Quantization and Bit Allocation for Channel State Feedback for Relay-Assisted Wireless Networks
ABSTRACT This paper investigates quantization of channel state information (CSI) and
bit allocation across wireless links in a multi-source, single-relay
cooperative cellular network. Our goal is to minimize the loss in performance,
measured as the achievable sum rate, due to limited-rate quantization of CSI.
We develop both a channel quantization scheme and allocation of limited
feedback bits to the various wireless links. We assume that the quantized CSI
is reported to a central node responsible for optimal resource allocation. We
first derive tight lower and upper bounds on the difference in rates between
the perfect CSI and quantized CSI scenarios. These bounds are then used to
derive an effective quantizer for arbitrary channel distributions. Next, we use
these bounds to optimize the allocation of bits across the links subject to a
budget on total available quantization bits. In particular, we show that the
optimal bit allocation algorithm allocates more bits to those links in the
network that contribute the most to the sum-rate. Finally, the paper
investigates the choice of the central node; we show that this choice plays a
significant role in CSI bits required to achieve a target performance level.
- Citations (18)
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ABSTRACT: We determine the capacity region of a degraded Gaussian relay channel with multiple relay stages. This is done by building an inductive argument based on the single-relay capacity theorem of Cover and El Gamal. For an arbitrary distribution of noise powers, we derive the optimal power distribution strategy among the transmitter and the relays and the best possible improvement in signal-to-noise ratio (SNR) that can be achieved from using a given number of relays. The time-division multiplexing operation of the relay channel in the wideband regime is analyzed and it is shown that time division does not achieve minimum energy per bit.IEEE Transactions on Information Theory 01/2005; · 2.62 Impact Factor - SourceAvailable from: psu.edu[Show abstract] [Hide abstract]
ABSTRACT: Coding strategies that exploit node cooperation are developed for relay networks. Two basic schemes are studied: the relays decode-and-forward the source message to the destination, or they compress-and-forward their channel outputs to the destination. The decode-and-forward scheme is a variant of multihopping, but in addition to having the relays successively decode the message, the transmitters cooperate and each receiver uses several or all of its past channel output blocks to decode. For the compress-and-forward scheme, the relays take advantage of the statistical dependence between their channel outputs and the destination's channel output. The strategies are applied to wireless channels, and it is shown that decode-and-forward achieves the ergodic capacity with phase fading if phase information is available only locally, and if the relays are near the source node. The ergodic capacity coincides with the rate of a distributed antenna array with full cooperation even though the transmitting antennas are not colocated. The capacity results generalize broadly, including to multiantenna transmission with Rayleigh fading, single-bounce fading, certain quasi-static fading problems, cases where partial channel knowledge is available at the transmitters, and cases where local user cooperation is permitted. The results further extend to multisource and multidestination networks such as multiaccess and broadcast relay channels.IEEE Transactions on Information Theory 10/2005; · 2.62 Impact Factor - [Show abstract] [Hide abstract]
ABSTRACT: In this paper, we propose a limited feedback scheme to improve outage performance for a wireless cooperative decode-and-forward network. Specifically, based on the instantaneous conditions of the source-destination and relay-destination channels, the destination will allocate the transmission time of the source and relay and feed back the allocation result to the source. Both limited and full (or infinite) rate feedback are considered. Under the practical assumption that only imperfect channel estimation is available at the receiver, we analyze the outage performance by deriving upper bounds on the outage probabilities. It will be demonstrated that, even with only one-bit feedback, the proposed feedback scheme can outperform the no feedback case. Furthermore, the outage performance can approach the optimality by exploiting limited (only a small number of bits) uniformly quantized feedback from the destination.IEEE Transactions on Communications 04/2010; · 1.75 Impact Factor
Page 1
arXiv:1112.0711v1 [cs.IT] 4 Dec 2011
Quantization and Bit Allocation for Channel
State Feedback in Relay-Assisted Wireless
Networks
Ehsan Karamad, Student Member, IEEE, Behrouz Khoshnevis, Member, IEEE,
and Raviraj S. Adve, Senior Member, IEEE
Abstract
This paper investigates quantization of channel state information (CSI) and bit allocation across
wireless links in a multi-source, single-relay cooperative cellular network. Our goal is to minimize
the loss in performance, measured as the achievable sum rate, due to limited-rate quantization of CSI.
We develop both a channel quantization scheme and allocation of limited feedback bits to the various
wireless links. We assume that the quantized CSI is reported to a central node responsible for optimal
resource allocation. We first derive tight lower and upper bounds on the difference in rates between the
perfect CSI and quantized CSI scenarios. These bounds are then used to derive an effective quantizer
for arbitrary channel distributions. Next, we use these bounds to optimize the allocation of bits across
the links subject to a budget on total available quantization bits. In particular, we show that the optimal
bit allocation algorithm allocates more bits to those links in the network that contribute the most to the
sum-rate. Finally, the paper investigates the choice of the central node; we show that this choice plays
a significant role in CSI bits required to achieve a target performance level.
