Error Exponent Regions for Gaussian Broadcast and Multiple-Access Channels

Abstract

In modern communication systems, different users have different requirements for quality of service (QoS). In this work, QoS refers to the average codeword error probability experienced by the users in the network. Although several practical schemes (collectively referred to as unequal error protection schemes) have been studied in the literature and are implemented in existing systems, the corresponding performance limits have not been studied in an information-theoretic framework. In this paper, an information-theoretic framework is considered to study communication systems which provide heterogeneous reliabilities for the users. This is done by defining individual probabilities of error for the users in the network and obtaining the fundamental tradeoffs of the corresponding error exponents. In particular, we quantify the reliability tradeoff by introducing the notion of error exponent region (EER), which specifies the set of error exponent vectors that are simultaneously achievable by the users for a fixed vector of users' rates. We show the existence of a tradeoff among the users' error exponents by deriving inner and outer bounds for the EER. Using this framework, a system can be realized, which can provide a tradeoff of reliabilities among the users for a fixed vector of users' rates. This adds a completely new dimension to the performance tradeoff in such networks, which is unique to multiterminal communication systems, and is beyond what is given by the conventional performance-versus-rate tradeoff in single-user systems. Although this is a very general concept and can be applied to any multiterminal communication system, in this paper we consider Gaussian broadcast and multiple-access channels (MACs).

Full text

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Error Exponent Region for Gaussian Multiple Access
Channels and Gaussian Broadcast Channels
Lihua Weng, Achilleas Anastasopoulos, and S. Sandeep Pradhan
Electrical Engineering and Computer Science Dept.
University of Michigan, Ann Arbor, MI 48109-2122
{lweng,anastas,pradhanv}@umich.edu
1Introduction
Errorexponentsforsingle-useradditivewhiteGaussiannoise(AWGN)channelsprovidebounds
on the rate of exponential decay of the average probability of error as a function of the block
length of the random codebooks. In this paper, we only consider Gallager-type lower bounds
of error exponents [1,2]. For simplicity, we use the term “error exponent” to mean the maxi-
mum of random coding exponent and expurgated exponent throughout this paper, though ac-
tually should be called a lower bound of error exponents. The concept of error exponents was
extended to Gaussian MAC channels [3, 4], where random coding exponents were derived.
Recently, Zheng et al. considered error exponents, in high signal-to-noise ratio (SNR), for
single-user wireless multi-input-multi-output (MIMO) channels [5], and for wireless MIMO
multiple access channels [6]1. Conceptually, the error exponent is a function of the channel
capacity C and the transmission rate R in a single user channel (see Fig. 1(a)). However,
error exponents for a multi-user network is quite different from the error exponent for a single-
user channel. Each user in a multi-user network is associated with his own error exponent.
Therefore, there are multiple error exponents for a given multi-user channel. In contrast to all
previous works, however, we make the following observations. Consider the capacity region
of a two-user multiple access channel as shown in Fig. 1(b). As expected, the error exponents
for the two users are functions of both the transmission rate point A and the channel capacity.
However, unlike the case in a single user channel where channel capacity boundary is a single
point, in a multi-user channel we have multiple points on the capacity boundary (e.g. A1,A2in
Fig. 1(b)). Thus it is expected that one can get different error exponents depending on which
particular point on the capacity boundary is considered. Furthermore, it might be possible to
trade off error exponents between users by considering different points on the capacity bound-
ary. For instance, consider a rate point A with respect to a boundary point A1in Fig. 1(b). It
is intuitive to expect that the error exponent for user 1 is smaller than that of user 2. On the
other hand, if we consider point A with respect to point A2, we then expect error exponent
for user 1 is larger than that of user 2. Therefore, tradeoff of error exponent between users
might be possible by considering different points on the capacity boundary. Similar behavior
can be observed in other multi-user channels, e.g. the broadcast channels in Fig. 1(c). It is our
intention in this paper to formalize these ideas and show that such tradeoff indeed exists and
suggest a constructive strategy to achieve it.
1Zheng et al. use the term “diversity gain”, but it’s just another form of the error exponent in high SNR.

