ITW 2007, Lake Tahoe, California, September 2 - 6, 2007

Optimnal Transminssion Strategy and CapacityRegion for Broadcast Z Channels

Bike Xie, Miguel Griot, Andres I. Vila Casado and Richard D. WeselDepartment of Electrical Engineering, University of California, Los Angeles, CA 90095-1594

Email: xbk@dee.ucla.edu, mgriot@Qee.ucla.edu, avila@Qee.ucla.edu, wesel@dee.ucla.edu

Abstract This paper presents an optimal transmissionstrategy, with simple encoding and decoding, for the two-user broadcast Z channel. This paper provides an explicit-form expression for the capacity region and proves that theoptimal surface can be achieved by independent encoding.Specifically, the information messages corresponding to eachuser are encoded independently and the OR of these twostreams is transmitted. Nonlinear turbo codes that pro-vide a controlled distribution of ones and zeros are used todemonstrate a low-complexity scheme that works close tothe optimal surface.

Index Terms broadcast channel, broadcast Z channel, ca-pacity region, turbo codes.

I. INTRODUCTION

Degraded-broadcast channels were first studied by Coverin [1] and a formulation for the capacity region was estab-lished in [2], [3] and [4]. The key idea to achieve the opti-mal surface of the capacity region for degraded-broadcastchannels is superposition coding. In superposition codingfor degraded-broadcast channels, the data sent to the userwith the most degraded channel is encoded first. Giventhe encoded bits for that user, an appropriate codebookfor the second most degraded channel user is selected, andso forth. Hence superposition coding is in general a jointencoding scheme. However, combining independently en-coded streams of each user is an optimal scheme for somebroadcast channels including broadcast Gaussian channels[1] and broadcast binary-symmetric channels[1] [2].This paper focuses on the study of broadcast Z channels.

The Z channel is the binary-asymmetric channel shown inFig. l(a). Fig. l(b) shows a two-user broadcast Z chan-nel. This paper provides an explicit-form expression forthe capacity region of the two-user broadcast Z channeland proves that independent encoding with successive de-coding can achieve this capacity region.

This paper is organized as follows. In Section II, somedefinitions and notation for broadcast channels are intro-duced. The proof that independent encoding can achievethe optimal surface of the capacity region for the two-userbroadcast Z channel is presented in Section III. Nonlinear-turbo codes, designed to achieve the optimal surface, arepresented in Section IV and simulation results are shownin Section V. Section VI delivers the conclusions.

x y1 0--:7w 1

o oZ ~

Y1

x

0(a) (b)

Fig. 1. (a) Z channel. (b) Broadcast Z channel.

II. DEFINITIONS AND PRELIMINARIES

A. Degraded broadcast channels

The general representation of a discrete memorylessbroadcast channel is given in Fig. 2. A single signal X isbroadcast to M users through M different channels. Chan-nel A2 is a physically degraded version of channel A1 andbroadcast channel X -) Y, Y2 is physically degraded ifP(Yi, Y2 1X) = P(Yi IX)P(Y2 1I) [5]. A physically degradedbroadcast channel with M users is shown in Fig. 3. Sinceeach user decodes its received signal without collaboration,we only need to consider the marginal transition probabil-ities p(Yi IX), P(Y2 IX), ,p(yM Ix) of the component chan-nels A1, A2,... , AM. Since only the marginal distributionsaffect receiver performance, a weaker notion of a stochas-tically degraded broadcast is defined in [2] and [5].

Let A1 and A2 be two channels with same input alphabetX, output alphabet Y3 and Y2, and transition probabilityP1 (y1 I x) and P2 (Y2 Ix) respectively. A2 is a degraded versionof A1 if there exists a transition probability q(y21/Y) suchthat

P2 (Y2 IX) =E q(Y2 IYI)PI (Y1I X).yi EYi

A broadcast channel with receivers Y¾, . , YM is astochastically degraded broadcast channel if every com-ponent channel Ai is a degraded version of Ai1- for alli = 2,... M [2]. Since the marginal transition probabil-ities P(YI IX), P(Y2 IX),. , P(YM IX) completely determine astochastically degraded broadcast channel, we can modelany stochastically degraded broadcast channel as a physi-cally degraded broadcast channel with the same marginaltransition probabilities.

