sd if sdmax{ sr, rd}
sd+max 2[0,1]minn
A( ), ( sr sd)o
otherwise
(2.6) where
A( ) := (1 ✓⇤( )) log 1
1 ✓⇤( ) +✓⇤( ) logL 1
✓⇤( ),
✓⇤( ) := 1 max
⇢1
L, , L:= 2( rd sd),
and is achieved with random switch and correlated non-uniform input bits at the relay. Moreover, a scheme with deterministic switch and independent and identically distributed (i.i.d.) Bernoulli(1/2)bits at the relay is at most 1 bit from the capacity in (2.6).
2.3 The gDoF for the Gaussian HD relay channel
In this section, by adapting known bounds for the general memoryless FD relay channel [87] to the HD case with the methodology introduced by [18], we derive the gDoF of the Gaussian HD relay channel in (2.1).
2.3.1 Cut-set upper bounds
We now prove a number of upper bounds that we shall use for the converse part of Theorem 1. From the cut-set bound we have:
Proposition 1. The capacity C(HD RC) of the Gaussian HD relay channel is upper bounded as
C(HD RC)
minn
I(Xs, Xr, Sr;Yd), I(Xs;Yr, Yd|Xr, Sr)o
(Xs,Xr,Sr)⇠P⇤Xs,Xr,Sr (2.7a)
max minn
H( ) + I1+ (1 )I2, I3+ (1 )I4o
=:r(CS HD) (2.7b)
2 + log (1 +S)
✓
1 + (b1 1)(b2 1) (b1 1) + (b2 1)
◆
, (2.7c)
where:
2.3 The gDoF for the Gaussian HD relay channel 31
• In (2.7a): the distribution P⇤Xs,Xr,Sr is the one that maximizes the cut-set upper bound, i.e.,
P⇤Xs,Xr,Sr := arg max
PXs,Xr,Sr
minn
I(Xs, Xr, Sr;Yd), I(Xs;Yr, Yd|Xr, Sr)o .
• In (2.7b): the parameter :=P[Sr = 0]2[0,1]represents the fraction of time the relay node listens, H( ) is the binary entropy function
H( ) := log( ) (1 ) log(1 ), (2.8) the maximization is over the set
2[0,1], (2.9a)
|↵1|1, (2.9b)
(Pu,0, Pu,1)2R4+: Pu,0+ (1 )Pu,1 1, u2{s, r}, (2.9c) and the mutual information terms I1, . . . , I4 are defined as
I1:= log (1 +S Ps,0), (2.10)
I2:= log⇣
1 +SPs,1+IPr,1+ 2|↵1|p
SPs,1 IPr,1⌘
, (2.11)
I3:= log (1 + (C+S)Ps,0), (2.12)
I4:= log 1 + (1 |↵1|2)S Ps,1 . (2.13)
• In (2.7c): the termsb1 and b2 are defined as b1 :=
log⇣ 1 + (p
I+p S)2⌘
log (1 +S) >1 sinceI >0, (2.14) b2 := log (1 +C+S)
log (1 +S) >1 sinceC >0. (2.15) Proof. The proof can be found in Appendix 2.A.
The upper bound in (2.7a) will be used to prove that PDF with random switch achieves the capacity to within 1 bit, the one in (2.7b) to prove that PDF with deterministic switch also achieves the capacity to within 1 bit and for numerical evaluations (since we do not know the distribution P⇤Xs,Xr,Sr
that maximizes the cut-set upper bound in (2.7a)), and the one in (2.7c) for analytical computations such as the derivation of the gDoF. With the upper bound in Proposition 1 we can show:
Proposition 2. The gDoF of the Gaussian HD relay channel is upper bounded by the Right Hand Side (RHS) of (2.4).
Proof. The proof can be found in Appendix 2.B.
2.3.2 PDF lower bounds
In this section we prove a number of lower bounds that we shall use for the direct part of Theorem 1. From the achievable rate with PDF we have:
Proposition 3. The capacity of the Gaussian HD relay channel is lower bounded as
C(HD RC) minn
I(U;Yr|Xr, Sr) +I(Xs;Yd|Xr, Sr, U), I(Xs, Xr, Sr;Yd)o
(2.16) max minn
I0(PDF)+ I5+(1 )I6, I7+(1 )I8o
=:r(PDF HD) (2.17) log (1 +S)
✓
1 + (c1 1)(c2 1) (c1 1) + (c2 1)
◆
, (2.18)
where:
• In (2.16): we fix the input PU,Xs,Xr,Sr to evaluate the PDF lower bound; in particular we set PXs,Xr,Sr to be the same distribution that maximizes the cut-set upper bound in (2.7a) and we choose either U =Xr or U =XrSr+Xs(1 Sr).
