Example 2: HD relay network with N = 1 relay equipped

We consider the network in Figure 3.5, which consists of a single-antenna source (Tx), a single-antenna destination (Rx) andN = 1 relay (RN) equipped withm_r= 2 antennas. For readability, we use here a di↵erent convention for the subscripts compared to the rest of the chapter and indicate the input-output relationship as

y_r =

(1 S1)hrs,1

(1 S₂)h_rs,2 x₀+z_r, (3.52a) y_d=⇥

h_ds h_dr,1 h_dr,2⇤ 2 4

x₀ S1x1

S₂x₂ 3

5+z_d, (3.52b)

where: (i)x0andxr= [x1; x2] are the signals transmitted by the source and the relay, respectively; (ii)y_r= [y₁; y₂] andy_dare the signals received at the relay and destination, respectively; (iii) z_r = [z₁; z₂] and z_d are the noises at the relay and destination, respectively; (iv) sr = [S1; S2] is the state of

Tx RN Rx h

_dr,1

h

rs,2

h

_dr,2

h

_rs,1

h

_ds

Figure 3.5: Example of network with N = 1 relay with m_r = 2 antennas, and single-antenna source and destination.

the relay antennas; (v) the inputs are subject to the power constraints E[|x₀|²] = X

s2[0:1]²

sE[|x₀|²|s_r=s] = X

s2[0:1]²

sP_0|s1, (3.53a) E⇥

kx_rk²⇤

= Tr 2 4 X

s2[0:1]² sE⇥

x_rx^H_r |s_r =s⇤3 5

= Tr 2 4 X

s2[0:1]² s

 P_1|s ⇢_sp

P_1|sP_2|s

⇢^⇤_sp

P_1|sP_2|s P_2|s 3

51, (3.53b)

where⇢_s:|⇢_s|2[0,1] is the correlation coefficient among the relay antennas in states2[0 : 1]² andP_k|sis the power allocated on x_k,k2[0 : 2], in state s2[0 : 1]².

In what follows we consider two di↵erent possible switching strategies at the relay: (i) s_r 2 [0 : 1]²: the m_r = 2 antennas at the relay are switched independently of one another, and (ii) sr = S12 :S 2 [0 : 1]: the mr = 2 antennas at the relay are used for the same purpose, either transmit or receive. We now analyze these two cases separately.

1. Case (i): independent use of the relay antennas. For the cut-set upper bound, two cuts must be considered, namely, A = ; (the relay is in the cut of the source) and A = {1} (the relay is in the cut of the

3.6 Network examples 97

destination). In this case the capacity C_{case (i)} is upper bounded as Ccase (i) max

Px0,xr ,sr

min{I(x₀,x_r,s_r;y_d), I(x₀;y_d,y_r|x_r,s_r)}

H(s_r) + max

Px0,xr

min{I(x₀,x_r;y_d|s_r), I(x₀;y_d,y_r|x_r,s_r)}, where the last inequality follows since I(s_r;y_d)  H(s_r)  2 bits.

Note that, in general, Gaussian inputs are not optimal for Gaussian networks with HD relays since useful information can be conveyed to the destination through random switch [18]. However, as seen in Re-mark 10, to within a constant gap a fixed switching policy between receive and transmit states is optimal, in which case a Gaussian input for each state is optimal. Moreover, the optimal choice of the correla-tion coefficients is ⇢₀₀=⇢₀₁=⇢₁₀= 0 and ⇢₁₁ = e^{j\(hHdr,1hdr,2)}. With this we have

I(x₀,x_r;y_d|s_r)I_;^(fix)

:=m_dlog(2) + ₀log 1 +|h_ds|²P_0|00 + ₁log 1 +|h_ds|²P_0|01+|h_dr,2|²P_2|01 + ₂log 1 +|h_ds|²P_0|10+|h_dr,1|²P_1|10 + ₃log

✓

1+|h_ds|²P_0|11+⇣q

|h_dr,1|²P_1|11+q

|h_dr,2|²P_2|11⌘2◆

, (3.54) where the term m_dlog(2) (with m_d being the number of antennas at the destination) accounts for the loss of considering independent inputs atTxand at RN. Similarly, we have

I(x₀;y_d,y_r|x_r,s_r)I_{^(fix)_1}

:= ₀log 1 + (|h_ds|²+|h_rs,1|²+|h_rs,2|²)P_0|00 + ₁log 1 + (|h_ds|²+|h_rs,1|²)P_0|01

