디지털 통신 (EE341-00)
교재: Digital communications, B. Sklar
전자공학과 홍 인기
ekhong@khu.ac.kr
031-201-2982, 010-5293-5203
Digital Communication 1
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Chapter 1
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Digital Communication?
 통신 (국어사전)
소식을 전함
 우편이나 전신, 전화 따위로 정보나 의사를 전달함
 신문이나 잡지에 실을 기사의 자료를 보냄. 또는 그 자료
 정보 전달을 다루는 과학기술, 정보를 모아 전류나 전기장으로
바꾼 다음 전기적 계통이나 공간을 통하여 다른 지점에 전달하면
수신자가 이해할 수 있도록 다시 바꾸는 기술

 人 (사람 인) 間(사이간)
 Why?

Smart Phone, T-Map, Facebook, Autonomous Driving, DNA
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Chapter 1
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Topics Covered
Chap 1. Signals and Spectra
Chap 2. Formatting and Baseband Modulation
Chap 3. Baseband Demodulation/Detection
Chap 4. Bandpass Modulation and Demodulation/Detection
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Chapter 1
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Outline
 Background for Digital Communications
Random Variables & Processes
 Signals

 Digitization
Sampling
 Quantization

 Base Band Transmission
 Modulation and Demodulation
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Chapter 1
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Chapter 1:
Signal and Spectra
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Chapter 1
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Contents
 Analog vs. Digital
 Advantages of Digital Communications
 Functions of Digital Communication Systems
 Definitions
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Chapter 1
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A General Communication System
Source: Speech, Video, Data, etc.
 Transmitter: Conveys information
 Channel: Invariably distorts signals
 Receiver: Extracts information signal
 User: Utilizes information

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Chapter 1
Analog vs. Digital
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 Analog systems have an alphabet which is
uncountably infinite

Example
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Chapter 1
Analog vs. Digital (con’d)

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Digital systems transmit signals from a discrete alphabet

Example: Binary digital communication systems
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Chapter 1
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Advantages of Digital Communications
 Resistant to distortion and interference
 Easily reproducible designs

Reduced cost
 High reliability through error detection and correction
 Easy to process different types of signals such as
speech, video, etc.
 Better encryption and privacy
 Facilitate data compression
What is the disadvantage?
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Chapter 1
Advantages of Digital Communications
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 Easy to regenerate ?
Digital
Analog
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Chapter 1
A General Digital Communication
System
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Chapter 1
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Key Functions of a DCS
 Formatting

Transforms the source information into bits
 Modulation
Bits are converted to waveforms for transmission over the
channel
 Baseband modulation and bandpass modulation

 Synchronization

Carrier recovery, timing recovery
 Demodulation

The recovery of a baseband signal
 Detection

Decision of the transmitted bits based on the demodulator output
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Supplementary Functions
 Source coding

Removes the redundant information in the source data
 Channel coding

Adds redundancy for reliable transmission
 Multiplexing

Combines signals from different sources
 Multiple access

Shares communication resources (spectrum, time, code)
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Definitions and Examples
 Information source

Analog  digital
Sampling & Quantization
 Bit (Binary digit) : 0 or 1
 Symbol (digital message)
M=2k
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Chapter 1
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Definitions
 Digital waveform
A waveform representing a symbol
 A pulse for baseband
 A sinusoid for bandpass

s1(t)
s2(t)
s3(t)
 Data rate
The transmission rate in bits/s
 R=k/T=(1/T)log2 M

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Chapter 1
Digital Communications:
Chapter 1
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Chapter 1
Signals and Spectra
 Classification of Signals
 Fourier Transform
 Spectral Density and Autocorrelation
 Random Signals
 Noise in Communication Systems
 Linear Systems
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Chapter 1
Classifications of Signals
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 Deterministic vs. Random Signals
 Periodic vs. Nonperiodic Signals
 Analog vs. Discrete Signals
 Energy vs. Power Signals
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Chapter 1
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Deterministic and Random Signals
 Deterministic

There is no uncertainty with respect to its value at any time
 Random

There is some degree of uncertainty before the signal actually
occurs
불확실성!!!
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Periodic and Nonperiodic Signals
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 Periodic in time
x(t ) = x(t + T0 )
T0 : Period

T0 : A minimum nonzero period
 Nonperiodic in time

T0 is infinite
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Analog and Discrete Signals
 An analog signal x(t) is a continuous function of time.
 A discrete signal x(kT) exists only at discrete times.

