Optimization of Energy Harvesting MISO Communication System with Feedback

0.31 MB·English

by  Rajeev Gangula

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Optimization of Energy Harvesting MISO
Communication System with Feedback
Rajeev Gangula∗, David Gesbert∗, and Deniz Gündüz §
∗Mobile Communications Department, Eurecom, France
§Dept. of Electrical and Electronic Engineering, Imperial College London, UK
Email: {gangula,gesbert}@eurecom.fr, [email protected]
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Abstract—Optimization of a point-to-point (p2p) multiple- nication systems under different assumptions regarding the
1
inputsingle-output(MISO)communication system isconsidered node’s knowledge about the underlying EH process. Offline
0
when both the transmitter (TX) and the receiver (RX) have
2 optimization framework deals with systems in which non-
energy harvesting (EH) capabilities. The RX is interested in
causal knowledge of the EH process is available. Within this
n feeding back the channel state information (CSI) to the TX to
a helpimprovethetransmissionrate.Theobjectiveistomaximize frame work, optimal transmission schemes are studied for the
J the throughput by a deadline, subject to the EH constraints p2p fading channel [3], broadcast channel [4], [5], [6] and
8 at the TX and the RX. The throughput metric considered is relay channel [7], [8]. See [9] for an extensive overview.
an upper bound on the ergodic rate of the MISO channel To the best of our knowledge, a common aspect of all
] withbeamformingandlimitedfeedback.Feedbackbitallocation prior works on EH communication networks is that the TX
T and transmission policies that maximize the upper bound on
I the ergodic rate are obtained. Tools from majorization theory is assumed to have access to perfect CSI. Knowledge of
. are used to simplify the formulated optimization problems. the CSI at the TX is beneficial in designing the optimal
s
c Optimal policies obtained for the modified problem outperform channel adaptation techniques and the TX filters in multi-
[ the naive scheme in which no intelligent management of energy antenna systems. However, recent studies have demonstrated

is performed.
1 that, although feedback enhances the system performance,
v Index Terms—Energy harvesting, Limited feedback, MISO, feedbackresources,namelypowerandbandwidth,arelimited,
8 Offline optimization. and must be spent wisely [10]. As a result, an important
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question arises: How do the EH constraints affect the design
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I. INTRODUCTION of feedback enabled wireless networks?
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0 Powering up terminals in communication networks by In this paper, we study the optimization of a feedback
. renewable ambient energy reduces the carbon footprint of enabledEHMISOchannel,wherefeedbackisusedtoimprove
1
0 the information and communication technologies, which can the rate through array gain. The system model and the main
5 no longer be neglected with the exponential growth in the assumptions in this paper are given in Section III. In Section
1 number of communication devices. Another advantage of EH IV, we considerthe optimizationof the feedbackpolicyunder
v: technology is that, it increases the autonomy of battery-run EH constraints at the RX, while the TX is assumed to have
i communicationdevices.Intraditionalwirelessnetworksnodes a constant power supply. The motivation is to address the
X
gettheirenergyfromthepowergridbyalwaysorperiodically following: In the case of EH, the available energy at the RX
r
connectingtoit.Whileitiseasytoconnecttheterminalstothe variesovertime.ShouldtheRXfeedbacksamequalityofCSI
a
grid in some networks, in others, such as sensor networks, it at all times? If so, can the CSI feedback quality be improved
cannotbe done once after the deployment.Therefore,in such by using more bandwidth in the low energy scenario? In the
networks a node’s lifetime, and hence, the network lifetime, secondpartofthispaper(SectionV),weassumethatboththe
is constrained by the limited initial energy in the battery. TX and the RX harvest energy. In this case, the transmission
Providing EH capabilities to the communication nodes is an power policy and the feedback policy are coupled, and need
attractive solution to the network lifetime problem [1]. An tobejointlyoptimized.Resultsfrommultivariatemajorization
EH node can scavenge energy from the environment (typical theory are used to devise simple algorithms. We start by
sources are solar, wind, vibration, thermal, etc.) [2]. With EH giving a brief preliminary description of majorization theory
nodesinthenetwork,inprinciple,onecanguaranteeperpetual inSectionII.Numericalresultsarepresentedin SectionVIto
lifetime without the need of replacing batteries. validatetheanalysis.Finally,SectionVIIconcludesthepaper.
However, EH poses a new design challenge as the energy Notation: Boldface letters are used to denote matrices and
sourcesaretypicallysporadicandrandom.Themainchallenge vectors.ThetransposeandconjugatetransposeofmatrixAis
liesinensuringtheQualityofService(QoS)constraintsofthe denoted by AT and AH, respectively. We use d to denote
i,j
network given the random and time varying energy sources. the elementat the i-th row and j-th columnof matrix D, and
Thiscallsfortheintelligentmanagementofvariousparameters |S| to denote the cardinality of the set S. The set of integers
involved in a communication system. frommton,m<n,isrepresentedby[m:n].Thealgorithm
Recently, a significant number of papers have appeared withname“Algo”is representedas[outputarguments]=Algo
studying the optimal transmission schemes for EH commu- (input arguments). A circularly-symmetric complex Gaussian
2
distributedrandomvariableη withzeromeanandvarianceσ2 Harvested Harvested
is denoted by η ∼CN(0,σ2). Energy Energy

II. PRELIMINARIES Energy Energy
buffer
buffer
Inthissection,thebasicnotionofmajorizationisintroduced
and some important inequalities on convex functions that are Tx (cid:1829)(cid:1860)(cid:1853)(cid:1866)(cid:1866)(cid:1857)(cid:1864)(cid:2190)(cid:3009) Rx
used in this work are stated. The readers are referred to
Feedback channel
[11], [12] for a complete reference. We start by stating the
C
Edmundson-Madansky’sinequality.