+ ₂log 1 + (|h_ds|²+|h_rs,2|²)P_0|10

+ 3log 1 +|h_ds|²P_0|11 . (3.55) Note that to determine the NNC achievable rate it suffices to remove the term I(y_r; ˆy_r|x₀,x_r,s_r, y_d) = m_rlog(1 + 1/ ²) from I_; and the termI(x₀;y_r|yˆ_r, y_d,x_r,s_r)log(1 + ²) fromI_{1}, with ² being the variance of the quantization noise. We let ² = 1 for simplicity. Note also that the expressions for I_;^(fix) andI_{^(fix)_1} should be optimized with

respect to the power allocation across the relay states, which makes the optimization problem non-linear in _s, s 2 [0 : 1]^mr. As pointed out in Remark 11 (see also the assumption in item 3 in Theorem 4), in order to apply Theorem 4 (see also Remark 14) we must further bound the two expressions so that to obtain a new optimization problem with constant powers across the relay states, i.e., we need to obtain a LP in { s}. In Appendix 3.D we showC_{case (i)}GAP+C⁰_{case (i)} where

and where GAP8 bits to account for deterministic switch, indepen-dent inputs at the source and at the relay, constant power allocation across the states and NNC transmission strategy. Now, by applying Theorem 4 (see also Remark 14) C⁰_{case (i)}, which can be straightfor-wardly cast into a LP as in (3.36), has at most N + 1 = 2 active states.

2. Case (ii): same use of the relay antennas. In this case the m_r = 2 antennas at the relay are used for the same purpose so it suffices to set ₁= ₂ = 0 inC⁰_{case (i)} and optimize over ₀ = 1 ₃= 2[0,1]. by equating the two expressions within the max min.

3.6 Network examples 99 We now show through some simple examples that not only C⁰_{case (i)} C⁰_{case (ii)}, i.e., independently switching the antennas at the relay brings achiev-able rate gains compared to using the antennas for the same purpose, but that the di↵erence between the two can be unbounded. In other words, at high SNR C⁰_{case (i)} and C⁰_{case (ii)} have di↵erent pre-logs / multiplexing gains / degrees of freedom.

Example 1: let |h_ds|=|hrs,2|=|h_dr,1|= 0 and |hrs,1|² =|h_dr,2|² = >0 in Figure 3.5. With this choice of the channel parameters we get

C⁰_{case (i)}= max

s

min{ 1log (1 + ) + ₃log (1 + ),

0log (1 + ) + ₁log (1 + )}= log (1 + ),

where the last equality follows since the optimal choice of _s is given by

0 = ₂ = ₃ = 0 and ₁ = 1, i.e., there is 1< N+ 1 = 2 active state. For C⁰_{case (ii)} the optimal is 1/2 and

C⁰_{case (ii)}= log (1 + )

2 .

From the two expressions above not only we haveC⁰_{case (i)} >C⁰_{case (ii)},8 >0, but independently switching the m_r = 2 antennas also provides a pre-log factor that is twice of the one provided by using the antennas for the same purpose. This can be interpreted as follows. By independently switching the m_r= 2 antennas at the relay, the achievable rateC⁰_{case (i)} equals (to within a constant gap) the capacity of a single-antenna relay channel with a FD relay with the source-relay and relay-destination channel gains of strength equal to . On the other hand, by using them_r = 2 antennas for the same purpose, the achievable rate C⁰_{case (ii)} reduces to the capacity of a single-antenna HD relay channel.

Example 2: let|h_ds|= 0 and|h_rs,1|² =|h_rs,2|² =|h_dr,1|² =|h_dr,2|²= >

0 in Figure 3.5. With this choice of the channel parameters we get C⁰_{case (i)} = max

s

min{ 1log (1 + ) + ₂log (1 + ) + ₃log (1 + 4 ),

0log (1 + 2 ) + ₁log (1 + ) + ₂log (1 + )}

(a)= max

⇢

log (1 + ), log (1 + 2 ) log (1 + 4 ) log (1 + 2 ) + log (1 + 4 )

(b)=

( log (1 + ) if 0.752

log(1+2 ) log(1+4 )

log(1+2 )+log(1+4 ) otherwise , (3.56)

0 1 2 3 4 5 6 7 0.5

1 1.5 2 2.5 3

γ

Achievable rate [bits/ch.use]

Case (i) − bounded Case (ii) − bounded Case (i) − WF Case (ii) − WF

Figure 3.6: C⁰_{case (i)},C⁰_{case (ii)},C⁰⁰_{case (i)},C⁰⁰_{case (ii)} versus di↵erent values of . where the equality in (a) follows since among the ten possible (approxi-mately) optimal simple schedules _s (six possible _s with two active states plus four possible _swith one active state), it is easy to see that only the two cases _s = [0,0,1,0] and _s = [ ,0,0,1 ], with = log(1+2 )+log(1+4 )^{log(1+4 )} , have to be considered and the equality in (b) follows from numerical eval-uations. Thus, if 0.752 the (approximately) optimal schedule has 1 < N + 1 = 2 active state (i.e., ₂ only), otherwise it has N + 1 = 2 active states (i.e., ₀ and ₃).

