Analysis and Optimization of a Frequency-Hopping Ad Hoc ...

Analysis and Optimization of a Frequency-HoppingAd Hoc Network in Rayleigh Fading

S. Talarico 1 M. C. Valenti 1 D. Torrieri 2

1West Virginia UniversityMorgantown, WV

2U.S. Army Research LaboratoryAdelphi, MD

May 30, 2012

Matthew C. Valenti (shortinst)Analysis and Optimization of FHAH Network in Rayleigh Fading May 30, 2012 1 / 31

Outline

1 Frequency-Hopping Ad Hoc Networks

2 Outage Probability

3 Transmission Capacity

4 Modulation Constraints

5 Optimization Results

6 Conclusions


Frequency-Hopping Ad Hoc Networks

Outline






6 Conclusions



Ad Hoc Networks

Reference transmi-er (X0)

Reference receiver

Mobile transmitters are randomly placed in 2-D space.A reference receiver is located at the origin.Xi represents the ith transmitter and its location.

X0 is location of the reference transmitter.M interfering transmitters, X1, ..., XM.|Xi| is distance from ith transmitter to the reference receiver.

Spatial modelsInterferers placed independently and uniformly over the network area.M can be fixed (BPP) or variable (PPP).



Frequency Hopping

To manage interference, frequency hopping (FH) is used:

W B

Each mobile transmits with duty factor d.

Likelihood of a transmission is d ≤ 1.

Each transmitting mobile randomly picks from among L frequencies.

Probability of a collision is p = d/L = 1/L′, where L′ = L/d.



SINR

The performance at the reference receiver is characterized by thesignal-to-interference and noise ratio (SINR), given by:

γ =g0Ω0

Γ−1 +

M∑i=1

IigiΩi

(1)

where:

Γ is the SNR at unit distance.

gi is the power gain due to Rayleigh fading (an exponential variable).

Ii is a Bernoulli variable that indicates a collision.

P [Ii = 1] = p = 1/L′.

Ωi = PiP0||Xi||−α is the normalized receiver power from transmitter i.

Pi is the power of transmitter i, assumed to be constant for all i.α is the path loss.||X0|| = 1 to normalize distance.


Outage Probability

Outline






6 Conclusions


Outage Probability

Outage Probability

An outage occurs when the SINR is below a threshold β.β depends on the choice of modulation and coding.

The outage probability is

ε = P[γ ≤ β

]. (2)

Substituting (1) into (2) and rearranging yields

ε = P[β−1g0Ω0 −

M∑i=1

IigiΩi︸︷︷︸Z

≤ Γ−1].

The outage probability is related to the cdf of Z,

ε = P[Z ≤ Γ−1

]= FZ(Γ−1).


Outage Probability

Outage Probability Derivation

The cdf conditionedon the network geometry, Ω:

εΩ = P[γ ≤ β

∣∣Ω] = FZ(Γ−1∣∣Ω)


Outage Probability



εΩ = P[γ ≤ β

∣∣Ω] = FZ(Γ−1∣∣Ω)

The cdf conditionedon the number of interferers, M :

εM = E [εΩ] =FZM (Γ−1) =

∫fΩ(ω)FZ(Γ−1

∣∣ω)dω

fΩ(ω) =

M∏i=1

fΩi(ωi)

is the pdf of Ω


Outage Probability



εΩ = P[γ ≤ β

∣∣Ω] = FZ(Γ−1∣∣Ω)

The cdf conditionedon the number of interferers, M :

εM = E [εΩ] =FZM (Γ−1) =

∫fΩ(ω)FZ(Γ−1

∣∣ω)dω

fΩ(ω) =

M∏i=1

fΩi(ωi)

is the pdf of Ω

The unconditional cdf:ε = E [εM ] =

FZ(Γ−1) =

∞∑m=0

pM (m)FZm(z)

pM (m)is the pmf of M


Outage Probability

Conditional Outage Probability

The outage probability conditioned on the network geometry:

FZ(z∣∣Ω) = 1− e−βΩ−1

0 zM∏i=1

[(1− pi)βΩ−1

0 + Ω−1i

βΩ−10 + Ω−1

i

](3)

−30 −20 −10 0 10 20 3010

−2

10−1

100

Γ (in dB)

ε Ω

β= −10dB

β= 0 dB

β= 10dB

Example:

M = 50 interferers.

Annular network.

rex = 0.25 inner radius.

rnet = 2 outer radius.

α = 3.

L′ = 200.

Analytical curves aresolid, while • representssimulated values.


Outage Probability

Average Outage Probability for a BPP: Results

In a binomial point process (BPP), a fixed number of interferers areindependently placed according to a uniform distribution.

The cdf with a BPP network is:

FZM (z) =

∫fΩ(ω)FZ(z

∣∣ω)dω (4)

where

fΩi(ω) =2ω

2−αα

α(r2net − r2

ex

) for rαex ≤ ω ≤ rαnet.

