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Probability Density Function of Reliability Metrics in BICM with Arbitrary Modulation: Closed-form through Algorithmic Approach Leszek Szczecinski, Senior Member, IEEE, Rolando Bettancourt, and Rodolfo Feick, Senior Member, IEEE
Abstract—In the popular bit-interleaved coded modulation (BICM) the output of the channel encoder and the input of the modulator are separated by a bit-level interleaver. From the decoder’s point of view, the modulator, the transmission channel, and the demodulator (calculating bits’ reliability metrics) become a memoryless BICM channel with binary inputs and real outputs. In unfaded channels, the BICM channel’s outputs (reliability metrics) are known to be Gaussian for binary- or quaternary phase shift keying but their probability density function (PDF) is not known for higher-order modulation. We fill this gap by presenting an algorithmic method to calculate closed-form expressions for the PDF of reliability metrics in BICM with arbitrary modulation and bits-to-symbol mapping when the so-called max-log approximation is applied. Such probabilistic description of BICM channel is useful to analyze, from an informationtheoretic point of view, any BICM constellation/mapping design. Index Terms—Logarithmic likelihood ratios, L-values, bitinterleaved coded modulation, probability density functions, softinput decoding, LLR, BICM, PDF.
I. I NTRODUCTION N this paper we propose a method to obtain the parametric description of probability density functions (PDF) of the reliability metrics for bit-interleaved coded modulation (BICM) in unfaded channels. BICM is a flexible modulation/coding scheme in which the selection of the modulation constellation is decoupled from the choice of the coding rate [1][2]. It was devised [1] as a competing alternative to trellis coded modulation (TCM) [3] in fading channels. However, its versatility is the reason for its popularity also in unfaded channels [2] particularly because it is accompanied by a small performance penalty when compared to less flexible coded modulation approaches such as TCM.
I
Paper approved by S. A. Jafar, the Editor for Wireless Communication Theory and CDMA of the IEEE Communications Society. Manuscript received March 27, 2006; revised August 28, 2006. This work was supported by NSERC, Canada (under research grant 249704-02), by Fundacion Andes, Chile, and by Conicyt, Chile (under project PBCT-ACT-11/2004). This work was presented at the IEEE Global Telecommunications Conference 2006, San Francisco, USA, Nov. 2006. L. Szczecinski is with the Institut National de la Recherche Scientifique, INRS-EMT, University of Quebec, Place Bonaventure 800, Gauchetiere W. Suite 6900 Montreal, H5A 1K6, Canada (e-mail:
[email protected]). R. Bettancourt was with the Universidad T´ecnica F´ederico Santa Mar´ıa, Valpara´ıso, Chile. He is now with VTR Globalcom, Reyes Lavalle 3340, Las Condes, Santiago, Chile (e-mail:
[email protected]). R. Feick is with the Department of Electronics Engineering, Universidad Tecnica Federico Santa Mar´ıa, Avenida Espana 1680, Valpara´ıso, Chile (email:
[email protected],
[email protected]). Digital Object Identifier 10.1109/TCOMM.2008.060169.
Lak
BICM channel
ENC
y
Π
η
yk
μ−1
+
μ x
r
Π−1
L
DEC
Lk
Fig. 1. Model of the BICM transmission where demapping uses a priori L-values : interleaver Π, bits-to-symbols mapper μ{·}, metrics (L-values) calculator μ−1 {·}, and deinterleaver Π−1 , may be modelled as a BICM channel with binary inputs y and real outputs L.