(cid:1860)(cid:1853)(cid:1866)(cid:1866)(cid:1857)(cid:1864)(cid:1857)(cid:1871)(cid:1872)(cid:1861)(cid:1865)(cid:1853)(cid:1872)(cid:1857)(cid:2190)(cid:3553)
Theorem1:[11]Iff isaconvexfunctionandxisarandom Figure 1. MISO channel with feedback, where both the TX and the RX
variable with values in an interval [a,b], then harvestandstoreambient energy.
b−µ µ−a
E[f(x)]≤ f(a)+ f(b),
b−a b−a , EH Interval , ,
(cid:2202) (cid:2200) (cid:2202) (cid:2200) (cid:2202) (cid:2200)
where µ is the mean of x. (cid:2187)(cid:2778)(cid:2187)(cid:2778) (cid:2187)(cid:2779)(cid:2187)(cid:2779) (cid:2187)(cid:2167)(cid:2187)(cid:2167)
Majorization theory formalizes the notion that the compo- 1 2 L 1 -th EH interval L
nentsof a vectorx are “less spread out”than the components (cid:2167)
of a vector y. (cid:594)(cid:2869)T (cid:594)(cid:2869)T (cid:594)(cid:2869) (cid:594)(cid:3012) (cid:594)(cid:3012)
Definition 1: Let x = [x ,...,x ],y = [y ,...,y ],
1 n 1 n Figure2. Energyharvesting timeframestructure.
x,y ∈ Rn and let x denote the i-th largest component of
(i)
x. Then x is said to be majorized by y, denotedbyx(cid:22)y, if
A. Energy Harvesting Model
l l
x ≤ y , ∀l ∈[1:n−1] ThetotalobservationtimeisdividedintoK equallengthEH
(i) (i)
intervals.Atthebeginningofthek-thEHinterval,k∈[1:K],
i=1 i=1
Xn Xn energy packets of size et,er units arrive at the TX and the
k k
x(i) = y(i). RX,respectively.Ateachnode,thisenergyisfirststoredinan
i=1 i=1 infinite size energy buffer, and used only for communication
X X
Definition2:[12,2.A.1] Ann×nmatrixDwithelements purposes, i.e., TX sending data, and the RX feeding back the
d is doubly stochastic if CSI.Weassumethatallet,er’sareknowninadvancebyboth
i,j k k
terminals. This model is suitable for an EH system in which
d ≥0, ∀i,j ∈[1:n],
i,j the time-varyingharvestedenergycanbe accuratelypredicted
n n
[9].
d =1, ∀j ∈[1:n] and d =1, ∀i∈[1:n].
i,j i,j
i=1 j=1
X X
B. Communication System Model
Theorem2:[12,4.A.1,4.B.1]Forx,y ∈Rn,thefollowing
conditions are equivalent: Each EH interval consists of L data frames, each of length
T channeluses.Weassumeablockfadingchannelmodel.The
• x(cid:22)y.
• x=yD for some doubly stochastic matrix D. channel is constant during T channel uses of each frame, but
• For all continuous concave functions g : R → R, changes in an independent and identically distributed (i.i.d.)
n g(x )≥ n g(y ). fashionfromoneframetoanother.Thetimeframestructureis
i=1 i i=1 i shown in Fig. 2. The TX has M >1 antennas, while the RX
Definition 3: [12, 15.A.2] Let X and Y be m×n real
P P has a single antenna. The received signal in a given channel
matrices. Then X is said to be majorized by Y, written X(cid:22)
use is given by
Y,ifX=YD,wherethen×nmatrixDisdoublystochastic.
y =hHws+η, (1)
Theorem 3: [12, 15.A.4] Let X and Y be m × n real
matrices. Then, X(cid:22)Y if and only if whereh∈CM×1 representsthevectorofchannelcoefficients
n n fromTX to the RX with i.i.d. CN(0,1) elements,w ∈CM×1
g(xc)≥ g(yc), denotesthebeamformingvector,theinputsymbolmaximizing
i i
i=1 i=1 the achievable ergodic rate in the k-th EH interval is s ∼
X X
forallcontinuousconcavefunctionsg :Rm →R;herexc and CN(0,pk), and η ∼CN(0,1) represents the noise at the RX.
i
yc denote the i-th column vector of X and Y, respectively.
i
C. Feedback Model
III. SYSTEMMODEL We assume that the RX perfectly estimates the channel
Weconsiderap2pMISOfadingchannelasshowninFig.1, state at the beginning of each data frame, and feeds back
where both the TX and the RX harvest energy from the the quantized CSI to the TX within the same frame. In the
environment.Eachnodeisequippedwithanindividualenergy k-th EH interval, the frame structure is as follows: The RX
buffer, i.e., a rechargeable battery, that can store the locally in τ channel uses sends the CSI through a feedback channel
k
harvested energy. (uplink)whichismodeledasanadditivewhiteGaussiannoise
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(AWGN) channel. In the remaining T − τ channel uses, D. Optimization Problem
k
TX sends data to the RX (downlink) exploiting the obtained
Theproblemof maximizingthe sum throughputbythe end
CSI.ThefeedbackmodelrepresentstheTime-DivisionDuplex of the K-th EH interval can be formulated as
(TDD) system in which uplink and downlink use the same
band in a time-sharing fashion, but the communication de-
K
vicesare notself-calibrated,andhence,inducenon-reciprocal max R (5a)
k
effects [13], [14]. In the above model, although the feedback pk,qk,τk
k=1
X
overhead incurs a cost in the downlink bandwidth, a similar
l l
trade-off in the resource allocation between the CSI feedback s.t. L q ≤ er, ∀l ∈[1:K], (5b)
i i
qualityanduplinkdataratealsoariseinaFrequency-Division
i=1 i=1
X X
Duplex (FDD) system [14]. Hence, the analytical results l l
obtained in this paper are applicable in general settings, and LT p ≤ et, ∀l∈[1:K], (5c)
i i
forinstance,canbeusedto addressthetrade-offbetweenCSI i=1 i=1
X X
quality and effective data rate in an FDD system. τ ∈[0,T), p ≥0, and q ≥0,∀k ∈[1:K].
k k k
(5d)
In the k-th EH interval, quantization of the channel state
is performed using a codebook C known at both the TX The constraints (5b) and (5c) guarantee the energy neutrality
k
and RX. The receiver uses Random Vector Quantization of the system, i.e., at each node, energyconsumedcan not be
(RVQ).ThecodebookconsistsofM-dimensionalunitvectors more than the energy harvested till that time. Also note that
Ck , {f1,...,f2bk}, where bk is the number of bits used τk impacts the achievable rate Rk in each EH interval.
for quantization. The RX chooses the beamforming vector Comingupwithsimplealgorithmstosolvetheoptimization
according to w = arg max |h˜Hf|2, where h˜ , h . problemisdesirableinEHnetworksasthenodesmaynothave
k ||h||
f∈Ck the computational and energy resources for running complex
We assume that the length of the EH interval is very large
optimizationalgorithms.However,the ergodicrate expression
comparedtothechannelcoherencetime(i.e.,Lisverylarge).
used in the above optimization problem is not in closed form
Asaresult,theachievableergodicrateinthek-thEHinterval
and offers little insight into the convexity of the problem
is given by
which is required to reduce the complexity of optimization.