• In (2.17): the parameter :=P[Sr= 0]2[0,1] represents the fraction of time the relay node listens, the maximization is over the set (2.9a)-(2.9c)as for the cut-set upper bound in (2.7b), the mutual information terms I5, . . . , I8 are
I5:=I1 given in (2.10), (2.19)
I6:=I2 given in (2.11), (2.20)
I7:= log (1 + max{C, S}Ps,0)I3 given in (2.12), (2.21)
I8:=I4 given in (2.13), (2.22)
and I0(PDF):=I(Sr;Yd) is computed from the density fYd(t) =
⇡v0
e |t|2/v0+1
⇡v1
e |t|2/v1, t2C, (2.23) with v0 = 2I5 where I5 is given in (2.19), and v1 = 2I6 where I6 is given in (2.20).
2.3 The gDoF for the Gaussian HD relay channel 33
• In (2.18): the termsc1 and c2 are c1 := log (1 +I+S)
log (1 +S) >1 sinceI >0, (2.24) c2 := log (1 + max{C, S})
log (1 +S) >1 sinceC >0. (2.25) Proof. The proof can be found in Appendix 2.C.
The lower bound in (2.16) will be compared to the upper bound in (2.7a) to prove that PDF with random switch achieves capacity to within 1 bit, the one in (2.17) with the one in (2.7b) to prove that PDF with deterministic switch also achieves capacity to within 1 bit and for numerical evaluations, and the one in (2.18) will be used for analytical computations such as the evaluation of the achievable gDoF.
Remark 1. In Appendix 2.C, we found that the approximately optimal schedule for PDF with deterministic switch is
PDF⇤ := (c1 1)
(c1 1) + (c2 1) 2 [0,1],
wherec1 is given in (2.24) andc2 is given in (2.25). The expression for PDF⇤ can be understood as follows. Suppose that min{C, I} S, otherwise the relay is not used in the transmission and setting either PDF⇤ = 0 or PDF⇤ = 1 is approximately optimal. Notice that PDF⇤ is a decreasing function in C and increasing in I. This implies that the stronger C compared to I the lesser the time the relay needs to listen to the channel to (partially) decode the source message. On the other hand, if C < I, more time is needed to learn the message and less time to convey the message to the destination.
With the lower bound in Proposition 3 we can show:
Proposition 4. The gDoF of the Gaussian HD relay channel is lower bounded by the RHS of (2.4).
Proof. The proof can be found in Appendix 2.D.
Propositions 2 and 4 prove that the gDoF of the Gaussian HD relay channel is given by (2.4) and that PDF achieves the gDoF.
Figure 2.3 shows the di↵erence between the gDoF of the Gaussian FD relay channel in (2.5) and that of the Gaussian HD relay channel in (2.4) as a function of sr and rd, where without loss of generality we fixed sd= 1.
This di↵erence is zero when min{ rd, sr} sd = 1, in which case both
0 0.5
1 1.5
2 2.5
0 0.5 1 1.5 2 2.5
0 0.2 0.4 0.6 0.8
βrd βsr
dFD−dHD
Figure 2.3: Di↵erence between the gDoF of the Gaussian FD and of the Gaussian HD relay channels, for sd= 1, as a function of sr and rd. the FD and the HD channels are gDoF-wise equivalent to a point-to-point channel without relay. When min{ rd, sr} > sd = 1, the point-to-point communication channel is outperformed by the relay channel since now using the relay to convey the information is optimal. Moreover, as expected, the di↵erence is always greater than or equal to zero because in the Gaussian FD relay channel the relay can simultaneously listen and transmit; therefore, the Gaussian FD relay channel represents an outer bound for the Gaussian HD relay channel. The largest di↵erence occurs when rd = sr := sd↵ in which case d(FD RC)
sd = max{1,↵},while d(HD RC)
sd = max 1,1+↵2 ,in other words, for↵>1 the rate di↵erence between FD and HD grows unboundedly as SNR increases. This might motivate the use of more expensive FD relays in future wireless networks in this regime.