EE140 Introduction to Communication Systems Review

3/4/2018

1

EE140 Introduction to

Communication Systems

Review

1

Architecture of a (Digital) Communication System

2

Source A/Dconverter

Sourceencoder

Channelencoder Modulator

Channel

DetectorChanneldecoder

Sourcedecoder

D/Aconverter

User

Transmitter

Receiver

Absent ifsource isdigital

Noise

3/4/2018

2


3

Source A/Dconverter

Sourceencoder


Channel


Sourcedecoder

D/Aconverter

User

Transmitter

Receiver


Noise

Deterministic signals

– Classification of signals

– Review of Fourier Transform

– Frequency-domain properties

– Time-domain properties

– Vector space and orthogonality

4

3/4/2018

3

Random signals

– Review of probability and random variables

– Random processes: basic concepts

– Gaussian white processes

5


6

Source A/Dconverter

Sourceencoder


Channel


Sourcedecoder

D/Aconverter

User

Transmitter

Receiver


Noise

3/4/2018

4

Analog Modulation

– Amplitude modulation

• DSB

• SSB

• VSB

– Pulse modulation

– Angle modulation (phase/frequency)

7


8

Source A/Dconverter

Sourceencoder


Channel


Sourcedecoder

D/Aconverter

User

Transmitter

Receiver


Noise

3/4/2018

5

Sampling and Quantization

• Sampling

– periodic sampling

– frequency domain representation of sampling

– Reconstruction of a bandlimited signal from samples

– change the sampling rate using discrete-time processing

• Quantization

– Uniform quantizer

– Non-uniform quantizer

– Vector quantization

– Sigma-delta modulation

9


10

Source A/Dconverter

Sourceencoder


Channel


Sourcedecoder

D/Aconverter

User

Transmitter

Receiver


Noise

3/4/2018

6

Contents

• Information Theory

• Source Coding

11


12

Source A/Dconverter

Sourceencoder


Channel


Sourcedecoder

D/Aconverter

User

Transmitter

Receiver


Noise

3/4/2018

7

Contents

• Signal Propagation

• Channel Models

• Channel Capacity

13

Contents

• Signal Waveforms

• Inter-symbol interference

• Signal design with zero ISI

• Signal design with controlled ISI

• Tx/Rx filter design with channel response

• Equalization

14

3/4/2018

8

Signal Design for Bandlimited Channel Zero ISI

• Nyquist condition for Zero ISI for pulse shape 1 0 0 0

or ∑ T

• With the above condition, the receiver output simplifies to

15

Nyquist Condition: Ideal Case• Nyquist’s first method for eliminating ISI is to use

1 | |

0

/

/

• = Nyquist bandwidth

• The minimum transmission bandwidth for zero ISI. A channel with bandwidth can support a max. transmission rate of 2 symbols/sec

16

3/4/2018

9

Practical Solution: Raised Cosine Spectrum• is made up of 3 parts: passband, stopband,

and transition band. The transition band is shaped like a cosine wave.

17

Equalizer Configuration

• Overall frequency response

• Nyquist criterion for zero-ISI

• Ideal zero-ISI equalizer is an inverse channel filter with

∝1

| | 1/218

3/4/2018

10

19

More about Equalization

Zero-Forcing Equalizer• : received pulse from a channel to be equalized

• ∑ 1, 00, 1, … ,

To suppose 2N adjacent interference terms

20

3/4/2018

11

Zero-Forcing Equalizer (cont’d)• Rearrange to matrix form

21


22

Source A/Dconverter

Sourceencoder


Channel


Sourcedecoder

D/Aconverter

User

Transmitter

Receiver


Noise

3/4/2018

12

Content• Signal Space • Coherent reception -- binary modulations

– BPSK– BFSK– BASK

• Noncoherent reception– Noncoherent detection of BFSK– Differential PSK (DPSK)

23

Signal Space• Basic Idea: If a signal can be represented by n-

tuple, then it can be treated in much the same way as a n-dim vector.