It is characterized by a sequence of numbers defined for each
time, kT, where k is an integer and T is a fixed time interval
x(t )
T
0
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x(2T ) x(4T ) x(6T )
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t
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Energy and Power Signals
 Energy signals
 Power signals

Energy is nonzero but finite

Finite energy but zero
 Finite power but infinite
average power
Exclusive!! energy

Both deterministic and
nonperiodic signals
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

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Power is nonzero but finite
Periodic signals and
random signals
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Signal Energy

Since we often think of signal as a function of varying amplitude through time,
it seems to reason that a good measurement of the strength of a signal
would be the area under the curve. However, this area may have a negative
part. This negative part does not have less strength than a positive signal of
the same size (reversing your grip on the paper clip in the socket is not going
to make you any more lively). This suggests either squaring the signal or
taking its absolute value, then finding the area under that curve. It turns out
that what we call the energy of a signal is the area under the squared signal.

The energy of this signal is the shaded region.
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Signal Power

Our definition of energy seems reasonable, and it is. However, what if the
signal does not decay? In this case we have infinite energy for any such signal.
Does this mean that a sixty hertz sine wave feeding into your headphones is as
strong as the sixty hertz sine wave coming out of your outlet? Obviously not.
This is what leads us to the idea of signal power.
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 Are all signals either energy or power signals?
No. Any infinite-duration, increasingmagnitude function will not be either. (eg f(t) =t
is neither)
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Chapter 1
Physically Realizable Waveforms
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 Have time duration
 Occupy finite frequency spectrum
 Are continuous
 Have finite peak value
 Are real-valued
 All real-world signals will have these properties, although
sometimes we use mathematical models which violate
these conditions
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Chapter 1
Unit Impulse (Dirac Delta) Function
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 Infinitely large amplitude at the point where the argument
is zero
 Zero pulse width
 Unity weight
i)
∞
∫ δ (t )dt = 1
−∞
ii) δ (t ) = 0 for t ≠ 0
iii) δ (t ) is unbounded at t = 0
∞
vi) ∫ x(t )δ (t − t0 )dt = x(t0 )
δ (t )
t
0
−∞
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Chapter 1
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Fourier Series
 A periodic continuous signal with period T0 has the
frequency components f 0 = 1/ T0 , 2 f 0 , 3 f 0, 
∞
x(t ) = ∑ cn e j 2πnf 0t
n = −∞
1 T0 / 2
cn = ∫
x(t )e − j 2πnf 0t dt =| cn | e jθ n
T0 −T0 / 2
 For a real-valued periodic continuous signal
c− n = cn*
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| c− n |=| cn |, θ − n = −θ n
29
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Fourier Series
합성
f
3f
5f
7f
10
f
f
7
3f
3f
5
5f
5f
3
7f
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7f
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Fourier Transform
 For nonperiodic continuous signals having finite energy

Fourier transform
∞
X ( f ) = F{x(t )} = ∫ x(t )e
−∞

− j 2πft
dt
Inverse Fourier transform
∞
x(t ) = F { X ( f )} = ∫ X ( f )e j 2πft df
−1
−∞
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Chapter 1
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Fourier Transform
 The properties of X ( f )

X ( f ) is complex
X ( f ) =| X ( f ) | e jθ ( f )

x(t ) is real
X (− f ) = X * ( f )
The magnitude | X ( f ) | : even function
 The phase θ ( f ) : odd function

