Accessible Capacity of Secondary Users

Transcription

1 Accessible Capacity of Secondary Users Xiao Ma Sun Yat-sen University November 14th, 2010 Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

2 This talk is based on: Xiao Ma, Xiujie Huang, Lei Lin and Baoming Bai, Accessible Capacity of Secondary Users, to submit to IEEE Trans. Inform. Theory. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

3 Outline 1 Introduction Motivations Backgrounds New Problem Formulation 2 Basic Definitions and Problem Statements Interference-Free AWGN Channels Gaussian Interference Channels Relation of the Accessible Capacity to the Capacity Region 3 Bounds on the Accessible Capacity 4 The Evaluation of the Upper and Lower Bounds 5 Conclusions and Future work Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

4 Introduction 1. Introduction Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

5 Introduction Motivations 1.1 Backgrounds 1) The Gaussian Interference Channel (GIFC) was first mentioned by Shannon [Shannon61]. 2) Ahlswede [Ahlswede74] gave simple but fundamental inner and outer bounds on the capacity region of GIFC. 3) Carleial [Carleial78] proposed a standard GIFC model as { Y1 = X 1 + a 21 X 2 + Z 1 Y 2 = a 12 X 1 + X 2 + Z 2, (1) where X i X i, Y i Y i, Z i R for i = 1, 2. Power:P 1 X 1 Z 1~N(0,1) Y 1 a12 a21 X 2 Power:P 1 Y 2 Z2~N(0,1) Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

6 Introduction Motivations 1.1 Backgrounds 4) For very strong GIFC with a P 2 and a P 1, Carleial [Carleial78] proved that the capacity region is a rectangle. 5) For strong GIFC with a and a2 21 1, Han and Kobayashi[Han81], and Sato [Sato78] obtained the capacity region independently by utilizing a compound multiple-access channel. 6) For the class of deterministic GIFCs [ElGamal82], where Y 1 and Y 2 are deterministic functions of X 1 and X 2, its capacity region was investigated by El Gamal. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

7 Introduction Motivations 1.1 Backgrounds 4) For very strong GIFC with a P 2 and a P 1, Carleial [Carleial78] proved that the capacity region is a rectangle. 5) For strong GIFC with a and a2 21 1, Han and Kobayashi[Han81], and Sato [Sato78] obtained the capacity region independently by utilizing a compound multiple-access channel. 6) For the class of deterministic GIFCs [ElGamal82], where Y 1 and Y 2 are deterministic functions of X 1 and X 2, its capacity region was investigated by El Gamal. For general GIFC, the determination of the capacity region is still open. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

8 Introduction Motivations 1.1 Backgrounds 7) For general GIFC, only various of inner and outer bound are presented. 7.1) The best inner bound on the capacity region is that put forth by Han and Kobayashi (HK) [Han81]. 7.2) The HK inner bound has been simplified by Karmer [Kramer06] and Chong et al. [Chong08] independently. 7.3) Kramer [Kramer04] derived two outer bounds on the capacity region based on the work of Sato, Carleial and Costa. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

9 Introduction Motivations 1.1 Backgrounds 7) For general GIFC, only various of inner and outer bound are presented. 7.1) The best inner bound on the capacity region is that put forth by Han and Kobayashi (HK) [Han81]. 7.2) The HK inner bound has been simplified by Karmer [Kramer06] and Chong et al. [Chong08] independently. 7.3) Kramer [Kramer04] derived two outer bounds on the capacity region based on the work of Sato, Carleial and Costa. 8) A fresh approach Approximation Method. 8.1) Etkin, Tse and Wang showed that the HK inner bound is within one bit of the capacity region [Etkin08]. 8.2) Bresler, Parekh and Tse investigated the capacity regions of many-to-one and one-to-many GIFCs and showed that the capacity regions can be determined to within a constant gap [Bresler10]. However, these results are particularly relevant in the high SNR regime. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

10 Introduction Motivations 1.2 New Problem Formulation Z 1 Power:P 1 A W 1 1 Encoder X 1 a 12 Y 1 Decoder Wˆ 1 B a 21 2 C Encoder Decoder W 2 X 2 Y 2 Wˆ 2 Power:P 2 D Z 2 Figure: 1. The model of the Gaussian interference channel. Original problem: For any pair of rates (R 1, R 2 ), is there a pair of codes (C 1, C 2 ) such that (R 1, R 2 ) is achievable? Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