• Consider a signal x(t) and suppose that

• If every signal can be written as above, then are the basis functions

24

N

iii )t(x)t(x

1

φ

Nii )t( 1φ

3/4/2018

13

Basis Functions for a Signal Set• Consider a set of M signals (M-ary symbol)

with finite energy. That is

• Then, we can express each of these waveforms as weighted linear combination of orthonormal signals

where N ≤ M is the dimension of the signal space, and are called the orthonormal basis functions

25

Mii )t(s 1

dt)t(si

2

N

jjiji )t(s)t(s

1

φ

Njj )t(1

φ

Gram Schmidt Orthogonalization (GSO) Procedure

• Suppose a signal set is given

• Find the orthogonal basis functions for this signal set

where

26

Mii )t(s 1

Njj )t(1

φ

3/4/2018

14

Binary Phase-Shift Keying (BPSK)• Modulation

“1” cos 2π

“0” cos 2π π cos 2π

– 0 , bit duration– : carrier frequency, ≫ 1/

– : transmitted signal energy per bit, i.e.

• The pair of signals differ only in a 180-degree phase shift

27

Signal Space Representation for BPSK• The basis function for BPSK

∅2cos 2 0

• Then ∅ and ∅

• A binary PSK system is characterized by signal space that is one-dimensional (i.e. N=1), and has two message points (i.e. M =2)

28

0 ∅ 2

3/4/2018

15

Decision Rule of BPSK• Assume that the two signals are equally likely, i.e.

0.5

• The optimum decision boundary is the midpoint of the line joining these two message points

• Decision rule:– Guess signal (or binary 1) was transmitted if the

received signal point r falls in region (r>0)– Guess signal (or binary 0) was transmitted otherwise

(r 0)29

0 ∅

- rr

Region Region

Probability of Error for BPSK• The conditional probability of the receiver deciding

in favor of given that is transmitted is0

1 2

• Due to symmetry 0

30

3/4/2018

16

for BPSK (cont’d)• Since the signals and are equally likely to

be transmitted, the average probability of error is

0.5 0.52

• This ratio is normally called bit energy to noise density ratio (or SNR/bit)

31

depends on ratio

Probability of Error and the Distance Between Signals

• In general,

32

2

BPSK BFSK BASK

, 2 , 2 , 2

2

3/4/2018

17

Content• M-ary modulations

– MASK– MPSK– MFSK– MQAM

• Performance trade-off

33

M-ary Phase-Shift Keying (MPSK)• The phase of the carrier takes on M possible values:

2 1 / , 1,… ,• Signal set:

2cos 2

2 1

– Energy per symbol– ≫

• Basis functions

∅2cos 2

∅2sin 2

34

1,… ,0

0

3/4/2018

18

MPSK (cont’d)• Signal space representation

s2

cos 22 1

2cos 2 cos

2 1 2sin 2 sin

2 1

cos2 1

∅ sin2 1

∅

cos sin

1,… ,

35

• Euclidean distance

2 12

• The minimum Euclidean distance is

2 12

2

– plays an important role in determining error performance (union bound)

• In the case of PSK modulation, the error probability is dominated by the erroneous selection of either one of the two signal points adjacent to the transmitted signal point

• Consequently, an approximation to the symbol error probability is

2/

/=2

36

MPSK

3/4/2018

19

M-ary Quadrature Amplitude Modulation(MQAM正交幅度调制)

• In MPSK, in-phase and quadrature components are interrelated in such a way that the envelope is constant (circular constellation)

• If we relax this constraint, we get M-ary QAM

37

MQAM• Signal set:

2cos 2

2sin 2 0

– is the energy of the signal with the lowest amplitude– , are a pair of independent integers

• Basis functions:

∅2cos 2 ∅

2sin 2 0

• Signal space representation

38

3/4/2018

20

Error Performance of MQAM• Upper bound of the symbol error probability

43

1 for 2

• Exercises:Determine the increase in required to maintain the same error performance if the number of bits per symbol is increased from k to k+1, when k is large.

39

Fundamental Tradeoff:Bandwidth Efficiency and Energy Efficiency

• To see the ultimate power-bandwidth tradeoff, we need to use Shannon’s channel capacity theorem:– Channel Capacity is the theoretical upper bound for the

maximum rate at which information could be transmitted without error (Shannon 1948)

– For a bandlimited channel corrupted by AWGN, the maximum rate achievable is given by

log 1 log 1

• Note that

• Thus2 / 1

40

3/4/2018

21

Content• Optimal Receive

– Concept– MAP, ML rules– Optimal receive in AWGN channel

• Correlator-type demodulator• Matched-filter-type demodulator

• Decision• Performance analysis

41

MAP Decision Criterion (cont’d)• By Bayes’ Rule:

|

• Since our criterion is to minimize the probability of detection error given , we deduce that the optimum decision rule is to choose if and only if is maximum for .

