We here note that for the outer (lower) bound to hold, we will make the com-mon assumption that the current channel state must be independent of the previous channel-estimates and estimation errors, conditioned on the current estimate (there is no need for the channel to be i.i.d. in time). We will also make the common assumption that the channel is drawn from a continuous er-godic distribution such that all the channel matrices and all their sub-matrices are full rank almost surely.
To lower bound T, we rst consider the easier problem where we want to serve s≤K dierent les to s users, each with access to all caches. We also consider that we repeat this (easier) last experimentbNsctimes, thus spanning a total duration of TbNsc (and up to bNscsles delivered). At this point, we transfer to the equivalent setting of the s-user MISO BC with delayed CSIT and imperfect current CSIT, and a side-information multicasting link to the receivers, of capacity dm (les per time slot). Under the assumption that in this latter setting, decoding happens at the end of communication, and once we set
dmTbN
sc=sM (3.49)
3.7. Appendix - Proof of Lemma 3.1 (Lower bound on T∗) (which guarantees that the side information from the side link, throughout the communication process, matches the maximum amount of information in the caches), we have that
TbN
scd0Σ(dm)≥ bN
scs (3.50)
where d0Σ(dm) is any sum-DoF upper bound on the above s-user MISO BC channel with delayed CSIT and the aforementioned side link. Using the bound
d0Σ(dm) =sα+ s Hs
(1−α+dm) from Lemma 3.2 (see below), and applying (3.49), we get
TbN
sc sα+ s
Hs(1−α+ sM TbNsc)
≥ bN
s cs (3.51)
and thus that
T ≥ 1
(Hsα+ 1−α)(Hs− M s
bNsc) (3.52)
which implies a lower bound on the original s-user problem. Maximization over all s, gives the desired bound on the optimal T∗
T∗ ≥ max
s∈{1,...,min(N,K)}
1
(Hsα+ 1−α)(Hs− M s
bNsc) (3.53) required for the originalK-user problem. This concludes the proof of Lemma 3.1.
3.7.1 Sum-DoF bound for the s-user MISO BC, with delayed CSIT, α-quality current CSIT, and additional side information
We begin with the statement of the lemma, which we prove immediately below.
Lemma 3.2 For the s-user MISO BC, with delayed CSIT, α-quality current CSIT, and an additional parallel side-link of capacity that scales as dmlogP, the sum-DoF is upper bounded as
dΣ(dm)≤sα+ s
Hs(1−α+dm). (3.54) Proof. Our proof traces the proof of [69], adapting for the additionalα-quality current CSIT.
Consider a permutationπ of the set E={1,2,· · ·, s}. For any user k, k∈ E, we provide the received signals yk[n] as well as the message Wk of user kto
userk+ 1, k+ 2,· · · , s. We usey0[n] to denote the output of the side-link and we also dene the following notations
Ω[n]:={h[n]k }sk=1, Ωˆ[n]:={hˆ[n]k }sk=1, U[n]:={Ω[n],Ωˆ[n]}, where (3.55) follows from Fano's inequality, where (3.56) holds due to the fact that the messages are independent, and where the last two steps use the basic chain rule. Note thatW0 = 0.
3.8. Appendix - Bounding the gap to optimal where (3.58) follows from the linearity of the summation, where (3.59) holds since the received signal is independent of the future channel state information, where (3.60) uses the fact that conditioning reduces entropy, and where (3.62) is from the fact that Gaussian distribution maximizes dierential entropy under the covariance constraint and from [23, Lemma 2]. From (3.57), we then have
s where the second inequality is from (3.62), where the last inequality follows from the fact that entropy is non-negative and k+11 − 1k ≤ 0. Dividing by nlogP and lettingP → ∞gives which completes the proof of Lemma 3.2.
3.8 Appendix - Bounding the gap to optimal
Our aim here is to show that
T(γ, α >0) T∗(γ, α >0) <4
and we will do so by showing that the above gap is smaller than the gap we calculated above for α = 0 in Appendix in the Chapter 2, which was again
bounded above by 4. For this, we will use the expression8 T(γ, α >0) = (1−γ)(HK−HΓ)
α(HK−HΓ) + (1−α)(1−γ) (3.66) from Theorem 3.2, and the expression
T∗(γ, α >0)≥ max
s∈{1,...,bMNc}
1
(Hsα+ 1−α)(Hs− M s bNsc) from Lemma 3.1. Hence we have
T T∗ ≤
(1−γ)(HK−HKγ) α(HK−HKγ)+(1−α)(1−γ)
max
s∈{1,...,bMNc}
1
(Hsα+1−α)(Hs− bM sN
sc) (3.67)
≤
(1−γ)(HK−HKγ) α(HK−HKγ)+(1−α)(1−γ)
1
(Hscα+1−α)(Hsc−M sbNc scc)
| {z }
g(sc,γ)
(3.68)
where s = sc ∈ {1, . . . ,bMNc}, but where this sc will be chosen here to be exactly the same as in the case ofα = 0. This will be useful because, for that case of α = 0, we have already proved that the same specic sc guarantees that
HK−HKγ Hsc−bM sNc
scc
<4 (3.69)
for the appropriate ranges ofγ. This will apply towards bounding (3.68).