Substituting (3) into (4), the cdf of ZM is

FZM (z) = 1− e−βΩ−10 z

[Ψ (rαnet)−Ψ (rαex)

r2net − r2

ex

]M(5)

where

Ψ(x) = x2α

[1− p+ 2p

α+2 ·x

2+αα

βΩ−10

· 2F 1

([1, α+2

α

]; 2α+2

α ,− xβΩ−1

0

)].


Outage Probability

Average Outage Probability for a BPP: Example

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

M

ε M

L’ = 10

L’ = 20

L’ = 50

L’ = 100

L’ = 200

Example:

rex = 0.25.

rnet = 2.

α = 3.

β = 3.7 dB.

Γ = 10 dB.

Figure: Outage probability εM as a function of M for five values of L′ when the

location of the nodes is drawn from a BPP. Analytical curves are solid, while •represents simulated values.


Outage Probability

Average Outage Probability for a PPP: Results

In a Poisson point process (PPP), the number of interferers M is aPoisson variable.

The cdf with a PPP network is:

FZ(z) =

∞∑m=0

pM (m)FZm(z) =

∞∑m=0

(λA)m

m!e−λAFZm(z) (6)

whereλ is the density of the interferers per unit area;A = π(r2

net − r2ex) is the area of the network;

substituting (5) into (6), the average outage probability is:

FZ(z) = 1− e−βΩ−10 ze−πλ·r2net−r2ex−[Ψ(rαnet)−Ψ(rαex)] (7)

When rnet →∞, rex = 0, and p = L′ = 1:

FZ(z) = 1− e−β0ze−2πλα

β2α0 Γ( 2

α)Γ(1− 2α)

The same expression is obtained in literature by Baccelli et al. [4].


Outage Probability

Average Outage Probability for a PPP: Examples

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

λ

ε

L’ = 10

L’ = 50

L’ = 200

α = 3

α = 3.5

α = 4

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

rnet

= 10

rnet

= 2

λ

ε

Infinite Network Finite Network

Parameters:

rex = 0.25;rnet = 2;β = 3.7 dB;Γ = 10 dB.

Parameters:

rex = 0;α = 3;β = 3.7 dB;Γ = 10 dB.L′ = 1;


Transmission Capacity

Outline






6 Conclusions



Transmission Capacity: Definition

The transmission capacity is the spatial spectral efficiency.

If there are λ mobiles per unit area, then the number of successfultransmissions per unit area is

τ = λ(1− ε)

If the outage probability ε is constrained to not exceed ζ, then thetransmission capacity is

τc(ζ) = ε−1(ζ)(1− ζ) (8)

where ε−1(ζ) is the maximum mobile density such that ε ≤ ζ.



Transmission Capacity: Results

For the BPP case, ε−1(ζ) is found be solving ε = FZM (Γ−1) = ζ forλ = M/A and then substituting into (8):

τc(ζ) =(1− ζ)

[log (1− ζ) + βΩ−1

0 Γ−1]

A log(r2net − r2

ex

)−1[Ψ (rαnet)−Ψ (rαex)]

In the PPP case, ε−1(ζ) is found be solving ε = FZ(Γ−1) = ζ for λand then substituting into (8):

τc(ζ) =(1− ζ)

[log (1− ζ)−1 − βΩ−1

0 Γ−1]

πr2net − r2

ex − [Ψ (rαnet)−Ψ (rαex)] .

Asymptotically for a PPP, with rex = 0 and p = 1, as rnet →∞:

τc(ζ) =(1− ζ)

[log (1− ζ)−1 − βΩ−1

0 Γ−1]

πβ2α0

2πα csc

(2πα

)The same expression is obtained in literature by Weber et al. [14].



Transmission Capacity: Examples

10−1

100

10−3

10−2

10−1

ζ

τ c(ζ)

Γ = 10 dB

Γ = 0 dB

Γ = −10 dB

10−3

10−2

10−1

100

10−5

10−4

10−3

10−2

10−1

ζ

τ(ζ)

rnet

= 2

rnet

= 10

rnet

= ∞

Parameters:

rex = 0;rnet = 2;L′ = 1.β = −10 dB;α = 3;

Parameters:

rex = 0;Γ = 10 dB.L′ = 1;β = −10 dB;α = 3;


Modulation Constraints

Outline






6 Conclusions



Accounting for Modulation

Until now, we have picked the SINR threshold β arbitrarily.

β depends on the choice of modulation.

For ideal (Gaussian-input) signaling

C(γ) = log2(1 + γ)

β is the value of γ for which C(γ) = R (the code rate),

β = 2R − 1

For other modulations, the modulation-constrained capacity C(γ) mustbe used, which can be found by measuring the mutual information be-tween channel input and output.

Additionally, the code rate R and the spectral-efficiency of the mod-ulation η can be taken into account to give transmission capacity inunits of bps/Hz/m2.



Noncoherent Binary CPFSK

FH systems often use noncoherent continuous-phase frequency-shiftkeying, whose capacity & bandwidth depends on the modulation index h.

