P. P. Bergmans, "Random coding theorem for broadcast channels with degraded components," IEEE Transactions on Information Theory, vol. 19, pp. 97 -- 20, 1973.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... has been done in studying the performance limits of broadcast channels [9] The capacity region of the broadcast channel was studied by Cover in 1972[8] The direct theorem for the degraded broadcast channel was proved by Gallager in 1974 [12] and the converse was established by Bergmans in 1973 [1, 2]. The general (nondegraded) broadcast channel capacity remains an open problem, with the best known achievable region due to Marton in 1979 [16] In this paper, we are interested in providing a constructive coding framework for broadcast coding. Our framework is inspired by the ....

P. P. Bergmans. Random coding theorem for broadcast channels with degraded components. IEEE Transactions on Information Theory, IT-19:197--207, March 1973.

....of CDMA systems that we now describe in some detail. The Gaussian broadcast channel: The set of achievable rates t i for an M user Gaussian broadcast channel with noise powers # 1 # 2 . #M in di#erent links and total transmit power P tot (assuming unit total bandwidth) is given by [3, 4] # i j i P j , i = 1, M P tot , P i 0, i = 1, M (3) where P i is the transmit power allocated to link i. Here the communications variables are r = P 1 , PM ) and the constraints (3) are exactly in the generic form (2) This model can be used for ....

P. P. Bergmans. Random coding theorem for broadcast channels with degraded components. IEEE Trans. Inform. Theory, 19:197--207, 1973.

....the capacity of source coding with side information. Also used by Gel fand Pinsker [14] to show the capacity of channel coding with side information. The latter was extended to include the broadcast channel by Marton [19] and El Gamal [12] Possibly also used earlier by Bergmans and Gallager in [1], 2] and [11] Interestingly also used to show an achievable region for interference channels by Carleial in [4] Probably also by Han. Theorem 2.1. Slepian Wolf: Let (X, Y ) be iid r.v. s from a joint distribution. Recall the notion of joint typicality. The achievable region is given by: ....

....region, along with a conjecture of the achievable region for the more general degraded broadcast channel. Remark 4.3. Note that due to the construction of the degraded broadcast channel, the stronger user always decodes the weaker user s message first before decoding its own message. Bergmans [1] gave the proof of the achievable region of a general degraded broadcast channel. The converse was given by Bergmans [2] and Gallager [11] The Bergmans proof applies to the Gaussian broadcast channel, i.e. one with power constraint. Gallager s proof on the other hand uses an auxiliary variable U ....

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P. P. Bergmans, Random coding theorem for broadcast channels with degraded components, IEEE Transactions on Information Theory IT-19 (1973), 197--207.

....Proof See Appendix D. 2 It is worth noting that the achievable rate vector regions given by Theorem 4. 1 for the U user Gaussian multiple access channel and the U user Gaussian broadcast channel are 18 exactly the capacity de ning achievable rate regions given for these channels in [1, 21] and [4, 14], respectively. We next extend Theorems 3.1 and 4.1 so as to allow di erent information to be sent over di erent ow graphs between a source destination pair (s; d) The basic idea is to colocate virtual source destination pairs at the same pair of nodes s and d, in the above scheme. We ....

P. Bergmans, \Random coding theorem for broadcast channels with degraded components, " IEEE Trans. Inform. Theory, vol. IT-19, no. 1, pp. 197-207, Jan. 1973.

....uELij(q) k mq) u) Proof See Appendix D. It is worth noting that the achievable rate vector regions given by Theorem 4. 1 for the U user Gaussian multiple access channel and the U user Gaussian broadcast channel are exactly the capacity defining achievable rate regions given in [1, 17] and [3, 12] respectively for these channels. We next extend Theorems 3.1 and 4.1 so as to allow different information to be sent over different flow graphs between a source destination pair (s, d) The basic idea is to colocate virtual source destination pairs at the same pair of nodes s and d, in the above ....

P. Bergmans, "Random coding theorem for broadcast channels with degraded compo- nents," IEEE Trans. Inform. Theory, vol. IT-19, no. 1, pp. 197-207, Jan. 1973.