In BICM, the output of the channel encoder and the input to the modulator are separated by a bit-level interleaver, as shown in Fig. 1. At the receiver, the reliability metrics calculated for the coded bits according to the maximum likelihood principle, are deinterleaved and passed to the softinput decoder. These metrics are most often expressed as logarithmic likelihood ratios (LLR), or simply : L-values. To calculate them, the so-called max-log approximation is frequently considered [1][2][4][5] as a way to alleviate the computational complexity of the metrics’ calculation. From the decoder’s point of view, the modulator, the transmission channel, and the demodulator (calculating the L-values) may be seen as one entity equivalent to a binary input – real output BICM channel [6] which, when assuming perfect interleaving, may be considered as memoryless [1]. The probabilistic description of the L-values fully defines the BICM channel so knowing their probability density function (PDF) we may evaluate the capacity, cutoff rate, and/or any other information-theoretic parameter. Even though some of these parameters may be obtained through Gaussian quadrature or Monte-Carlo integration (e.g., the channel’s capacity [2]), the possibility of using an analytical form of the PDFs would enormously simplify calculations. Moreover, for accurate evaluation of the performance of the soft input decoder, knowledge of the PDFs is necessary [6][7]. To deal with this problem, a Gaussian approximation is sometimes used [6][8] because, in unfaded channels, L-values are indeed Gaussian for binary- and quaternary phase-shift keying (BPSK and QPSK) [9][10]. However, it is quite clear from the observations of the histograms (e.g., [6][8]) that the Gaussian model is not appropriate for higher-order modulations. Although the analytical description of the PDFs is fundamental to effectively characterize BICM channels, this prob-
c 2008 IEEE 0090-6778/08$25.00
SZCZECINSKI et al.: PROBABILITY DENSITY FUNCTION OF RELIABILITY METRICS IN BICM WITH ARBITRARY MODULATION
lem was considered difficult [6] and very little work has been reported on how to explicitly address this issue. Only recently, an exact expression for the PDF of L-values in M -ary quadrature amplitude modulation (QAM) with Gray mapping was obtained. It was shown to be a sum of truncated Gaussian functions [11]. To the best of our knowledge, there is no published work showing the form of the PDF for other modulations. We note that the developments of [11] took advantage of the particularity of the Gray mapping in M -QAM, and were based on one-dimensional considerations. The objective of this paper and its main contribution is to generalize the approach of [11] to arbitrary complex constellations. Therefore, our results extend those presented in [11], and we demonstrate that the PDF is a sum of truncated Gaussian functions weighted with shifted and scaled complementary error functions. With no constraints on the modulation or mapping, the parameters of the sought closed-form expressions of the PDFs are obtained through well defined algorithmic steps; this explains the title of this paper. For an even more general framework, we consider also the case when a priori L-values of the modulating bits are available. Such a scenario resembles the case of iterative (turbo) detection [12], [13] or it may be encountered when non-uniform signaling is to be considered [14]. This paper is organized as follows. A model of the system is shown in Section II. In Section III we present the method to obtain closed form expressions for the PDF of the Lvalues. We show in Section IV numerical examples contrasting the developed analytical expressions with the histograms of the LLRs obtained from simulated data. The conclusions are presented in Section V. II. S YSTEM M ODEL Consider a baseband model of the BICM transmission where coded bits yk (n) are interleaved, gathered in codewords of length m, y(n) = [y0 (n), . . . , ym−1 (n)] ∈ B and mapped into symbols x(n) = μ{y(n)} ∈ X , where n denotes the discrete time, M = 2m , and B = {b0 , . . . , bM−1 } is a set of all codewords labelling the symbols from the zero-mean and unitary energy constellation M−1 M−1 X2 = {a0 , . . . , aM−1 }, i.e., 1 a = 0 and k k=0 k=0 |ak | = 1. The probability of M generating zeros and ones are equal, i.e., Pr{yk (n) = 1} = 12 . Aiming at a fully general formulation, we do not restrain X or μ{·} to be of any particular form. The case of μ{·} implementing Gray mapping for X being M -QAM, treated in [11], is a particular instance of our model. In what follows, to simplify the notation, we omit showing the dependence on n because all the operations are memoryless. Transmission over additive white Gaussian noise (AWGN) channel results in r = x + η, where η ∈ C is a zero-mean, complex, white Gaussian noise with variance N0 (i.e., its real and imaginary parts are independent Gaussian variables each with variance N20 ), thus the average received signal-to-noise ratio (SNR) per symbol is given by γ = N10 . The a posteriori reliability metrics for the k-th transmitted bits are calculated as logarithmic likelihood ratios (L-values)
[4]
Pr{yk = 1|r} Pr{y = 0|r} k p r|yk = 1 + Lak = log p r|yk = 0
Lap k = log
737
(1)
