Multi-Antenna Coded Caching

0.52 MB·

by  Seyed Pooya Shariatpanahi

Multi-Antenna Coded Caching
1 2 3
Seyed Pooya Shariatpanahi , Giuseppe Caire , Babak Hossein Khalaj
1: School of Computer Science, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran.
2: Technical University of Berlin, 10587 Berlin, Germany.
3: Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran.
(emails: [email protected], [email protected], [email protected])
Abstract—Inthispaperweconsiderasingle-celldownlinksce- the transmitterto design max-minfair multicasttransmissions
7 nario where a multiple-antennabase station delivers contents to to receiver groups. The second approach we consider is em-
1 multiplecache-enableduserterminals.Basedonthemulticasting
ployingthetransmitantennas’multiplexinggaintoenlargethe
0 opportunitiesprovidedbytheso-calledCodedCachingtechnique,
2 we investigate three delivery approaches. Our baseline scheme sizeofmulticastgroups,followingideasin[6].Weinvestigate
employs the coded caching technique on top of max-min fair this approach by assuming coded file chunks formed in both
n
multicasting. The second one consists of a joint design of Zero- the signal domain (complex field) and the data domain (finite
a
J Forcing (ZF) and coded caching, where the coded chunks are field). For all the aforementionedschemes we derive delivery
formed in the signal domain (complex field). The third scheme
1 rate at finite SNR.
is similar to the second one with the difference that the coded
1 It should be noticed that a conventional approach using
chunks are formed in the data domain (finite field). We derive
closed-form rate expressions where our results suggest that the caching in the non-cooperative traditional way would simply
]
T latter two schemes surpass the first one in terms of Degrees cacheafractionofthelibraryateachuser,andserveanumber
I of Freedom (DoF). However, at the intermediate SNR regime of users (smaller than the number of transmit antennas) via
s. forming coded chunks in the signal domain results in power spatial multiplexing and single-user codes. In contrast to our
loss, and will deteriorate throughput of the second scheme. The
c proposed schemes, such solution does not benefit from the
[ main message of our paper is that the schemes performing well
intermsofDoFmaynotbedirectlyappropriateforintermediate cooperative caching gain making it non-scalable for large

1 SNR regimes, and modified schemes should be employed. networks.
v While there are other works which have considered coded
9 I. INTRODUCTION caching in wireless scenarios such as [7]–[10] our paper
7
Cachingcontentnearend-usersisoneofthemostpromising addressestheproblematfiniteSNR. SuchfiniteSNRanalysis
9
2 solutionsproposedfornextgenerationwirelessnetworkssuch proposes important design guidelines not revealed in earlier
0 as 5G [1]. The main reason of the effectiveness of this (high SNR) DoF analysis of wireless coded caching papers.
. approach is that a significant portion of mobile traffic is Moreover, although the paper [11] considers a finite SNR
1
0 multimedia-like which makes demands predictable [2]. On setup, in contrast to our paper, their scheme is designed
7 the other hand, memory hardware is much cheaper than for a massive MIMO scenario with more transmit antennas
1 bandwidth, and is abundantly available at mobile devices. than users. In this case, if full Channel State Information
:
v Thus, using a technique which turns memory into bandwidth at the Transmitter (CSIT) is available, there is no need for
i (i.e., Caching) is of high interest. multicasting opportunities provided by coded caching.
X
The surprising finding of [3] shows that by careful caching The rest of the paper is organized as follows. In Section
r
a at end-users the delivery bandwidth burden can be greatly II we describe the model. In Section III we describe the
relaxed by multicasting coded file chunks to these users. The baselineschemeofMax-MinFairMulticasting.InSectionsIV
main idea of this so-called Coded Caching approach is to and V we analyze Multi-AntennaCoded Caching. Section VI
cache non-identical file chunks among different users, which comparesthethreeschemes,andfinally,SectionVIIconcludes
will provide coded multicasting opportunities. This idea has the paper.
