Introduction:
Digital Communication
2008.09.02
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Biography
Min-Goo Kang
-’82-’86 B.S.
-’87-’89 M.S.
-’89-’94 Ph.D. in the Dept. of Electronic engineering, Yonsei University
- Professor in the Dept. of Inform. & Telecomm.,
Hanshin University, Osan, Korea, from 2000
-’85-’87 Researcher in Samsung
-’97-’98 Post Doc. in Osaka University, Japan
-’06-’07 Visiting Scholar in Queen’s University, Canada
- Research interests Mobile Telecomm. & DTV.
- E-mail : [email protected]
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Contents

Communication System Model

Analog-to-Digital (A/D) Conversion : Sampling, Quantization,Coding
Pulse-Code Modulation (PCM) : Line Coding(RZ, NRZ etc.)
CODEC : Souce CODEC(JPEG, MPEG, H.264, MP3,…)
Channel Coding(Viterbi, Turbo, …)  CH3,4,9




Filter : Lowpass Filter(LPF), Highpass Filter(HPF), Bandpass Filter(BPF)
Fourier Series, Fourier Transform :FFT(Frequency Analysis)

Modulation(MODEM) AM(ASK), FM(FM), PM(PSK)  CH11,14
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Communication System Model

Message signal m(t)
▶
▶

Transmitter
▶

modifies the baseband signal for efficient transmission
Channel
▶
▶

analog or digital
baseband signal
Channel is a medium through which the transmitter output is sent
example: wire, coaxial cable, optical fiber, radio link
Receiver
▶
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reprocesses the received signal by undoing the signal modifications
made at the transmitter and the channel
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Typical Digital Communication System
Analog
message
A/D
conversion
Source
coding
Channel
coding
Digital
message
Modulation
Multiple
access
Transmitter
From other
source
Channel
To other
sink
Digital data
output
Analog data
D/A
output
conversion
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Source
decoding
Channel
decoding
Demodulation
Multiple
access
Receiver
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Analog-to-Digital (A/D) Conversion

Sampling
▶
▶
▶

Sampling makes signal discrete in time
Sampling theorem says that bandlimited signal can be sampled without
introducing distortion
The sample values are still not digital
Quantization
▶
Quantizer makes signal discrete in amplitude
▶
Quantizer introduces some distortion (“quantization noise”)
▶
Good quantizers are able to use few bits and introduce small distortion
 Inherently digital information (e.g. computer files) do not require sampling
or quantization.
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Sampling
m(t )
sampler
ms (t )  m(t ) 
m(t )
0
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ms (t )
t
0

  (t  nT )
n 
s
t
Ts
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Quantization
binary
code
quantized
level
m(t )를 양자화한 표본
3.5
2.5
110
1.5
101
0.5
100
-0.5
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m(t )
111
Ts
011
-1.5
010
-2.5
001
-3.5
000
t
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Pulse-Code Modulation (PCM)

Multi-amplitude pulse code
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Channel Effects
 Distortion
attenuation, noise, fading
 Simple channel model: additive white Gaussian noise (AWGN)
▶
transmitted signal
received distorted signal (without noise)
received distorted signal (with noise)
regenerated signal (delayed)
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Source Coding and Channel Coding

Source Coding
▶

Compression of digital data to eliminate redundant information
Channel Coding
▶
Provides protection against transmission errors by selectively inserting
redundant data
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SNR, Bandwidth, Rate of Communication


Fundamental parameters that control the rate and quality of
information transmission are the channel bandwidth B and the
signal power S.
Channel Bandwidth B
▶
Range of frequencies that the channel can transmit with reasonable fidelity
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SNR, Bandwidth, Rate of Communication

Signal Power S
▶
▶
Signal power is related to the quality of transmission
Increasing S reduces the effects of channel noise, and the information is
received with less uncertainty
▶
In any event, a certain minimum SNR is necessary for communication
▶
Channel bandwidth B and signal power S are exchangeable;
- We can trade S for B, or vice versa.
- One may reduce B if one is willing to increase S.
▶
Example: PCM with 16 quantization levels
- multi-amplitude scheme
- binary scheme
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SNR, Bandwidth, Rate of Communication

Rate of Information Transmission C
▶
▶
Channel capacity : maximum number of bits that can be transmitted per
second with a probability of error arbitrarily close to zero
Shannon’s limit
C  B log 2 (1  SNR) [bits/sec]
▶
▶
The channel capacity is related to channel bandwidth and signal power.
It is impossible to transmit at a rate higher than channel capacity without
incurring errors.
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Modulation