I. INTRODUCTION
It is well established that using relays can significantly improve the communication capacity
and reliability of wireless networks [1]. Based on approaches suggested in [2], the work in [3]
and [4] analyzes different relaying strategies such as the decode-and-forward (DF) and amplify-
and-forward (AF) relaying techniques. Our focus is on DF, wherein the relay must decode
and then re-encode the source data. The potential gains associated with the relay systems, and
The authors are with the Edwards S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto,
Toronto, Ontario, Canada M5S 3G4 (email: {ekaramad,bkhoshnevis,rsadve}@comm.utoronto.ca).
Page 2
2
cooperative diversity in general, has attracted a great deal of research into the optimization of
the relay-assisted network performance. For wireless networks, the optimization is mainly in
terms of resource allocation, specifically power and/or bandwidth allocation, relay routing, and
selection of relaying strategy [5]–[7].
Although the literature on relay network optimization shows that there are significant perfor-
mance improvements to be had, most of the analysis is based on the crucial assumption that
some central node has exact knowledge of the network-wide channel state information (CSI).
This assumption, however, is impossible to satisfy in a practical system implementations due to
the limited resources available for CSI training and feedback. As a result, in practice, network-
wide CSI is not known perfectly; only a quantized version of the information may be available via
feedback. Since the performance of resource allocation algorithms in cooperative relay networks
depends heavily on the availability of CSI, it becomes essential to investigate the performance
of resource allocation schemes under the assumption of limited-rate quantization of CSI.
This paper takes a step in this direction by investigating the uplink in a relay-assisted wireless
cellular network (much of our analysis can also be applied to the downlink as well). Our system
model considers multiple sources communicating to a base-station (BS) with the help of a single
relay. We assume that while the receiver end in any link has perfect CSI of that link, a quantized
version of all channels is available at a central node responsible for resource allocation. The
main goal here is to (i) optimize the quantization of CSI and, (ii) given a constraint on the
number of bits for CSI feedback, allocate those bits across all links in the network.
The performance of cooperative networks with limited CSI has been addressed in the available
literature for several communication scenarios. As a general result, it is shown that providing even
a few bits of quantized CSI significantly improves the performance of cooperative systems [8]–
[13]. The authors of [8], in particular, investigate optimal temporal resource allocation between
the source and the relay under quantized CSI using a DF relaying strategy and show that even
with a single bit of feedback provides significant gains in bit error rate. The work in [9] shows
similar results for diversity gains for both AF and DF relaying strategies.
The authors of [13] investigate the optimum throughput in a cooperative network using DF.
This work maximizes an upper bound on overall throughput, thereby deriving a suboptimal
resource allocation scheme. The authors of [10] investigate optimal relay selection in multi-
relay AF cooperative networks. This work investigates network performance with quantized and
December 6, 2011DRAFT
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statistical CSI; again a few CSI bits are shown to provide significant performance gains.
The works mentioned so far do not specifically investigate how to quantize CSI or allocate bits.
The authors of [11] consider a cooperative communication system in a cellular network with
inter-cell interference. The paper adopts zero-forcing beamforming and finds an approximate
expression for the received signal-to-interference-plus-noise ratio (SINR), based on which bit
allocation is optimized across the network links. The authors also determine the minimum number
of CSI bits that are required to outperform a non-cooperative network. The authors of [12] design
quantization codebooks for the transmit power vectors in a single-relay network with DF. For the
design, however, they adopt the Lloyd algorithm with Euclidian-distance as the design metric.
This paper takes a different tack by analyzing the loss due to quantization in a cooperative
wireless network. We start by introducing the performance loss as the loss in the maximum
achievable sum rate due to CSI quantization. To the best of our knowledge, an analysis of
this communication scenario under quantized CSI has not been presented before. Our analysis
includes proposing bounds on the performance loss and, then, using these bounds to formulate
and optimize quantization schemes for the network-wide CSI. Our main contributions are:
• derivation of a tight upper bound on the performance loss due to quantized CSI for the
sum-rate maximization problem in cellular networks;
• using the upper bound to formulate the optimal CSI quantizer design problem. By using
the proposed quantizer, the bound on the performance loss is shown to grow extremely
slowly with respect to the average link signal-to-noise ratio (SNR) and, as is more common,
decreases exponentially with the number of quantization bits;
• investigating the optimal allocation of CSI quantization bits across the wireless channels
to minimize the performance loss. It is shown that most of the quantization bits should be
used for the links that contribute the most to the sum-rate;
• a discussion of the choice of the central node to show that this choice can have a significant
effect on the CSI required to achieve a given performance target.
