Feedback Can Double the Prelog of Some Memoryless Gaussian Networks

0.31 MB·English

by  Michael Gastpar

1
Feedback Can Double the Prelog of Some
Memoryless Gaussian Networks
Michael Gastpar, Amos Lapidoth, Yossef Steinberg, and Miche`le Wigger
Abstract—We exhibit two memoryless Gaussian networks In these cases the prelog1 (the factor in the sum-capacity
where the capacity-gains afforded by feedback are unbounded high-SNR expansion in front of 1log(1+SNR)) is 2 and
in the signal-to-noise ratio (SNR). The networks are instances 2
exceeds the number of transmit antennas. This is in sharp
2 of the Gaussian broadcast channel and the two-user Gaussian
contrast to the case where the received signals are corrupted
1 interference channel. To demonstrate the capacity-gains we
0 propose and analyze a novel feedback coding scheme. by independentAWGN sequences, a case in whichthe prelog
2 For the broadcast channel with two receivers it is shown that (with or without feedback) cannot exceed the number of
if the noise sequences at the two receivers are perfectly anticor- transmit (or receive) antennas [1].
n related, then, at high SNR, feedback asymptotically doubles the
Todemonstratethesegainsweproposeandanalyzeanovel
a sum-capacity.Thesameholdsifthenoisesequencesareperfectly
J feedback coding scheme, similar to the schemes in [4], [5],
correlated provided that they are of unequal variances. This

5 result extends to the multi-receiver broadcast channel: if the and [6]. (The gains for anti-correlated AWGN sequences are
2 noisesequencesarealldifferentandhavearank-onecovariance also achieved by the Ozarow-Leung scheme in [8], [9] if
matrix,then,athigh-SNR,feedbackasymptoticallymultipliesthe the scheme’s parameter are chosen carefully, see [10], [11].)
] sum-capacity by the number of receivers. However, as we show,
T Our proposed coding scheme also allows us to determine
these multiplicative gains collapse when the feedback is noisy.
I For the two-receiver Gaussian broadcast channel with noise- the high-SNR asymptotics of the sum-capacity of the two-
.
s free feedback we also derive the high-SNR asymptotic sum- receiver AWGN BC with noise-free feedback and arbitrary
c capacity. The expansion is exact in the sense that, as the SNR noise correlation. The expansionis exact in the sense that the
[
atesnydmspttootiincfinexitpyr,etshseiodniffteernednscetobeztewroe.enIfththeesunmoi-sceapsaecqiutyenacnedsoaurre differencebetween it and the sum-capacityCBC,Σ vanishesas
2 the SNR tends to infinity:
perfectly anticorrelated or if they are perfectly correlated and
v
of unequal variances, then the asymptotic expansion is as if the When the two AWGN sequences are perfectly anticorre-
2 •
8 transmittercommunicatedtothetworeceivers over twoparallel lated (i.e, have correlation coefficient ρz = 1) or when
0 Gaussian channels. Otherwise, the asymptotic expansion is the they are perfectly correlated and have unequ−al variances
same as if the receivers could cooperate.
6 σ2 =σ2,
. For the two-user interference channel it is shown that if the 1 6 2
3 noises experienced by the two receivers are perfectly correlated
0 orperfectlyanticorrelated,thenformostchannel-gainsfeedback 1 P 1 P
lim C log 1+ + log 1+
10 doubles the high SNR sum-capacity. P→∞" BC,Σ−(cid:18)2 (cid:18) σ12(cid:19) 2 (cid:18) σ22(cid:19)(cid:19)#
: =0. (1)
v
i I. INTRODUCTION Thus, the sum-capacity approaches the sum-capacity of
X
We present two memoryless Gaussian networks where the two parallel AWGN channels where the transmitter can
r
a capacitygainsaffordedby noise-freefeedbackareunbounded communicate with power P over each of them. In this
in the signal-to-noiseratio (SNR). The networksare instances case, noise-free feedback doubles the prelog and thus
of the real, scalar, additive white Gaussian noise (AWGN) provides gains in the sum-capacity that are unbounded
broadcast channel (BC) and the real, scalar, AWGN interfer- in the SNR.
ence channel (IC). When the AWGN sequences are neither perfectly anti-
For the two-user AWGN BC we prove that, when the • correlated nor perfectly correlated,
AWGN sequences corrupting the outputs at the two receivers
are perfectly anti-correlated (i.e, have correlation coefficient 1 P(σ2+σ2 2ρ σ σ )
lim C log 1+ 1 2 − z 1 2
aρnzd=o−f 1u)noeqruwalhennoitsheeyvaarreianpceersfec(tσly2 c=orreσl2a)t,edat(ih.ei,ghρzS=NR1), P→∞" BC,Σ− 2 (cid:18) σ12σ22(1−ρ2z) (cid:19)#
1 6 2 =0. (2)
noise-free feedback approximately doubles the sum-capacity
from 1log(1+SNR) to approximately 2 1log(1+SNR). and the sum-capacity approaches the sum-capacity of
2 · 2
Thus,inthesecasesthenoise-freefeedbackincreasesthesum- a setup where the two receivers can cooperate. Here
capacitybyapproximately 1log(1+SNR)whichisunbounded feedback does not increase the prelog, and its benefit at
2
ρisneqtuheenSc1eN+sRa.r1eTsh)u,efofisrcawimehneetlnhyotchldleosysaealrsteoospueevfrfiefnceicewtnlyhtleyanncttlihocesoertrwteoloapteAerdWfe(ciG.telN.y, ThihgihsSgNaiRn iisstbhoeuandddeidtivientethrme p21olwoegr(cid:16), mbσua12tx+{iσσt1222c,−σa222nρ}z(b1σ−e1σρ2v2z)e(cid:17)ry.
z ≈− SNR
correlated (i.e., ρz ≈ 1− SN1R) and have different variances 1Theprelogisoftenalsoreferredtoasthemultiplexing gainordegreesof
(σ2 =σ2). freedom.
1 6 2
2
significant if ρz is close to −1 or if it is close to 1 with D✛
σ2 =σ2. Z
For1th6eK2>2userAWGNBCourproposedcodingscheme ✲1,t ✐❄ Y1✲,t Receiv. 1 Mˆ1✲
❄
allowsustoprovethatiftheAWGNsequencescorruptingthe M ,M X
1 2✲ t
K receivedsignalsarealldifferentbuthavecovariancematrix Trans.
Z
ofrank1,thenaprelogofK isachievable.Forarelatedrecent ✻ ✲2,t ✐❄ Y2✲,t Receiv. 2 Mˆ2✲
result see [12].
