1

Introduction to

Mobile Communications

Lecture 4 (Diversity)

Dong In Kim

College of Info/Comm Engineering

Sungkyunkwan University (SKKU)

2

Introduction to Diversity

Basic Idea

Send same bits over independent fading paths

Independent fading paths obtained by time, space,

frequency, or polarization diversity

Combine paths to mitigate fading effectsTb

tMultiple paths unlikely to fade simultaneously

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Combining Techniques

Selection Combining

Fading path with highest gain used

Maximal Ratio Combining

All paths cophased and summed with optimal weighting to

maximize combiner output SNR

Equal Gain Combining

All paths cophased and summed with equal weighting

Array/Diversity gain

Array gain is from noise averaging (AWGN and fading)

Diversity gain is change in BER slope (fading)

4

Maximal Ratio Combining

h3

Combined signalChannel gains Combiner weights

5

Maximal Ratio Combining

1 1 2 2 1 1 2 2

total noise

2 2 2 2 2*

1 2

2 2 2

1 22 2 2

1 2

array gain

Combined output ( )

where / and .

M M M M

w

i i M i o

M

M

h h h x w w w

h h h h E w E w N B

h h hh h h SNR M SNR

diversity gain

2 2 2

1 2

2 2 2

1 2

div

1 1 Pr[ 1] Pr

!

where the is defined as the ersity order expone of inverse / .

Note: 1 (no fading) as

nt

out M M

s o

M

M

P h h hSNR M SNR

SNR E N

h h h

M

with probability one!M

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Equal Gain Combining

h3

Combined signalChannel gains Combiner weights

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Diversity Performance

Selection Combining (SC) Combiner SNR is the maximum of the branch SNRs.

CDF easy to obtain, pdf found by differentiating.

Diminishing returns with number of antennas.

Can get up to about 20 dB of gain.

Maximal Ratio Combining (MRC)

Optimal technique (maximizes output SNR)

Combiner SNR is the sum of the branch SNRs.

Distribution of SNR hard to obtain.

Can use MGF approach for simplified analysis.

Exhibits 20-40 dB gains in Rayleigh fading.

1

1M

i i

1

M

ii

at high SNRM

bP

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MRC and its Performance

With MRC, =i for branch SNRs i

Optimal technique to maximize output SNR Yields 20-40 dB performance gains Distribution of hard to obtain

Standard average BER calculation

Hard to obtain in closed form Integral often diverges

MGF Approach

1 2 1 2( ) ( ) ( ) ( ) ( ) ( )b b b M MP P p d P p p p d d d

/ 2

210

;sin

M

b i i

i

gP d

M

/ 2

202 exp with

sinb b b

gP Q g d P E P

1 1 1, , 1 1, , ( ) ( )

M MM MP P P

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EGC and Transmit Diversity

EGQ simpler than MRC Harder to analyze Performance about 1 dB worse than MRC

Transmit diversity With channel knowledge, similar to receiver

diversity, same array/diversity gain Without channel knowledge, can obtain

diversity gain through Alamouti scheme: works over 2 consecutive symbols

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Main Points

Diversity typically entails some penalty in terms of rate, bandwidth, complexity, or size.

Techniques trade complexity for performance.

MRC yields 20-40 dB gain, SC around 20 dB.

Analysis of MRC simplified using MGF approach

EGC easier to implement than MRC: hard to analyze. Performance about 1 dB worse than MRC

Transmit diversity can obtain diversity gain even without channel information at transmitter.

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Time Diversity

• Time diversity can be obtained by interleaving and coding over

symbols across different coherence time periods.

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Example: GSM

• Amount of time diversity limited by delay constraint and how fast

channel varies.

• In GSM, delay constraint is 40ms (voice).

• To get full diversity of 8, needs v > 30 km/hr at fc = 900Mhz.

500kHz and sub-channel 200KHz

Total bandwidth of uplink/downlink 25MHz

cB

5ms, interleaving of 40ms diversity order 8!cT

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Antenna Diversity

Receive Transmit Both

SIMO MISO MIMO

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Receive Diversity

Same as repetition coding in time diversity,

except that there is a further power gain.

Optimal reception is via matched-filtering (MRC!)

(receive beamforming).

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Transmit Diversity

If transmitter knows the channel, send:

maximizes the received SNR by in-phase addition of

signals at the receiver (transmit beamforming).

Reduce to scalar channel:

same as receive beamforming.

What happens if transmitter does not know the channel?

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Space-Time Codes

• Transmitting the same symbol simultaneously at the

antennas doesn’t work.

• Using the antennas one at a time and sending the same

symbol over the different antennas is like repetition

coding.

• More generally, can use any time-diversity code by

turning on one antenna at a time.

• Space-time codes are designed specifically for the

transmit diversity scenario.

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Alamouti Scheme

Over two symbol times:

Projecting onto the two columns of the H matrix yields:

• double the symbol rate of repetition coding.