The remainder of this paper is organized as follows. Section II presents the system and CSI
quantization models. Section III derives an upper bound on the performance loss due to quantiza-
tion. Section IV then formulates the optimization problem for quantizer design and presents the
corresponding performance analysis. This is followed by Section V, which investigates optimal
bit allocation to minimize the upper bound on the performance loss due to quantized CSI and
December 6, 2011DRAFT
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discusses the selection of the central node. Finally, Section VI concludes the paper.
II. SYSTEM MODEL
The network model comprises NS source nodes (S1...SNS) communicating with a single
destination D through a single relay node R. To avoid multiuser interference, each source is
allotted an orthogonal channel. This model most closely represents the uplink of a relay-assisted
cellular network, where due to the unavailability of direct source-destination (Si-D) links, a relay
node is deployed between the source nodes (mobile users) and the destination (base station) to
facilitate communication.
Transmission occurs in two consecutive time slots: the first time slot is dedicated to source-
relay transmission, while in the second time slot, the relay, using DF, forwards the source
messages to the destination. We further assume that the receiver, in any specific link, knows the
CSI of that specific link exactly, e.g., via adequate training at the start of each the transmission
phase. Such channel estimation is generally necessary to demodulate and is not an additional
requirement imposed by the resource allocation process.
We assume there exists a central node that collects the quantized network-wide CSI. This node
is responsible for optimal power allocation at the relay by using the available quantized CSI.
Since the relay uses DF, only the channel magnitudes are required. Specifically, to calculate the
optimal power allocation required at R, the central node needs the magnitudes of the R-D and
all the Si-R channels.
We assume that the long-term average channel powers of all links are known a priori at the
central node. These average powers are functions of the large scale fading parameters of the
links that vary slowly as compared to the instantaneous channel values. The channel for a link
between a transmitter X and receiver Y is denoted by gXY and the corresponding normalized
channel power is defined as hXY = |gXY|2/E[|gXY|2]. Here, E[ · ] denotes expectation. Since the
average power, E[|gXY|2], is known at the central node, we focus on quantizing the normalized
channel power, hXY. The probability density functions (pdf) of all the normalized channels are
assumed identical for all links. For the random normalized channel power h, fH(h) and FH(h)
denote, respectively, the pdf and cumulative distribution function (cdf). Finally, we assume fH(h)
is bounded and has a bounded derivative almost everywhere.
The central node is to be given some knowledge of the channel powers of all the links in the
December 6, 2011DRAFT
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network. In the case of quantized CSI for the link X-Y with log2N bits for quantization, the
quantization rule q[hXY] is implemented as follows:
• the range [0,∞) is divided into N + 1 disjoint quantization intervals defined by their
boundaries {qn}N
n=−1where q−1 = 0 and qN = inf{h ≥ 0 : FHXY(h) = 1}, i.e., the
maximum possible value of h (for many pdfs, qN= ∞.). Note that N - and so the boundaries
(and associated intervals) - may be different for different channels.
• The receiver node, Y , observes its instantaneous normalized channel power hXY and when
this value falls within the n-th interval, i.e., hXY ∈ [qn−1,qn), the index n is fed back to
the central node.
• The central node then assumes the quantized channel power as q[hXY] = qn−1, i.e., the
most conservative value is chosen so the resulting sum rate obtained can be guaranteed.
On receiving the network-wide CSI, the central node calculates the power allocation (or
equivalently the rate allocation) at the relay node for all the sources. The relay has a power
constraint of PR. The resource allocation problem for sum-rate maximization is:
max
P
NS
?
i=1
Ri
(1)
subject to: 1TP ≤ PRD,
(2)
where Riis the rate achieved by source Siand 1 is a length-Nsvector of ones. In (2), PRD=
|gRD|2(PR/σ2) is the SNR at the destination (σ2denotes the noise variance and PRthe power
at the relay). Accounting for the R-D channel gain within the power constraint simplifies the
notation in the upcoming analysis. The optimization is over the vector P = [P1,...,PNS]Twhich
also includes the R-D channel gain. Pithen denotes the receive SNR (at the destination) that
the relay node provides to the source node Si. This SNR is the actual power allocated by the
relay to source Simultiplied by the factor of |gRD|2/σ2.
Let PSiR= hSiR(PS/σ2), where PSis the source transmit power. Then Riis given by [3]:
Ri= min(C(PSiR),C(Pi)),
(3)
with C(p) = ln(1 + p), i.e., rate is measured in nats.
To further simplify the notation, we express PRDand PSiRin terms of the normalized channel
powers, by writing PRD= γRDhRDand PSiR= γSiRhSiR, where γRD= (PR/σ2)E[|gRD|]2and
γSiR= (PS/σ2)E[|gSiR|2] are the average SNR for the R-D and Si-R links, respectively.
