Optimal Transmission Strategy and Explicit Capacity Region for Broadcast Z Channels

Abstract

This paper provides an explicit expression for the capacity region of the two-user broadcast Z channel and proves that the optimal boundary can be achieved by independent encoding of each user. Specifically, the information messages corresponding to each user are encoded independently and the OR of these two encoded streams is transmitted. Nonlinear turbo codes that provide a controlled distribution of ones and zeros are used to demonstrate a low-complexity scheme that operates close to the optimal boundary.

Full text

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arXiv:0810.3226v1 [cs.IT] 17 Oct 2008
Optimal Transmission Strategy and Explicit
Capacity Region for Broadcast Z Channels
Bike Xie, Student Member, IEEE, Miguel Griot, Student
Member, IEEE, Andres I. Vila Casado, Student Member,
IEEE and Richard D. Wesel, Senior Member, IEEE
Abstract—This paper provides an explicit expression for the
capacity region of the two-user broadcast Z channel and proves
that the optimal boundary can be achieved by independent
encoding of each user. Specifically, the information messages
corresponding to each user are encoded independently and the
OR of these two encoded streams is transmitted. Nonlinear turbo
codes that provide a controlled distribution of ones and zeros are
used to demonstrate a low-complexity scheme that operates close
to the optimal boundary.
Index Terms—broadcast channel, broadcast Z channel, capac-
ity region, nonlinear turbo codes, turbo codes.
I. INTRODUCTION
Degraded broadcast channels were first studied by Cover in
[1] and a formulation of the capacity region was established
in [2], [3] and [4]. Superposition encoding is the key idea
to achieve the optimal boundary of the capacity region for
degraded broadcast channels [5]. With superposition encoding
for degraded broadcast channels, the data sent to the user
with the most degraded channel is encoded first. Given the
encoded bits for that user, an appropriate codebook for the
second most degraded channel user is selected, and so forth.
Hence superposition encoding is, in general, a joint encoding
scheme. However, combining independently encoded streams,
one for each user, is an optimal scheme for some broadcast
channels including broadcast Gaussian channels [1] and broad-
cast binary-symmetric channels [1] [2].
Successive decoding is a natural decoding scheme for
superposition encoding [1] [2] [5]. With successive decoding
for degraded broadcast channels, each receiver first decodes
the data sent to the user with the most degraded channel.
Conditioning on the decoded data for that user, each receiver
determines the codebook for the user with the second most
degraded channel and decodes that data, and so forth until the
desired user’s data is decoded. The performance of successive
decoding for degraded broadcast channels is very close to
optimal decoding under normal operating conditions.
Turbo codes [6] and Low-Density Parity-Check (LDPC)
codes [7] perform close to the Shannon limit. LDPC and
turbo coding approach for broadcast channels were studied
in [8] and [9] respectively. In [8], LDPC codes provided
reliable transmission over two-user broadcast channels with
additive white Gaussian noise (AWGN) and fading known at
the receiver only. In [9], a superposition turbo coding scheme
This work was supported by the Defence Advanced Research Project
Agency SPAWAR Systems Center, San Diego, California under Grant
N66001-02-1-8938. This paper was presented in part at the Information
Theory Workshop 2007.
The authors are with the Electrical Engineering Department, University
of California, Los Angeles, CA 90095 USA (e-mail:xbk@ee.ucla.edu;
mgriot@ee.ucla.edu; avila@ee.ucla.edu; wesel@ee.ucla.edu).
X?
Y?1?
1?
0?
1?
a?1?
Y?2?
0?
1?
a?2?
X? Y?
1?
0?0?
1?
a?
(?a?)?
(?b?)?
}?
0?}?
Fig. 1.(a) Z channel. (b) Broadcast Z channel.
performs within 1dB of the capacity region boundary for
broadcast Gaussian channels. Both of these approaches are
designed specifically for broadcast Gaussian channels and used
linear codes. For multi-user binary adder channels, nonlinear
trellis codes were studied and designed in [10].