Theorem 1 ([2] [4]) The capacity region for the two-userdegraded broadcast channel X -) Y¾ - Y2 is the convexhull of the closure of all (R1, R2) satisfying

This work was supported by the Defence Advanced ResearchProject Agency SPAWAR Systems Center, San Diego, California un-der Grant N66001-02-1-8938 and by the state of California and STMicroelectronics through UC Discovery Grant 03-10142.

R2 <I(X2;Y2)

for some joint distribution p(X2)P(X X2)P(Y, z X), where the

1-4244-1564-0/07/$25.00 ©2007 IEEE

R1 < I(X; Y IX2), (1)

Y2

390

Al YM

A2

Fig. 2. Broadcast channel.

A1Al x) D2 P3r92YI)

Fig. 3. Physically degraded broadcast channel.

auxiliary random variable X2 has cardinality bounded byIX21 <_min {XI, lYll, IY21}.

B. The broadcast Z channel

The Z channel is a binary-asymmetric channel withPr{y = 0lx = 1} = 0 (see Fig. 1(a)). If symbol 1 is trans-mitted, symbol 1 is received with probability 1. If symbol 0is transmitted, symbol 1 is received with probability av andsymbol 0 is received with probability 1 -o. We can con-sider a Z channel as the OR operation of the channel inputX and Bernoulli noise N with parameter av (see Fig. 4(a)).The diagram of a two-user broadcast Z channel is shown inFig. l(b). Because broadcast Z channels are stochasticallydegraded, we can model any broadcast Z channel as a phys-ically degraded broadcast Z channel as shown in Fig. 4(b),where cva = 12 and cv2 > al

III. OPTIMAL TRANSMISSION STRATEGY FOR THETwo-USER BROADCAST Z CHANNEL

The communication system for the two-user broadcastZ channel is shown in Fig. 5. In a general scheme, thetransmitter jointly encodes the independent messages Wiand W2. The receivers decode the noisy signals withoutcollaboration. Since the broadcast Z channel is stochasti-cally degraded, its capacity region can be found directlyfrom Theorem 1. The capacity region for the broadcast Zchannel X -- Y- Y2 (see Fig. 6) is the convex hull of theclosure of all (R1, R2) satisfying

R2 < 2 =I(X2;Y2)= H((p27 + q2ql) (1

P2H(7y(1 -a2))-

RI < 11 = I(X; Yl IX2)= P2 (H(y(1 -al)) -

q2 (H(q1 (1 - a,))

- a2)) -

q2H(q1(1 -c2)),

- yH(1 cva)) +

-qiH(l- a,)),

(2)

(3)

NPr(N= 1)=a

X OR y

(a)

x Y1 Y21 1

0 ((b)

Fig. 4. (a) OR operation view of Z channel. (b) Physically degradedbroadcast Z channel.

Z channel 1> Decoder I 4 1

E~~~~~~~Z~~~chainnel

Yr-*" Decoder2 r* W^

Fig. 5. Communication system for 2-user broadcast Z channel.

A. An independent encoding scheme

The previously defined broadcast communication systemhas a joint encoder which is potentially quite complex. Wepropose an independent encoding scheme for the broadcastZ channel, which can achieve the optimal surface of the ca-pacity region. Fig. 7 shows the system diagram of the inde-pendent encoding scheme. First the messages WT and W2are encoded separately and independently. Xl and X2 aretwo binary random variables with PrfXj = l} = pj andPrfXj= 0} = qj. Thus pj +qj = 1, j = 1, 2. The transmit-ter broadcasts X, which is the OR of Xl and X2. We willshow that by appropriately choosing the distribution of Xland X2, i.e. q, and q2, we can achieve any transmissionrate pair in the optimal surface of the capacity region. Thecorresponding information theoretic diagram is Fig. 6 withy = 0. Letting ' = 0 in equation (2) and (3), the achievableregion for the broadcast Z channel X -- Y- Y2, with theindependent encoding scheme of Fig. 7, is the closure of all(R1, R2) satisfying

R2 < 2 = H(q2q, (1 -a2))R1 <'1 = q2H(qi (1 -ca,))

q2H(q1(1- a2)),-q2q1H(I- a,),

(5)(6)

for some 0 < ql,q2 < I.Let us prove that any transmission rate pair in the op-

timal surface of the capacity region can be achieved usingthe independent encoding scheme of Fig. 7 with appropri-ate distributions of Xi and X2.

