Achievable Rates for Shaped Bit-Metric Decoding

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1 Achievable Rates for Shaped Bit-Metric Decoding Georg Böcherer, Meber, IEEE Abstract arxiv: v6 [cs.it] 8 May 06 A new achievable rate for bit-etric decoding (BMD) is derived using rando coding arguents. The rate expression can be evaluated for any input distribution, and in particular the bit-levels of binary input labels can be stochastically dependent. Probabilistic shaping with dependent bit-levels (shaped BMD), shaping of independent bit-levels (bit-shaped BMD) and uniforly distributed independent bit-levels (unifor BMD) are evaluated on the additive white Gaussian noise (AWGN) channel with Gray labeled bipolar aplitude shift keying (ASK). For 3-ASK at a rate of 3.8 bits/channel use, the gap to 3-ASK capacity is db for shaped BMD, 0.46 db for bit-shaped BMD, and.4 db for unifor BMD. These nuerical results illustrate that dependence between the bit-levels is beneficial on the AWGN channel. The relation to the LM rate and the generalized utual inforation (GMI) is discussed. Index Ters probabilistic shaping, bit-etric decoding, bit-interleaved coded odulation (BICM), achievable rate, aplitude shift keying (ASK), binary labeling I. INTRODUCTION Bit-interleaved coded odulation (BICM) cobines higher order odulation with binary error correcting codes [], [3]. This akes BICM attractive for practical application and BICM is widely used in standards, e.g., in DVB-T/S/C. At a BICM receiver, bit-etric decoding (BMD) is used [4, Sec. II]. For BMD, the channel input is labeled by bit strings of length. The bit-levels are treated independently at the decoder. Let B = (B, B,..., B ) denote a vector of binary rando variables B i, i =,,...,, representing the bit-levels with joint distribution P B on {0, }. Consider the channel p Y B with output Y and input distribution P B. Define R BMD (P B ) := [ H(B) + H(B i Y )] () where [ ] + := ax{0, }, and where H( ) denotes entropy. For independent bit-levels, we have P B = P B i and R BMD ( P B i ) = I(B i ; Y ) () where I( ; ) denotes utual inforation. Martinez et al. showed in [4] that () with independent and uniforly distributed bit-levels is achievable with BMD. We call this ethod unifor BMD. Guillén i Fàbregas and Martinez [5] generalized the result of [4] to non-uniforly distributed independent bit-levels. We call this ethod bit-shaped BMD. An iportant tool to assess the perforance of decoding etrics is the generalized utual inforation (GMI) [6, Sec..4]. An interpretation of unifor BMD and bit-shaped BMD as a GMI are given in [4] and [5], respectively. In [7, Sec. 4..4], the GMI is evaluated for a bitetric. It is observed that the GMI increases when the bits are dependent. We call this approach shaped GMI. Another ethod to evaluate decoding etrics is the LM rate, see [8] and references therein. The LM rate is applied to BICM in [9]. A part of this work has been presented at ISIT 04 in Honolulu []. The author is with the Institute for Counications Engineering, Technical University of Munich, Munich D-80333, Gerany (e-ail: georg.boecherer@tu.de).

