Extension to the general Gaussian CCIC

d^S(↵, )<d(↵, ), in very strong interference and strong cooperation, i.e., ↵ 2 and 1. This is due to the fact that the information for the PRx can no longer be routed through the CTx since p

Ice^j✓c = 0.

• As for the Z-channel, also the capacity region of the S-channel, dif-ferently from that of the symmetric Gaussian CCIC, does not have bounds of the type 2R_p+R_c andR_p+ 2R_c. In other words, when the link between the CTx and the PRx is absent, unilateral cooperation allows for a full utilization of the channel resources, i.e., there are no

‘resource holes’ in the system [56].

4.7 Extension to the general Gaussian CCIC

In this section we seek to extend our gap results to the general Gaussian CCIC, which is more complex to analyze due to the fact that one has to deal with 5 di↵erent channel parameters. Following the naming convention of the non-cooperative IC, we say that the general Gaussian CCIC has strong in-terference if {SpIp, ScIc}, weak interference if{Sp>Ip, Sc>Ic}, and mixed interference otherwise. Moreover, we say that the general Gaussian CCIC has strong cooperation ifC>Spand weak cooperation otherwise. As we shall see in what follows, this section provides a capacity characteriza-tion to within a constant gap for the general Gaussian CCIC when, roughly speaking, the two receivers do not experience weak interference simultane-ously. In particular,

• Case A: When CSp, i.e., weak cooperation regime, we can further upper bound (4.14) as

O^{Case A} : Rplog(1 +Sp) + log(2), (4.42a)

Rclog(1 +Sc), (4.42b)

R_p+R_clog

✓

1+ Sp

1 +Ip

◆

+log (1+Sc+Ip)+2 log(2), (4.42c) R_p+R_clog

✓

1 + Sc

1 +Ic

◆

+log (1 +Sp+Ic)+log(2). (4.42d)

Figure 4.7: Regime identified asCase A, withGAP2 bits.

The bounds in (4.42) are to within 1.5 bits of

I^{Case A}: Rplog(1 +Sp), (4.43a)

Rclog(1 +Sc), (4.43b)

R_p+R_clog⁺

✓1 +Sp

1 +Ip

◆

+ log(1 +Sc+Ip), (4.43c) R_p+R_clog⁺

✓1 +Sc

1 +Ic

◆

+ log(1 +Sp+Ic), (4.43d) which is achievable to within 1 bit for the non-cooperative IC [12]

when ScSp(1 +Ip)(1 +Ic). Thus, for this regime, depicted in yellow in Figure 4.7, we have GAP2.5 bits.

• Case B: When Sp<CIp, we further upper bound (4.14) as

O^{Case B}: (4.44a)

R_plog(1 +C) + log(2), (4.44b)

Rplog (1 +Sp+Ic) + log(2), (4.44c)

Rclog(1 +Sc), (4.44d)

R_p+R_clog (1 +Sc+Ip) + 2 log(2), (4.44e) R_p+R_clog

✓

1 + Sc

1 +Ic

◆

+ log (1 +Sp+Ic) + log(2). (4.44f)

4.7 Extension to the general Gaussian CCIC 167 In this regime, unilateral cooperation helps increasing the rate of the primary user. In the symmetric case, this regime corresponds to part of the blue region in Figure 4.4 with 1< ↵; we therefore consider the generalization of the achievable scheme we used for the symmetric case in this regime to the case of general channel gains. Here the PTx takes advantage of the strong cooperation link and sends its message with the help of the CTx. The sum-rate upper bound in (4.44e) sug-gests that the CRx should decode the PTx’s message in addition to its intended message, that is, the PTx should use a (cooperative) common message only. The sum-rate upper bound in (4.44f), suggests that the PRx should decode the CTx’s message only when Ic > Sc, that is, the CTx should use both a cooperative) common and a (non-cooperative) private message. This is exactly the strategy described in Appendix 4.E.1 with U₁ = T₁ = ; (i.e., no non-cooperative mes-sages for PTx) and the resulting achievable region is given in (4.51).