A naturally ensuing question is whether these gains in D✛
capacity can also be attained when the feedback is noisy. In
this paper we concentrate on how feedback noise affects the
Fig.1. Thetwo-userAWGNBCwithnoise-free feedback.
reportedgainsin the prelog.We showthatwhen the feedback
links are corruptedby independentAWGN sequences, then—
irrespective of the positive feedback-noise variances and of thesetofpositiverealnumbersbyR+,andthesetofpositive
the correlation of the forward noise-sequences—the prelog integers by Z+. The abbreviation IID stands for independent
of the two-user AWGN BC setup equals one (as in the and identically distributed.
absence of feedback). The proof of this result is based on The paper is organized as follows. In Sections II–V we
a genie argumentinspired by the work of Kim, Lapidoth,and present the channel models and the main results for the
Weissman [13]. two-user AWGN BC with noise-free feedback, the two-user
The second network we consider is the two-user scalar AWGN BC with noisy feedback, the K 2 user AWGN BC
AWGN IC with noise-free one-sided feedback where each of with noise-free feedback, and the two-u≥ser AWGN IC with
the two transmitters communicates with a different intended one-sided noise-free feedback. In Section VI we present a
receiver, and each transmitter observes feedback from its cor- coding scheme for the two-user AWGN BC with noise-free
respondingreceiveronly.Ourproposedcodingschemeproves feedback, and prove our results for this setup. In Section VII
that when the two AWGN sequences experienced at the two we prove our results for the AWGN BC with noisy feedback.
receivers are perfectly anticorrelated or perfectly correlated, In Sections VIII and IX we present coding schemes and
then for most channel gains noise-free feedback doubles the prove our results for the K-user AWGN BC with noise-free
prelog from 1 to 2. Thus, noise-free feedback allows to feedback and the two-user AWGN IC with one-sided noise-
approximately double the sum-capacity at high SNR and can free feedback, respectively.
provide unbounded gains. (When the interference channel is
symmetric,theprelog2resultcanalsobeshownusingaslight
II. TWO-USERBROADCAST CHANNEL WITHNOISE-FREE
generalization(toaccountforthecorrelationbetweenthenoise
FEEDBACK:SETUP AND MAIN RESULTS
sequences) of Kramer’s memoryless LMMSE-scheme [14].)
Previously, a prelog of 2 was known to be achievable for The real, scalar, AWGN BC with noise-free feedback is
the two-userscalar AWGN IC onlywhenthe two transmitters depicted in Figure 1. Denoting the time-t transmitted symbol
(or the two receivers) could fully cooperate [15] in the sense by xt R and the time-t received symbols by Y1,t and Y2,t
∈
that both transmitters can compute their channel inputs as respectively, we have
a function of both messages. Our result shows that limited
Y = x +Z , (3a)
cooperation through feedback can be sufficient. 1,t t 1,t
For the two-user AWGN IC we do not consider noisy and
feedback. Rate-limited feedback for this setup has recently
been studied in [18]. Y2,t = xt+Z2,t, (3b)
Thefactthatfeedbackcanincreasethecapacityofmemory-
where the sequence of noise pairs (Z ,Z ) is drawn IID
lessnetworkswithoutboundswasfirstreportedbytheauthors { 1,t 2,t }
according to a centered Gaussian distribution of covariance
in [2] for the two-user AWGN BC and for the symmetric
matrix
two-user AWGN IC where the individual noise sequences σ2 ρ σ σ
corrupting the outputs at the two receivers are perfectly anti- Kz = ρ σ1σ zσ12 2 . (4)
correlated.The same conclusionwas later also reachedin [3], (cid:18) z 1 2 2 (cid:19)
[16] (also based on the scheme proposedin [17]) forthe two- We assume that both noise variances σ2,σ2 are positive (i.e.,
1 2
user AWGN IC when the noise sequences are independent. not equal to 0).
MultiplicativegainsfortheAWGNICwithindependentnoises The transmitter wishes to send Message M to Receiver 1
1
atmoderateSNRwerealreadyreportedin[14,SectionVI-B]. andan independentmessageM to Receiver2. Themessages
2
We concludethissectionwithsomenotationandanoutline M and M are assumed to be uniformly distributed over
1 2
oftherestofthepaper.Throughoutthepaperthelogarithmis the sets , 1,..., 2nR1 and , 1,..., 2nR2 ,
1 2
M { ⌊ ⌋} M { ⌊ ⌋}
takenwithrespecttothebase2andforconveniencewedefine wherendenotestheblocklengthandR andR therespective
1 2
log0= . We will use the shorthand notation log+(x) for rates of transmission.
− ∞
max 0,log(x) . Also, we denote by An and an the tuple of It is assumed that the transmitter has access to noise-free
{ }
randomvariablesA ,...,A andtheirrealizationsa ,...,a , feedback from both receivers, i.e., that after sending x it
1 n 1 n t 1
respectively. The set of real numbers will be denoted by R, learns both outputs Y and Y . The transmitter−can
1,t 1 2,t 1
− −
3
thus compute its time-t channel input as a function of both Our main result in Theorem 1 is the asymptotic high-SNR
messages and all previous channel outputs: sum-capacityof the AWGN BC with feedback.Wheneverthe
BCisnotphysicallydegraded,theasymptotichigh-SNRsum-
Xt =fB(nC),t M1,M2,Y1t−1,Y2t−1 , t∈{1,...,n}, (5) capacity with feedback is strictly larger than without.
where the enc(cid:0)oding function f(n) is(cid:1)of the form Definition 1: Let CHighSNR(P,σ12,σ22,ρz) be defined as fol-
BC,t lows.
f(n) : Rt 1 Rt 1 R. (6) For physically degraded channels, i.e., ρ σ1,σ2 ,
BC,t M1×M2× − × − → • z ∈ σ2 σ1
Thechannelinputsaresubjecttoanexpectedaverageblock- 1 nP o
power constraint P > 0. Thus, in (5) we only allow for CHighSNR(P,σ12,σ22,ρz), 2log 1+ min σ2,σ2 .
encoding functions fB(nC),t n such that (cid:18) { 1 2}(cid:19)(15)
t=1
n o
n For not physically-degraded channels with fully corre-
1
nE"t=1Xt2#≤P. (7) • lated noises, i.e., |ρz|=1 and ρz ∈/ σσ21,σσ12 ,
X n o
Afterthen-thchanneluseeachreceiverdecodesitsintended C (P,σ2,σ2,ρ ), 1log P + 1log P . (16)
message based on its observed channel output sequence. HighSNR 1 2 z 2 σ2 2 σ2
1 2
Receiver 1 produces the estimate
Finally, for not physically degraded channels with par-
•
Mˆ1 = φ(1n)(Y1n), (8) tially correlated noises, i.e., ρz ∈ (−1,1) and ρz ∈/
σ1,σ2 ,2
and Receiver 2 the estimate σ2 σ1
n o
Mˆ = φ(n)(Yn), (9) CHighSNR(P,σ12,σ22,ρz)
2 2 2
1 P(σ2+σ2 2ρ σ σ )
where the decoding functions φ(1n) and φ(2n) are of the form , 2log 1σ2σ22(1− ρz2)1 2 . (17)
(cid:18) 1 2 − z (cid:19)
φ(n): Rn 1,..., 2nR1 , (10) Theorem 1 (Asymptotic High-SNR Sum-Capacity): For all
1 →{ ⌊ ⌋}
φ(2n): Rn →{1,...,⌊2nR2⌋}. (11) σ12,σ22 >0 and ρz ∈[−1,1]
A rate pair (R1,R2) is said to be achievable if for every Plim CBC,Σ(P,σ12,σ22,ρz)−CHighSNR(P,σ12,σ22,ρz) =0.
block-length n there exists a set of n encoding functions →∞ (18)
(cid:0) (cid:1)
n
f(n) as in (6) satisfying the power constraint (7) and Proof:Whenthechannelisphysicallydegraded,i.e.,ρz
BC,t ∈
t=1 σ1,σ2 ,theresultfollowsfrom(14).Fortheothercasesthe
tnwodeocodingfunctionsφ(n) andφ(n) asin(10)and(11)such σ2 σ1
that 1 2 nresult isoproved in Section VI-C.
