ETN644 Lec8 Handouts

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Wireless Communication Techniques

(Fall 2011)

Lecture 8

Diversity &

Equalization

Dr Shurjeel Wyne

DIVERSITY

(CH7 - Rappaport,2e)

(CH13- Molisch)

Error Performance of Digital Modulationin Slow Flat-Fading Channels

To obtain average error performance under fading

, we must viewthe AWGN error performance at some arbitrary received SNR

as a conditional probability where the condition is that α is fixed.

To obtain the average error performance when α is random, wemust average the (conditional) AWGN error performance over theprobability density function of ,i.e., we evaluate the integral:

Where is the probability density function of

0

2

N

E b

d pdf PP AWGN ee

0

,

pdf

BER in Fading Channels…

Fading causes serious BER problems !

0 5 10 15 20 25 3010

-6

10-4

10-2

100

Eb/No [dB]

A v e r a g e B E R

BPSK with Coherent Detection

Rayleigh Fading

AWGN (non-fading)



2

Diversity techniques are used to combat Fading

• Macro-diversity: mitigates the effects of shadowing frombuildings and objects

– Combining signals received by several base stations – Requires coordination among involved Base stations – Needs protocols in infra-structure based networks(we will not study this further)

• Micro-diversity: mitigate the effects of multipath fading

(we will only focus on Micro-diversity in the course)

DIVERSITY TECHNIQUES

The principle of diversity is to transmit the

same information on M statisticallyindependent channels.

By doing this, we increase the chance thatthe information will be received properly.

An example of one such arrangement isantenna diversity.

DIVERSITY ARRANGEMENTSThe Diversity Principle

Diversity is used to combat Fading ! !

(Note: correlated channels will givediminished performance relative touncorrelated channels).

DIVERSITY ARRANGEMENTSSmall Scale Fading Pattern

DIVERSITY ARRANGEMENTSGeneral Improvement Trend

0 5 10 15 20 25 3010

-6

10-5

10-4

10-3

10-2

10-1

100

Eb/No [dB]

A v e r a g e B E R

SER of BPSK with Coherent Detection

Rayleigh Fading

Rayleigh Fading (4 branch Diversity)

AWGN (non-fading)

2



3

DIVERSITY ARRANGEMENTSSome Techniques

Our focus will be on spatial(antenna) diversity.

M M

Figure: General Block Diagram for M-branch Antenna Diversity

Antenna (Space) Diversity Techniques

Antenna Diversity Techniques can be classified into four Categories:

1. Selection Diversity2. Maximal Ratio Combining (MRC)3. Equal Gain Diversity4. Feedback Diversity

CombinerOutput

:

Diversity Advantages

Increase in average SNR (at combiner

output) relative to the single-branch SNR

Improvement in the outage probability of the SNR at the combiner output relative tothe single-branch SNR

ANTENNA DIVERSITY Selection Diversity

• Among the M r eceiver branches (antennas), the reciever branch havingthe highest in stantaneous SNR is connected to the demodulator

-10 -5 0 5 1010

-4

10-3

10-2

10-1

100

/ [dB]

P r o b .

[ /

a b s c i s s a ]

Pout

of Selection Co mbining in Rayleigh Fading

M=1

M=2

M=3

M=4

M=5

M=6

Diminishing returns withincreasing number ofantennas, M

Fig: RSSI-driven 2-Branch selection Diversity



4

ANTENNA DIVERSITY Maximum Ratio Combini ng ( MRC) Diversity

• The signals from each of M branches are co-phased and weightedaccording to respective signal to noise ratios and then summed

• MRC produces an output SNR equal to the sum of the input SNRs•This technique gives best statistical reduction of fading among alllinear diversity combiners

MRC

M M

M G

ANTENNA DIVERSITY Equal Gain Comb inin g Diversit y

• The variable weighting capability of MRC diversity is replaced withfixed branch weights, set to unity for all M branches