been extended to other scenarios such as hierarchical caching In this paper we use the following notations. We use (.)H
[4], D2D networks [5], and multi-server setup [6]. to denote the Hermitian of a complex matrix. Let C and N
Since the coded caching scheme is not originally designed denote the set of complex and natural numbers and . be
||||
for wireless scenarios (such as 5G) this approach should the norm of a complex vector. Also [1 : m] denotes the set
carefully be adapted to specific characteristics of wireless of integer numbers 1,...,m , and represents addition in
{ } ⊕
channels. In order to achieve this goal we consider a multi- the correspondingfinite field. For any vectorv, we definev⊥
antenna transmitter (e.g., a base station) delivering contents such that vHv⊥ =0.
to multiple single-antenna receivers (e.g., user terminals),
i.e., cache-enabled MISO broadcast. Since the coded caching II. MODEL
approachisdesignedtoprovidemulticastingopportunities,the Weconsiderdownlinktransmissionfromamultiple-antenna
baseline schemewe consideris usingthe multipleantennasat basestation(BS)withLantennashavingaccesstoalibraryof
N files W ,...,W ,each ofsize F bits. Thetransmission a common message to all users in subset S. It follows that
1 N
{ }
is to K single antenna User Terminals (UT). The wireless the total transmission time necessary to deliver all multicast
channel from the BS to the UTs is represented by the ma- subcodewords is given by
trix HH CK×L (complex baseband discrete-time channel
∈ (US)F
model). In addition, we consider H to be constant over large T = L .
C(SNR,S,H)
blocks of B ≫1 channel uses in the time-frequencydomain. S⊂X[1:K]
Also, we assume full CSIT is available, and K L.
SincetimeT isnecessaryforeachusertobeabletodecode
≥
Let us represent data by m-bit symbols in the finite field
its own request file of F bits, the system symmetric rate can
F2m. Consider a one-to-one map ψ from F2m to n complex be defined as the “goodput” (useful bits per second) at which
numbersbelongingtoaGaussiancodebook ,i.e.,ψ :F2m eachuser is served.Since each useris able to decodea file of
Cn, which constitute the transmit signal fCrom BS antenn→as
F bits after time T, the per user symmetric rate is given by
passed through n channel uses. We also assume a power
−1
constraintonthecodebookasfollows:Ifxisanm-bitsymbol,
F (U )
then we should have E |ψ(x)|2 ≤ n×SNR. The operator Rsym = T = C(SLNR,SS,H) . (3)
ψ encodes a vector of(cid:2)symbol(cid:3)s element-wise. For ease of S⊂X[1:K]
presentation we denote the encoded version of file W with  
III. CODED CACHING WITH MAX-MIN FAIR
ψ(W)=W˜ throughout the paper.
MULTICASTING
Consider n channel uses. Then, the received signal at user
In this case, the cache content placement works exactly as
k is given by
in Maddah-Ali and Niesen scheme [3]. For the case of t =
y =hHX+z , (1)
k k k MK N, each file is partitioned into K non-overlapping
N ∈ t
where y ,z C1×n denote the received signal sequence subfiles as:
k k ∈ (cid:0) (cid:1)
(over n channel uses) at receiver k and the corresponding
W = W :τ [1:K], τ =t , n [1:N].
n n,τ
additivewhiteGaussiannoisesequence,withi.i.d.components { ⊂ | | } ∀ ∈
(0,1), h is the k-th column of H, and X CL×n is Each user k stores in its cache all subfiles such that k τ.
k
t∼heCsNpace-time block of transmitted coded signal c∈ollectively These are K−1 , such that the cache memory is comple∈tely
t−1
transmitted by the BS over the n channel uses. Also, we used since NF K−1 =MF.
(cid:0)(K) (cid:1)t−1
consider the total transmit power constraint t
In the content delivery phase, let d = (d ,...,d ) be the
(cid:0) (cid:1) 1 K
1 current demand vector. For all subsets S [1 : K] of size
E X 2 SNR. (2) ⊆
nL | i,j| ≤ S =t+1 (denoted in the following as (t+1)-subsets), the
i,j | |
X (cid:2) (cid:3) BS forms the coded message
In the system at hand, the users are equipped with a cache
U = W . (4)
memory of capacity MF bits. Before the network operations S ⊕k∈S dk,S\{k}
(e.g., at off-peak times, or at home, downloading from a In Maddah-Ali and Niesen proposal [3] the BS communi-
home INTERNET access) each user k has stored in its cache cates to all users simultaneously through an error free link of
a message Zk = Zk(W1,...,WN), where Zk() denotes capacityC bitsperunittime.Inthiscase,thecodedmulticast
·
a function of the library files with entropy not larger than codeword consists simply of the concatenation of the coded
MF bits. This operation is referred to as the cache content messages
placement, and it is performed once and at no cost. During
the network operation, users place requests for files in the X = US :S [1:K], S =t+1 .
