Channel Modeling

the receiving antenna elements in the relay experience identical but exponentially correlated fading.

Analytical formulations have been carried out with integer values of m which represents severity of fading. Performance of such system, has been evaluated including the effects of relay placement, num- ber of antennas installed on relay node, correlation among antenna elements, different transmission rate and various fading conditions. Parts of this work have been reported and published in open literature.

In [KB09e], closed form expressions for outage probability and average error rate have been derived, when multi-antenna cooperative relay network operates in correlated Nakagami-mfading channel and both the relay and destination perform MRC combining of signals.

3.2 Channel Modeling

As shown in Fig. 2.1, the regenerative cooperative relay network considered here consists of multi- antenna relay (r), placed between source (s) and destination (d). Relay node is assumed to be equipped with one transmitting antenna andnreceiving antennas¹. Due to half-duplex nature of relay, the source transmits to the relay and destination in first hop (i.e. s → r and s → d). The multi-antenna relay receives signal from the source through multiple links and it either coherently combines them or selects the best link. If received signal is above the threshold of the decoder, the message is decoded and retransmitted to the destination (i.e. r→ d). If this condition is not fulfilled, only direct path is used.

Signals received via direct path and via relay path are either coherently combined or best one of them is selected byd. Here, SNR (γ_ij) gamma distributed [SA05, Eq.(2.21)]

fγij(γ) = λ^m_ij^ijm^m_ij^ij

Γ (m_ij) γ^mij−1exp (−λijmijγ), (3.2.1) where, i∈ {s, r_o}, j ∈ {r_k, d}, λ_ij = (2`<sub>ij</sub>/`_sd)^ψ

.

ω, ψ is path loss exponent,`<sub>ij</sub> represents distance between nodeiandj, which is normalized by reference distance`sd/2,ωis the SNR at reference point, m_ij is severity of fading parameter. In this chapter, independent and identically distributed (i.i.d.) fading between s → r (i.e. ∀λ_srk = ¯λ_sr and ∀m_srk = m_sr) have been considered. So the PDF of received SNR atrwhen it is equipped withnantennas and perform MRC [YA05]

f_γ^mrc_sr (γ) =

¡¯λsrmsr

¢_nmsr

Γ (nm_sr) γ^nmsr−1exp¡

−λ¯_srm_srγ¢

. (3.2.2)

1As defined earlier, symbolrorepresents transmitting antenna ofrandrkrepresents thek^threceiving antenna ofr.

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3.2. CHANNEL MODELING 42 In case when r perform SC, PDF of received SNR at r can be calculated with the help of [KK08b, Eq.(6)]

f_γ^sc_sr(γ) = nλ¯^m_sr^srm^m_sr^sr

Γ (m_sr) γ^msr−1exp¡

−λ¯_srm_srγ¢( Γ¡

msr,λ¯srmsrγ¢ Γ (m_sr)

)_n−1

. (3.2.3)

For the case when antenna elements of relay experience identical but exponentially correlated fading and performs MRC, PDF of received SNR (γ_sr) atrcan be given as [SA05, Eq.(9.215)]

f_γ^corr−mrc_sr (γ) = γ^msrn2/^rρ−1exp

³

−^msr_r^¯λ_ρ^srnγ

´

Γ

³msrn² rρ

´ ³ rρ

msrλ¯srn

´(^msrn2/^rρ), (3.2.4)

hererρ=n+₁₋^2√√ρ_ρ

³

n−^1−ρ₁₋^√n/2_ρ

´

,ρis exponential power correlation coefficient. In first hop, if channel condition betweens→rlink (after coherent addition in case of MRC or selection of best link for SC) is above the particular threshold, then relay is assumed to decode the message received from the source.

In this condition, mutual information (I), transmitted by the source is greater than target data rate R (spectral efficiency) [LW03]:

I =







1 2log₂

·

1 + Pⁿ

k=1

γ_srk

¸

> R f or MRC

1

2log₂[1 + max (γsrk)]> R f or SC.