2 bits/channel use log ( + SNR) ASK capacity C shaped BMD shaped GMI [7, Sec. 4..4] bit-shaped BMD [5] unifor ASK unifor BMD [4] SNR in db Fig.. Achievable rates for bipolar ASK with 3 equidistant signal points, see Sec. IV. At 3.8 bits/channel use, bit-shaped BMD is 0.46 db less energy efficient than shaped BMD. Our ain contribution is to show that R BMD in () with arbitrarily distributed bit-levels is achievable with BMD. In particular, the bit-levels can be dependent, in which case R BMD is not equal to (). We call our ethod shaped BMD. For exaple, consider the additive white Gaussian noise (AWGN) channel with bipolar aplitude shift keying (ASK), see Sec. IV for details. We display inforation rate results for 3- ASK in Fig.. At a rate of 3.8 bits/channel use, the gap to ASK capacity of shaped BMD is db, the gap for shaped GMI is 0. db, the gap for bit-shaped BMD is 0.46 db, and the gap is.4 db for unifor BMD. Dependence between the bit-levels is thus beneficial on the AWGN channel. The rate expression () is used in [0] to construct surrogate channels, which are used to design low-density parity-check codes for shaped BMD. In [], R BMD is used to estiate achievable rates in fiber-optic transission experients. This paper is organized as follows. We state our ain result in Sec. II. We relate R BMD to the LM rate and the GMI in Sec. III and we discuss its application to the AWGN channel in Sec. IV. Sec. V concludes the paper and the appendix provides technical results. II. MAIN RESULT Let p Y B be a eoryless channel with input B = (B, B,..., B ). Let C be a codebook with code words b n = (b, b,..., b n ) with b i {0, }. We denote the ith bit-level of a code word by b n i = (b i, b i,..., b in ). A bit-etric decoder uses a decision rule of the for argax b n C q i (y n, b n i ) (3)

3 3 where y n is the decoder s observation of the channel output and where for each bit-level i, the value of the bit-etric q i (y n, b n i ) depends on P B p Y B only via the arginal P Bi (b)p Y Bi (y b) = p Y B (y a)p B (a). (4) a {0,} : a i =b Theore. Let P B be a distribution on {0, } and let p Y B be a eoryless channel. The rate R BMD (P B ) can be achieved by a bit-etric decoder. Proof: The theore follows by Lea, see Sec. III. For channels with finite output alphabets, we give a proof by typicality in Appendix B. A. Dependent Bit-Levels Can Be Better We develop a siple contrived exaple to show that dependent bit-levels can be better than independent bit-levels. Consider the identity channel with input label B B and transition probabilities Consider the input cost function f satisfying P Y B B (ab ab) =, ab {00, 0, 0, }. f(00) = f() =, f(0) = f(0) = 0 and suppose we ipose the average cost constraint E[f(B B )] <, where E[ ] denotes expectation. For independent bit-levels B and B, this constraint can be achieved only by P B (0) = P B () = or P B () = P B (0) =. In both cases, we have H(B B ) H(B i Y ) = 0 = R BMD (P B P B ). (5) We next choose P B B (0) = P B B (0) = /, which akes the bit-levels dependent. The average input cost is zero and we have H(B B ) H(B i Y ) = = R BMD (P B B ). (6) We conclude that for the considered input-constraint channel, no positive rate is achievable with independent bit-levels and any rate below one is achievable with dependent bit-levels. B. H(B) H(B i Y ) Can Be Negative Consider the erase-all channel with output alphabet {e} and transition probabilities P Y B B (e ab) =, ab {00, 0, 0, }. For the input distribution P B B (0) = P B B (0) = /, we copute Thus, R BMD (P B B ) = [ ] + = 0. H(B B ) H(B i Y ) = =. (7)

