Lecture on Communication Theory
Chapter 2. Representation of signals and systems
2.1 Introduction
Deterministic signals : A class of signals where
waveforms are defined exactly as function of time.
2.2 F.T.
1) Let g(t) = non-periodic deterministic signal
FT of g(t)
G(f) =



g (t ) exp(  j 2ft)dt

where f = frequency
inverse F.T.
g(t) =


G( f ) exp( j 2ft)d f

2) To exist FT of g(t), sufficient condition (not necessary)
Dirichlet’s conditions

 g(t) is single-valued, with a finite number of maxima and minima
in any finite time interval
 g(t) has a finite # of discontinuities in any finite time interval
 g(t) is absolutely integrable, that is,



g (t ) dt  
3) Physical realizability  existence of F.T.
4) All energy signal  Fourier transformable
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Lecture on Communication Theory
2.2 Fourier Transform
1. Notation
t : time [sec]
f : frequency [Hertz]
w= 2f : angular frequency [radians/sec]
식 G(f) = F [g(t)]
식 g(t) = F-1 [G(f)]
F, F-1 : linear operator
 g(t)  G(f)



2. Continuous Spectrum
1) Continuous Spectrum
G(f) = |G(f)| exp[j(f)]
|G(f)| : Continuous amplitude spectrum of g(t)
(f) : Continuous phase spectrum of g(t)
2) For real-valued g(t)
G(f) = G*(f) = |G(-f)| = | G(f)|
(-f) = - (f)
Complex conjugation = conjugate symmetric
Real : y 축 대칭, symmetric
Imaginary : 원점 대칭, anti-symmetric
2

Lecture on Communication Theory
Ex1) Rectangular pulse
AT
A
-1/T
T/2
-T/2
1/T
0
Def) rect(t) = 1 -1/2 < t < 1/2
0 |t| > 1/2
 g(t) = A rect (t/T)
 sin( fT ) 

G ( f )  AT 

fT




Def) sinc function
sinc( λ) 
sin(  )


   G(f) = AT sinc (fT)
 A rect(t/T)  AT sinc (fT)
3) Real Symmetric  Real Symmetric
FT
Ex2) exp(-at) u(t) 
exp(at) u(-t) 
1
a  j 2f
1
a  j 2f
3

Lecture on Communication Theory
2.3 Properties of the Fourier Transform
1. Linearity (Superposition)
Let g1(t)  G1(f) and g2(t)  G2(f).
Then for all constants C1 and C2, we have
C1 g1(t) + C2 g2(t)  C1 G1(f)+ C2 G2(f)
2. Time Scaling
Let g(t)  G(f). Then g(at)  1/|a| G(f/a)
when a=-1, g(-t)  G(-f)
3. Duality
If g(t)  G(f), then G(t)  g(-f)
4. Time shifting
If g(t)  G(f), then
g(t-t0)  G(f)exp(-j2ft0)
5. Frequency shifting
If g(t)  G(f), then exp(j2fct)g(t)  G(f-fc)
where fc is a real constant
 modulation theorem
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Lecture on Communication Theory
Ex5) RF pulse
g(t) =a rect(t/T)cos(2fct)
G(f) = AT/2 {sinc[T(f-fc)] + sinc[T(f+fc)]}
|G(f)|
g(t)
-fc
T
fc
2
T
6. Area under g(t)
If g(t)  G(f), then
7. Area under G(f)
If g(t)  G(f), then

 g (t )dt  G (0)


g (0)   G( f )df

8. Differentiation in the Time Domain
Let g(t)  G(f), and assume that the 1st derivative of g(t) is
Fourier transformable. Then
d
g (t )  j 2fG ( f )
dt
n-th generalization
dn
g (t )  ( j 2f ) n G ( f )
n
dt
Ex6) Gaussian pulse
g(t) = exp(-pt2)  exp(-pf2)
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Lecture on Communication Theory
9. Integration in the Time Domain
Let g(t)  G(f), then provided that G(0)=0, we have
 g ( )d 
t
1
G( f )
j 2f
10. Conjugate Functions
If g(t)  G(f), then for a complex-valued time function g(t),
we have
g*(t)  G*(-f),
when the asterisk denotes the complex conjugate operation
<Corollary> g*(-t)  G*(f)
11. Multiplication in the Time Domain
(Multiplication Theorem)
Let g1(t)  G1(f) and g2(t)  G2(f), Then
g1(t)g2(t) 
 G ( )G ( f   )d