(3.2.5)

In equation (3.2.5), logarithm is multiplied with 1/2 because such system operates in two time-slots and utilize only1/2part of channel.

3.2.1 Probability of relay in inactive mode

Probability that the relay would not transmit (inactive mode) is given by

P[I ≤R] =





 P

·Pn

k=1

γsrk ≤χ

¸

F or MRC P[max (γ_srk)≤χ] F or SC,

(3.2.6)

where χ = 2^2R− 1 is the threshold. So, (3.2.6) can be evaluated with the help of (3.2.2), (3.2.3) and [GR07, Eq.(3.381.1)]

P [I ≤R] =







Γ(^{nmsr,¯λsrmsrχ})

Γ(nmsr) F or M RC

½

Γ(^{msr,¯λsrmsrχ})

Γ(msr)

¾_n

F or SC (3.2.7)

whereΓ (·,·), Γ (·)are upper incomplete gamma function [GR07, Eq.(8.350.1)] and gamma function [GR07, Eq.(8.310.1)] respectively. In case antenna elements of relay are effected with identical but

3.2. CHANNEL MODELING 43 exponentially correlated fading and perform MRC, the probability that the relay would not transmit (inactive mode), can be evaluated from (3.2.4)

P[I ≤R] = Zχ

0

f_γ^corr−mrc_sr (γ)dγ, (3.2.8)

Evaluation of (3.2.8) can be carried out with the help of [GR07, Eq.(3.381.1)]

P [I ≤R] = Γ¡

m_srn²±

r_ρ, m_srnλ¯_srχ± r_ρ¢

Γ (m_srn²/r_ρ) , (3.2.9)

Probability that relay would transmit (active mode)

P [I > R] = 1−P [I ≤R]. (3.2.10)

3.2.2 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 relay is inactive, then conditional PDF of RVΘis given as

f_Θ|I≤R(θ) =δ(θ). (3.2.11)

For the case when the relay is active, conditional PDF of RVΘcan be written as f_Θ|I>R(θ) = λ^m_ro^rod_d m^m_ro^rod_d

Γ (m_rod) θ^mrod−1exp (−λ_rodm_rodθ). (3.2.12)

3.2.3 PDF of total received SNR

From the theorem on total probability, PDF of received SNR through relay link can be expressed as f_Θ^srd(θ) =f_Θ|I≤R(θ)P [I ≤R] +f_Θ|I>R(θ)P[I > R]. (3.2.13) The MGF²off_Θ^srd(θ), through relay link can be written as

Msrd(s) =P [I ≤R] +P[I > R]

µ λrodmrod

s+λ_rodm_rod

¶_mrod

. (3.2.14)

Similarly, MGF of direct link can be given as M_sd(s) =

µ λsdmsd

s+λ_sdm_sd

¶_msd

. (3.2.15)

2M GF =R_∞

0 f_Θ^srd(θ) exp (−sθ)dθ

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3.2. CHANNEL MODELING 44 MRC at Destination

For the case, when channels betweens→r→dands→dare assumed to be independent, MGF of equivalent link is given by

MT(s) = Msrd(s)Msd(s), (3.2.16)

M_T(s)can be simplified as [GR07, Eq.(2.102)]

M_T(s) = P [I ≤R]

³ msdλsd

s+msdλsd

´_msd

+P [I > R]

×

"

mPsd

α=1 Aα

(s+msdλsd)^α +

mP_rod

β=1

Bβ

(s+m_rodλ_rod)^β

#

, (3.2.17)

whereAmsd−k+1 = ^ψ(k−1)A _(k−1)!^(−msdλsd),ψA(s) = ^(msdλ_(s+m^{sd)msd(mrodλrod)mrod}

rodλ_rod)^mrod ,Bm_rod−k+1 = ^{ψ(k−1)B (−m}_(k−1)!^rodλrod), ψ_B(s) = ^(msdλsd_(s+m^{)msd(mrodλrod)mrod}

sdλsd)^msd . Details for partial fraction of the expression is given in section A.2.