4 4 A. Rando Coding III. R BMD, LM RATE, AND GMI Consider a eoryless channel p Y B with input alphabet {0, } and real-valued output Y. Let C {0, } n be a codebook with block length n over the alphabet {0, } of size C = nr. The decision rule of a axiu likelihood (ML) decoder is ˆb n = argax b n C n p Y B (y i b i ) (8) where y n is the decoder s observation of the channel output. If ore than one code word axiizes the likelihood function, the argax operator selects one axiizing code word randoly. For the enseble of codes whose code words are drawn iid according to PB n, the following holds (see, e.g., [, Chap. 7]): the average probability of erroneous ML-decoding (the average is both over the codes in the enseble and the code words) approaches zero for n approaching infinity if R < I(B; Y ) (9) which shows that I(B; Y ) is an achievable rate for ML-decoding. A isatched decoder [8, Sec. II] uses a etric q : {0, } R R instead of p Y B and the isatched decoding rule is n ˆb n = argax q(y i, b i ). (0) b n C B. LM Rate and R BMD Let s 0 be a non-negative scalar and let r : {0, } R be a real-valued function defined on the input alphabet {0, }. Define [ ] q(y, B) s r(b) R(P B, q, s, r) = E log () b supp P B P B (b)q(y, b) s r(b) where supp P B := {b {0, } : P B (b) > 0} is the support of P B. By [8, Theore ], the LM rate R LM (P X, q) = ax s 0,r R(P B, q, s, r) () is achievable by the isatched decoder (0). This iplies that for each s 0 and function r, the rate [R(P B, q, s, r)] + is also an achievable rate for the isatched decoder. The next lea states that R BMD is an instance of [R(P B, q, s, r)] + for a particular choice of q, s, r. Lea. Consider the eoryless channel p Y B with input distribution P B and define q BMD (y, b) = p Y Bi (y b i ) (3) ) We have s BMD = (4) r BMD (b) = P B i (b i ). (5) P B (b) R BMD (P B ) [R(P B, q BMD, s BMD, r BMD )] + (6) with equality if and only if P B is strictly positive, i.e., if P B (b) > 0 for all b {0, }. ) R BMD is an achievable rate for the isatched decoder (0) for q = q BMD.

5 5 3) R BMD is less or equal to the LM rate, i.e., R BMD (P B ) R LM (P B, q BMD ). (7) Proof: We prove stateent ) in Appendix C. Stateents ) and 3) now follow by ) and (). Reark. The achievability of R BMD for channels with finite output alphabets is show in Appendix B by the following code construction, which is different fro the construction in Sec. III-A: The codebook C is generated iid according to a unifor distribution on the alphabet {0, }. The transitter only transits code words that are P B typical (for the foral definition of typicality, see Appendix A), which results in the signaling set S = {b n C : b n is P B typical}. (8) The receiver uses a bit-wise typicality decoder, i.e., it outputs b n, if it is the only code word whose bit levels b n i are jointly P Bi Y typical with the observed channel output y n, for i =,,...,. Thus, the decoder evaluates its decoding etric for the decoding set D = {b n C : b n i is P Bi typical, for i =,,..., }. (9) In the case of dependent bit-levels, the signaling set S and the decoding set D are not equal. This codebook isatch is also present in the practical ipleentation of shaped BMD [3]. A related work on codebook isatch is [4]. C. Generalized Mutual Inforation By setting r( ) = and axiizing over s, we obtain the GMI R GMI (P B, q) = ax s 0 R(P B, q, s, ). (0) The achievability of R GMI by the isatched decoder (0) now follows by observing that R GMI (P B, q) R LM (P B, q) or by invoking [6, Sec..4]. In [7, Sec. 4..4], dependent bit-levels are used together with the etric q BMD, which yields the shaped GMI rate R sgmi (P B ) := R GMI (P B, q BMD ). () In the next section, we will see that for bipolar ASK on the AWGN channel, R BMD is larger than R sgmi. IV. -ASK MODULATION FOR THE AWGN CHANNEL The signal constellation of bipolar ASK is given by X ASK = {±, ±3,..., ±( )}. () The points x X ASK are labeled by a binary vector B = (B,..., B ). We use the Binary Reflected Gray Code (BRGC) [5]. The labeling influences the rate that is achievable by BMD, see, e.g., [3, Sec. VI.C]. To control the transit power, the channel input x B is scaled by a positive real nuber. The input-output relation of the AWGN channel is Y = x B + Z (3) where Z is zero ean Gaussian noise with variance one. The input is subject to an average power constraint P, i.e., and P B ust satisfy E[( x B ) ] P. The ASK capacity is C = ax I(B; Y ). (4),P B : E[( x B ) ] P The optial paraeters, PB can be calculated using the Blahut-Arioto algorith [6], [7] and they can be approxiated closely by axiizing over the faily of Maxwell-Boltzann distributions [8],

