ReducedComplexity PowerEfficient Wireless OFDMA using an Equally Probable CSI Quantizer
ABSTRACT Emerging applications involving lowcost wireless sensor networks motivate well optimization of multiuser orthogonal frequencydivision multiple access (OFDMA) in the powerlimited regime. In this context, the present paper relies on limited rate feedback (LRF) sent from the access point to terminals to acquire quantized channel state information (CSI) in order to minimize the total average transmitpower under individual average rate and error probability constraints. Specifically, we introduce two suboptimal reducedcomplexity schemes to: (i) allocate power, rate and subcarriers across users; and (ii) design accordingly the channel quantizer. The latter relies on the solution of (i) to design equally probable quantization regions per subcarrier and user. Numerical examples corroborate the analytical claims and reveal that the power savings achieved by our reducedcomplexity LRF designs are close to those achieved by the optimal solution.

Conference Paper: Optimal stochastic dual resource allocation for cognitive radios based on quantized CSI.
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ABSTRACT: The present paper deals with dynamic resource management based on quantized channel state information (CSI) for multicarrier cognitive radio networks comprising primary and secondary wireless users. For each subcarrier, users rely on adaptive modulation, coding and power modes that they select in accordance with the limitedrate feedback they receive from the access point. The access point uses CSI to maximize the sum of generic concave utilities of the individual average rates in the network while respecting rate and power constraints on the primary and secondary users. Using a stochastic dual approach, optimum dual prices are found to optimally allocate resources across users per channel realization without requiring knowledge of the channel distribution.Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2008, March 30  April 4, 2008, Caesars Palace, Las Vegas, Nevada, USA; 01/2008
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ReducedComplexity PowerEfficient Wireless
OFDMA using an Equally Probable CSI Quantizer
Antonio G. Marques∗, Fadel F. Digham†, Georgios B. Giannakis (contact author)†, and F. Javier Ramos∗
∗Dept. of Signal Theory and Communications. Universidad Rey Juan Carlos. Fuenlabrada, 28943 Madrid (SPAIN)
Email: {antonio.garcia.marques, javier.ramos}@urjc.es
†Dept. of Electrical and Computer Engineering. University of Minnesota. Minneapolis, 55455 MN (USA)
Email: {georgios, fdigham}@ece.umn.edu
Abstract—Emerging applications involving lowcost wireless
sensor networks motivate well optimization of multiuser orthog
onal frequencydivision multiple access (OFDMA) in the power
limited regime. In this context, the present paper relies on limited
rate feedback (LRF) sent from the access point to terminals
to acquire quantized channel state information (CSI) in order
to minimize the total average transmitpower under individual
average rate and error probability constraints. Specifically, we
introduce two suboptimal reducedcomplexity schemes to: (i)
allocate power, rate and subcarriers across users; and (ii) design
accordingly the channel quantizer. The latter relies on the
solution of (i) to design equally probable quantization regions
per subcarrier and user. Numerical examples corroborate the
analytical claims and reveal that the power savings achieved by
our reducedcomplexity LRF designs are close to those achieved
by the optimal solution.
I. INTRODUCTION
Orthogonal frequencydivision multiplexing (OFDM) is the
most common modulation for bandwidth limited wireline
and wireless transmissions over frequencyselective multipath
channels. OFDM transmissions over wireline or slowly fading
wireless links have traditionally relied on deterministic or per
fect (P) channel state information at the transmitters (CSIT)
to adaptively load power, bits and/or subcarriers so as to either
maximize rate (capacity) for a prescribed transmitpower, or,
minimize power subject to instantaneous rate constraints [9].
