Multi-antenna Relay Cooperation in Absence of Direct Link

5.3. MULTI-ANTENNA RELAY COOPERATION IN ABSENCE OF DIRECT LINK 76

5.3.1 System Model

As shown in Fig. 5.4, a system model has been analyzed in which communication between source (s) and destination (d) is supported by two multi-antenna relays (r₁, r₂). Here, r₁, r₂ are assumed to be equipped with m, n receiving antennas, respectively and one transmitting antenna each. Due to half-duplex nature of relays,stransmits tor₁ andr₂ in first time-slot (i.e. s→r₁ ands→r₂). Relay r₁ receives the signal fromsthrough multiple links and performs MRC. If received SNR is above the threshold of the decoder, the message is decoded byr₁ and retransmitted tor₂ andd(i.e. r₁ →r₂ and r₁ →d) in the second time slot. In case the received SNR atr₁ is below the threshold, r₁ will remain silent in second time slot. Through multiple receiving antennas, relayr₂ receives the signal fromsand r₁ in first and second time slots, respectively. After coherent combination, if received SNR is above a threshold, r₂ retransmit the signal to d in third time slot. In this work, fading envelop is assumed to be Rayleigh distributed, so the PDF of received SNR (γ_ij) whenj^thnode perform MRC of signals transmitted fromi^thnode [YA05]

f_γij(γ_ij) = λ^η_ijγ_ij^η−1

(η−1)! exp (−λ_ijγ_ij), (5.3.1) herei∈ {s, r1, r2}, j ∈ {r1, r2, d}, λij = (2`ij/`sd)^ψ

.

ω, ψ is path loss exponent, `ij represents dis- tance between nodei and j, which is normalized by reference distance`_sd/2, ω is SNR at reference point. Value of η will bem, n, 1, if j is assigned tor₁, r₂, d, respectively. Since relays operates in half-duplex mode they cannot transmit and receive the signal simultaneously. Relaysr1 andr2 decode the message if the receive SNR is above the particular threshold. In this condition, mutual information (I) should be grater than target data rateR(spectral efficiency) [ZAL05]:

I = 1

3log₂(1 +SN R)> R. (5.3.2) In equation (5.3.2), we multiply logarithm with1/3 because such system operates in three time-slots and utilize only1/3part of channel.

5.3.2 Mathematical Analysis

In this section, we first outline the steps involved in finding out the PDF of total received SNR based on the link conditions. With the PDF of received SNR known, outage probability expression has been derived next.

5.3. MULTI-ANTENNA RELAY COOPERATION IN ABSENCE OF DIRECT LINK 77 Probability of relayr₁ remaining inactive

Relayr₁ will not transmit (inactive mode), if maximum possible transmitted information between s→r₁ (i.e.I_sr1) is less thanR, so its probability can be modeled as

P£

r^inactive₁ ¤

=P [I_sr1 ≤R] =P [γ_sr1 ≤χ], (5.3.3) here,χ= 2^3R−1is the threshold. So, (5.3.3) can be written as [GR07, Eq.(3.381.1)]

P£

r^inactive₁ ¤

= Γ (m, λ_sr1χ)

(m−1)! . (5.3.4)

Probability that relay will transmit (active mode) can be given as P£

r^active₁ ¤

= 1−P £

r^inactive₁ ¤

. (5.3.5)

Probability of relayr₂ remaining inactive

Ifr₁is inactive then conditional probability of maximum possible transmitted information between s→r₂ (i.e.I_sr2) is less thanRis given as [GR07, Eq.(3.381.1)]

P £

I_sr2 ≤R|r^inactive₁ ¤

= P [γ_sr2 ≤χ]

= Γ (n, λ_sr2χ)

(n−1)! . (5.3.6)

For the case whenr1 is in active mode,r2 will coherently combined the signal received fromsandr1. In such scenario,r₂is in inactive mode, if maximum possible transmitted information betweens→r₂ andr₁ →r₂(i.e. I_sr2+I_r1r2) is less thanR. Conditional probability thatr₂ will not decode the message can be calculated with the help of (5.3.1) and [GR07, Eq.(2.102)]

P £

Isr2 +Ir1r2 ≤R|r₁^active¤

=P [γ_sr2 +γ_r1r2 ≤χ] (5.3.7)

= Xn

α=1

A_αΓ (α, λ_sr2χ) λ^α_sr2(α−1)! +

Xn

α=1

B_αΓ (α, λ_r1r2χ) λ^α_r1r2(α−1)! , hereA_µ−κ+1 = ^ψ

(κ−1) A (^−λsr2)

(κ−1)! (λ_sr2)^µ(λ_r1r2)^µ,B_µ−κ+1 = ^ψ

(κ−1)

B (^−λr1r2)

(κ−1)! (λ_sr2)^µ(λ_r1r2)^µ,ψ_A(s) = ¹ (^s+λr1r2)^µ andψ_B(s) = (^s+λ1sr2)^µ.