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rate point, the errorexponent region consists all achievable errorexponents when the channel is
transmitted at that rate point. For example, the error exponent region for a single user channel
with transmission rate R is a line segment from the origin to the error exponent E(R) (see
Fig. 2(a)). For a multiple access channel operated at rate point A (see Fig. 1(b)), the error
exponent region is a two-dimensional region which depends on rates R1and R2(see Fig. 2(b)).
The concept of the error exponent region is very similar to the channel capacity region (CCR).
In the EER, it is possible to increase user 1 error exponent by decreasing user 2 error exponent.
This is similar to increasing user 1 transmission rate by reducing user 2 transmission rate in the
CCR. However, there is a fundamental difference between CCR and EER. For a given channel,
there is only one CCR. One the other hand, an EER depends on the channel operating rate
point, and for a given channel, there are numerous EERs depending on which operating point
we consider. Therefore, when we refer to an EER, we need to specify both the channel and the
channel operating rate point.
The rest of the paper is structured as follows. In section 2, we review error exponents
for a two-user Gaussian MAC channel and we compare the error exponent region derived by
superposition with the the error exponent regions derived by other schemes. We extend this
result to Gaussian broadcast channels in section 3, and conclude our work in section 4.
A(R?1?,R?2?)?
A?1?
A?2?
rate 2?
0?
0?
R?
single-user channel?multiple access channel?
0?
A?1?
A?2?
broadcast channel?
C?
rate 1?rate 1?
rate 2?
A(R?1?,R?2?)?
(? a? )?(b? )?(c? )?
Figure 1: Capacity region for single user , multiple access , and broadcast channels.
2Error exponent region for Gaussian MAC channels
Error exponent for a single-user AWGN channel is the maximum of random coding exponent
and expurgated exponent [1,2]. Consider a two-user Gaussian MAC channel
Y
= X1+ X2+ Z,
(1)
where X1and X2are channel inputs for user 1 and user 2 and Z is white Gaussian noise.
Gallager derived random coding exponent for this channel using maximum-likelihood joint
decoding [3]. There are three types of error using jointly decoding. Type 1 error denotes
when user 1 codeword is decoded wrong, but user 2 codeword is decoded correctly. Type
2 error denotes when user 2 codeword is decoded wrong, but user 1 codeword is decoded
correctly. Type 3 error denotes when both users’ codewords are decoded as wrong codewords.
Denote SNR1and SNR2as the signal-to-noise ratios for user 1 and user2, and SNR =

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0?
(E?1?,E?2?)?
(E'?1?,E'?2?)?
error exponent region associate with?
rate point A(R?1?,R?2?) (multi-user)?
exponent 2?
0?
E(R)?
E?
error exponent region associate?
with rate R (single-user)?
EER(R?1?,R?2?)?
error?
exponent 1?
(? a? )?(b? )?
Figure 2: Error exponent region.
SNR1+ SNR2(noise power is normalized to 1). We use the notation E(R,SNR) to denote
the error exponent for a single user channel at rate R with power constraint SNR. Similarly,
we use the notation Et3(R1+ R2,SNR1,SNR2) to denote the error exponent for type 3 error
in a two-user MAC channels2. Therefore, the error exponents for user 1 and user 2 using
superposition encoding are
Es
Es
1
= min{E(R1,SNR1),Et3(R1+ R2,SNR1,SNR2)}
= min{E(R2,SNR2),Et3(R1+ R2,SNR1,SNR2)}.
An explicit expression for Et3is given in the appendix. For an important case when SNR1=
SNR2, there is closed form formula for Et3(hence Es
Et3need to be solved numerically.
The capacity region of a two-user Gaussian MAC channel can be divided into four regions
R12, R13, R23, and R3depending whether type 1 error exponent Et1, type 2 error exponent
Et2, or type 3 error exponent Et3dominates (see Fig. 3). In the region R12, Et1 ≤ Et3and
Et2 ≤ Et3. The error exponent for user 1 is Es
E(R2,SNR2). This is the region where the appearance of the second (first) user doesn’t affect
the error exponent of the first (second) user. In this region, we can not increase error exponents
for either of the users. In the region R13, Et1 ≤ Et3 ≤ Et3. The error exponent for user 1
is Es
although we can not increase first user’s error exponent by reducing the second user’s error
exponent, it is possible to increase the second user’s error exponent by reducing the first user
errorexponentbecausethedominanterrorforuser2isatype 3error. Asimilarresultalsoholds
for region R23by changing the role of user 1 and user 2 in the R13region. In the region R3,
Et3≤ Et1and Et3≤ Et2. The error exponents are Es
In this region, type 3 error is dominant over both type 1 and type 2 errors. It is possible
to increase the first (second) user’s error exponent by reducing the second (first) user’s error
exponent.
As explained in section 1, error exponent is a function of both the transmission rate and
the channel capacity. For a multiple access channel, one might get different error exponents
(2)
(3)
2
1and Es
2). In general (SNR1?= SNR2),
1= E(R1,SNR1) and for user 2 is Es
2=
1= E(R1,SNR1) and for user 2 is Es
2= Et3(R1+ R2,SNR1,SNR2). In this region,
1= Es
2= Et3(R1+ R2,SNR1,SNR2).
2Error exponent functions for type 1 error and type 2 error are the same as the single-user error exponent
function, i.e. Et1(R1,SNR1) = E(R1,SNR1) and Et2(R2,SNR2) = E(R2,SNR2).