December 6, 2011DRAFT
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Let R∗
idenote the optimal transmission rate of source Siobtained by solving (1) assuming
perfect CSI. Similarly, let Rq∗
i denote the solution to the same problem using quantized CSI, i.e.,
the solution to (1) when one replaces PSiRand PRDwith q[PSiR] = γSiRq[hSiR] and q[PRD] =
γRDq[hRD]. Our main goal is to investigate the performance loss due to quantization, i.e. the
difference between the sum-rate found by solving (1) with perfect and quantized CSI. We address
this problem in the next section by deriving tight bounds on the performance loss.
III. UPPER BOUND ON PERFORMANCE LOSS
Throughout this paper, the term performance loss or simply loss refers to the difference
between the optimal sum-rate for the perfect and quantized CSI scenarios. In this section, we
provide an upper bound on this loss in terms of the quantization levels and CSI statistics. This
bound is then used in Section IV to optimize the quantizer and eventually derive the optimal bit
allocation across the links in Section V. The performance loss is defined as
∆ =
NS
?
i=1
∆i=
NS
?
i=1
(R∗
i− Rq∗
i),
(4)
where ∆irepresents the rate loss seen by source Si. We are interested in the expectation of this
loss, i.e., the expected value of (4) over the channel variables. For each node i define
E[∆i] = E [R∗
i− Rq∗
i] = E [min(C(PSiR),C(P∗
i)) − min(C(q[PSiR]),C(Pq∗
i))].
(5)
In (5), P∗
iand Pq∗
i
are, respectively, the optimal power (including the channel gain) allocated
by the relay to source Siin the perfect CSI and quantized CSI cases.
Due to the function min(·,·) in (5) the integration region is divided into four distinct sets. In
order to distinguish these sets, for the source Si, define Ai= {h : PSiR≤ P∗
Bi= {h : q[PSiR] ≤ Pq∗
hS1R,hS2R...hSNSR,hRD
be quantized. The sets Aiand Biare, respectively, the regions where the source-relay channel
i} and similarly,
i}. Here, h =
??T
is the vector of variables to
capacity is the bottleneck for the perfect and quantized CSI scenarios. By definition, the capacity
function C(·) is increasing and (5) can be expressed as
E[∆i] =
?
?
h∈Ai∩Bi
(C(PSiR) − C(q[PSiR]))fH(h)dh +
?
h∈Ac
i∩Bi
(C(P∗
i) − C(q[PSiR]))fH(h)dh
+
h∈Ai∩Bc
i
(C(PSiR) − C(Pq∗
i))fH(h)dh +
?
h∈Ac
i∩Bc
i
(C(P∗
i) − C(Pq∗
i))fH(h)dh,
(6)
December 6, 2011DRAFT
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where Ac
iand Bc
irepresent the complements of Aiand Bi. ¿From the definitions of Aiand Bi,
C(P∗
i) ≤ C(PSiR) ∀h ∈ Ac
i∩ Bi,
(7)
C(PSiR) ≤ C(P∗
i) ∀h ∈ Ai∩ Bc
i.
(8)
Now from (6), (7), and (8) we have the following upper bound on the performance loss
E[∆i] ≤ ∆SiR+ ∆RD,i,
(9)
where∆SiR=
?
?
h∈Bi
(C(PSiR) − C(q[PSiR]))fH(h)dh,
(10)
and∆RD,i=
h∈Bc
i
(C(P∗
i) − C(Pq∗
i))fH(h)dh.
(11)
Equation (10) is an upper bound on the average performance loss due to quantization of the link
Si-R and is found by merging the first two terms of (6) using (7) ; similarly, (11) defines an
upper bound the performance loss due to the power allocated to source Sibased on quantization
of the link R-D derived from the third and fourth terms of (6) using (8).
A similar analysis can be proposed to derive a lower bound on (6) with an expression
resembling (9) where the integration region is replaced by the set Ai and Ac
i. Since we are
focusing on the achievable rate regions for the proposed system model, we will continue with
the upper bound on performance loss which ultimately leads to a lower bound on the achievable
rates. In the next section, we further bound the terms in (9).
A. Loss due to the Quantization of Si-R Links
In this section we focus on analyzing (10). From our assumption that all channels are mutually
independent, the joint pdf of h, fH(h), has a product form. However, in (10), the region
of integration is coupled across the channel variables making the integration complicated. To
overcome this problem we define a larger region, Bt
i, which includes Bi, and results in a product
form for the integral region in (10). The set Bt
iis defined as
Bt
i= {h : q[PSiR] ≤ PRD} = {h : γSiRq[hSiR] ≤ γRDhRD}.
