4.6.1 Proof of Corollary 4.3
Our aim here is to show that for large K, and when γ 1, the achievable T (both from Theorem 4.1 corresponding to the case of Γ<1, but also from (4.5)), has a gap to optimal that is at most 2, for all α. We rst consider the scenario where α= 0, and note that
T(Γ≥1, α= 0) =HK−HKγ ≤log(K) +2−log(Kγ)
≤log(1 γ) +2
T(Γ<1, α= 0) =HK−Kγ≤log(K) +2
≤log(1 γ) +2
and thus note that in both cases, we have that T ≤log(1
γ) +2,∀Γ≥0 which means that
T
T∗ ≤ log(γ1) +2
max
s∈{1,...,bMNc}
Hs−bsMN sc
≤ log(γ1) +2
Hsc−bM sNc scc
(4.44)
for any sc ∈ {1, . . . ,bMNc}. Now let us choose sc = bq
1
γc, and note that N ≥M s2c sc, which means that bNscNc
sc → 1. Consequently, from (4.44), for both cases, we have
K→∞lim T
T∗ ≤ lim
K→∞
log(1γ) +2 log(sc)−γs2c
= lim
K→∞
2 log(sc) +2
log(sc)−1 = 2 (4.45)
proving the tighter gap to optimal, which is at most 2, for the case ofα= 0.
The additional case for α > 0,Γ < 1 is handled in the extended version in [75], while the case of α > 0,Γ ≥ 1 (but still with γ 1) can be found in [74].
4.6.2 Proof of Lemma 4.1
We need to show that αb,η is increasing in η, and that T0,η is decreasing in η. The rst follows by noting that
αb,η+1−αb,η= (HK−Hη)
(HK−Hη−1)+Γη − (HK−Hη+1) (HK−Hη+1−1)+η+1Γ
= HK−Hη+ η−Γη+1
((HK−Hη−1)+ηΓ)((HK−Hη+1−1)+η+1Γ ) >0 (4.46) which holds becauseη≥Γ.
To see that T0,η decreases in η, after simplifying notation by letting Dη := (K−η) +α(η+K(HK−Hη −1))
Dη+1 = (K−η−1) +α(η+ 1 +K(HK−Hη+1−1))
to denote the denominators ofT0,η and ofT0,η+1 respectively, we see that T0,η−T0,η+1
K−Γ = HK−Hη
Dη −HK−Hη+1 Dη+1
= (K−ηη+1 +Hη−HK)(1−α) DηDη+1
= (K−ηη+1 −(η+11 +η+21 +· · ·+K1))(1−α)
DηDη+1 >0 (4.47)
which holds because η ≤ K and η+11 ≥ η+i1 ,∀i ∈ [1, K −η]∩Z. The above inequality is strict when η >Γ. This completes the proof.
Chapter 5
Coded caching for reducing CSIT-feedback in wireless communications
In the sameK-user symmetric MISO BC setting as before, the work explores the role of caching content at receiving users for the purpose of reducing the need for current feedback. Emphasis is entirely on current CSIT quality, and this chapter does not consider delayed CSIT. In this setting, we show how caching, when combined with a rate-splitting broadcast approach, can not only improve performance, but can also reduce the need for CSIT, in the sense that the identied cache-aided optimal DoF associated to caching and perfect CSIT, can in fact be achieved with reduced-quality CSIT. These CSIT savings can be traced back to an inherent relationship between caching, perfor-mance, and CSIT; caching improves performance by leveraging multi-casting of common information, which automatically reduces the need for CSIT, by virtue of the fact that common information is not a cause of interference. At the same time though, too much multicasting of common information can be detrimental, as it does not utilize existing CSIT. Our caching method builds on the Maddah-Ali and Niesen coded caching scheme, by properly balancing multicast and broadcast opportunities, and by combing caching with rate-splitting communication schemes that are specically designed to operate un-der imperfect-quality CSIT. The observed achievable CSIT savings here, are more pronounced for smaller values of K users and N les.
5.1 Introduction
5.1.1 Cache-aided K-user broadcast channel
We continue to explore the same wireless communication setting where a K-antenna transmitter communicates information to K single-antenna receiv-ing users (see Fig 2.1). The entire information content at the transmitter consists of N distinct les W1, W2, . . . , WN, each of size f bits. Each user k = 1,2, . . . , K has a cache Zk of size M f bits, where M < N. After each transmission, the corresponding received signals at receiverk, can be modeled as
yk=hTkx+zk, k= 1, . . . , K (5.1) wherex∈CK×1 denotes the transmitted vector which satises a power con-straintE(|x|2) ≤P, where hk ∈CK×1 denotes the vector fading coecients, and wherezk represents unit power AWGN noise at user k.
We rst proceed with motivating examples that provide insight on dierent elements of the problem.