= Lk + Lak where p(·|·) is the conditional PDF of the channel output and Lak is the a priori L-value for the k-th bit. The inclusion of this term generalizes our approach and has practically no impact on the complexity of the proposed algorithm. On the other hand, it might be exploited in the context of the so-called BICM with iterative decoding (BICM-ID) [12][15] or for transmission with non-uniform signaling [14]. If the a priori L-values are of no interest, their effect may be simply omitted by setting Lak to zero. The term Lk in (1), is known as the extrinsic L-value, and for the AWGN channel, it is calculated as [16][4]
2 a∈Xk,1 exp(−γ|r − a| )Pr{x = a} − Lak , Lk = log 2 )Pr{x = a} exp(−γ|r − a| a∈Xk,0 (2) where Xk,b is the set of symbols having the k-th labelling bit equal to b; its cardinality is given by |Xk,b | = M 2 . Deinterleaved extrinsic L-values Lk are denoted by L. In this paper we assume that the so-called max-log approximation is used by the receiver, i.e. Lk = min γ|r − a|2 − Λk {a} a∈Xk,0 (3) − min γ|r − a|2 − Λk {a} a∈Xk,1
where Λk {a} =
m−1 j=0 j=k
βj {a}Laj includes the effect of the
a priori L-values1 and βj {a} denotes the j-th bit of the codeword labelling the symbol a. We note that (3), considered already in the early works on BICM [1][2], is a frequently used approximation for (2) whose application is justified by a relatively small performance loss (as will be also discussed at the end of this paper) and a significant complexity reduction – an important issue in an industrial context, e.g., [5]. Thus, the metrics’ calculator implementing (3) is an element of the BICM channel, which justifies the analysis we conduct in this paper. As we will see, the use of approximation (3) is also key to obtain the exact expressions for the metrics’ PDF which, otherwise, are difficult to treat analytically. Deriving the PDF for (2) seems to be a much harder problem. III. PDF OF L- VALUES We must find an expression for the conditional PDF of the L-values of the k-th bit given by d d Fk (λ|b) = Pr{Lk < λ|yk = b}, (4) dλ dλ where Fk (λ|b) is the cumulative distribution function (CDF) for the bit at the k-th position (or simply – for the k-th bit) of pk (λ|yk = b) =
1 The bits y are assumed independent, so using a priori L-values we obtain k Pr{x = a} = m j=1 Pr{yj = βj {a}}.
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the label y. Knowing that Pr{x = a} = |X1k,b | = M 2 due to the assumed equiprobable generation of the bits yk , Fk (λ|b) is calculated as Fk (λ|b) = Pr{Lk < λ|yk = b} 2 1 Pr{Lk < λ|x = a} = Pk (λ|a), = |Xk,b | M a∈Xk,b
a∈Xk,b
(5)
where Pk (λ|a) also has the meaning of a CDF, but conditioned on sending the symbol a. Since we do not assume any particular form of X or μ{·}, all the required terms in (5) must be calculated explicitly. Based on the assumption of perfect interleaving, the PDF averaged over the bits’ positions yields p(λ|y = b) =
1 m
m−1
pk (λ|yk = b).
2 1.5
Z1,2
1 0.5 0 −0.5
1010 1010
0000
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a1,0,2
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1000
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0010 0010
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0100
1100 1100
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1001 1001
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−1.5 −2 −2
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−1
−0.5
0
0.5
a)
1
1.5
2
(6)
k=0
2
To arrive at the desired solution we will use the following general steps • First, we will divide the observation space into regions, within which the expression (3) is a linear form of the channel output r. • Next, we will calculate the CDF of the L-values (5), identifying explicitly the terms which are dependent on λ. • Finally, we will differentiate the CDF with respect to λ, to obtain the desired expression for the PDF.
1.5 1010 1
0010
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0000
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Z1,4
1000
0110 0110
1110
a1,0,1
0100
1100 1100
a1,0,2 a1,0,3 1111 1101 1101 Z1,3 0101 a1,1,3 a1,1,4
Z1,1 0111 0111 a1,1,1 0011
0001 1001
1011 1011
−1
a1,1,2 −1.5
Z1,2
A. Linearization and related problems
−2 −2
We start with the following representation of (3)
Lk (r) = γ|r − ak,0 (r)|2 − Λk {ak,0 (r)} − γ|r − ak,1 (r)|2 − Λk {ak,1 (r)} ∗ =2γ [r · ak,1 (r) − ak,0 (r) ] + |ak,0 (r)|2 − |ak,1 (r)|2 + Λk {ak,1 (r)} − Λk {ak,0 (r)}
(7)
where [·] denotes the real part and the terms ak,b (r), b ∈ {0, 1} are non-linear functions of r given by ak,b (r) = arg min {γ|r − a|2 − Λk {a}} b = 0, 1. a∈Xk,b
(8)
This yields the symbols from Xk,b “closest” to the observation r in the sense of a metric that combines the squared Euclidean distance and a priori L-values. The non-linearity in (8) is difficult to deal with directly, so for each bit k, we divide the observation space into Tk non-empty regions Zk,t = {r : ak,0 (r) = ak,0,t ∧ ak,1 (r) = ak,1,t }
t = 1, . . . , Tk (9)
within which ak,b (r) are constant and equal to symbols ak,b,t ∈ Xk,b , b ∈ {0, 1}, cf. Fig. 2; the operator ∧ here denotes a logical AND. The division defined in (9) is crucial for our development because for r ∈ Zk,t the symbols ak,b (r) do not depend on r, so the L-value Lk (r) then becomes a locally (i.e., within Zk,t ) linear form of r, cf. (7).