....y k x) broadcast channel is said to be physically degraded if p(y 1 , y 2 , y k x) p(y 1 x)p(y 1 ) p(y K K 1 ) 2) for some appropriate ordering of the outputs. The problem of capacity of broadcast channels has been considered in an early paper by Cover [2] Bergmans ([1]) specializes the results of [2] to the case of degraded broadcast channels and proves the coding theorem for such channels. Gallager [4] provides the converse to the coding theorem of [1] The Gaussian broadcast channel is a degraded broadcast channel and the capacity region of this channel is ....

....The problem of capacity of broadcast channels has been considered in an early paper by Cover [2] Bergmans ( 1] specializes the results of [2] to the case of degraded broadcast channels and proves the coding theorem for such channels. Gallager [4] provides the converse to the coding theorem of [1]. The Gaussian broadcast channel is a degraded broadcast channel and the capacity region of this channel is given by: N 1 N 2 # 1 P (3) R k C # # # # # k P N k P # # # RK C # # # #KP NK P K 1 # # where P is the total transmitted power; # k is the ....

Patrick P. Bergmans. Random coding theorem for broadcast channels with degraded components. IEEE Transactions on Information Theory, IT19: 197--207, March 1973.

....region for the AWGN broadcast channel using CD with successive decoding. Proof: Applying Lemma 3. 2 to (21) directly yields (26) For the two user degraded AWGN broadcast channel with noise variances n 1 B and n 2 B (n 1 B n 2 B) the capacity region for CD with successive decoding is [8]: CCD = 27) Therefore, R on the boundary of the capacity region (27) also satisfies (26) 2 3.4.2 CD without Successive Decoding minimum required total power to support R in a fading state n = n 1 ; n 2 ) is : 28) If R is on the boundary surface of C ....

P. P. Bergmans, "Random coding theorem for broadcast channels with degraded components," IEEE Trans. Inform. Theory, vol. IT-19, pp. 197--207, Mar. 1973.

....dreamt of. Thus it is that one turns to information theory for an answer to the question: How much information can wireless networks transport It is a triumph of information theory that the capacity regions for some systems have been characterized, as for example the Gaussian broadcast channel [10, 11, 12, 13] shown in Figure 1, and the Gaussian multiple access channel [14, 15] shown in Figure 2. Recently, for a network with a single source destination pair, the asymptotic rate has been characterized as the number of nodes in a bounded domain is increased, while excluding them from open neighborhoods ....

P. Bergmans, \Random coding theorem for broadcast channels with degraded components, " IEEE Trans. Inform. Theory, vol. 19, pp. 197-207, 1973.

....the M users and we denote them as S 0 ; Delta Delta Delta ; SN Gamma1 , the probabilities of which are p(S 0 ) Delta Delta Delta ; p(SN Gamma1 ) respectively. In each joint state, the channel can be viewed as a time invariant AWGN broadcast channel, the capacity region of which is known [19]. Given a block length n, we then design an encoder decoder pair for the M users in each state S k (0 k N Gamma 1) with codewords of average power P j (S k ) for each user j which achieve rate R j; k C j;S k , 1 j M , where C j;S k is the maximum achievable rate for User j on the ....

P. P. Bergmans, "Random coding theorem for broadcast channels with degraded components," IEEE Trans. Inform. Theory, vol. 19, no. 2, pp. 197--207, Mar. 1973.

....using CD with successive decoding. Proof: Applying Lemma 2 to (21) directly yields (26) For the two user degraded AWGN broadcast channel with noise 1108 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 3, MARCH 2001 variances and , the capacity region for CD with successive decoding is [10], 11] 27) Therefore, if is on the boundary of the capacity region (27) i.e. all the equalities in (27) are achieved, then which means that also satisfies (26) 2) CD Without Successive Decoding: For a two user broadcast channel with fading, given rate vector ,we know by (12) that the ....

P. P. Bergmans, "Random coding theorem for broadcast channels with degraded components," IEEE Trans. Inform. Theory, vol. IT-19, pp. 197--207, Mar. 1973.