5.1.2 Motivating examples Example - (M = 0)
Let us rst consider the no-caching case whereM = 0, which again without loss of generality can be taken to correspond to a delivery phase that is strictly of a broadcast nature. In this broadcast channel, the sum DoF with perfect CSIT, is equal toK, which means that theKrequested les1will take 1 time slot to deliver, corresponding to an optimalT∗(M = 0, α= 1) =T∗(0) = 1. Applying the new result by Davoodi and Jafar [63], which states that the maximum DoF ofKcan only be achieved in the presence ofα= 1, immediately reveals that to achieve the optimalT∗(0) = 1, we need α=αth= 1.
Example - (N =K = 2)
Let us now oer a more involved example, which reveals an interesting achiev-able tradeo betweenT, M and α. Specically let us consider the case where N = K = 2, and let 0 ≤ M ≤ 1. There are two les, which we relabel as W1 =A, W2 =B, each of size f = logP bits.
In this example, for the placement phase, we rst split both lesA and B into three subles, i.e, A= (A1, A2, A3), B = (B1, B2, B3), where the subles Ai, Bi, i = 1,2 are each of size M f2 bits, and where A3 and B3 are each of size (1−M)f bits. We ll up the caches as follows Z1 = (A1, B1) and Z2 =
1Recall that the request considered over the delivery phase, is one where each user requests a dierent le.
5.1. Introduction (A2, B2), so that each user has in their cache, an equal part of clean information for each le.
For the delivery phase and again focusing on the request WF1 =W1 = A, WF2 = W2 = B we see that to complete the task, user 1 needs suble A2 (which is available in the cache of user 2) as well asA3, and user 2 needs subleB1(available at the cache of user 1), as well asB3. As a result,A2⊕B1
(containing M f2 bits) has information that can be useful to both users, whileA3
andB3has private information for user 1 and 2 respectively. The challenge will be to communicate this information, as eciently as possible, over a channel with imperfect feedback, corresponding to some α < 1. Towards this, let us consider a scheme that sends a single transmission of the form
x=wc+ ˆh⊥2a1+ ˆh⊥1a2 (5.2) where x∈C2×1, where w ∈C2×1 is a randomly chosen precoder, and where hˆ⊥k ∈C2×1 is a precoder that is orthogonal to the estimate hˆk for hk. Addi-tionally, the above symbolsc, a1, a2are respectively allocated power as follows2 given by
P(c) =˙ P, P(a1) =˙ P(a2) =˙ Pα and are allocated rate as follows
r(c) = (1−α)f, r(a1) =r(a2)=αf.
In particular, a1 is loaded with αlogP bits from A3, and a2 is loaded with αlogP bits from B3, whilecis loaded with the M f2 bits ofA2⊕B1, as well as with the max{2((1−M)f −T αf),0} bits ofA3, B3 that did not t insidea1
and a2. The precoders, power-allocation and rate-allocation are known to all nodes. As a result the received signals at the two users, take the form
y1=hT1wc
2We here use =˙ to denote exponential equality, i.e., we write f(P) ˙=PB to denote
Plim→∞
logf(P) logP =B.
At this point, user 1 can decode common symbolcby treating all other signals as noise. Consequently, user 1 removes hT1wc from y1 and decodes private symbol a1. From c, user 1 can recover A2 ⊕B1, which combined with Z1 allows for recovery ofA2. Finally fromcanda1, user 1 can recoverA3. Given thatA1 is already available in its cache, user 1 can thus reconstruct A. User 2 similarly obtainsB.
To calculate the achieved T for this example, we note that the total of
M f
2 + (1 −M)f + (1−M)f = (4−3M2 ) logP bits in A2 ⊕B1, A3, B3, was sent at an achievable rate (provided by this specic rate-splitting scheme) of (1−α+α+α)f = (1 +α) logP bits per time slot. Hence the corresponding achievable duration is
T(M, α) = 4−3M
2(1 +α). (5.4)
As we will show later using basic cut-set bound arguments, the optimalT∗(M) associated to perfect CSIT takes the form T∗(M) = 1− MN = 1−M2 . Hence equating
T(M, α) = 4−3M
2(1 +α) =T∗(M) = 1−M 2 and solving forα, gives that any α bigger than
αth= 1− M
2−M, 0≤M ≤1 (5.5)
suces to achieve the optimal T∗(M, α = 1) = 1−M2 . A few simple obser-vations include the fact that, as expected, αth reduces with M (Fig. 5.1), as well as the fact that forM = 0, we have αth = 1, which correctly reects the discussion in the previous example, which reminded us that in the broadcast channel, in the limit of highP, perfect CSIT is necessary for DoF optimality, i.e., perfect CSIT is necessary to transmit one le to each user within a time duration that is asymptotically optimal. On the other hand, we see that hav-ing M ≥ 1, leads to αth = 0 because, as we have seen in [2], when M ≥ 1 (N =K= 2), a simple transmission of a common message, suces to achieve T∗(M) = 1− M2 . Since the transmission is limited to a common symbol, it does not require CSIT.
Example - (N =K = 2, M = 1/2). Modifying coded caching for the BC
Let us now look at the interesting instance of N = K = 2, M = 1/2, which showcases some of the dierences between the multicast case in [2] and the broadcast approach here, and which motivates caching specically for the multi-antenna wireless setting, as compared to caching for the multicast case with a single shared medium with only serial multicast possibilities, that was