1
2
 G1 ( f )  G2 ( f )
12. Convolution in the Time Domain
(Convolution Theorem)
Let g1(t)  G1(f) and g2(t)  G2(f), Then
g1(t)  g2(t)  G1(f)G2(f)
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Lecture on Communication Theory
2.4 Rayleigh’s Energy Theorem
1. Rayleigh’s Energy Theorem


E =  g (t ) dt  G( f ) df
Ex9) E = A 2
2

2

linear transform
A 2 
f
2
sinc
(
2
Wt
)
dt

(
)
rect
(
)df

2W 
2W
A 2 W
(
)  df
W
2W
A2

2W
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Lecture on Communication Theory
2.5 The inverse relationship btw time and frequency
1. Time과 frequency 관계
1) linear transform of FT
2) Time 
Frequency
wide

low
narrow 
high
cannot specify arbitrary function of time and frequency.
3) Strictly limited in time  infinite in freq.
infinite in time  strictly band-limited in freq.
cannot be strictly limited in both time and freq.
2. Bandwidth
1) def) extent of significant spectral content of the signal for
signal positive frequencies
2) strictly band-limited 경우가 아닐 경우의 definitions
 BW = Main lobe bounded by well-defined nulls
Base-band
Pass-band
modulation
-1/T
1/T
0
0
BW = 2/T
BW = 1/T
2 배 차이
8
fc
fc + 1/T
Lecture on Communication Theory
 3 dB Bandwidth : 1/ 2 of its peak value
 rms(root mean square) BW
Wrms
  f 2 G ( f ) 2 df

 
   G( f ) 2 df
 
1/ 2





(장점) mathematical evaluation
(단점) not easily measurable in lab
3. Time-Bandwidth Product
1) for base-band sync function
= (duration T) • (BW of main lobe = 1/T) =1
2) Trms Wrms  1/4
“ = “ when Gaussian pulse
2.6 Dirac Delta Function
1. Definition
(t) = 0, t  0
  (t )dt  1


 (t) is even function of t
2. Shifting property
 g (t ) (t  t )dt  g (t )


0
0
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Lecture on Communication Theory
3. Replication property g(t)  (t) = g(t)
 g ( ) (t   )d  g (t )



4. Fourier Transform F [(t)] = 1
g(t)
G(f)
1.0
t
f
5. Applications of the Delta Function
1) dc signal
1  (f) 

 exp(  j 2ft )dt
  cos( 2ft ) dt




2) Complex Exponential Function
exp(j2fct)  (f-fc)
3) Sinusoidal Functions
cos (2fct) = 1/2 [exp (j2fct) + exp (-j2fct)]
 1/2 [(f-fc) + (f+fc)]
f
Sin(2fct) 1/2j [(f-fc) - (f+fc)]
f
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Lecture on Communication Theory
4) signum Function
sgn[t] =
g(t)
+1, t > 0
0, t = 0
-1, t < 0
+ 1.0
t
0
- 1.0
 j 4f
a  ( 2f ) 2
exp(-|a|t)sgn(t) 
 F[sgn(t)] = lim
a 0
|G(f)|
2
 j 4f
1

a 2  (2f ) 2
jf
5) Unit step function
u(t) =
f
0
g(t)
1,
t>0
1/2, t = 0
0, t < 0
t
|G(f)|
 u(t) = 1/2 [sgn(t) + 1]
u(t) 
1/2
1
1
 (f )
j 2f 2
f
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Lecture on Communication Theory
6) Integration in the Time Domain (Revisited)
Let
y (t )   g ( ) d   g ( )u (t   ) d g (t )  u (t )
1
1
 Y ( f )  G ( f )[
  ( f )]
j 2f
2
1
1
Y ( f ) 
G ( f )  G (0) ( f )
j 2f
2
t

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Lecture on Communication Theory
2.7 Fourier Transforms of periodic signals
Let a periodic signal be gTo(t) with period T0
complex exponential Fourier series

g T (t )   cn exp( j 2nf 0t )
0
1
n  
cn 
1 T /2
g To (t ) exp( j 2nf0 t )dt

T0 T / 2
2
0
0
where f0 = 1/T0 ; fundamental freq.
Generating function g(t)
gTo(t) , - T0/2  t  T0/2
0, elsewhere
g(t) =
from
cn  f 0  g (t ) exp(  j 2nf 0t )dt  f 0G (n f 0 )

