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8/2/2019 Communication System Overview and Random Process
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Communication System Overview
Gwo-Ruey Lee
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Wireless Access Tech. Lab.
CCUWireless Access Tech. Lab.
Outlines
Communication System
Digital Communication System
Modulation
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Communication System
Input Transducer
Transmitter
Channel Receiver
Output Transducer
Input Transducer ChannelTransmitter Receiver Output Transducer
Input
Message
Message
Signal
Transmitted
Signal
Received
Signal
Output
Signal
Output
Message
Carrier
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Communication System
Input transducerMessages can be categorized as analog (continuous
form)or digital (discrete form).
The message produced by a source must be converted by a
transducer to a form suitable for the particular type ofcommunication system employed.
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Communication System
Transmitter The purpose of the transmitter is to couple the message
to the channel.
Modulation
For ease of radiation to reduce noise and interference
For channel assignment
For multiplexing or transmission of several message over asingle channel
To overcome equipment limitation
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Communication System
Channel Different forms
The signal undergoes degradation from transmitter toreceiver
Noise, fading, interference
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Communication System
Receiver The receiver is to extract the desired message from the
received signal at the channel output and to convert it to aform suitable for the output transducer
Demodulation
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Communication System
Output Transducer The output transducer completes the communication
system
The device converts the electric signal at its input into
the form desired for the system user
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Digital Communication System
Information
Source
&Input Transducer
Source Encodre Channel Encoder
Outpot
TransducerSource Decoder Channel Decoder
Digital
Modulator
Digital
Demodulator
ChannelSynchroniz
ation
Output
Signal
Transmitted
Signal
Received
Signal
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Digital Communication System
Source Encoder/ Decoder The purpose of source coding is to reduce the number of
bits required to convey the information provided by theinformation source.
The task of source coding is to represent the sourceinformation with the minimum of symbols.
High compression rates (Good compression rates) make beachieved with source encoding with lossless or little lossof information.
Source Coding Fixed-length coding
Pulse-code modulation (PCM) Differential Pulse-code modulation (DPCM)
Variable-length coding Huffman Coding/ entropy coding
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Digital Communication System
Channel Encoder/ Decoder A way of encoding data in a communications channel that
adds patterns of redundancy into the transmission path inorder to lower the error rate.
The task of channel coding is to represent the sourceinformation in a manner that minimizes the errorprobability in decoding.
Error Control Coding
Error detection coding Error correct coding
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Digital Communication System
Error Control Coding Linear block code
Convolutional code
RS code
Modulation Coding Trellis code
Turbo code
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Digital Communication System
Synchronization Symbol/ Timing synchronization
Frequency synchronization
Carrier frequency synchronization Sampling frequency synchronization
Two basic types of synchronization Data-aid algorithm
Training sequences
Preambles
Non-data-aid algorithm Blind
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Digital Communication System
Channel Estimation A channel estimate is only a mathematical estimation of
what is truly happening in nature.
Allows the receiver to approximate the effect of the
channel on the signal. The channel estimate is essential for removing inter
symbol interference, noise rejection techniques etc.
Two basic types of channel estimation methods Data-aid algorithm
Training sequences
pilots
Non-data-aid algorithm Blind
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Modulation
Analog Modulation AM FM PM
Pulse Modulation PAM / PPM / PCM / PWM
Digital Modulation ASK FSK PSKQAM
Amplitude Frequency Phase
cos 2 cfc t tA Carrier:
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Modulation
Mapping The process of mapping the information bits onto the
signal constellation plays a fundamental role in determiningthe properties of the modulation
Modulation type Phase shift keying (PSK)
Quadrature Amplitude Modulation (QAM)
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Modulation
M-ary Phase Shift Keying Consider M-ary phase-shift keying (M-PSK) for which the
signal set is
where is the signal energy per symbol, is the symbolduration, and is the carrier frequency.
This phase of the carrier takes on one of the Mpossiblevalues, namely, , where .
2 12
cos 2 0 , 1,2,...,s
i c s
s
iEs t f t t T i M T M
sE sTc f
2 1i i M 1,2,...,i M
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Modulation
An example of signal-space diagram for 8-PSK
sE
2m
3m
4m
5m
6m
7m
8m
Decision
boundary
2
message
point
sE
sE
d
d
M
M 1m
Decision
region
1sE
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Modulation
Phase shift keying BPSK
QPSK with Gray code
M-ary PSK
where
0
sinseEp erfcN M
,1
2e BPSKp erfc
,1
2e QPSKp erfc
22 exp( )x
erfc x z dz
:SNR
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Modulation
BER versus SNR curves in AWGN channel using BPSK, QPSK,8-PSK,16-PSK .