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00.20.40.60.811.21.4
0
0.2
0.4
0.6
0.8
1
1.2
R12
R13
R23
R3
R1 (nats)
R2 (nats)
Figure 3: Regions where Etidominates; SNR = 20.
depending on which particular point on the capacity boundary is considered. For a Gaussian
MAC channel (or a wireless MIMO MAC channel), however, the capacity boundary happen
to be achieved by the same input distributions. That’s why all previous works in Gaussian
MAC channels and wireless MIMO MAC channels derived only one single point on the er-
ror exponent region (i.e. error exponents for user 1 and user 2 are fixed by the data rates for
user 1 and for user 2). That single point derived by previous works in fact implies a rectan-
gle error exponent region achieved by superposition encoding. Consider a two-user Gaussian
MAC channel with equal power constraint for user 1 and user 2 (i.e. SNR1 = SNR2). In
Fig. 4(a)), the dashed square is the achievable error exponent region (EER) by superposition.
Although the (EER) is still unknown for Gaussian MAC channels, we do propose a simple
scheme (time-division) to enlarge the achievable EER beyond what has been achieved by su-
perposition encoding. The error exponents for user 1 and user 2 by time-division are
Etd
1
= αE
?R1
α,SNR1
?
α
R2
1 − α,SNR2
?
(4)
Etd
2
= (1 − α)E
1 − α
?
,
(5)
where 0 < α < 1. In Fig. 4(a), the solid curve is the achievable EER by time-division. The
union of the superposition achievable EER and the time-division achievable EER is an inner
bound for the EER of a Gaussian MAC channel. We summarize in the following theorem.
Theorem 1: For a two-user Gaussian MAC channel with power constraints SNR1and SNR2
for user 1 and user 2 (noise power is normalized to one), the achievable EER(R1,R2) (inner
bound)isEERs(R1,R2)∪EERtd(R1,R2), wheresuperpositionachievableregion EERs(R1,R2)
and time-division achievable region EERtd(R1,R2) are
EERs(R1,R2) = {(e1,e2) : e1≤ Es
EERtd(R1,R2) = {(e1,e2) : e1≤ αE(R1
where Es
1,e2≤ Es
α,SNR1
2}
(6)
α
),e2≤ (1 − α)E(
R2
1 − α,SNR2
1 − α)}, (7)
1and Es
2are defined in (2), (3).

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for rate points inside region R12(see Fig. 3). What’s the maximum rate region we can increase
EERsby time-division? If we adjust α to make Etd
get the largest error exponent Etd
the shaded region is the region where Etd
where Etd
region where EERscan be enlarge by time-division.
Error exponents for user 1 and user 2 by time-sharing method are
2arbitrary small (but still positive), then we
1,maxfor user 1 by time-sharing method (see (4),(5)). In Fig. 5,
1,max> Es
2by reducing Etd
1. Similarly, we can obtain another region
1close to zero. The union of these two regions is the rate
2,max> Es
Ets
1
= αE
?R1
α,SNR1
?
?
(8)
Ets
2
= (1 − α)E
R2
1 − α,SNR2
?
,
(9)
where 0 < α < 1. We can also enlarge EERsby time-sharing instead of time-division, but
the shaded region by time-sharing in Fig. 5 is smaller than that by time-division.
00.05 0.10.150.20.250.30.35
0
0.05
0.1
0.15
0.2
0.25
0.3
E1
E2
(a) Error exponent achievable regions by su-
perposition and time-division
00.050.10.150.2 0.25 0.3 0.35
0
0.05
0.1
0.15
0.2
0.25
0.3
E1
E2
(b) Inner bound for the error exponent region
Figure 4: Error exponent achievable region (R1= 0.5, R2= 0.5) by time-division and superposition;
SNR = 20.
3Error exponent region for Gaussian Broadcast channels
Consider a two-user Gaussian broadcast channel
Y1 = X + Z1
Y2 = X + Z2,
(10)
(11)
where X is the channel input, and Y1and Y2are the channel outputs for user 1 and user 2. As-
sume noise power for Z1is σ2
malized). In contrast to Gaussian MAC channels, capacity boundary for Gaussian broadcast
channels is achieved by different input distributions. In Fig. 1(c), the rate point A1is achieved
1= 1 (normalized) and noise power for Z2is σ2
2= βσ2
1= β (nor-