2


m  
n  
3
 g T 0 (t )   g (t  mT0 )  f 0  G (nf 0 ) exp( j 2nf 0t )
Use 1

3
Poisson’s sum formula

 g (t  mT )  f  G (nf ) ( f  nf )
m  
0
0
n  
0
0
where G(f)  g(t)
(Observations) periodicity in time domain
 discrete spectrum defined at nf0
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Lecture on Communication Theory

ex11)

  (t  mT )  f   ( f  nf )
m  
0
0
n  
14
0
Lecture on Communication Theory
g(t)
0
sampling 1
 T (t )
T
0 T/2 T
2
G(f)*  2 ( f )
T
0
2/T
0
2/T
2(f)
T
G(f)
sampling 2
T
0
T/8
2
RECT 2 ( f )
T
0
2/T
0
2/T
G(f)* RECT 2 ( f )
T
<HW> 2.1, 2.4, 2.8, 2.17, 2.18
15
t
t
f
f
f
0
rectT (t )
2T
t
f
f
Lecture on Communication Theory
2.8 Transmission of signals through linear system
Def) linear system : a system which holds the principle of
superposition

x1(t)
h(t)
y1(t)
x2(t)
h(t)
y2(t)
c1x1(t) + c2x2(t)
h(t)
c1y1(t) + c2y2(t)
1. Time Response
1) Impulse Response : the response of system to (t)
(t)
h(t)
h(t)
h(t)
y(t)
2) Convolution
x(t)
y(t) = x(t)  h(t)  h(t)  x(t)
  x( )h(t   )d

  h( ) x(t   )d

excitation time 
response time t
system - memory time t- 
; a weighted integral over the past history of the input signal,
weighted according to the impulse response of the system.
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Lecture on Communication Theory
ex) x(KT)
0 1 2
3
KT
1
h(KT )
h(KT)
-1
y (2T )   x( KT )h(2T  KT )dk
0 1 2
KT
2
h( KT )
-2 -1
0 1
KT
3
KT
4
KT
5
KT
6
KT
7
h(2T  KT )
-4 -3 -2 -1
 y (2T ) 
1
 4
0
0
h(T  KT )
 y (T ) 
 y ( 0) 
 y (T ) 
1
1
1
 5
 6
 7
1
3
4
 y (2T )  5
 y (3T )  4
KT
 y (4T )  2
 y (5T )  1
KT
y(kt)
 y (6T )  0
-1 0
17
1 2 34 5
KT
Lecture on Communication Theory
t
T
T
*
*

t
T
-2T

2T
t
t
T

 x(t ) y (t )dt

x(t)
1
y(t)
T
t

*
* correlation
t
남자
여자
0
-1
18
1
0
-1
2T
Lecture on Communication Theory
ex12) Tapped-Delay-line Filter
assumptions)
1
h(t) = 0 for t < 0
2
h(t) = 0 for t  Tf
 y (t )  0T h( ) x(t   )d
f
sampling with , t=n, =k
N 1
 y(n )   h(k ) x(n  k )
k 0
where N  = Tf
let wk = h(k ) 
N 1
y(n ) =  wk x(n  k )
k 0
= w0x[n ] + w1 x[n  - ] + ..... + wN-1 x[n -(N-1) ]
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Lecture on Communication Theory
2. Causality & stability
1) Causal if a system does not respond before the excitation is
applied
h(t) = 0, t < 0  causality
 Real time으로 동작 하는 system  must be causal
 Memory 기능이 있는 것  can be non-causal
2) Stable if the output signal is bounded for all bounded input
signals (BIBO)
cf, BIBO: bounded input-bounded output
BIBO stability criterion
BIBO stability 

 h(t ) dt  
3. Frequency Response
1)
x(t)
y(t)
h(t)
y(t) = x(t)  h(t)
Y(f) = X(f)H(f)
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Lecture on Communication Theory
2) H(f)의 (freq. domain에서의) 표현
 H(f) = |H(f)| exp(j(f))
|H(f)| : Amplitude response
(f) : phase response
 h(t)가 real  H(f) : conjugate symmetry
|H(f)| = |H(-f)| even
(f) = - (-f) odd
 polar form
ln H(f) = ln |H(f)| + j (f)
= a(f) + j (f)
a(f):gain[nepers]
(f):[radians] )
 a’(f) = 20 log10 |H(f)| [dB] decibels
 a’(f) = 8.69 a(f)  1 neper = 8.69 dB
3) BW - 3 dB BW
B
-B
-fc
0
fc-B
fc+B
4. Paley-Wiener Criterion : freq. domain equivalent of
causality requirement
a(f) : gain of causal filter