....(u) q) j;u;k 1: Proof See Appendix D. 2 It is worth noting that the achievable rate vector regions given by Theorem 4. 1 for the U user Gaussian multiple access channel and the U user Gaussian broadcast channel are exactly the capacity de ning achievable rate regions given in [1, 17] and [3, 12] respectively for these channels. We next extend Theorems 3.1 and 4.1 so as to allow di erent information to be sent over di erent ow graphs between a source destination pair (s; d) The basic idea is to colocate virtual source destination pairs at the same pair of nodes s and d, in the above ....

P. Bergmans, \Random coding theorem for broadcast channels with degraded components, " IEEE Trans. Inform. Theory, vol. IT-19, no. 1, pp. 197-207, Jan. 1973.

....noise density of each user into states. Therefore, there are joint channel states of the users and we denote them as , the probabilities of which are , respectively. In each joint state, the channel can be viewed as a time invariant AWGN broadcast channel, the capacity region of which is known [19]. Given a 1096 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 3, MARCH 2001 block length , we then design an encoder decoder pair for the users in each state with codewords of average power for each user which achieve rate , where is the maximum achievable rate for User on the equivalent ....

P. P. Bergmans, "Random coding theorem for broadcast channels with degraded components," IEEE Trans. Inform. Theory, vol. IT-19, pp. 197--207, Mar. 1973.

....capacity region for the AWGN broadcast channel using CD with successive decoding. Proof: Applying Lemma 2 to (21) directly yields (26) For the two user degraded AWGN broadcast channel with noise variances n 1 B and n 2 B (n 1 B n 2 B) the capacity region for CD with successive decoding is [10], 11] CCD = ae R : R 1 B log 1 P 1 n 1 B ; R 2 B log 1 P 2 n 2 B P 1 ; 8P 1 P 2 P Psi : 27) Therefore, if R is on the boundary of the capacity region (27) i.e. all the equalities in (27) are achieved, then P = P 1 P 2 = 2 R1=B Gamma 1)n 1 B ....

P. P. Bergmans, "Random coding theorem for broadcast channels with degraded components," IEEE Trans. Inform. Theory, vol. 19, no. 2, pp. 197--207, Mar. 1973.

....user on channel with less noise, and is the fraction of allocated to the user on channel with more noise. Equivalently, power is used for the cloud centers of the superposition code designed for the th component degraded broadcast channel, while power is used for the fine points within the clouds [12]. Note that in (24) and (25) we have scaled both the noise and signal power by . Thus, from (23) the set must satisfy the power constraint (26) In Section VI we give numerical examples of the optimal power allocation within and between parallel degraded broadcast channels for several examples. ....

P. Bergmans, "Random coding theorem for broadcast channels with degraded components," IEEE Trans. Inform. Theory, vol. IT-19, pp. 197--207, Mar. 1973.

....can be found in [19] 22] 23] 26] 35] 62] 69] 98] 99] 100] 107] and [108] We first consider sending independent information over a degraded broadcast channel (Fig. 2) at rates to and to . The capacity region, conjectured in [16] was proved to be achievable by Bergmans [9], and the converse was established by Bergmans [10] and Gallager [41] Theorem 1: The capacity region for the degraded broadcast channel is the convex hull of the closure of all satisfying for some joint distribution , where the auxiliary random variable has cardinality bounded by . ....

....scheme described in [16] Choose Gaussian codewords independent and identically distributed (i.i.d. For each of these codewords , generate satellite Gaussian codewords of power and add them to form codewords . Thus, the fine information is superimposed on the coarse information . Bermans [9], 10] proved the converse. The achievability of the region in Theorem 1 for general degraded broadcast channels was established by Bergmans [9] There followed a year of intense activity trying to prove the converse, i.e. to prove that the natural achievable rate region, was indeed the capacity ....

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P. P. Bergmans, "Random coding theorem for broadcast channels with degraded components," IEEE Trans. Inform. Theory, vol. IT-19, pp. 197--207, Mar. 1973.