• 3dB loss of received SNR compared to transmit

beamforming (penalty because of no CSIT!).

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Alamouti Scheme

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Space-Time Code Design

A space-time code is a set of matrices

Full diversity is achieved if all pairwise differences

have full rank.

Coding (Array) gain determined by the determinants of

Time-diversity codes have diagonal matrices and the

determinant reduces to squared product distances.

4

det[( )( ) ]

L

A B L

A B A B

PSNR

X XX X X X

*( )( )A B A B X X X X

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Cooperative Diversity

• Different users can form a distributed antenna array

to help each other in increasing diversity.

• Distributed versions of space-time codes may be

applicable.

• Interesting characteristics:

– Users have to exchange information and this

consumes bandwidth.

– Operation typically in half-duplex mode.

– Broadcast nature of the wireless medium can be

exploited.

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Cooperative Diversity

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Frequency Diversity

• Resolution of multipaths provides diversity.

• Full diversity is achieved by sending one symbol every

L symbol times.

• But this is inefficient (like repetition coding).

• Sending symbols more frequently may result in

intersymbol interference.

• Challenge is how to mitigate the ISI while extracting the

inherent diversity in the frequency-selective channel.

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Approaches

• Time-domain equalization (eg. GSM)

• Direct-sequence spread spectrum (eg. IS-95 CDMA)

• Orthogonal frequency-division multiplexing OFDM

(eg. 802.11a, Flash-OFDM)

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ISI Equalization

• Suppose a sequence of uncoded symbols are

transmitted.

• Maximum-likelihood sequence detection is performed

using the Viterbi algorithm.

• Can full diversity be achieved?

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Reduction to Transmit Diversity

0 1 2. . 3 { , , }e g L h h h

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MLSD Achieves Full Diversity

Space-time code matrix for input sequence

Difference matrix for two sequences first differing at

is full rank.

A( ) or ( )A B B x X x X

x

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Direct-Sequence Spread Spectrum

• Information symbol rate is much lower than chip rate

(large processing gain).

• Signal-to-noise ratio per chip is low.

• ISI is not significant compared to interference from other

users and matched-filtering (Rake) is near-optimal.

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Frequency Diversity via Rake

• Considered a simplified situation (uncoded).

• Each information bit is spread into two pseudorandom sequences xA and xB (xB= -xA).

• Each tap of the matched-filter is a finger of the Rake.

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ISI vs Frequency Diversity

• In narrowband systems, ISI is mitigated using a complex

receiver.

• In asynchronous CDMA uplink, ISI is there but small

compared to interference from other users.

• But ISI is not intrinsic to achieve frequency diversity.

• The transmitter needs to do some work too!

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OFDM: Basic Concept

• Most wireless channels are underspread

(delay spread << coherence time) .

• Can be approximated by a linear time invariant channel

over a long time scale.

• Complex sinusoids are the only eigenfunctions of linear

time-invariant channels.

• Should signal in the frequency domain and then

transform to the time domain.

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OFDM

01 2

0

1

0

1[ ] ( )

( / , )

b b

N j nf t

nt mT n t mT

b

d m d t d eN

f W N W T

0 1

cyclic convolut[ [ ], , [ 1]] ( )

[ , , ,0, ,0] , [ [0], , [ 1]

ion

]

t

t t

L

y L y N L

h h d d N

y h d w

h d

0

DFT( ) DFT( ) DFT( ) DFT( )

Note: ( )!n

n n n n n

n n n

y N

h

h H nf

d w

h d w h d w

32

cont …

OFDM transforms the communication problem into the

frequency domain:

a bunch of non-interfering sub-channels, one for each

sub-carrier.

Can apply time-diversity techniques.

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Cyclic Prefix Overhead

• OFDM overhead

= length of cyclic prefix / OFDM symbol time

• Cyclic prefix dictated by delay spread.

• OFDM symbol time limited by channel coherence time.

• Equivalently, the inter-carrier spacing should be much

larger than the Doppler spread.

• Since most channels are underspread, the overhead can

be made small.

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Example: Flash OFDM (Flarion)

• Bandwidth = 1.25 Mz

• OFDM symbol = 128 samples = 100 m s

• Cyclic prefix = 16 samples = 11 m s delay spread

• 11 % overhead.

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Channel Uncertainty

• In fast varying channels, tap gain measurement errors

may have an impact on diversity combining performance.

• The impact is particularly significant in channel with

many taps each containing a small fraction of the total

received energy. (eg. Ultra-wideband channels)

• The impact depends on the modulation scheme.

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Summary

• Fading makes wireless channels unreliable.

• Diversity increases reliability and makes the channel

more consistent.

• Smart codes yields a coding gain in addition to the

diversity gain.

• This viewpoint of the adversity of fading will be

challenged and enriched in later parts of the course.