0
sin
2
0
=−= ∫∫
∞
πidk
k
rk
idz
z
e
ABCDEFGHA
zi ( )
2
sin
0
π
=∫
∞
dk
k
kr
Complex Variables
102. SOLO Complex Variables
Cauchy’s Theorem
C
x
y
R
Proof:
( ) 0=∫C
dzzf
If f (z) is analytic with derivative f ‘ (z) which is continuous at all points inside
and on a simple closed curve C, then:
( ) ( ) ( )yxviyxuzf ,, +=Since is analytic and has continuous
first order derivative
( )
y
u
i
y
v
x
v
i
x
u
zd
fd
zf
iyzxz
∂
∂
−
∂
∂
=
∂
∂
+
∂
∂
==
==
'
y
u
x
v
y
v
x
u
∂
∂
−=
∂
∂
∂
∂
=
∂
∂
& Cauchy - Riemann
( ) ( ) ( ) ( ) ( )
0
00
=
∂
∂
−
∂
∂
+
∂
∂
−
∂
∂
−=
++−=++=
∫∫∫∫
∫∫∫∫
RR
dydx
y
v
x
u
idydx
y
u
x
v
dyudxvidyvdxudyidxviudzzf
CCCC
q.e.d.
Augustin Louis Cauchy
)1789-1857(
103. SOLO Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Evaluation of Integrals
Jordan’s Lemma
If |F (z)| ≤ M/Rk
for z = R e iθ
where k > 0 and M are constants, then
where Γ is the semicircle arc of radius R, center at origin, in the
upper part of z plane, and m is a positive constant.
( ) 0lim =∫Γ
→∞
zdzFe zmi
R
x
y
Γ
R
Proof:
( ) 0lim =∫Γ
→∞
zdzFe zmi
R
using:
q.e.d.
( ) ( )∫∫
=
Γ
=
π
θθ
θ
θ
θ
0
deRieRFezdzFe iieRmi
eRz
zmi i
i
( ) ( ) ( )
( ) ∫∫∫
∫∫∫
−
−
−
−
−
−
=≤=
=≤
2/
0
sin
1
0
sin
1
0
sin
0
sincos
00
2
π
θ
π
θ
π
θθ
π
θθθθ
π
θθ
π
θθ
θθθ
θθθ
θθ
dRe
R
M
dRe
R
M
dReRFe
deRieRFedeRieRFedeRieRFe
Rm
k
Rm
k
iRm
iiRmRmiiieRmiiieRmi ii
2/0/2sin πθπθθ ≤≤≥ for
π2/π
1
θsin
πθ /2 θ
( ) ( )Rm
k
Rm
k
Rm
k
iieRmi
e
R
M
de
R
M
de
R
M
deRieRFe
i
−−
−
−
−
−=≤≤ ∫∫∫ 1
222
2/
0
/2
1
2/
0
sin
1
0
π
π
π
θ
π
θθ
θθθ
θ
( ) ( ) 01
2
limlim
0
=−≤ −
→∞→∞ ∫
Rm
kR
iieRmi
R
e
R
M
deRieRFe
i
π
θθ
θ
θ
Marie Ennemond Camille Jordan
1838 - 1922
104. SOLO Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Evaluation of Integrals
Jordan’s Lemma Generalization
If |F (z)| ≤ M/Rk
for z = R e iθ
where k > 0 and M are constants, then
for Γ a semicircle arc of radius R, and center at origin:
( ) 00lim >=∫Γ
→∞
mzdzFe zmi
R
x
y
Γ
R
where Γ is the semicircle, in the upper part of z plane.
1
( ) 00lim <=∫Γ
→∞
mzdzFe zmi
R
x
y
Γ
R
where Γ is the semicircle, in the down part of z plane.
2
( ) 00lim >=∫Γ
→∞
mzdzFe zm
R x
y
Γ
R
where Γ is the semicircle, in the right part of z plane.
3
( ) 00lim <=∫Γ
→∞
mzdzFe zm
R
where Γ is the semicircle, in the left part of z plane.