The Z channel is the binary-asymmetric channel shown in
Fig. 1(a). The capacity of the Z channel was studied in [11].
Nonlinear trellis codes were designed to maintain a low ones
density for the Z channel in [12] [14] and parallel concatenated
nonlinear turbo codes were designed for the Z channel in [13].
This paper focuses on the study of the two-user broadcast Z
channel X → Y1,Y2shown in Fig. 1(b). This paper provides
an explicit expression of the capacity region for the two-user
broadcast Z channel and shows that independentencoding with
successive decoding can achieve the boundary of this capacity
region.
This paper is organized as follows. Section II introduces
definitions and notation for broadcast channels. Section III
provides the explicit expression of the capacity region for
the two-user broadcast Z channel and proves that independent
encoding can achieve the optimal boundary of the capacity
region. Section IV presents nonlinear-turbo codes designed
to achieve the optimal boundary, and Section V provides the
simulation results. Section VI delivers the conclusions.
II. DEFINITIONS AND PRELIMINARIES
A. Degraded broadcast channels
The general representation of a discrete memoryless broad-
cast channel is given in Fig. 2. A single signal X is broadcast
to M users through M different channels A1,··· ,AM. If
p(yi,yi+1|x) = p(yi|x)p(yi+1|yi), then channel Ai+1 is a
physically degraded version of channel Ai (and thus the
broadcast channel X → Yi,Yi+1 is physically degraded)
[5]. A physically degraded broadcast channel with M users
is shown in Fig. 3. Since each user decodes its received
signal without collaboration, only the marginal transition
probabilities p(y1|x),p(y2|x),··· ,p(yM|x) of the component
channels A1,A2,··· ,AMaffect receiver performance. Hence,
the stochastically degraded broadcast channel is defined in [2]
and [5] as follows:
Let Aibe a channel with input alphabet X, output alphabet
Yi, and transition probability pi(yi|x). Let Ai+1 be another
channel with the same input alphabet X, output alphabet Yi+1,
and transition probability pi+1(yi+1|x). Ai+1 is a stochas-
tically degraded version of Ai if there exists a transition
1

Page 2

X?
A?1?
A?M?
A?2?
.? ?.? ?.?
Y?1?
Y?2?
Y?M?
Fig. 2.Broadcast channel.
A?M?:? p?M?(?y?M?|? x?)?
A?2?:? p?2?(?y?2?|? x?)?
X?
A?1?
p?1?(?y?1?|?x?)?
D?2?
q?(?y?2?|?y?1?)?
Y?1?
Y?2?
D?3?
D?M?
.? ?.? ?.?
Y?3?
Y?M?-? 1?
Y?M?
Fig. 3. Physically degraded broadcast channel.
probability q(yi+1|yi) such that
pi+1(yi+1|x) =
?
yi∈Yi
q(yi+1|yi)pi(yi|x).
(1)
A broadcast channel with receivers Y1,Y2··· ,YM is a
stochastically degraded broadcast channel if every component
channel Ai is a stochastically degraded version of Ai−1 for
all i = 2,··· ,M [2]. Since the marginal transition proba-
bilities p(y1|x),p(y2|x),··· ,p(yM|x) completely determine
a stochastically degraded broadcast channel, we can model
any stochastically degraded broadcast channel as a physically
degraded broadcast channel with the same marginal transition
probabilities.
Theorem 1 ([2] [4]): The capacity region for the two-user
stochastically degraded broadcast channel X → Y1→ Y2 is
the convex hull of the closure of all (R1,R2) satisfying
R2≤ I(X2;Y2)R1≤ I(X;Y1|X2),
(2)
for some joint distribution p(x2)p(x|x2)p(y1,y2|x), where the
auxiliary random variable X21has cardinality bounded by
|X2| ≤ min{|X|,|Y1|,|Y2|}.
B. The broadcast Z channel
The Z channel, shown in Fig. 1(a), is a binary-asymmetric
channel with the transition probability matrix
T =
?1α
01 − α
?
