디지털통신 주요식 정리
유상조 교수 (인하대학교 정보통신대학원)
Chapter 2 Signal and Linear System Analysis
2.2 Signal classification
E  lim
- energy
T

T  T
2

xt  dt   xt  dt
2

2
1 T


x
t
dt
T   2T T
- energy signal 0  E  , P  0
- power P  lim
- power signal 0  P  ,
E
2.3 Generalized Fourier series
xa t   n1 X nn t , t 0  t  t 0  T
N
- approximation of x(t) :
X n :coefficient,  n t  = linearly independent basis function
cn  
t 0 T
t0
 n t  n * t dt
2
- approximation error (integral squared error ISE)
 N   xt   x z t  dt
T
 Xn 
- to minimizes the ISE
- ISE=0를 만족하는 (즉
 x t 
1
cn
 xt  t dt
*
n
T
xt   xa t   n1 X nn t  ),
N
(complete orthogonal function) 구성 조건

T
xt  dt  n 1 cn X n
N
2
2
 Parseval’s theorem
2.4 Fourier series
- complete exponential Fourier series
xt   n   X n e jnw0t

where X n 
1
T0

t 0  t  t 0  T0
t 0 T0
t0
xt e  jnw0t dt
2.5 Fourier transform




X  f    x e  j 2f d , xt    X  f e  j 2ft df
- energy

E   xt  dt  
2


X  f  df
2

(Rayleigh’s energy theorem)
- energy spectral density (units of energy density) G  f   X  f
2

xt   x1 t   x2 t    x1 t x2 t   d
- convolution

2.6 Power spectral density and correlation
1
T  2T
P  lim
2

xt  dt   S  f df
T

T

( S(f): power spectral density)
- time average autocorrelation function for energy signal
    x   x    lim
 x x   d
T
T  T
- time average autocorrelation function for power signal
R   xt xt     lim
1
T  2T
 xt xt   dt
T
T
Wiener-Khinchine theorem

S  f    R e  j 2f d ,


R    S  f e j 2f df

2.7 Signal and linear systems
- linear system
yt   1 x1 t    2 x2 t   1x1 t    2 x2 t   1 y1 t    2 y2 t 
- time invariant system
yt  t 0   xt  t 0 
- impulse response
ht    t 
- output of the system
Fourier transform
- Gy  f   Y  f

yt    x ht   d

Y  f   H  f X  f 
 2  H  f X  f  2  H  f  2 G x  f 
for energy signal
- Sy  f   H f
 2 Sx  f 
for power signal
- distortionless transmission: 입력신호의 전 주파수 요소에 대해 동일한 감쇄와 지연
을 겪는 시스템
yt   H 0 xt  t 0 
H  f   H 0 e  j 2ft0
 distortionless system 조건
2.8 Sampling theory
- 만약 신호 x(t)의 주파수가 f=W 이하로 제한되었을 때, 원 신호를 Ts<1/2W로 샘플링
된 신호를 B (W<B<fs-W)의 대역을 갖는 LPF를 통과 시킴으로써 원 신호를 완벽히
재생할 수 있다.
2.9 Hilbert transform
- 원 신호의 모든 주파수 성분을
 1 2 만큼 phase-shift 시키는 변환
1, f  0

H  f    j s gn f , s gn f   0, f  0
 1, f  0

2.10 Discrete Fourier transform
X k  n 0 xn e  j 2nk / N , k  0,1,, N  1
N 1
xn 
1 N 1
X k e j 2nk / N , k  0,1,, N  1

k 0
N
Chapter 3 Basic Modulation Techniques
3.1 Linear modulation
1) Double-sideband modulation (DSB)
** modulator
xc t   Ac mt  cos wc t
Xcf 
:
m(t)=message signal
1
1
Ac M  f  f c   Ac M  f  f c 
2
2
** demodulator : *2coswct  LPF
d t   2 Ac mt  c o swc t c o swc t  Ac mt   Ac mt c o s2wc t d
** power 관점에서 효율적이나, 복조 시 modulation시의 carrier 와 동일한 phase를
갖는 2coswct 가 필요 (구현에서는 수신신호의 square를 수행함으로써 가능)
2) Amplitude modulation (AM)
** modulation
xc t   Ac 1 amn t cos wc t , mn(t)= m(t)의 최소값이 -1 이 되도록 정규화
a=modulation index
3) Single sideband modulation (SSB) : 한쪽의 sideband를 전송 전에 제거
xc t  
1
1
Ac mt  cos wc t  Ac mˆ t  cos wc t
2
2
3.2 Angle modulation
1) general form
xc t   Ac c o swc t   t 
instantaneous phase :
 i t   wc t   t 
instantaneous frequency
wi t  
d i
d
 wc 
dt
dt
2) phase modulation (PM)
 t   k p mt  ,
kp= deviation constant
3) frequency modulation (FM)
t
d
 2f d mt ,  t   2f d  m d , fd= frequency deviation constant
0
dt
- angle modulation power
x c2 t  
1 2
Ac
2
- FM의 demodulation: 수신신호의 미분 후 envelope 검출기 통과
- PM의 demodulation: FM 수신기에 적분기를 하나 더 달면됨
Chapter 4 Probability and Random Variables
4.1 Probability
- Bayes’ rule
P  B | A 
PB P A | B 
P  A
4.2 Random variables
Px1  X  x2   FX x2   FX x1    f X x dx
x2
x1
f X x dx  Px  dx  X  x  FX ,Y  P X  x, Y  y 
- independent random variables
FXY x, y   FX x FY  y ,
f X ,Y x, y   f X x  f Y  y 
- not independent random variables
f X ,Y x, y   f X x  f Y | X  y | x 
4.3 Statistical average
X  EX    j 1 x j Pj discrete random variable
M
EX    xf X x dx

continuous random variable



 X2  E X  E  X 2  EX 2   E 2 X 
 XY 
 XY
EXY   EX EY 

 XY
 XY
variance
correlation coefficient
4.4 Useful pdf
- Binomial distribution
n
PK  k   Pn k     p k q n k
k 
- Geometric distribution
Pk   pq k 1
- Gaussian distribution
nm X ,  X  
1
2 X
2

e xp x  m X  / 2 X
2
2

Chapter 5 Random Signal and Noise
5.1 Corelation and power sepectral density
RX    EX t X t   

S X  f    R e 2f d

Fourier transform pairs
5.2 Linear systems and random process

2
S y  f   H  f  S x  f , R y     H  f  S x  f e j 2f df
2


Rxy     hu R  u du  h   Rx    S xy  f   H  f S x  f 