(12)
To see Bi⊆ Bt
inote that ∀i and h ∈ Biwe have q[PSiR] ≤ Pq∗
i
≤ q[PRD] ≤ PRD⇒ h ∈ Bt
i.
December 6, 2011 DRAFT
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8
Since Bi⊆ Bt
i, we achieve an upper bound on the term ∆SiR. From (10) and (12):
∆SiR≤
?
?∞
h∈Bt
i
?
(C(PSiR) − C(q[PSiR]))fH(h)dh
=
0
h′≥
γSiR
γRDq[h]
hSiR+ γ−1
q[hSiR] + γ−1
(C(γSiRh) − C(γSiRq[h]))fHRD(h′)fHSiR(h)dh′dh
= E
?
ln
?
SiR
SiR
?
(1 − FHRD(αiq[hSiR]))
?
,
(13)
with αi= γ−1
SiRγRDand using C(p) = ln(1 + p). The expectation is over hSiR.
The term 1−FHRD(αiq[hSiR]) in (13) shows that, in general, the quantization of one Si-R link
depends on the distribution of the R-D link channel power. This renders the quantization opti-
mization intractable. We therefore upper-bound (13) by dropping the term 1−FHRD(αiq[hSiR]):
∆SiR≤ Eh
?
ln
?
h + γ−1
q[h] + γ−1
SiR
SiR
??
.
(14)
The upper-bound in (14) can be minimized with respect to the quantization rule q[·]. This
ultimately leads to the optimal quantization levels for hSiR, the normalized Si-R channel power.
Crucially, by using (14) as the objective function to optimize the quantization, the quantization
levels found for quantization of Si-R link depend on the statistics the Si-R channel power only.
As is shown in the following sections, adopting the upper bound in (14) also leads to separable
problems for the optimal quantization design and bit allocation.
After finding the optimal quantization levels based on (14) in Section IV, we will return to (13)
in Section V-A to discuss the optimal bit allocation across the wireless channels.
B. Loss due to the Quantization of R-D Link
The analysis for the R-D link follows the same approach as that of Section III-A. Follow-
ing (11) we define the R-D loss component ∆RDas
∆RD=
NS
?
i=1
∆RD,i=
NS
?
i=1
?
Bc
i
(C(P∗
i) − C(Pq∗
i))fH(h)dh.
(15)
The problem of evaluating (15) for a general distribution function is intractable. Therefore, similar
to the analysis in the previous section, we extend the region of integration in (15), resulting in
an upper bound on ∆RD. To this end, define B = ∪NS
i=1Bc
iwhich readily yields Bc
i⊂ B. Since
December 6, 2011DRAFT
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9
the integrand in (15) is positive, we have
∆RD,i=
?
Bc
i
(C(P∗
i) − C(Pq∗
i))fH(h)dh ≤
?
B
(C(P∗
i) − C(Pq∗
i))fH(h)dh
(16)
⇒ ∆RD≤
?
i}NS
B
NS
?
i=1represent the optimal power allocation maximizing the
i=1
(C(P∗
i) − C(Pq∗
i))fH(h)dh.
(17)
The set of power variables {Pq∗
sum-rate under the quantized CSI; therefore
NS
?
?NS
i=1
C(Pq∗
i) ≥
NS
?
i=1
C
?q[PRD]
PRD
P∗
i
?
,
(18)
which is true since
?
q[PRD]
PRDP∗
i
i=1is a valid, and likely suboptimal, solution to the max sum-rate
problem satisfying the power constraint. From (17) and (18) it follows that
∆RD≤
?
h∈B
NS
?
NS
?
i=1
?
C(P∗
i) − C
?q[PRD]
PRD
P∗
i
??
fH(h)dh
≤
?
h∈B
i=1
ln
?
1 + P∗
1 +q[PRD]
i
PRDP∗
i
?
fH(h)dh.
(19)
For any channel power h ∈ B we have?NS
in (19) is a concave function of P∗
i=1P∗
i≤ PRD. On the other hand, the integrand
ifor Pi≥ 0 and all i. Using Jensen’s inequality, we have
1
NS
NS
?
i=1
ln
?
1 + P∗
1 +q[PRD]
i
PRDP∗
i
?
≤ ln
?
1 +PRD
1 +q[PRD]
NS
NS
?
.
(20)
Using (19) and (20), we have
∆RD≤ NS
?
h∈B
ln
?
1 +PRD
1 +q[PRD]
NS
NS
?
fH(h)dh = NS
?
h∈B
ln
?
NS
γRD+ hRD
NS
γRD+ q[hRD]
?
fH(h)dh.
(21)
In order to proceed with the evaluation of (21), we need to simplify the definition of set B.
This can be achieved by applying the following lemma.