Converts digital data to a continuous waveform suitable for
transmission over channel
▶
Baseband (usually square waveform): “line coding”
▶
Bandpass (usually sinusoidal waveform): “bandpass modulation”
Carrier Modulation (bandpass modulation)
▶
▶
Information is transmitted by varying one or more parameters of the
carrier waveform: amplitude, frequency, phase
A carrier is a sinusoid of high frequency, and one of its parameters
(amplitude, frequency, phase) is varied in proportion to the baseband
signal (message signal).
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Line Coding
1
(a)
1
1
0
0
1
1
0
1
0
0
Unipolar NRZ
(b)
Polar NRZ
(c)
Unipolar RZ
(d)
Polar RZ
Tb
(e)
Bipolar RZ
(AMI)
Tb
2
(f)
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Manchester
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Modulation
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Modulation
Examples of Digital Modulation
1) Amplitude Shift Keying (ASK) or
ON/OFF Keying (OOK):
1  A cos(2f ct ) 0  0
2) Phase Shift Keying (PSK):
1  A cos(2f ct )
0  A cos(2f ct   )   A cos(2f ct )
3) Frequency Shift Keying (FSK):
0  A cos(2f 0t )
1  A cos(2f1t )
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Bit Error Probability for Binary Systems
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Good Communication System

Large data rate (measured in bits/sec), R

Small bandwidth (measured in Hertz), B

Small required signal power (measured in Watts or dBW), or
equivalently small required Eb/N0

Low distortion (measured in S/N or probability of bit error)

Large system utilization: large number of users with small delay

Low system complexity, computational load, and system cost
• With digital communications, high complexity does not always result in
high cost

In practice, there are tradeoffs made in achieving these goals
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Data Rate vs. Bandwidth

Data rate R ↑  data pulse width ↓  bandwidth B ↑

This tradeoff cannot be avoided - however, some systems use
bandwidth more efficiently than others.

Define Bandwidth Efficiency as the ratio of data rate R to
bandwidth B: hB = R / B

We want large bandwidth efficiency hB
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BER vs. Signal Power

One way to get low probability of error would be to use large
signal power to overcome the effect of noise.

Some types of modulation achieve low probability of error at
lower power than others.

Define hE for Energy Efficiency : hE = E b / N 0 |P = 10

We want small hE
- 5
b
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Tradeoff in System Design

Tradeoff between bandwidth efficiency and energy efficiency
• M-ary modulation
 Binary
modulation sends only one bit per use of the channel.
 M-ary modulation
sends multiple bits, but is more vulnerable to noise.
• Error correction coding
 Inserting redundant bits improves BER
performance, but increases
bandwidth
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Why Digital Communication?

Any noise introduces distortion to an analog signal. Since a digital receiver
needs only distinguish between two waveforms it is possible to exactly
recover digital information.

Many signal processing techniques are available to improve system
performance: source coding, channel coding, equalization, encryption.

Digital ICs are inexpensive to manufacture: a single chip can be mass
produced at low cost, no matter how complex.

Digital communication allows integration of voice, video, and data on a single
system.

Digital communication systems provide a better tradeoff of bandwidth
efficiency and energy efficiency than analog systems.
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Fourier Series

Example (pulse train)
 n  
sin  o 
 2 
cn 
n
 n  
sin  o 
1 n o
 2 



n o
n
2
2

 n  
 o  Sa  o 
2
 2 

1
  
  Sa 

T
 2    n0
 Line spectrum
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Fourier Series
cn
/T
T/
n
0
cn
/T
2
0
0
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Fourier Series

Example (pulse train): in terms of frequency
 n  
sin  o 
 2 
cn 
n
sin  nf o 
1

  nf o 
n
 nf o
cn
/T
 f o  sinc  nf o 
1
   sinc  f  
T
1
f  nf 0
f
0
f0
 Line spectrum
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Fourier Series
cn

T0
 / T0
sinc( f  )
1


  T0 / 4
4
T0
f
0 f0 2 f0 3 f0
cn
 / T0

T0
sinc( f  )
1

 f0
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
0 f0 2 f0 3 f0
2
T0
f
  T0 / 2
Fourier Series

Sampling function and sinc function
Sa(t)
1
sin t
Sa(t ) 
t
4
3
2

0

2
3
4
2
3
4
t
sinc ( t )
1
sinc(t ) 
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sin  t
 Sa( t )
t
4
3
2

0

t
Fourier Series
cn

T0
 / T0
sinc( f  )
1


  T0 / 4
4
T0
f
0 f0 2 f0 3 f0
cn
 / T0

T0
sinc( f  )
1

 f0
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
0 f0 2 f0 3 f0
2
T0
f
  T0 / 2
Fourier Series