11 Introduction Motivations 1.2 New Problem Formulation New problem: We make the following assumptions. 1) The primary users (User A and B) employ a pair of encoder C 1 and decoder that were originally designed to satisfy a given error performance requirement ǫ without any interferences. 2) The secondary users (User C and D) attempt to access the same medium. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

12 Introduction Motivations 1.2 New Problem Formulation New problem: We make the following assumptions. 1) The primary users (User A and B) employ a pair of encoder C 1 and decoder that were originally designed to satisfy a given error performance requirement ǫ without any interferences. 2) The secondary users (User C and D) attempt to access the same medium. What is the maximum transmission rate (accessible capacity) of the secondary users under the following two constraints (a) and (b)? (a) The primary encoder (not the decoder) is kept unchanged. (b) The secondary users should not affect the error performance requirement ǫ. Equivalently, for any rate R 2, is there a code C 2 such that R 2 is accessible when the code C 1 of rate R 1 is fixed? Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

13 Introduction Motivations 1.2 New Problem Formulation These assumptions are reasonable at least in the following scenarios. 1 It is not convenient to change the encoder C 1 at User A. For example, User A is located in a place (say the Space Station) that can not be reached easily. 2 It is not economic to change the encoder C 1 at User A which has been widely implemented according to international standards. 3 User A is weak in the sense that it can only afford the simple encoders C 1. For example, User A is an energy-limited wireless sensor that collects and transmits data to the powerful data center (User B). Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

14 Basic Definitions and Problem Statements 2. Basic Definitions and Problem Statements Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

15 Basic Definitions and Problem Statements Interference-Free AWGN Channels Generalized Trellis Code (GTC) Without loss of generality, almost all coded modulation schemes can be described by the generalized trellis code (GTC). A GTC can be specified by a time-invariant trellis (trellis section) as: a) A state set is indexed bys={0, 1,..., S 1}. b) A branch set is denoted byb. Each branch inbis characterized by b = (s (b), u(b), c(b), s + (b)) as s - (b) u(b) / c(b) s + (b) (For more details of this representation, see [Ma03].) c) Assume that the average energy emitted from each state is normalized, i.e., 1 M b:s (b)=s c(b) 2 = n for all s. Ma03 X. Ma and A. Kavčić, Path partition and forward-only trellis algorithms, IEEE Trans. Inform. Theory, vol. 49, no. 1, pp , Jan Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

16 Basic Definitions and Problem Statements Interference-Free AWGN Channels Examples of GTCs (I) 0 0 s - (b) u(b) c(b) s + (b) Figure: 2. Uncoded BPSK 0 0 s - (b) u(b) c(b) s + (b) Figure: 3. Repetition coded BPSK Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

17 Basic Definitions and Problem Statements Interference-Free AWGN Channels Examples of GTCs (II) (c) 0 0 s - (b) u(b) c(b) s + (b) Figure: 4. Extended Hamming coded BPSK Remark: A conventional block code of size M can be regarded as a GTC with only one state and M parallel branches, which correspond to M codewords, resp.. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

18 Basic Definitions and Problem Statements Interference-Free AWGN Channels Examples of GTCs (III) s - (b) u(b) c(b) s + (b) Figure: 5. Convolutional coded BPSK Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

19 Basic Definitions and Problem Statements Interference-Free AWGN Channels Primary Link without Interferences Encoding: Let w 1 = (u 1, u 2,...,u N ) M 1 be a data sequence, drawn from an independent and uniformly distributed source, to be transmitted. 1) At time t = 0, the state of the encoder is initialized as s 0 S. 2) At time t = 1, 2,..., the message u t is input to the encoder and drives the encoder from state s t 1 to s t. In the meantime, the encoder delivers a coded signal c t such that (s t 1, u t, c t, s t ) forms a valid branch. 3) Suppose that the available power is P 1. Then the signal x 1,t = P 1 c t at time t is transmitted. The transmitted signal sequence is denoted by x 1. The collection of all coded (or transmitted) sequences is denoted by C 1. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

20 Basic Definitions and Problem Statements Interference-Free AWGN Channels Primary Link without Interferences AWGN Channel: The channel is assumed to be an AWGN channel and the received signal sequence is denoted by y 1, which is statistically determined by y 1 = x 1 + z 1 (2) where z 1 is a sequence of samples from a white Gaussian noise of variance one per dimension. Decoding: Upon on receiving y 1, User B can utilize, in principle, the Viterbi algorithm, the BCJR algorithm or other trellis decoding algorithms to estimate the transmitted messages. Assume that a decoderψ 1 is utilized and ŵ 1 = (û 1, û 2,...,û N ) is the estimated message sequence after decoding. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