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Chapter 1
Fourier Transform Properties
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 Linearity
F{αx1 (t ) + βx2 (t )} = αF{x1 (t )} + βF{x2 (t )}
 Time shifting
x(t ) ↔ X ( f ) : F {x(t − t0 )} = X ( f )e − j 2πft0
 Frequency shifting
x(t ) ↔ X ( f ) : F{x(t )e
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j 2 πf0t
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} = X ( f − f0 )
Chapter 1
Fourier Transform Properties
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 Modulation (mixing) theorem
1
1
x(t ) cos(2πf 0t ) ↔ X ( f − f 0 ) + X ( f + f 0 )
2
2

Note)
[
1
x(t ) cos(2πf 0t ) = x(t )e j 2πf 0t + x(t )e − j 2πf 0t
2
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]
Chapter 1
Fourier Transform Properties
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 Convolution
F{x(t ) ∗ y (t )} = X ( f )Y ( f )
F{x(t ) y (t )} = X ( f ) * Y ( f )
 Parseval’s theorem
∞
∞
∫
∫ | x(t ) | dt = ∫ | X ( f ) | df
−∞
x(t ) y (t )dt = ∫ X ( f )Y * ( f )df
∞
−∞
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*
−∞
2
∞
2
−∞
35
Chapter 1
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Fourier Transform Examples
 Example 1
∞
F{δ (t )} = ∫ δ (t )e − j 2πft dt = 1
−∞
 Example 2
τ /2
x(t ) = rect (t / τ )
F{rect (t / τ )} = ∫
−τ / 2
1
τ /2
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(
1
e − j 2πf (τ / 2 ) − e j 2πf (τ / 2 )
− j 2πf
− 2 j sin(πfτ ) τ sin(πfτ )
=
=
− 2πfj
πfτ
= τsinc( fτ )
=
−τ / 2
e − j 2πft dt
t
36
Chapter 1
)
Fourier Series with Fourier Transform
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∞
x(t ) = ∑ cn e j 2πnf 0t
n = −∞
∞
X ( f ) = ∑ cnδ ( f − nf 0 )
n = −∞
c−1
c0
− 2 f0 − f0
0
c− 2
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c1
f0
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c2
2 f0
c3
3 f0
f
Chapter 1
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Spectral Density
 The distribution of the signal’s energy or power in the
frequency domain
 Energy spectral density (ESD) for energy signals
∞
∞
E x = ∫ x(t ) dt = ∫ | X ( f ) |2 df
−∞
2
−∞
ESD
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Chapter 1
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Spectral Density
 Power spectral density (PSD) for periodic power signals
PSD
 PSD for nonperiodic power signals

Truncated version : x (t ) = x(t ), t ∈ ( −T / 2, T / 2)

This definition will be used for the random process
T
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Chapter 1
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Spectral Density
 Example
x(t ) = A cos(2πf 0t )

Average power using time averaging

Average power using PSD
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Chapter 1
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Autocorrelation
 Autocorrelation
The matching of a signal with a delayed version of itself
 Cf) Cross-correlation

Real-valued energy signal
∞
Rx (τ ) = ∫ x(t ) x(t + τ )dt
−∞
Real-valued periodic power signal
1 T0 / 2
Rx (τ ) = ∫
x(t ) x(t + τ )dt
T0 −T0 / 2
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Chapter 1
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Autocorrelation
 Properties
Real-valued energy signal
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Real-valued periodic power signal
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Chapter 1
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Autocorrelation
 Conceptual relation between the autocorrelation and
signal frequency
A(t )
RA (τ )
t
B(t )
t
C (t )
RB (τ )
RC (τ )
RD (τ )
D(t )
t
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τ
t
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Chapter 1
Why Are Random Processes
Important?
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 Random variables and processes let us talk about
quantities and signals which are unknown in advance
 The data sent through a communication system is
modeled as random
 The noise,interference, and fading introduced by the
channel can all be modeled as random processes
 Even the measure of performance (PE) is expressed in
terms of a probability
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Chapter 1
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Random Events
 When we conduct a random experiment, we can use set
notation to describe outcomes.
 Example: Roll a six-sided die
Possible outcomes: S = {
}
 An event is any subset of possible outcomes: A={
 The complementary event: Ā=S-A={
}
 The subset of all outcomes is the certain event:
 The set of all outcomes is the sample space:
 The null event:

}
 Transmitting a data bit is also a random experiment.
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Chapter 1
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Probability
 The probability P(A) is a number which measures the
likelihood of the event A.
 Axioms of probability
 All other probability laws follow from these axioms
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Chapter 1
Relationships Between Random
Events
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 Joint probability: P( A, B) = P( A ∩ B)

Probability that both A and B occur
 Conditional probability:

Probability that A will occur given that B has occurred
P( A | B) =
P( A, B)
P(B)
 Statistical independence

Events A and B are statistically independent iff:
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Chapter 1
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Random Variables
 A random variable X(A) is a real-valued function of the
underlying event space.
A
X (A)
A1, A2, A3
…, Ai
x
R
 A random variable may be
Discrete-valued: range is finite (e.g., {0,1}) or countably infinite
(e.g., {1,2,3,…})
 Continuous-valued: range is uncountably infinite (e.g., R)

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Chapter 1
Probability Mass Function (pmf)
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 For a discrete random variable, we usually use
probability mass functions (pmf):
pX ( x ) = PX ( X = x )
1. 0 ≤ p X ( x )
2. ∑ p X ( x ) = 1
X
b
3. P ( a ≤ X ≤ b) =
∑ p X ( x)
x=a
=
 Mean : m
X
∑ x ⋅ p ( x)
X
x
 Variance : σ X =
2
2
(
x
−
m
)
⋅ p X ( x)
∑
X
x
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Chapter 1
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Example. Binary Distribution
 pmf
1 / 2, x = 0
pX ( x ) = 
1 / 2, x = 1
 This is frequently used to model the binary data
 Mean ?
 If X1 and X2 are independent binary random variables,
p X 1 X 2 (0,0) =
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Chapter 1
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Binomial Distribution
 Let Y =
n
∑X
i =1
i
where {Xi, i=1,…,n} are independent binary RV’s with
1 − p, x = 0
pX ( x ) = 
x =1
 p,
 n y
n!
n−y  n 




p
(
y
)
p
(
1
p
)
,
=
−
=
 Then
Y
y 
 y  y!(n − y )!
 
 
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Binomial Distribution (Cont.)
 Suppose that we transmit a 31 bit sequence with error
correction capable of correcting up to 3 errors.

If the probability of a bit error is p=0.001, what is the probability
that the codeword is received in error?

If no error correction is used, the error probability is:
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Chapter 1
Cumulative Distribution Function (cdf)
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 Definition: FX ( x ) = P( X ≤ x )
 Properties
1. 0 ≤ FX (x ) ≤ 1
2. FX (x1 ) ≤ FX ( x2 ) if x1 ≤ x2
3. FX (−∞) = 0, FX (∞) = 1
4. P(a < X ≤ b) = FX (b) − FX (a)
 While the cdf completely defines the distribution of a
random variable, we will usually work with the pdf or pmf
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Chapter 1
Probability Density Function (pdf)
 Definition:
pX ( x ) =
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dFX ( x )
dx
 Interpretations

The pdf measures how fast cdf is increasing or how likely a
random variable is to lie at a particular value
 Properties
1. 0 ≤ pX (x )
∞
2. ∫ pX (x ) dx = 1
−∞
b
3. P(a < X ≤ b) = ∫ pX (x ) dx
a
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Chapter 1
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Mean or Expected Value
 Expected values are a shorthand way of describing a
random variable
m X E=
 Mean:=
[X ]
 Variance:
=
=
σ
var(X)
2
X