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00.20.40.60.811.21.4
0
0.2
0.4
0.6
0.8
1
1.2
R1 (nats)
R2 (nats)
Figure 5: Rate region where Etd
1,max> Es
1; SNR = 20.
by input distributions N(0,α1SNR) and N(0,(1 − α1)SNR), but the point A2is achieved
by another input distributions N(0,α2SNR) and N(0,(1 − α2)SNR) (0 < α1< α2< 1).
Therefore, we expect the error exponents for rate point A evaluated with respect to A1are
different from those evaluated with respect to A2. We can also derive achievable EER by su-
perposition encoding for a two-user Gaussian broadcast channels. We summarize the result in
the following theorem.
Theorem 2: For a two-user Gaussian broadcast channel with power constraint SNR (noise
powers are normalized to 1 and β), the achievable EER(R1,R2) (inner bound) by superposi-
tion is
{(e1,e2) : e1≤ min{E(R1,αSNR),Et3(R1+ R2,αSNR,(1 − α)SNR)},
e2≤ min{E(R2,(1 − α)SNR
β
0 ≤ α ≤ 1}.
(12)
),Et3(R1+ R2,αSNR
β
,(1 − α)SNR
β
)},(13)
(14)
4Summary and Conclusion
In this paper, we consider error region for Gaussian MAC channels and Gaussian broadcast
channels. For a Gaussian MAC channel, Gallager-type superposition encoding derives only
one single point inside the error exponent region. For a Gaussian broadcast channel, however,
the channel capacity boundary corresponds to an error-exponent curve (instead of one single
point). A simple scheme (time-division) is used to increase achievable error exponent region
by superposition for Gaussian MAC channels. The concept of error exponent region is general
and it’s possible to extend the results in this paperto other channel models, like wireless MIMO
MAC channels and wireless MIMO broadcast channels.
5Appendix
Type 3 error exponent Et3(R3,SNR1,SNR2) is the maximum of type 3 random coding expo-
nent Et3,r(R3,SNR1,SNR2) and type 3 expurgated exponent Et3,ex(R3,SNR1,SNR2). The

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Et3,r(R3,SNR1,SNR2) =max
ρ,θ1,θ2{Et3,0(ρ,θ1,θ2) − ρR3}
?e√θ1θ2
1 + ρ
+ρ
2ln
θ1
(15)
Et3,0(ρ,θ1,θ2) = (1 + ρ)ln
?
+SNR2
−θ1+ θ2
2
?
1 +SNR1
θ2
?
,
(16)
where maximization is over 0 ≤ ρ ≤ 1, and 0 < θ1,θ2 ≤ 1 + ρ [3]. The expression for
Et3,ex(R3,SNR1,SNR2) is
Et3,ex(R3,SNR1,SNR2) =max
ρ,r1,r2{Et3,x(ρ,r1,r2) − ρR3}
(17)
Et3,x(ρ,r1,r2) = 2ρ(r1SNR1+ r2SNR2) + ρln[(1 − 2r1SNR1)(1 − 2r2SNR2)]
+ρ
2ln
?
1 +
SNR1
2ρ(1 − 2r1SNR1)+
2SNR1, and0 ≤ r2<
SNR2
2ρ(1 − 2r2SNR2)
2SNR2. Et3(R3,SNR1,SNR2)
?
,
(18)
wheremaximizationisover ρ ≥ 1, 0 ≤ r1<
has a closed form solution (i.e. no need to maximize over dummy parameters ρ, θ1, etc.) when
SNR1= SNR2
1
1
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