In (4.51), we further set |b₁|= 1 (common cooperative message only for PTx carried by V₁) and |b₂|² = _1+I¹

c so that the private message of the CTx (carried by T2) is received below the noise level at the PRx in the spirit of [12]. With these choices the achievable rate region in (4.51) is contained into

I^{Case B}: R_plog (1 +C), (4.45a)

Rplog (1 +Sp+Ic) log(2), (4.45b)

Rclog (1 +Sc), (4.45c)

R_p+R_clog(1 +Sc+Ip), (4.45d) R_p+R_clog

✓ 1+ Sc

1+Ic

◆

+log(1+Sp+Ic) log(2). (4.45e)

By comparing the upper bound in (4.44) with the achievable region in (4.45) we conclude that the capacity region is known to within 2 bits for a general Gaussian CCIC where the channel gains satisfy Sp < C  Ip (see blue regime Figure 4.8). Notice that we did not impose any condition on the strength of Ic compared to Sc, i.e., in other words this gap result holds regardless of whether the interference at PRx is strong (Ic Sc) or weak (Ic<Sc).

S

_p

I

_p

S

_c

C

weak

S

c

(1+S

p

) I

_c

GAP  2 bits

GAP  2 bits

Figure 4.8: Blue and green regimes identified as Case B and Case C, respectively withGAP2 bits.

• Case C: When max{Sp,Ip} < C and Sc  Ic, we further upper bound (4.14) as

O^{Case C} : R_plog(1 +C) + log(2), (4.46a)

R_clog(1 +Sc), (4.46b)

R_p+R_clog

✓

1 + C 1 +Ip

◆

+ log (1 +Sc+Ip) + 2 log(2), (4.46c) R_p+R_clog (1 +Sp+Ic) + 2 log(2). (4.46d) In this regime, unilateral cooperation helps increasing both the rate of the primary user and the sum-capacity. In the symmetric case, this regime corresponds to part of the blue region in Figure 4.4 with 1 <

↵ < . Here the PTx takes advantage of the strong cooperation link and sends its message with the help of the CTx. The sum-rate upper bound in (4.46d) suggests that PRx should decode the CTx’s message in addition to its intended message, that is, CTx should use a (non-cooperative) common message only; this is so because the condition Sc  Ic corresponds to strong interference at the PRx. The sum-rate upper bound in (4.46c), suggests that PTx should use both a (cooperative) common and a (cooperative) private message; this is so because here we do not specify which one among Sp and Ip is the

4.7 Extension to the general Gaussian CCIC 169 largest, and therefore the interference at CRx could be either strong or weak. This is exactly the strategy described in Appendix 4.E.2 with T1=T2 =;, i.e., no private non-cooperative messages. The achievable region is given in (4.53). In (4.53), we further set|a₁|=|d₁|=|c₂|= 0 (i.e., no private non-cooperative messages) and|c₁|² = _1+I¹

p and|a₂|² =

1

1+Sc inspired by [12]. Moreover, we do not pre-encodeU₂ against the private (cooperative) message of PTx, i.e., _U = 0. With these choices we have k1 = log⇣ _1+S achievable rate region in (4.53) is contained into

I^{Case C}: Rplog (1 +C), (4.47a)

eq(4.52c) + eq(4.52g)eq(4.52h)()

eq(4.49r) + eq(4.49g)eq(4.49f) + eq(4.49t)() I(Y_c;U₂|Q, V₁)I(Y_p;U₂|Q, V₁)()

By comparing the upper bound in (4.46) with the inner bound in (4.47) it easy to see that they are at most 2 bits away from one another. Thus, when max{Sp,Ip} < C, Sc Ic and if Sc1+Ip+Sp

1+2Ip  Ic, then GAP  2 bits (see green regime Figure 4.8).

Remark 16. The analysis above provides the characterization of the capac-ity to within a constant gap for the general Gaussian CCIC when, roughly speaking, the two receivers do not su↵er weak interference simultaneously.

Although not a trivial task (since there are 5 di↵erent channel gains), char-acterizing the capacity (to within a constant gap) in the regimes left open in this chapter is an interesting future research direction.

4.8 Conclusions and future directions

In this chapter we considered the two-user CCIC where the CTx is con-strained to operate in FD mode. We first derived two novel outer bounds of the type 2R_p+R_c andR_p+ 2R_con the capacity region of the ISD chan-nel with independent noises at the two destinations. We then designed a transmission strategy based on binning and superposition encoding, PDF relaying and jointly decoding and we derived its achievable rate region. We evaluated the outer and lower bounds on the capacity for the Gaussian noise case and we proved that these bounds are a constant number of bits (uni-versally over all channel gains) apart from one another for the symmetric case (i.e., the two direct links and the two interfering links are of the same strength) and for the case where one interfering link is absent (i.e., Z-channel and S-channel). In particular, we showed that, for the symmetric case, the two novel outer bounds of the type 2R_p+R_candR_p+2R_care active in weak interference when the cooperation link is weaker than the direct link. We also considered the general Gaussian CCIC and we proved a constant gap result when, roughly speaking, the two receivers do not experience weak in-terference simultaneously. Finally, we identified the set of parameters where causal cooperation achieves the same gDoF of the non-cooperative IC and of the ideal non-causal CIC.