a( f )
1 f
2
df  
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Lecture on Communication Theory
2.9 Filters
1. Ideal Low Pass Filter
1, -B  f  B
0, |f| > B
 h(t) = 2B sinc[2B(t-t0)]
(f) = -2f t0 : linear phase
|H(f)| =
t0 =0
1/B
t0 >0
t0
To make a causal filter, | sinc[2B(t-t0)] | << 1 for t < 0
If making digital filter, non-causal is O.K.
2. computer experiment I pulse response of ideal LPF
X(t)
-T/2
T/2
H(f)
B
-B
f
B=5/T, BT=5
BT=10
1/T
10/T
5/T
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Lecture on Communication Theory
(observations)
overshoot =  9 %
 overshoot is independent of B
 frequency of ripple is B
BT
5
10
20
100
Gibb’s phenomenon
Ringing
oscillation freq.
5 Hz
10 Hz
20 Hz
100 Hz
overshoot (%)
9.11
8.98
8.99
9.63
3. Fig 27. B = 1 Hz 일 때 f0 = 0.1(T=5), 0.25(T=2), 0.5(t=1),
1(T=0.5) Hz 입력이 들어갔을 때의 output
conclusion) BT  1 이 되어야 recognizable output이 나온다
4. Design of Filters
1) Basic Design steps
 approximation of a prescribed frequency response by a realizable
transfer function
 Realization of the approximating transfer function by a physical
device
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Lecture on Communication Theory
2) Stable system : BIBO
 complex frequency s = j2f plane 상에서 표시
H’(s) is a rational function
H’(s) = H(f) | j2f=s
= k
( s  z1 )( s  z 2 )...( s  z m )
( s  p1 )( s  p2 )...( s  pn )
SI
z1 z2, ... zm : zeros
p1 p2, ... pn : poles
SR
stability  Re[pi] < 0 for all i
 Minimum-phase systems
Re[pi] < 0, Re[zi] < 0
 Non-minimum phase systems
Re[pi] < 0, - Re[zi]  
3) Butterworth filters
Chebyshev filters
4) Implementation of filters
 Analog filters : (a) L, C
(b) C, R, OP-amp
 Discrete-time filters : switched-capacitor filters
(SAW) surface accoustic wave filters
 Digital filters : FIR, IIR
(장점) programmable, flexibility
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Lecture on Communication Theory
2.10 Hilbert Transform
1. Frequency selective filters
Phase selective filters
180º : ideal transformer
± 90 º : Hilbert Transform
2. Def. of Hilbert Transform
1  g ( )
gˆ (t )  
d
 t 
1  gˆ ( )
g (t )   
d
 t 
sgn(f)
1
And 1/t  -jsgn(f)
f
-1
 Gˆ ( f )   j sgn( f )G( f )
3. Applications
1) SSB Modulation
2) Mathematical basis for the representation of band-pass
signals
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Lecture on Communication Theory
ex13)
g (t )  cos( 2f c t )
1
G ( f )   ( f  f c )   ( f  f c )
2
ˆ
G ( f )   j sgn( f )G ( f )
j
   ( f  f c )   ( f  f c )sgn( f )
2
1
 ( f  f c )   ( f  f c )

2j
 gˆ (t )  sin( 2f c t )
 sin( 2f c t ) Hillbert

  cos( 2f c t )
4. Properties of Hilbert transform
Assumption) g(t) is real-valued
property
1) g (t ) & gˆ (t ) have the same amplitude spectrum
2) if g (t ) Hillbert

 gˆ (t )
then gˆ (t ) Hillbert

 - g (t )
pf )
 j sgn( f )
2
 1
3) g(t) and gˆ(t) are orthogonal
pf)
G(f)
f
x0
jG(f)
f
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Lecture on Communication Theory
2.11 Pre - envelope
1. Pre-envelope for positive frequency.
For a read-valued signal g(t)
pre-envelope of g(t)
g (t)  g(t)  jĝ(t )
G ( f )  g( f )  j j sgn( f )G( f )
FT.
, f 0
, f 0
, f 0
 2G( f )

 G( 0 )
0

G(f)
-W
W
G+(f)
f
W
-W
2. Pre-enveloped for negative frequency.
g (t)  g(t)  jĝ(t )
FT.
g (t)  g (t)



G ( f )  g( f )  j j sgn( f )G( f )
 0

  G( 0 )
 2G( f )