The proof is broken in two cases, corresponding to γ ∈ [361,K−1K ], and γ ∈[0,361].
Case 1 (α >0, γ∈[361,K−1K ])
As whenα= 0 (cf. [71]), we again set s= 1, which reduces (3.68) to T(α >0, γ)
T∗(α >0, γ) ≤
(1−γ)(HK−HKγ) α(HK−HKγ)+(1−α)(1−γ)
1−γ .
For this case whenαwas zero, and when we chose the sames= 1 we have already proved that T(α=0,γ)1−γ <4.As a result, sinceT(α >0, γ)< T(α= 0, γ), and since 1−γ ≤ T∗, we conclude that T(α>0,γ)T∗ < 4, γ ∈ [361,K−1K ] which completes this part of the proof.
8We note that the here derived upper bound on the gap corresponding to the T in Theorem 3.2, automatically applies as an upper bound to the gap corresponding to theT from Theorem 3.1, because the latterT is smaller than the former.
3.8. Appendix - Bounding the gap to optimal Case 2 (α >0, γ∈[0,361])
Going back to (3.68), we now aim to bound
g(sc, γ),
(1−γ)(HK−HKγ) α(HK−HKγ)+(1−α)(1−γ)
1
(Hscα+1−α)(Hsc−M sbNc
scc) <4. (3.70) We already know from the case of α= 0 (cf. (3.69)) that
HK−HKγ Hsc−bM sNc
scc
<4 (3.71)
holds. Hence we will prove that
g(sc, γ)≤ HK−HKγ Hsc−M sbNc
scc
(3.72)
to guarantee the bound. We note that (3.72) is implied by Hsc ≤ HK−HKγ
1−γ (3.73)
which is implied by
log(sc)≤ log(1/γ)
1−γ −6, 6 =H6−log(6) (3.74) because Hsc ≤ log(sc) +6,∀sc ≥ 6,∀γ ∈ [0,361],∀K. Furthermore (3.74) is implied by
1 2log(1
γ)≤ log(1/γ)
1−γ −6 (3.75)
because γ ∈ [(s 1
c+1)2,s12
c]means that sc ≤q
1
γ. Since 1−γ1 ≥1, then (3.75) is implied by
1 2log(1
γ)≤log(1
γ)−6. (3.76)
It is obvious that (3.76) holds since γ ≤ 361. Towards this, by proving (3.76), we guarantee (3.70) and the desired bound. This completes the proof.
Chapter 4
Feedback-Aided Coded Caching with Small Caches
This work continues to explore coded caching in the symmetricK-user cache-aided MISO BC with imperfect CSIT-type feedback, but for the specic case where the cache size is much smaller than the library size. Building on the recently explored synergy between caching and delayed-CSIT, and building on the tradeo between caching and CSIT quality, the work proposes new schemes that boost the impact of small caches, focusing on the case where the cumulative cache size is smaller than the library size. This small-cache setting places an additional challenge due to the fact that some of the library content must remain entirely uncached, which forces us to dynamically change the caching redundancy to compensate for this. Our proposed scheme is near-optimal, and the work identies the optimal cache-aided degrees-of-freedom (DoF) performance within a factor of 4.
4.1 Introduction
Our aim here is to explore the ideas in the previous chapters, and to further our understanding of the joint eect of coded caching now with small caches and (variable quality) feedback, in removing interference and in eventually improving performance. The connections between these ingredients (caches and feedback) will prove particularly crucial here, in boosting the eect of otherwise insuciently large caches.
4.1.1 K-user feedback-aided symmetric MISO BC with small caches
The model remains the same as in Chapter 3, and it corresponds to the sym-metricK-user wireless MISO BC with aK-antenna transmitter, andK single-antenna receiving users. Our emphasis here will be on the small cache regime where the cumulative cache size is less than the library size (Kγ ≤ 1, i.e., KM ≤ N), and which will force us to account for the fact that not all con-tent can be cached. We will also touch upon the general small-cache setting where the individual cache size is much less than the library size (M N, i.e.,γ 1). As before, we will consider the mixed CSIT model.
Related work on smaller-sized caches can be found in [73] which considered coded caching in the single stream case withK ≥N in the presence of very small caches withKM ≤1, corresponding to the case where pooling all the caches together, can at most match the size of a single le.