While the assumptions of PCSI at the transmitters and
receiver render analysis and design tractable, they may not be
as realistic due to wireless channel variations and estimation
errors, feedback delay, bandwidth limitation, and jamming
induced errors [6]. These considerations motivate a limited
rate feedback (LRF) mode, where only quantized (Q) CSIT
is available through a (typically small) number of bits fed
back from the receiver to the transmitters; see e.g., [10]. Q
CSIT entails a finite number of quantization regions describ
ing different clusters of channel realizations [7], [10]. Upon
estimating the channel, the receiver feeds back the index of
the region individual uplink channels belong to (channel code
word), based on which each terminal adapts its transmission
parameters accordingly. This LRFbased mode of operation
Work in this paper was supported by the US ARL under the CTA Program,
Cooperative Agreement No. DAAD190120011; by the USDoD ARO grant
No. W911NF0510283; and by the Government of C.A. Madrid under grant
No. PTIC0002230505.
fulfills two requirements: (i) the feedback is pragmatically
affordable in most practical wireless links, and (ii) the QCIST
is robust to channel uncertainties since transmitters adapt to a
few regions rather than individual channel realizations.
Resource allocation in orthogonal frequencydivision mul
tiple access (OFDMA) minimizing the transmitpower per
symbol based on PCSIT was first studied in [9]. Relying
on fixed (as opposed to adaptive) QCSIT, recent works deal
with optimization of power or rate performance per OFDMA
symbol [2], [5]. Different from these works, here we jointly
adapt power, rate, and subcarrier resources based on QCSIT
to minimize the average transmitpower. Our focus is on
allocation algorithms with negligible online computational
complexity. Moreover, we rely on the optimal allocation for
designing a novel noniterative channel quantizer that enforces
equally probable quantization regions per user and subcarrier.
The rest of the paper is organized as follows. After intro
ducing preliminaries on the setup we deal with (Section II),
for a given quantizer design, we derive suboptimal subcarrier,
power, and bit OFDMA allocation (Section III). Once the
allocation is characterized, we capitalize on it for designing
a quantizer with equally probable regions (Section IV). Nu
merical results and comparisons that corroborate our claims
are presented (Section V), and concluding remarks finish this
paper (Section VI)1.
II. PRELIMINARIES AND PROBLEM STATEMENT
We consider a wireless OFDMA system (see Fig. 1) with
M users, indexed by m ∈ [1,M], sharing K subcarriers
(subchannels), indexed by k ∈ [1,K]. The instantaneous (per
symbol) power and rate user m loads on subcarrier k are
denoted by pk,mand rk,m, respectively. With these as entries
we form K×M instantaneous power and rate matrices P and
R, that is [P]k,m:= pk,mand [R]k,m:= rk,m. For a given
1Notation: Lower and upper case boldface letters are used to denote
(column) vectors and matrices, respectively; (·)Tdenotes transpose; [·]k,l
the (k,l)th entry of a matrix, and [·]kthe kth entry of a vector; X ≥ 0
means all entries of X are nonnegative; FN stands for the normalized FFT
matrix with entries [FN]n,k= e−j2π
denotes the joint probability density function (PDF) of a matrix X; likewise,
fx(x) denotes the PDF of a scalar x; EX[·] stands for the expectation operator
over X; ?·? (?·?) denotes the floor (ceiling) operation; I{·}is short for the
indicator function; i.e., I{x}= 1 if x is true and zero otherwise; and LHS(x)
denotes the left hand side of equation (x).
Nkn, n,k = 0,...,N − 1; fX(X)
Page 2
Fig. 1.System block diagram.
feedback update, we consider a time sharing user access per
subcarrier; i.e., time division multiple access (TDMA)2. This
sharing process is described by the K × M weight matrix
W whose (k,m)th entry wk,m represents the percentage of
time the kth subcarrier is utilized by the mth user. Clearly,
?M
is pk,mwk,mand rk,mwk,mfor the kth subcarrier of user m.
Each user’s discretetime baseband equivalent impulse
response of the corresponding frequencyselective fading
channel is hm
:= [hm,0,...,hm,Nm]T, where: Nm
?Dm,max/Ts? denotes the channel order, Dm,max the max
imum delay spread, Ts the sampling period, and Nmax :=
maxm∈[1,M]Nm,max. As usual in OFDM, we suppose K ?
Nmax. For notational convenience, we collect the M impulse
response vectors in a K×M matrix H := [h1,...,hM], where
the length of each column is increased to K by padding an
appropriate number of zeros.