,
where 0 ≤ α ≤ 1. If symbol 1 is transmitted, symbol 1 is
received with probability 1. If symbol 0 is transmitted, symbol
1 is received with probability α and symbol 0 is received with
probability 1 − α. We can model the Z channel as the OR
operation of the channel input X and Bernoulli noise N with
parameter α as shown in Fig. 4. In an OR Multiple Access
Channel, each user appears to transmit over a Z channel
when the other users are treated as noise [13]. Thus, in an
OR network with multiple transmitters and multiple receivers,
each transmitter transmitting to more than one receiver sees
1U was used as the auxiliary random variable in [2] [4]. In this paper, we
use X2 instead of U because the auxiliary random variable corresponds to
the second user’s encoded stream.
N
Pr(N = 1) =
XY
OR
?
Fig. 4.OR operation view of Z channel.
X
Y1
?1
Y2
??
1
0
1
0
Fig. 5.Physically degraded broadcast Z channel.
a broadcast Z channel if other transmitters transmitting to
those receivers are treated as noise. The two-user broadcast
Z channel with the marginal transition probability matrices
T1=
?1α1
01 − α1
?
T2=
?1α2
01 − α2
?
is shown in Fig. 1, where 0 ≤ α1 ≤ α2 ≤ 1. Because
broadcast Z channels are stochastically degraded, we can
model any broadcast Z channel as a physically degraded
broadcast Z channel as shown in Fig. 5, where
α∆=α2− α1
1 − α1
.
(3)
III. OPTIMAL TRANSMISSION STRATEGY FOR THE
TWO-USER BROADCAST Z CHANNEL
Since the broadcast Z channel is stochastically degraded, its
capacity region can be obtained directly from Theorem 1. The
capacity region for the broadcast Z channel X → Y1→ Y2
as shown in Fig. 6 is the convex hull of the closure of all
(R1,R2) satisfying
R2≤ I2= I(X2;Y2)
= H?(¯ µ2γ + µ2µ1)(1 − α2)?
− ¯ µ2H?γ(1 − α2)?− µ2H?µ1(1 − α2)?,
R1≤ I1= I(X;Y1|X2)
= ¯ µ2
?H(γ(1 − α1)) − γH(1 − α1)?
+ µ2
?H(µ1(1 − α1)) − µ1H(1 − α1)?,
for some probabilities µ1,µ2,γ, where µ1= Pr(x = 0|x2=
0), µ2 = Pr(x2 = 0), γ = Pr(x = 0|x2 = 1), H(·) is the
binary entropy function, ¯ µ1= 1 − µ1, ¯ µ2= 1 − µ2and
(4)
(5)
α2= Pr{y2= 1|x = 0} = 1 − (1 − α1)(1 − α∆).
(6)
Each particular choice of (µ1,µ2,γ) in Fig. 6 specifies
a particular transmission strategy and a rate pair (I1,I2).
The optimal boundary of a capacity region is the set of all
Pareto optimal points (I1,I2), for which it is impossible to
increase rate I1 without decreasing rate I2 or vice versa. A
transmission strategy is optimal if and only if it achieves
a rate pair point on the optimal boundary. We call a set

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X
Y1
?1
Y2
??
X2
?
2
?
2
?
1 ?
1 ?
Fig. 6.Information theoretic diagram of the system.
of transmission strategies sufficient if all rate pairs on the
optimal boundary can be achieved by using these strategies
and time sharing. Furthermore, a set of transmission strategies
is strongly sufficient if these strategies can achieve all rate pairs
on the optimal boundary without using time sharing. Equations
(4) and (5) give a set of pentagons that yield the capacity
region through their convex hull, but do not explicitly show
the optimal transmission strategies or derive the boundary of
the capacity region.
A. Optimal transmission strategies
The following theorem identifies a set of optimal trans-
mission strategies and provides an explicit expression of the
boundary of the capacity region.