21 Basic Definitions and Problem Statements Interference-Free AWGN Channels Primary Link without Interferences Error Performance Criterion: For all t 1, define error random variables as E t = { 0 if Ût = U t 1 if Û t U t. (3) In order to characterize the performance of the (de)coding scheme in a unified way, we introduce the following random variables Θ N = ΣN t=1 E t, for N = 1, 2,... (4) N and consider the limit superior in probability [Han93] of the sequence{θ N }. Han93 T. S. Han and S. Verdú, Approximation theory of output statistics, IEEE Trans. Inform. Theory, vol. 39, no. 3, pp , May Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

22 Basic Definitions and Problem Statements Interference-Free AWGN Channels Primary Link without Interferences Definition 1 Letǫ be a real number in the interval (0, 1). A GTC is said to be ǫ-satisfactory under the decoderψif the limit superior in probability of{θ N } is not greater thanǫ, that is, } p- lim supθ N = inf {α lim Pr{Θ N >α} = 0 ǫ. (5) N N Equivalently, lim Pr{Θ N>ǫ} = 0. (6) N Remark: For a block code,{e t } under commonly-used decodersψ s are i.i.d.. Then, by WLLN, for anyδ>0, lim N Pr{ Θ N ε 1 δ} = 1, whereε 1 = Pr(E 1 = 1). Hence, Definition 1 for a block code is consistent with thatε 1 ǫ. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

23 Basic Definitions and Problem Statements Gaussian Interference Channels 2.2 GIFC with Primary and Secondary Users User C attempts to send message to User D by accessing the same medium. Encoding: 1) The encoding function at User A is 2) The encoding function at User C is φ 1 : M 1 R nn w 1 x 1 =φ 1 (w 1 ). (7) φ 2 : M 2 R nn w 2 x 2 =φ 2 (w 2 ). (8) 3) The coding rates (bits/dimension) are R 1 = log M n and R 2 = log M 2 nn. 4) The coded sequences are required to satisfy the power constraints E X 1 2 nnp 1 and E X 2 2 nnp 2, respectively. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

24 Basic Definitions and Problem Statements Gaussian Interference Channels 2.2 GIFC with Primary and Secondary Users Gaussian Interference Channels: Assume that User A and C transmit synchronously x 1 and x 2, respectively. The received sequences at User B and D are y 1 and y 2, respectively. For the standard Gaussian interference channel as shown in Fig. 1, we have y 1 = x 1 + a 21 x 2 + z 1 y 2 = a 12 x 1 + x 2 + z 2, (9) where z 1 and z 2 are two sequences of samples drawn from an AWGN of variance one per dimension, and a = (a 12, a 21 ) is the real interference coefficient vector. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

25 Basic Definitions and Problem Statements Gaussian Interference Channels 2.2 GIFC with Primary and Secondary Users Decoding: 1) The decoding function at User B is ψ 1 : R nn M 1 y 1 ŵ 1 = ψ 1 (y 1 ), (10) which can be different from the decoderψ 1 used in the case when no interference exists. 2) The decoding function at User D is ψ 2 : R nn M 2 y 2 ŵ 2 =ψ 2 (y 2 ). (11) Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

26 Basic Definitions and Problem Statements Gaussian Interference Channels 2.2 GIFC with Primary and Secondary Users Error Performance Criteria: 1) For primary users, we define the similar criteria p- lim sup N ΘN. 2) For secondary users, we defineε (N) 2 = Pr{Ŵ 2 W 2 }. Definition 2 A rate R 2 is achievable for the secondary users if for anyδ>0, there is a pair of coding/decoding functions (φ 2,ψ 2 ) of coding rate R 2 δ such that lim N ε (N) 2 = 0. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

27 Basic Definitions and Problem Statements Gaussian Interference Channels 2.2 GIFC with Primary and Secondary Users Definition 3 A rate R 2 is accessible if R 2 is achievable for the secondary users and there exists a decoder ψ 1 such that the GTC C 1 isǫ-satisfactory, that is, p- lim sup N ΘN ǫ. Definition 4 The accessible capacity for the secondary users is defined as C 2 = sup{r 2 : R 2 is accessible}. (12) Problem formulation: The problem is then formulated as given C 1 andǫ, find C 2. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

