Mathematical Analysis

.

ω, `_ij represents distance between nodeiandj.

4.3 Mathematical Analysis

Probability of relay transmission

Probability of relay in active mode can be given as P[I > R] = P [γ_sr> χ] = 1−

Z _χ

0

f_sr(γ)dγ, (4.3.1)

whereχ = 2^2R−1is the threshold. Substituting (4.2.1) in (4.3.1) and solving integral the following expression is obtained

P [I > R] = 1− Γ (m_sr, λ_srm_srχ)

Γ (m_sr) (4.3.2)

Probability of relay being in inactive mode is given by

P [I ≤R] = 1−P[I > R]. (4.3.3) PDF of received SNR based on link condition

Let, random variableΘmodels the received SNR atdvias→r→dlink, which take-care the fading on both side of relay link. When the relay is inactive, conditional PDF of received SNR is given by

f_Θ|I≤R(θ) =δ(θ). (4.3.4)

For the case when relay is active, conditional PDF of received SNR can be written as f_Θ|I>R(θ) = λ^nm_rd ^rdm^nm_rd ^rd

Γ (nm_rd) θ^nmrd−1exp (−λ_rdm_rdθ) (4.3.5) PDF of total received SNR

The PDF of the received SNR can be expressed as [BH06]

f_Θ^srd(θ) =f_Θ|I≤R(θ)P [I ≤R] +f_Θ|I>R(θ)P[I > R]. (4.3.6) The MGF¹off_Θ^srd(θ), through relay link can be written as

M_srd(s) = P [I ≤R] +P[I > R]

µ mrdλrd

s+m_rdλ_rd

¶_nmrd

(4.3.7)

1M GF =R_∞

0 f_Θ^srd(θ) exp (−sθ)dθ

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4.3. MATHEMATICAL ANALYSIS 60 On assuming the channels betweens→r→dands→dare independent, MGF of equivalent link can be written as

M_T(s) = M_srd(s)M_sd(s), (4.3.8)

whereM_sd(s) = {m_sdλ_sd/(s+m_sdλ_sd)}^nmsd is MGF of direct link. For the case when m_sd, m_rd are assumed to be integer then MGF of total received SNR can be simplified as

M_T(s) = P [I ≤R]

µ m_sdλ_sd s+msdλsd

¶_nmsd

+P [I > R]

×

"_nm Xsd

α=1

Aα

(s+m_sdλ_sd)^α +

nmXrd

α=1

Bα

(s+m_rdλ_rd)^α

#

(4.3.9)

here Anmsd−α+1 = ^ψ(α−1)A _(α−1)!^(−msdλsd), ψA(s) = ^(msdλ_(s+m^sd)nmsd_rdλ^(m_rd)^rd_nmrd^λrd)nmrd, Bnmrd−α+1 = ^ψ(α−1)B _(α−1)!^(−mrdλrd), ψ_B(s) = ^(msdλ_(s+m^{sd)nmsd(mrdλrd)nmrd}

sdλsd)^nmsd . PDF of the received SNR atd can be calculated, by taking inverse Laplace transform ofM_T(s), using [GR07, Eq.(2.102)]

f_ΘT(θ) = (m_sdλ_sd)^nmsdP [I ≤R]

Γ (nmsd) θ^nmsd−1exp (−m_sdλ_sdθ) +P [I > R]

"_nm Xsd

α=1

A_αθ^α−1

(α−1)!exp (−msdλsdθ) +

nmXrd

β=1

B_βθ^β−1

(β−1)!exp (−m_rdλ_rdθ)

#

(4.3.10)

For the case when severity of fading parameter is non-integer but identical betweens-d, r-dlink (i.e.

msd=mrd=m), PDF of received SNR can be calculated with the help of (4.3.8), [Pou99, Table(2.3.141)]

f_ΘT (θ) = (mλ_sd)^nmP [I ≤R]

Γ (nm) θ^nm−1exp (−mλ_sdθ) +P [I > R] (m²λsdλrd)^nm√

πθ^nm−0.5 Γ (nm){m(λ_sd −λ_rd)}^nm−0.5

×exp µ

−m(λsd+λrd)

2 θ

¶

Inm−0.5

½m(λsd−λrd)

2 θ

¾

(4.3.11)

4.3. MATHEMATICAL ANALYSIS 61 whereI_ν(·) isν^th order modified Bessel functions of the first kind. Simplification of (4.3.11) can be done with the help of [GR07, Eq.(8.445)]

fΘT (θ) = (mλsd)^nmP [I ≤R]

Γ (nm) θ^nm−1exp (−mλsdθ) +P [I > R](m²λsdλrd)^nm√ π Γ (nm)

× X∞

α=0

{m(λ_sd −λ_rd)}^2αθ^2α+2nm−1exp n

−^m(λsd₂^+λrd)θ o

4^2α+nm−0.5α!Γ (α+nm+ 0.5) (4.3.12)

Outage Probability

Outage probability (Pout), can be evaluated as [SA05, Eq.(1.4)]

P_out = Z _χ

0

f_ΘT(θ)dθ, (4.3.13)

here, χ is the threshold. Expression for Pout can be established with the help of (4.3.10), (4.3.13) and [GR07, Eq.(3.381.1)]

P_out = P [I ≤R]Γ (nm_sd, m_sdλ_sdχ)

Γ (nm_sd) +P[I > R]

× (_nm

Xsd

α=1

A_αΓ (α, m_sdλ_sdχ) (m_sdλ_sd)^α(α−1)! +

nmXrd

β=1

B_βΓ (β, m_rdλ_rdχ) (m_rdλ_rd)^β(β−1)!

