Communication System Overview and Random Process

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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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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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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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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Channel Different forms

The signal undergoes degradation from transmitter toreceiver

Noise, fading, interference

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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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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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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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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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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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Error Control Coding Linear block code

Convolutional code

RS code

Modulation Coding Trellis code

Turbo code

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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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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

Wir l A T h Lab


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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

Wireless Access Tech Lab


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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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Wi eless Access Tech. Lab.


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


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


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


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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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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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

Documents

Communication System Overview and Random Process