nlim Pr (M1,M2)6=(Mˆ1,Mˆ2) =0. degNroatdeed1,:tIhfe|sρuzm| -=capa1ciatyndatthheighchSanNnRelisisansoitf pthheyrseicwalalys
The closure→of∞thehset of all achievable raite pairs (R ,R ) a separate, non-interfering link from the transmitter to each
1 2
is called the capacity region. The supremum of the sum of the two receivers and the transmitter could communicate
R1+R2 over all achievable rate pairs (R1,R2) is called its with power P over both these links. If ρz ∈(−1,1), then the
sum-capacity, and we denote it by C (P,σ2,σ2,ρ ). In sum-capacityathighSNRisasifthetworeceiverscouldfully
BC,Σ 1 2 z
this paper we are particularly interested in the prelog, which cooperate in their decoding.
characterizes the logarithmic growth of the sum-capacity at Note 2: Given σ22,σ12 >0, define γ: (−1,1)→R+ as
high powers, and is defined as: σ2+σ2 2ρ σ σ
γ(ρ ), 1 2 − z 1 2. (19)
C (P,σ2,σ2,ρ ) z σ2σ2(1 ρ2)
lim BC,Σ 1 2 z . (12) 1 2 − z
P→∞ 12log(1+P) By Theorem 1, γ(ρz) describes the behavior of the sum-
The sum-capacity of our setup CBC,Σ is unknown except capacity in the asymptotic high-SNR regime when ρz ∈
for the physically degraded case where (−1,1).Noticethatγ(ρz)→∞asρz →−1,andifσ12 6=σ22,
then also γ(ρ ) as ρ +1. Moreover, γ(ρ ) is
σ σ z → ∞ z → z
ρ 1, 2 . (13) strictlydecreasingoverρ 1,min σ1,σ2 andstrictly
z ∈(cid:26)σ2 σ1(cid:27) z ∈ − σ2 σ1
increasing over ρ min(cid:16) σ1,σ2 n,1 . Itoth(cid:17)us takes its
Inthiscase thecapacityregionis notincreasedwith feedback z ∈ σ2 σ1
[19], and thus, [20], [21] minimum at ρ =min(cid:16)σ1,σn2 wheore γ(cid:17)(ρ )= 1 .
C (P,σ2,σ2,ρ )= 1log 1+ P . (14) Figure 2 shozws the tnypσi2calσ1boehavior of γ(zρz). min{σ12,σ22}
BC,Σ 1 2 z 2 min σ2,σ2 From Theorem 1 we obtain the following corollary.
(cid:18) { 1 2}(cid:19)
The AWGN BC with feedback is in particular physically 2Notice that for physically degraded channels with partially correlated
degradedif ρz =1 andσ12 =σ22,in whichcasebothreceivers noises, i.e., for ρz ∈ (−1,1) and ρz ∈ nσσ12,σσ12o, the definitions in (15)
observe exactly the same output sequence. and(17)coincide.
4
40 same observation can be made when ρz =1−ǫ if σ12 6=σ22.
Theorem 3 and Corollary 4 ahead explore the relationship
30 betweenP andthe requiredcorrelationρz. Sincethe required
correlation depends on the transmit power P, we make the
γ(ρz)20 dependence explicit and denote the correlation by ρz(P).
Theorem 3 and Corollary 4 thus characterize the generalized
10 prelog where the channel parameters (here the noise correla-
tion ρ ) vary with the power P.
z
0 0.8 0.4 0 0.4 0.8 Let noise variances σ12,σ22 > 0 and for each power P > 0
− −
ρz a correlation coefficients ρz(P) be given. Define
aFnigd.σ22.2 =Th0e.2f5u.ncTtihoenmγi(nρimz)umis pisloatttedpo1v/e8r ρ≈z0∈.3[5−306.,9a,n0d.9t5h]efofurnσct12io=n i2s ζ−1 ,Plim −logl(o1g+(Pρ)z(P)), (20)
strictlydecreasingover −1, 1/8 andstrictlyincreasingover 1/8,1 . →∞
(cid:0) p (cid:1) (cid:0)p (cid:1) log(1 ρ (P))
ζ , lim − − z , (21)
+1
P log(P)
22 →∞
where recall that we defined log0 , . Notice that
1.8 − ∞
ζ > 0 only if lim ρ (P) = 1, and ζ > 0 only
P) if−l1im ρ (P)=1P.→∞ z − +1
R(P)Σlog(1+ 11..64 depTehnePdos→reo∞mn3wz(hGetehneerrathliezendoiPserevloagri)a:ncTehseσ2g,eσn2er>aliz0eadreperqeluoagl
/2 1 2
1 or different. If σ2 =σ2, the generalized prelog is
1.2 1 2
C (P,σ2,σ2,ρ (P))
1 101 102 103 104 Plim BC,Σ1log(11+2P)z =min{1+ζ−1,2}. (22)
→∞ 2
and if σ2 =σ2, it is
Fig. 3. The sum-rate RΣ(P) achieved by the scheme in Section VI-A 1 6 2
Pnorm>al0iz.edThebyno12iselogva(r1ia+ncesσP12σ)2is=pσlo2tte=d 1asarae ffiuxnecdti,oanndofthethedifpfeorwenert lim CBC,Σ(P,σ12,σ22,ρz(P))
curves correspond (in increasing1 order2) to correlation coefficients ρz = P→∞ 12log(1+P)
−0.85,−0.95,−0.99,−0.999−0.9999.
=min 1+max ζ ,ζ ,2 . (23)
1 +1
−
n n o o
Proof: See Section VI-C.
Corollary 2 (Prelog): The prelog of the AWGN BC with
Corollary 4 (Generalized Prelog): Let ρ (P) be of the
noise-free feedback is 2 if ρ = 1 or if ρ =+1 and σ2 = z
z − z 1 6 form
σ2, otherwise it is 1.
2
ǫ(P)
theNotrtaen3sm: iOtteurr hreassulotsnlyforonρez-s∈ide{d−n1o,i1s}e-frreemeafineevdablaicdkw, ih.een., ρz(P)=± 1− Pζ , ν ∈{1,2}, ζ ∈[0,1]
(cid:18) (cid:19)
when the transmitter for example only observes the outputs where
Y but not Y . This holds because for ρ 1,1 log ǫ(P)
1,t 2,t z
{ } { } ∈ {− } lim =0.