Theorem 2: For a broadcast Z channel with 0 < α1< α2<
1, the set of the optimal transmission strategies (µ1,µ2,γ),
which satisfy
γ = 0,
(7)
1
(1 − α1)(eH(1−α1)/(1−α1)+ 1)≤ µ1≤ 1,
(8)
and
?H(µ1(1 − α1)) − µ1H(1 − α1)?· ln(1 − µ1(1 − α2))
=?H(µ1(1 − α2)) − µ1(1 − α2)ln1 − µ2µ1(1 − α2)
· ln(1 − µ1(1 − α1)),
µ2µ1(1 − α2)
?
(9)
are strongly sufficient. In other words, all rate pairs on the
optimal boundary of the capacity region can be achieved by
using exactly the transmission strategies described in (7-9)
without the need of time sharing. Furthermore, applying (7-
9) to (4) and (5) yields an explicit expression of the optimal
boundary of the capacity region.
Before proving Theorem 2, we present and prove some
preliminary results. From (4) and (5), we can see that the
transmission strategies (µ1,µ2,γ) and (γ,1−µ2,µ1) have the
same transmission rate pairs. Therefore, we assume γ ≤ µ1
in the rest of the section without loss of generality.
Theorem 3: For a broadcast Z channel with 0 < α1< α2<
1, any transmission strategy (µ1,µ2,γ) with 0 < µ2< 1,0 <
γ < µ1is not optimal.
The proof is given in Appendix A.
Corollary 1: The set of all the transmission strategies with
γ = 0 is sufficient for any broadcast Z channel with 0 < α1<
α2< 1.
Proof: From Theorem 3, we know that the transmission
strategy (µ1,µ2,γ) is optimal only if at least one of these
four equations µ2 = 0, µ2 = 1, γ = µ1, γ = 0 is
true. Hence the set of all the transmission strategies with
µ2 = 0, µ2 = 1, γ = µ1 or γ = 0 is sufficient. When
µ2 = 0, µ2 = 1 or γ = µ1, the transmission rate for the
second user, I2 in equation (4), is zero. This optimal rate
pair is the point B in Fig. 7(a). Since this point can also be
achieved by the transmission strategy with γ = 0, µ2= 1 and
µ1= argmax(H(x(1 −α1)) − xH(1 − α1)), all the optimal
rate pairs on the optimal boundary of the capacity region can
be achieved by using the transmission strategies with γ = 0
and time sharing. Thus, the set of all the transmission strategies
with γ = 0 is sufficient. Q.E.D.
From Corollary 1, we can set γ = 0 in Fig. 6 without losing
any part of the capacity region and so the designed virtual
channel X2→ X is a Z channel. Since we can consider the
output of a Z channel as the OR operation of two Bernoulli
random variables, an independent encoding scheme that works
well for the broadcast Z channel will be introduced later in
this paper.
Applying γ = 0 to (4) and (5) yields
R2≤ I2= H(µ2µ1(1 − α2)) − µ2H(µ1(1 − α2)),
R1≤ I1= µ2H(µ1(1 − α1)) − µ2µ1H(1 − α1).
(10)
(11)
By Corollary 1, the capacity region is the convex hull of the
closure of all rate pairs (R1,R2) satisfying (10) and (11)
for some probability µ1,µ2. However, not all transmission
strategies of (µ1,µ2,γ = 0) achieve the optimal boundary of
the capacity region. Since any optimal transmission strategy
maximizes I1+ λI2 for some nonnegative λ, we solve the
optimization problem of maximizing I1+ λI2 for any fixed
λ ≥ 0 in order to find the constraints on µ1and µ2for optimal
transmission strategies. Theorem 4 provides the solution to this
maximization problem.
Theorem 4: The optimal solution to the maximization prob-
lem
maximize
subject to
I1+ λI2
I2= H(µ2µ1(1 − α2)) − µ2H(µ1(1 − α2))
I1= µ2H(µ1(1 − α1)) − µ2µ1H(1 − α1)
0 ≤ µ2≤ 1,0 ≤ µ1≤ 1,
(12)
is unique and it is given below for any fixed λ ≥ 0.
Define
ϕ(x) =ln(1 − (1 − α1)x)
ln(1 − (1 − α2)x)
(13)
and
ψ(x) =
1
xeH(x)/x+ x.