B. Optimal transmission strategy

Each particular choice of (Pl,P2, ) in Fig. 6 gives a par-ticular transmission strategy and a rate pair (I1, 12). We

X2 x Y1 Y2P2

a,,aq2 O!-qI

Fig. 6. Information theoretic diagram of the system.

for some probabilities P1, P2,7Y, where H(p) is the binaryentropy function, q, = l-P1, q2 = 1-P2 and

ca2 = Pr{y2 = X == 0} I1- (1 )(l -aA). (4)

W,X EnoerI xi/ Y,[ Enecodc OR Decoder 1-11

X2 IFr j,.; ,-- Y2L1 Encoder 2 OR OR - R - Decoder 2 0-I*v

Fig. 7. Optimal transmission strategy for broadcast Z channels.

391

say that the optimal surface of a capacity region is the setof all Pareto optimal points (I,I 12), which are points forwhich it is impossible to increase rate A1 without decreasingrate 12 or vice versa. A transmission strategy is optimal ifand only if it achieves a rate pair in the optimal surface.

Theorem 2: Every rate pair in the optimal surface of thecapacity region for a broadcast Z channel witho < av1 < cv2 < 1 can be achieved with the independentencoding scheme shown in Fig. 7. In other words, everyrate pair in the optimal surface of the capacity region canbe achieved with y = 0 in Fig. 6.

Before proving Theorem 2, we present and prove somepreliminary results. From equations (2) and (3), we cansee that the transmission strategies (1- 'y, 1 -P2, 1 -P1)and (P1,P2, -y) have the same transmission rate pairs. Sowe can assume -y < 1-P1 in the rest of the section withoutloss of generality.

Theorem 3: For a broadcast Z channel witho < OvI <cv2 < 1, any transmission strategy (P1,P2, ay) witho < P2<l,O <1 y < 1 -P1 is not optimal.The proof is given in Appendix A.Corollary 1: The capacity region for the broadcast Z

channel X -- Y- Y2 is the convex hull of the achiev-able region with the independent encoding scheme.

Proof: From Theorem 3, we know that the transmissionstrategy (P1,P2,7) is optimal only if at least one of thesefour equations P2 = 0, P2 = 1, -y = -P1, y = 0 is true.When P2 = 0, P2 = 1 or -y = 1-P1, the transmissionrate for the second user, 12 in equation (2), is zero, whichmeans that the only optimal rate pair that can be achievedis point B in Fig. 13. Point B can also be achieved bythe independent encoding scheme with q 0, P2 0 andP1 = argmax(H((1 -x)(1 -O)) -(1x)H(1- ci)).Thus, all the optimal rate pairs in the optimal surface of thecapacity region can be achieved by using the independentencoding scheme with y = 0 and time sharing.The achievable region given by equations (5) and (6)

is not in an explicit form. In order to find the explicitform of the achievable region, we consider the followingoptimization problem: maximize A1 + (1 -A)12 for anyfixed A C [0,1].

Theorem 4: The optimal solution to the maximizationproblem

maximize Al1 + (1 -A)12 (7)subject to 12 H(q2qi (1 -a2)) -q2H(qi (I - 2))

II = q2H(q(1- a,)) -q2qiH(1- a,)0 < q2 < 1, 0 < qi < 1,

is unique and it is given below for any fixed A C [0,1].Define

p(x) l 1og(1- (1 -oa2)X)log(1 (1 cvl)x) + log(1 (1 a2)x) (8)

1'>~ xeH(x)lx +xx

Case 1: if o(b(1-ci)) < A < 1, then the optimal solutionis q* = 1, q* = (1 -cv) and the corresponding rate pair

R2

x

z

y R1

(a) (b)

Fig. 8. (a) The capacity region and two upper bounds. (b) Point Zcan not be on the boundary of the capacity region.

