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National Chiao Tung University

Mobile Communication

Channel Coding, Equalization & Diversity

Wen-Shen Wuen

Trans Wireless Technology Laboratory

National Chiao Tung University

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RF Transceivers

Outline

Channel Coding

Equalization

Diversity

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Introduction to Channel Coding

Goal: to achieve reliable communication over a noisy

wireless channel

Approach: add redundancy to minimize error rate

01 0 1111 0

Transmitter Receiver

1 1 0 1

00 0 1 111 1

Source

Channel

encoder

Sink

Channel

decoderChannel

1 1 0 1

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Channel Coding

Mapping an incoming data sequence into an output

data sequence in such a way that the overall effect of

channel noise on the system is minimized

Introduce redundant (parity) bits in the channel

encoder so as to reconstruct the original data sequence as accurately as possible

Reduces the likelihood that all of the original data will be wiped out during a single transmission

The addition of parity bits will generally increase transmission bandwidth requirements or message delay

(or both).

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Channel Coding Theorem

Channel Coding Theorem

– If a discrete memoryless channel has capacity C and

a source generates information at a rate less than C,

then there exists a coding technique such that the output of the source may be transmitted over the

channel with an arbitrary low probability of symbol error.

– Channel capacity C is a fundamental limit for reliable (error free) transmission

– Asserts the existence of good codes, but how to find

good codes?

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Error Controlled Coding Techniques

Forward error-correction (FEC) codes (Channel codes)– Use of redundancy in the transmitted codeword for

both detection and correction of errors

– One-way link between the TX and RX

Automatic-repeat request (ARQ) schemes– Use of redundancy merely for error detection

– When detecting an error, RX requests a repeat transmission of the codeword in question

– Thereby necessitating the use of a return path(feedback channel)

– Retransmission à Latency à not suitable for speech communication

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Forward Error Correction Codes

Digital

Information

Source

(n,k)

EncoderData bits

Modulator

Digital

Information

Sink

(n,k)

DecoderDemodulator

kddd ,...,, 21 nuuu ,...,, 21

Code bits

Noisy

Channel

kddd ˆ,...,ˆ,ˆ21 nuuu ˆ,...,ˆ,ˆ 21

Data bits

estimates

Code bits

estimates

Code Rate:n

kR = Redundancy:

n

kn −

Large redundancy is advantageous because it reduces the likelihood that all of the original data will be wiped out during a single transmission.

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Example (6,3) Block Code

(6,3)

Block

Encoder

010 011 110

Message Sequence

101010 000011 000110

Message bits

Parity bits

Codeword

10-2

10-4

Uncoded

Coded

Coding

Gain

SNR [dB]

Bit

Err

or

Ra

te

8 12

3 bits/block 6 bits/block

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Block Codes vs Convolutional Code

Block Code– Input: subdivide data bits into blocks with k bits/block

– Encode: add (n-k) redundant bits

– Output: n code bits (n>k)

Convolutional Code– Encoding process: viewed as the discrete-time

convolution of input sequence with the impulse response of the encoder

– Duration of impulse = memory of the encoder

(n,k)

Block

Encoder

010 011 110 101010 000011

k bits/block n bits/block

Rs bits/secRo=(n/k)Rs bits/sec

Channel data rate

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Block Codes

Hamming distance– the number of different bits between two codes

Euclidean distance– geometric distance between two code vectors

Minimum distance

Weight– Number of nonzero bits

Systematic codes– Information bits and parity check bits are at fixed

locations

∑ −=n

nnH yxd ),( yx

( )),(minmin yxHdd =

[ ] { }1,0 21 ∈= nn xxxx �x

22 ),( ∑ −=n

nnE yxd yx

k k k k m m m(7,4)

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Block Codes

Linear codes (group codes)

– The sum of any codewords gives another valid

codeword

– Properties

• The all-zero word is a valid codeword

• All codewords (except the all-zero word) have a weight equal to or larger than dmin

• Distribution of distance is equal to the weight distribution of the code

• All codewords can be represented by a linear combination of basic codewords (generator words)

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Block Codes

Cyclic codes

– Any cyclic shift of a codeword in the code is also a

codeword

– Extremely well suited for error detection

• Can be designed to detect many combinations of likely errors

• Implementation of encoding and error-detecting circuit is very simple

– A cyclic code used for error detection is referred to

as a cyclic redundancy check (CRC) code

[ ]012345 010110)(

010110

xxxxxxxX ⋅+⋅+⋅+⋅+⋅+⋅==x

Generation polynomial 1)( 12345 +++++= xxxxxxX

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Example: (7,4) Hamming Code

[ ]

[ ] [ ]101101

1111000

1100100

1010010

0110001

1101

1101

1111000

1100100

1010010

0110001

=

⋅==

==

=

sGx

s

sGx

G

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Encoding and Decoding

Encoding

– kxk identity metrix

– n-k columns parity check bits

Decoding

– To check whether the received codeword is a valid codeword, multiply it by a parity check matrix H

– Resultant n-k dimensional syndrome vector ssynd.