28 Basic Definitions and Problem Statements Relation of the Accessible Capacity to the Capacity Region 2.3 Relation of the Accessible Capacity to the Capacity Region For a conventional GIFC, R = Convex hull of { (R 1, R 2 ) : (R 1, R 2 ) is achievable }, (13) where (R 1, R 2 ) is achievable means that, there exists a pair of codes (C 1, C 2 ) of rates (R 1, R 2 ) such that the decoding error probability converges to zero. For the considered GIFC, C 2 = sup { R 2 : there exists C 2 such that R 2 is accessible }, (14) where the code C 1 of rate R 1 is fixed. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

29 Basic Definitions and Problem Statements Relation of the Accessible Capacity to the Capacity Region 2.3 Relation of the Accessible Capacity to the Capacity Region For largeǫ, the pair (R 1, C 2 ) may lie in the outside of the capacity region of the GIFC. Whenǫ 0, the pair (R 1, C 2 ) must lie in the inside of the capacity region. In particular, the pair (R 1, C 2 ) must be on the boundary of the capacity region, where C 2 is defined as C 2 = lim sup {C 2 (C 1,ǫ)}. (15) R 2 * R 2 ǫ 0{C 1 (R 1,ǫ)} * * " R 1 * ' R 1 R 1 R 1 Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

30 Bounds on the Accessible Capacity 3. Bounds on the Accessible Capacity Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

31 Bounds on the Accessible Capacity 3. Bounds on the Accessible Capacity Rewrite the considered system in (9) in terms of random variables as where Y 1 = X 1 + a 21 X 2 + Z 1 Y 2 = a 12 X 1 + X 2 + Z 2, (16) X 1 is a random sequence of length nn with probability mass function p 1 (x 1 ) = 1 2 nnr 1 for x 1 C 1 and p 1 (x 1 ) = 0 otherwise; Z 1 N(0, 1)and Z 2 N(0, 1) ; X 2 is a random sequence of length nn whose distribution is to be determined. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

32 Bounds on the Accessible Capacity 3. Bounds on the Accessible Capacity Define: and Let and C (N) L R (N) 21 R (N) 22 C (N) U = 1 nn I(X 2; Y 1 ) (17) = 1 nn I(X 2; Y 2 ). (18) = max R (N) (19) 22 {p(x 2 )} = max min{r (N) 21, R(N) 22 } (20) {p(x 2 )} where the set{p(x 2 )} consists of all possible pmfs p(x 2 ) such that E[ X 2 ] nnp 2. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

33 Bounds on the Accessible Capacity 3.1 Two Lemmas Lemma 5 Both the links X 2 Y 2 and X 2 Y 1 can be viewed as noncontrollable finite-state channels (FSCs) [Vontobel08,Huang09a,Huang09b] where the processes of channel states are Markovian. Outline of the proof of Lemma 5: Consider the link X 2 Y 2. (1) The transmitted sequence x 1 corresponds to a path through the trellis of the GTC C 1, denoted by B 1, B 2,, B N. The sequence{b t } forms a Markov chain such that the left state of B 1 is s (B 1 ) = s 0 and { 1 Pr(B t = b t B t 1 = b t 1 ) = M, if s (b t ) = s + (b t 1 ). (21) 0, otherwise (2) Given the channel state b t, we have 1 p(y 2,t b t, x 2,t ) = (2π) n/2 exp{ y 2,t a 12 x 1,t x 2,t 2 } (22) 2 where x 2,t X 2 and x 1,t = P 1 c(b t ). Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

34 Bounds on the Accessible Capacity 3.1 Two Lemmas Outline of the proof of Lemma 5: (3) The channel states{b t } evolve freely in the sense that the states do not depend on the input x 2 and hence the link X 2 Y 2 is a noncontrollable FSC. (4) Specifically, we have p(y 2 x 2 ) = p(b, y 2 x 2 ) (23) b B N = p(b)p(y 2 b, x 2 ) (24) b B N = b B N t=1 N p(b t b t 1 )p(y 2,t b t, x 2,t ) (25) (5) Similarly, the link X 2 Y 1 is also a noncontrollable FSC with the Markov process{b t } as states. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