Each user applies a Kpoint inverse fast Fourier transform
(IFFT) to each snapshot of Ksymbol streams, and subse
quently inserts a cyclic prefix (CP) of size Nmax to obtain
a block of K + Nmax symbols (i.e., one OFDM symbol),
which are subsequently multiplexed and digital to analog
(D/A) converted for transmission. These operations along with
the corresponding FFT and CP removal at the receiver convert
each user’s frequencyselective channel to a set of K parallel
flatfading subchannels, each with fading coefficient given by
the frequency response of this user’s channel evaluated on the
corresponding subcarrier. Consider the K × M matrix˜ H :=
(1/√K)FKH, whose mth column comprises the frequency
response of user m’s channel.
With the multiuser channel matrix˜ H acquired (via training
symbols), the receiver has available a noisenormalized chan
nel power gain matrix G, where [G]k,m:= [˜ H]k,m2/σ2
m=1wk,m ≤ 1,∀k, and the average power and rate over
the transmission period between successive feedback updates
:=
k,m,
2Orthogonal access schemes other than TDMA are also possible. But as we
will see later, the one chosen is not particulary important because the optimal
choice will typically correspond to no sharing; i.e., each subcarrier will be
owned by a single user.
with σ2
additive white Gaussian noise (AWGN) at the receiver. We
will use gk,m := [G]k,m to denote the instantaneous noise
normalized channel power gain for the kth subchannel of the
mth user. Likewise, letting¯G := EG[G], its generic entry
¯ gk,m:= [¯G]k,m shall denote the average gain of the (k,m)
subcarrieruser pair. Having (practically perfect) knowledge of
each G realization, the access point (AP) allocates subcarriers
to users after assigning entries of G to appropriate quantization
regions they fall into. Using the indices of these regions, the
receiver feeds back the codeword c = c(G) for the users to
adapt their transmission modes (power, rate and subcarriers)
from a finite set of mode triplets.
Our work relies on the following assumptions:
(as1) Different user channels are uncorrelated; i.e., the
columns of G are uncorrelated.
(as2) Each user’s subchannels are allowed to be correlated,
and complex Gaussian distributed; i.e., gk,mobeys an expo
nential PDF fgk,m(gk,m)= (1/¯ gk,m)exp(−gk,m/¯ gk,m).
(as3) Subchannel states (regions) remain invariant over at
least two consecutive OFDM symbols.
(as4) The feedback channel is errorfree and incurs negligible
delay.
(as5) Symbols are drawn from quadrature amplitude modula
tion (QAM) constellations so that the resulting instantaneous
BER can be approximated as (κ1= 0.2, κ2= 1.5)
k,mdenoting the known variance of the zeromean
?(pk,m,gk,m,rk,m) ? κ1exp
(as6) A realization of each gk,mgain falls into one of Lk,m
disjoint regions {Rk,ml}Lk,m
Since users are sufficiently separated in space (as1) is gen
erally true; (as2) corresponds to fading amplitudes adhering to
the commonly encountered Rayleigh model but generalizations
are possible; (as3) allows each subchannel to vary from one
OFDM symbol to the next so long as the quantization region it
falls into remains invariant; errorfree feedback under (as4) is
guaranteed with sufficiently strong error control codes (espe
cially since data rates in the feedback link are typically low);
the accuracy of (as5) is widely accepted; see e.g., [4]; and
(as6) represents a practical and low complexity quantization.
The ultimate goal in this paper is twofold: (G1) design
a channel quantizer to obtain c; and (G2) given c, find
appropriate allocation matrices P, R, and W. We want to
design P, R, W, and {Rk,ml}Lk,m
average power¯P is minimized under prescribed average rate
¯ r0 := [¯ r0,1,..., ¯ r0,M]Tand average bit error rate (BER)
¯ ?0:= [¯ ?0,1,...,¯ ?0,M]Tconstraints across users.