.... and Wolf [93] had conducted seminal research on lossless multiterminal source coding problems analogous to the multiple access channel models of Ahlswede [90] and Liao [91] Berger and Wyner agreed that research should be done on a lossy source coding analog of the novel Cover Bergmans [88] [89] theory of broadcast channels. Gray and Wyner were the first to collaborate successfully on such an endeavor, authoring what proved to be the first of many papers in the burgeoning subject of multiterminal lossy source coding [92] C. The Wyner Ziv Rate Distortion Function The seminal piece of ....

....the special case in which and are Bernoulli and statistically related as if connected by a BSC of crossover probability and . for this case is shown in (17) at the top of this page, where and is such that the sraight line segment for is tangent to the curved segment for . Berger had used Bergmans [89] theory of satellites and clouds to show that (17) was an upper bound to for this binary symmetric case. The major contribution of Wyner and Ziv s paper resided in proving a converse to the unlikely effect that this performance cannot be improved upon, and then generalizing to (17) for arbitrary ....

P. Bergmans, "Random coding theorem for broadcast channels with degraded components," IEEE Trans. Inform. Theory, vol. IT-19, pp. 197--207, 1973.

....v OE(k;1) v OE(k;2) Delta Delta Delta, v OE(k;M) and transmit power P 1 (S k ) P 2 (S k ) Delta Delta Delta ; PM (S k ) of the M users. Note that according to (49) v OE(k;j) OE(k;j) m , 81 j M . For a given n, let k = bnp(S k )c = np(S k ) for n sufficiently large. From [13], we know that for R j; k = B log 1 P j (S k ) v OE(k;j) B P M i=1 P i (S k )1[v OE(k;j) v OE(k;i) j = 1; 2; Delta Delta Delta ; M; there exists a sequence of i (2 k R 1; k ; 2 k R 2; k ; Delta Delta Delta ; 2 k RM; k ) k j codes fxw k [i]g k i=1 , w k ....

.... It is easy to see that there exists ff 1 ; ff 2 2 [0; 1] for which exp 2 n H(Z 2 jW 0 ; W 1 ; Z 1 ; V) 2e(ff 2 fi P N 21 ) 69) exp 2 n Gamma n H(Y 1 jW 0 ; W 2 ; Y 2 ; V) 2e(ff 1 fi P N 11 ) 70) Combining (69) and (70) with the conditional entropy inequalities [13] H(Y 2 jW 0 ; W 1 ; Z 1 ; V) n 2 ln ae exp 2 n H(Z 2 jW 0 ; W 1 ; Z 1 ; V) 2eN 22 oe ; H(Z 1 jW 0 ; W 2 ; Y 2 ; V) n Gamma n 2 ln ae exp 2 n Gamma n H(Y 1 jW 0 ; W 2 ; Y 2 ; V) 2eN 12 oe ; Li Goldsmith: Capacities of Fading Broadcast Channels: Part ....

P. P. Bergmans, "Random coding theorem for broadcast channels with degraded components," IEEE Trans. Inform. Theory, vol. 19, pp. 197--207, Mar. 1973.

.... capacity region is given by [7] Czero ( P ) P2F n2N CCD (n; P) 1) where CCD (n; P) is the capacity region of the time invariant Gaussian broadcast channel consisting of all rate vectors R satisfying R j B log 1 P j (n) n j B P M i=1 P i (n)1[n j n i ] 81 j M [8]. Here 1[ Delta] denotes the indicator function (1[x] 1 if x is true and zero otherwise) Therefore, by denoting ( Delta) as the permutation such that n (1) n (2) Delta Delta Delta n (M) the minimum required total power P min (R; n) that can support R in state n is [7] P min (R; ....

P. P. Bergmans, "Random coding theorem for broadcast channels with degraded components," IEEE Trans. Inform. Theory, vol. 19, pp. 197--207, Mar. 1973.