This motivates the use of convex bounds on (4) as the
objective function in the following optimization problems.
τ p Solving these modified problemsprovidesan upper bound on
R = 1− k E log 1+ k khk2ν ,
k T ||h||2,νk" 2 1− τTk k!# the throughput. Since the constraints in the original and the
(cid:16) (cid:17) (2) modifiedoptimizationproblemsare the same, the solutionfor
whereν =|h˜Hw |2. Note thatν and(cid:0)||h||2 a(cid:1)reindependent the modified problem is also feasible in the original problem,
k k k
[15]. By using the AWGN feedback channel model, the and if used in evaluating the exact rate expression in (4), we
number of feedback bits b can be related to the energy used obtain a lower boundon the throughput.In some settings, we
k
by the RX, q , and the numberof channeluses τ as follows: show that the bounds used are very close to the ergodic rate.
k k
Before tackling the above problem, first, we consider a
special case in which only the RX harvests energy. Later, the
general case with both the TX and the RX harvesting energy
q
bk =τklog2 1+ τ σk2 , (3) is studied.
(cid:18) k (cid:19)
IV. EH RECEIVER
In this setting, the RX harvests energy from the environ-
where σ2 is the noise variance in the uplink. For analytical
ment, whereas the TX is connected to the power grid so that
tractability, we neglect the practical constraint that b should
k
it has a fixed power supply at all times. Therefore, there are
be an integer. Using the ergodic rate expression given in [15,
Equation (27)] and (3), the ergodic rate R , R(p ,q ,τ ) no EH constraints at the TX, and constraints (5c) can be
k k k k
ignored. However, there is now a constraint on the average
is found to be
transmissionpowerat eachdataframeof thek-thEH interval
i.e., p ≤p,∀k. The expected value ν is given by [15], [16]
k k
Rk = 1− τTk log2e eρkM−1El+1(ρk)− E[νk]=1−2bkβ 2bk,MM−1 , (6)
(cid:18) (cid:19)
(cid:16) (cid:17) Xl=0 where β(x,y) denotes the beta function. Using the quantiza-
1
1−(1−νk)M−1 Nk Me(cid:16)ρνkk(cid:17)EM+1 ρk dνk tion error bound in [16, Lemma 6], (6) can be bounded as1
ν ν 
νkZ=0 (cid:16) (cid:17) k (cid:18) k(cid:19) (4) E[νk]≤νku ,1− MM−1 2M−b−k1. (7)
w∞heree−xρtkx−=nd(cid:16)t1i−spkτtTkhe(cid:17),n-Nthko=rde(cid:16)r1ex+poτnkqkσen2t(cid:17)iaτkl,inatnegdraEl.n(x) , zat1ioTnhiosfbaonunisdotirsopuinciavlelrysadlisitnribthuetesdenvsee(cid:18)cttohra,tniottanpep(cid:19)cleiesssatroilyanlyimbitke-dbittoqRuVanQti.-
1
R
4
Applying Jensen’s inequality on (2), substituting (7) and (3), where p is the power constraint at the transmitter.
and using the fact that Ekhk2 = M, an upper bound on the As the objective function is monotonic in q and p ,
k k
ergodic rate Ru ,Ru(p ,q ,τ ) is obtained as the constraint in (15b) must be satisfied with equality for
k k k k
l=K, and the first constraint in (15c) must be satisfied with
−τk
Ru =t log 1+ pkM 1− M −1 1+ qk M−1 , equality, i.e., pk = p,∀k; otherwise, we can always increase
k k 2 tk  M (cid:18) τkσ2(cid:19)  qK,pk, and hence, the objective function, without violating
anyconstraints.Nowit remainsto optimizeoverthevariables
(8)
  
where tk , 1− τTk . qkTahnedfτeka.sible set is represented as
We now illustrate the tightness of the upper bound. Ap-
(cid:0) (cid:1)
plying the Jensen’s inequality on (2), Rku−Rk can be lower F={q,τ|qk,τk satisfy (15b), (15d) ∀k}, (16)
bounded as
where q = [q ,...,q ] and τ = [τ ,...,τ ]. To show that
p 1 K 1 K
Ru−R ≥t log 1+ kMνu − theaboveproblemisaconvexoptimizationproblem,wemake
k k k 2 t k
(cid:18) k (cid:19) (9) use of the following lemma.
tkE||h||2log2(cid:18)1+ ptkk khk2E[νk](cid:19). andLemg(mya,z1): I:f tRh2+e fu→nctiRon+fis(xc,ot)nc:aRve2+a→ndRm+oinsoctoonniccaavlley,
Since(2)isaconcavefunctionofν andν ∈[0,1],applying increasing in each argument, then the function h(x,y,t) =
k k
Theorem 1 on (2), we have 1− t g y ,f(x,t) isconcave∀(x,y)∈R2,t∈[0,T).
T 1−t 1−t +
T T
Rk ≥tkE||h||2log2 1+ ptk khk2 E[νk] (10) (cid:0) Pro(cid:1)of:(cid:16) The proof(cid:17)is similar to that of showing the
(cid:18) k (cid:19) perspective of a concave function is concave. See Appendix.
Now using (10), Ru−R can be upper bounded as
k k
Proposition 1: The objective function of the optimization
p
Rku−Rk ≤tklog2 1+ tkMνku − problem (15) is concave.
(cid:18) k (cid:19) (11) Proof: See Appendix.
p
t E log 1+ k khk2 E[ν ] Since the objective function in (15) is concave and the
k ||h||2 2 t k
(cid:18) k (cid:19) constraintsarelinear,ithasauniquemaximizer[17].Usingthe
Since both lim νu = 1 and lim E[ν ] = 1 [15], concavity of the objective function, we show that the optimal
bk→∞ k bk→∞ k
and using (9) and (11), we have, energyallocationvectoris themostmajorizedfeasible energy
vector.