0 5 10 15 20 25 3010
-6
10-5
10-4
10-3
10-2
10-1
100
SNR Vs BER
SNR
BER
BPSK theoretical result
BPSK simulation
QPSK theoretical resultQPSK simulation
8PSK approximate result
8PSK simulation
16PSK approximate result
16PSK simulation
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Quadrature Amplitude Modulation The transmitted M-ary QAM signal for symbol n can be
expressed as
where E is the energy of the signal with the lowestamplitude, and , and are amplitudes taking onthe values
Note that Mis assumed to be a power of 4.
The parameter a can be related to the average signalenergy ( ) by
2 2
cos 2 sin 2 , 0 , 0, 1, 2,....n n c n cE E
s t a f t b f t t T nT T
2, , 3 , , log 1n na b a a M a n
a nb2
E a
3
2 1
sEaM
sE
Modulation 7/10
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An example of signal-space diagram for 16-squareQAM.
Modulation 8/10
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QAM
2
,
2
1 11 4 | 4 2 | 2 |
loge M QAMp p c M p c M p c
M M
3
2 1
sE
a M
2
2
0
2| 1
ap c Q
N
2 2
0 0
2 2| 1 2 1a a
p c Q QN N
2
2
0
2| 1 2
ap c Q
N
Modulation 9/10
3aa- a- 3a
a
3a
- a
- 3a
na
nb
: I part
: II part
: III part
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BER versus SNR curves in AWGN channel using BPSK/QPSK,16QAM, 64QAM, 256QAM.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
10-5
10-4
10-3
10-2
10-1
100
Eb/N0 Vs BER
Eb/N0
BER
BPSK/QPSK theorem
BPSK/QPSK simulation
16QAM theorem
16 QAM simulation64 QAM simulation
64 QAM theorem
256 QAM simulation
256 QAM theorem
Modulation 10/10
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Communication System Overview
Readings Any book about communications
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Random Process/ Stochastic Process
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Outlines
Basic Concepts
Stationary Process
Transmission over Linear Time-Invariant (LTI)
Systems
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Basic Concepts
Why study random processes? Due to the uncertainty of 1. noise and 2. the unpredictable
nature of information itself.
Information signal usually is randomlikeWe can not predict the exact value of the signal
Signal must be distributed by its statistical properties. Ex: mean, variance..
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Basic Concepts
Random Variable (r.v.) Consider an experiment with sample space . The element
of are the random outcomes, , of the experiment. Ifto every , we assign a real value , such a rule
is called a random variable (r.v.)
SS
X x
S
Real line
X x
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Basic Concepts
Random Process (r.p.) A random process is the mapping of the outcomes in
into a set of real valued functions of time, called samplefunction . , iX t
1
S
2
n
1,X t
t
2,X t
t
,n
X t
t
S
0t t
0 1,X t
0 2,X t
0, nX t r.v.
1. : ensemble
2. : sample function
(or a realization)
3. : r.v.
4. : numerical value
,i
X t
2
,X t
20 ,X t
0 , iX t
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Basic Concepts
Classification of random process From the perspective of time
Random process:
for , t has a continuous of values
Random sequence: for , t can take on a finite or countably infinite
number of values
From the perspective of the valueof
Continuous: can take on a continuous of values
Discrete : Values of are countable
X t
X t
X t X n
X t
X t
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Basic Concepts
Classification of random process
Continuous random process
Discrete random process
Continuous random sequence Discrete random sequence
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Basic Concepts
1st-order distributions function It describes the instantaneous amplitude distribution of a
random process
Mean:
2nd-order distributions function It distributes the structure of the signal in the time
domain Autocorrelation Function (A.F.)
1
,
,
x X
n
i i
i
m t E X t x f x t dx
X t p
1 2 1 2
1 2 1 2 1 2 1 2
1 2
1
,
, , ,
, ,
XX
X
n
i i i
i
R t t E X t X t
x x f x x t t dx dx
X t X t p
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Basic Concepts
Autocovariance
Cross-correlation
If and are orthogonal
If and are statistically uncorrelated
1 2 1 1 2 2
1 2 1 2
,
,
XX X X
XX X X
C t t E X t m t X t m t
R t t m t m t
1 2 1 2 1 2
1 2
1
, , , ,
, , ,
XY XY
n
i i XY i i
i
R t t E X t Y t x y f x y t t dxdy
X t Y t p
0XYR X t Y t
XY X YR m m X t Y t
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Basic Concepts
Crosscovariance
The autocorrelation function of a real WSS process is
1 2 1 1 2 2
1 2 1 2
,
,
XY X Y
XY X Y
C t t E X t m t Y t m t
R t t m t m t
X t
,XX XXR t t E X t X t R
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Basic Concepts
The cross-correlation function of two real WSS process
and is
If and are orthogonal If and are statistically uncorrelated
Power Spectral Density (PSD) PSD represents the distribution of signal strength (ie,
energy or power) with frequency The PSD of WSS process is the Fourier transform (FT)
of the A.F.