tchoeinccaidpea.ciItny rfaegcti,onins wthiethsoetnuep-siwdietdh aonnde-tswidoe-dsidfeeeddfbeaecdkbathcke P→∞ lo(cid:0)g(P)(cid:1)
Unless σ2 =σ2 and lim ρ (P)=1,
transmitter can locally compute the missing outputs Y2,t (or 1 2 P→∞ z
Y1,t): lim CBC,Σ(P,σ12,σ22,ρz(P)) =1+ζ. (24)
σ2 P 1log(1+P)
Y2,t =ρz (Y1,t Xt)+Xt, t 1,...,n . →∞ 2
σ − ∈{ }
1
Theorem 1 and Corollary 2 show that when ρ 1,1 III. TWO-USERBROADCAST CHANNEL WITHNOISY
z
and the channel is not physically degraded, feedbac∈k{a−pprox}- FEEDBACK: SETUPAND RESULTS
imately doubles the high-SNR sum-capacity. In Section VI-A InthissectionwestudytheAWGNBCwithnoisyfeedback,
we present a coding scheme with this desired performance. whichisdepictedin Figure4. Thegoalofthecommunication
(When ρz = 1 also the Ozarow-Leung scheme [8], [9] is the same as in the previoussection. That is, the transmitter
−
achieves such sum-rates [10], [11].) In Figure 3 we have wishes to convey Message M to Receiver 1 and Message
1
plotted the relationshipbetween the sum-rate achieved by our M to Receiver 2 by communicating over the AWGN BC
2
scheme in Section VI-A and the transmitted power P for describedin(3).Thetransmitterhasaccesstonoisyfeedback.
various values of the correlation ρz. It shows that for large Thus,insteadofobservingthechanneloutputsY1,t andY2,t as
powers P (i.e., P 100), feedback can nearly double the inthe previoussection,itobservesthe noisyfeedbackoutputs
≥
capacity not only when the correlation ρ is exactly -1, but
z
alsowhenρ issufficientlycloseto-1,(i.e.,whenρ = 1+ǫ V1,t =Y1,t+W1,t,
z z
−
for sufficiently small ǫ > 0 depending on the power P). The V =Y +W .
2,t 2,t 2,t
5
W
D ✐❄✛1,t convergenceisslowerthanP−ξ forallξ >0.Moreprecisely,
Z if σ2 ,σ2 depend on P in a way that
❄ ✲ ❄✐1,t Y1,t✲Receiv.1 Mˆ1✲ W1 W2 log σ2
M ,M X lim − Wν 0, ν 1,2 ,
1 2✲Trans. t Z P→∞ log(cid:0)(P) (cid:1) ≤ ∈{ }
✻ ✲ ❄✐2,t Y2,t✲Receiv. 2 Mˆ2✲ then the prelog is 1, irrespective of the noise correlationρz ∈
[ 1,1].
✛ ✛ −
D ✐ Proof: See Section VII-B.
✻W
2,t
Fig.4. Thetwo-userAWGNBCwithnoisyfeedback. IV. K-USERBROADCAST CHANNEL WITH NOISE-FREE
FEEDBACK:SETUP AND MAIN RESULTS
We generalize the setup in Section II to K 2 users.
The feedback-noise sequences (W ,W ) are assumed ≥
{ 1,t 2,t } Thus,here,thesingletransmitterwishestocommunicatewith
to be independent of the messages (M ,M ) and of the
1 2 K receivers. The goal of the communication is that each
noise sequences on the forward path (Z ,Z ) and IID
{ 1,t 2,t } Receiver k 1,...,K learns its desired Message Mk,
according to a zero-mean bivariate Gaussian distribution of ∈ { }
where the messages M ,...,M are independent, and each
σ2 0 1 K
diagonal3 covariancematrix W01 σ2 . We shallassume Mk is uniformly distributed over Mk ,{1,...,⌊2nRk⌋}.
(cid:18) W2(cid:19) The channel model is described as follows. Given that
throughout this section that the feedback noise variances are
the transmitter sends the time-t input symbol x , Receiver k
positive t
observes the time-t output symbol
σ ,σ >0. (25)
W1 W2
Y =x +Z , t 1,...,n , (29)
k,t t k,t
In this setup the transmitter computes its channel inputs as ∈{ }
where (Z ,...,Z ) n is a sequence of IID centered
Xt =fB(nC)Noisy,t M1,M2,V1t−1,V2t−1 , t∈{1,...,n}, Gaussia{n ve1,cttors of cKo,vta}rtia=n1cematrix Kz and independentof
(26) the messages (M ,...,M ).
where the encodin(cid:0)g function f(n) i(cid:1)s of the form The transmitter1 has noiKse-free feedback from all channel
BCNoisy,t
outputs, and thus can compute its channel inputs as
fB(nC)Noisy,t: M1×M2×Rt−1×Rt−1 →R. (27) Xt =fK(n-B)C,t(M1,...,MK,Y1t−1,...,YKt−1)
The channel input sequence (X ,...,X ) is again subject
1 n
to an expected average block-power constraint P > 0 as in for some encoding function fK(n-B)C,t, such that the input se-
(7). quence (X1,...,Xn) satisfies the expected average block-
Decoding rules, achievable rates, capacity region, sum- power constraint in (7).
capacityandprelogaredefinedasintheprevioussection.The An error occurs in the communication whenever
sum-capacityforthis setup with noisyfeedbackis denotedby (M ,...,M )=(Mˆ ,...,Mˆ ). (30)
C (P,σ2,σ2,ρ ,σ2 ,σ2 ). 1 K 6 1 K
BCNoisy,Σ 1 2 z W1 W2
Noisy feedback does not increase the prelog. Achievable rate-tuples, capacity region, and sum-capacity are
Theorem 5: Irrespectiveofthecorrelationρ [ 1,1],the defined in analogy to the setup with two users. The sum-
z
prelog is one: ∈ − capacity of this setup is denoted by CK-BC,Σ(P,Kz). We are
interested in its prelog.
lim CBCNoisy,Σ(P,σ12,σ22,ρz,σW2 1,σW2 2) =1. (28) Theorem 6: If the covariance matrix Kz has rank 1, then
P 1log(1+P) the prelog is
→∞ 2
Proof: See Section VI. lim CK-BC,Σ(P,Kz) =n . (31)
Thus,withnoisyfeedbacktheprelogequals1alsowhenthe P 1log(1+P) α
→∞ 2
noise correlation ρ 1,1 . This result assumes that the
feedback-noise varizan∈ce{s−σ2 },σ2 > 0 are fixed. If instead wherenα denotesthenumberofrowsinKz thataredifferent.
W1 W2 Proof: See Section VIII-B.
they tend to 0 as the power P , then for ρ 1,1
→ ∞ z ∈ {− } Corollary 7: When K is of rank 1 and all its rows are
the (generalized) prelog may be larger than 1, depending z
different, the prelog is K.
on the speed of convergence of the limits σ2 ,σ2 0.
W1 W2 → The achievability of prelog K 3 was first proved in [12]
The followingnote examinesthe generalizedprelog when the ≥
for the complex memorylessGaussian broadcastchannel. The
feedback-noise variances tend to 0 slower than P ξ for any
− prelog 3 result in [12] however requires that the real and
ξ >0. ≥
imaginary parts of the noise symbols are correlated, and thus
Note 4: Theorem 5 remains valid if the feedback-noise
does not include our result in Theorem 6 as a special case.
variances σ2 ,σ2 tend to 0 as the power P , if the
W1 W2 → ∞ Note 5: Theorem6remainsvalidalsowhenthetransmitter
hasfeedbackonly froma single receiver.This can be seen by
3Forsimplicity, wedonottreat setups with correlated feedback noises or
analogy to Note 3.
setupswithfeedback noisesthatarecorrelated withtheforwardnoises.