•The signals from each branch are still co-phased

•The performance of Equal Gain Combining Diversity is marginallyinferior to MRC, and superior to selection diversity

1

1

ANTENNA DIVERSITY Scanning/ Feedback Diversity

• Instead of always using the best of M signals, the M signals arescanned in a fixed sequence until one is found to be above a pre-determined threshold

• This signal is then received until it falls below the pre-determinedthreshold, and the scanning is intiated again

ANTENNA DIVERSITY Performance Comparison of Selection & MRC Diversity

Normalized SNR [dB]



5

Q. Assume 4-branch selection diversity isused, where each branch receives anindependent Rayleigh fading signal. If the average branch SNR is 20 dB,determine the probability that theinstantaneous SNR will drop below 10dB. Compare this with the case of asingle receiver without diversity

EQUALIZATION

(CH16 - Molisch)

EQUALIZATION

Inter-symbol interference

Discrete Time Channel Model

Linear equalizers

Decision-feedback equalizers

Maximum-likelihood sequence estimation

INTER SYMBOL INTERFERENCEBackground

Even if we have designed the basis pulses of our modulation to beinterference free in time, i.e. no leakage of energy betweenconsecutive symbols, multi-path propagation in our channel willcause a delay-spread and inter-symbol int erference (I SI ).

ISI will degrade performance of our receiver, unless mitigated bysome mechanism. This mechanism is called an equalizer .



6

ASK

ASK Matched Filter shouldnow be matched tocascade of Tx pulse& channel filter

INTER SYMBOL INTERFERENCEIncluding a Channel Impulse Response

Assume Amplitude Shift Keying (ASK) modulation wit h Matched Filtering:

AWGN Channel Model

Matched Filter

MF output samples

ISI-free for properly

shaped pulses g(t)

These output samplesno longer ISI-free & noise is not white

I ncluding im pulse response of a Dispersive Channel, h(t )

The discrete time equivalent of the “new” system is:

where we can say that F(z) represents the basis pulse and channel, whileF*(z-1) represents the matched filter.

We can now achieve white noise quite easily, if (the not unique) F(z) ischosen wisely, i.e., F*(z-1) has a stable inverse:

k c

k c

k n k

k nk

uk

INTER SYMBOL INTERFERENCEThe Discrete Time Channel Model

)( zF

)( zF

)( 1* zF

)( 1* zF )(1

1* zF

k c k

nk u

)( zF

L

j

k jk jk nc f u0

INTER SYMBOL INTERFERENCEThe Discrete Time Channel Model…

With the application of a noise-whitening filter, we arrive at a discrete-time model

where we have ISI and white additive noise, in the form

The coefficients f j represent the causal impulse response of the discrete-timeequivalent of the channel F (z ), with an ISI that extends over L symbols.

k u

k c

L

j

k jk jk nc f u0

INTER SYMBOL INTERFERENCEThe Discrete Time Channel Model…

+

k n

What is the the operation indicated by the summation?



EQUALIZATION

Compensates for intersymbol interference (ISI)created by multipath time dispersive channels

Compensates for the average range of expectedchannel amplitude and delay characteristics

Must be adaptive since the channel is generallyunknown and time varying

Adaptive equalizer operation includes trainingand tracking

Must balance ISI reduction with minimizing noiseenhancement

Classification of Equalizers

Apply an equalizer filter E (z ) at the receiver to mitigate the effect of ISI:

LINEAR EQUALIZER Principle

Now we have two different strategies:

1) Design E(z) so that the ISI is totally removed:

2) Design E(z) so that we minimize the mean squared-error:

k k k cc ˆ

Zero-forcing

MSE

Equalizer Filter

1)()( z E zF

LINEAR EQUALIZER Zero Forcing Equalizer

zero-forcing equalizer aims to remove the ISI completely, i.e., 1)()( z E zF

k c)(

1)( zF

z E k cˆ

k u

k n

)( zF

ZF Equalizer



A serious problem with the zero-forcing equalizer is the noise

enhancement , which can result in infinite noise power spectral densitiesafter the equalizer.

