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General Outline

• Part 0: Background, Motivation, and Goals.

• Part I: Some Basics.

• Part II: Diversity Systems.

• Part III: Co-Channel Interference.

• Part IV: Multi-Hop Communication Systems.

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Outline - Part III: Co-Channel Interference

1. Co-Channel Interference (CCI) Analysis

• Effect of Shadowing

• Effect of Multipath Fading

– Single Interferer

– Multiple Interferers

– Minimum Desired Signal Requirement

– Random Number of Interferers

2. CCI Mitigation

• Diversity Combining

• Optimum Combining/Smart Antennas

• Optimized MIMO Systems in Presence of CCI

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Effect of Shadowing

• Finding the statistics of the sum of log-normalrandom variables.

• No known exact closed-form available.

• Several analytical techniques have been devel-oped over the years.

• As an example, we cover the Farley bounds onthe sum of log-normal random variables.

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Effect of Multipath Fading (1)

• Single Interferer

– Let the carrier-to-interference ratio (CIR)

λ =sd

si,

where sd = α2d (with average Ωd) and si = α2

i(with average Ωi) are the instantaneous fadingpowers of desired and interfering users.

– Outage probability

Pout = Prob[λ ≤ λth]

=

∫ ∞

0psd(sd) Prob

[si ≥

sd

λth|sd

]dsd

=

∫ ∞

0psd(sd)

∫ ∞

sd/λth

psi(si) dsi dsd·

– Exp: Rician (Rician factor Kd)/Rayleih case

Pout =λth

λth + bexp

(− Kb

λth + b

),

where

b =Ωd

Ωi(Kd + 1).

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Effect of Multipath Fading (2)

• NI independent identically distributed interferers

– Let the carrier-to-interference ratio

λ =sd

sI,

where sI =∑NI

i=1 si and all the si have thesame average fading power Ωi.

– Outage probability

Pout = Prob[λ ≤ λth]

=

∫ ∞

0psd(sd)

∫ ∞

sd/λth

psI(sI) dsI dsd·

– Exp: Nakagami/Nakagami scenario

Pout = Ix(m,mNI)

where

x =

(1 +

Ωd

Ωiλth

)

and

Ix(a, b) =Γ(a + b)

Γ(a)Γ(b)

∫ x

0ta−1 (1− t)b−1 dt

is the incomplete Beta function ratio.

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Effect of Multipath Fading (3)

• NI independent non-identically distributed inter-ferers

– Let the carrier-to-interference ratio

λ =sd

sI,

where sI =∑NI

i=1 si and all the si can havedifferent average fading power Ωi.

– Outage probability

Pout = Prob[λ ≤ λth]

= Prob

sd − λth

NI∑

i=1

si ≤ 0

.

– Define α = λth∑NI

i=1 si − sd.

∗ α ≥ 0 corresponds to an outage.

∗ α ≤ 0 corresponds to satisfactory transmis-sion.

– Find characteristic function of α and then useGil-Palaez lemma.

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Gil-Palaez Lemma and Application

• Let X be a random variable with cumulative dis-tribution function (CDF) PX(x) = Prob[X ≤x] and characteristic function (CF)

φX(t) = E[ejXt

]then

PX(x) =1

2− 1

π

∫ ∞

0

Im[φX(t)e−jXt

]

tdt.

• Application to outage probability

Pout = Prob[α ≥ 0] = 1− Prob[α ≤ 0]

= 1− Pα(0) =1

2+

1

π

∫ ∞

0

Im [φα(t)]

tdt,

where for the Nakagami/Nakagami fading case

φα(t)=φsd(−t)

NI∏

i=1

φsi(λtht)

=

(1 +

jtΩd

md

)−mdNI∏

i=1

(1− jtΩiλth

mi

)−mi

.

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Random Number of Interferers

• Depending on traffic conditions, the number ofactive interferers nI is a random variable from 0to NI which is the maximum number of activeinterferers.

• Outage probability is given by

Pout =

NI∑

nI=0

Prob[nI ] Pout[nI ].

• Assume Nc available channels per cell each withactivity probability pa.

• The blocking probability

B = pNca .

• PDF of nI is Binomial

Prob[nI ]=

(NInI

)pnIa (1− pa)NI−nI , nI = 0, · · · , NI .

=

(NInI

)BnI/Nc

(1−B1/Nc

)NI−nI.