−10 −5 0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

Es/No in dB

Mut

ual I

nfor

mat

ion

h=0.2

h=1

h=0.4

0.6

h=0.8dashed line

(a) channel capacity versus ES/N0

h (modulation index)

Ban

dwid

th B

(Hz/

bps)

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

4

q=2q=4

q=8

q=16q=32

q=64

99% Power Bandwidth

(b) bandwidth versus modulation index

[9] S. Cheng, R. Iyer Sehshadri, M.C. Valenti, and D. Torrieri, “The capacity ofnoncoherent continuous-phase frequency shift keying”, in Proc. Conf. on Info. Sci. andSys. (CISS), (Baltimore, MD), Mar. 2007.



Modulation-Constrained Transmission Capacity

The modulation-constrained transmission capacity is

τ ′ = λ(1− ε)(Rη(h)

L′

)where

R is the rate of the channel code.h is the modulation index.η(h) the modulation’s spectral efficiency (bps/Hz).ε = P [C(γ) ≤ R] = P [γ ≤ C−1(R)].L′ is the effective number of hopping channels.λ is the density of interferers.τ ′ has units of bps/Hz/m2.

For a given λ, Γ, and spatial model, there is a set of (L′, R, h) thatmaximizes τ ′.



Influence of Parameters: BPP

0 10 20 30 40 50 60 70 80 90 1000

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

L’

τ’ opt(L

)

α=4α=3.5α=3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

R

τ’ opt(R

)

α=4

α=3.5

α=3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

h

τ’ opt(h

)

α=4α=3.5α=3 Parameters:

rex = 0.25;rnet = 2;Γ = 10 dB.M = 50.



Influence of Parameters: PPP

0 10 20 30 40 50 60 70 80 90 1000

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

L’

τ’ opt(L

)

λ = 5

λ = 2

λ = 0.5

λ = 0.1

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8

0.0146

0.0148

0.015

0.0152

0.0154

0.0156

0.0158

0.016

0.0162

0.0164

0.0166

R

τ’ opt(R

)

λ = 5

λ = 2

λ = 0.5

λ = 0.1

0.4 0.45 0.5 0.55 0.6 0.650.0125

0.013

0.0135

0.014

0.0145

0.015

0.0155

0.016

0.0165

h

τ’ opt(h

)

λ = 5

λ = 2

λ = 0.5

λ = 0.1

Parameters:

rex = 0.25;rnet = 2;Γ = 10 dB.α = 3.


Optimization Results

Outline






6 Conclusions



Optimization Objectives

For a given network model (rex, rnet, α, λ) and channel SNR Γ, thevalues of (L′, R, h) that maximize the modulation-constrained trans-mission capacity τ ′ are found.

Optimization is through exhaustive search or a gradient-search strategy.

Performed for both BPP and PPP spatial models.



Example of optimization for a BPP

rnet rex α L′ R h τ ′opt2 0.25 3 32 0.62 0.59 0.01590

3.5 30 0.62 0.59 0.016884 28 0.62 0.59 0.01792

0.5 3 30 0.61 0.59 0.016413.5 29 0.61 0.59 0.017524 26 0.59 0.59 0.01871

4 0.25 3 12 0.54 0.59 0.009833.5 10 0.55 0.59 0.011874 8 0.54 0.59 0.01395

0.5 3 11 0.52 0.59 0.010243.5 9 0.52 0.59 0.012524 8 0.55 0.59 0.01484

Table: Results of the Optimization for an annular network area where the interferersare drawn from a BPP. The number of interferers is fixed to M = 50.



Example of optimization for a PPP

rnet rex α L′ R h τ ′opt2 0.25 3 8 0.62 0.59 0.01597

3.5 7 0.62 0.59 0.016974 7 0.62 0.59 0.01801

0.5 3 7 0.61 0.59 0.017313.5 7 0.61 0.59 0.018454 6 0.59 0.59 0.01973

4 0.25 3 12 0.52 0.59 0.009773.5 10 0.54 0.59 0.011804 8 0.54 0.59 0.01387

0.5 3 11 0.52 0.59 0.010303.5 9 0.53 0.59 0.012584 8 0.55 0.59 0.01491

Table: Results of the Optimization for an annular network area where the interferersare drawn from a PPP. The intensity λ per unit area is fixed to λ = 1.


Conclusions

Outline






6 Conclusions


Conclusions

Conclusions

The performance of frequency-hopping ad hoc networks is a functionof many parameters.

Number of hopping channels L.Code rate R.Modulation index h (if CPFSK modulation).

These parameters can be jointly optimized.

Transmission capacity is the objective function of choice.The modulation-constrained TC quantifies the tradeoffs involved.

The approach is general enough to handle a wide variety of conditions

Can be extended to accommodate Nakagami fading and shadowing.Any spatial model, including repulsion models.Adjacent-channel interference due to spectral splatter.Adaptive code rates (R not fixed for all users).

Additional constraints can be imposed on the optimization.

Per-node outage constraint.Fixed or minimum data rates per user.


Conclusions

Thank You


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