6
V. TWO-USERINTERFERENCECHANNEL WITH VI. ASCHEME AND PROOFS FOR THEAWGN BC WITH
NOISE-FREE FEEDBACK: SETUPAND MAIN RESULTS NOISE-FREEFEEDBACK
In this section we consider the real scalar AWGN IC with
A. A coding scheme
noise-freefeedback,whichhastwotransmitters.Transmitter1
wishes to send Message M to Receiver 1, and Transmitter 2 We present a new coding scheme for the K = 2 user
1
MessageM toReceiver2.Assumingthatattimet Transmit- Gaussian BC with noise-free feedback, see also [7]. (The
2
ter 1 sends the real symbol x and Transmitter 2 sends the scheme is generalized to K 2 users in Section VIII-A.)
1,t ≥
real symbol x , Receiver 1 observes Our scheme is similar to the schemes proposed by Cover and
2,t
Pombra [4] for (non-white) Gaussian point-to-point channels
Y = a x +a x +Z , (32a)
1,t 1,1 1,t 1,2 2,t 1,t with noise-free feedback, by Lapidoth and Wigger [5] for the
and Receiver 2 observes two-user AWGN multi-access channel with noisy feedback,
and by Lapidoth, Steinberg, and Wigger [6] for the two-
Y2,t = a2,1x1,t+a2,2x2,t+Z2,t. (32b) user AWGN BC with one-sided noise-free feedback. We also
proposea choiceofparametersforourscheme.Despitebeing
The “channel gains”
suboptimal(see[7]forimprovedchoices),thischoicesuffices
a1,1,a1,2,a2,1,a2,2 =0, (33) to prove the achievability results in Theorems 1 and 3 when
6
the channel is not physically degraded.
are non-zero real constants and the noise sequences
Our scheme has the following parameters: the positive
(Z ,Z ) n are defined as in Section II.
{ 1,t 2,t }t=1 integernumberη;theη-by-η strictlylower-triangularmatrices
Each transmitter has access to noise-free feedback from its
B ,B ; the η-dimensional column-vectors u ,u ; and the η-
correspondingreceiver.Thus,eachofthetwotransmitterscan 1 2 1 2
dimensional row-vectors v ,v .
compute its time-t channel input as 1 2
1) Code Construction: Let the block-length n of our
Xk,t = fI(Cn,)k,t Mk,Ykt−1 , k ∈{1,2}, scheme be a multiple of η, i.e., n = ηn′ for some positive
integer number n.
for some encoding functio(cid:0)n fI(Cn,)k,t of(cid:1)the form We generate tw′o codebooks {Ck}2k=1, where Ck contains
fI(Cn,)k,t: Mk×Rt−1 →R, ν ∈{1,2}. ⌊w2enRdkra⌋wcoedaecwhoerdnstryofolfenegacthhnc′o.dTewoogrednerarantdeomthleycaonddebionodkes-
The two channel input sequences are subject to the same pendentlyaccordingto a standard Gaussian distribution.Both
average block-power constraint P >0: codebooksare revealed to the transmitter, and each codebook
to the corresponding receiver k.
1 n Ck
nE" Xk2,t#≤P, k ∈{1,2}. (34) cod2e)wEonrdcoidning:thFaotrcokrre∈spo{n1d,s2t}o,MleetssΞagke(MMk), adnednoletteΞthe
t=1 k k k,i
X C
be its i-th symbol.
Decoding rules, achievable rate pairs, capacity region, sum-
capacity, and prelog are defined as for the AWGN BC. We Also, for each i 1,...,n′ let Xi denote the η-length
∈ { }
denote the sum-capacity of the symmetric AWGN IC by column vector obtained by stacking the η input symbols
CIC,Σ(P,σ12,σ22,ρz). X(i−1)η+1,...,X(iη) on top of each other:
Without feedback, the prelog of the AWGN IC equals 1;
X ,(X ,...,X )T, (38)
withnoise-freefeedbackitcanbe2,dependingonthechannel i (i 1)η+1 (iη)
−
gains a ,a ,a ,a = 0 and on the noise parameters
1,1 1,2 2,1 2,2 6 andforeachk 1,2 ,lettheη-lengthvectorsY andZ
σ12,σ22,ρz. be defined simi∈lar{ly. } k,i k,i
Theorem 8: The prelog of the AWGN IC with noise-free
Theencodingprocedureisasfollows.Thetransmitterpicks
feedback satisfies the following three statements.
from each codebook , for k 1,2 , the codeword that
• Iρf ρσ2z,∈ (−1,1) or if ρz ∈ {−1,1} and aa21,,22 = aa11,,21 = corresponds to the mCekssage M∈k,{and}then sends in each
zσ1 C (P,σ2,σ2,ρ ) subblock i ∈ {1,...,n′} a linear combination of the i-th
lim IC,Σ 1 2 z =1; (35) symbols of these codewords and the subblock’s past noise-
P 1log(1+P) symbols4
→∞ 2
if ρ 1,1 and moreover, both a2,2 and a1,2 are
• diffezre∈nt{fr−om ρ}σ2, then a1,2 a1,1 Xi =Ξ1,iu1+Ξ2,iu2+B1Z1,i+B2Z2,i. (39)
zσ1
Plim CIC,1Σl(oPg,(σ112+,σP22,)ρz) =2; (36) tThhaetsthtreicfteleodwbearc-ktriiasnugsueldariotynloyfcthauesmalalytr.icesB1andB2assures
→∞ 2 The inputs satisfy the average block-power constraint (7)
otherwise
whenever Inequality (40) on top of the next page is satisfied.
•
C (P,σ2,σ2,ρ )
1 lim IC,Σ 1 2 z 2. (37)
≤P 1log(1+P) ≤ 4Thetransmitter cancomputeallthepastnoisesymbolsbecause, through
→∞ 2
thefeedback,itlearnsthepastchanneloutputsandbecauseitalsoknowsthe
Proof: See Section IX-B. pastchannel inputs.
7
u 2+ u 2+tr(B BT)σ2+tr(B BT)σ2+2tr(B BT)ρ σ σ ηP. (40)
k 1k k 2k 1 1 1 2 2 2 1 2 z 1 2 ≤
3) Decoding: In subblock i 1,...,n Receiver 1 first η 2 columns of the matrices B and (B +I), and to
′ 1 2
observes ∈ { } the vec−tor u , but not to u . Therefore,
1 2
Y1,i =Ξ1,iu1+Ξ2,iu2+(B1+I)Z1,i+B2Z2,i, (41a) P b2 η−1 a2 b1
I = 1 Ξ
and Receiver 2 observes 1,i vu2+2ab2121 (cid:18)−a2(cid:19) (cid:18) − b2 a1(cid:19) 1,i
u
Y =Ξ u +Ξ u +(B +I)Z +B Z . (41b) t b
2,i 1,i 1 2,i 2 2 2,i 1 1,i + 2 +a Z +Z +a Z (50a)
1 1,iη 1 1,iη 2 2,iη 1
To decode its desired message M , each Receiver k 1,2 (cid:18)−a2 (cid:19) − −
k
∈{ } η 1
forms the ”new outputs” P b1 − a1 b2
I = 1 Ξ
Ik,i ,vkYk,i, i∈{1,...,n′}, (42) 2,i vuu2+2ab2222 (cid:18)−a1(cid:19) (cid:18) − b1 a2(cid:19) 2,i
t b
and optimally decodes Mk based on (Ik,1,...,Ik,n′), e.g., + 1 +a2 Z2,iη 1+Z2,iη+a1Z1,iη 1. (50b)
using a maximum-likelihooddecoder. (cid:18)−a1 (cid:19) − −
4) Choice of Parameters: We describe a choice of the
We see that with our choice of parameters the last η 2
parameters B ,B , u ,u , v ,v for every positive integer −
1 2 1 2 1 2 channel uses in each subblock are used to iteratively can-
η, and then let η .