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Outline - Part III: Co-Channel Interference

1. Co-Channel Interference (CCI) Analysis

• Effect of Shadowing

• Effect of Multipath Fading

– Single Interferer

– Multiple Interferers

– Minimum Desired Signal Requirement

– Random Number of Interferers

2. CCI Mitigation

• Diversity Combining

• Optimum Combining/Smart Antennas

• Optimized MIMO Systems in Presence of CCI

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Diversity Combining

• Reduce the effect via selective (or switched) an-tenna diversity combining techniques.

• Consider a dual-antenna diversity system withone co-channel interferer. Let α11 denote thefading amplitude from desired user to antenna1, α12 denote the fading amplitude from desireduser to antenna 2, α21 denote the fading am-plitude from interfering user to antenna 1, andα22 denote the fading amplitude from interferinguser to antenna 2.

• Three main decision algorithms for selective di-versity.

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Decision Algorithms

• CIR algorithm: picks and process the informationfrom the antenna with the highest CIR. For thescenario describe previously:

Max

[(α11

α21

)2

,

(α12

α22

)2]

.

• Desired signal algorithm: picks and process theinformation from the antenna with the highestdesired signal, i.e.,

Max[α2

11, α212

].

• Signal plus interference algorithm: picks and pro-cess the information from the antenna with thehighest desired plus interfering signal, i.e.,

Max[α2

11 + α221, α

212 + α2

22

].

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Interference Mitigation

•More advanced interference mitigation techniques

– Optimum combining

– Optimized MIMO systems

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General Outline

• Part 0: Background, Motivation, and Goals.

• Part I: Some Basics.

• Part II: Diversity Systems.

• Part III: Co-Channel Interference.

• Part IV: Multi-Hop CommunicationSystems.

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Multi-Hop Communication Systems

• Advantages of transmission with relays:

– Broader coverage

– Lower transmitted power (higher battery lifeand lower interference)

– “Cooperative/Collaborative/Multi-user” diver-sity.

• “Pionnering” work on this topic:

– Sendonaris, Erkip, and Aazhang, [ISIT’98].

– Laneman, Wornell, and Tse [WCNC’00, Aller-ton’00, ISIT’01].

– Emamian and Kaveh, [ISC’01].

• Goal:Develop an analytical framework for the exactend-to-end performance analysis of dual-hop thenmulti-hop relayed transmission over fading chan-nels.

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Dual-Hop Systems

• Consider the following dual-hop communicationsystem with a relay

CA

B

α2α1

• Two relaying options:

– Non-regenerative relaying (known also as ana-log or amplify-and-forward relaying)

– Regenerative relaying (known also as digital ordecode-and-forward relaying)

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Non-Regenerative Systems

• Received signal at the relay input (B) is

rb(t) = α1s(t) + n1(t).

• Received signal at the destination (C) is

rc(t) = α2Grb(t) + n2(t)

= α2G(α1s(t) + n1(t)) + n2(t).

• Equivalent end-to-end SNR

γeq =α2

1α22G

2

α22G

2N01+ N02

=

α21

N01

α22

N02

α22

N02+ 1

G2N01

.

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Choice of the Relay Gain

• One possible choice of the relay gain is just chan-nel inversion, i.e.,

G2 = 1/α21,

– Resulting equivalent end-to-end SNR

γeq2 =γ1γ2

γ1 + γ2.

– Lower bound on the performance of practicalrelays

– Related to the Harmonic Mean of γ1 and γ2

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A Second Choice of the Relay Gain

• Another possible choice of the relay gain [Lane-man et al. ’00]

G2 =1

α21 + N0

.

– Limits the gain of the relay when first hop isdeeply faded

– Resulting equivalent end-to-end SNR

γeq1 =γ1γ2

γ1 + γ2 + 1,

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Monte Carlo Simulation

• Comparison of the outage probability for the twochoices of the relay gain

0 5 10 15 20 25 30 35 4010

−4

10−3

10−2

10−1

100

Normalized SNR [dB]

Out

age

Pro

babi

lity

Pou

t

Comparison of the Two Choices of Relay Gain Using Monte Carlo Simulaion

Relay Gain (Laneman)Relay Gain (Channel Inversion)

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Harmonic Mean

• Given two numbers X1, X2:

– Arithmetic Mean

µA(X1, X2) =X1 + X2

2

– Geometric Mean

µG(X1, X2) =√

X1X2

– Harmonic Mean

µH(X1, X2) =2X1X2

X1 + X2=

21

X1+ 1

X2

• Relation with end-to-end SNR

γeq2 =1

2µH(γ1, γ2) ≥ γeq1,

where γ1 and γ2 are the instantaneous SNRs ofhops 1 and 2, respectively.