∆R , lim Ru−R =t E log tk+pkM . Proposition 2: The global optimum of (15) is obtained at
k bk→∞ k k k ||h||2 2 tk+pkkhk2! (q∗,τ∗), where q∗ (cid:22) q,∀(q,τ) ∈ F, and τk∗ is the solution
(12) of the following equation
Further, for all feasible τ , in the low power regime,
k ∂Ru
k| =0, ∀k ∈[1:K]. (17)
lim ∆Rk =0, (13) ∂τk (qk∗,τk∗)
pk→0
Proof: Consider the following equivalent form of (15),
and in the high power regime,
where the optimization is performed in two steps.
lim ∆R =t log M −E log ||h||2
pk→∞ k k 2 ||h||2 2 (14) max U˜(q) s.t. ∀(q,τ)∈F, (18)
q
≤log(cid:0) M −E log ||h||2. (cid:1)
2 ||h||2 2
where U˜(q) is obtained by
From the aboveanalysis, it can be seen thatwhenthe RX has
enough harvested energy to send large number of feedback U˜(q)=max U(q,τ) s.t. ∀(q,τ)∈F. (19)
bits, in the low power regime the bound is tight, and in the τ
high power regime the difference is bounded by a constant. Since U is a concave function over the convex set F, the
For example, it is 0.1958 for M = 4, and also note that function U˜(q) is concave, where the domain of U˜ is the set
lim log M −E log ||h||2 =0. F˜ ={q|(q,τ)∈F}[17,3.2.5].U= K Ru iscontinuous,
M→∞ 2 ||h||2 2 k=1 k
Using (8) as the objective function, the modified optimiza- differentiable and concave in τ ∈ [0,T). Furthermore, for
k
tion problem can be written as follows, givenq ,Ru approacheslog (1+p)Pand0,asτ approaches
k k 2 k
0andT,respectively.Therefore,theuniquemaximizerof(19)
K
max U= Ru (15a) lies in [0,T), and it is obtained at
k
pk,qk,τk k=1 ∂U ∂Ru
s.t. L l Xq ≤ l er,∀l∈[1:K], (15b) ∂τk|τk∗ = ∂τkk|τk∗ =0, ∀k ∈[1:K]. (20)
i i From above, as τ∗ is only a function of q ,
i=1 i=1 k k
X X
pk ≤p, and pk ≥0, ∀k ∈[1:K], (15c) K
U˜(q)= R˜u (21)
τk ∈[0,T), and qk ≥0, ∀k ∈[1:K], (15d) k
k=1
X
5
Input : EH intervals K; Harvested energy {e }
i
7
SNR 10 dB Output: Energy allocation o⋆, Energy band indices
6 SNR 20 dB S= B0,B1,...B|S|
*τ)Number of channel uses (k2345 /Bfo/0r:fi=io=rn0i(k1;it)=:ioKa⋆lK(cid:8)l=di:ozP−a1kjt=k:−iB(iBo−Bin1−i+−111ej+,(cid:9)1l)∈do{Bi−1+1,...,k}
if l o⋆ ≤ l e ,l=1,...,K then
1 i=1 i i=1 i
B =k;
i
0 PSave {o⋆,·P·· ,o⋆}
0 10 20 30 40 50 60 70 80 1 k
Energy allocated in the k−th EH interval (q*k) break;
end
Figure3. Optimalnumberofchannelusesforsendingfeedback. end
if B ==K then
i
break;
where R˜u , R˜u(q ) = Ru(q ,τ∗(q )). Using (21) and
k k k k k end
Theorem 2, U˜(q∗) ≥ U˜(q),∀q ∈ F˜. Finding the optimal
end
energyallocationvectorq∗ undertheEHconstraintsturnsout
Algorithm1:OptimalEnergyAllocation(OEA)algorithm
beawellknownproblem,andthealgorithmtoconstructq∗ is
giveninvariousworks[18]–[20].Theproofthatthealgorithm
constructs the most majorized feasible energy vector is given
in [20]. Since the optimal energy allocation vector is q∗, the to send feedback in this interval. Therefore, without loss of
optimal τ∗ is obtained by (17). optimalitywe onlyconsiderEH profileswhereet >0.Other-
1
A brief description of the algorithm tailored to this work is wise,ifthereisanEHprofilesuchthatet =0,k∈[1:m−1],
k
givennext, while the details can be foundin [18]–[20]. There then p = 0,k ∈ [1 : m−1] due to the constraints in (5c).
k
is no closed form expression for the solution of (17), hence In these intervals the RX simply accumulates the harvested
weresorttonumericalmethodstoobtainτ∗.Fig.3showsthe energy, and without loss of optimality we can have a new
behavior of τ∗ as a function of the allocated energy q∗. EH profile with e˜t = et ,∀i ∈ [1 : K −m+1], and
k k 1 i+m−1
e˜r = m er and e˜r = er ,∀i ∈ [2 : K −m+1] for
1 k=1 k i i+m−1
further analysis.
A. Optimal Energy Allocation P
The ergodic rate upper bound in (8) is not concave, but
From Definition 1, we can see that the components of
concave in each variable given the other variables are fixed.
the most majorized energy vector are "less spread out" than
To obtain a simple algorithm and an upper bound on the
any other feasible energy vector. Therefore, the algorithm
throughput, we follow a similar approach as in the previous
essentially tries to make the energy vector as equalized as
section,anduseaconcaveupperboundon(8)astheobjective
possible. This is done by spreading the energy to future
function for throughput optimization.
intervals. However, note that the energy arriving in later
This bound is obtained by using a hypothetical system in
intervals cannot be spread to earlier intervals due to the EH
which the transmission power is 1 watt higher than the actual
constraints.TheOptimalEnergyAllocation(OEA) algorithm,
transmission power of the system, which is p /t . Plugging
givenin Algorithm1, dividesthe EH intervalsinto |S| energy k k
this into the upper bound in (8), a new upper bound Rub ,
bands whose indices form the set S = B0,B1,...B|S| , Rub(p ,q ,τ ) on the ergodic rate is obtained: k
where Bi < Bj,∀i < j, B0 = 0,(cid:8)and B|S| = K(cid:9). k k k
The i-th energy band contains the EH intervals with indices
p f
k ∈ [Bi−1+1 : Bi]. Moreover, the optimal allocated energy Rkub =tklog2 1+ 1+ tk tk , (22)
values in each EH interval belonging to the i-th energy band (cid:18) (cid:18) k(cid:19) k(cid:19)
are equal,anddenotedbyq∗ . The energyvectorq∗ obtained
by [q∗,Sr]=OEA(BKi,{eri/(Lei)r}), has the following properties: where tk ,1− τTk and fk ,M −(M −1) 1+ τkqkσ2 M−τ−k1.