{ ⊆ | | }
library. We let d [1: N] denote the request of user k and
k ∈ The transmission length of X is
d=(d ,...,d ) be the request vector.
1 K
K(1 M/N)
Upon a set of requests d at the content delivery phase,
(U )F = − F,
S
the BS transmits a coded signal, such that at the end of L 1+MK/N
S
X
transmissionalluserscanreliablydecodetheirrequestedfiles.
resulting in the symmetric rate
Notice that user k decoder, in order to produce the decoded
file W , makes use of its own cache content Z as well as C C(1+MK/N)
dk k Rsym = = (5)
of its own received signal from the wireless channel. SL(US) K(1−M/N)
Incthis work we focus on the worst-case (over the users) (consistently with our definition given before).
P
deliveryrate at whichthe system can serve anyusers request-
Now,itturnsoutthateachcodedmessageU isusefulonly
S
ing any file of the library. Consider the MIMO channel at
to the users in S. Therefore,each message U can be sent by
S
hand, and suppose that the coded multicasting codeword is
multicastingtothegroupofusersS.Inordertodothisweuse
formedbytheconcatenationofsubcodewordsUS,whereeach a beamforming vector w (where w 1) which results
S S
subcodeword is dedicated to a subset S [1 : K] of users, || || ≤
in the following common rate for group S:
⊆
of length (U )F bits each. Let C(SNR,S,H) (in bit/s)
S
denote theLmulticast rate at which the BS can communicate mk∈iSnlog 1+|hHk wS|2SNR , (6)
(cid:0) (cid:1)
Algorithm 1 Multi-Antenna Coded Caching - Complex Field Auxiliary Procedures.
Subfile Combination
1: procedure uT =BFV(S, T,H)
S
12:: procteduMreKD/ENLIVERY(W1,...,WN, d1,...,dK, H) 2: andDhesignuuTTSifsujch tTha,ta:nfdor aullTj ∈=S1, hj ⊥uTS if j 6∈T
3: IN←DEX-INIT 3: end pjro6⊥cedSure ∈ || S||
4: for all S [K], S =t+L do ————————-
⊆ | |
5: for all T ⊆S,|T|=t+1 do 4: procedure INDEX-INIT
uT =BFV(S, T,H)
S 5: for all T [K], T =t+1 do
6: end for 6: for all⊆r T|do|
7: for all ω =1,..., t+Lt−1 do 7: N(r,T∈ r ) 1
8: for all T ⊆S,|(cid:0)T|=t(cid:1)+1 do 8: end for \{ } ←
9: Gω(T)←Lωr∈T W˜dNr,(Tr,\T{\r{}r}) 9: end for
10: end for (cid:16) (cid:17) 10: end procedure
11: Xω(S)← T⊆S,|T|=t+1uTSGω(T) ————————-
12: end for 11: procedure INDEX-UPDATE
P
13: transmit X(S) = X1(S),...,X(t+L−1)(S) 12: for all T S, T =t+1 do
t ⊆ | |
with rate in (11). h i 13: for all r T do
∈
14: INDEX-UPDATE 14: N(r,T r ) N(r,T r )+1
\{ } ← \{ }
15: end for 15: end for
16: end procedure 16: end for
17: end procedure
which can be maximized by choosing
parts and in the cache content placement phase the caches
wS∗ = argmwaxmk∈iSn|hHk w|, (7) are filled as:
s.t. w 2 1.
|| || ≤ Z = A ,B ,C ,Z = A ,B ,C ,Z = A ,B ,C .
1 1 1 1 2 2 2 2 3 3 3 3
{ } { } { }
This optimization problem has been shown to be NP-Hard,
Then, the transmitter will send the following blocks sequen-
however, close-to-optimal solutions can be obtained by a
tially
Semidefinite Relaxation (SDR) approach [12].