(14)
Case 1: if 0 ≤ λ ≤ ϕ(ψ(1−α1)), then the optimal solution is
µ∗
corresponding rate pair is I∗
I∗
Case 2: if λ ≥ ϕ(1), then the optimal solution is µ∗
ψ(1 − α2),µ∗
corresponding rate pair is I∗
µ∗
Case 3: if ϕ(ψ(1−α1)) < λ < ϕ(1), then the optimal solution
2= 1,µ∗
1= ψ(1 − α1), which satisfies (8) and (9), and the
1= H(µ∗
1(1−α1))−µ∗
1H(1−α1),
2= 0.
2=
1= 1, which also satisfies (8) and (9), and the
1= 0,I∗
2H(1 − α2).
2= H(µ∗
2(1 − α2)) −

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U? p? p?e? r? ?b? o?u? n? d? ?1?
U? p? p? e? r? ? b? o?u? n? d? ?2?
(?a? )?(?b?)?
Fig. 7.
be on the boundary of the capacity region.
(a) The capacity region and two upper bounds. (b) Point Z cannot
given below also satisfies (8) and (9):
µ∗
1= ϕ−1(λ) =
eλ− 1
eλ(1 − α2) − (1 − α1)
(15)
and
?H(µ∗
=?H(µ∗
· ln(1 − µ∗
1(1 − α1)) − µ∗
1H(1 − α1)?· ln(1 − µ∗
1(1 − α2)ln1 − µ∗
1(1 − α2))
1(1 − α2)
1(1 − α2)
1(1 − α2)) − µ∗
2µ∗
µ∗
2µ∗
?
(16)
1(1 − α1)).
The proof is given in Appendix B. Combining Case 1,2
and 3, we conclude that (µ1,µ2) is a maximizer of (12) if
and only if the pair (µ1,µ2) satisfies (8) and (9). In other
words, if (µ1,µ2) doesn’t satisfy (8) or (9), (µ1,µ2) cannot
be a maximizer of (12), and thus the transmission strategy
(µ1,µ2,γ = 0) is not optimal. Since the set of the transmission
strategies with γ = 0 is sufficient by Corollary 1, the set of
all the transmission strategies satisfying (7-9) is also sufficient.
Therefore the capacity region is the convex hull of the closure
of all rate pairs (R1,R2) satisfying (10) and (11) for some
µ1,µ2which satisfy (8) and (9).
A sketch of the capacity region is shown with two upper
bounds in Fig. 7(a). From Case 1 in Theorem 4, the point B
corresponds to the largest transmission rate for the first user.
The first upper bound is the tangent of the capacity region at
the point B, and its slope is −1/ϕ(ψ(1 − α1)). From Case
2, the point A provides the largest transmission rate for the
second user. The second upper bound is the tangent of the
capacity region at the point A, and its slope is −1/ϕ(1). Case
3 gives us the optimal boundary of the capacity region except
the points A and B.
Given α1 and α2, which completely describe a two-user
degraded broadcast Z channel, the optimal boundary of the
capacity region can be explicitly described by (8-11). For any
µ1in the range of (8), the value of the unique associated µ2
follows from (9). The curve of the optimal boundary of the
capacity region is then the set of (I1,I2) pairs satisfying (10)
and (11) for these µ1 and associated µ2. For example, for
α1= 0.15 and α2= 0.6, the range of optimal µ1 values is
0.445 ≤ µ1≤ 1, the range of optimal µ2 values implied by
(9) is 0.392 ≤ µ2 ≤ 1, and the associated capacity region
boundary is plotted in Fig. 13.
Now we prove Theorem 2. Since we have proved that the set
of all the transmission strategies satisfying (7-9) is sufficient,
we only need to show that any rate pair on the optimal
X?
(? W?1?,? W?2?)?
E?n?c? o? d? e? r?
Z? ?c? h? a? n? n? e?l?
p?1?(?y?1?|? x?)?
D?e? c?o?d? e? r? ?1?
Y?1?
1?ˆ? w?