III. QUANTIZER AND TRANSMISSION MODE DESIGN
A. Problem Formulation
Given (as6), let Rk,ml:= {G : gk,m∈ Rk,ml} denote the
set of matrices G for which gk,mbelongs to the region Rk,ml.
Furthermore, let pk,mland rk,mldenote3respectively, the
?−pk,mκ2gk,m
(2rk,m− 1)
?
.
(1)
l=1.
l=1
∀k,m, so that the
3The subscript l here will be also written explicitly as l(G) in places that
this dependence must be emphasized.
Page 3
instantaneous power and rate loadings of user m on subcarrier
k given that G ∈ Rk,ml. Recall that wk,m(G) ≤ 1, and
thus the expected power and bit loadings for the G realization
over the time between successive feedback updates will be
pk,mlwk,m(G) and rk,mlwk,m(G), respectively.
Our goal is to minimize the average transmitpower
EG[pk,ml(G)wk,m(G)] over all subcarriers and users while
satisfying average rate and BER requirements. Specifically,
we want the average rate of any user (say the mth) across all
subcarriers to satisfy?K
instantaneous BER stay always below a prespecified BER4.
Then if ?−1
(1) w.r.t. pk,m and gmin
represent the worst channel gain, the power loading
k=1EG[rk,ml(G)wk,m(G)] ≥ [¯ r0]m.
To enforce the average BER requirement we will have the
Pdenotes the inverse function involved when solving
k,ml:= min{gk,m  gk,m ∈ Rk,ml}
pk,ml(G):= ?−1
P(rk,ml(G),gmin
k,ml(G),[¯ ?0]m)
(2)
will automatically fulfill the BER requirement.
Analytically, the constrained optimization problem we wish
to solve is:
where through the second constraint we enforce the total
utilization of any subcarrier by all users not to exceed one,
per G realization.
Althoughtheoptimization
convex,definingthe auxiliary
rk,ml¯ wk,ml,
fG(G)dG we can render it convex. Since ¯ wk,ml ?= 0
the resultant convex optimization problem using globally
convergent interior point algorithms [1]. The final formulation
now enjoys both reduced complexity and global convergence
in polynomial time thanks to convexity.
The objective in (3) is to minimize the average power
over all possible channel realizations. However, the constraints
involve different forms of CSI: while C2 and C4 needs to
be satisfied per channel realization (and thus will change
depending on G); C1 is an average requirement and C3
is unique per region, and therefore their values do not de
pend of the concrete channel realization G. In the following
subsection, we will derive the KarushKuhnTucker (KKT)
conditions associated with (3). These will lead us not only
to the expressions determining the optimal loading variables
but will also provide valuable insights about the structure of
the powerefficient resource allocation policies. At this point,
minR(G)≥0,W(G)≥0
?K
C1. −?K
C3. − rk,ml≤ 0,∀k,m,l, C4. − wk,m(G) ≤ 0,∀k,m,G,
¯P, where¯P :=
P(rk,ml(G),gmin
k=1
?M
k=1EG[rk,ml(G)wk,m(G)] + [¯ r0]m≤ 0,∀m,
?M
m=1EG[?−1
k,ml(G),[¯ ?0]k)wk,m(G)]
subject to :
C2.
m=1wk,m(G) − 1 ≤ 0, ∀k,G,
(3)
problemin (3)
¯ rk,ml
isnot
=
variable
?
with
¯ wk,ml
:=
G∈Rk,mlwk,m(G)
4Although this relaxation may lead to a suboptimal solution, it leads to a
less complex optimization problem that turns out to be convex. Numerical
results will show that solution of the relaxed problem attains comparable
performance to that reached by the optimum solution.
a remark is due on another aspect related to power efficiency
in OFDMA.
Remark 1: Althoughthe
(PAPR)plays animportant
consumption of OFDM systems, in (3) we did not impose
PAPR constraints. The underlying reason is that available
digital predistortion schemes can be applied to the users’
OFDM symbols to meet such constraints, see e.g., [8].
peaktoaveragepowerratio
role inpower(battery)
B. Optimal Policies
Let βr
Lagrange multipliers associated with C1C4, respectively.