−1.5
−1
−0.5
0
b)
0.5
1
1.5
2
Fig. 2. 16-APSK constellation (symbols shown as markers) and labelling used in numerical examples. The regions Zk,t , t = 1, 2, 3, 4 obtained for a) k = 0, and b) k = 3 are shown shaded. Following the notational convention of Fig. 3, the symbols ak,b,t are indicated by blank markers: squares for b = 0 and circles for b = 1.
To determine the form of Zk,t and to find the corresponding symbols ak,b,t we rewrite (9) using (8) Zk,t ={r : γ|r − ak,0,t |2 − Λk {ak,0,t } ≤ γ|r − a|2 − Λk {a} ∧ γ|r − ak,1,t |2 − Λk {ak,1,t } ≤ γ|r − a |2 − Λk {a }, a ∈ Xk,0 , a ∈ Xk,1 , a = ak,0,t , a = ak,1,t }. (10) The quadratic terms of r cancel in (10) so, for given ak,b,t , b = 0, 1, there are M − 2 linear inequalities defining the region Zk,t , but only Jk,t ≤ M − 2 are non-redundant, i.e., we may write Zk,t = {r : Lk,t,j (r) ≤ 0 c∗k,t,j ]
j = 1, . . . , Jk,t },
(11)
where Lk,t,j (r) ≡ [r · + dk,t,j are linear forms of r defining non-redundant inequalities obtained from (10). Henceforth, we will often use the symbol L (i.e., without the argument r ) to denote a set of parameters {c, d} defining the linear form L(r) = [r · c∗ ] + d, and we use −L ≡ {−c, −d}. A numerically efficient procedure adopted from the area of computational geometry, described in [17] may be used to find
SZCZECINSKI et al.: PROBABILITY DENSITY FUNCTION OF RELIABILITY METRICS IN BICM WITH ARBITRARY MODULATION
Lk,t,j and the vertices vk,t,f of the polygon directly from (10). Note, however, that although the problem (10) resembles the one defining the so-called decision regions in [17], the forms of both entities are entirely different, and cannot be deduced one from another. The region Zk,t may be i) a (convex) polygon with Vk,t = Jk,t vertices, ii) a region that extends to infinity (i.e., an “infinite” polygon) defined by Vk,t = Jk,t − 1 vertices or, iii) an empty set [if inequalities in (10) are contradictory, i.e. Jk,t = 0]. The last condition allows us also to reject those candidates for pairs of symbols ak,0,t and ak,1,t , which produce empty regions Zk,t . For completeness, we note that in some cases, the two dimensional constellation may be seen as a product of two, independent one-dimensional constellations. Such a particular case, which occurs, e.g., for 2m -QAM with Gray mapping, requires different (and simpler) considerations and results in a different form of PDF as shown in [11]. B. Calculating the CDF Because Zk,t are disjoint and their union covers the whole Tk observation space, i.e., t=1 Zk,t = C (C is the space of complex numbers), we can write the terms appearing in (5) as Pk (λ|a) = =
Tk t=1 Tk
Pr{Lk (r) < λ ∧ r ∈ Zk,t | x = a}
;;; ;;; ;;;
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Lk,t (r; λ) = 0
Lk,t,5 (r) = 0
Lk,t,4 (r) = 0 ak,1,t
Lk,t,1 (r) = 0
Lk,t,2 (r) = 0
Lk,t,3 (r) = 0
ak,0,t
a)
;;;; ;;;; ;;;; Lk,t (r; λ) = 0
Lk,t,3 (r) = 0 ak,1,t
Lk,t,1 (r) = 0
Lk,t,2 (r) = 0
ak,0,t
b)
Pr{Lk,t (r; λ) < 0 ∧ Lk,t,1 (r) ≤ 0 ∧
t=1
. . . ∧ Lk,t,Jk,t (r) ≤ 0|x = a}
(12)
where, using (7) we define Lk,t (r; λ) = [r · c∗k,t ] + dk,t (λ) = Lk,t (r; 0) −
λ 2γ
(13)
with ck,t = ak,1,t − ak,0,t 1 dk,t (λ) = [|ak,0,t |2 − |ak,1,t |2 ] 2 1 [Λk {ak,1,t } − Λk {ak,0,t } − λ]. + 2γ
(14)
so that Lk,t (r; λ) +
1 λ = Lk (r). 2γ 2γ
(15)
Again, using the introduced notation we may write Lk,t (λ) = {ck,t , dk,t (λ)} to emphasize that the linear form (13) depends on λ. From (13) and (14) we may also deduce that the line Lk,t (r; λ) = 0 is perpendicular to the line passing through ak,0,t and ak,1,t . The terms in (12) determine the probabilities of finding r in the region defined by Lk,t,j (r) ≤ 0, j = 1, . . . , Jk,t and Lk,t (r; λ) ≤ 0, conditioned on the transmission of x = a. We thus now consider two effectively different cases shown schematically in Fig. 3; the vectors normal to the lines defining the regions are shown pointing to the half-plane where the inequality is not satisfied [i.e., where Lk,t,j (r) > 0]. In the first case, cf. Fig. 3a, depending on the value of λ, the line Lk,t (r; λ) = 0 either intersects with the sides of two
Fig. 3.