From the theorem of total probability, probability thatr₂will not transmit (inactive mode) can be writ- ten as

P£

r^inactive₂ ¤

= P£

Isr2 ≤R|r₁^inactive¤ P £

r^inactive₁ ¤

(5.3.8) +P £

Isr2 +Ir1r2 ≤R|r₁^active¤ P £

r^active₁ ¤ . TH-993_06610202

5.3. MULTI-ANTENNA RELAY COOPERATION IN ABSENCE OF DIRECT LINK 78 Probability that relay will transmit (active mode)

P£

r^active₂ ¤

= 1−P £

r^inactive₂ ¤

. (5.3.9)

PDF of received SNR based on link condition

Let random variableΘmodels the received SNR atd. In caser₁andr₂are inactive, the conditional PDF of RVΘcan be written as

f_Θ|r^inactive

1 ,r^inactive₂ (θ) =δ(θ), (5.3.10)

whereδ(•)is dirac delta function. Whenr₁is active butr₂is inactive, the conditional PDF is given by f_Θ|r^active

1 ,r₂^inactive(θ) = λ_r1dexp (−λ_r1dθ). (5.3.11)

For the case whenr₁ is inactive butr₂is active, the conditional PDF is given by f_Θ|r^inactive

1 ,r₂^active(θ) = λ_r2dexp (−λ_r2dθ). (5.3.12)

When bothr₁andr₂ are active, the conditional PDF is given by f_Θ|r^active_{1 ,r}^active₂ (θ) = λ_r1dλ_r2d

λ_r2d−λ_r1d (5.3.13)

× {exp (−λ_r1dθ)−exp (−λ_r2dθ)}. End-to-end PDF of received SNR

From the theorem on total probability, end-to-end PDF of received SNR can be written as fΘT(θ) = P £

r₁^inactive, r^inactive₂ ¤

f_Θ|r^inactive_{1 ,r}^inactive₂ (θ) +P£

r^active₁ , r^inactive₂ ¤

f_Θ|r^active_{1 ,r2}^inactive(θ) +P£

r^inactive₁ , r^active₂ ¤

f_Θ|r^inactive_{1 ,r2}^active(θ) +P£

r^active₁ , r^active₂ ¤

f_Θ|r1^active_,r2^active(θ). (5.3.14)

In (5.3.14),r₁ andr₂ experience independent fading, so the event of relay participation will be inde- pendent

P[r₁^inactive, r^inactive₂ ] =P [r^inactive₁ ]P[r₂^inactive] P[r₁^active, r^inactive₂ ] =P [r^active₁ ]P[r₂^inactive] P[r₁^inactive, r^active₂ ] =P [r^inactive₁ ]P[r₂^active] P[r₁^active, r^active₂ ] =P [r^active₁ ]P [r^active₂ ].















(5.3.15)

5.3. MULTI-ANTENNA RELAY COOPERATION IN ABSENCE OF DIRECT LINK 79

0 20 40 60 80 100

0 20 40 60 80 100

0.2 0.4 0.6 0.8 1

Relay 2 position Relay 1 position

Outage Probability

Figure 5.5: Outage probability at d for multi- antenna cooperative relay network forn= 2.

0.36976

0.36976

0.49574 0.49574

0.49574

0.49574

0.62171

0.62171 0.62171

0.62171 0.74769

0.74769

0.74769

0.74769 0.87367

0.87367

0.87367

0.87367

0.87367

0.87367

Relay 2 position

Relay 1 position

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

Figure 5.6: Contour plot for outage performance of multi-antenna cooperative relay network for n = 2.

0 20

40 60

80 100

0 20 40 60 80 100

0 0.2 0.4 0.6 0.8 1

Relay 2 position Relay 1 position

Outage Probability

Figure 5.7: Outage probability at d for multi- antenna cooperative relay network forn= 4.

0.18489

0.18489

0.18489 0.34783

0.34783

0.34783

0.34783 0.51077

0.51077

0.51077 0.51077

0.67371 0.67371

0.67371 0.67371

0.67371 0.83665

0.83665

0.83665 0.83665

0.83665

Relay 2 position

Relay 1 position

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

Figure 5.8: Contour plot for outage performance of multi-antenna cooperative relay network for n = 4.

Outage Probability (P_out)

Outage occurs when received SNR falls below a certain specified threshold. Here, for the given thresholdχ, we can calculate outage probability as follows [SA05, Eq.(1.4)]

P_out = Z _χ

0

f_ΘT(θ)dθ. (5.3.16)

TH-993_06610202

5.4. PERFORMANCE OF MULTI-ANTENNA RELAYING EMPLOYING SC 80