D?e? c? o?d? e? r? ?2?
2?
ˆ? w?
Z? ? c?h?a? n? n? e? l?
p?2?(?y?2?|?x?)?
Y?2?
Fig. 8.Communication system for 2-user broadcast Z channels.
W?1?
W?2?
O?R?
X?
N?1?
O?R?
E?n? c? o?d? e? r? ?1?
E?n? c? o? d? e?r? ?2?
X?1?
X?2?
Y?1?
D?e? c? o? d?e? r? ?1?
1?ˆ? w?
N?1?
O?R?O?R?
N?2?
Y?2?
D?e? c? o? d?e? r? ?2?
2?
ˆ? w?
Fig. 9. Optimal transmission strategy for broadcast Z channels.
boundary of the capacity region can be achieved without using
time sharing.
Proof by contradiction: Suppose the point Z in Fig. 7(b)
is on the optimal boundary of the capacity region for the
broadcast Z channel and this point can only be achieved by
time sharing of the points X and Y , which can be directly
achieved by using transmission strategies satisfying (7-9).
Clearly, the slope of the line segment XY is neither zero nor
minus infinity. Denote −k,0 < k < ∞ as the slope of XY .
The points X and Y provide the same value of I1+1
Theorem 4, the optimal solution to the maximization problem
of max(I1+λI2) is unique, and so neither X nor Y maximizes
(I1+1
this point is on the right upper side of the line XY . Since
and the triangle △XY P is in the capacity region, the point
Z must not be on the optimal boundary of the capacity region
(contradiction). Q.E.D.
kI2. By
kI2). Thus, there exists an achievable point P such that
B. Independent encoding scheme
The communication system for the two-user broadcast Z
channel is shown in Fig. 8. In a general scheme, the trans-
mitter jointly encodes the independent messages W1and W2,
which is potentially too complex to implement. Theorem 2
demonstrates that there exists an independent encoding scheme
which achieves the optimal boundary of the capacity region.
Since γ = 0 is strongly sufficient, the designed channel
X2→ X is a Z channel. Thus, the broadcast signal X can be
constructed as the OR of two Bernoulli random variables X1
and X2. This construction of X is an independent encoding
scheme. The system diagram of the independent encoding
scheme is shown in Fig. 9. First the messages W1 and W2
are encoded separately and independently. X1 and X2 are
two binary random variables with Pr{Xj = 1} = ¯ µj and
Pr{Xj = 0} = µj, where ¯ µj + µj = 1 for j = 1,2.
The transmitter broadcasts X, which is the OR of X1 and
X2. From Theorem 2, this independent encoding scheme with
any choice of (µ1,µ2) satisfying (8) and (9) achieves a rate
pair (I1,I2) arbitrarily close to the optimal boundary of the
capacity region if the codes for X1and X2are properly chosen
and have sufficiently large block lengths.

Page 5

Π
1s
2s
3s
4s
1u
2 u
Look-up table
1
2
3
456
1
2
1
2
0n
Fig. 10.
trellis section.
16-state nonlinear turbo code structure, with k0= 2 input bits per
TABLE I
LABELING FOR CONSTITUENT TRELLIS CODES. RATES
R1= 1/6,R2= 1/6. ROWS REPRESENT THE STATE s1s2s3s4, COLUMNS
REPRESENT THE INPUT u1u2. LABELING IN OCTAL NOTATION.
User 1User 2
stateinput
01
20
40
04
10
01
02
21
42
02
01
10
04
14
05
40
20
stateinput
01
34
07
16
25
13
61
15
23
43
70
62
51
52
64
54
26
00
40
20
10
04
02
01
42
21
01
02
04
10
05
14
20
40
10
10
04
02
01
40
20
14
05
04
10
20
40
21
42
01
02
11
04
10
01
02
20
40
05
14
10
04
40
20
42
21
02
01
00
07
34
25
16
61
13
23
15
70
43
51
62
64
52
26
54
10
62
51
43
70
54
26
52
64
16
25
34
07
15
23
13
61
11
51
62
70
43
26
54
64
52
25
16
07
34
23
15
61
13
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
IV. NONLINEAR-TURBO CODES FOR THE TWO-USER
BROADCAST Z CHANNEL
In this section we show a practical implementation of the
transmission strategy for the two-user broadcast Z channel.