Sampling function and sinc function
Sa(t)
1
sin t
Sa(t ) 
t
4
3
2

0

2
3
4
2
3
4
t
sinc ( t )
1
sinc(t ) 
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sin  t
 Sa( t )
t
4
3
2

0

t
Fourier Series
x(t )

Pulse Signals
1
T0
1 T
x(t ) exp(  jn o t ) dt
T 0
1 T2
1
  (1) exp(  jn o t ) dt 
T 0
T
T
 0
2
cn 
1

exp(  jn o t )
 jn oT

T 2
0

T
T 2
( 1) exp(  jn o t ) dt
1

exp(  jn o t )
 jn oT
T
T 2
1
[{exp(  jn )  exp(0)}  {exp(  j 2n )  exp(  jn )}]
 jn 2
exp( j )  1, exp( j 2 )  1
 2
, n  odd

cn   jn
0, n  even

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0
1
T0
2
T0
t
Fourier Series
cn
 2
, n  odd

cn   jn
0, n  even

5w0
3w0
2w0
0
w0
w0
w
2w0
3w0
5w0
3w0
5w0
cn
90
w0
5w0
3w0
2w0
w0
90
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0
2w0
w
Fourier Series

impulse train
x(t )
2T0
3T0
T0
0
T0
2T0
t
3T0
cn
6 f 0 5 f 0 4 f 0 3 f 0 2 f 0  f 0 0
1
cn 
T0
x(t ) 
2008.09.02


1
1
 (t ) exp( jn ot ) dt   exp( jn ot ) 
2
T0
T0
t 0
T0 2
T0
f0 2 f0 3 f0 4 f0 5 f0 6 f0
f
1
exp( jn ot )

n  T0
Linear System & Fourier Series

System Analysis
For periodic input:
x(t ) 

ce
n 
jn 0t
h
n
y (t ) 


n 

H (n 0 )cn e jn0t

de
n 
jn 0t
n
d n  H (n0 )cn
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y (t )
Fourier Transform :FFT
(Fast Fourier Transform :Frequency Analysis)
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Fourier Transform

As the fundamental period of the time waveform increases, the
fundamental frequency of the Fourier series components making up
the waveform decreases and the harmonics become more closely
spaced.

In the limit, as the time between pulses approaches infinity, the
harmonic spacing becomes infinitely small and the spectrum is in
fact continuous and bounded by the sinc function as shown.
2008.09.02
Fourier Transform
x (t)
1
...
...
T
T
/
2
0
T
cn
/
2
t
T
/To
1
(a) T = T o
0
f
fo
cn
/2To
(b) T =2 T o
0
2008.09.02
1
f
/
fo
2
Fourier Transform
t
x(t )   

 1 t   / 2

 0 otherwise
x (t)
1
t
0
X (f )
1
0
X ( f )   sinc( f  )
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f
Fourier Transform

Fourier transform pair
1
x(t ) 
2



X ( )e
j t

dt  X ( )   x(t )e  jt dt



x(t )   X ( f )e j 2 ft dt
 X ( f )   x(t )e  j 2 ft dt


X ( )  X ( ) exp[ j ( )]
: (continuous) spectrum of x(t )
X ( )  Re{ X ( )}2  Im{ X ( )}2 : magnitude spectrum of x(t )
Im{ X ( )}
 ( )  X ( )  tan
Re{ X ( )}
1
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: phase spectrum of x(t )
Fourier Transform

Example 1: rectangular pulse
 
x(t )  rect (t /  )  X ( )   Sa 
 2

   sinc  f  

X(f )
x(t )

1

t
0
0

X ( )   x(t )e

 j t
dt  
 /2
 / 2
1
2
3



e jt dt
1  j / 2
2

  
j / 2
e

e

sin


Sa

  2


j
 2 
• A smaller  produces a wider main lobe (broad bandwidth)

2008.09.02
f
Fourier Transform

Example 2: exponential function
x(t )  e at u (t ), a  0  X ( f ) 
X(f )  

X(f ) 



e at e j 2 ft u (t )dt   e ( a  j 2 f )t dt
0
1
, a0
a  j 2 f
1
a 2  4 2 f 2
 2 f 

 a 
 ( f )   tan 1 
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1
a  j 2 f

1
a
1
 f 
1  4 2  
a
2
Fourier Transform

Example 2: exponential function
X(f )
1
a
1
X(f ) 
a 2
x(t )  e  at u (t )