With respect to the results presented in this chapter, interesting future research directions may include: (i) characterization of the capacity to within a constant gap for the general Gaussian CCIC in the left open regimes and (ii) derivation of tighter outer bounds and design of novel transmission strategies in order to reduce the gap.

Appendix

4.A Proof of the Markov chains in (4.8a) and (4.8b)

We prove the two Markov chains in (4.8a)-(4.8b) by using the FDG [102].

Figure 4.9 proves the Markov chain in (4.8a), while Figure 4.10 the one in (4.8b). The two proofs, without loss of generality, consider the time instant i= 3. According to [102], we proceed through the following steps.

4.A Proof of the Markov chains in (4.8a) and (4.8b) 171

Figure 4.9: Proof of the Markov chain in (4.8a) using the FDG.

Wp

Figure 4.10: Proof of the Markov chain in (4.8b) using the FDG.

1. Draw the directed graph G1, which takes into consideration the de-pendence between the di↵erent random variables involved in the ISD CCIC considered. In particular, we define

Z_fi^? =

Z_fi Z_ci

to consider the fact that the noises at the CTx and at the CRx can be arbitrally correlated and we have

Xpi=f(Wp), Ypi=f(Xpi, Tci), Tci=f(Xci, Zpi), Tfi=f(Xpi, Z_fi^?), X_ci =f(W_c, T_f^{i 1}), Y_ci =f(X_ci, T_pi), T_pi=f(X_pi, Z_fi^?),

where with f we indicate that the left-hand side of the equality is a function of the random variables into the bracket.

2. InG1, highlight all the di↵erent nodes / random variables involved in the two Markov chains in (4.8a)-(4.8b) we aim to prove. In particular, the random variables circled in magenta, given those circled in green, should be proved to be independent of those circled in grey.

3. From the graph G1, consider the subgraph G2 which contains those edges and vertices encountered when moving backwards one or more edges starting from the colored (magenta, green and grey) random variables. The edges of the subgraph G2 are depicted with dashed black lines in Figure 4.9 and Figure 4.10 and the vertices in G2 are all those touched by a dashed black line.

4. From the graphG2, remove all the edges coming out from the random variables in green (those which are supposed tod-separate the random variables colored in magenta and grey). In Figure 4.9 and Figure 4.10, this step is highlighted with red crosses on the edges which are re-moved. We let G3 be the subgraph obtained from G2 by removing all the edges with red crosses.

5. FromG3, remove all the arrows on the edges, and obtain the undirected subgraph G4. In G4 it is easy to see that, by starting from any grey node, it is not possible to reach any magenta node. This concludes the proof of the two Markov chains in (4.8a)-(4.8b).

4.B Proof of the sum-rate outer bound in (4.9f) 173

4.B Proof of the sum-rate outer bound in (4.9f)

By using the two Markov chains in (4.8a)-(4.8b) we can now derive the sum-rate outer bound in (4.9f). This bound was originally derived in [47] for the case of independent noises; here we extend it to the case when only the noises at the di↵erent source-destination pairs are independent, i.e., PYFc,Yc|Xp,Xc

in (4.2) is not a product distribution. By using Fano’s inequality and by providing the same genie side information as in [47], we have

N(R_p+R_c 2✏_N)

I W_p;Y_p^N +I W_c;Y_c^N

I W_p;Y_p^N, T_p^N, T_f^N +I W_c;Y_c^N, T_c^N, T_f^N

=H Y_p^N, T_p^N, T_f^N H Y_c^N, T_c^N, T_f^N|W_c +H Y_c^N, T_c^N, T_f^N H Y_p^N, T_p^N, T_f^N|W_p . We now analyze and bound the two pairs of terms. First pair:

H Y_p^N, T_p^N, T_f^N H Y_c^N, T_c^N, T_f^N|W_c

(a)= X

i2[1:N]

H Y_pi, T_pi, T_fi|Y_p^{i 1}, T_p^{i 1}, T_f^{i 1} H Y_ci, T_ci, T_fi|Y_c^{i 1}, T_c^{i 1}, T_f^{i 1}, W_c, X_cⁱ

(b)= X

i2[1:N]

H Ypi, Tpi, T_fi|Ypi 1, Tpi 1, T_f^{i 1} H Tpi, Tci, T_fi|Tpi 1, Tci 1, T_f^{i 1}, Wc, Xci (c) X

i2[1:N]