27
, f 0
, f 0
, f 0
f
Lecture on Communication Theory
G(f)
-W
W
G-(f)
f
-W
W
f
Pre-envelope for negative freq.
3. 용도: Useful in handling band-pass signals and systems
2.12 Canonical Representations of Band-Pass
signals
1. Base-band 신호와 Pass-band신호의 관계
Base-band signal ( 복소수 ) : complex envelope
 g I (t) : In - phase component
g~ (t )  g I (t )  jg Q (t )

 g Q(t) : Quadrature component
; low - pass signal
Pass-band signal ( 실수 ) : band-pass signal, narrow-band
signal
~(t )
g

Re( • )
exp( j 2f c t )
~ (t ) exp( j 2f t )
g (t )  Reg
c
 g I (t ) cos( 2f c t )  g Q (t ) sin( 2f c t )
; canonical form
28
g (t )
Lecture on Communication Theory
Pre-envelope of g(t)
~ (t ) exp( j 2f t )
g  (t )  g
c
~
G( f )
w
w
f
G( f )
 fc
fc
BW  2 W
BW  W
physical line
2 line(복소수)
G ( f )
1 line(실수)
spectral efficiency is same
29
f
fc
BW  2 W
Lecture on Communication Theory
2. Physical Implementation of pass-band signal
30
Lecture on Communication Theory
3. Expression in the polar form
g~(t )  a(t ) exp( j (t )) : base - band
a(t), (t) : real
Pass-Band
g (t )  a(t ) cos[ 2f c t   (t )]
a(t ) : natural envelope  amplitude mod
 (t ) : phase  angle mod .
4. Envelope 용어 정리
a(t) : envelope of g(t)
~
g (t) : complex envelope
g  (t) : pre - envelope
a(t)  g~ (t )  g (t)

31
Lecture on Communication Theory
t
~
ex14) g (t )  A rect( )
T
t
g (t )  A rect( ) cos( j 2f c t )
T
t
g  (t )  A rect( ) exp(  j 2f c t )
T
~ (t )  A rect( t )
a (t )  g
T
~(t )
g
g+(t)real
t
-T/2
T/2
g(t)
g +(t)image
~
G( f )
G ( f )
G( f )
f
0
-fc
0
0
fc
32
fc
Lecture on Communication Theory
2.13 band-pass systems
1.Pass-band 신호를 Base-band에서 표현
Base - band
Pass - band
~
Input
x (t )  X ( f ) x(t )  Re ~
x (t ) exp( j 2f ct )
~
~
Filter
h (t )  H ( f ) h(t )  Re h (t ) exp( j 2f ct )
Output ~
y (t )  Y ( f ) y(t )  Re ~y (t ) exp( j 2f ct )

Pass - band
Base - band

y (t )  x(t )  h(t )
~
2~
y (t )  ~
x (t )  h (t )
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Lecture on Communication Theory
2. Complex Filter의 구현
~
x ( t )  xI ( t )  j xQ ( t )
~
h ( t )  hI ( t )  j hQ ( t )
~y ( t )  y ( t )  j y ( t )
I
Q
~
x( t )  h ( t )
2 ~y ( t )  ~
 xI ( t )  j xQ ( t )hI ( t )  hQ ( t )
 xI ( t )  hI ( t )  xQ ( t )  hQ ( t )  jxI ( t )  hQ ( t )  xQ ( t )  hI ( t )
34
Lecture on Communication Theory
3. Band-pass system의 response를 구하는 과정 summary
1) x(t )  ~
x (t )
~
2) h(t )  h (t )
~
3) y (t )  ~
x (t )  h (t )
4) y (t )  Re~
y (t ) exp( j 2f t )
c
 결론  Pass - band filtering 은 (Base - band filtering  Modulation )
4. Computer Experiment II. Response of Ideal Base-pass
Filter to a pulsed RF Wave
BT=5일 경우의 RF파형
Base-band 파형 + Modulation
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Lecture on Communication Theory
2.14 Phase and Group Delay
A dispersive channel in a polar form
H ( f )  K exp  j ( f )
 ( f ) : nonlinear function of frequency
input signal x(t )
x(t )  m(t ) cos( 2f c t )
By expanding  ( f ) in a Taylor series
 ( f )   ( fc )  ( f  fc )
Define
p  
 ( f )
f
ff c
 ( fc )
2f c
1  ( f )
g  
2 f
f  fc
  ( f )  2f c p  2 ( f  f c ) g
 H ( f )  K exp  j 2f c p  j 2 ( f  f c ) g 
Equivalent low - pass filter
~
H ( f )  K exp j 2f c p  2f g 
36
Lecture on Communication Theory
Recevied signal
~
~
y (t ) 
 Y ( f )
1 ~
~
~
Y( f )  H ( f ) X ( f )
2
 K exp  j 2f c p exp  j 2f  g M ( f )
~
y (t )  K exp  j 2f  m(t   )
c
Finally
p
g
y (t )  Re~
y (t ) exp  j 2f c t

