Conclusion

In this work we explored the behavior of coded caching in the topological broadcast channel (BC), identifying the optimal cache-aided performance within a multiplicative factor of 8. Our proposed scheme uses a simple form of in-terference enhancement to alleviate the negative eect of having to multicast to both strong and weak links. By showing that the optimal performance can be achieved even in the presence of weaker links, the work reveals a new role of coded caching which is to partially balance the performance between weaker and stronger users, and to a certain degree without any penalty to the performance of the stronger users.

6.5 Appendix

6.5.1 Proving the gap to optimal

To prove the gap to optimal in Theorem 6.1, we rst recall from [40] (which corresponds to the case of τ = 1) that _T^T∗_(τ=1)^(K) ≤ 4. Let us consider the following three cases.

6.5. Appendix of capacity τ. This means that the lower bound in [40] holds, after simple normalization (division) by τ. At the same time, we know that for this case, the achievable performance here is ^T(W_τ ⁾. Given that the normalization of the lower bound, matches the normalization of the achievable performance, then the gap remains, as in [40], equal to _T^T∗ ≤4.

Combining the above three cases, yields the desired T

T^∗ ≤8 which completes the proof.

6.5.2 Proof of Corollary 6.1

From (6.4) we recall that forW < K(1−γ)thenτ_thr= 1−(^K−WKγ+1)

where the rst equation comes from expanding the binomial coecients ^K−W_Kγ+1 and _Kγ+1^K

. Since _K−i^wi is increasing with i, we have 0 ≤ _K−i^wi ≤ _K−Kγ^wKγ . Applying this inequality to the last equation above (cf.(6.27)), gives

(1−w− wγ

1−γ)^gmax ≤

K−W Kγ+1

K Kγ+1

≤(1−w)^gmax

which in turn gives the lower and upper bound ofτ_thr, in the form τ_thrLB = 1−(1−w)^gmax and τ_{thrU B} = 1−(1−w− _1−γ^wγ )^gmax. It is easy to show that the dierence between the upper and lower bound is not larger than γ^Kγ+1, which vanishes asγ decreases.

6.5.3 Proof of Corollary 6.2

Let us recall from (6.25) that whenτ¯_thr ≤τ ≤τ_thr then T(τ) = min{T(K−W) +T(W),τ_thrT(K)

τ } (6.28)

≤T(K−W) +T(W)≤2T(K) (6.29) which, together with the fact that G ≥ 2, implies that such a performance degradation (beyond a factor of 2), requires that τ <τ¯_thr, which in turn says that the achievable T(τ) takes the form T(τ) = ^T(W_τ ⁾. Applying this in the denition in (6.10), yields the presentedτthr,G.

6.5.4 Removing the integer relaxation constraint

To remove the aforementioned integer relaxation, we consider the extension of the centralized MN algorithm in [2], to any value of γ (not just when Kγ is an integer). This has already been addressed in [40] which plots the interme-diate values. For the sake of completeness we proceed to explicitly describe the corresponding performance, achieved here by the memory-sharing scheme described below. The following holds for anyγ and for τ = 1.

Proposition 6.1 In the K-user cache-aided SISO BC, with N ≥K les and cache size such thatKγ∈[t, t+ 1], t= 0,1,· · · , K−1, then

T⁰⁰(K) = (t+ 1)−KγK−t

t+ 1 + (Kγ−t)K−(t+ 1) t+ 2

= K−t

t+ 1 + (Kγ−t)(K+ 1)

(t+ 1)(t+ 2) (6.30)

is achievable and it has a gap from optimal T⁰⁰(K)

T^∗ ≤4 (6.31)

that is less than 4.

6.5. Appendix The above maintains the gap from optimal of4, simply because the inter-polation gives an improved performance over the case whereKγ∈[1,2, . . . , K]

(see also [40]). The expression coincides with the originalT(K)for integer val-ues of Kγ. The purpose of this proposition is to allow for the applicability of Theorem 6.1 without the integer relaxation assumption. WithT⁰⁰(L) in place, Theorem 6.1 can apply, simply now with slightly dierent values for τ¯_thr and τthr, which though are more complicated and which do not oer any additional insight and are thus omitted.

Below we briey describe the scheme.