Specifically, upon defining κ3,m := κ−1
setting the derivative of the dual Lagrangian function in (3)
with respect to (w.r.t.) the auxiliary variable ¯ rk,mlequal to
zero, yields after tedious but straightforward manipulations the
following KKT condition expressed in terms of the original
variable5
?(βr∗
m, βw
k, αp
k,ml, αr
k,ml, αw
k,mdenote the positive
2 ln(κ1/[¯ ?0]m) and
r∗
k,ml(G)= log2
m+ αr∗
k,ml)gmin
ln(2)κ3,m
k,ml(G)
?
.
(4)
Because KKT conditions for C3 dictate r∗
for r∗
condition gmin
order for the optimum rate loading in the region to be nonzero.
Intuitively, this condition eliminates from the optimum alloca
tion set the regions Rk,mlwith very poor channel conditions
(αr∗
Furthermore, it is worth noting that due to the logarithmic
expression of ?−1
P
[c.f. (1)], the rate loading is reminiscent of
the classical capacity waterfilling solution.
Before analyzing the optimality condition for wk,m(G), let
us define the power cost of user m utilizing subcarrier k as
Pk,m(G) :=(2r∗
gmin
k,ml(G)
Supposing that Rk,mlis active, using (5), and differenti
ating the Lagrangian of (3) w.r.t. wk,m(G), we find at the
optimum
k,mlαp∗
k,ml= 0 [1],
k,ml> 0, we need αr∗
k,ml(G)> ln(2)κ3,m(βr∗
k,ml= 0 in (4) and therefore the
m)−1has to be satisfied in
k,ml> 0), while for the remaining regions αr∗
k,ml= 0.
k,ml(G)− 1)κ3,m
− βr+∗
m r∗
k,ml(G).
(5)
Pk,m(G)fG(G)+βw∗
It is useful to check three things: (i) the LHS(6) does not
depend explicitly on w∗
multipliers βw∗
is common ∀m; and (iii) for the same subcarrier k and a given
realization G, the power cost Pk,m(G) is fixed and in general
different for each user m. Furthermore, for each k, the KKT
condition corresponding to C4 also dictates
k(G)−αw∗
k,m(G) = 0, ∀G, ∀m ∈ [1,M].
(6)
k,m(G) but only through the associated
k(G) and αw∗
k,m(G); (ii) the multiplier βw∗
k(G)
w∗
k,m(G)αw∗
k,m(G) = 0, ∀G, ∀m ∈ [1,M].
the set
{w∗
(7)
Proposition 1: In
w∗
for m
k,m(G)}M
m=1, wehave
k,mk(G) = 1 for a unique user mk, and w∗
?=
mk. Moreover, the user mk
k,m(G) = 0
assigned to
5Henceforth, x∗will denote the optimal value of x.
Page 4
utilize
mk= argminm{Pk,m(G)}M
Proof: Assume that w∗
and (6) implies that βw∗
w∗
k,m?
and (6) for m?
which is not true and thus w∗
(uniqueness). If now w∗
βw∗
can write αw∗
which is not true if Pk,mk(G) < Pk,m?
Notice that if for a given G the minimum value of
{Pk,m(G)}M
trary time sharing of the subcarrier k among them is optimum;
or we can simply pick one of them at random without altering
the power cost. Finally, if there exists a realization G such that
Pk,m(G) > 0 ∀m, then since βw∗
(6) that αw∗
allocate this subcarrier to any user. Therefore, introducing a
fictitious Pk,0(G) = 0 ∀k and G, we can express w∗
in compact form using the indicator function as
the
kth subchannel is theonethat satisfies:
m=1.
k,mk(G) > 0, then αw∗
k(G) = −Pk,mk(G)fG(G). If also
k(G) > 0 with m?
kbecomes Pk,mk(G) = Pk,m?
k,m?
k(G) = 1 for a m?
k(G) = −Pk,m?
k,mk(G) = [Pk,mk(G) − Pk,m?
k,mk(G) = 0
k?= mk then αw∗
k,m?