)

(4.3.14)

where, Γ (·,·)is upper incomplete gamma function [GR07, Eq.(8.350.1)]. In case of Rayleigh fading channel, (4.3.14) can be simplified as

P_out = P [I ≤R]Γ (n, λ_sdχ)

(n−1)! +P [I > R]

×

" _n X

α=1

A_αΓ (α, λ_sdχ) λ^α_sd(α−1)! +

Xn

β=1

B_βΓ (β, λ_rdχ) λ^β_rd(β−1)!

#

(4.3.15)

For non-integer but identical severity of fading parameter,P_outcan be evaluated from (4.3.12), (4.3.13) and [GR07, Eq.(3.381.1)]

P_out = P [I ≤R]Γ (nm, λ_sdmχ)

Γ (nm) +P [I > R](m²λ_sdλ_rd)^nm√ π Γ (nm)

× X∞

α=0

{m(λ_sd−λ_rd)}^2αΓ (2α+ 2nm, m(λ_sd+λ_rd)χ/2)

2^2α−1α!Γ (α+nm+ 0.5){m(λ_sd+λ_rd)}^2α+2nm (4.3.16)

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4.3. MATHEMATICAL ANALYSIS 62 For Rayleigh fading channel, (4.3.16) can be expressed as

P_out = P [I ≤R]Γ (n, λ_sdχ)

(n−1)! +P [I > R](λ_sdλ_rd)ⁿ√ π (n−1)!

× X∞

α=0

(λ_sd−λ_rd)^2αΓ (2α+ 2n,(λ_sd+λ_rd)χ/2)

2^2α−1α!Γ (α+n+ 0.5) (λsd+λrd)^2α+2n (4.3.17)

For the case whens communicates withddirectly (i.e. SIMO), P_outcan be evaluated with the help of (4.2.2), (4.3.13) and [GR07, Eq.(3.381.1)]

P_out = Γ (nm_sd, m_sdλ_sdχ_{SIM O})

Γ (nm_sd) , (4.3.18)

Here, SIMO operates in single time-slot, so the threshold valueχfor this case is taken asχSIM O. Average Symbol Error Rate

AverageSERfor M-PSK modulated signal is define as [SA05, Eq.(5.1)]

SER= Z _∞

0

Q

³√ cθ

´

f_ΘT (θ)dθ, (4.3.19)

where,c= 2 sin²(π/M). AverageSERfrom (4.3.19), can be evaluated as [EM95, Eq.(A8)]

SER = P [I ≤R]µ(m_sdλ_sd, nm_sd)

Γ (nm_sd) (m_sdλ_sd)^nmsd +P [I > R]

× (_nm

Xsd

α=1

A_αµ(m_sdλ_sd, α) (α−1)! +

nmXrd

β=1

B_βµ(m_rdλ_rd, β) (β−1)!

)

(4.3.20)

whereµ{x, y}= ^Γ(y)_xy

np _c

2x+c

(2x+c)^−yΓ(y+0.5) 2√

πΓ(y+1)(2x)^−y

o

2F₁(1, y+ 0.5;y+ 1;_2x+c^2x ),₂F₁(·,·;·;·)is Gauss hy- pergeometric function [GR07, eq.(9.100)]. In case of Rayleigh fading channel, (4.3.20) can be ex- pressed as

SER = P [I ≤R]

(n−1)! λⁿ_sdµ{λ_sd, n}+P [I > R]

×

"

Xn

α=1

A_αµ{λ_sd, α}

(α−1)! + Xn

β=1

B_βµ{λ_rd, β}

(β−1)!

#

(4.3.21)

4.3. MATHEMATICAL ANALYSIS 63 For non-integer but identical severity of fading parameter,SERcan be evaluated from (4.3.12), (4.3.19) and [GR07, Eq.(3.381.1)]

SER = P [I ≤R](mλ_sd)^nm

Γ (nm) µ(mλ_sd, nm) +P [I > R](m²λ_sdλ_rd)^nm√ π Γ (nm)

× X∞

α=0

{m(λ_sd−λ_rd)}^2αµ(m(λ_sd +λ_rd)/2,2α+ 2nm)

4^2α+nm−0.5α!Γ (α+nm+ 0.5) (4.3.22)

For Rayleigh fading channel, (4.3.22) can be written as SER = P [I ≤R]λⁿ_sdµ{λ_sd, n}

(n−1)! +P [I > R](λ_sdλ_rd)ⁿ√ π (n−1)!