35 Bounds on the Accessible Capacity 3.1 Two Lemmas Without loss of generality, we make the following assumptions: (a) The Markov chain{b t } is irreducible, (b) There exists a channel state b t for some time t> 1 such that p(b t b 1 )>0for any initial state b 1. Hence, both the links X 2 Y 1 and X 2 Y 2 are indecomposable. Then we have a lemma as follows. Lemma 6 The limits C L = lim C (N) and C N L U = lim C (N) exist. N U Proof: The proof is similar to that of Theorem in [Gallager68]. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

36 Bounds on the Accessible Capacity 3.2 The Main Theorem Theorem 7 The accessible capacity C 2 is bounded as C L C 2 C U. Outline of proof of Theorem 7: (1) Any rate R 2 > C U is not achievable for the link X 2 Y 2. From Fano s inequality and data processing inequality, we have Dividing by nn, nnr 2 = H(W 2 ) = H(W 2 Y 2 )+I(W 2 ; Y 2 ) 1+ε (N) 2 nnr 2 + I(X 2 ; Y 2 ). (26) R 2 1 nn +ε(n) R nn I(X 2; Y 2 ) 1 nn +ε(n) R C (N) U. (27) As N, we have R 2 C U since 1 nn 0 andε(n) 2 0. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

37 Bounds on the Accessible Capacity 3.2 The Main Theorem Outline of proof of Theorem 7: (2) Any rate R 2 < C L is accessible. (2.1) Any rate R 2 < C L is achievable. According to Theorem in [Gallager68], for anyε>0, there exists N(ε) such that for each N N(ε) and each R 2 0 there exists a block code C 2 with rate R 2 and codeword length nn such that where ε (N) ε (N) 1 exp{ N[E r,1 (R 2 ) ε]} ε (N) 2 exp{ N[E r,2 (R 2 ) ε]}, (28) 1 = Pr{ ψ 1,1 (Y 1 ) W 2 } andε (N) 2 = Pr{ψ 2 (Y 2 ) W 2 }; ψ 1,1 andψ 2 are the maximum-likelihood decoders at User B and User D, respectively; E r,1 (R 2 ) and E r,2 (R 2 ) are the random coding error exponents, which are strictly positive for R 2 < C L. Therefore, as N,ε (N) 1 0 andε (N) 2 0. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

38 Bounds on the Accessible Capacity 3.2 The Main Theorem Outline of proof of Theorem 7: (2.2) There exists a decoder ψ 1 such that the error performance requirementǫ is still fulfilled, that is, Pr( Θ N ǫ) 1. Consider the following two-stage decoder ψ 1. Step 1: Upon receiving y 1, User B utilizes the maximum-likelihood decoder ψ 1,1 to get an estimated message ŵ 2. For convenience, we introduce a random variable as { 0, Ŵ2 = W Υ 1 = 2. (29) 1, Ŵ 2 W 2 Thenε (N) = Pr(Υ 1 1 = 1) 0. Step 2: User B re-encodes ŵ 2 to get an estimated coded sequence ˆx 2 =φ 2 (ŵ 2 ). Then User B uses the primary decoderψ 1 to decode the sequence ỹ 1 = y 1 a 21 ˆx 2 to get a sequence of estimated messages ŵ 1 = (û 1, û 2,...,û N ). Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

39 Bounds on the Accessible Capacity 3.2 The Main Theorem Outline of proof of Theorem 7: (2.2 ) For this two-stage decoder ψ 1, the statistical dependence among the error random variables{ẽ t } becomes even more complicated because (a) the erroneously-decoding ŵ 2 at User B may cause burst errors in ŵ 1 at the second stage, (b) the correctly-decoding ŵ 2 at User B indicates that the link X 2 Y 1 is not that noisy, equivalently, that the sum of the transmitted codeword x 1 and the Gaussian noise sequence z 1 is not that strong. Fortunately, this complicatedness does not affect the correctness of p- lim sup N Θ N ǫ. Actually, Pr{ Θ N >ǫ} = Pr{ Θ N >ǫ,υ 1 = 0}+Pr{ Θ N >ǫ,υ 1 = 1} Pr{ Θ N >ǫ,υ 1 = 0}+ε (N) 1 Pr{Θ N >ǫ}+ε (N) 1 (30) 0, as N. (31) Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