k(G) = 0,
k(G) ∀G;
k(G) > 0 is not either
k?= mk, then
k,m?
k(G)fG(G) and using (6) for mk we
k(G)]fG(G) ≥ 0,
k(G) (minimum).
m=1is attained by more than one user, any arbi
k(G) ≥ 0 it follows from
k,m(G) ?= 0, ∀m; and the optimal solution will not
k,m(G)
w∗
k,m(G) = I{m=arg minm?{Pk,m?(G)}M
Since so far we have obtained conditions that the optimal
k,mland w∗
for p∗
substituting r∗
the optimization problem tractable we had enforced via (2) an
instantaneous BER constraint that is a more restrictive than the
original average BER constraint. Capitalizing on this fact, a
different approach to obtain the power per region (denoted by
p+∗
to satisfy
m?=0}.
(8)
r∗
k,m(G) should satisfy, what remains is a condition
k,ml. Although in principle p∗
k,mland w∗
k,mlcan be calculated by
k,m(G) into (2), recall that to render
k,ml), given r∗
k,mland w∗
k,m(G), consists of finding p+∗
k,ml
EG∈Rk,ml[?(p+∗
k,ml,gk,m,r∗
= [¯ ?0]mEG∈Rk,ml[w∗
k,ml)w∗
k,m(G)]
k,m(G)].
(9)
Using this approach we, guarantee that the expected BER
averaged over all channel realizations inside the region Rk,ml
satisfies the prespecified requirement [¯ ?0]m. Since p+∗
p∗
loading. Note also that, although (9) involves an integration
and p+∗
of the exponential inside the instantaneous BER in (1) allows
obtaining p+∗
method).
The following algorithm summarizes the main steps for
finding the optimal allocation of R∗, W∗and P∗.
k,ml<
k,ml, we will rely on (9) to calculate the final power
k,mlcan not be found in closedform, the monotonicity
k,mlthrough line search (using e.g., the bisection
Algorithm 1: Resource Allocation
(S1.0) Select a small positive number δ, and initialize βrusing
an arbitrary nonnegative vector.
(S1.1) For each (k,m,l) triplet per iteration:
(S2.1.1) Use [βr]mto determine rk,mlvia (4)(αr
(S2.1.2) Use [βr]mto determine wk,m(G) via (8).
k,ml= 0).
(S1.2) Check C1 (3) ∀m; if LHS(C1) < δ[¯ r0]m ∀m, then
go to (S1.3); otherwise, increase [βr]m for the users m
whose LHS(C1) > 0; decrease [βr]mfor the users m
whose LHS(C1) < 0; and go to (S1.1).
(S1.3) Once R∗,βr∗,W∗are obtained, use (9) to calculate the
finally allocated power.
The convexity of (3) enables efficient methods to update
βr[1]. For example, we can set the initial value of βr
equal to any small number and update each component [βr]m
separately ∀m by fixing [βr]m?, ∀m??= m from the previous
iteration. The adaptation of each [βr]m is then performed
using line search until the rate constraint for the mth user
is tightly satisfied. This simple algorithm has guaranteed con
vergence and facilitates computation distributed across users.
C. Codeword Structure
Given the quantizer design, we developed so far resource
allocation policies to assign rate, power and subcarriers across
users. Once the quantizer and resource allocation strategy are
designed, the AP quantizes each fading state and feeds back a
codeword that identifies the usersubcarrier assignment and the
region index each subchannel falls into per fading realization
G. Based on this form of QCSIT, each user node is informed
about its own subset of subcarriers (if any) and relies on the
region indices to retrieve the corresponding power and rate
levels from a lookup table. The following proposition describes
the construction of this codeword.