((PP12)) qTk∗he=eqn(∗tir)ie=sqP∗Ll(=BaBir−ie−B1si+t−r1i1c)tlly, m∀kon∈ot[oBnii−c1,i+.e.1,q:∗Bi]<. q∗ < Wthee lnoowwainlldushtriagthe tphoewteigrhrtengeismseosf. tFhoeruapllpefrea(cid:16)bsoibulnedτikn,p((cid:17)2k2)anind
(i) (1) (2)
...<q(∗|S|). qk, we can see that 0<tk ≤1 and 1≤fk ≤M. Consider
t2 +t f +p f
V. EH TRANSMITTERANDRECEIVER Rkub−Rku =tklog2 k t k+kp f k k −tklog2(tk)
Inthissection,weconsiderthegeneralcasewhereboththe (cid:18) k k k (cid:19) (23)
TXandtheRX harvestenergy.NotethatiftheTXissilentin Note that (23) is decreasing in p for fixed τ and q . Since
k k k
the k-th interval,i.e., p =0, thereis no incentivefor the RX τ ,f are bounded, for fixed τ and q , in the low power
k k k k k
6
regime where R˜ub ,R˜ub(p ,q )=Rub(p ,q ,τ∗(p ,q )).
k k k k k k k k
lim Rub−Ru =t log 1+ fk In order to get an insight on how the optimal solution of
pk→0 k k k 2(cid:18) tk(cid:19) (24) (27) may look like, consider a simple scenario in which there
≤log (1+M), isonlya sum powerconstraintatthe TX andthe RX, i.e.,the
2
constraintsin(26b),(26c)hastobesatisfiedforonlyl=K.In
and in the high power regime,
thiscase,byJensen’sinequality,the uniformpowerallocation
lim Rkub−Rku =−tklog2(tk)≤0.5. (25) at the TX and the RX is optimal2. However, due to the EH
pk→∞ constraints, this may not be feasible. Using this intuition, we
Fromtheaboveanalysis,itcanbeseenthat,(23) decreasesas
can see that the optimal policy tries to equalize the powersas
the power is increased, and it is boundedby a constant in the
much as possible, while satisfying the EH constraints. Next,
high power regime. By using (22), the modified throughput
we consider the case in which the EH profiles at the TX and
maximization problem is formulated as
the RX aresimilar, andshow thatthe optimizationproblemis
K considerably simplified.
max U = Rub (26a)
1 k
pk,qk,τk
k=1
X A. Similar EH Profiles
l l
s.t. L q ≤ er,∀l ∈[1:K], (26b) The EH profiles are similar in the sense that the most
i i
i=1 i=1 majorized feasible vectors obtained from the EH profiles of
X X
l l the TX and RX, p∗ and q∗, have the same structure, i.e., if
LT pi ≤ eti,∀l ∈[1:K], (26c) p∗i = c1,∀i ∈ [m : n], then qi∗ = c2,∀i ∈ [m : n] for some
i=1 i=1 constants c ,c ≥0. We now give a formal definition.
X X 1 2
τk ∈[0,T), pk ≥0, qk ≥0, and ∀k ∈[1:K]. Definition 4: By using the OEA algorithm, let [q∗,Sr] =
(26d) OEA(K,{er/L}) and [p∗,S ] = OEA(K,{et/LT}). EH
i t i
Since the objective function is monotonic in qk and pk, the profilesattheTXandtheRXaresaidtobesimilarifSr =St.
constraints in (26b) and (26c) must be satisfied with equality FromSection II, we can see thatthedefinitionof majoriza-
for l = K, otherwise, we can always increase q ,p , and tion for the vector case does not directly extend to the matrix
K K
hencetheobjectivefunction,withoutviolatinganyconstraints. case. If OEA algorithm is used at the TX and RX separately,
The feasible set is represented as we get the most individually majorized power vectors, which
in general may not be the optimal solution of (27). However,
J={(p,q,τ)|p ,q ,τ satisfy (26b), (26c) and (26d) ∀k},
k k k we now show that if the EH profiles are similar, the above
where p = [p ,...,p ], q = [q ,...,q ] and τ = mentioned approach is indeed optimal.
1 K 1 K
[τ ,...,τ ]. Proposition 4: If the EH profiles at the TX and the RX are
1 K
Proposition 3: The objective function in the optimization similarthen(q∗,p∗,τ∗)istheglobaloptimumof(26),where
problem (26) is concave. q∗ (cid:22)q,p∗ (cid:22)p, ∀(q,p,τ)∈J, and τ∗ is the solution of
k
Proof: See Appendix.
∂Rub
Since the objective function in (26) is concave and the con- k | =0, ∀k ∈[1:K]. (31)
straints are linear, it has a unique maximizer [17]. Consider ∂τk (p∗k,qk∗,τk∗)
the following equivalentform of (26), where the optimization Proof: See Appendix.
is performed in two steps.
max U˜ (p,q) s.t. ∀(p,q,τ)∈J, (27) B. Different EH Profiles
1
p,q
Unfortunately,wecouldnotfindasimplealgorithmtosolve
where U˜1(p,q) is obtained by (26)ina generalsettingwheretheEHprofilesarenotsimilar.
U˜ (p,q)=max U (p,q,τ) s.t. ∀(p,q,τ)∈J. (28) In(30),ifonevariableisfixed,optimizingovertheothervari-
1 1
τ able has a directional or staircase water-filling interpretation
SinceU isaconcavefunctionovertheconvexsetJ,thefunc- [3], [18], however, the difficulty lies in the fact that there is
1
tion U˜ is concave with domain J˜ = {(p,q)|(p,q,τ)∈J} noclosed formexpressionfor R˜ub. Nonetheless,basedonthe
1 k
[17, 3.2.5]. U = K Rub is continuous, differentiable and convexity of the objective function, some properties of the
1 k=1 k
concaveinτ ∈[0,T).Furthermore,forgivenp andq ,Rub optimal solution are given below.
k P k k k
approaches log (2+p ) and 0, as τ approaches 0 and T, Lemma2:Undertheoptimalpolicy,thetransmissionpower
2 k k
respectively. Therefore, the unique maximizer of (28), τ∗,∀k p , and the energy used to send the feedback q are non-
k k k
lies in [0,T), and it is obtained as decreasing in k,∀k ∈[1:K].
∂U ∂Rub Lemma 3: Under the optimal policy, at the time instants at
1|τ∗ = k |τ∗ =0, ∀k ∈[1:K]. (29) which Rub changes, the energy buffer of either the TX or the
∂τk k ∂τk k RX is emptied.