cel the previous noise symbols in this subblock. Therefore,
→∞
Choose q > 0 and δ / 1,0 to satisfy Equation (43)
the ”new output symbols” I or I depend only on the
∈ {− } 1,i 2,i
shown on top of the next page, and define
noise symbols Z ,Z ,Z ,Z but not on the
1,iη 1 1,iη 2,iη 1 2,iη
− −
a ,q (44) previous noise symbols. Also, Receiver 1’s ”new outputs”
1
a , δ2q (45) I1,1,...,I1,n′ do not depend on the non-desired codeword
2 − Ξ2,1,...,Ξ2,n′, andthustheycan beviewedastheoutputsof
b1 , δ(1+δ)q2 (46) a point-to-point channel where the transmitter only sends the
−
b2 , δ2(1+δ)q2. (47) codeword Ξ1,1,...,Ξ1,n′, but not Ξ2,1,...,Ξ2,n′. Similarly
−
for Receiver 2’s ”new outputs”.
For a given positive integer η we choose the η η matrices
5) Achievable Rates: By (44)–(49d) and (50) and by stan-
B andB Toeplitzwithnon-zeroentriesonlyon×thefirstand
1 2 dard arguments, our scheme achieves all rate pairs (R ,R )
1 2
second diagonals below the main diagonal:
that satisfy
0 0 0 0 0 0
ak 0 0 ······ 0 0 0 R 1 log 1+ 2+2Pq(21δ+2(δ1)+2δ)2 q2(1+δ)2 η−1 (51a)
Bk =b00...k a.b.kk. a.00.k. ·.b0·.·. ·a.0·.·. 00...... 00......, (48) R12 ≤≤ 221ηη log1+ 2P+(21(q+q2(1(2q1/δ2+δ2δ2δ)2+)22+1(cid:0)q)1(cid:0)σ2)δσ12222+(+1q+2qδ2δ4σ)σ1222(cid:1)2(cid:1)η−1. (51b)
 ··· k k 
0 0 b a 0  
 ··· ··· k k  We have the following lemma.
 
and we choose the η-dimensional vectors Lemma 1: Let ξ,ζ be positive real numbers. If 1+ ζ
ξ, then the mapping η Z+ 1 log(1+ξη 1ζ), has i≥ts
P T ∈ 7→ 2η −
u = 1 b1 0 ... 0 (49a) maximumat η =1;otherwiseit hasits supremumat η .
1 vuut2+2ab2121 (cid:16) a1 (cid:17) By LPermoomf:aS1e,eηA=ppe1ndmixaxAim.izes the constraints in (5→1) ∞for
u = P 1 b2 0 ... 0 T (49b) smallpowersP andη maximizesthemforlargepowers
2 v2+2b22 a2 P. Letting η , we→co∞nclude the following.
uu a22 (cid:16) (cid:17) Proposition→9:∞Allnonnegativerate-pairs(R ,R )thatsat-
t η 1 η 2 1 2
v1 = b2 − b2 − ... b2 1 (49c) isfy
−a2 −a2 −a2
(cid:18) (cid:19)
v2 = (cid:16)−ab11(cid:17)η−1 (cid:16)−ab11(cid:17)η−2 ... −ab11 1 . (49d) R1 ≤ 21log+ q2(1+δ)2 (52a)
By (43)(cid:18)–((cid:16)48), f(cid:17)or ever(cid:16)y η t(cid:17)his choice satisfies (cid:19)the power R2 1log+(cid:0)q2δ2(1+δ(cid:1))2 , (52b)
≤ 2
constraint (40). Moreover, the vector v is orthogonal to the
1 (cid:0) (cid:1)
first η 2 columns of the matrices (B +I) and B , and to for some δ / 1,0 and q such that (43) holds, are
1 2
thevec−toru ,butnottou .Similarly,v isorthogonaltothe achievable. ∈ {− }
2 1 2
8
q2(σ2+δ4σ2 2δ2ρ σ σ )+q4(1+δ)2δ2(σ2+δ2σ2+2δρ σ σ ) P. (43)
1 2 − z 1 2 1 2 z 1 2 ≤
When ρ (P) ( 1,1), the choice Letting η we thus obtain the following corollary for
z
∈ − → ∞
δ = σ1 σ1−ρzσ2 (53a) |ρzC|o=ro1ll.ary 11: If ρ =1,andthechannelisnotphysically
σ2 · σ2 ρzσ1 | z|
− degraded, i.e., ρ / σ1,σ2 , then all nonnegativerate-pairs
(1 ǫ)P z ∈ σ2 σ1
q2 =sδ2(1+δ)2(σ12+−δ2σ22+2δρzσ1σ2) (53b) (R1,R2) that satisfyn o
1 P
satisfies thepowerconstraint(43) forallǫ>0andallpowers R log+ (59a)
1 ≤ 2 σ2
P exceeding some threshold P0(ǫ,σ12,σ22,ρz). Plugging this (cid:18) 1(cid:19)
1 P
choice into the achievablerates in Proposition9 results in the R log+ (59b)
following corollary. 2 ≤ 2 σ2
(cid:18) 2(cid:19)
Corollary 10: For ρz ( 1,1) and every ǫ (0,1) there are achievable.
∈ − ∈
exists a positive real number P0(ǫ,σ12,σ22,ρz) such that the Note 6: For ρz =1andwhenthechannelisnotphysically
sum-rate | |
degraded our scheme achieves prelog 2 when η . For
R +R = 1log+ (1−ǫ)P(σ12+σ22−2ρzσ1σ2) (54) finite η our scheme achieves prelog 2η−η1, see (→58)∞. Thus,
1 2 2 σ2σ2(1 ρ2) η = 3 suffices to increase the prelog compared to the non-
(cid:18) 1 2 − z (cid:19)
feedback setup.
is achievable for all powers P >P (ǫ,σ2,σ2,ρ ).
0 1 2 z Note 7: To gain intuition about the scheme and our choice
When ρ =1, the choice
| z| ofparameters,itisinstructivetoconsiderthefollowingscheme
σ
δ = ρ 1 (55a) for the AWGN point-to-point channel with feedback. We
z
− σ2 consideranAWGNchannelwithnoisevarianceσ2 andpower
P constraintP. The scheme is obtainedby specializingour BC-
q = (55b)
2 scheme to K = 1. We describe our choice of parameters
vσ2 1 σ1ρ
u 1 − σ2 z through an explicit description of the encoding and decoding
u
satisfies (43), and the tmatric(cid:16)es B and(cid:17)B defined through strategies.