× X∞

α=0

(λsd−λrd)^2αµ{(λsd+λrd)/2,2α+ 2n}

4^2α+n−0.5α!Γ (α+n+ 0.5) (4.3.23)

In case of SIMO, averageSERcan be evaluated from (4.2.2), (4.3.19) and [EM95, Eq.(A8)]

SER= µ(m_sdλ_sd, nm_sd)

Γ (nm_sd) (m_sdλ_sd)^nmsd (4.3.24) Average received SNR

Average received SNR (¯γ), can be evaluated as [SA05, Eq.(1.1)]

¯ γ = 1

2 Z _∞

0

θf_ΘT (θ)dθ. (4.3.25)

In (4.3.25), the factor 1/2 is used, because average received SNR at multi-antenna destination is re- ceived in two time-slots. For integer but non-identical severity of fading parameter,γ¯can be evaluated from (4.3.10), (4.3.25) and [GR07, Eq.(3.326.2)]

¯ γ = 1

2

"

n

λ_sdP [I ≤R] +P [I > R]

(_nm Xsd

α=1

αAα

(m_sdλ_sd)^α+1 +

nmXrd

β=1

βBβ

(m_rdλ_rd)^β+1 )#

(4.3.26) In case of Rayleigh fading channel, (4.3.26) can be simplified as

¯ γ = 1

2

"

n λsd

P [I ≤R] +P [I > R]

( _n X

α=1

αA_α (λsd)^α+1 +

Xn

β=1

βB_β (λ_rd)^β+1

)#

(4.3.27) For non-integer but identical severity of fading parameter, γ¯ can be evaluated from (4.3.12), (4.3.25) and [GR07, Eq.(3.326.2)]

¯ γ = 1

2

· n λsd

P [I ≤R] +P[I > R](m²λ_sdλ_rd)^nm√ π m^(2nm+1)Γ (nm)

× X∞

α=0

Γ (2α+ 2nm+ 1) (λsd −λrd)^2α2^(2−2α) Γ (α+nm+ 0.5) (λsd+λrd)^(2α+2nm+1)α!

#

(4.3.28)

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4.3. MATHEMATICAL ANALYSIS 64 For Rayleigh fading channel, (4.3.28) reduces to

¯ γ = 1

2

· n λsd

P [I ≤R] +P[I > R](λ_sdλ_rd)ⁿ√ π (n−1)!

× X∞

α=0

Γ (2α+ 2n+ 1) (λsd−λrd)^2α2^(2−2α) Γ (α+n+ 0.5) (λ_sd +λ_rd)^(2α+2n+1)α!

#

(4.3.29)

In case of SIMO,γ¯can be evaluated with the help of (4.2.2), (4.3.25) and [GR07, Eq.(3.326.2)]

¯ γ = n

λsd

. (4.3.30)

Average Capacity

Average capacity is define as [SA05, eq.(5.1)]

C = 1 2

Z _∞

0

log₂(1 +γ)f_ΘT (γ)dγ (4.3.31) Average capacity from (4.3.31), can be evaluated with the help of (4.3.10) and [GR07, Eq.(4.337.5)]

C = 1 2

·(m_sdλ_sd)^nmsd

Γ (nm_sd) Ω (msdλsd, nmsd)P [I ≤R] +P [I > R]

× (_nm

Xsd

α=1

A_αΩ (m_sdλ_sd, α) (α−1)! +

nmXrd

β=1

B_βΩ (m_rdλ_rd, β) (β−1)!

)#

(4.3.32)

Here,Ω (α, β) = _α^(β−1)!βlog_e2 β−1P

µ=0 1 (β−µ−1)!

h

(−1)^β−µ−2α^β−µ−1exp (α)Ei(−α)+^β−µ−1P

ν=1

(ν−1)! (−α)^{β−µ−ν−1}

¸ , Ei(·)is exponential integral [GR07, Eq.(27.5.3)]. In case of Rayleigh fading channel, (4.3.32) reduces to

C = 1 2

· λⁿ_sd

(n−1)!Ω (λsd, n)P[I ≤R] +P [I > R]

× (Xn

α=1

A_αΩ (λ_sd, α) (α−1)! +

Xn

β=1

B_βΩ (λ_rd, β) (β−1)!

)#

(4.3.33)

For integer and identical severity of fading parameter, average capacity can be evaluated with the help of (4.3.12) and [GR07, Eq.(4.337.5)]

C = 1 2

·

P[I ≤R](mλ_sd)^nm

Γ (nm) Ω (mλ_sd, nm) +P [I > R](m²λ_sdλ_rd)^nm√ π Γ (nm)

× X∞

α=0

{m(λsd −λrd)}^2αΩ{m(λsd+λrd)/2,2α+ 2nm}

4^2α+nm−0.5α!Γ (α+nm+ 0.5)

#

(4.3.34)

4.4. NUMERICAL RESULTS AND DISCUSSION 65