SimilarlyM_T (s)/scan be simplified as [GR07, Eq.(2.102)]

M_T(s)

s = P [I ≤R]

"

1 s +

msd

X

α=1

C_α (s+λ_sdm_sd)^α

#

+P[I > R]

(3.2.18)

×

"

1 s +

msd

X

α=1

D_α

(s+λsdmsd)^α+

mX_rod

β=1

E_β (s+λ_rodm_rod)^β

#

hereC_msd−k+1 = ^ψC(k−1)_(k−1)!^(−λsdmsd)(λ_sdm_sd)^msd,ψ_C(s) = ¹_s,D_msd−k+1 = ^ψ(k−1)D _(k−1)!^(−λsdmsd)

×(λsdmsd)^msd(λrodmrod)^mrod,ψD(s) = _s(s+λ ¹

rodm_rod)^mrod,Em_rod−k+1= ^{ψE(k−1)(−λ}_(k−1)!^rodmrod)

×(λ_sdm_sd)^msd(λ_rodm_rod)^mrod,ψ_E(s) = _s(s+λ ¹

sdmsd)^msd. From (3.2.17), PDF of received SNR atdcan be calculated, by taking inverse Laplace transform ofM_T(s)i.e.

f_Θ⁽¹⁾(θ) = (m_sdλ_sd)^msdP[I ≤R]

Γ (m_sd) θ^msd−1exp (−msdλsdθ) +P [I > R]

"_m Xsd

α=1

Aαθ^α−1

(α−1)! exp (−msdλsdθ) +

mX_rod

β=1

B_βθ^β−1

(β−1)!exp (−mrodλrodθ)

#

(3.2.19)

3.2. CHANNEL MODELING 45 SC at Destination

CDF of received SNR through direct link can be calculated from (3.2.1), [GR07, Eq.(3.381.1)], [GR07, Eq.(8.352.1)]

F_sd(γ) = 1−exp (−m_sdλ_sdγ)

mXsd−1

α=0

(m_sdλ_sdγ)^α

α! . (3.2.20)

Similarly, CDF of received SNR through relay link can be calculated from (3.2.13) F_srd(γ) = P [I ≤R] +P [I > R]

×

"

1−exp (−m_rodλ_rodγ)

mXrd−1

β=0

(mrodλrodγ)^β β!

#

(3.2.21)

So, CDF of total received SNR can be given as

F (γ) = F_sd(γ)F_srd(γ). (3.2.22)

From (3.2.20),(3.2.21) and (3.2.22), CDF of total received SNR can be simplified as F(γ) = P [I ≤R]

"

1−

mXsd−1

α=0

(msdλsd)^α

α! γ^αexp (−m_sdλ_sdγ)

#

+P [I > R]

"

1−

mXsd−1

α=0

(msdλsd)^α

α! γ^αexp (−msdλsdγ)

−

mX_rod−1

β=0

(m_rodλ_rod)^β

β! γ^βexp (−m_rodλ_rodγ) +

mXsd−1

α=0

mX_rod−1

β=0

(m_sdλ_sd)^α(m_rodλ_rod)^β

α!β! (3.2.23)

×γ^α+βexp{−(m_sdλ_sd +m_rodλ_rod)γ}¤

Differentiating (3.2.23) with respect toγ gives PDF of total received SNR atd f_Θ⁽²⁾(γ) = P [I ≤R]

"

(−1)

mXsd−1

α=0

(m_sdλ_sd)^α

α! ξ(m_sdλ_sd, α)] +P [I > R]

×

"

(−1)

mXsd−1

α=0

(m_sdλ_sd)^α

α! ξ(msdλsd, α) + (−1)

mX_rod−1

β=0

(m_rodλ_rod)^β

β! ξ(mrodλrod, β) +

mXsd−1

α=0

mX_rod−1

β=0

(msdλsd)^α(mrodλrod)^β

α!β! ξ{(m_sdλ_sd+m_rodλ_rod), α+β}] (3.2.24)

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3.3. SYSTEM PERFORMANCE 46