As τ∗ is only a function of q and p , (27) can be written as
k k k The proofs of the above lemmas are given in Appendix.
K
max U˜ = R˜ub s.t. ∀k,(p ,q )∈J˜, (30) 2 In this section, with slight abuse of terminology we use the terms RX
1 k k k
pk,qk powerandRXenergyinterchangeably.
k=1
X
7
30 6
upper bound
5.5 OEA
25
greedy
5
OEA with bit rounding
HPN [dB]1250 ate [bits/s/Hz]34..455 greedy with bit rounding
10 ge r 3
a
er
5 Av2.5
2
0
5 10 15 20
EH interval index (time of day) [hours] 1.5
1
Figure4. Modelforasolarenergyharvesting profile. −2 0 2 4 6 8 10 12 14
SNR [dB]
Figure5. ErgodicratewithonlyanEHRX,andM =4.
VI. NUMERICAL RESULTS
We start by considering the case in which the RX harvests
12
energy,whiletheTXhasaconstantpowersupply.Weassume
OEA
that the RX is equipped with a solar EH device. Following Greedy
10 OEA with
[21], solar irradiance data is taken from the database reported bit rounding
in [22].EachEH intervalis ofduration∆=1 hour,T =200 Greedy with
ms,resultinginL=18000frames.Theharvestedpowerfrom bits] 8 bit rounding
theirradiancedatacanbecalculatedas,pharv =I[Watt/m2]× ack [ 6
Area[m2]×ρ, where ρ is the efficiency of the harvester. A edb
e
hypotheticalsolar panel of variablearea is assumed. The area F 4
ofthepanelisadjustedsuchthatwehavetheEHprofileshown
in Fig. 4 at the RX. In Fig. 4, the harvested power to noise 2
ratio (HPN) in each EH interval erk is shown.
∆σ2 0
Using this EH profile, throughput of different feedback 0 5 10 15 20
EH interval index (Time of Day) [hrs]
policies is shown in Fig. 5. In Fig. 5, OEA represents the
proposed policy in which the energy vector is obtained by Figure6. Feedback loadatdownlink SNRof10dB,M =4.
using the OEA algorithm, and then the optimal time span
of feedback τ∗ is obtained by solving (20). In the greedy
k
scheme,theconsumedenergyisequaltotheharvestedenergy the allocated energy is equal to the harvested energy in that
in that interval, i.e., q = er/L, and then optimization is interval, i.e., at the TX p = et/LT, at the RX q = er/L,
k k k k k k
performed only over τ , given q . The performance of the and then optimization is performed only over τ , given p
k k k k
above policies when the feedback bits are rounded to the and q . The difference in throughputbetween the greedy and
k
largest previous integer is also shown. We can see that the OEA is small when the average HPN is low, and it increases
proposed approach outperforms the greedy policy by 1.6 dB with the HPN. In contrast to the OEA scheme, using the
atarateof4bits/s/Hz.Alsotheratelossduetobitroundingis greedyapproachwith the solar EH profile resultsin some EH
negligible. In Fig. 6, feedback bit allocation is shown for the intervals being allocated zero energy, and therefore does not
abovementionedpoliciesforadownlinkSNRof10dB.From scale by increasing the harvester area. This particularly hurts
Fig. 6, we can see that with the proposed strategy, feedback the greedypolicy’sthroughputin the highHPN regimeas the
bit allocation is equalized as much as possible. multiplexing gain (pre-log factor) is reduced.
We now consider the case in which both the TX and the Finally, we consider a case with non-similar EH profiles,
RX harvest energy, with similar EH profiles. The same EH where the EH profiles are generated independently at the
profileinFig.4isseparatelyusedatboththeRX andtheTX, TX and the RX, and they are i.i.d. with exponential distri-
hence the EH profiles are similar. In Fig. 7, the throughput bution. EH profiles are verified not to be similar according
of differentschemes is shown at variousmean HPN values at to Definition 4. Similarly to Fig. 7, in Fig. 8, the mean
the TX. The mean HPN at the TX is varied by increasing the HPN at the TX is varied by multiplying the EH profile by
harvester area at the TX, i.e., the EH profile is multiplied by a constant, while keeping the same shape. Since we could
a positive number (area), while keeping the same shape and not find a simple algorithm in this case, CVX solver is used
efficiency. In Fig. 7, OEA represents the proposed policy in to solve the optimization problem [17], and is denoted as
which the energy vector at the TX and the RX is obtained by CVX in Fig. 8. As we can see, the heuristic of using the
using the OEA algorithm, and then the optimal time span of OEA approach performs quite well even in the non-similar
feedbackτ∗ isobtainedbysolving(29).Inthegreedyscheme, EH profile scenario. The energy allocation at the TX and the
k
8
Energy allocation at the RX
9 2
CVX
ec/Hz]678 Energy [linear]01..155 OEA
s
s/
bit5 0
ate [4 0 2 4 6 EH8 interval i1n0dex 12 14 16 18
e r Energy allocation at the TX
Averag3 upper bound 235000
12 OgrEeAedy Energy [linear]112050000
−05 0 5 10 15 20 25 50
Average HPN per data frame at the TX [dB] 0
0 2 4 6 8 10 12 14 16 18
EH interval index
Figure7. ErgodicrateforsimilarEHprofiles,M =4.
Figure9. Energyallocation attheTXandtheRX,M =4.
8
remarkthattheresultobtainedinProposition4isgeneral,and,
upper bound
7 CVX for example, it can be used in a network setting in which a
OEA
concaveutilityistobemaximizedinthepresenceofEHnodes
z]6 greedy
H withsimilarharvestingprofilesandinfinitesizeenergybuffers.
c/
se5 Numerical results show that the proposed policies not only
s/
bit outperform the greedy policy, but also achieve performances
e [4
at very close to the theoretical upper bound. Our work sheds
e r
ag3 lightonthedesignoffeedback-enabledmulti-antennasystems
er
Av2 when the nodes depend on EH devices for their energy.
1 APPENDIX
A. Proof of Lemma 1
0
−5 Averag0e HPN per dat5a frame at the 1T0X [dB] 15 Let X1 =[x1 y1 t1]T,X2 =[x2 y2 t2]T, we have
h(λX +(1−λ)X )
1 2
Figure8. Ergodicratefornon-similarEHprofiles,M =4.
λy +(1−λ)y f x,t
1 2
=Θg ,
Θ Θ
(cid:0) (cid:1)!