Proof of Proposition 6.1

LetΓ = ^KM_N ∈[t, t+ 1],for somet= 0,1,· · ·, K−1. Let us start by splitting each leW_ninto two parts, where the rst partWn⁽¹⁾ has size (t+ 1)−Kγ

f and the second part Wn⁽²⁾ has size (Kγ−t)f. Split each cache Z_k into two parts, Z_k,1, Z_k,2 such that ^|Z_|Z^k,1_k,2^|_| = ^(t+1)−Kγ

(Kγ−t) . Focusing on the rst part, apply the original MN algorithm, where now the library is {Wn⁽¹⁾}^N_n=1, the caches are {Z_k,1}^K_k=1, and caching is performed as though Kγ = t, i.e., by splitting each half-le W_n⁽¹⁾ into ^K_t

equally-sized sublesW_n,τ⁽¹⁾, τ ∈Ψ_t (each suble now has size((t+1)−Kγ)f / ^K_t

), and by lling the caches according to Z_k,1 ={Wn,τ⁽¹⁾}n∈[N],τ∈Ψ_t,k∈τ. Then simply create the sequence of _t+1^K (where now each XOR is intended fort+1users), the delivery of which requiresXORs

T⁽¹⁾ = (t+ 1−Kγ)

K t+1

K t

. (6.32)

We then do the same for the second half of the les (second library{Wn⁽²⁾}^Nn=1) except that now we substitutetwitht+ 1, to get a corresponding duration of

T⁽²⁾ = (Kγ−t)

K t+2

K t+1

. (6.33)

Combining the two cases yields the whole duration of the delivery phase to be T =T⁽¹⁾+T⁽²⁾ = K−t

t+ 1 +(Kγ−t)(K+ 1)

(t+ 1)(t+ 2) (6.34) which completes the proof.

Chapter 7

Achieving the DoF Limits with Imperfect-Quality CSIT

In this chapter, we investigate the problem of how to exploit imperfect CSIT without involving caching. This is done here in the setting of the two-user single-input single-output X channel. In this setting, recent works have ex-plored the DoF limits in the presence of perfect CSIT, as well as in the presence of perfect-quality delayed CSIT. Our work shows that the same DoF-optimal performance previously associated to perfect-quality current CSIT can in fact be achieved with current CSIT that is of imperfect quality. The work also shows that the DoF performance previously associated to perfect-quality delayed CSIT, can in fact be achieved in the presence of imperfect-quality delayed CSIT. These follow from the presented sum-DoF lower bound that bridges the gap as a function of the quality of delayed CSIT between the cases of having no feedback and having delayed feedback, and then another bound that bridges the DoF gap as a function of the quality of current CSIT between delayed and perfect current CSIT. The bounds are based on novel precoding schemes that are presented here and which employ imperfect-quality current and/or delayed feedback to align interference in space and in time.

7.1 Introduction

We consider the two-user Gaussian single-input single-output (SISO) X chan-nel (XC), with two single-antenna transmitters and two single-antenna re-ceivers, where each transmitter has an independent message for each of the

Rx1

Rx2

Tx1

Tx1

W₁₁,W₁₂

W₂₁,W₂₂

y

z

Figure 7.1: 2-user SISO X channel.

two receivers. The corresponding channel model takes the form y_t=h⁽¹⁾_t x⁽¹⁾_t +h⁽²⁾_t x⁽²⁾_t +m_t

zt=g_t⁽¹⁾x⁽¹⁾_t +g⁽²⁾_t x⁽²⁾_t +nt (7.1) where at any timet,h⁽ⁱ⁾_t ,g_t⁽ⁱ⁾ denote the scalar fading coecients of the chan-nel from transmitter i to receiver 1 and 2 respectively, where m_t, n_t denote the unit-power AWGN noise at the two receivers, and where x⁽ⁱ⁾_t , i = 1,2 denotes the transmitted signals at transmitteri, satisfying a power constraint E(|x⁽ⁱ⁾_t |²) ≤ P. Naturally each x⁽ⁱ⁾_t may include some private information originating from transmitter i intended for receiver 1, and some private information intended for receiver 2.

In this setting, for a quadruple of achievable rates R_ij, i, j = 1,2 cor-responding to communication from transmitter i to receiver j, we adopt the high-SNR DoF approximation

d_ij = lim

P→∞

Rij

logP, i, j= 1,2 (7.2) to describe the limits of performance over the XC, particularly focusing on the sum DoF measured_Σ:=d₁₁+d₂₁+d₁₂+d₂₂.

In this context, the challenge originates from the fact that each transmitter is both an interferer as well as an intended transmitter to both receivers.

Crucial in addressing this challenge is the role of feedback and specically of CSIT which can allow for separation, at each receiver, of the intended and the interfering signals. In particular, while the optimal sum DoF without CSIT has been shown to bed_Σ= 1 (cf. [25]), the DoF increases tod_Σ= ⁶₅ in the presence of perfect-quality delayed CSIT (see [98] which proved that this performance is optimal over all linear schemes), and the DoF further increases to an optimal sum-DoF ofd_Σ= ⁴₃ (see [99]) in the presence of perfect-quality and instantaneously available CSIT (perfect current CSIT), other related work can be see in [100102].

7.1.1 Feedback quality model

Motivated by practical settings of limited feedback links, we here consider the case where feedback can be of imperfect-quality, and potentially also

de-7.2. DoF performance with imperfect-quality current and delayed CSIT