Lemma 1: If the solution to the sum rate maximization problem in (1) for some channel vector
h leads to P∗
i≤ PSiRfor some source Si, then the following inequality is valid
NS
?
i=1
PSiR≥
NS
?
i=1
P∗
i.
(22)
Proof: See Appendix A.
December 6, 2011DRAFT
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10
According to the definition of set B, we have Pq∗
i
< q[PSiR] for at least one i. Then by
Lemma 1,?NS
B = ∪NS
i=1q[PSiR] ≥ q[PRD]. This leads to an alternative representation of the set B as
i=1Bc
i= {h : ∃ i such that q[PSiR] ≥ Pq∗
i} =
?
h :
NS
?
i=1
q[PSiR] ≥ q[PRD]
?
.
(23)
According to (23), the integration region in (21) is defined in terms of the quantized values of
channel powers. Due to the complexity of working with quantized random variables, we will
introduce a slightly larger set B′described by the true Si-R channel powers. Define B′as
B′=
?
h :
NS
?
i=1
PSiR≥ q[PRD]
?
⊃ B.
(24)
By defining Y =?NS
i=1PSiR=?NS
∆RD≤ NS
i=1γSiRhSiRit follows for (21) that
?
?
h + NSγ−1
q[h] + NSγ−1
h + NSγ−1
RD
q[h] + NSγ−1
RD
?
?
?
?∞
?∞
?∞
h∈B
ln
NS
γRD+ hRD
NS
γRD+ q[hRD]
?
?
fH(h)dh
≤ NS
h∈B′ln
NS
γRD+ hRD
NS
γRD+ q[hRD]
?
fH(h)dh
= NS
0
?
y≥γRDq[h]
?
ln
RD
RD
?
fHRD(h)fY(y)dydh
= NS
0
ln
?
?
(1 − FY(γRDq[h]))fHRD(h)dh,
≤ NS
0
ln
h + NSγ−1
q[h] + NSγ−1
RD
RD
fHRD(h)dh,
(25)
where the first inequality uses B ⊂ B′and the next separates out the integral into an integral
over Y and and hRD. The next step completes the integration over Y . The final inequality drops
the (1 − FY(γRDh)) term as in the previous section.
This series of steps leaves us with the same objective function as that of (13) used for the
optimal quantization problem. Therefore, based on our analysis, the same quantization structure
is optimal for the upper bounds derived for all links across the network. After investigating the
optimal quantizer in the following section, we return to (14) and (25) in Section V-B for the
analysis of the optimal bit allocation.
IV. DERIVATION OF THE OPTIMAL QUANTIZER
The general structure of a quantizer requires quantization intervals followed by a choice of
quantization levels. As described in Section II, our approach requires the quantization level to
December 6, 2011DRAFT
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11
be the lowest value of the chosen quantization interval. The quantizer is, therefore, completely
characterized by the quantization vector q = [q0,q1,...,qN−1], i.e., a vector comprising the N
non-zero quantization levels (note that by definition, q−1= 0). Then, according to the results
from the previous section, for both the Si-R and R-D channels, the optimal quantizer is the one
which minimizes δ(q) where
δ(q) = E
?
ln
?
h + γ−1
q[h] + γ−1
??
=
?∞
0
ln
?
h + γ−1
q[h] + γ−1
?
fH(h)dh.
(26)
In (26), γ is the average SNR and equals γSiRfor the case of the Si-R channel and γRDfor the
R-D channel. Moreover, the expectation in (26) is with respect to h, the instantaneous channel
power of the corresponding link, whose distribution is that of hSiR or hRD for the Si-R and
R-D links, respectively. For the function δ(q) in (26) we have
δ(q) =
n=N−1
?
n=−1
In=
n=N−1
?
n=−1
?qn+1
qn
ln
?h + γ−1
qn+ γ−1
?
fH(h)dh,
(27)
i.e., Inis as the component of the expectation integral over the interval [qn,qn+1). Note that, as
defined earlier, we set two fixed quantization levels q−1= 0 and qN= +∞ (or qN= inf{h ≥
0 : FH(h) = 1} if the pdf has finite support).
In our model we only consider channels distributions with finite average power. More specif-
ically, we assume E[h] = 1. This assures δ(q) < ∞ and consequently, from the continuity of
fH(·), the function δ(·) is differentiable with respect to the quantization levels qn, 0 ≤ n ≤ N−1.
Therefore, the optimal quantization level qnsatisfies the following
∂
∂qnδ(q) =
∂
∂qn(In−1+ In) = 0.
(28)
The following theorem presents the fundamental iterative relation between the optimal quanti-
zation levels and is a key contribution of this paper.
Theorem 1: The quantization levels of the optimal quantizer minimizing δ(q) in (26) satisfy
(qn+ γ−1)ln
?
qn+ γ−1
qn−1+ γ−1
?