4
x
yΓ
R
105. SOLO Complex Variables
The Residue Theorem, Evaluations of Integral and Series
Evaluation of Integrals
Integral of the Type (Bromwwich-Wagner) ( )∫
∞+
∞−
jc
jc
ts
sdsFe
iπ2
1
The contour from c - i ∞ to c + i ∞ is called Bromwich Contour
Thomas Bromwich
1875 - 1929
x
y
0<
Γt
R
c
x
y
0>Γt
R c
( ) ( ) ( ) ( )
( )
( )
( )
<
>
==
+==
∫
∫∫∫ Γ
∞+
∞−
→∞
∞+
∞−
0
0
2
1
lim
2
1
2
1
tzFeRes
tzFeRes
zdzF
i
sdsFesdsFe
i
sdsFe
i
tf
tz
planezRight
tz
planezLeft
ts
ic
ic
ts
R
ic
ic
ts
π
ππ
where Γ is the semicircle, in the right part of z plane, for t < 0.
where Γ is the semicircle, in the left part of z plane, for t > 0.
This integral is also the Inverse Laplace Transform.
Notes de l'éditeur
R.N. Bracewell, “The Fourier Transform and Its Applications”, McGraw Hill, 2nd Ed., 1978
Al. Spãtaru, “Teoria Transmisiunii Informaţiei” (romanian), Editura Technicã, Bucureşti, 1965
R.N. Bracewell, “The Fourier Transform and Its Applications”, McGraw Hill, 2nd Ed., 1978
Al. Spãtaru, “Teoria Transmisiunii Informaţiei” (romanian), Editura Technicã, Bucureşti, 1965
R.N. Bracewell, “The Fourier Transform and Its Applications”, McGraw Hill, 2nd Ed., 1978
Al. Spãtaru, “Teoria Transmisiunii Informaţiei” (romanian), Editura Technicã, Bucureşti, 1965
R.N. Bracewell, “The Fourier Transform and Its Applications”, McGraw Hill, 2nd Ed., 1978
Al. Spãtaru, “Teoria Transmisiunii Informaţiei” (romanian), Editura Technicã, Bucureşti, 1965
R.N. Bracewell, “The Fourier Transform and Its Applications”, McGraw Hill, 2nd Ed., 1978
Al. Spãtaru, “Teoria Transmisiunii Informaţiei” (romanian), Editura Technicã, Bucureşti, 1965
R.N. Bracewell, “The Fourier Transform and Its Applications”, McGraw Hill, 2nd Ed., 1978
Al. Spãtaru, “Teoria Transmisiunii Informaţiei” (romanian), Editura Technicã, Bucureşti, 1965
R.N. Bracewell, “The Fourier Transform and Its Applications”, McGraw Hill, 2nd Ed., 1978
Al. Spãtaru, “Teoria Transmisiunii Informaţiei” (romanian), Editura Technicã, Bucureşti, 1965
R.N. Bracewell, “The Fourier Transform and Its Applications”, McGraw Hill, 2nd Ed., 1978
Al. Spãtaru, “Teoria Transmisiunii Informaţiei” (romanian), Editura Technicã, Bucureşti, 1965
R.N. Bracewell, “The Fourier Transform and Its Applications”, McGraw Hill, 2nd Ed., 1978
Al. Spãtaru, “Teoria Transmisiunii Informaţiei” (romanian), Editura Technicã, Bucureşti, 1965
Minkoff, J., “Signals, Noise, and Active Sensors”, John Wiley & Sons, 1992, pp.72-74
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Minkoff, J., “Signals, Noise, and Active Sensors”, John Wiley & Sons, 1992, pp.72-74
François Le Chevalier, “Principes De Traitement Des Signaux Radar et Sonar”, Masson, 1989, pp.39 et 75
Ralph Deutsch, “System Analysis Techniques”, Prentice-Hall, Inc., 1969, § 4.7, “Effective Bandwidth”, pp.126-138
Athanasios Papoulis, “Signal Analysis”, McGraw-Hill, 1977, § 8-2, Uncertainty Principle and Sophisticated Signals, pp.273-278
Minkoff, J., “Signals, Noise, and Active Sensors”, John Wiley & Sons, 1992, pp.72-74