is I = H(q (1-1 ovi))- q H(1( i), I2* = U.Case 2: if 0 < A < o(1), then the optimal solution isq* =b(1-c2), q= 1 and the corresponding rate pair isI* = O, 12* =H(q2*(I-a2)- q*H(1-I )Case 3: if (o(1) < A < o(0b(1 -ci)), then the optimalsolution has

A log(1-q1cv(i-a)) = (1 -A) log(1- q1( c-2)), (10)

H(q1(l-ac2)) q1 a-2) 10g 1 q2q (1-a2)log(I-q*(1-012)q21(i)~2

Iog(I-q*(l-o,j)) IIHq*l°1)q*(-g)-(1The proof is given in Appendix B. The achievable re-

gion is shown with two upper bounds in Fig. 8(a). Fromcase 1, we can see that point A corresponds to the largesttransmission rate for the second user. The first upperbound is the tangent of the achievable region in pointA, and its slope is -(1)/(1-(1)). From case 2, weshow that point B provides the largest transmission ratefor the first user. The second upper bound is the tan-gent of the achievable region in point B, and its slope is-(F(b(1c-vl))I(1-F((1-cv))). Case 3 gives us the op-

timal surface of the achievable region except points A andB.

Given cvi and av2, which completely describe a two-userdegraded broadcast Z channel, the capacity region can beexplicitly described. Cases 1 and 2 identify the cornerpoints of the capacity region. From Cases 1, 2, and 3 above,the rest of the curve is described by the following range ofqi values:

(12)

The associated q2 values follow from (11). The curve of thecapacity region boundary follows from using the qi and q2values in (5) and (6). For example, for ci = 0.15 andcv2 = 0.6, the range of qi values is 0.445 < qi < 1 and theassociated capacity region boundary is plotted in Fig. 12.

Finally, we prove Theorem 2.Proof by contradiction: Suppose the point Z in Fig. 8(b)

is on the boundary of the capacity region for the broad-cast Z channel but not in the achievable region with theindependent encoding scheme. Thus, it can be achievedonly by time sharing of points X and Y, which is in theachievable region. Clearly, The slope of the line segmentXY is neither zero nor infinity. Suppose the slope of XYis -k,0 < k < oc, so points X and Y provide the samek .R1 -+ R2. From Theorem 4, the optimal solution tothe maximization problem max(AIl + (1 -A)12)) is unique,therefore neither X nor Y maximizes

392

o(I ozl) < q, < 1.

1) /, (,)

q2 I

P 1~~~Iqa e

Iql1

00

0q

0

(a) User 1: Z channel with erasures (b) User 2: Z channel

Fig. 9. Perceived channel by each decoder.

Y, - IDecoderI .X

X2

Fig. 11. Decoder structure for user 1.

02

Capacity surfacex simulated rates0 optimal rates

0.15

0.1 k

0.05

Fig. 10. 16-state nonlinear turbo code structure, with ko = 2 inputbits per trellis section.

(k. I +12). Thus, there exists an achievable point P on theright upper side of the line XY and the triangle AXYP isin the capacity region. So the point Z must not be on theboundary of the capacity region (contradiction).L

IV. NONLINEAR-TURBO CODES FOR THE TWO-USERBROADCAST Z CHANNEL

In this section we show a practical implementation of thetransmission strategy for the two-user broadcast Z channel.As proved in Section III, the optimal surface is achievedby transmitting the OR of the encoded data of each user,provided that the density of ones of each of these encodedstreams is chosen properly. Hence, a family of codes thatprovides a controlled density of ones and zeros is required.We propose the use of nonlinear turbo codes, introduced in[6]. Nonlinear turbo codes are parallel concatenated trelliscodes, with ko input bits and no output bits per trellissection. A look-up table assigns the output label of eachbranch of the trellis, so that the required ones density isachieved. Each constituent encoder for the turbo code inthis paper is a 16-state trellis code, with ko = 2, with trellisstructure shown in Fig. 10. The output labels are assignedvia a constrained search that provides the required onesdensity for each case, using the tools presented in [6] forthe Z Channel. Also, the tools presented in [6] were generalenough to be applied to the Z Channel with erasures asperceived by user 1.