• Equivalently,

• This decision rule is known as maximum a posterior or MAP decision criterion

42

3/4/2018

22

ML Decision Criterion• If ⋯ , i.e., the signals { } are

equiprobable, finding the signal that maximizes is equivalent to finding the signal that

maximizes |• The conditional pdf | is usually called the

likelihood function. The decision criterion based on the maximum of | is called the Maximum-Likelihood (ML) criterion.

• ML decision rule:

• In any digital communication systems, the decision task ultimately reverts to one of these rules

43

Correlation Type Demodulator• The received signal r(t) is passed through a parallel

bank of N cross correlators which basically compute the projection of r(t) onto the N basis functions

∅ , 1,… ,

44

3/4/2018

23

Matched-Filter Type Demodulator• Alternatively, we may apply the received signal r(t)

to a bank of N matched filters and sample the output of filters at . The impulse responses of the filters are

∅ , 0

45

What is Matched Filter ?• The matched filter (MF) is the optimal linear filter

maximizing the output SNR.• Derivation of the MF

– Input signal component ↔

– Input noise component with PSD /2

– Output signal component

46

3/4/2018

24

Solution of Matched Filter• When the max output SNR 2 / is achieved, we

have∗

∗

∗

• Transfer function of the matched filter: complex conjugate of the input signal spectrum

• Impulse response: time-reversal and delayed version of the input signal

47

Determining the Optimum DecisionRegions

• In general, boundaries of decision regions are perpendicular bisectors of the lines joining the original transmitted signals

• Example: three equiprobable2-dim signals

48

3/4/2018

25

Probability of Error using DecisionRegions

• Suppose is transmitted and is received• Correct decision is made when ∈ with

probability∈ |

• Averaging over all possible transmitted symbols, we obtain the average probability of making correct decision

∈

• Average probability of error

1 1 ∈

49

Example: analysis• Now consider our example with binary data

transmission− Given is transmitted, then

∈

− Since is Gaussian with zero mean and variance /2

2 2ln

1/2

50

3/4/2018

26

Example: analysis (cont’d)• Note that when

2

/2

/2 22

51

Union Bound• Conditional error probability

→2

• Finally, with equally likely messages, the average probability of symbol error is upper bounded by

1 12

• The most general formulation of union bound

52

3/4/2018

27

Union Bound (cont’d)• Let denote the minimum distance, i.e.

min, ,

• Since · is a monotone decreasing function

21

2

• Consequently, we may simplify the union bound as

1

• Simplified form of union bound

53

Synchronization

• Synchronization

– Carrier synchronization

– Symbol/Bit synchronization

– Frame synchronization

– Network Synchronization

54

3/4/2018

28


55

Source A/Dconverter

Sourceencoder


Channel


Sourcedecoder

D/Aconverter

User

Transmitter

Receiver


Noise

Contents

• Linear block code introduction

• Hamming codes

• Linear block codes

• Decoding of LBC

• Cyclic code, BCH code, R-S code

56

3/4/2018

29

Linear Block Code• : A block code of block length over an

alphabet χ is a non-empty set of -tuples of symbols from χ.– , . . . , , . . . , , . . . ,

– The -tuples of the code are called codewords.

• Rate:– channel alphabet: symbols– information symbols: tuples– number or codewords:

– code length: – rate

1log

– , code57

Hamming Distance• The Hamming distance between -tuples is the

number of components in which the n-tuples differ

, ∑ , , where ,1, if0, if

– , 0 with equality if and only if (nonnegativity)– , , (symmetry)– , , , (triangle inequality)– Hamming distance is a coarse or pessimistic measure of

difference.

• Other useful distances in error control coding:– , distance on a circle, is applicable to phase

shift coding.– ,is used with sensewords in .

58

3/4/2018

30

Minimum Distance• The minimum (Hamming) distance ∗ of a block

code is the minimum distance between any two codewords:

∗ min , ∶ , arecodewordsand

• Properties of minimum distance:– ∗ 1since Hamming distance between distinct codewords

is a positive integer.– ∗ if code has two or more codewords.– ∗ 1or ∗ ∞for the useless code with only one

codeword. (This is a convention, not a theorem.)– ∗ ∗ if ⊆ —smaller codes have larger (or

equal) minimum distance.