=FH(qn+1) − FH(qn)
fH(qn)
,0 ≤ n ≤ N − 1.
(29)
Proof: Refer to Appendix B.
The quantization levels proposed in Theorem 1 are optimal for a variety of distributions,
including the uniform distribution. We first investigate the structure of the optimal quantizer for
the uniform distribution and then extend the results to more general distributions of channel
power.
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A. Optimal Quantization for the Uniform Distribution
In this section we focus on the uniform distribution for the channel power and present the
optimal quantization vector q which minimizes δ(q) in (26). For the uniform distribution and
from the assumption E[h] = 1 we have fH(h) =1
2for 0 ≤ h ≤ 2 and fH(h) = 0 for h > 2.
Since the pdf is a constant, for any 0 ≤ qn< qn+1≤ 2, we have
FH(qn+1) − FH(qn) = fH(qn)(qn+1− qn).
(30)
This will simplify the optimality condition proposed in Theorem 1. Essentially it follows from (29)
and (30) that
(qn+ γ−1)ln
?
qn+ γ−1
qn−1+ γ−1
?
= qn+1− qn,0 ≤ n ≤ N − 1,
(31)
where qN= inf{h ≥ 0 : FHXY(h) = 1} = 2. Setting n = N − 1 and adding (qN−1+ γ−1) to
both sides leads to
?qN−1+ γ−1
Theorem 2 now specifies all N non-zero quantization levels for the uniform distribution.
(qN−1+ γ−1)
?
1 + ln
qN−2+ γ−1
??
= 2 + γ−1.
(32)
Theorem 2: The n-th quantization level of the optimal quantizer for the uniform distribution
is given by
qn=
?i=n
i=0ri
γ
− γ−1, 0 ≤ n ≤ N
(33)
where riis an iterated logarithmic sequence defined as
ri= 1 + lnri−1, 1 ≤ i ≤ N.
(34)
Finally, the optimal value of r0satisfies
N
?
n=0
rn= 2γ + 1.
(35)
Proof: Refer to Appendix C.
Note that in the high-SNR regime, we can ignore the γ−1term and we have ri= qi/qi−1,0 ≤
i ≤ N, i.e., the ratio of consecutive quantization levels.
While Theorem 2 clearly defines the optimal quantizer for the uniform distribution, this
distribution is impractical. Therefore, in the next section, we extend this quantizer to a general
distribution function. This is an intractable problem for arbitrary N and we focus on the case
of asymptotically large N.
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B. Asymptotically Optimal Quantization for General Distributions
The iterated logarithm in (34) is a direct result of (30) which holds exactly for the uniform
distribution. For a general distribution function fH(h), (30) is the result of a first-order Taylor-
series approximation of FH(qn+1) at h = qn. This approximation becomes accurate for large N,
i.e., FH(qn+1)−FH(qn) → fH(qn)(qn+1− qn) as qn→ qn+1. As N → ∞, we have qn+1−qn→
0,n = 0,1,...,N − 2. Then the optimality condition for the quantization levels q0,q1,...,qN−2
presented in (29) is consistent with that of the uniform quantizer in (31). As a result, the quantizer
structure presented through Theorem 2 is asymptotically optimal.
However, this statement is not true for all n, specifically, n = N−1. For a general distribution,
qN= ∞ and the Taylor series approximation cannot be applied to the last interval, [qN−1,∞).
Therefore, for a general distribution, qN−1 remains unspecified. The value of qN−1 should be
chosen such that IN−1 in (27) is small, in turn making δ(q) → 0 with N → ∞. We call a
quantizer consistent if δ(q) → 0 as N → ∞. Unfortunately, using (33)-(35) results in a bounded
value for qN−1even if N → ∞, in turn making IN−1always non-zero. This would, therefore,
lead to an inconsistent quantizer. In this respect, (35) needs to be modified.
The key is to realize that by using (33) and (34) finding the proper value of qN−1is equivalent to
choosing an appropriate r0. This choice is based on the behavior of fH(h) at large h; specifically,
the value of r0needs to increase with N to guarantee that all Inapproach zero for large N.
This can be achieved by replacing (35) with the following
N
?
n=0
rn= κNγ + 1,
(36)
where the constant 2 in (35) is replaced with κN. Here κN increases with N to ensure that the
quantizer is consistent - for large N, h is quantized such that Inbecomes sufficiently small ∀n.
The following theorem develops an appropriate choice of κNfor a general distribution function.
Theorem 3: Consider the proposed quantizer with r0found from (36). For a general distribu-
tion function fH(h) with cdf FH(h), the quantization loss, defined in (26), is bounded by
δ(q) ≤ O
?ln(κ∗
N)
N
?
,
(37)
where κ∗
N= F−1
Proof: Refer to Appendix D.
H(1 − N−1).