X t Y t
,XY XYR t t E X t Y t R
0XYR X t Y t
constant
XY X YR m m
X t Y t
2
1 2
j f
XX XX XX
j f
XX XX XX
S f R R e d
R S f S f e df
X t
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Stationary Process
Stationary A random process whose statistical properties do not
change over time
Stationary Process Strictly-Sense Stationary (SSS)
Wide-Sense Stationary (WSS)
Strictly-Sense Cyclostationary
Wide-Sense Cyclostationary
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Stationary Process
Strictly-Sense Stationary (SSS) A nth-order strictly-sense stationary process is a process
in which for all , all , and all
Note: Mth-order stationary of the above equation holdsfor all .
Example: 2nd-order SSS process 1st-order SSSprocess
1 2
1 2
1 2, ,...,
1 2, ,...,
, ,...,, ,...,
n
n
nX t X t X t
nX t k X t k X t k
f x x xf x x x
k 1 2, ,..., nt t tn
n M
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Stationary Process
A example of 2nd-order stationary
2t t1t t
1
S
2
n
1
,X t
t
2,X t
t
,n
X t
t
2t t k
1t t k
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W A
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Stationary Process
Wide-Sense Stationary (WSS) A random process is wide-sense stationary process
(WSS) if
Its mean is constant
Its A.F. depends only on the time difference.
constantxm t E X t
X t
1 2 2 1
,xx xx xxR t t R t t R
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A
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Stationary Process
The relationship between SSS and WSS SSSWSS (True)
SSSWSS (Fault)
1st-order SSS
2nd-order SSS
For Gaussian process : SSSWSS
Since the joint-Gaussian pdf is completely specified by itsmean and A.F.
constantE X t 1 2 2 1,xx xx xxR t t R t t R
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A
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Stationary Process
Strictly-Sense Cyclostationary A nth-order strictly-sense cyclostationary process is a
process in which for all , all , and integer m
( mTis integer multiples of period T) 1 2, ,..., nt t tn
1 2
1 2
1 2, ,...,
1 2, ,...,
, ,...,
, ,...,
n
n
nX t X t X t
nX t mT X t mT X t mT
f x x x
f x x x
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Stationary Process
Wide-Sense Cyclostationary A random process with and is wide-sense
cyclostationary if
Its mean satisfies
Its a.F. satisfies
X t
x xm t mT m t
1 2 1 2
, ,XX XX XX
R t mT t mT R t t R
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Stationary Process
Ergodic Process A random process is strictly ergodic process if all time
and ensemble (statistical) average are interchangeableincluding mean, A.F. PSD, etc.
A random process is wise-sense ergodic if it it ergodic inthe mean and the A.F. mean ergodic
A.F. ergodic
lim X XTT
m m
lim XX XXTT R R
2
2
1T
TX TT
X
X t m X t dtT
m E X t
2
2
1
XXT T
T
T
XX
X t X t R
X t X t dtT
R E X X t X t
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Stationary Process
The relationship between ergodic and stationary Ergodic stationary (True)
Ergodic stationary (Fault)
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Transmission over LTI Systems
Linear Time-Invariant (LTI) Systems
LTI System
h t
x t y t
*
*
y t x h t x t h t
h x t h t x t
Y f X f H f
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Transmission over LTI Systems
Assumptions:
and are real-valued and is WSS.
The mean of the output
The cross-correlation function
x t h t x t
0x xE y t m h d m H
y t
YX XX
XY XX
R E Y t X t h R
R E X t Y t h R
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Transmission over LTI Systems
The A.F. of the output
The PSD of the output
YY
YX
XY
XX
XX
R E Y t Y t
R h
R h
h R h
R h h
2
YY XX S f S f H f
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Random Process/ Stochastic Process
Readings Communication Systems, 4thedition, Simon Haykin, Wiley
Chapter 1 1.1 ~1.7, 1.8