• The minimum distance of a code determines both error-detecting ability and error-correcting ability.

59

Error-detecting Ability• Suppose that a block code is used for error

detection only.– If the received -tuple is not a codeword, a detectable error

has occurred– If the received -tuple is a codeword but not the

transmitted codeword, an error has occurred that cannot be detected.

• : The guaranteed error-detecting ability is ∗ 1.

• The error-detecting ability is a worst case measure of the code.

60

3/4/2018

31

Error-correcting Ability (cont’d)• : Using nearest-neighbor decoding, errors of

weight t can be corrected if and only if 2 ∗. (For Hamming distance, equivalently, 2 1 ∗).

• : The “spheres” of radius surrounding the codewords do not overlap. Otherwise, there would be two codewords 2 distant. Therefore when errors occur, the decoder can unambiguously decide which codeword was sent.

61

Minimum Weight• The Hamming weight is the number of

nonzero components of .• Facts:

– 0,

– 1, 2 1 2 2 1

– 0if and only if 0

• Definition: The minimum (Hamming) weight of a block code is the weight of the nonzero codewordwith smallest weight:

min ∗ min ∶ ∈ , 0

• Examples of minimum weight:– Simple parity-check codes: ∗ 2.– Repetition codes: ∗ .– (7,4) Hamming code: ∗ 3. 62

3/4/2018

32

Minimum Distance = Minimum Weight• Theorem: For every linear block code, ∗ ∗.

Proof : We show that ∗ ∗ and ∗ ∗.– (≥) Let be a nonzero minimum-weight codeword. the 0

vector is a codeword, so∗ 0, ∗

– (≤) Let be two closest codewords. Then is a nonzero codeword, so

∗ , ∗

Combining these two inequalities, we obtain ∗ ∗.

• It is easier to find minimum weight than minimum distance because the weight minimization considers only a single parameter.

63

Syndrome (校正子) Decoding• Encoding for LBC is vector-matrix multiplication.• Maximum-likelihood decoding: maximize |• Decoding is inherently nonlinear. Fact: linear

decoders are very weak.• However, several steps in the decoding process are

linear:– syndrome computation– correction after error pattern and location have been found– extracting estimated message from estimated codeword

• Definition: The error vector or error pattern is the difference between the received -tuple and the transmitted codeword :

≜ ⇒ 64

3/4/2018

33

Syndrome Decoding (cont’d)• Multiply both sides of the equation by :

≜ 0 – The syndrome of the senseword is defined to be .– The syndrome of (known to receiver) equals the

syndrome of the error pattern e (not known to receiver but must be estimated).

• Decoding consists of finding the most plausible (貌似真实的) error pattern such that

• “Plausible” depends on the error characteristics:– For binary symmetric channel, most plausible means

smallest number of bit errors. Decoder picks error pattern of smallest weight satisfying .

– For bursty channels, error patterns are plausible if the symbol errors are close together.

65

Syndrome Decoding (cont’d)• Syndrome table decoding:

1. Calculate syndrome of received -tuple.2. Find most plausible error pattern with .3. Estimate transmitted codeword: .4. Determine message from the encoding equation .

• Only step 2 requires nonlinear operations.• For small values of , lookup tables can be used

for step 2.• Step 4 is not needed for systematic encoders, since

1: .

66

3/4/2018

34

Contents

• Convolutional code

– Review

– Viterbi decoder

– Examples of CC

• Construct long codes

• Turbo code

• LDPC code

67

CC Example• Rate-1/2 convolutional code

– The rational function is also element of the field 68

3/4/2018

35

Finite-state Transition Diagram (cont’d)

69

Viterbi Decoder Example• Rate-1/2 convolutional code

70

2 2( ) 1 1G D D D D

10,10, 00,01, 11, 01, 11

3/4/2018

36


71

Source A/Dconverter

Sourceencoder


Channel


Sourcedecoder

D/Aconverter

User

Transmitter

Receiver


Noise

Contents

• Multiple access

– TDMA

– FDMA (OFDMA)

– CDMA

– SDMA

72

Documents

EE140 Introduction to Communication Systems Review