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Harmonic Mean of Exponential Variates

• Theorem :Let X1 and X2 be two independent exponentialvariates with parameters β1 and β2 respectively.Then, the PDF of X = µH(X1, X2), pX(x), isgiven by

pX(x)=1

2β1β2xe−

x2(β1+β2)

[(β1 + β2√

β1β2

)K1

(x√

β1β2

)

+2K0

(x√

β1β2

)]U(x),

where Ki(·) is the ith order modified Besselfunction of the second kind and U(·) is the unitstep function.

• The CDF and MGF of the harmonic mean of twoindependent exponential variates are also avail-able in closed-form.

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Validation by Monte-Carlo Simulations

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Monte Carlo Simulation for pΓ(γ)

Pro

babi

lity

Den

sity

Fun

ctio

n, p

Γ(γ)

γ

Analysis Simulation

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Derivation of the CDF of X = µH(X1, X2)

• Let

Z =1

X=

1

2

(1

X1+

1

X2

).

• The CDF of X, PX(x), is given by

PX(x) = Pr (X < x)

= Pr

(1

X>

1

x

)= Pr

(Z >

1

x

)

= 1− Pr

(Z <

1

x

)= 1− PZ

(1

x

),

where PZ(·) is the CDF of Z.

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Derivation of the CDF of X = µH(X1, X2)(Continued)

• If X is an exponential random variable with pa-rameter β then the MGF of Y = 1/X can beshown to be given by

MY (s) = E[e−sY

]= 2

√βs K1

(2√

βs)

.

• Using the differentiation property of the Laplacetransform, PZ(z) can be written as

PZ(z) = L−1(MZ(s)

s

)

= 1− L−1(2√

β1β2K1

(2√

β1s)

K1

(2√

β2s))|z=1

x,

which is a tabulated inverse Laplace transformleading to

PX(x)=1− PZ

(1

x

)

=1− x√

β1β2e−x

2(β1+β2)K1

(x√

β1β2

).

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Derivation of the PDF of X = µH(X1, X2)

• Taking the derivative of the CDF of X with re-spect to x results in

d

dx(PX(x)) = −

[√β1β2e

−x2(β1+β2)K1

(x√

β1β2

)

+ x√

β1β2

(−1

2(β1 + β2) e−

x2(β1+β2)

×K1

(x√

β1β2

)+ e−

x2(β1+β2)

d

dx

[K1

(x√

β1β2

)])].

• Using

zd

dzKv(z) + vKv(z) = −zKv−1(z)

leads to the final desired result

pX(x) =1

2β1β2xe−

x2(β1+β2)

[(β1 + β2√

β1β2

)

K1

(x√

β1β2

)+ 2K0

(x√

β1β2

)].

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Formulas for the Outage Probability

• For non-regenerative systems, Pout is given by

Pout = 1− 2γth√γ1γ2

K1

(2γth√γ1γ2

)e−γth

(1γ1

+ 1γ2

),

where γ1 and γ2 are the average SNRs of hops1 and 2, respectively.

• For regenerative systems, Pout is given by

Pout = 1− e−γth

(1γ1

+ 1γ2

).

• Both formulas are equivalent at high average SNRsince for small x

K1(x) ' 1

x.

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Outage Probability: Numerical Example

0 5 10 15 20 25 3010

−3

10−2

10−1

100

Normalized SNR [dB]

Out

age

Pro

babi

lity

Pou

t

Comparison of Outage Probability for Regenerative and Non−Regenerative Systems

Non−Regenerative System,γ1=γ

2Regenerative System,γ

1=γ

2Non−Regenerative System,γ

1=2γ

2Regenerative System,γ

1=2γ

2

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Outage Probability of CollaborativeSystems

• Consider a wireless communication system withone direct link and L collaborating paths.

• Assume direct link with average SNR γ0 and thatthe two hops in collaborating path l have thesame average SNR γl.