1 2
(44)–(48) and (55) satisfy LetΞ1,...,Ξn′ denotethe symbolsofthe single codeword
chosen by the transmitter. In the first channel use of each
B σ +B ρ σ =B, (56)
1 1 2 z 2 subblock i 1,...,n the transmitter sends the codeword
′
∈ { }
where B is a Toeplitz matrix with all zero entries except on symbolΞi scaled by √P, and in allother channeluses ofthe
the first diagonal below the main diagonal where the entries subblock it sends the previous noise symbol scaled by P :
σ2
equal √P:
q
X =√PΞ (60)
0 0 0 0 0 (i 1)η+1 i
··· −
√P 0 0 0 0 P
B= 0... √..P. .0.. ··.··.··. 0... 0... . (57) X(i−1)η...+2 ==r...σ2Z(i−1)η+1 (61)
 
 0 0 √P 0 0  P
 ···  X = Z (62)
 0 ··· 0 0 √P 0  iη rσ2 iη−1
 
Since ρz 1,1 thisimpliesthat—similarlyto thescheme where Zt denotes the Gaussian noise symbol corrupting the
∈{− }
for the AWGN point-to-pointchannelwith feedbacksketched outputattimet.Thereceiverusesthelastη 1channeloutputs
−
in Note 7 ahead—in each channel use the transmitter only of each subblockto iterativelycancel the noise corruptingthe
sendsthepreviousnoisesymbol,butnottheonesbefore.This first output in the subblock
is unlike our scheme for ρ ( 1,1) where the transmitter
sends the two previous noisze∈sym−bols. I , η P η−ℓY =√P P η−1Ξ +Z , (63)
Specializingtherateconstraintsin(51)tothechoicein(55) i rσ2 (i−1)η+ℓ rσ2 i iη
ℓ=1
results in X
and then decodes its desired message M by applying
R 1 log 1+ P η−1 1 ρ σ1 2 P/σ12 a maximum-likelihood decoder to the resulting sequence
1 ≤ 2η (cid:18)σ12(cid:19) (cid:18) − zσ2(cid:19) 2+2P/σ22! I1,...,In′. Letting η → ∞, the described scheme achieves
(58a) all rates satisfying
R 1 log 1+ P η−1 1 ρ σ2 2 P/σ22 . 0 R 1log+ P , (64)
2 ≤ 2η (cid:18)σ22(cid:19) (cid:18) − zσ1(cid:19) 2+2P/σ12! ≤ ≤ 2 (cid:18)σ2(cid:19)
(58b) and is thus optimal in the asymptotic high-SNR regime.
9
Noticethatforfiniteη theperformanceoftheschemecould C. Proof of Theorem 3
be improved, if instead of as in (63) the symbol Ii is chosen Recall that here ρ (P) depends on the power P.
z
as the linear minimum-meansquare estimate (LMMSE)of Ξi We first prove the converse to (22) where σ2 =σ2. In this
given Y η . However, in the limit η both 1 2
choices{a(cih−i1e)vηe+tℓh}eℓ=s1ame rates. → ∞ case, Upper bound (66) specializes to
Note 8: The simple scheme in Note 7 illustrates that with CBC,Σ(P,σ12,σ12,ρz(P))
feedback one can achieve a rate R = 21log+ σP2 over an 1 P
AWGN point-to-pointchannel by sending most(cid:0)of t(cid:1)he time— ≤ 2log 1+ σ12(1+ρ (P))!, ρz(P)∈(−1,1).(67)
in fact, in all channel uses except for the first channel use of 2 z
each subblock—the previous noise-symbol scaled to satisfy In view of (14) and since we defined log(0) = , Upper
− ∞
the power constraint P. bound (67) holds also for ρ (P) 1,1 , and thus for all
z
∈ {− }
Now, in our scheme for the AWGN BC with noise correla- ρ (P) [ 1,1]. Therefore, by the definition of ζ in (20),
z 1
tion ρ =1 we applythe same idea with the only difference ∈ − −
that |wze| wish to convey two different codewords (one for lim CBC,Σ(P,σ12,σ12,ρz(P)) 1+ζ . (68)
each receiver) and that we send their corresponding symbols P→∞ 12log(1+P) ≤ −1
over the first two channel uses of each subblock. Sending the On the other hand, by (65), irrespective of ρ (P) ,
z P>0
codesymbols during two channel uses allows us to choose { }{ }
theirsignalingdirections(u1 andu2, respectively)orthogonal lim CBC,Σ(P,σ12,σ12,ρz(P)) 2. (69)
to the beamforming direction at their non-intended receiver P 1log(1+P) ≤
→∞ 2
(v and v , respectively).When the channelis not physically
2 1 Combining (68) and (69) establishes the converse to (22).
degraded, the beamforming directions at the two receivers We now prove the converse to (23) where σ2 = σ2. By
are different, and each receiver observes essentially the same Upperbound(66) and since σ2+σ2 2ρ (P)σ σ1 6<2(2σ2+
channel outputs as if the transmitter was using the point-to- σ2) and 1 ρ2(P) 1 ρ (1P) f2o−r allzρ (P)1 2( 1,1)1,
point scheme in Note 7. Therefore, in this case, our scheme 2 − z ≥ −| z | z ∈ −
achieves the rate pair R1 = 21log+ σP12 , R2 = 21log+ σP22 CBC,Σ(P,σ12,σ22,ρz(P))
(see Corollary 11). (cid:16) (cid:17) (cid:16) (cid:17) 1 P
The special case |ρz| = 1 illustrates very nicely that the ≤ 2log1+ σ12σ22 (1 ρ (P)), ρz(P)∈(−1,1).
intuition put forth in [23], [9], [24], [25] applies also to our 2(σ12+σ22) −| z |
scheme: feedback is helpful for a memoryless BC because it   (70)
allowsthetransmittertoidentifyandtransmitinformationthat
Now,sincewedefined log(0)= ,Upperbound(70)holds
atthesametimeis usefulforbothreceivers,andthereforethe − ∞
also for ρ (P) 1,1 , and hence for all ρ (P) [ 1,1].
BC can be used in a more efficient way. z ∈{− } z ∈ −
Moreover,since by the definitionsof ζ and ζ in (20) and
1 +1
(21), −
B. Proof of Theorem 1
log(1 ρ (P))
lim − −| z | =max ζ ,ζ , (71)
Forphysicallydegradedchannelswhereρ σ2,σ1 the P log(P) { −1 +1}
z ∈ σ1 σ2 →∞
result is proved in [19] and [20]. n o we conclude that
When |ρz| = 1 and ρz ∈/ σσ21,σσ12 , the achievability lim CBC,Σ(P,σ12,σ22,ρz(P)) 1+max ζ ,ζ . (72)
follows from Corollary 11, and tnhe conveorse by applying the P 1log(1+P) ≤ { −1 +1}
cutset bound with two individualcuts between the transmitter →∞ 2
Combining (72) with (69) establishes the converse to (23).