H Tpi, T_fi|Tpi 1, T_f^{i 1} H Tpi, T_fi|Tpi 1, T_f^{i 1}, Wc, Tci 1, Xci

| {z }

= 0 because of (4.8b)

+ X

i2[1:N]

H Ypi|Tpi, T_fi H Tci|Tpi, T_fⁱ, Wc, Tci 1, Xci, Xpi (d)= X

i2[1:N]

H Y_pi|T_pi, T_fi H Y_pi|T_pi, T_fi, X_pi, X_ci ,

where: the equality in (a) follows by applying the chain rule of the entropy and since, for the ISD CCIC, the encoding functionXci(Wc, Y_Fc^{i 1}) is equiv-alent to X_ci(W_c, T_f^{i 1}); the equality in (b) is due to the fact that Y_c is a deterministic function of (X_c, T_p), which is invertible givenX_c; the inequal-ity in (c) is due to the conditioning reduces entropy principle; the equalinequal-ity in

(d) follows because of the ISD property of the channel and since the channel is memoryless. Second pair:

H Y_c^N, T_c^N, T_f^N H Y_p^N, T_p^N, T_f^N|W_p

(e)= X

i2[1:N]

H Y_ci, T_ci, T_fi|Y_c^{i 1}, T_c^{i 1}, T_f^{i 1} H Y_pi, T_pi, T_fi|Y_p^{i 1}, T_p^{i 1}, T_f^{i 1}, W_p, X_pⁱ

(f)= X

i2[1:N]

H Y_ci, T_ci, T_fi|Y_c^{i 1}, T_c^{i 1}, T_f^{i 1} H T_ci, T_pi, T_fi|T_c^{i 1}, T_p^{i 1}, T_f^{i 1}, W_p, X_pⁱ

(g) X

i2[1:N]

H Tci|Tci 1, T_f^{i 1} H Tci|Tci 1, T_f^{i 1}, Wp, Tpi 1, Xpi

| {z }

= 0 because of (4.8a)

+ X

i2[1:N]

H(T_fi|Tci) H T_fi|Tci, T_f^{i 1}, Wp, Tpi 1, Xpi, Xci

+ X

i2[1:N]

H(Y_ci|T_ci, T_fi) H T_pi|T_cⁱ, T_fⁱ, W_p, T_p^{i 1}, X_pⁱ, X_cⁱ

(h)= X

i2[1:N]

H(T_fi|T_ci) H T_fi|T_ci, X_pi, X_ci

+ X

i2[1:N]

H(Y_ci|T_ci, T_fi) H Y_ci|T_ci, T_fi, X_pi, X_ci ,

where: the equality in (e) follows by applying the chain rule of the entropy and since, given W_p, X_p is uniquely determined; the equality in (f) is due to the fact thatY_p is a deterministic function of (X_p, T_c), which is invertible given Xp; the inequality in (g) is due to the conditioning reduces entropy principle; the equality in (h) follows because of the ISD property of the channel and since the channel is memoryless.

By combining all the terms together, by introducing the time sharing random variable uniformly distributed over [1 :N] and independent of ev-erything else, by dividing both sides byN and taking the limit forN ! 1 we get the bound in (4.9f). We finally notice that by dropping the time sharing we do not decrease the bound.

4.C Proof of the outer bound in (4.7) 175

4.C Proof of the outer bound in (4.7)

By Fano’s inequality, by considering that the messages Wp and Wc are in-dependent and by giving side information similarly to [47], we have

N(Rp+ 2Rc 3✏N)

I Wp;YpN + 2I Wc;YcN

I Wp;YpN, TpN, TfN +I Wc;YcN +I Wc;YcN, TcN, TfN|Wp

H YcN H YcN, TcN, TfN|Wp, Wc

+H YcN, TcN, TfN|Wp H YpN, TpN, TfN|Wp

+H YpN, TpN, TfN H YcN|Wc .

We now analyze each pair of terms. In particular, we proceed similarly as we did to prove the outer bound 2R_p+R_c in (4.6).

First pair:

H Y_c^N H Y_c^N, T_c^N, T_f^N|W_p, W_c

= X

i2[1:N]

H Y_ci|Y_c^{i 1} H Y_ci, T_ci, T_fi|W_p, W_c, Y_c^{i 1}, T_c^{i 1}, T_f^{i 1}, X_pⁱ, X_cⁱ

 X

i2[1:N]

H(Y_ci) H Y_ci, T_ci, T_fi|W_p, W_c, Y_c^{i 1}, T_c^{i 1}, T_f^{i 1}, X_pⁱ, X_cⁱ

= X

i2[1:N]

H(Y_ci) H Y_ci, T_ci, T_fi|X_pi, X_ci .