This will result in the symmetric rate 1 h⊥ h⊥ h⊥
X = B˜ +A˜ 3 + B˜ +C˜ 1 + A˜ +C˜ 2
−1 1 √6(cid:20)(cid:16) 1 2(cid:17)|h⊥3| (cid:16) 3 2(cid:17)|h⊥1| (cid:16) 3 1(cid:17)|h⊥2|(cid:21)
F 1/ K X = 1 B˜ +A˜ h⊥3 + C˜ B˜ h⊥1 A˜ +C˜ h⊥2
Rs(y1)m = T =S|S⊂X|=[1t:+K1]log(cid:0)1+mink∈S(cid:0)|th(cid:1)Hk wS∗|2SNR(cid:1)(8) wh2ere√X61(cid:20)(cid:16)and1 X22(cid:17)ar|eh⊥3L| ×(cid:16)F3mn2−com3p(cid:17)le|xh⊥1b|lo−ck(cid:16)s,3requi1r(cid:17)in|gh⊥2|(cid:21)
Fn channel uses. The first user will receive hHX + z(1)
where w∗ is the solution to (7). 3m 1 1 1
S and hHX +z(2) sequentially, and with the help of its cache
1 2 1
contents, the first user extracts
IV. MULTI-ANTENNACODED CACHING,
LINEARCOMBINATION IN THECOMPLEX FIELD
hHh⊥
ticTashtiengbaosipcpoidrteuaniotifesthteoltahset sweicrteiolensswcahsantonealdvaipat mthaexm-muiln- √13U |1h0⊥33| h|H1h0⊥h⊥2| (cid:18) AA˜˜23 (cid:19)+ zz(1(112)) !,
2
fair beamforming. However, in the above scheme the spatial  
multiplexinggainoftransmitantennasisnotexploited.Inthis where U is the unitary matrix
section we introducea new scheme which exploitsmultiplex-
ing and caching gains simultaneously. This scheme borrows 1 1 1
U= .
the idea of simultaneous Zero-Forcing and Coded Caching √2 1 1
(cid:18) − (cid:19)
in the delivery phase from [6], which is here adapted to the
wireless scenario. Let us first explain the main idea through This user multiplies its received signal by UH, and then can
an example. decode its desired messages if encoding rate of subfiles A2
and A is less than:
3
Example 1. HereweconsiderL=2transmitantennas,K =
3users,N =3files{W1,W2,W3}={A,B,C},andM =1. log 1+ 1min |hH1 h⊥2|2,|hH1 h⊥3|2 SNR .
In the first phase, each file is divided into three equal-sized 3 h⊥ 2 h⊥ 2
(cid:18) (cid:18) | 2| | 3| (cid:19) (cid:19)
Algorithm 2 Multi-Antenna Coded Caching - Finite Field then modulated to complex numbers. Let us first revisit our
Subfile Combination example:
1: procedure DELIVERY(W1,...,WN, d1,...,dK, H) Example 2. Here the problem setup and the cache content
2: t MK/N placement is the same as Example 1, but the second phase is
←
3: INDEX-INIT different. In the delivery phase, the transmitter will send X=
4: for all S [K], S =t+L do
⊆ | | 1 h⊥ h⊥ h⊥
65:: forualTSl T=B⊆FVS,(S|T,|T=,Ht)+1 do √3(cid:20)ψ(B1⊕A2)|h⊥33| +ψ(B3⊕C2)|h⊥11| +ψ(A3⊕C1)|h⊥22|(cid:21)
7: end for where X is a L Fn complex block, requiring Fn channel
× 3m 3m
8: for all T S, T =t+1 do uses.LetusfocusonthefirstuserwhowillreceivehHX+z
9: G′(T)⊆←⊕|r∈|TWdNr,(Tr,\T{\r{}r}) as follows 1 1
10: end for uT hHh⊥ hHh⊥
11: X(S) S ψ(G′(T)) hHX+z =ψ(B A ) 1 3 +ψ(A C ) 1 2 +z .