....region for the AWGN broadcast channel using CD with successive decoding. Proof: Applying Lemma 3. 2 to (21) directly yields (26) For the two user degraded AWGN broadcast channel with noise variances n 1 B and n 2 B (n 1 B n 2 B) the capacity region for CD with successive decoding is [8, 9]: CCD = ae R 1 B log 1 P 1 n 1 B ; R 2 B log 1 P 2 n 2 B P 1 ; 8P 1 P 2 P oe : 27) Therefore, if R is on the boundary of the capacity region (27) i.e. all the equalities in (27) are achieved, then P = P 1 P 2 = 2 R1=B Gamma 1)n 1 B (2 R2=B ....

P. P. Bergmans, "Random coding theorem for broadcast channels with degraded components," IEEE Trans. Inform. Theory, vol. 19, pp. 197--207, Mar. 1973.

....law for U defines a joint probability law for ( U ; Y 1 Delta Delta Delta YN ) Given a U 2 U (i.e. satisfying (2.5) define S U = f(R 1 ; RN : for 1 k N; 0 R k I(U k Gamma1 ; Y k j U k )g : 2.6) Now define the set S = U2U S U : 2. 7) It is known [4, 2, 3, 5], that the achievable rate region R = convex hull of S. Let us remark that from (2.5) for U 2 U , since UN j 0, the information I(UN Gamma1 ; YN j UN ) which appears in (2.6) I(UN Gamma1 ; YN ) Also since U 0 j X , I(U 0 ; Y 1 j U 1 ) I(X ; Y 1 j U 1 ) Thus, for example, the ....

....that from (2.5) for U 2 U , since UN j 0, the information I(UN Gamma1 ; YN j UN ) which appears in (2. 6) I(UN Gamma1 ; YN ) Also since U 0 j X , I(U 0 ; Y 1 j U 1 ) I(X ; Y 1 j U 1 ) Thus, for example, the achievable rate region for the common broadcast channel with N = 2 (see [4, 3]) is R = convex hull of [ U2U SU ; where U satisfies U ffi X ffi Y 1 ffi Y 2 , and SU = f(R 1 ; R 2 ) 0 R 1 I(X ; Y 1 j U) 0 R 2 I(U ; Y 2 )g : Now let us turn to the special case of the our erasure channel as defined above. We first establish an identity involving I(U ....

P. Bergmans, "Random coding theorem for broadcast channels with degraded components, " IEEE Trans. Inform. Theory, vol. IT--19, pp. 197--207, 1973.

....our knowledge there is no known capacity region for any broadcast channel with memory. The most common example of a memoryless broadcast channel with a known capacity region is the degraded broadcast channel. This channel was studied by Cover in [2] with the capacity region obtained by Bergmans [6] and Gallager [7] This model was extended to a set of parallel degraded broadcast channels by Hughes Hartogs [8] who obtained an achievable rate region for a parallel set of degraded broadcast channels without common information. This region was shown to equal the capacity region by Poltyrev ....

....k with more noise, and (1 Gamma ff k ) is the fraction of P k allocated to the user with less noise. Equivalently, power ff k P k is used for the cloud centers of the superposition code designed for the kth component channel, while power ff k P k is used for the cloud points of this code [6]. Numerical examples of the optimal power allocation for several example channels are given in Section 6. 14 k X k Z k k W k Y V X Z W Y V X Z k W Y V 1 1 1 1 1 n n n n Figure 3: Equivalent Channel Model for the n CGBC. 15 The capacity region with common information (73) 75) is ....

P. Bergmans. Random coding theorem for broadcast channels with degraded components. IEEE Transactions on Information Theory, IT-19(2), pp. 197--207, Mar. 1973.

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P. P. Bergmans, "Random coding theorem for broadcast channels with degraded components," IEEE Transactions on Information Theory, vol. 19, pp. 97 -- 20, 1973.

No context found.

P. P. Bergmans, "Random coding theorem for the broadcast channels with degraded components," IEEE Trans. Inform. Theory, vol. IT-19, pp. 197--207, Mar. 1973.

No context found.

P. P. Bergmans. Random coding theorem for the broadcast channels with degraded components. IEEE Trans. on Info. Theory, 19(2):197-- 207, March 1973.

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