As proved in Section III, the optimal boundary is achieved
by transmitting the OR of the encoded data of each user,
provided that the density of ones of each of these encoded
streams is chosen properly. Hence, a family of codes that
provides a controlled density of ones is required. We use
the nonlinear turbo codes, introduced in [13], to provide the
needed controlled density of ones. Nonlinear turbo codes are
parallel concatenated trellis codes with k0 input bits and n0
output bits per trellis section. A look-up table assigns the
output label for each branch of the trellis so that the required
ones density is achieved. Each constituent encoder for the
turbo code in this paper is a 16-state trellis code with k0= 2
and the trellis structure shown in Fig. 10. The output labels
are assigned via a constrained search that provides the required
ones density for each case, using the tools presented in [13]
for the Z Channel. The output labels for the codes with rate
pair (R1= 1/6,R2= 1/6), which is simulated on a broadcast
Z channel with α1= 0.15,α2= 0.6, are listed in Table I.
Receiver 1 uses successive decoding as shown in Fig. 11.
Denote asˆ X2 the decoded stream corresponding to user 2.
Since the transmitted data is x = x1(OR)x2, whenever a bit
x2 = 1, there is no information about x1, and x1 can be
considered an erasure. Hence, the input stream to Decoder 1
1Y
e
Decoder 2
Decoder 1
1ˆX
2
ˆX
1Y
?
Fig. 11.Decoder structure for user 1.
1
0
1
0
e
1
00
1
(b) User 2: Z channel
2
?
(a) User 1: Z channel with erasures
12
1(1 )
????
12
(1)
???
2
?
2
?
21
? ?
21
(1)
???
Fig. 12.Perceived channel by each decoder.
is
ˆ y1= e(y1, ˆ x2) =
?
y1
if
if
ˆ x2= 0,
ˆ x2= 1.e
(17)
Therefore, Decoder 2 sees a Z Channel with erasures as
shown in Fig. 12. The tools presented in [13] were general
enough to be applied to the Z Channel with erasures. Note
that if α1is much smaller than α2we can use hard decoding
in Decoder 2 instead of soft decoding without any loss in
performance. Since the code for user 2 is designed for a Z
Channel with 0-to-1 crossover probability 1−(1−α2)µ1, and
the channel perceived by Decoder 2 in user 1 is a Z-Channel
with crossover probability 1 − (1 − α1)µ1< 1 − (1 − α2)µ1,
the bit error rate of ˆ x2is negligible compared to the bit error
rate of Decoder 1. In fact, in all the simulations shown in
Section V, which include 100 frame errors of user 1, none of
the errors were produced by Decoder 2.
V. RESULTS
We simulate the transmission strategy for the two-user
broadcast Z channel with crossover probabilities α1 = 0.15
and α2= 0.6, using nonlinear turbo codes, with the structure
shown in Fig. 10. Fig. 13 shows the capacity region for the
broadcast Z channel and identifies the simulated rate pairs. It
also shows the optimal rate pairs, which are used to compute
the ones densities of each code. The output labels for the codes
with each simulated rate pair are listed at [15]. For each of
these four simulated rate pairs, the loss in mutual information
from the associated optimal rate is only 0.04 bits or less in
R1and only 0.02 bits or less in R2. Table II shows bit error
rates for each rate pair, the ones densities ¯ µ1 and ¯ µ2, and
the interleaver lengths K1 and K2 used for each code. For
simplicity, we chose K1and K2so that the codeword length
n would be the same for user 1 and user 2, except for rate
pairs R1= 1/2 and R2= 1/22, where one codeword length
of user 2 is twice the length of user 1.
VI. CONCLUSIONS
This paper presented an optimal transmission strategy for
the broadcast Z channel with independent encoding and suc-
cessive decoding. We proved that any point on the optimal