Examples of the regions Zk,t : a) polygon, b) “infinite” polygon.
regions or has no intersection at all. These conditions may be encountered when the region Zk,t is a polygon or an “infinite” polygon. In the second case, independently of the value of λ, the line Lk,t (r; λ) = 0 intersects with only one of the region’s sides. This may happen only when the region is an “infinite” polygon, cf. Fig. 3b. Calculation of (12) requires integration of the Gaussian PDF over the region defined by linear constraints. This problem was already treated in [17], where it was demonstrated that each of such regions (shown hashed in Fig. 3) may be represented as a union and subtraction of the so-called wedges. By definition, the wedge W(Lx , Ly ) is a region where two arbitrary linear inequalities Lx (r) ≤ 0 and Ly (r) ≤ 0 are satisfied. The probability that r belongs to a wedge, conditioned on sending x = a, is given by [17][18] Pr{Lx (r) < 0 ∧ Ly (r) < 0|x = a} ≡ I Lx , Ly | a [cx c∗y ] Lx (a) Ly (a) 2γ, 2γ; =Q (16) |cx | |cy | |cx ||cy | where the two-dimensional Q-function is defined as [19] ∞ ∞ v2 −2vuρ+u2 1 − 2(1−ρ2 ) e dvdu. Q(t, s; ρ) = 2π 1 − ρ2 t s (17) In what follows we will also use the one-dimensional Q∞ 2 function Q(t) = √12π t exp (− τ2 )dτ ; the difference in
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notation with (17) is clear through the number of arguments. Since the hashed regions in Fig. 3 are union/subtraction of wedges, the integral of the Gaussian distribution over these regions can be expressed in terms of the sum/subtraction of the integrals over wedges. Thus, the terms in (12) may be evaluated as Pr{Lk,t (r; λ) < 0 ∧ r ∈ Zk,t | x = a} = ⎧ ⎪ ⎨I Lk,t (λ), Lk,t,jα |a Jk,t cf. Fig. 3a −I Lk,t (λ), −Lk,t,jω |a + ϕ {Lk,t,j }j=1 ⎪ Jk,t ⎩ I Lk,t (λ), Lk,t,jα |a + ϕ {Lk,t,j }j=1 cf. Fig. 3b (18) where jα , jω are indices of the polygon’s sides intersecting with the line Lk,t (r; λ) = 0 (in Fig. 3a jα = 3 and jω = 5; in Fig. 3b jα = 2). Integrals over the wedges indicated by the darkened arcs in Fig. 3 correspond to I Lk,t (λ), ·|a , while ϕ(·) contains integrals over the wedges indicated by blank arcs.