François Le Chevalier, “Principes De Traitement Des Signaux Radar et Sonar”, Masson, 1989, pp.39 et 75
Ralph Deutsch, “System Analysis Techniques”, Prentice-Hall, Inc., 1969, § 4.7, “Effective Bandwidth”, pp.126-138
Minkoff, J., “Signals, Noise, and Active Sensors”, John Wiley & Sons, 1992, pp.72-74
François Le Chevalier, “Principes De Traitement Des Signaux Radar et Sonar”, Masson, 1989, pp.39 et 75
Ralph Deutsch, “System Analysis Techniques”, Prentice-Hall, Inc., 1969, § 4.7, “Effective Bandwidth”, pp.126-138
Athanasios Papoulis, “Signal Analysis”, McGraw-Hill, 1977, § 8-2, Uncertainty Principle and Sophisticated Signals, pp.273-278
Minkoff, J., “Signals, Noise, and Active Sensors”, John Wiley & Sons, 1992, pp.72-74
François Le Chevalier, “Principes De Traitement Des Signaux Radar et Sonar”, Masson, 1989, pp.39 et 75
Ralph Deutsch, “System Analysis Techniques”, Prentice-Hall, Inc., 1969, § 4.7, “Effective Bandwidth”, pp.126-138
Athanasios Papoulis, “Signal Analysis”, McGraw-Hill, 1977, § 8-2, Uncertainty Principle and Sophisticated Signals, pp.273-278
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http://en.wikipedia.org/wiki/Discrete-time_Fourier_transform
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http://en.wikipedia.org/wiki/Discrete_Fourier_transform
Oppenheim, A.V., Schafer, R.W., “Discrete Time Signal Processing”, Prentice Hall, 1989, Ch. 8
http://en.wikipedia.org/wiki/Discrete_Fourier_transform
Oppenheim, A.V., Schafer, R.W., “Discrete Time Signal Processing”, Prentice Hall, 1989, Ch. 8
http://en.wikipedia.org/wiki/Discrete_Fourier_transform
Oppenheim, A.V., Schafer, R.W., “Discrete Time Signal Processing”, Prentice Hall, 1989, Ch. 8
http://en.wikipedia.org/wiki/Discrete_Fourier_transform
Oppenheim, A.V., Schafer, R.W., “Discrete Time Signal Processing”, Prentice Hall, 1989, Ch. 8
http://en.wikipedia.org/wiki/Discrete_Fourier_transform
Oppenheim, A.V., Schafer, R.W., “Discrete Time Signal Processing”, Prentice Hall, 1989, Ch. 8
http://en.wikipedia.org/wiki/Discrete_Fourier_transform
Oppenheim, A.V., Schafer, R.W., “Discrete Time Signal Processing”, Prentice Hall, 1989, Ch. 8
http://en.wikipedia.org/wiki/Discrete_Fourier_transform
Oppenheim, A.V., Schafer, R.W., “Discrete Time Signal Processing”, Prentice Hall, 1989, Ch. 8
http://en.wikipedia.org/wiki/Discrete_Fourier_transform
Oppenheim, A.V., Schafer, R.W., “Discrete Time Signal Processing”, Prentice Hall, 1989, Ch. 8
M.A. Richards, “Modern Radar Control”, GeorgiaTech Course, 2004
http://en.wikipedia.org/wiki/Discrete_Fourier_transform
Oppenheim, A.V., Schafer, R.W., “Discrete Time Signal Processing”, Prentice Hall, 1989, Ch. 8
M.A. Richards, “Modern Radar Control”, GeorgiaTech Course, 2004
http://en.wikipedia.org/wiki/Discrete_Fourier_transform
Oppenheim, A.V., Schafer, R.W., “Discrete Time Signal Processing”, Prentice Hall, 1989, Ch. 8
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Poularikas, A.,D., Ed., “The Transforms and Applications Handbook”, IEEE Press, CRC Press, 1996, pp. 450-451
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