6 6 see also [3, Sec. III]. We evaluate R BMD (shaped BMD) and R sgmi (shaped GMI) in, PB. In Fig., we plot for 3 signal points ( = 5) the ASK capacity C and the inforation rate curves of shaped BMD and shaped GMI together with the corresponding rate curves that result fro unifor inputs. Since we noralized the noise power to one, the signal-to-noise ratio (SNR) in db is given by E[( x B ) ] SNR = 0 log 0. (5) The gap between the 3-ASK capacity C and the shaped BMD rate R BMD is negligibly sall over the considered SNR range. At 3.8 bits/channel use, the gap between C and R BMD is db and the gap of sgmi is 0. db. For coparison, we calculate the bit-shaped BMD rate. The optiization proble is axiize I(B i ; Y ) P B, (6) subject to P B = P Bi, E[( x B ) ] P. This is a non-convex optiization proble [9], [0] so we calculate a solution by exhaustive search over the bit distributions with a precision of ± The resulting rate curve is displayed in Fig.. We observe that bit-shaped BMD (independent bit-levels) is 0.46 db less energy efficient than shaped BMD (dependent bit-levels) at 3.8 bits/channel use. V. CONCLUSIONS The achievable rate in () allows dependence between the bit-levels while the achievable rate in () (see [4], [5]) requires independent bit-levels. We have shown that on the AWGN channel under bit-etric decoding, dependent bit-levels can achieve higher rates than independent bit-levels. Interesting research directions are to study codebook isatch (see Reark ) and to study error exponents for shaped BMD. ACKNOWLEDGMENT This work was supported by the Geran Federal Ministry of Education and Research in the fraework of an Alexander von Huboldt Professorship. APPENDIX A TYPICAL SEQUENCES We use letter-typical sequences as defined in [, Sec..3]. Consider a discrete eoryless source (DMS) P X with a finite alphabet X. For x n X n, let N(a x n ) be the nuber of ties that letter a X occurs in x n, i.e., N(a x n ) = {i: x i = a}. (7) We say x n is ɛ-letter-typical with respect to P X if for each letter a X, ( ɛ)p X (a) N(a xn ) ( + ɛ)p X (a), a X. (8) n Let Tɛ n (P X ) be the set of all sequences x n that fulfill (8). The sequences (8) are called typical in [, Sec. 3.3], [3, Sec..4] and robust typical in [4, Appendix]. We next list the properties of typical sequences that we need in this work and whenever possible, we refer for the proofs to the literature. Define µ X := in a supp P X P X (a). (9)

7 7 Lea (Typicality, [, Theore.], [4, Lea 9]). Suppose 0 < ɛ < µ X. We have ( δ ɛ (n, P X )) n( ɛ) H(X) T n ɛ (P X ) (30) where δ ɛ (P X, n) is such that δ ɛ (P X, n) n exponentially fast in n. The definitions and properties of typicality apply in particular when the rando variable X stands for a tuple of rando variables, e.g., X = (Y, Z). If x n = (y n, z n ) is typical, then y n and z n are called jointly typical. Lea 3 (Marginal Typicality, [4, Lea ], [, Sec..5]). Joint typicality iplies arginal typicality, i.e., Tɛ n (P Y Z ) Tɛ n (P Y ) Tɛ n (P Z ). Lea 4 (Misatched Typicality). Suppose ɛ > 0, X n is eitted by the DMS P X and supp P X supp P X. We have ( δ ɛ (P X, n)) n[d(p X P X ) ɛ log (µ Xµ X )] Pr[X n T n ɛ (P X)]. (3) Proof: For x n T n ɛ (P X), we have Now, we have Pr[X n T n ɛ (P X)] = (3) P n X(x n ) = = x n T n ɛ (P X) x n T n ɛ (P X) a supp P X a supp P X a supp P X P X (a) N(a xn ) n(+ɛ)p P X (a) X(a) n(+ɛ)p X(a) log P X (a). (3) P n X(x n ) (33) a supp P n(+ɛ)p X(a) log P X (a) X (30) ( δ ɛ (n, P X)) n( ɛ) H( X) a supp P n(+ɛ)p X(a) log P X (a) X = ( δ ɛ (n, P X)) n[d(p X P X ) ɛ H( X)+ɛ a supp P X P X(a) log P X (a)] (34) (35) (36) ( δ ɛ (n, P X)) n[d(p X P X )+ɛ log (µ Xµ X )]. (37) Define now the set Note that T n ɛ (P XY x n ) = if x n / T n ɛ (P X ). { } Tɛ n (P XY x n ) := y n : (x n, y n ) Tɛ n (P XY ). (38) Lea 5 (Conditional Typicality, [, Theore.], [4, Lea, 4]). Suppose 0 < ɛ < ɛ < µ XY. For any x n X n, we have T n ɛ (P XY x n ) n(+ɛ ) H(Y X). (39) Suppose x n T n ɛ (P X ) and that (X n, Y n ) is eitted by the DMS P XY. We have δ ɛ,ɛ (P XY, n) Pr[Y n T n ɛ (P XY x n ) X n = x n ] (40) where δ ɛ,ɛ (P XY, n) approaches zero exponentially fast in n.