12: transm←itPX(TS⊆)S,w|Tit|=htr+a1teqin(tt++(1L16)). 1 1 1⊕ 2 √3|h⊥3| 3⊕ 1 √3|h⊥2| 1
Since User 1 is interested in decoding both ψ(B A ) and
13: INDEX-UPDATE 1⊕ 2
ψ(A C ), and their encoding is done independently, this
14: end for 3 ⊕ 1
can be considered as a MAC channel. Let us define
15: end procedure
1 hHh⊥ 2 hHh⊥ 2
R(1) = log 1+ | 1 3| + | 1 2| SNR ,
Sum 3 h⊥ 2 h⊥ 2
Thus, consideringthe minimum rate ofallthe three users, the (cid:18) (cid:18) | 3| | 2| (cid:19) (cid:19)
1 hHh⊥ 2
common transmission rate to all three users should be less R(1) = log 1+ | 1 3| SNR ,
{1,2} 3 h⊥ 2
than (cid:18) (cid:18) | 3| (cid:19) (cid:19)
1 hHh⊥ 2
log1+ 31i,j∈mi{6=1ij,n2,3} |h|Hih⊥jh|⊥j2|2!SNR. TRh{(e11n),,3}if w=e oploerga(cid:18)te1t+he3M(cid:18)A|C|1hch⊥2a|2n2n|e(cid:19)l aStNanRe(cid:19)qu.al rate po(1in2t),
  user 1 will receive useful information with the effective sum
This results in the symmetric rate:
rate:
R(1) =min R(1) ,2R(1) ,2R(1) . (13)
3 1 hHh⊥ 2 Eff Sum {1,2} {1,3}
Rsym = 2log1+ 3i,j∈mi{6=1ijn,2,3} | |ih⊥j |j2| !SNR. (9) Similarly, effective su(cid:16)m rate for other users w(cid:17)ill be RE(2ff) and
  R(3). Then, the symmetric rate will be:
The scheme in Example 1 can be extended to the general Eff
caseofK,N,L,andM,followingthesameguidelinesin[6], R = 3min R(1),R(2),R(3) . (14)
adapted to the wireless scenario.This generalizationis shown sym 2 Eff Eff Eff
in Algorithm 1. Theorem 1 characterizes the symmetric rate (cid:16) (cid:17)
The generalization of Example 2 to general K, N, L, and
of this algorithm. M is shown in Algorithm 2. Theorem 2 characterizes the
Theorem1. Algorithm1willresultinthefollowingsymmetric symmetric rate of Algorithm 2.
rate
Theorem2. Algorithm2willresultinthefollowingsymmetric
−1 rate
K K−t−1
R(2) = t L−1  R(2)(S) −1 , −1
sym t+L−1 C K K−t−1
(cid:0) (cid:0)(cid:1)(cid:0) t (cid:1) (cid:1)|SS⊆|X=[1t:+KL](cid:16) (cid:17)  (10) Rs(y3)m = (cid:0)t(cid:1)t(cid:0)+LtL−−11 (cid:1)S⊆X[1:K](cid:16)RC(3)(S)(cid:17)−1 ,
(cid:0) (cid:1) |S|=t+L 
where   (15)
where
RC(2)(S)=log1+ St+NRL |TTm|=⊆itnS+1mr∈iTn|hHr uTS|2. (11) RC(3)(S) = mr∈iSnmin log 1+ t+1L |hHr uTS|2SNR ,
Proof. Please refer to the Appendix A.  h (cid:16) t+1 |TTX|=⊆tS+1 (cid:17)
(cid:0) (cid:1) r∈T
LINVE.AMRUCLOTMI-BAINNTAETNIONNAICNOTDHEEDFCINAICTHEIFNIGE,LD t+Lt−1!|TTm|=⊆itnS+1log(cid:16)1+ tt++1L1 |hHr uTS|2SNR(cid:17)i.
In this section we analyze a similar scheme, but here the r∈T
(cid:0) (cid:1)
coded caching messages are developed in the finite field, and (16)
Fig. 2: High SNR Rate Comparison. Here we have K = 3,
Fig.1:LowandIntermediateSNRRateComparison.Herewe
L=2, and t=1.
have K =3, L=2, and t=1.
the same time. We have derived finite SNR rate expressions
Proof. Please refer to the Appendix B.
for the proposed scheme. Although the proposed scheme is
Remark1. Itshouldbenotedthatthemaindifferencebetween shownto surpassa max-minfair multicastbaselineschemein
Algorithm 1 and 2 is that G(T) in Algorithm 1 is a linear terms of DoF performance,careful modificationsare required
combination of subfiles in the complex field, while G′(T) in for proper performance at low SNR values. .
Algorithm2islinearcombinationofsubfilesinthefinitefield.