Proposition 2: Given the quantizer design and the optimal
allocation parameters (P∗, R∗, W∗(βr∗)) returned by Al
gorithm 1, the AP broadcasts to the users the codeword
c∗(G) = [c∗
allocation for the current fading state, where c∗
[m∗
1) m∗
(pick randomly any user m∗
occur); and
2) l∗
The structure of c∗(G) in Proposition 2 encodes in
formation pertinent to each subcarrier (namely, its region
and assigned user) which is more efficient in terms of
the number of feedback bits relative to encoding each
user’s individual information, yielding a codeword length of
??K
P) in (3) are involved only in average quantities. Hence,
R∗and P∗are computed offline and only the subcarrier
user assignment (involved in instantaneous constraints) and the
indexing of the corresponding entries of these matrices need to
be fed back online. Thus, almost all the complexity is carried
out offline (Algorithm 1), while only a light computation
(Proposition 2) has to be carried out online.
IV. QUANTIZER DESIGN
In the previous section we addressed our objective (G2) to
derive optimum subcarrier, rate, and power allocation policies
1(G),...,c∗
K(G)] specifying the optimal resource
k(G) =
k(G),l∗
k(G)]Tis determined ∀k as:
k(G) = argmin
m{Pk,m(G,P∗,R∗,βr∗,{β?∗}Lk,m
kwhen multiple minima
l=1)}M
m=1
k(G) = { l  G ∈ Rk,m∗
k(G),l, l = 1,...,Lk}.
k=1log2
We conclude this section by emphasizing that R (and thus
??M
m=1(Lk,m)
??
bits.
Page 5
assuming the quantization regions, Rk,ml, are given. In this
section, we will address (G1) by deriving a quantizer that
enforces equally probable quantization regions.
A. Equally Probable Region Quantizer
To design the quantizer, we first solve the optimal resource
allocation problem supposing CSI is available without quanti
zation (i.e., Lk,m= ∞), and subsequently calculate τk,mlto
satisfy
?τk,ml+1
τk,m1
with τk,m1
probabilities can be computed, solving (10) yields thresholds
{τk,ml}Lk.m
probability Pr(wk,m = 1,gk,m) into regions of equal area;
hence the term equally probable region quantizer. Intuitively
speaking, this quantizer design tries to maximize the entropy
in the feedback link.6
To evaluate Pr(wk,m= 1,gk,m) needed in (10), we apply
Bayes’ rule to rewrite it as Pr(wk,m= 1gk,m)fgk,m(gk,m)
and recall that fgk,i(gk,m) is known per (as2). To calculate
Pr(wk,m = 1gk,m), we will need to first solve the optimal
resource allocation problem assuming no quantization. Clearly,
as Lk,m→ ∞, we have gmin→ g in (5). Letting P∞
βr∞∗
m
denote, respectively, the cost indicator and the Lagrange
multiplier when Lk,m→ ∞, we can write
P∞
gk,m
Becauseequation (8) establishes
I{m=arg minm?{P∞
then gk,m> ln(2)κ3,m/βr∞∗
m
user m to be active. Taking also into account that channels of
different users are uncorrelated [cf. (as1)], we can write
τk,ml
Pr(wk,m= 1,gk,m)dgk,m
Pr(wk,m= 1,gk,m)dgk,m/Lk,m,
=?τk,mLk,m
(10)
= 0 and τk,mLk,m+1
= ∞. If the joint
l=1
per subcarrier k and user m that divide the joint
k,mand
k,m(gk,m) =βr∞∗
m
ln(2)−κ3,m
− βr∞∗
m
log2
?
that
gk,mβr∞∗
(κ3,mln(2))
wk,m(G)
m
?
. (11)
=
k,m?(G)}M
m?=0}, if P∞
is a necessary condition for the
k,m(gk,m) < P∞
k,0= 0,
Pr(wk,m= 1gk,m) = I{gk,m>
×?M
Interestingly,for
P∞
log2
(ln(2)κ3,m)
,
I{gk,m>
m
that P∞
can find unique channel gains γk,µ, ∀µ ?= m, such that
P∞
It is then clear that P∞
and P∞
quently, Pr(P∞
ln(2)κ3,m
λr∗
m
k,µgk,m).
users
?βr∞∗
∂P∞
}
µ=1,µ?=mPr(P∞
the
=
I{gk,m>
gk,mβr∞∗
m
?κ3,m
k,m< P∞
active
(12)
itholdsthat
k,m(gk,m)
?
ln(2)κ3,m
βr∞∗
m
therefore
?