8 8 APPENDIX B PROOF OF THEOREM We prove Theore by rando coding arguents. In the following, let 0 < ɛ < ɛ < µ BY, where by (9), µ BY = in P BY (by). b,y : P BY (by)>0 Code Construction: Choose n(r+ R) code words U n (w, v), w =,,..., nr, v =,,..., n R of length n by choosing the n n(r+ R) sybols independent and uniforly distributed according to the unifor distribution P U on {0, }. Let C be the resulting codebook. Encoding: Given essage w {,, 3,..., nr }, try to find a v such that U n (w, v) Tɛ n (P B ). If there is such a v, transit U n (w, v). If there is no such v, declare an error. Decoding: We define the bit-etric { q i (y n, b n, (b n i, y n ) Tɛ n i ) = (P Bi Y ) (4) 0, otherwise. The corresponding decoding etric is q(y n, b n ) = Define the set ˆB(y n ) := {b n C : q(y n, b n ) = }. The decoder output is { b n, if ˆB(y n ) = {b n } error, otherwise. q i (y n, b n i ). (4) Analysis: Suppose essage w should be transitted. The first error event is (43) E := { v : U n (w, v) T n ɛ (P B ) }. (44) Suppose now E did not occur and for soe v, U n (w, v) = b n with b n T n ɛ (P B ). The vector b n is transitted. The second error event can occur at the decoder, and it is given by E := { b n / ˆB(Y n ) U n (w, v) = b n}. (45) Suppose next that the second error event did not occur. This iplies in particular that Y n Tɛ n (P Y ). Suppose that Y n = y n for soe y n Tɛ n (P Y ). The third error event is now First error event: By (3), we have E 3 := { w, ṽ : w w and U n ( w, ṽ) ˆB(y n ) Y n = y n}. (46) Pr[U n T n ɛ (P B )] [ δ ɛ (P B, n)] n[d(p B P U ) ɛ log (µ B µ U )] where by (9), µ B = in P B(b) and µ U = in P U(u). Note that since P U is unifor on b: P B (b)>0 u: P U (u)>0 {0, }, we have µ U =. For large enough n, we have δ ɛ (P B, n) /. The probability to generate n R sequences U n (w, v), v =,,..., n R, that are not in Tɛ n (P B ) is thus bounded fro above by ( ) [ n R n[d(p B P U ) ɛ log (µ B µ U )] exp ] n[d(p B P U ) ɛ log (µ B µ U )] n R (48) where inequality in (48) follows by ( r) s exp( rs). This probability tends to zero if (47) R > D(P B P U ) + ɛ log µ B µ U. (49)