∞
∫ x ⋅ p ( x) dx
X
−∞

=
E (X-m
X) 
2
∫
∞
−∞
( x-m X ) 2 ⋅ p X ( x) dx
The randomness of a random variable
 The expectation operator works with any function
E =
g ( X ) 
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∞
∫ g ( x) ⋅ p ( x) dx
−∞
X
55
Chapter 1
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Example. Uniform pdf
1 / 10, 0 ≤ x ≤ 10
pX ( x ) = 
elsewhere
 0,
pX (x )
1 / 10
0
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10
56
x
Chapter 1
Example. Uniform pdf (Cont.)
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 Mean:
 Variance:
 Probability calculation
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Chapter 1
Example. Gaussian pdf
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 A Gaussian random variable is completely determined
by its mean and variance
p X ( x) =
−
1
2πσ
2
x
( x − mx ) 2
e
2σ x2
 Gaussian distribution
X ~ G (mx , σ x2 )
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Chapter 1
Example. Gaussian pdf (Cont.)
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pX (x )
x
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Chapter 1
Example. Rayleigh pdf
 Let R =
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X12 + X 22
where X1 and X2 are Gaussian distributed with mean 0
and variance σ2
 Then R has a Rayleigh distribution with pdf
r2
r
2
pR (r ) = 2 e 2σ
σ
−
 Rayleigh pdf’s are frequently used to model fading when
non-line of sight signal is present.
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Chapter 1
Example. Rayleigh pdf (Cont.)
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pR (r )
r
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Chapter 1
Random Processes
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 A random variable has a single value. However, actual
signals change with time.
 Random variables model unknown events.
 Random processes model unknown signals.
 A random process is just a collection of random
variables.
 If X(t) is a random process, then X(1), X(1.5), and
X(100.3) are all random variables for any specific time t.
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Chapter 1
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Random Processes
 A random process X ( A, t )
 N sample functions
A function of an event A
and time t
 For a specific time t 0,

X ( A, t0 )
is a random variable
 For a specific event A ,
j
X ( A j , t ) = X j (t )
is a single time function
Ensemble: A set of all sample functions
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Chapter 1
Mean and Autocorrelation
 Mean of the random process
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X (t )
 Autocorrelation function of the random process

A measure of the degree to which two time samples are related
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Chapter 1
Stationarity of Random Processes
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 Strict-sense Stationary (SSS)

Its cdf is not affected by time
 Wide-sense Stationary (WSS)
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Chapter 1
Ergodic Processes
Time averages
=
{ X j (t1 ), X j (t 2 ),..., X j (t N )}
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Ensemble averages
{x1 (t k ), x2 (t k ),..., x N (t k )}
The statistical properties of the process is determined by time
averaging over a single sample function of the process
 A random signal is ergodic in the mean and autocorrelation if it
satisfies

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Chapter 1
Relationship of Random Processes
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Random processes
SSS
WSS
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Chapter 1
Terminology Describing Random
Processes
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 A stationary random process has statistical properties
which do not change at all time (i.e., all joint pdfs do not
change).
 A wide sense stationary (WSS) process has a mean and
autocorrelation function which do not change with time
(this is usually sufficient).
 A random process is ergodic if the time average always
converges to the statistical average.
 Unless specified, we will assume that all random
processes are WSS and ergodic in the mean and
autocorrelation.
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Chapter 1
Random Process Examples
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 Ergodic process 1:
X i (t ) = k m for mT ≤ t ≤ (m + 1)T
where k m = 0 (1) when the coin flipping
at the mth time is tail (head)
 Ergodic process 2:
X j (t ) = cos(2πft + θ j )
where θ j is a uniform r.v. over [0, 2π )
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Chapter 1
Random Process Examples
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 Stationary but not ergodic
Dice rolling results : n
X 1 (t ) = 1
X 2 (t ) = −1
X n (t ) = cos(2πnt ), n = 3,4,5,6
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Random Process Examples
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 Non-stationary
 − 1, if n = 1,2,3
, if 0 ≤ t < t1

 1, if n = 4,5,6
X n (t ) =  n, if t1 ≤ t < t2
 cos(2πnt ), if t ≤ t < ∞
2


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Autocorrelation RX(τ)
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 Only a function of the time difference
Autocorrelation measures how a random process changes with
time.
 Intuitively, X(1) and X(1.1) will be more strongly related than X(1)
and X(1000). The autocorrelation function quantifies this.