December 6, 2011DRAFT
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14
Note that, using (33) and (36), the suggested choice of κ∗
Nensures that the probability of the
channel falling in the final interval, i.e., Pr{(H > qN−1}, is almost 1/N.
Corollary 1: For any channel with E[h] = 1, κ∗
N< N and therefore the quantizer is consistent.
Corollary 2: Under Rayleigh fading, i.e., H ∼ e−h, and N → ∞, we have κ∗
N= lnN and
δ(q) ≤ O
?lnlnN
N
?
.
(38)
Corollary 3: For channel power uniformly distributed in [0,2), for any N > 1, κ∗= 2 and
δ(q) ≤
c
N
(39)
for some fixed constant c > 0.
C. Performance loss for High Average SNR
In the previous section we considered the asymptotic N → ∞ case, but for finite SNR. In this
section we consider the reverse and investigate how the quantizer levels (equivalently ratios) must
change as a function of SNR; specifically we investigate the high-SNR regime. However, we do
not assume that N → ∞. We are, therefore, interested in the limiting behavior of δ(q) defined
in (26) for the optimal quantizer vector q designed using Theorems 2 and 3 when γ → ∞.
To illustrate the importance of this analysis we first consider the performance of a fixed
quantizer, denoted by q′, for Rayleigh fading. The quantization levels in q′do not change with
SNR, specifically q′
0is constant with γ. Then using q′
−1= 0 and the concavity of ln(·) we have
the following lower bound on I−1(defined in (27))
?q′
From (40) we see that δ(q′) ≥ O(lnγ) as γ → ∞. The loss in sum-rate would, therefore, be
I−1=
0
0
ln(γh + 1)fH(h)dH ≥1
2q0e−q0ln(γq0+ 1) ≃q0
2lnγ.
(40)
at least O(lnγ). On the other hand, as γ → ∞, the overall sum rate is also O(lnγ). Therefore
for a fixed quantizer, at least a fixed percentage of the transmission rate is lost due to CSI
quantization. For small values of N this loss becomes quite significant.
The key contribution of this section is to show that, in the limit as γ → ∞, our proposed
quantizer results in a δ(q) that grows at a pace much slower than lnγ. As a consequence, the
relative rate loss due to CSI quantization tends to zero as the SNR grows. This is true for a wide
class of channel distributions and even for small values of N. The next theorem summarizes the
main results on the high SNR behavior of δ(q) for the proposed quantizer.
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Theorem 4: In the high SNR regime, where γ → ∞, the quantizer described by Theorem 2
leads to a quantization loss δ(q) (defined in (26)) which scales as the N-th order iterated
logarithm of the average SNR, ln(N+1)γ, where ln(0)x = x and ln(n)x is defined as
ln(n)x = 1 + ln
?
ln(n−1)x
?
, n > 0,x ≥ 1.
(41)
Proof: Refer to Appendix E.
Theorem 4 shows that the loss, δ(q), approaches infinity extremely slowly - at a rate of
O(ln(N+1)γ). This result is valid for any finite valued channel distribution function and as long
as N > 1, the relative performance loss vanishes with γ → ∞.
D. Numerical Validation
To validate our analysis we investigate the loss defined in (26) through computer simulations.
Fig. 1 plots the objective function δ(q) in bits (as opposed to nats) for two quantizers: the
first adapts to the average SNR by setting the quantization levels according to Theorem 2. The
second quantizer is similar but the levels are optimized for an average SNR of 10dB and then
kept fixed. Note that δ(q) is, for any specific link, the upper bound on the loss in the rate of
that link - see (26). This figure is obtained by numerically generating channel powers drawn
from the uniform distribution - we use Theorem 2 to obtain the quantizer - and then averaging
the resulting δ(q) over many channel realizations. As Fig. 1 shows, even for N = 3 there is a
significant difference between the performance of adaptive and fixed quantizers. Importantly, as
suggested in Section IV-C, with a fixed quantizer the loss is linear in average SNR (measured
in dB), while, for the optimal quantizer the loss grows, but very slowly.
In a second test scenario, we simulate a network comprising two source nodes and a relay
node. We assume Rayleigh fading with the same average SNR for all links. Also all channels
deploy the same kavg-bit quantizer (N + 1 = 2kavg). We compare the optimal quantizer and the
max-entropy quantizer [14], i.e., the quantizer which maximizes the entropy of CSI messages
by creating equi-probable quantization intervals. Fig. 2 illustrates the performance loss for the
two quantizers. The curves in Fig. 2 show the percentage of the perfect CSI rate which is lost
to quantization as a function of average SNR. As predicted, this fraction goes to zero for the
optimal quantizer while, for the max-entropy quantizer, it increases as a function of average
SNR and converges (from below) to a constant.
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- Available from Ehsan Karamad · Jun 1, 2014
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