• Assume that the strongest path is selected at anygiven time.

• Resulting outage probability

Pout =

(1− e

−γthγ0

)

×L∏

l=1

(1− 2γth

γle−2γth

γl K1

(2γth

γl

)).

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Formulas for the Average BER

• The MGF of γeq, E(e−γs), for identical and in-dependent faded hops, i.e. γ1 = γ2 = γ, is givenby

MΓ(s) =

√γ4s

(γ4s + 1

)+ arcsinh

(√γ4s

)

2√

γ4s

(γ4s + 1

)32

• For non-regenerative systems with DPSK the av-erage BER is

Pb(E) =1

2Mγ(1).

• For regenerative systems with DPSK over iden-tical and independent faded hops

Pb(E) =1 + 2γ

2 (1 + γ)2.

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Average BER: Numerical Example

0 5 10 15 20 25 3010

−3

10−2

10−1

100

Average SNR per Hop [dB]

Bit

Err

or R

ate

Pb(E

)

Comparison of Bit Error Rates for Regenerative and Non−Regenerative Systems

Non−Regenerative SystemRegenerative Systems

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Average BER with Collaboration

• Consider one direct link and L i.i.d. faded col-laborating paths.

• Using maximal-ratio combining at the receiver,the overall SNR can be written as

γt = γ0 +

L∑

l=1

γl.

• Under these conditions the MGF of the overallcombined SNR γt is given by

Mγt(s) = Mγ0(s)

L∏

l=1

Mγl(s).

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Diversity Gain due to Collaboration

0 5 10 15 20 25 30 35 4010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Effect of Collaborative Diversity on Average BER Performance

Ave

rage

Bit

Err

or R

ate

Average SNR per bit

L=0L=1L=2L=3L=4

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Formulas for the Shannon Capacity

• For non-regenerative systems

C/W = log2(1 + γeq)bps/Hz

– Capacity PDF is given by

pC(c)=2c+1 ln 2(2c − 1)e

−(2c−1)(

1γ1

+ 1γ2

)

γ1γ2

×[(

γ1 + γ2√γ1γ2

)K1

(2(2c − 1)√

γ1γ2

)+2K0

(2(2c − 1)√

γ1γ2

)].

• For regenerative systems

Ceq = min(C1, C2)

– Capacity PDF is given by

pC(c)= ln 2

(1

γ1+

1

γ2

)2ce

−(2c−1)(

1γ1

+ 1γ2

).

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Capacity: Numerical Example

0 5 10 15 20 25 3010

−3

10−2

10−1

100

Average SNR per Hop [dB]

Cap

acity

Out

age

Comparison of Capacity Outage for Regenerative and Non−Regenerative Systems

Non−Regenerative SystemRegenerative Systems

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Extension to Systems with N Hops

• Analog relaying with channel inversion of the pre-vious link

γeq2 =

N∑

n=1

1

γn

−1

– Related to the harmonic mean of the hop’sSNRs.

• Analog relaying with bounded relay gains

γeq1 =

N∏

n=1

(1 +

1

γn

)− 1

−1

.

– Example of a triple-hop system:

1

γeq1=

1

γ1+

1

γ2+

1

γ3+

1

γ1γ2+

1

γ1γ3+

1

γ2γ3.

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Where to Regenerate ?

0 2 4 6 8 10 12 14 16 18 2010

−3

10−2

10−1

100

Effect of the Regeneration Position

Normalized Average SNR per Hop [dB]

Out

age

Pro

babi

lity

Non−regenerative SystemRegeneration after hop 1 or 3Regeneration after hop 2

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Increasing the Number of Hops

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−4

10−3

10−2

10−1

100

Effect of Number of Hops on Outage Probability

Normalized Average SNR per Hop [dB]

Out

age

Pro

babi

lity

N=1 N=10

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Analog versus Digital Relaying

5 10 15 20 25 30 35 4010

−3

10−2

10−1

100

Number of Hops

Out

age

Pro

babi

lity

Comparison Between Regenerative and Non−regenerative Systems

Non−regenerative SystemRegenerative System

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Other Topics of Interest

• Analog relaying with “fixed” relay gain.

• Variable-power and/or variable rate relays.

• Latency and delay associated with multi-hop sys-tems.

• Global optimization versus local optimization.

• Power consumption and fairness issues.