40 Bounds on the Accessible Capacity 3.3 Two Corollaries An immediate consequence of Theorem 7 is the following corollary, which is related to a similar case when interferences are strong. Corollary 8 If R (N) 22 R(N) 21 holds for any pmfs p(x 2), then C 2 = C L = C U. For block codes, we have a corollary whose condition is slightly weaker than that of Corollary 8. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

41 Bounds on the Accessible Capacity 3.3 Two Corollaries For block codes, a special class of GTCs, we have and The bounds are then reduced to R (N) 21 = 1 n I(X 2; Y 1 ) = R 21 (32) R (N) 22 = 1 n I(X 2; Y 2 ) = R 22. (33) C U = max R 22 {p(x 2 )} and C L = max min{r 21, R 22 }, (34) {p(x 2 )} where the set{p(x 2 )} consists of all possible pmfs overx 2 such that E[ X 2 ] np 2. Assume that Q 1 2 = arg max {p(x 2 )} n I(X 2; Y 2 ). (35) Then we have the following corollary. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

42 Bounds on the Accessible Capacity 3.3 Two Corollaries Corollary 9 Suppose that the GTC C 1 represents a block code. If R 22 R 21 holds for the pmf Q 2, then C 2 = C L = C U, that is, C 2 = 1 n I(X 2; Y 2 ) X2 Q 2. (36) Proof: It is obvious. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

43 The Evaluation of the Upper and Lower Bounds 4. The Evaluation of the Upper and Lower Bounds Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

44 The Evaluation of the Upper and Lower Bounds 4.1 The Method to Evaluate Bounds To make the computations tractable, we assume that the transmitted sequence is i.u.d. overx 2. In the following, we evaluate the upper bounds by using a method similar to those introduced in [Arnold01,Pfister01,Arnold06]. C U = lim R (N) N 22 1 = lim N nn I(X 2; Y 2 ) 1 = lim N nn h(y 1 2) lim N nn h(y 2 X 2 ) 1 = lim N nn h(y 1 2) lim N nn h(a 12X 1 + Z 2 ). (37) The above two entropy rates can be computed by the BCJR algorithm. We take the computation of lim N 1 nn h(y 2) as an example. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

45 The Evaluation of the Upper and Lower Bounds 4.1 The Method to Evaluate Bounds Firstly, from the Shannon-McMillan-Breiman theorem, with probability 1, lim 1 N nn log f(y 1 2) = lim N nn h(y 2), (38) since the sequence Y 2 is a stationary stochastic process. We also have N N log f(y 2 ) = log f(y 2,t y (t 1) ) 2 = log f(y 2,t y (t 1) ). (39) 2 t=1 Secondly, we can represent the link X 2 Y 2 by a time-invariant trellis, which is based on the mentioned trellis representation of the GTC C 1. We introduce an example in the next page. t=1 Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

46 The Evaluation of the Upper and Lower Bounds 4.1 The Method to Evaluate Bounds s - (b) u(b) c(b) x 2(b) s + (b) s - (b) u(b) c(b) x 2(b) s + (b) Figure: 6. A trellis section of the link X 2 Y 2 where the GTC C 1 represents the (2, 1, 2) convolutional coded BPSK. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

47 The Evaluation of the Upper and Lower Bounds 4.1 The Method to Evaluate Bounds To compute f(y 2,t y (t 1) ), we introduce 2 ( α t (s t ) = p Then s t y (t) 2 ), t = 0, 1,..., N. (40) f(y 2,t y (t 1) ) = α 2 t 1 (s t 1 )f(y 2,t s t 1 ), (41) s t 1 where f(y 2,t s t 1 ) = p(u t ) p(x 2,t ) p(y 2,t u t, x 2,t ) b t :s =s t = exp { y } 2,t a 12 P1 c t x 2,t 2, (42) M X 2 (2π) n/2 2 b t :s =s t 1 1 and s ( t 1 p s t 1 y (t 1) ) p(st, y 2 2,t s t 1 ) α t (s t ) = s ( t 1,s t p s t 1 y (t 1) ) p(st, y 2 2,t s t 1 ). (43) Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

48 The Evaluation of the Upper and Lower Bounds Remark: The method to compute entropy rates is applicable to any i.i.d. pmf p(x 2 ) overx 2. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