1
a
2
a
2
0
f
1
a
1
 2 f 
1 

 a 
2
( f )

0
t
2

4

2008.09.02
a

2  4


2
a
2
f
 2 f 

 a 
 ( f )   tan 1 
Properties of Fourier Transform

Linearity
ax1 (t )  bx2 (t )  aX 1 ( )  bX 2 ( )
aX 1 ( f )  bX 2 ( f )

Time shifting; shifting in time changes only the phase
y(t )  x(t  t0 )  Y ( )  e jt0 X ( )
Y ( f )  e j 2 ft0 X ( f )
 Y( f )  X ( f ) ,
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Y ( f )  Y ( f )  (2 t0 ) f
Properties of Fourier Transform

Time scaling
x (at ) 
1  
X 
a a
1  f 
X 
a a
• expanding in time (|a|<1) leads to slow variation
(deemphasizing high frequency components)
• a fast replay (|a|>1) of man’s voice may be heard like girl’s voice
x (t)
1
t
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0
1
t
t
Properties of Fourier Transform

Time scaling
X(f )
x(t )
1




2
2
t

f

1
1


X(f )
x(t )
2
1

2008.09.02

t

f

1
2
1
2
Properties of Fourier Transform

Frequency shifting and modulation
e j t x(t )  X (   0 )
0
e j 2 f0t x(t )  X ( f  f 0 )
e  j0t x(t )  X (   0 )
1
 X (  0 )  X (  0 )
2
1
sin( 0 t) 
 X (  0 )  X (  0 )
2j
cos( 0 t) 
Example: rect (t /10) cos(1000 t )
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Properties of Fourier Transform

Frequency shifting and modulation
X(f )
x(t )

1


2
0
t


f
0

2
1
1


Y( f )
y (t )  x(t ) cos(2 f 0t )



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

2
2
t
2
 f0
0
f0
2

f
Properties of Fourier Transform

Duality
X (t )  2 x( )
X (t )  x(  f )

pf) 2 x(t )   X ( )e

j t

d 
 2 x( )   X (t )e  j t dt
t 

Example:
e
a t
2a
2a
a 
 2



2

e
a 2
a2  t 2
t
x(t )  rect 

2008.09.02


t

 

X
(

)


sinc


sinc

rect

 
2
2
2

 
Properties of Fourier Transform

Duality
X(f )
x(t )

1



0

t
f

2
2
y (t )
1 0
1


Y( f )

1

t

2008.09.02
1 0
1




2
0

2
f
Properties of Fourier Transform

Differentiation
d
x(t )  j X ( ) (obtained by differentiating both sides w.r.t. t )
dt
j 2 f X ( f )
x ( n ) (t )  ( j ) n X ( )
Example:
y(t )  2 y(t )  y (t )  x(t )  x(t ) 
( j )2 Y ( )  2( j )Y ( )  Y ( )  ( j ) X ( )  X ( )

Integration
t


x( )d 
X ( )
+ X (0) ( )
j
X ( f ) X (0)
+
(f )
j 2 f
2
2008.09.02
Properties of Fourier Transform

Example : triangular pulse
x(t )
t
x(t )    
 
1

d 2x 1
  (t   )  2 (t )   (t   ) 
dt 2 
t

dx
dt
1



1


d 2x
dt 2
1
1





2008.09.02
t

2

d 2x 
1
F  2  = e j 2 f   2  e  j 2 f  
 dt  
2
4
  cos(2 f  )  1   sin 2 ( f  )
t

4 sin 2 ( f  )
 dx 
F  
 j 2 f
 dt 
4 sin 2 ( f  )
sin 2 ( f  )
F  x(t )  

  sinc2 ( f  )
2
2
 ( j 2 f )
( f  )
Properties of Fourier Transform

Parseval’s Theorem
E


pf)



1
x(t ) dt 
2
2

X ( ) d  
2





Energy Spectral Density (ESD) :
X(f )
• the frequency distribution of total energy
2008.09.02

2
X ( f ) df
 1  

 j t
x(t ) dt   x(t ) x (t ) dt   x(t ) 
X
(

)
e
d

 dt




2




1  
1  
 j t

X ( )  x(t )e dt d 
X ( ) X ( )d

2 
2 
2




2
Properties of Fourier Transform

Convolution (in time)
  (t /  )

 (t /  )
 (t /  )
1
1




2
0
t


2

2
0

t


t
2
(a)
 sinc2 (f  )
(t /  )

1



t

(b)
2008.09.02
1
1

0

f
Properties of Fourier Transform

Multiplication (in time) : dual of convolution property
x(t )m(t ) 
2008.09.02
1
X ( )  M ( )
2
X ( f )M ( f )
Application of Fourier Transform