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APPENDIX A Lemma 1. Consider a (t + L)-subset of users S. Then, if
the common transmission rate to this subset according to
Proof of Theorem 1. For the general K, L, N, and M case, Algorithm 1 is less than
theplacementprocedureisthesame as[3].However,herewe
edqivuiadleseizaechassufobl-lpoawcsk:et further into (cid:0)KL−−t−11(cid:1) mini-packets of RC(S) = log1+ St+NRL mr∈iSnTm⊆inS|hHr uTS|2
r∈T
K t 1  SNR 
Wn,τ =(cid:26)Wnj,τ :j =1,...,(cid:18) L−−−1 (cid:19)(cid:27). = log(cid:18)1+ t+L Tm⊆inSmr∈iTn|hHr uTS|2(cid:19), (20)
The delivery phase is based on the ideas proposed in [6], thenalltheuserscandecodetheirrequiredmini-filessuccess-
adapted to the wireless medium, which is represented in fully.
Algorithm 1.
In order to prove Lemma 1 let us consider user r S in a
Considera(t+L)-subsetofusersnamedS.Thissubsethas ∈
q = t+L number of (t+1)-subsets which we call T ,i = fixed (t+L)-subset S. This user will receive
t+1 i
1,...,q. Our aim is to deliver
(cid:0) (cid:1) hHr Xω(S) = hHr uTSGω(T)
Gω(Ti) = Lωr∈Ti W˜dr,Ti\{r} |TTX|=⊆tS+1
= σdω(cid:16)r,Ti\{r}W˜dr(cid:17),Ti\{r}, (18) (=a) hHr uTS σdωi,T\{i}W˜di,T\{i}
rX∈Ti |TTX|=⊆tS+1 Xi∈T
ttioonth.eUspuobnsestuTcic.eHssefruelLreωrc∈eTpitimoneaonfstahirsanlidnoemarlcinoemarbicnoamtiobninaa-t (=b) hHr uTS σdωi,T\{i}W˜di,T\{i}
T⊆S i∈T
Ti, each user r Ti can, with the help of its cache contents, |TX|=t+1 X
decode W ∈ . r∈T
The madinr,Tidi\e{ar}in Algorithm 1 is that with the help of the (=c) hHr uTSσdωr,T\{r}W˜dr,T\{r}, (21)
sZiemrou-ltFaonrecoinugslyte.cThhneiqumeasinwechcaalnlednegleiveisrGthωat(Ttih)etomaelmlbTeir⊆s oSf |TTX|=⊆tS+1
r∈T
S T are not interested in receiving G (T ). Thus, to avoid
i ω i
in\terference, G (T ) is zero-forced at these L 1 users. For for all ω = 1,..., t+L−1 . In (21), (a) follows from (18),
ω i t
doingthesameideaforallTi ⊆S thetransmitt−ershouldsend f(ob)llofwolslofwrosmfrtohme ftahc(cid:0)etftahcatttt(cid:1)hhaist husHreruTSha=s c0acihfedr,6∈anTd,thaunsdc(acn)
Xω(S)= uTSGω(T), (19) eliminate W˜di,T\{i} for all i6=r.
Let us define
T⊆S,X|T|=t+1
W˜
jwbyh6∈etrTheeawpnredochdeejdsu6⊥igrneuBTSuFTSiVfsjuin∈chATtl,hgaaotnr:ditfh|oumrTSa1|ll,=wj1h∈.eTreShi,istshtajdsek⊥taisiulsdTSoanrieef WrS = W˜dr,T...1\{r} ,
 dr,Tv\{r} 
shown in Auxiliary Procedures.  