by jα and jω . Changing value of λ, the line Lk,t (r; λ) = 0 moves along the line linking the points ak,0,t and ak,1,t (shown dashed in Fig. 3). In particular, by increasing λ we move Lk,t (r; λ) = 0 towards ak,1,t . It is obvious that the indices jα and jω of the “active” sides do not change with λ as long as the value of λ is such that the line Lk,t (r; λ) = 0 does not pass through a vertex of the polygon. Therefore, the coefficients of the PDF are constant for λ belonging to the interval limited by L-values λ calculated at the vertices of the polygon: λk,t,j = Lk (vk,t,j ), cf. (7). The limits of the intervals are schematically shown in Fig. 3 as orthogonal projections of the vertices on the line passing through ak,0,t and ak,1,t , (L-values calculated for such a projection and for the vertex itself are equal). Sorting λk,t,j , we obtain the limits of the intervals over which the parameters of the function defined in (20) remain constant. Since there are Vk,t vertices, there will be Vk,t + 1 of such disjoint intervals. Differentiating (12) and applying (18) and (20) we obtain the PDF conditioned on the sent symbol T
k d Pk (λ|a) ≡ Pk (λ|a) = Ψk,t λ|a dλ t=1
C. Calculating the PDF To obtain (6), we need to differentiate (12) with respect to λ. This implies differentiation of (18) as well, and is the reason why the expression of ϕ(·) in (18) is not shown here2 : it is independent of λ, and thus will be eliminated after the differentiation. First, differentiation of (17) yields d Q(t(λ), s; ρ) dλ 2 −1 d t (λ) s − t(λ)ρ = √ exp − t(λ) (19) Q 2 2π 1 − ρ2 dλ so I Lk,t (λ), Lk,t,j | a is also easily differentiated with respect to λ d I Lk,t (λ), Lk,t,j |a ≡ f (λ; Ak,t,j (a), Bk,t,j , Ωk,t (a), Σk,t ) dλ 2 Ωk,t (a) − λ 1 exp − = 2 · Σk,t · γ 2πΣk,t · γ √ · Q γ(Ak,t,j (a) + λBk,t,j ) (20) where
Σk,t = 2|ck,t | [ck,t c∗k,t,j ] . ρk,t,j = |ck,t ||ck,t,j |
Vk,t +1
f (λ; Ak,t,jα,k,t,v (a), Bk,t,jα,k,t,v , Ωk,t (a), Σk,t )
v=1
− f (λ; −Ak,t,jω,k,t,v (a), −Bk,t,jω,k,t,v , Ωk,t (a), Σk,t ) · wk,t,v (λ) (27)
and the windowing function 1 if λ ∈ (λk,t,v−1 , λk,t,v ) wk,t,v (λ) = 0 otherwise
(28)
(25)
IV. N UMERICAL EXAMPLES
Lk,t,j (a) 2 Lk,t (a; 0) − ρ (21) k,t,j 1 − ρ2k,t |ck,t,j | |ck,t | ρk,t,j (22) = |ck,t |γ 2(1 − ρ2k,t,j ) 2
Ψk,t λ|a =
(24)
Ωk,t (a) = 2γLk,t (a; 0)
where
determines the interval of λ over which the function f (·) contributes to Pk (λ|a). For convenience, we use here λk,t,0 = −∞ and λk,t,Vk,t +1 = ∞. The indices jα,k,t,v and jω,k,t,v correspond to the values of jα and jω obtained in the v-th interval of λ in the region Zk,t . Finally, grouping will define the complete form the results 2 P (λ|a), which further should be of pk (λ|b) = M a∈Xk,b k used to obtain p(λ|b) via (6). We note that if the extrinsic L-values Lk are calculated for Lak = 0 (i.e., without a priori information), the form of the regions Zk,t does not change with γ, cf. (10). Therefore, once the parameters of the function f (·) in (20) are known, the PDF can be calculated for any value of SNR γ without geometric considerations, i.e., practically without computational overhead. From a computational complexity point of view, this compares favorably with the histogram-based approach.
Ak,t,j (a) = Bk,t,j
(26)
(23)
We note that the coefficients of the PDF (A, B, Ω, Σ) will depend on the “active” polygon’s sides, i.e., those indicated 2 But it is straightforward to obtain as shown in [17]. For example, in Fig. 3a, ϕ({Lk,t,j }5j=1 ) = −I Lk,t,5 , −Lk,t,1 |a − I Lk,t,1 , −Lk,t,2 |a + I −Lk,t,2 , −Lk,t,3 |a
As an example we consider a 16-ary amplitude phase shift keying (APSK) constellation whose form and labelling are defined in [20, Sec. 5.4.3] and reproduced in Fig. 2. The ratio of the outer and inner constellation radii R2 /R1 was taken equal to the standard value 3.15 [20, Sec. 5.4.3]; this constellation was designed to work with a coding rate ρ = 23 .
SZCZECINSKI et al.: PROBABILITY DENSITY FUNCTION OF RELIABILITY METRICS IN BICM WITH ARBITRARY MODULATION
0.45
0.45
0.4
0.4
0.35
0.35 0.3
P0 (λ|1)
P0 (λ|0)
0.3
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0.25 0.2
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0.15 0.1
0.1
0.05
0.05
0 −25
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a)
−5
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Fig. 4. PDF Pk (λ|b) obtained from the proposed analytical formulas (lines) and estimated using histograms (markers) for a) k = 0, b = 0, b) k = 0, b = 1, c) k = 2, b = 0, and d) k = 2, b = 1. The solid lines denote results obtained when γ = 4dB and dashed ones when γ = 9dB.