and each of the two receivers:
We next prove that for arbitrary σ2,σ2:
1 2
R +R max I(X;Y )+I(X;Y )
1 2 ≤X1:E[X2]≤P{P 11 2P} Plim CBC,Σ1(Plo,gσ(121,+σ22P,ρ)z(P)) ≥min{1+ζ−1,2}. (73)
= log 1+ + log 1+ , (65) →∞ 2
2 (cid:18) σ12(cid:19) 2 (cid:18) σ22(cid:19) Since a generalized prelog of 1 is achievable even without
feedback [20], [21] the interesting case is ζ > 0. In the
wheretheequalityfollowsbecauseaGaussianlawmaximizes 1
following, assume that ζ > 0, which impli−es the existence
the differential entropy under a variance constraint [22]. 1
−
of an increasing unbounded sequence P such that
When ρz ∈ (−1,1) and ρz ∈/ σσ21,σσ21 , the achievability { ℓ}∞ℓ=1
follows from Corollary 10 and thne conveorse by applying the lim −log(1+ρz(Pℓ)) =ζ >0, (74)
cutset bound with a single cut between the transmitter and ℓ→∞ 12log(Pℓ) −1
both receivers:
and in particular
R +R max I(X;Y ,Y )
1 2 1 2 lim ρ (P )= 1. (75)
≤X:E[X2]≤P ℓ z ℓ −
1 P(σ2+σ2 2σ σ ρ ) →∞
log 1+ 1 2 − 1 2 z . (66) For each ℓ we choose parameters qℓ and δℓ and show that
≤ 2 (cid:18) σ12σ22(1−ρ2z) (cid:19) Inequality(73)followsfromProposition9specializedtothese
10
parameters. Let ǫ (0,1) be a small positive number, and (75) by lim ρ (P)=1, and (80) by
ℓ z
∈ →∞
define the limit (not necessarily finite)
2
σ κ
κ,ℓlim Pℓ σ12+δℓ2σ22+2δℓρz(Pℓ)σ1σ2 . (76) σ12(cid:18)1− σ21(cid:19) β(cid:18)1+βσ22(cid:19)=1. (87)
→∞
We choose (cid:0) (cid:1) The assumption of non-equal noise variances σ2 = σ2 is
1 6 2
needed here to conclude that (83) holds and that (87) has a
δℓ =(−σσ21ρz·(σσP12ℓ−−)ρρσσzz12((PPℓℓ))σσ21 iiff ρρzz((PPℓℓ))∈∈({−−11,,11)}, (77) finCiteomsobliuntiinognfifonralβly.(85)with (73) establishestheachievabil-
and depending on κ, we choose q >0 as follows. ity of (22) and (23) and concludes the proof.
ℓ
If κ= , we choose q >0 such that
ℓ
• ∞ VII. PROOFSFORTHEAWGNBCWITHNOISYFEEDBACK
(1 ǫ)P
q2 = − ℓ ; A. Proof of Theorem 5
ℓ sδℓ2(1+δℓ)2(σ12+σ22δℓ2+2ρz(Pℓ)δℓσ1σ2) The interesting part is the converse result, which we prove
(78)
using a genie-argumentsimilar to [13].
if κ [0, ), we choose q >0 such that
• ∈ ∞ ℓ The following three steps establish the desired converse
q2 =β(1 ǫ)P (79) result. 1.) We introduce a genie-aided AWGN BC without
ℓ − ℓ
feedback and show that its sum-capacity upper bounds the
where β >0 is a solution to
sum-capacityof the originalAWGN BC with noisy feedback.
σ 2 κ 2.) We introduce a less noisy AWGN BC with neither genie-
σ2 1+ 1 β 1+β =1. (80)
1 σ σ2 informationnorfeedbackandshowthatitssum-capacitycoin-
(cid:18) 2(cid:19) (cid:18) 2(cid:19)
cideswiththesum-capacityofthegenie-aidedAWGNBC.3.)
Notice that for any ǫ > 0 there exists a positive integer
We show that the prelog of the less noisy AWGN BC equals
ℓ (ǫ,σ2,σ2,κ)suchthatourchoice(δ ,q )satisfiesthepower
0 1 2 ℓ ℓ 1, irrespective of the noise variances σ2,σ2,σ2 ,σ2 > 0
constraint (43) for all ℓ>ℓ (ǫ,σ2,σ2,κ). 1 2 W1 W2
0 1 2 and the correlation coefficient ρ [ 1,1].
Moreover, if κ [0, ), then specializing the rates in z ∈ −
∈ ∞ We elaborate on these three steps starting with the first
Proposition 9 to the choices in (77) and (79) proves that
one. The genie-aided AWGN BC is defined as the original
lim CBC,Σ(Pℓ,σ12,σ22,ρz(Pℓ)) 2. (81) AWGN BC without feedback, but with a genie that prior to
ℓ→∞ 12log(1+Pℓ) ≥ transmission reveals the sequences {(Z1,t + W1,t)}nt=1 and
(Z +W ) n tothetransmitterandbothreceivers.Notice
If κ = , then for all sufficiently large ℓ the correlation { 2,t 2,t }t=1
∞ that with this genie information, after each channel use t, the
coefficient ρ (P ) ( 1,1), and by (77) the choice in (78)
z ℓ ∈ − transmitter can locally compute the missing feedback outputs
evaluates to
V and V :
1,t 2,t
(σ σ ρ (P ))2
q2 = 2− 1 z ℓ V =X +(Z +W ), (88)
ℓ sδℓ2(1+δℓ)2σ12(σ22+σ12−2ρz(Pℓ)σ1σ2)(1+|ρz(Pℓ)|) V1,t =Xt+(Z1,t+W1,t). (89)
2,t t 2,t 2,t
(1 ǫ)P
− ℓ (82)Wecanthusconcludethatthesum-capacityofthegenie-aided
·s1−|ρz(P)| AWGN BC is at least as large as the sum-capacity of the
Notice that by (75) original AWGN BC with feedback.
Wenextelaborateonthesecondstep.ThelessnoisyAWGN
lim log(cid:18)δℓ2(1+δℓ)2σ12(σ22+(σσ212−−σ21ρρzz((PPℓℓ))σ)21σ2)(1+|ρz(Pℓ)|)(cid:19) =0, BC is described by the channel law
ℓ→∞ 12log(P) Y1′,t =xt+Z1′,t, (90)
(83)
Y =x +Z , (91)
2′,t t 2′,t
and therefore, specializing the rates in Proposition 9 to the
choice in (77) and (82) proves that where the noise sequences are defined as
ℓl→im∞CBC,Σ(12Plℓo,gσ(112,+σ22P,ℓρ)z(Pℓ)) ≥1+ζ−1. (84) ZZ12′′,,tt ,,ZZ12,,tt−−EE[[ZZ12,,tt||((ZZ11,,tt++WW11,,tt)),,((ZZ22,,tt++WW22,,tt))]],, ((9923))
Combining (84) with (81) establishes (73).
and are of variances
In a similar way we can also prove that when σ2 =σ2
1 6 2 σ2 σ2(1 ρ2)+σ2 σ2
ToPtl→himis∞eCndB,C,iΣt12(sPulof,figσ(c121e,s+σi22fP,iρn)z(tPhe))pr≥oomf tion{(713+) wζ+e1r,e2p}la.ce(ζ8−51) VVaarr(cid:0)ZZ21′′,,tt(cid:1)==σσ2122((σσ1122σ++WW2σσ21WW22σ12211())1((σσ−−2222ρ++2zz)σσ+WW22 22σ))WW2−−21σσσWW21122σσ122222ρρ2z2z,. ((9945))
by ζ+1, (74) by By the(cid:0)follo(cid:1)wing two observations, the sum-capacity of this
lim −log(1−ρz(Pℓ)) =ζ >0, (86) less noisy AWGN BC coincides with the sum-capacity of
ℓ 1log(P) +1 the genie-aided AWGN BC. Firstly, the sum-capacity of
→∞ 2