9 9 We conclude that if R > D(P B P U ) (50) then for sall enough positive ɛ and large enough n, we have U n (w, v) Tɛ n (P B ) for soe v {,,..., n R} with high probability. Second error event: By (40), the probability Pr[(b n, Y n ) Tɛ n (P BY ) U n (w, v) = b n ] = Pr[Y n Tɛ n (P BY b n ) U n (w, v) = b n ] approaches one for n. By Lea 3, joint typicality iplies arginal typicality, so also Pr[(b n i, Y n ) Tɛ n (P Bi Y ) U n (w, v) = b n ] approaches one for i =,,...,. This shows that Pr[E ] n 0. Third error event: By (39), we have The size of ˆB(y n ) is thus bounded as Furtherore, by our rando coding experient, we have We have T n ɛ (P Bi Y y n ) n H(B i Y )(+ɛ ). (5) ˆB(y n ) n H(B i Y )(+ɛ ). (5) Pr[U n ( w, ṽ) = b n ] = n = n H(U), b n {0, } n. (53) [ Pr[E 3 ] = Pr w w ṽ [ = Pr w w ṽ U n ( w, ṽ) ˆB(y n ) Y n = y n ] U n ( w, ṽ) ˆB(y ] n ) (54) (55) ( nr ) n R Pr[U n ˆB(y n )] (56) < n(r+ R) Pr[U n ˆB(y n )] (57) n(r+ R) = Pr[U n = b n ] (58) (53) n(r+ R) = b n ˆB(y n ) b n ˆB(y n ) n H(U) (59) (5) n(r+ R) n H(B i Y )(+ɛ ) n H(U) (60) where equality in (55) follows because U n (w, v) was transitted, so Y n and U n ( w, ṽ) are independent for w w. Inequality in (56) follows by the union bound. By (60), the probability Pr[E 3 ] approaches zero for n if R + R + H(B i Y )( + ɛ ) H(U) < 0. (6) By choosing R according to (49), condition (6) becoes R < R H(B i Y )( + ɛ ) + H(U) (6) < H(B i Y )( + ɛ ) + H(U) D(P B P U ) ɛ log µ B µ U (63) = H(B) H(B i Y )( + ɛ ) ɛ log µ B µ U (64)

10 0 where equality in (64) follows because U is uniforly distributed, thus D(P B P U ) = H(U) H(B). Suppose now H(B) H(B i Y ) > 0. Then, for any positive R < H(B) H(B i Y ), we can find sall enough positive ɛ < ɛ so that both condition (49) and (6) are fulfilled. By choosing n large enough, the probability of the three error events approaches zero. If H(B) H(B i Y ) 0, we let the transitter transit a duy sequence corresponding to a rate of zero, which can be achieved on any channel. This shows that R BMD as defined in () can be achieved by BMD. APPENDIX C PROOF OF LEMMA We have [ R(P B, q BMD, s BMD, r BMD ) = E log P ] [ ] B i (B i ) + E log P B (B) p Y Bi (Y B i ) }{{} =H(B) }{{} H(B i) = h(y B i) [ ( )] E log P B (b) P B i (b i ) p Y Bj (Y b j ) P B (b) b supp P B j= }{{} ( ) where h( ) denotes differential entropy in bits. For the ter ( ), we have [ ( )] ( ) = E log P Bi (b i )p Y Bi (Y b i ) E log = E log = E = [ log b supp P B b {0,} ( b {0,} p Y (Y ) (65) (66) P Bi (b i )p Y Bi (Y b i ) (67) ] P Bi (b)p Y Bi (Y b)) (68) (69) h(y ) (70) with equality in (67) if and only if P B is strictly positive. Using (70) in (65), we have [ R(P B, q BMD, s BMD, r BMD ) H(B) H(B i ) + h(y ) h(y Bi ) ] (7) = H(B) = H(B) = H(B) with equality in (7) if and only if P B is strictly positive. H(B i ) + H(B i ) + I(B i ; Y ) (7) [ H(Bi ) H(B i Y ) ] (73) H(B i Y ) (74)

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