The forms of regions Z0,t and Z3,t , shown in Fig. 2a and Fig. 2b, respectively, are obtained for Lak ≡ 0 (i.e., without a priori information). Regions Zk,t are notably irregular and could not have been predicted by simple inspection. Thus, their efficient and automated calculation, using techniques of [17], is helpful for the implementation of the proposed method. Further, using γ = 4dB and γ = 9dB, and applying the proposed algorithm, we calculate the conditional PDFs Pk (λ|b) for k = 0, 2 and b = 0, 1 and compare them in Fig. 4 to the histograms of the L-values obtained transmitting 5 · 105 randomly generated symbols. First of all, we note the perfect match between histograms (markers) and analytical formulas (lines), as well as the fact that the L-values are clearly not well characterized by a Gaussian PDF. We may also appreciate that the distributions are not symmetric for k = 0. Note that increased SNR causes not only a shift of the PDF away from zero (which reflects the higher reliability of the bits’ metrics) but also makes the form of the PDF change significantly. Finally, as an additional consistency check, and using the developed expressions for PDF, we calculated mutual information (MI) between the L-values L and the bits y (for 16APSK modulation and a wide range of SNR γ). Since MI has the meaning of the BICM channel capacity, it may be directly compared to the capacity obtained using the method of [2]. Note, that the latter corresponds to calculating the Lvalues by the means of the exact formula (2). On the other hand, the former takes into account the use of the max-log approximation (3) and then, to obtain the capacity curve, the effort (related to calculation of the parameters of the PDF) is not high and must be deployed only once. When the
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parameters are calculated, the capacity is rapidly evaluated via a one-dimensional numerical integration (to calculate the expectation in the MI formulation) for any value of SNR. We do not show here the graphical representation of the obtained results because both MI curves practically overlapped. At a coding rate ρ = 23 , for which the modulation 16APSK was designed, the SNR loss of the max-log approximation was less than 0.05dB. This fully justifies applications of the approximation to alleviate the computational complexity of the metrics calculation. V. C ONCLUSIONS In this paper we develop a method for the calculation of the parameters of the probability density function for the reliability metrics (L-values) in BICM. For the case of unfaded channels, closed-form expressions for these PDFs can be obtained assuming that the so-called max-log approximation is applied to calculate the metrics. Our method works for any modulation and bits-to-symbols mapping. A generalization which deals with a priori L-values is also considered. The advantage of the proposed method in the analysis and design of the constellation and/or mapping for BICM transmission is twofold. First, the PDFs of the L-values may be used to efficiently calculate (i.e., via one-dimensional integration) information-theoretic properties of the BICM channel. This offers an advantage when compared to two-dimensional or Monte-Carlo integrations used, e.g., to calculate the capacity [2]. Second, the PDF form, necessary to estimate the coded performance is available so, the proposed method may lead to simplified and/or improved accuracy coded performance analysis. R EFERENCES [1] E. Zehavi, “8-PSK trellis codes for a Rayleigh channel,” IEEE Trans. Commun., vol. 40, no. 3, pp. 873–884, May 1992. [2] G. Caire, G. Taricco, and E. Biglieri, “Bit-interleaved coded modulation,” IEEE Trans. Inform. Theory, vol. 44, no. 3, pp. 927–946, May 1998. [3] G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE Trans. Inform. Theory, no. 1, pp. 55–67, Jan. 1982. [4] B. Hochwald and S. ten Brink, “Achieving near-capacity on a multipleantenna channel,” IEEE Trans. Commun., vol. 51, no. 3, pp. 389–399, Mar. 2003. [5] Ericsson, Motorola, and Nokia, “Link evaluation methods for high speed downlink packet access (HSDPA),” TSG-RAN Working Group 1 Meeting #15, TSGR1#15(00)1093, Tech. Rep., Aug. 2000. [6] A. Martinez, A. Guill´en i F`abregas, and G. Caire, “Error probability analysis of bit-interleaved coded modulation,” IEEE Trans. Inform. Theory, vol. 52, no. 1, pp. 262–271, Jan. 2006. [7] A. Abedi and A. K. Khandani, “An analytical method for approximate performance evaluation of binary linear block codes,” IEEE Trans. Commun., vol. 52, no. 2, pp. 228–235, Feb. 2004. [8] A. Guill´en i F`abregas, A. Martinez, and G. Caire, “Error probability of bit-interleaved coded modulation using the Gaussian approximation,” in Proc. Conference on Information Sciences and Systems, 2004. [9] S. ten Brink, “Convergence behaviour of iteratively decoded parallel concatenated codes,” IEEE Trans. Commun., vol. 49, no. 10, pp. 1727– 1737, Oct. 2001. [10] M. T¨uchler, R. Koetter, and A. C. Singer, “Minimum mean squared error equalisations using a priori information,” IEEE Trans. Signal Processing, vol. 50, no. 3, pp. 673–683, Mar. 2002. [11] M. Benjillali, L. Szczecinski, and S. Aissa, “Probability density functions of logarithmic likelihood ratios in rectangular QAM,” in Proc. Twenty-Third Biennial Symposium on Communications, Kingston, Canada, May 2006, pp. 283–286. [12] X. Li and J. A. Ritcey, “Bit-interleaved coded modulation with iterative decoding,” IEEE Commun. Lett., vol. 1, no. 6, pp. 169–171, Nov. 1997.