 Properties:
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Power Spectral Density GX(f)
 GX(f) tells us how much power at each frequency is
 Power spectral density and autocorrelation are a Fourier
transform pair.
GX(f) = F{RX(τ)}
 Properties
G X (f ) ≥ 0
G X (f ) = G X (−f )
∞
PX = ∫ G X (f ) df
−∞
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Example
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Example
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Noise in Communication Systems
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 Noise

Unwanted signals that are always present in electrical systems
 Noise sources

Man-made noise
 Spark-plug ignition noise
 Switching transients
 Radiating electromagnetic signals

Natural noise
 Atmosphere disturbances, the sun, thermal noise
 Thermal noise
Caused by electron’s thermal motion
 Zero-mean Gaussian random process

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Noise in Communication Systems
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 Additive noise model
n(t )
s (t )

r (t ) = s (t ) + n(t )
n(t): a Gaussian process
 At any arbitrary time t, n = n(t ) has the Gaussian pdf
 1  n 2 
1
p ( n) =
exp −   
σ 2π
 2  σ  
 r = s+n
 1  r − s 2 
1
p (r ) =
exp − 
 
σ 2π
 2  σ  
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Noise in Communication Systems
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 Gaussian pdf
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Noise in Communication Systems
 The Gaussian distribution is used as the system noise
model
 Central limit theorem

The sum of N statistically independent random variables
approaches the Gaussian distribution as N  ∞ even if the
individual distribution function is not Gaussian!
N
Z = ∑ Zi
i =1
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Gaussian pdf
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Noise in Communication Systems
 White noise
Power spectral density of thermal noise is the same for all
frequencies (0 to 1012 Hz)
Gn ( f )
 Two-sided power spectral density

N
Gn ( f ) = 0 W/Hz
2
White noise
N0
Rn (τ ) = E{n(t )n(t + τ )} =
δ (τ )
2
Rn (τ )
N0 / 2
N0 / 2
f
τ
 Any two different samples of white noise are uncorrelated
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Linear System
 Impulse response
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Computing Output of Linear Systems
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 Deterministic signals
Time domain: y(t) = h(t) ∗ x(t)
 Frequency domain: Y(f) = F{y(t)} = X(f)·H(f)

 For a random signals, we can still relate the statistical
properties of the input and output signal

Time domain:
RY (τ) = RX (τ) ∗ h(τ) ∗ h(−τ)

Frequency domain:
GY (f ) = G X (f )⋅ | H(f ) |2
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Distortionless Transmission
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 Condition
 System transfer function
Constant magnitude response
 Linear phase shift

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Ideal Filter
Lowpass
Bandpass
Highpass
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Ideal Lowpass Filter
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 Transfer function
 Impulse response
h(t ) = ∫
fu / 2
− fu / 2
e − j 2πft0 e j 2πft df
= 2 f u sinc2πf u (t − t0 )
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Ideal Lowpass Filter
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 Effect of an ideal filter on white noise
Y (t )
n(t )
Gn ( f )
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Practical Filter
 RC Filter
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Practical Filter
 Effect of an RC filter on white noise
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Bandwidth
 Baseband vs. Bandpass
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WDSB Chapter
= 2 f 1m
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Bandwidth
 Bandwidth dilemma
Strictly bandlimited signals are not realizable
 Tradeoff between the time and frequency domain

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Definitions of Bandwidth
 Half-power bandwidth

The interval between the frequencies at which Gx(f) has dropped
to half-power (3dB)
 Equivalent rectangular or noise equivalent bandwidth

Gx(fc)WN=Px
 Null-to-null bandwidth

The width of the spectral lobe
 Fractional power constrained bandwidth

The x% ( 99%) of the power is inside the occupied band (FCC,
USA)
 Bounded power spectral density

Everywhere outside the specified band, Gx(f) must be fallen to a
certain level below the peak value (for example, 40dB in FCC)
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Bandwidth
Half-power
Noise-equivalent
Null-to-null
X% power
35 dB bounded
50 dB bounded
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