The noise is enhanced (amplified) at frequencies where the channelhas a high attenuation.

Another, related, problem is that the resulting noise is colored, whichmakes an optimal detector quite complicated.

By applying the minimum mean squared-error criterion instead, wecan remove the noise-enhancement effects.

LINEAR EQUALIZER Zero Forcing Equalizer…

LINEAR EQUALIZER MSE Equalizer

k c)( zF

0

22

1*2

)(

)(

N zF

zF

s

s

k cˆk u

k nThe MSE equalizer is designed to minimize the error variance

The MSE equalizer removes the most problematic noiseenhancements as compared to the ZF equalizer. The noise power

spectral density cannot go to infinity any more.

This improvement from a noise perspective comes at the cost of nottotally removing the ISI.

The noise is still colored after the MSE equalizer which, in combinationwith the residual ISI, makes an optimal detector quite complicated.

LINEAR EQUALIZER MSE Equalizer…

We have seen that taking care of the ISI using only a linear filter willcause (sometimes severe) noise coloring.

A slightly more sophisticated approach is to subtract the interferencecaused by already detected data (symbols).

This principle of detecting symbols and using feedback to remove theISI they cause (before detecting the next symbol), is calleddecision feedback Equalization (DFE).

DECISION FEEDBACK EQUALIZER Principle



k c k c

k n

)( zF )( z E

)( z D

This part shapesthe signal towork well withthe decision

feedback.

This part removesISI on “future”symbols from thecurrently detected

symbol.

a wrongdecisionfrom thedecisiondevice willincreasethe ISIinstead of minimizingit

DECISION FEEDBACK EQUALIZER Principle…

DECISION FEEDBACK EQUALIZER Zero Forcing DFE

k c)( zF )( z E

k ck n

)( z D

In the design of a ZF-DFE, we want to completely remove all ISIbefore the detection.

This enforces a relation between the E (z ) and D (z ), which is (we assumethat we make correct decisions!)

As soon as we have chosen E (z ), we can determine D (z ).

1 z D z E zF

Like in the linear ZF equalizer, forcing the ISI to zero before thedecision

device of the DFE will cause noise enhancement.

Noise enhancement can lead to high probabilities for making the wrongdecisions ... which in turn can cause error propagation, since we mayadd ISI instead of removing it in the decision-feedback loop.

DECISION FEEDBACK EQUALIZER Zero Forcing DFE…

k c)( zF )( z E

k ck n

)( z D

To limit noise enhancement problems, we can concentrate on minimizingmean squared-error (MSE) before the decision device instead of totallyremoving the ISI.

The overall strategy for minimizing the MSE is the same as for t helinear MSE equalizer (again assuming that we make correct decisions).

DECISION FEEDBACK EQUALIZER MSE DFE



By concentrating on minimal MSE before the detector, we can reducethe noise enhancements in the MSE-DFE, as compared to the ZF-DFE.

The performance of the MSE-DFE equalizer is (in most cases) higherthan the previous equalizers ... but we still have the error propagationproblem that can occur if we make an incorrect decision.

DECISION FEEDBACK EQUALIZER MSE DFE

k c k ck n

)( zF k u

The optimal equalizer that correctly detects the transmitted sequence with the highestprobability, is the maximum- likelihood sequence estimator (MLSE) .

The principle is the same as for the Maximumi Likelihood symbol detector (receiver)but with the difference that we now look at the entire sequence of transmitted symbols.

MLSE:Compare the received

noisy sequence uk

With all possible noise freereceived sequences andselect the closest one

For sequences of length N symbols, this requires comparison with 2N

different noise free sequences.Brut e force MLSE is too complex

to implement practically complex

MAXIMUM LIKELIHOOD SEQUENCE ESTIMATOR Principle

T T T T

T

T

T

T 2

T T T 3T 2

T 3

Single pulse at input of linear equalizer filter:

Compensated pulse at output of linear filter:

ISI to neighboring pulses

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