RX are shown in Fig. 9 for the above mentioned policies at
(a) λy +(1−λ)y λf +(1−λ)f
an average per frame HPN of 0.5 dB at the TX. Different ≥ Θg 1 2, 1 2
from Fig. 7, in Fig. 8 the rate scaling with average HPNs is (cid:18) Θ Θ (cid:19) (32)
sameforboththegreedyandtheOEApolicies.Forthegreedy Θ1y1 Θ2y2 Θ1f1 Θ2f2
=Θg + , +
policy, the allocated energy in an EH interval scales with the Θα Θα Θα Θα
(cid:18) 1 2 1 2 (cid:19)
increasing mean HPN, in contrast to the solar EH profile, for (b) y f y f
1 1 2 2
which the allocated energy is zero in some intervals. ≥ Θ1g , +Θ2g ,
α α α α
(cid:18) 1 1(cid:19) (cid:18) 2 2(cid:19)
=λh(X )+(1−λ)h(X ),
1 2
VII. CONCLUSION
where x , λx + (1−λ)x , t , λt + (1−λ)t ,
1 2 1 2
In contrastto the existingliteratureon the designof energy f , f(x ,t ), f , f(x ,t ), Θ , λ 1− t1 and
1 1 1 2 2 2 1 T
harvesting communication systems, we have assumed in this Θ , (1−λ) 1− t2 ,Θ = Θ + Θ , α , 1− t1 ,
paper that the perfect channel state information is available α2, 1− t2 . Here T 1 2 1 (cid:0) (cid:1) T
only at the receiver side; and we have studied the problem 2 T (cid:0) (cid:1) (cid:0) (cid:1)
(a) followsfromthefactthatf(x,t)isconcave,andg(y,z)
of CSI feedback design in a p2p MISO channel under EH (cid:0) (cid:1)
is monotonically increasing in each argument,
constraintsatboththeTXandtheRX.Sincetheexactexpres-
(b) follows from the fact that Θ1 + Θ2 = 1, and g(y,z) is
sions of throughput are complicated, concave upper bounds Θ Θ
concave.
have been used in the optimization problems. We have first
consideredthecaseinwhichonlytheRXharvestsenergy,and
B. Proof of Proposition 1
optimizedthefeedbackpolicyunderEHconstraints.Later,the
Reproducingtheergodicrateboundin(8)withp =P,∀k,
generalcase,inwhichboththeTXandtheRXharvestenergy, k
we have
is analyzed. We have shown that, if EH profiles are similar, Pf
Ru(q ,τ )=t log 1+ k , (33)
the optimization problem can be considerably simplified. We k k k 2 t
(cid:18) k (cid:19)
9
where tk , 1− τTk,fk , (M −(M −1)(1+ τkqkσ2)M−τ−k1). whereBi’saretheenergybandindicesasexplainedinSection
Since b in (3) is concave in q and τ , it can be easily seen IV-A.
k k k
that 2−Mb−k1 = 1+ τkqkσ2 M−τ−k1 is convex, and hence, fk is getApplying(39) fori=1,andrememberingthatB0 =0,we
concave.Using L(cid:16)emma1 w(cid:17)ith g(y,z)=log2(1+z)andfk, B1 B1 K
we can see that Ru is concave. Since the objective function qˆ = q∗d ≤V . (40)
k j i i,j 1
in (15) is the summation of Ru’s, it is also concave.
k Xj=1 Xj=1Xi=1
By (P1) and (P2) in Section IV-A, q∗ =q∗ +L , where
i (1) i
C. Proof of Proposition 3
L =0 ∀i∈[1:B ],
First,weshowthatg(y,z)=log (1+(1+y)z),(y,z)∈ i 1 (41)
2
R2 is concavefor y ≥0,z ≥1. The Hessian of g is givenby Li >0 ∀i∈[B1+1:K].
+
1 −z2 1 From (40) and (41)
J= , (34)
β 1 −(1+y)2 B1 K B1 K
(cid:18) (cid:19) q∗ d + L d ≤V . (42)
where β = log 2(1+(1+y)z)2 > 0. Consider uTJu = (1) i,j i i,j 1
e Xj=1Xi=1 Xj=1i=XB1+1
−β1 a2z2+b2(1+y)2−2ab , where u = [a b]T ∈ Using the fact that D is doubly stochastic and by (P1),
R2.(cid:16)It can be easily seen t(cid:17)hat uTJu ≤ 0 for ab ≤ B q∗ =V , and we have
0. For ab > 0, since z(1+y) ≥ 1, uTJu = 1 (1) 1
−1 (az−b(1+y))2+2ab(z(1+y)−1) ≤ 0. As Hes- B1 K
β L d ≤0. (43)
sianhis negativesemidefinite, g(y,z)is concaive.Reproducing i i,j
the ergodic rate bound in (22), we have Xj=1i=XB1+1
From (41) and (43), we get
p f
Rub =t log 1+ 1+ k k , (35)
k k 2(cid:18) (cid:18) tk(cid:19) tk(cid:19) di,j =0, ∀i∈[B1+1:K], ∀j ∈[1:B1]. (44)
where tk and fk are as defined before. As D is doubly stochastic, using (P1) and (44),
By following the similar steps in Proposition 1, f can be
k
shown to be concave. Using Lemma 1 with g(y,z) and f , B1 B1
k qˆ = q∗ d =q∗ =q∗,∀j ∈[1:B ]. (45)
we can see that Rub is concave. Since the objective function j (1) i,j (1) j 1
k
in (26) is the summation of Rub’s, it is also concave. Xi=1 Xi=1
k Since D is doubly stochastic, using (44), we get
D. Proof of Proposition 4 B1 K B1
d =B , d =1, ∀j ∈[1:B ]. (46)
First, (p∗,q∗) is shown to be the solution of (30) and then i,j 1 i,j 1
i=1j=1 i=1
τ∗ is obtained by (31). Before solving (30), we prove that XX X
We can rewrite (46) as
K
(p∗,q∗)=arg max g(pk,qk) B1 K B1 B1 B1 K
g,pk,qk k=1 (36) di,j = di,j + di,j, (47)
X
s.t. ∀k,(pk,qk)∈J˜,g ∈C, Xi=1Xj=1 Xi=1Xj=1 Xi=1j=XB1+1
from which it follows that
whereCisthesetofallcontinuousconcavefunctions.As(30)
is a special case of (36), (p∗,q∗) is also the solution of (30). B1 K
Before starting, we note that the notations and properties di,j =0, (48)
of the OEA algorithm discussed in Section IV-A are used Xi=1j=XB1+1
throughouttheproof.Bycontradiction,letusassumethatthere and hence,
exists a [pˆT qˆT]T 6= [p∗T q∗T]T and (pˆ,qˆ) be the solution
d =0, ∀i∈[1:B ],∀j ∈[B +1:K]. (49)
of (36). Then, by Theorem 3 we have, i,j 1 1
pˆT qˆT T (cid:22) pT qT T, ∀(p,q)∈J˜. (37) Then applying (39) for i=2,
Since (p∗,q(cid:2)∗)∈J˜,(cid:3)by (3(cid:2)7) and D(cid:3)efinition 3, B2 qˆ = B2 K q∗d ≤V . (50)
j i i,j 2
pˆT qˆT T = p∗T q∗T TD. (38) j=XB1+1 j=XB1+1Xi=1
By (P1) and (P2), we have q∗ =q∗ +L , where
By the feasibility(cid:2) constr(cid:3)aint inh(26b), i i (2) i
L <0 ∀i∈[1:B ],
i 1
Bi Bi
qˆ ≤V = er/L, (39) Li =0 ∀i∈[B1+1:B2], (51)
j i j
j=BXi−1+1 j=BXi−1+1 Li >0 ∀i∈[B2+1:K].