Receiver 1 uses sequential decoding as shown in Fig. 11.Denote as X2 the decoded stream corresponding to user 2.Since the transmitted data is x = xl(OR)x2, whenever abit X2 = 1, there is no information about xl, and xi can beconsidered an erasure. Hence, the input stream to Decoder1 is

Yl = e(Y1,X2) I if x2

if x-~2

0,1. (13)

Therefore, Decoder 2 sees a Z Channel with erasures asshown in Fig. 9. Note that if a,i is much smaller than cv2 wecan use hard decoding in Decoder 2 instead of soft decoding

00 01 02 03 04 05 06 07

R1

Fig. 12. Broadcast Z channel with crossover probabilities oal = 0.15and a2 = 0.6 for receiver 1 and 2 respectively: achievable capac-ity region, simulated rate pairs (Rl, R2) and their correspondingoptimal rates.

without any loss in performance. Since the code for user 2 isdesigned for a Z Channel with 0-to-i crossover probability1 -(1 -o2)ql, and the channel perceived by Decoder 2 inuser 1 is a Z-Channel with crossover probability 1 -(1-aI)qI < 1 -(1 - 2)ql, the bit error rate of x2 is negligiblecompared to the bit error rate of Decoder 1. In fact, inall the simulations shown in Section V, which include 100frame errors of user 1, none of the errors were produced byDecoder 2.

V. RESULTS

We have simulated the transmission strategy for thetwo-user broadcast Z channel with crossover probabilitiesaz1 = 0.15 and aZ2 = 0.6, using nonlinear turbo codes, withthe structure shown in Fig. 10. Fig. 12 shows the achiev-able region of the rate pairs (RI, R2) on this channel, andthe simulated rate pairs. It also shows the optimal ratepairs used to compute the ones densities of each code. Foreach of these four simulated rate pairs, the loss in mutualinformation from the associated optimal rate is only 0.04bits or less in R1 and only 0.02 bits or less in R2. Table Ishows bit error rates for each rate pair, the ones densitiesP1 and P2, and the interleaver lengths K1 and K2 used foreach code. For simplicity, we chose K1 and K2 so that thecodeword length n would be the same for user 1 and user2, except for rate pairs R1 = 1/2 and R2 = 1/22, whereone codeword length of user 2 is twice the length of user 1.

VI. CONCLUSIONS

This paper presented an optimal transmission strategyfor the broadcast Z channel with simple encoding and de-coding. We proved that any point in the optimal surface ofthe capacity region can be achieved by independently en-coding the messages corresponding to different users and

393

TABLE I

BER FOR TWO-USER BROADCAST Z CHANNEL WITH CROSSOVER PROBABILITIES al = 0.15 AND a2 = 0.6.

TR1 R2 Pi1/12 1/5 0.1061/6 1/6 0.1961/3 1/9 0.3361/2 1/22 0.463

P20.560.5

0.37390.1979

K14800204846085632

K2 BER11700 2.54 x 10-2048 7.01 x 10-1536 7.13 x 10-1024 9.27 x 10-

transmitting the OR of the encoded signals. Also, the dis-tributions of the outputs of each encoder that achieve theoptimal surface were provided. Nonlinear-turbo codes thatprovide a controlled distribution of ones and zeros in thecodes were used to demonstrate a low-complexity schemethat works close to the optimal surface.

APPENDICES

Appendisx A

Here we prove Theorem 3. In (2) and (3), denote

11(P1,P2,7) = I(X;Yi X2) P1,P2<Y

I2 (P1, P2<, Y) = I(X2; Y2) IP1,P2,TYI1,2 (P1, P2,'Y) = (1, 2) Pl,P2,tY

(14)

(15)

(16)

The strategy (P1, P2, -y) has the rate pair I1,2 (P1, P2, 'Y). Thetheorem is true if we can increase both I and 12 when

° <P2 < 1,0 < -Y <I1 -PI1Firstly we compare the strategies (P1,P2,-y) and

(PI -P21, P2, -Y q2 61) for a small positive number 51.