49 The Evaluation of the Upper and Lower Bounds 4.2 Numerical Results Consider the GTCs represent those familiar codes shown in the following Table. Table: Comparison of the Chosen Codes Code Rate P 1 BER Uncoded-BPSK e 2 [2, 1, 2]-RCBPSK 1/ e 3 [8, 4, 4]-EHCBPSK 1/ e 4 (2, 1, 2)-CCBPSK 1/ e 7 (3, 1, 2)-CCBPSK 1/3 6 < 0.63 e 7 We evaluate the upper and lower bounds C U accessible capacity C 2 and C L on the Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

50 The Evaluation of the Upper and Lower Bounds 4.2 Numerical Results Assume that the power constraint at User A is P 1 = R C U : (2,1,2)-CCBPSK C L : (2,1,2)-CCBPSK 0.6 C U : [8,4,4]-EHCBPSK C L : [8,4,4]-EHCBPSK 0.5 C U : [2,1,2]-RCBPSK C L : [2,1,2]-RCBPSK P 2 Figure: 7. The interference coefficients of the considered GIFC are a 2 12 = a2 21 = 1.5. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

51 The Evaluation of the Upper and Lower Bounds 4.2 Numerical Results R C U : (2,1,2)-CCBPSK C L : (2,1,2)-CCBPSK 0.6 C U : [8,4,4]-EHCBPSK C L : [8,4,4]-EHCBPSK 0.5 C U : [2,1,2]-RCBPSK C L : [2,1,2]-RCBPSK P 2 Figure: 8. The interference coefficients of the considered GIFC are a 2 12 = a2 21 = 1.0. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

52 The Evaluation of the Upper and Lower Bounds 4.2 Numerical Results R C U : (2,1,2)-CCBPSK 0.5 C L : (2,1,2)-CCBPSK C U : [8,4,4]-EHCBPSK 0.4 C L : [8,4,4]-EHCBPSK 0.3 C U : [2,1,2]-RCBPSK C L : [2,1,2]-RCBPSK P 2 Figure: 9. The interference coefficients of the considered GIFC are a 2 12 = a2 21 = 0.5. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

53 The Evaluation of the Upper and Lower Bounds 4.2 Numerical Results 1 R C U : (3,1,2)-CCBPSK C L : (3,1,2)-CCBPSK C U : (2,1,2)-CCBPSK C L : (2,1,2)-CCBPSK C U : Uncoded-BPSK C L : Uncoded-BPSK P 2 Figure: 10. The interference coefficients of the considered GIFC are a 2 12 = a2 21 = 1.5. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

54 The Evaluation of the Upper and Lower Bounds 4.2 Numerical Results R C U : (3,1,2)-CCBPSK C L : (3,1,2)-CCBPSK C U : (2,1,2)-CCBPSK C L : (2,1,2)-CCBPSK C U : Uncoded-BPSK C L : Uncoded-BPSK P 2 Figure: 11. The interference coefficients of the considered GIFC are a 2 12 = a2 21 = 1.0. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

55 The Evaluation of the Upper and Lower Bounds 4.2 Numerical Results 1 R C U : (3,1,2)-CCBPSK C L : (3,1,2)-CCBPSK C U : (2,1,2)-CCBPSK C L : (2,1,2)-CCBPSK C U : Uncoded-BPSK C L : Uncoded-BPSK P 2 Figure: 12. The interference coefficients of the considered GIFC are a 2 12 = a2 21 = 0.5. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

56 The Evaluation of the Upper and Lower Bounds 4.2 Numerical Results R C U : [8,4,4]-EHCBPSK C L : [8,4,4]-EHCBPSK 0.5 C U : [2,1,2]-RCBPSK C L : [2,1,2]-RCBPSK P 2 Figure: 13. The interference coefficients of the considered GIFC are a 2 12 = 0.5, a 2 21 = 1.5. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

57 Conclusions and Future work 5. Conclusions and Future work Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

58 Conclusions and Future work 5.1 Conclusions 1) We present a new problem formulation of the GIFC with primary users and secondary users. 2) We define the accessible capacity for the considered GIFC. For a given code C 1 of rate R 1, C 2 = sup { R 2 : there exists C 2 such that R 2 is accessible }. (44) 3) We reveal the relation of the accessible capacity to the capacity region. 4) We derive upper and lower bounds on the accessible capacity. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

59 Conclusions and Future work 5.1 Conclusions 5) We evaluate the bounds for some special cases and obtain the following results from the numerical simulation. a. Primary users with lower transmission rates may allow higher accessible rates. b. Better primary encoders guarantee not only higher quality of the primary link but also higher accessible rates of the secondary users. c. More interestingly, the accessible capacity does not always increase with the transmission power of the secondary users. Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