Second pair:

H Y_c^N, T_c^N, T_f^N|W_p H Y_p^N, T_p^N, T_f^N|W_p

= X

i2[1:N]

H Y_ci, T_ci, T_fi|Y_c^{i 1}, T_c^{i 1}, T_f^{i 1}, W_p, X_pⁱ H Y_pi, T_pi, T_fi|Y_p^{i 1}, T_p^{i 1}, T_f^{i 1}, W_p, X_pⁱ

= X

i2[1:N]

H Yci, Tci, Tfi|Yci 1, Tci 1, Tfi 1, Wp, Xpi

H Tci, Tpi, Tfi|Tci 1, Tpi 1, Tfi 1, Wp, Xpi

 X

i2[1:N]

H Tci|Tci 1, T_f^{i 1} H Tci|Tci 1, Tpi 1, T_f^{i 1}, Wp, Xpi

| {z }

= 0 because of (4.8a)

+ X

i2[1:N]

H T_fi|Tci, Xpi H T_fi|Tci, Tpi 1, T_f^{i 1}, Wp, Xpi

+ X

i2[1:N]

H Y_ci|T_ci, T_fi, X_pi H T_pi|T_cⁱ, T_p^{i 1}, T_fⁱ, W_p, X_pⁱ, X_cⁱ

= X

i2[1:N]

H T_fi|Tci, Xpi H T_fi|Tci, Xpi

+ X

i2[1:N]

H Y_ci|T_ci, T_fi, X_pi H Y_ci|T_ci, T_fi, X_ci, X_pi

= X

i2[1:N]

H Yci|Tci, T_fi, Xpi H Yci|Tci, T_fi, Xci, Xpi . Third pair: since

H Y_c^N|W_c

= X

i2[1:N]

H Y_ci|Y_c^{i 1}, W_c X

i2[1:N]

H Y_ci|Y_c^{i 1}, W_c, T_f^{i 1}, X_cⁱ

= X

i2[1:N]

H T_pi|T_p^{i 1}, W_c, T_f^{i 1}, X_cⁱ

= X

i2[1:N]

H Tpi|Tpi 1, T_f^{i 1} , where the last equality follows because of (4.8b), then

H Y_p^N, T_p^N, T_f^N H Y_c^N|W_c

 X

i2[1:N]

H Y_pi, T_pi, T_fi|Y_p^{i 1}, T_p^{i 1}, T_f^{i 1} H T_pi|T_p^{i 1}, T_f^{i 1}

 X

i2[1:N]

H Ypi, T_fi|Tpi .

By combining everything together, by introducing the time sharing ran-dom variable uniformly distributed over [1 :N] and independent of every-thing else, by dividing both sides byN and taking the limit for N ! 1 we get the bound in (4.7). We finally notice that by dropping the time sharing we do not decrease the bound.

4.D Evaluation of the outer bounds in (4.9),(4.6) and (4.7)for the Gaussian CCIC177

4.D Evaluation of the outer bounds in (4.9), (4.6) and (4.7) for the Gaussian CCIC

By defining E[X_pX_c^⇤] :=⇢ :|⇢|2[0,1] we obtain: from the cut-set bounds From the bounds in (4.9d)-(4.9e) we get

R_p+R_clog 1 +(Sp+C) 1 |⇢|²

where the inequality in (a) follows by evaluating the first logarithm in|⇢|= 0 and the second logarithm in ⇢ = e ^j✓p and the inequality in (b) follows by

evaluating the first logarithm in|⇢|= 0 and the second logarithm in⇢= e^j✓c. Finally, from the bound in (4.9f) we obtain

Rp+Rclog 1 +Sp+Ic+ 2p

where the inequality in (c) follows by: (i) evaluating the first term of the first logarithm in ⇢ = e^j✓c and the second term of the first logarithm in

|⇢| = 0; (ii) evaluating the first term of the second logarithm in ⇢ =

We now evaluate the new outer bounds in Theorem 8 and we get 2R_p+R_clog⇣

4.D Evaluation of the outer bounds in (4.9),(4.6) and (4.7)for the Gaussian CCIC179

where the inequality in (d) follows by (i) evaluating the first logarithm in

⇢ = e^j✓c, (ii) evaluating the second logarithm in |⇢| = 0, (iii) evaluating the first term of the third logarithm in ⇢ = e ^j✓p and the second term of the third logarithm in|⇢|= 0 and (iv) since log

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