A IIj = Ij(PI-P261,P2, a-y-q206) --Ij(PI, P2, -Y)Ij (PI -P261, P2,Y - q261 )

0Dc1 61=

1)'q2p2(1 -caj) (log 1 1-y(1 - aj)'-Y(l ai)

+log 1 qi(1 aj) ) 1,j = 1,2. (17)

The small change of the rate pair (A1I11, A112) is shownFig. 13. Point A is the rate pair of the strategy (P1,P2, Y),the arrow A1 shows the small movement of the rate pair(Al1l,, A12)

Secondly we compare the strategies (P1, P2, -y) and

(P1- (-y q1)2,P2 -q262, -Y) for a small positive number

62.

A2-1j =Ij (pi -(-y ql 2, P2 -q22, Y)

--Ij(PI, P2, -Y)DIj (PI- (y -q1 )2, P2- q22, Y)

622062 62=0

(-1)jq262 f{Y(1 - aj) log q, +

1 _qi(1 j)}

(-1)q2(2D(y() l- aj) 11 ql (l-aj)),j= 1,2. (18)

A'

B R1

Fig. 13. Capacity region and the changes of rate pairs.

where D(p 11 q) is the relative entropy between distribu-tion p and q. The arrow A2 in Fig. 13 shows the smallmovement of the rate pair (A211, A212).Now we show that

A112 < A/2I2 < 0All, A211

A112 < A212All, A211

D(y(1-02)jjql(l- 2))+l0g 1-'(1 -2)ql 1( 2)

D(y(1 oi) ||ql (1-o))+log 1-I(I-a1 i)

> D(-y(I-a2) |lql (I1-2))D (-y(I-o,) ||lql ( 1-ol) )

D(-y(I-ol) llql (1-ol)) >D(y(I-12) llql (1-02))log 1 (1 1 ) log 1- q(1"2)

X f(x) >=Dl (7Xlllq)lzis monotonically increasinglog 1-ql

(19)

in {xlO < < ifX4 f'(x) = (log -x log 1l_j (log 1-Th5)2

+log x ( - 1 ))'Y(log I jl )-2 > 0. (20)

Letu= 1-yxandv = -q1x. Sowehave0< v < u < 1

and need to prove that

g(u, v) = log U log 1 U-(log )2+log 1 .

(21)Since g(v,v) = 0,VO < v < 1, we just need to show

that 09(U,v) > 0 VO < v < u < 1. For a fixed u, weaucan consider zu(v) =9g(u) as a function of v. Because

g)=u v0)0u VO < u < 1, we only need to provethat 02g9(uvV) < 0. It is easy to check that VO< v < u < 1

ouOv

2g(u, v)

Duav(u v)2

u2v2(1 -u)( -v)< 0. (22)

Thus, the inequality (19) is true, which means that theslope of A1 is smaller than that of A2 in Fig. 13. The

394

6--

-6

BER21.24 x 10-55.33 x 10-66.70 x 10-63.27 x 10-6

_ _

__

__

_

achievable shaded region is on the upper right side of pointA. Therefore, we can increase the rate pair I1,2(p2, Yy,pI)together and the strategy (P2, -Y,PI) is not optimal.Li

Appendigx B

Here we prove Theorem 4. In problem (7), the objectivefunction Al1 + (1 -A)12 is bounded and the domaino < ql, q2 < 1 is closed, so the maximum exists and can beattained. First we discuss some possible optimal solutionsand then we show that only one of them is the optimumfor any fixed A between 0 and 1.Case 0: If qi = Oorq2 = Oorq1 = q2 = 1, thenl1 = 12 = 0and so it can not be the optimum.Case 1: If q2 = l and 0 < qi < 1, then 12 = 0 and

D1q,9q, (1- a,) log 1qj(1 i) -H(1- a,) = 0 (23)qi (I1 a,)

1=:' q I

(1 -ol)(eH(l-aj)/(1-c) + 1)-

Case 2: If q1 =land < q2 <1, then I f=l and

0-I2Oq2 (1-a2) 109o q2(I a2) H(1- a2) = O (25)(1a2)logq2 (I1 a2)

=:>q-(1 a2)(eH(I-a2)/(1 a2) + 1) (26)

Proof: From Lemma 2, the solution q* to equation (27)is unique if Ao(1) < A < io(b(1- al)). From (28)

m(q2) = (H(q*(l a-a2))- q1( a-a2)log 1q2q-l(1 2)) *I I ~~~~~~q2q*(1 Oa2)}

log(1- q1(-a,)) -(H(q* (1- a,))- qH(1- a1)) *log(1- q(1( a-2)) = 0 (29)