 Km(t   g ) cos2f c (t   p )
Conclusion
1)  p  
2)  g  
 ( fc )
; phase delay, carrier delay
2f c
즉, carrier is delayed by  p .
1  ( f )
2 f
; envelope delay, group delay
f  fc
즉, m(t ) is delayed by  g
(해석) if  g  constant  no ploblem
if  g  function of f  group delay variation 을
규정해야 한다.
37
Lecture on Communication Theory
2.15 Numerical Computation of the Fourier Transform
1. DFT & IDFT
1) Nyquist Sampling Theorem
Sampling Rate should be greater than twice the highest frequency
component of the input signal to avoid aliasing.
t
 fs
2
T
fs
2
T=NTs
fs
fs
fs=Nf
2) DFT & IDFT
{g0,g1,•••,gN-1}
Given finite data sequence
ex) analog gn=g(nTs)
DFT of gn
N 1
Gk   g n exp( 
n 0
IDFT of Gk
j 2
kn) where k=0,1,•••,N-1
N
1 N 1
j 2
g n   Gk exp(
kn) where n=0,1,•••,N-1
k

0
N
N
{G0,G1,•••,GN-1} : transform sequence
k=0,•••, N-1 : frequency index
38
Lecture on Communication Theory
3) Physical meaning of DFT
N 1
G0   g n
DC
n 0
2n
)
n 0
N
N 1
2 2n
G0   g n exp( 
)
n 0
N
N 1
G0   g n exp( 
1cycle/N
2cycle/N
•••
•••
2. Interpretation of the DFT and the IDFT.
DFT & IDFT: Gk and gn must be periodic.
DFT & IDFT: are linear.
39
Lecture on Communication Theory
3.FFT Algorithms.
DFT of gn
N 1
Gk   g nW nk
n 0
where
(1)
k = 0, 1, 2, , , , , , ,N-1
j 2
W  exp( 
)
N
W n  1, W N / 2  1
W ( k lN )( n mN )  W kn
m,l  0,1,2,  
i.e. Wkn : periodic with period N.
Assume N=2L, where L is integer
N
1
2
(1)
N 1
Gk   g nW   g nW nk
nk
n 0
n
N
1
2
N
1
2
n 0
n 0
N
2
  g nW nk   g N W
N
1
2
n
 k k
 n 
 2
2
kN
2
  ( g n  g N W )W kn
n 0
n
2
k  0,1,  , N  1
N
1
2
  ( g n  (1) k g
n 0
N
n
2
40
)W kn
Lecture on Communication Theory
1
N
k  even , i.e, k  2l , l  0,1,    1
2
N
2
1
G2 l   ( g n  g
n 0
n
2 ln
)(
W
)
N
2
N
1
2
  xn (W 2 ) ln ; N  po int
2
FFT
2
n 0
k  odd , i.e, k  2l  1, l  0,1,   N  1
2
N
1
2
G2 l 1   ( g n  g
n 0
n
( 2 l 1) n
)
W
N
2
N
1
2
  ( y nW n )(W 2 ) ln
n 0
N point FFT = 2  [
;
N
-point
2
41
N/2-point FFT
FFT]
Lecture on Communication Theory
ex) Fig 2.38
ex) Fig
2.38
Butterfly.
42
Lecture on Communication Theory
# of computation of DFT
# of computation of FFT
N2complex(x)+N(N-1)complex(+)
N
log 2 N if decimation in freq.
2
one complex(x) + 2complex(+)
ex) N=1024=1k
DFT
1M
200배 차이
FFT
5K
단 FFT는 2N개로 해야 함
Other algorithm : Decimation-in-time algorithm
 N log2N computations
4. Computation of IDFT
1 N 1
g n   GkW kn
N k 0
N 1
Ng n   Gk W kn


n  0,1,  , N  1
k 0
GK


Ng n complex Ng n
complex GK
FFT
conjugate
conjugate
where  : complex conjugate
43
N
qn
Lecture on Communication Theory
<HW1> 2.30, 2.32
문제) N=16(g0~g15)일 때의 Decimation-in-freq. Algorithm을
그림으로 그려라.
<Computer HW> 2.1 (see book)
Bit reversed
index.
44