}
m
ln(2)−κ3,m
gk,m
−βr∞∗
m
??
}
and
k,m(gk,m)
∂gk,m
=
ln(2)κ3,m
βr∞∗
k,m(gk,m) is monotonically decreasing. Therefore, we
gk,m−βr∞∗
m
ln(2)
κ3,m
gk,m
≤ 0, which implies
k,m(gk,m) = P∞
k,m(gk,m) ≤ P∞
k,µ(gk,µ) if gk,µ > γk,µ. And conse
k,m< P∞
k,µ(γk,µ).
(13)
k,µ(gk,µ) if gk,µ≤ γk,µ,
k,m(gk,m) > P∞
k,µgk,m) = Pr(gk,µ< γk,µgk,m).
6The simplicity and efficiency of using an equally probable quantizer were
first pointed out in [7] for the basic case of a single user and subcarrier. We
here largely expand the scope of this quantizer and apply it to the challenging
multipleuser/multiplesubcarrier scenario.
12345
0
0.5
1
fgk,m(gk,m)
12345
0
0.5
1
Pr(wk,m= 1gk,m)
12345
0
0.02
0.04
0.06
Pr(wk,m= 1,gk,m)
Normalized Subcarrier Power Gain gk,m/gk,m
Fig. 2.Equally Probable Regions Quantizer (K = 64, M = 6, L = 5).
Excluding the case P∞
wk,m= 0 and solving (13) w.r.t. γk,µyields
k,m(gk,m) > 0, which amounts to
γk,µ(gk,m) = −
κ3,µln(2)/βr∞∗
e−1?eln(2)κ3,m
µ
fW
?
−2
−
κ3,m
gk,mβr∞∗
µ
gk,mβr∞∗
m
?βr∞∗
m
µ
βr∞∗
?,
(14)
where fW[x] = y is the realvalued Lambert’s fWfunction
which solves the equation yey= x for −1 ≤ y ≤ 0 and
−1/e ≤ x ≤ 0 [3].
Using (12)(14), we express Pr(wk,m= 1,gk,m) in (10) as
Pr(wk,m= 1,gk,m) = I{gk,m>
×?M
Since (15) depends on {βr∞∗
optimal allocation problem as Lk,m → ∞. The thresholds
{τk,ml}Lk,m
Notice that we can also take advantage of the condition
P∞
An example illustrating this quantizer is given in Fig.
2 which depicts Pr(wk,m
=
Pr(wk,m= 1,gk,m) versus gk,m/¯ gk,mfor M = 6, Lk,m= 5,
equal average subcarrier gains, and equal rate constraints. The
first subplot in this figure, Pr(wk,m= 1gk,m), reveals that the
better the channel the more likely the corresponding user is
to be selected. Coupling this observation with the exponential
behavior of fgk,m(gk,m) in the second subplot, the bellring
characteristic of the joint PDF, Pr(wk,m = 1,gk,m), results
naturally in the third subplot (after pairwise multiplication of
the functions in the first two subplots), where the quantization
thresholds (and regions) resulting from (10) are also identified.
Remark 2: With the method presented in this section, calcu
lation of τk,mlbased on (10) has to be executed only once
(to solve (G1)). With the thresholds available, the resource
allocation can be easily obtained through Algorithm 1 (to
solve (G2)). Numerical results in the next section will show
that this low complexity noniterative design exhibits power
consumption similar to that of the PCSIT solution.
ln(2)κ3,µ
βr∞∗
m
}
e−gk,m/¯ gk,m
¯ gk,m
µ=1,µ?=m
?1 − e−γk,µ(gk,m)/¯ gk,µ?.
µ
}M
(15)
µ=1, we need to solve the
l=2
are obtained by solving (10) using a line search.
k,m(gk,m) < 0, by setting τk,m,2≥ κ3,mln(2)/βr∞∗
m
.
1gk,m), fgk,m(gk,m) and