Communication system : modulation and demodulation
Modulation
m(t )  cos  0t 
1
2
e
j0t

 e  j0t  M ( )    (   0 )   (   0 )

y (t )  x(t )m(t )  Y ( ) 
2008.09.02
1
 ( f  f 0 )   ( f  f 0 )
2
1
1
X ( )  M ( )   X (   0 )  X (   0 )
2
2
1
Y ( f )  X ( f )  M ( f )   X ( f  f 0 )  X ( f  f 0 )
2
Application of Fourier Transform

Communication system : modulation and demodulation
Demodulation
z (t )  y (t )m(t )  Z ( ) 
2008.09.02
1
1
Y ( )  M ( )  Y (   0 )  Y (   0 )
2
2
1
  X (  2 0 )  2 X ( )  X (  2 0 )
4
1
Z ( f )   X ( f  2 f 0 )  2 X ( f )  X ( f  2 f 0 )
4
Application of Fourier Transform

Multiplexing (frequency division multiplexing)
2008.09.02
Useful Fourier Transform Pairs

x(t )   (t )
Unit impulse
 (t )  1
 (t  t0 )  e jt ,

Constant
1  2 ( ),
e
 j 2 ft0
(f )
0
1
1
(f )
2
j 2 f
Exponential
1
e u (t ) 
,
a  j
2008.09.02
1
a  j 2 f
f
0
X(f ) ( f )
1
Unit step function
 at
t
0
x(t )  1
1
u (t )   ( ) 
,
j

1

0

X ( f ) 1

t
0
f
Useful Fourier Transform Pairs

Complex exponential
e j0t   (  0 ),
e j0t   (  0 )

Sinusoidal functions
cos  0t 
1
2
e
j0t
e j 2 f0t   ( f  f 0 )
e j 2 f0t   ( f  f 0 )

 e  j0t    (   0 )   (   0 )
1
 ( f  f 0 )   ( f  f 0 )
2
 j  (   0 )   (   0 )
cos 2 f 0t 
sin  0t 
1
2j

e j0t  e  j0t

sin 2 f 0t 
2008.09.02
1
 ( f  f 0 )   ( f  f 0 )
2j
Useful Fourier Transform Pairs

Rectangular pulse
t
x(t )   
T


  X ( f )  T sinc( fT )

Triangular pulse
t
x(t )      X ( f )  T sinc 2 ( fT )
T 

Impulse train
x(t ) 

  (t  nT )  X ( f )  f 0
n 
2008.09.02

  ( f  nf 0 )
n 
f0 
1
T
Fourier Transform of Periodic Signals

Impulse train
x(t ) 

  (t  kT )
: periodic with period T
k 
x (t )
...
2T
T
0
...
T
2T
f0 
1
T
i) Fourier series
x(t ) 


n 
cn e j 2 nf0t ,
1
1
 j 2 nf 0t
cn    (t )e
dt 
T
T
T
2008.09.02
3T
Energy Spectral Density (ESD)

Energy transmission through LTI system
X(f )
x( f )
2008.09.02
H( f )
Y( f )  H ( f )X ( f )
y ( f )  H ( f ) x( f )
2
Power Spectral Density (PSD)

Power Spectral Density (PSD) function
• Average power
1
P  lim
T  T

T /2
T / 2
x(t ) dt

  S x ( f )df

• Power Spectral Density Sx(f)
S x ( f )  lim
T 
XT ( f )
2
T
where
 x(t )
xT (t )  
0
2008.09.02
T / 2  t  T / 2
otherwise
2
Power Spectral Density (PSD)

Power transmission through LTI system
X(f )
Sx ( f )
2008.09.02
H( f )
Y ( f )  H ( f )X ( f )
S y ( f )  H ( f ) Sx ( f )
2
Filter

Ideal Lowpass Filter (LPF)
 f 
H( f )  K 

 2W 
 h ( f )  H ( f )  (2 t0 ) f , for  W  f  W
H( f )
h(t )  2 KW sinc(2W (t  t0 ))
K
2KW
W
f
0
W
t0 
f
1
2W
t
0
slope  2 t0
2008.09.02
1
2W

 h ( f )  (2 t0 ) f
0
t0 
t0
Filter

Ideal Highpass Filter (HPF) and Bandpass Filter (BPF)
W
H( f )
H( f )
K
K
f
0
W
h ( f )
f
 f0
0
2008.09.02
W
h ( f )
slope  2 t0
0
f0
slope  2 t0
f
0
f