– If ssynd has all-zero entries, then the received codeword is valid

=

1111000

1100100

1010010

0110001

G

k x k identity metrix

parity checkbits

k x n generator metrix

sGx =

T

synd ˆHxs =T

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Syndrome for Hamming Code

Received information: 1000

Received parity check bits: 101

Computed parity check bits: 110 (parity of the

transmitted codeword)

[ ]110ˆ T

synd == HxsT

=1001110

0101101

0011011

H[ ]1010001ˆ =x

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How to Find Parity Check Matrix H

Relation to be hold: 0GH =⋅ T

[ ][ ] [ ] [ ]T

T

T

TT

T

PHIH

0PHHP

IHHPIHHGH

0GH

12

21

−==

=+=

==⋅

=⋅

and

2121

Let à The relation holds

=

111

110

101

011

P

=1110

1101

1011TP

=1001110

0101101

0011011

H

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Convolutional Code

Cell 1 Cell 2

n1

n2

n3

xnU

K=1N=3L=3

R=1/3

U1=1à X1=1,1,1

U1=1 U2=1 à X2=1,0,0

Initial: all zero state State Cell 1 Cell 2A 0 0B 1 0C 0 1D 1 1

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Trellis Diagram

A 00

001011110011111

00101Input bit sequence

Output coded

000 000 000 000 000

B 10

C 01

D 11

111

011

110

001

0

1

101 101 101

111111

111111

001

001

011011

011

010010

010

100

100 100

100

110110

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Maximum-Likelihood Decoding

Maximum-Likelihood Decoding of Convolutional

Codes

– Message vector: m

– Code vector: c

– Received vector: r

– Estimate for message:

– Esitmate for code:

Maximum Likelyhood Decoder

– Choose the estimate for which the log-likelihood function is maximum

m

c

( )cr|log p

c

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Maximum-Likelihood Decoding

( )

( ) ( )

( )pNp

pd

pdNpdp

crp

crpcrpcrpp

crpp

ii

ii

iiii

N

i

N

i

ii

−+

−=

−−+=

=−≠

==

=

∑

∏

=

=

1log1

log

)1log()(log)ˆ|(log

ˆ1

ˆˆ|,ˆ|log)ˆ|(log

ˆ|)ˆ|(

1

1

cr

cr

cr

d: Hamming distance between r and c

For binary symmetric channel, maximum likelihood decoderreduces to a minimum distance decoder.

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Equalization

Compensates for intersymbol interference (ISI)

created by multipath time dispersive channels

Compensates for the average range of

expected channel amplitude and delay

characterisitics

Must be adaptive since the channel is generally

unknown and time varying

Adaptive equalizer operation include training

and tracking

Must balance ISI reduction with minimizing

noise enhancement

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Equalizer Noise Enhancement

Channel

H(f)

Equalizer

Heq(f)Demod

s(t)

n(t)

y(t) s(t)+n’(t)

[ ]

2

0

0

)|(|2)('

)(')()()()()(

)(

1)(

2)(

)()()()(

fH

NPSDfN

fNfSfHfNfHfS

fHfH

NPSDfN

fNfHfSfY

eq

eq

=⇒

+=+

=

=

+=

For some frequency H(f)=0, noise becomes infinite!!