10
From (50) and (51), Sincetheobjectivefunctionisconcave,byJensen’sinequality,
the new policy strictly increasesthe objective.Finally consid-
B2 K B2 K
L d + q∗ d ≤V . (52) ering the case where pk <pk+1,qk >qk+1, we can construct
i i,j (2) i,j 2 another feasible policy,
j=XB1+1Xi=1 j=XB1+1Xi=1
Since D is doubly stochastic, by (P1), we obtain p˜k =pk, p˜k+1 =pk+1,
(61)
(B −B )q∗ = V , and using (49) and (51) in (52), we q˜ =q , q˜ =q .
2 1 (2) 2 k k+1 k+1 k
get
The function Rub with variables p,q,τ can be written as,
B2 K
L d ≤0,L >0. (53)
i i,j i 1 p
Rub(p,q,τ)=tlog 1+ + f , (62)
j=XB1+1i=XB2+1 2 t t2
(cid:18) (cid:18) (cid:19) (cid:19)
From (51) and (53) it can be concluded that
−τ
where f , M −(M −1) 1+ q M−1, t , 1 − τ and
τσ2 T
di,j =0, ∀i∈[B2+1:K], ∀j ∈[B1+1:B2]. (54) 0≤τ <T.ThesecondorderpartialderivativeofRub(p,q,τ)
(cid:0) (cid:1)
is given by,
As D is doubly stochastic, using (P1) together with (49) and
(54), we have ∂2Rub ∂f
= ∂q . (63)
B2 ∂p∂q t(1+f/t+pf/t2)2
qˆ =q∗ d =q∗ =q∗,∀j ∈[B +1:B ]. (55)
j (2)i=XB1+1 i,j (2) j 1 2 Since f is monotonicin q, (63) is positive. As ∂∂2pR∂uqb >0, by
the definition of derivative,
Again, since D is doubly stochastic, using (49) and (54),
Rub(p,q,τ)+Rub(p+δ,q+α,τ)>
B2 K
d =B −B , Rub(p+δ,q,τ)+Rub(p,q+α,τ), δ,α>0.
i,j 2 1
(64)
i=XB1+1Xj=1 (56)
B2 Since (64) holds for all 0≤τ <T, we have
di,j =1, ∀j ∈[B1+1:B2]. R˜ub(p,q)+R˜ub(q+δ,q+α)>
i=XB1+1 R˜ub(p+δ,q)+R˜ub(p,q+α), (65)
We can rewrite (56) as
where R˜ub is obtained by,
B2 K B2 B2 B2 K
di,j = di,j + di,j. R˜ub(p,q)=max Rub(p,q,τ). (66)
i=XB1+1Xj=1 i=XB1+1j=XB1+1 i=XB1+1j=XB2+1 τ
(57)
Finally, using (61) and (65) we can see that the newly
From (57) we can see that
constructed policy strictly increases the objective.
B2 K
d =0, (58)
i,j F. Proof of Lemma 3
i=XB1+1j=XB2+1
Let us assume that the transmission rates in the k-th
and hence, and the k +1-th intervals are different, i.e., R˜ub(p ,q ) 6=
k k
di,j =0,∀i∈[B1+1:B2] and ∀j ∈[B2+1:K]. (59) bR˜uufbfe(rpsko+f1T,qXk+a1n)d.BtheefoRrXethareek∆+1-,thintkervaelr,−thLeenekrgyqinatnhde
r i=1 i i=1 i
qˆ C=onqti∗n.uiSnignctehitsheapEprHoapchrofifolersia=re3s,i.m..i,l(a|rS,ri|.e−.,1S),rw=e gSett, ∆thatt,∆r ≤ki=∆1etti.−WLeTcankic=o1npsitr,urcetspaencoPttivheelryf.eWas.ilb.ol.esP,pwoleicayssume
replacing qˆ by pˆ and er by et/T in the above proof, we P P
j j
reach the similar conclusion for pˆ, i.e., pˆ = p∗. Therefore, p˜k =pk+δ, p˜k+1 =pk+1−δ,
(67)
[pˆT qˆT]T =[p∗T q∗T]T. q˜k =qk+δ, q˜k+1 =qk+1−δ,
where δ is chosen such that δ < ∆ and q˜ < q˜ . Now,
r k k+1
E. Proof of Lemma 2 (67) can be written as
Assume that at least one of the pk,qk is not monotonically p˜k =αpk+(1−α)pk+1, p˜k+1 =(1−α)pk+αpk+1,
increasingink.Withoutlossofgenerality(w.l.o.s)weconsider q˜ =αq +(1−α)q , q˜ =(1−α)q +αq ,
k k k+1 k+1 k k+1
thecasesinwhichp >p ,q ≥q andp <p ,q > (68)
k k+1 k k+1 k k+1 k
qk+1. In the case of pk >pk+1,qk ≥qk+1, we can construct where α=1−δ/(qk+1−qk). Using Jensen’s inequality
a new feasible policy,
k+1 k+1
p˜ =p˜ = pk+pk+1, R˜ub(p˜j,q˜j)> R˜ub(pj,qj), (69)
k k+1
2 (60) Xj=k Xj=k
q +q
k k+1
q˜k =q˜k+1 = . which concludes the proof.
2