ForC⁰_{case (ii)} we obtain that the optimal = log(1+2 )+log(1+4 )^{log(1+4 )} and C⁰_{case (ii)}= log (1 + 2 ) log (1 + 4 )

log (1 + 2 ) + log (1 + 4 ). (3.57) It hence follows thatC⁰_{case (i)} >C⁰_{case (ii)},8 0.752, as can also be observed from Figure 3.6 (blue dashed line for C⁰_{case (i)} versus red dashed line for C⁰_{case (ii)}). Moreover, in the high-SNR regime, the pre-log factor forC⁰_{case (i)} = log (1 + ) is again twice of the one ofC⁰_{case (ii)}⇡ ¹₂log (1 + ). This example (as also Example 1) highlights the importance of smartly switching the relay antennas in order to fully exploit the available system resources. Figure 3.6 also shows the achievable ratesC⁰⁰_{case (i)} = max _smin{I_;^(fix), I_{^(fix)_1} }(solid blue

3.6 Network examples 101 line) andC⁰⁰_{case (ii)}(solid red line) obtained by optimizing the powers inI_;^(fix) in (3.54) andI_{^(fix)_1} in (3.55) across the di↵erent states by Water Filling (WF), as described in Appendix 3.E. In particular, under the channel conditions considered in this example, from Appendix 3.E we get that the optimal power allocation can be found by solving

C⁰⁰_{case (i)}= max

which is represented by the blue solid line in Figure 3.6. For case (ii) it suffices to set = 0 inC⁰⁰_{case (i)}; with this we obtain

C⁰⁰_{case (ii)}= 1

2log (1 + 4 ), (3.59)

which is represented by the red solid line in Figure 3.6.

From Figure 3.6 we observe that the highest rates are achieved by opti-mizing the powers across the di↵erent states (solid lines versus dashed lines).

However, as also highlighted in Remark 11 (see also the assumption in item 3 in Theorem 4), with optimal power allocation there are no guarantees that the (approximately) optimal schedule is simple. This is exactly what we observe in this example for which the optimal 2 [0,1] that maximizes C⁰⁰_{case (i)} in (3.58) is neither zero nor one, i.e., the schedule has 3> N+ 1 = 2 active states. From Figure 3.6 we also notice that the di↵erence between the solid lines (obtained by optimizing the powers across the states) and the dashed lines (obtained with a constant / fixed power allocation) is at most 0.1977 bits for case (i) (blue lines) and 0.2636 bits for case (ii) (red lines).

These di↵erences are far smaller than the 3 bits computed analytically in Appendix 3.D, showing that the theoretical gap of 3 bits is very conservative, at least for this choice of the channel parameters.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.8

2 2.2 2.4 2.6 2.8 3 3.2

d

Achievable rate [bits/s/Hz]

Average C’_case(i) Average C’_case(ii)

Figure 3.7: Eh

C⁰_{case (i)}i

(solid curve) andEh

C⁰_{case (ii)}i

(dashed curve) versus di↵erent values ofd2[0,1].

Example 3: we consider the case of Rayleigh fading, whereh_ds ⇠N 0, _ds² , h_rs,i ⇠ N 0, ²_rs and h_dr,i ⇠ N 0, _dr² with i2 [1 : 2] in Figure 3.5 are assumed to be constant over the whole slot (block-fading model) and we let _ds² = Eh

|h_ds|²i

= ₁^c↵, ²_rs = Eh

|h_rs,i|²i

= _d^c↵ and _dr² = Eh

|h_dr,i|²i

=

c

(1 d)^↵, wherecis a constant,d2[0,1] is the distance between the source and the relay and (1 d) is the distance between the relay and the destination, and↵ 2 is the path loss exponent.

Figure 3.7 shows the averageC⁰_{case (i)}(solid curve) and the averageC⁰_{case (ii)} (dashed curve) versus d 2 [0,1], with fixed ↵ = 3 and c = 1. The av-erage was taken over 5·10⁴ di↵erent realizations of the channel gains for each value of d2 [0,1]. From Figure 3.7 we observe again that in general Eh

C⁰_{case (i)}i

>Eh

C⁰_{case (ii)}i

, with a maximum di↵erence of around 0.6 bits at d= 0.5. Note, in fact, that for d= 0.5 we have _ds² = 1 and ²_rs= _dr² = 8.

Under these channel conditions, by independently switching the m_r = 2 antennas at the relay we (approximately) achieve the FD performance, i.e., Eh

C⁰_{case (i)}i

⇡log _rs² = 3 bits/s/Hz, while by using the m_r = 2 antennas for the same purpose the rate performance reduces to the capacity of a

single-3.7 Applications of Theorem 6 103

Dans le document Cooperation strategies for next generation cellular systems (Page 116-124)