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Equalizer Category

noise enhancement

Most common optimal

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Adaptive Equalizer in Communication Systems

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Adaptive Equalizer in Communication Systems

1)()(

)()()()(

without )()(ˆ

)()(

)()()()(

)()()()()()(ˆ

)()()()(

*

*

*

*

=−⇒

=⊗=

=

−=

⊗+⊗=

⊗+⊗⊗=

+⊗=

∑

fFfH

tthtftg

(t)ntxtd

nTtcth

thtntgtx

thtnthtftxtd

tntftxty

eq

eq

b

n

neq

eqb

eqbeq

b

δ

δ

Equalizer is actually an inverse filter of the channel

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Basic Linear Equalizer During Training

k

T

kk

T

k

N

n

nknkk ywd

ywwy ==

=∑=

−0

ˆ

[ ]TNkkkkk wwww �210=w

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Basic Linear Equalizer During Training

k

T

kkk

T

kkkkkkk xxdxdde ywwy −=−=−=−= ˆˆ

Compute mean square error:

[ ] [ ] [ ] [ ] k

T

kkk

T

kk

T

kkk

k

T

kkk

T

kk

T

kkk

xEExEeE

xxe

wywyyw

wywyyw

2

2

22

22

−+=

−+=

Cross- correlation vector p

Input correlation matrix, input covariance matrix R

[ ] [ ]TNkkkkkkkkkk yxyxyxyxExE −−−== �21yp

=

−−−−−−

−−−−−−

−−−

2

21

121

2

11

21

2

NkkNkkNkkNk

Nkkkkkkk

Nkkkkkkk

yyyyyyy

yyyyyyy

yyyyyyy

�

�����

�

�

R

Mean square error: [ ] wpRww TT

kxE 22 −+=ξ

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Basic Linear Transversal Equalizer Structure

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Nonlinear Equalization

Decision Feedback Equalizer

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Nonlinear Equalization

Predictive Decision Feedback Equalizer

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Nonlinear Equalization

Maximum Likelihood Sequence Estimation

(MLSE)

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Diversity

Rayleigh fading

Log-normal shadowing

Diversity Combining

– Combining of independent signal paths

Deep signal fades à SNR degrade

Signals

fading path 1

fading path 2

fading path N

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Diversity

Micro-diversity à mitigate the effects of

multipath fading

Macro-diversity à mitigate the effects of

shadowing from buildings and objects

– Combining signals received by several base stations

– Requiring coordination among these BS’s

– Need protocols in infrastructure based networks

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Realization of Independent Paths

Approach 1: Multiple antennas (antenna array)

– Space diversity

– RX space diversity: independent fading paths are

realized w/o an increase in TX signal power / BW

– Increase SNR à array gain

– Diversity gain

• Change in slope of the error probability resulting from the diversity combining

– Maximizing diversity gain for TX or RX space diversity requires the separation between antennas

• Such that fading amplitude corresponding to each antenna are approximately independent

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Space Diversity

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Realization of Independent Paths

Approach 2: Using Two TX antennas or Two

RX antennas with different polarization (V,H)

waves

– Random reflection and scattering angles àindependent of different polarization antennas

– Disadvantages

• At most two diversity

• Loses effectively half power because TX/RX power is divided between two polarized antennas

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Realization of Independent Paths

Approach 3: Directional Antennas

– Angle/Directional diversity

– Restrict the RX antenna beamwidth to a given angle

– As small angle as possible such that at most one of

the multipath rays will fall within the RX beamwidth

– Requires a sufficient number of directional antennas

– Or requires a single antenna whose directivity can

be steered to the arrival angle of one multipathcomponent

– SNR may be decreased due to the loss of mulitpathcomponents that fall outside the RX beamwidth

– Smart antennas à antennas with adjustable phase

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Realization of Independent Paths

Frequency Diversity

– Transmit the same narrowband signal at different

carrier frequency

Time Diversity

– Transmit the same narrowband signal at different

time, where time differece is greater than channel coherence time (1/Dopper spread)

– Doesn’t increase transmit power

– Lower data rate due to latency

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Combining of Independent Fading Paths

Linear Combining

– The output of the combiner is just a weighted sum of different fading paths or branches

Co-phasing

– Combining more than one branch need co-phasing

– Remove θi by multiply αi

Multiplication by αi

– Pre-detection

– Post-detection

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The Received SNR of Receiver Diversity

MN

E

N

EN

N

E

aN

raS

M

i

s

M

i

s

M

i

i

M

i

ii

⋅=

=

=∑

∑

∑

∑

=

=

=

=Σ

0

1 0

0

2

1 0

1

2

0

2

1γ

00

N

ra

N

EEr i

is

isi === γAssume

Array Gain

γγ Σ=gA

And in the absence of fading

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Diversity Combining Considerations

Selection Combining

Threshold Combining

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Diversity Combining Considerations

Maximum Ratio Combining

Equal Gain Combining