60 Conclusions and Future work 5.2 Future work We will focus on the method to construct matched information rate codes (MIRCs) [Kavcic05] for the considered GIFC system. Kavcic05 A. Kavčić, X. Ma, and N. Varnica, Matched information rate codes for partial response channels, IEEE Trans. Inform. Theory, vol. 51, no. 3, pp , Mar Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

61 Some References I Shannon61 Ahlswede74 C. E. Shannon, Two-way communication channels, in Forth Berkeley Symp. on Math. Statist. and Prob., J. Neyman, Ed., vol. 1. Statist. Lab. of the University of California, Berkely: University of California Press, Jun Jul. 30, , pp R. Ahlswede, The capacity region of a channel with two senders and two receivers, The Annals of Probability, vol. 2, no. 5, pp , Oct Carleial78 A. B. Carleial, Interference channels, IEEE Trans. Inform. Theory, vol. IT-24, no. 1, pp , Jan Han81 Sato78 ElGamal82 Kramer06 Chong08 Kramer04 Etkin08 Bresler10 T. S. Han and K. Kobayashi, A new achievable rate region for the interference channel, IEEE Trans. Inform. Theory, vol. IT-27, no. 1, pp , Jan H. Sato, On degraded Gaussian two-user channels, IEEE Trans. Inform. Theory, vol. IT-24, no. 5, pp , Sep A. A. El Gamal and M. H. M. Costa, The capacity region of a class of determinsitic interference channels, IEEE Trans. Inform. Theory, vol. IT-28, no. 2, pp , Mar G. Kramer, Review of rate regions for interference channels, in Int. Zurich Seminar on Communications (IZS), Zurich, Feb , pp H.-F. Chong, M. Motani, H. K. Garg, and H. E. Gamal, On the Han-Kobayashi region for the interference channel, IEEE Trans. Inform. Theory, vol. 54, no. 7, pp , Jul G. Kramer, Outer bounds on the capacity of Gaussian interference channels, IEEE Trans. Inform. Theory, vol. 50, no. 3, pp , Mar R. H. Etkin, D. N. C. Tse, and H. Wang, Gaussian interference channel capacity to within one bit, IEEE Trans. Inform. Theory, vol. 54, no. 12, pp , Dec G. Bresler, A. Parekh, and D. N. C. Tse, The approximate capacity of the many-to-one and one-to-many Gaussian interference channels, IEEE Trans. Inform. Theory, vol. 56, no. 9, pp , Sep Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

62 Some References II Gallager68 R. G. Gallager, Information Theory and Reliable Communication. New York: John Wiley and Sons, Inc, Vontobel08 Huang09a Huang09b P. O. Vontobel, A. Kavčić, D. M. Arnold, and H.-A. Loeliger, A generalization of the Blahut-Arimoto algorithm to finite-state channels, IEEE Trans. Inform. Theory, vol. 54, no. 5, pp , May X. Huang, A. Kavcic, X. Ma and D. Mandic, Upper bounds on the capacities of non-controllable finite-state channels using dynamic programming methods, ISIT2009, Seoul, Korea, June 28-July 3, 2009, pp X. Huang, A. Kavcic, X. Ma, Upper bounds on the capacities of non-controllable finite-state channels with/without feedback, Submitted to IEEE Trans. Inform. Theory, Mar. 2009, and Revised Aug Arnold01 D. M. Arnold and H.-A. Loeliger, On the information rate of binary-input channels with memory, in Proc IEEE Int. Conf. Commun., vol. 9, Helsinki, Finland, Jun. 2001, pp Pfister01 Arnold06 H. D. Pfister, J. B. Soriaga, and P. H. Siegel, On the achievable information rates of finite state ISI channels, in Proc. IEEE GLOBECOM01, vol. 5, San Antonio, Texas, Nov , 2001, pp D. M. Arnold, H.-A. Loeliger, P. O. Vontobel, A. Kavčić, and W. Zeng, Simulation-based computation of information rates for channels with memory, IEEE Trans. Inform. Theory, vol. 52, no. 8, pp , Aug Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60

63 Acknowledgements Acknowledgements This work is supported by NSF of China and Guangdong Province (No. U ). The authors are grateful to the group of pre-973 project... Thank You for Your Attention! Xiao Ma (Sun Yat-sen Univ.) Accessible Capacity 11/14/ / 60