Clearly, m(q2) is monotonically increasing and

lim m(q2)q2 -*0

oTo < O.

q(()< A < i()(l- a))=,~> q* >n ()-0,)

=:~m(1) > O. (31)

That means the unique solution q* to equation (28) is in

t24he domain 0 < q2 < 1. Furthermore, when io(1) < A <(24) ino(b(1- a,)), case 1 and case 2 can not be the optimum

because

D(A1l + (1- A)12) <Dq2 q2 iqi $(1 al,)

D(A1l + (1- A)12) < 0Dql qi 1,q2 (1-2)

(32)

(33)

Case 3:when

If 0 < ql, q2 < 1, then the optimum is attained

D(A1 + (1 -A)12) + (Al1 + (1- A)12)Dq2 + q, =q 0 be(N9q2 Oqi Ca

=A log(1- q (1- a,)) (1-A) log(1- q (1 a2)), (27) cas

and in

O(XIi+(1 X)12) =0 the0q2

(1- A) (H(q (I a2)) -q* (I a2) log 1 q2q1(1-2) noq2 q*(1 L2) I opI

-A(H(q(t-1 a,))- qH(1- a,)) = 0 ]

=' H(q* (1- a2))- q (1- a2) log 1q2 q1(1 02)1og(1-q*(1-a2))q2q(1 W2

logl-q(l-el) *(H(ql*(I t1)ql*Hf(l-t1 (28)1og(1-q*(1 (Y2))

Now we are going to find which case is optimal for dif- [2]ferent A.Lemma 1: Function l(ql) lg(1-qj(1- 2)) is monotoni- [3]

cally decreasing in the domain 0 < q1 < 1 when oz, < (Y2Lemma 2: The solution to equation (27) exists in (0,1) [4]

and is unique for any A with io(1) < A < io(O).Proof: Equation (27) is equivalent to l(q*) = A/(1 -A). [5]

From Lemma 1, l(ql) is monotonically decreasing. There- [6]fore, when 1(1) < A/(1 -A) < 1(0), i.e. pn() < A < pD(O),the solution q -*is unique and ql* E (0, 1).LIILemma 3: The unique solution (q*, q*) to equation (27)

and (28) in case 3 is the optimum if io(1) < A < io(0(1-al1)) -

Lerefore, case 3 is the optimum.iLemma 4: The unique solution (q. = 1, q* =(1- ai))case 1 is the optimum if o(b(1 - a,)) < A < 1.Proof: When o(ob(1 -a,)) < A < 1, case 3 is not optimalcause there is no solution qi C (0, 1) to equation (27).ise 2 is not optimal because inequality (33) holds. Sose 1 is the optimum.LLemma 5: The unique solution (q* = (1- a2), q* = 1)case 2 is the optimum if 0 < A < io(1).Proof: When 0 < A < in(1), case 3 is not optimal becauseere is no solution q2 C (0,1) to equation (28). Case 1 isit optimal because inequality (32) holds. So case 2 is theitimum.DFrom Lemma 3,4 and 5, Theorem 4 is immediatelyoved. Lii

REFERENCEST. M. Cover. Broadcast channels. IEEE Trans. Inform. Theory,IT-18:2-14, January 1972.P. P. Bergmans. Random coding theorem for broadcast channelswith degraded components. IEEE Trans. Inform. Theory, IT-19:197-207, March 1973.P. P. Bergmans. A simple converse for broadcast channels withadditive white Gaussian noise. IEEE Trans. Inform. Theory, IT-20:279-280, March 1974.R. G. Gallager. Capacity and coding for degraded broadcast chan-nels. Probl. Pered. Inform., 10:3-14, July-Sept. 1974.T. M. Cover. Comments on Broadcast channels. IEEE Trans.Inform. Theory, 44:2524-2530, October 1998.M. Griot, A.I. Vila Casado, and R.D. Wesel. Non-linear turbocodes for interleaver-division multiple access on the OR channel.In GLOBECOM '06. IEEE Global Telecomm. Conf., 27 Nov. - 1Dec. 2006.

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