The index ω in Lωr∈T ensures that, with appropriate choice and the v×v matrix LSr such that
of coefficients, each user will receive independent linear
LS(ω,i)=hHuTiσω
combinations of its desired mini-files. The index N(.,.) in r r S dr,Ti\{r}
Algorithm 1 ensures that each user will not receive the same for all i,ω = 1,..., t+L−1 , where T ,...T S,r T
sub-file more than once. The Procedures INDEX-INIT and for v = t+L−1 . Then, tthe observation1s of thvis∈user ca∈n bei
INDEX-UPDATE will take care of that concern in Algorithm t (cid:0) (cid:1)
shown as
1, where their details are shown in Auxiliary Procedures. (cid:0) (cid:1)
LSWS,+z ,
It has been shown in [6] that, in a noise-free scenario the r r r
Algorithm 1 will succeed in delivering the requested files to
whereitis easytoverifythatthe coefficientsσ canbechosen
all the users. However, in the current finite SNR wireless
such that we have:
scenario we should limit the transmit rate of all the mini-files
encoders to guarantee successful delivery at all the receivers. hHuT1 ... 0
r S
Thefollowinglemmacharacterizesthemaximumtransmission LS = t+Lt−1 U ... ... ... ,
rthateeirtoreeqaucihresdubmseintio-fifluesserssu,cactewsshfuiclhly.alltheseuserscandecode r vuut(t(cid:0)+1)(cid:0)tt++(cid:1)L1(cid:1)  00 ...... hHr0uTSv 
 (22)
whereUisa t+L−1 t+L−1 unitarymatrix.Alsowehave where
t × t
used the fact(cid:0)that if a(cid:1)ll t(cid:0)he coef(cid:1)ficients satisfy RE(rff) = min log 1+ t+1L |hHr uTS|2SNR ,
σ 2 1 , h (cid:16) t+1 |TTX|=⊆tS+1 (cid:17)
| | ≤ t+L (t+1) (cid:0) (cid:1) r∈T
t+1
theTnhtuhse, tursaenrsmrictapnowdeecrocdoen(cid:0)satlrlaii(cid:1)ntst irneq(u2i)reids sdaattiasfiiefdt.he trans- t+Lt−1!|TTm|=⊆itnS+1log(cid:16)1+ tt++1L1 |hHr uTS|2SNR(cid:17)i,
r∈T
mission rate is less than (cid:0) (cid:1)
this transmission will be successful.
InordertoproveLemma2, letusfocusonuser r whowill
SNR
log1+ min hHuT 2. receive
t+L T⊆S | r S|
 |Tr|=∈tT+1  hHr X(S)+zr = hHr uTS ψ(G′(T))+zr
  t+L
Considering the fact that we need all the users in S to |TTX|=⊆tS+1 t+1
q
successfully decode, they should have the common rate h(cid:0)HuT(cid:1)
= r S ψ(G′(T))+z .
r
t+L
TX⊆S t+1
SNR |T|=t+1
log1+ min min hHuT 2. (23) r∈T q(cid:0) (cid:1)
t+L r∈S T⊆S | r S|
 r∈T  Considering the fact that User r is interested in decoding
 |T|=t+1  all the t+L−1 terms in above summation with equal rates,
  t
and using achievable capacity region of MAC channels [13],
It should be noted that, based on Lemma 1, each user in S
(cid:0) (cid:1)
the lemma proof is complete.
can receive data with rate (23). Since each user in S decodes
Now, since each user in S decodes
t+L−1
t F t+L−1
K K−t−1 t F
t(cid:0) L−1(cid:1) K K−t−1
t(cid:0) L−1(cid:1)
bits after transmission t(cid:0)o S(cid:1)(cid:0)is conc(cid:1)luded, the symmetric rate bits after transmission t(cid:0)o S(cid:1)(cid:0)is conc(cid:1)luded, the symmetric rate
will be equal to (10), and the proof is complete. willbeequalto(15),andtheproofofTheorem2iscomplete.
APPENDIX B
Proof of Theorem 2. Algorithm 2 is the same as Algorithm
1 in the cache content placement phase. The only difference
is in the delivery phase. In Algorithm 1 the coded chunks
are formed in the signal domain (complex field), while in
Algorithm 2 they are formed in the data domain (finite field).
Thus, let us first form the coded chunks as
G′(T )= W . (24)
i ⊕r∈Ti dr,Ti\{r}
Then, in Algorithm 2 the transmitter will send
1
X= uT ψ(G′(T)). (25)
S
t+L
TX⊆S t+1
|T|=t+1
q
(cid:0) (cid:1)
Itcanbeeasilycheckedthatthetransmitsignalin(25)satisfies
thetransmitpowerconstraintin(2).Then,thefollowinglemma
characterizes the rate for successful transmission of X.
Lemma 2. Consider a (t + L)-subset of users S. Then, if
the common transmission rate to this subset according to
Algorithm 2 is less than
R (S)=minR(r), (26)
C Eff
r∈S