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[13] S. ten Brink, “Convergence of iterative decoding,” IEE Electron. Lett., vol. 35, no. 10, pp. 806–808, May 1999. [14] G. Takahara, F. Alajaji, N. C. Beaulieu, and H. Kuai, “Constellation mappings for two-dimensional signaling of nonuniform sources,” IEEE Trans. Commun., vol. 51, no. 3, pp. 400–408, Mar. 2003. [15] F. Schreckenbach, N. G¨ortz, J. Hagenauer, and G. Bauch, “Optimization of symbol mappings for bit-interleaved coded modulation with iterative decoding,” IEEE Commun. Lett., vol. 7, no. 12, pp. 593–595, Dec. 2003. [16] G. Caire, G. Taricco, and E. Biglieri, “Capacity of bit-interleaved channels,” IEE Electron. Lett., vol. 32, no. 12, pp. 1060–1061, June 1996. [17] L. Szczecinski, S. Aissa, C. Gonzalez, and M. Bacic, “Exact evaluation of bit- and symbol-error rates for arbitrary 2-D modulation and nonuniform signaling in AWGN channel,” IEEE Trans. Commun., vol. 54, no. 6, pp. 1049–1056, June 2006. [18] S. Park and D. Yoon, “An alternative expression for the symbol-error probability of MPSK in the presence of I/Q unbalance,” IEEE Trans. Commun., vol. 52, no. 12, pp. 2079–2081, Dec. 2004. [19] M. K. Simon, “A simpler form of the Craig representation for the twodimensional joint Gaussian Q-function,” IEEE Commun. Lett., vol. 6, no. 2, pp. 49–51, Feb. 2002. [20] ETSI EN 302 307, “Digital video broadcasting (DVB), second generation framing structure, channel coding and modulation systems for broadcasting, interactive services, news gathering and other broadband satellite applications,” Jan. 2004.
Leszek Szczecinski (M’98-SM’07), received M.Eng. degree from the Technical University of Warsaw in 1992, and Ph.D. from INRSTelecommunications, Montreal, Canada in 1997. From 1998 to 2000, he was Assistant Professor at the Department of Electrical Engineering, University of Chile. From 2001 to 2007, he had been Assistant Professor, and since 2007, Associate Professor at INRS-EMT, University of Quebec, Canada. His research interests are in the area of digital signal processing, communication theory, wireless communications and analysis and design of iterative (turbo) processing algorithms
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 5, MAY 2008
Rolando Bettancourt was born in 1982 in Valpara´ıso, Chile. He received the B.S. and M.Sc. in Electronic Engineering degrees in 2003 and 2006 respectively, from Universidad T´ecnica Federico Santa Mar´ıa, Valpara´ıso, Chile. From 2005 to 2006, he conducted research in collaboration with Institut National de la Recherche Scientifique, Monreal, QC, Canada, in the field of iterative (turbo) receivers. Since February, 2007, he has been working with VTR GlobalCom, Santiago, Chile. His current research interests are wireless and mobile communications, advanced coding techniques, iterative processing and data communications and networking.
Rodolfo Feick (S’71-M’76-SM’95) obtained the degree of Ingeniero Civil Electr´onico at Universidad T´ecnica Federico Santa Mar´ıa, Valpara´ıso, Chile in 1970 and the Ph.D. degree in Electrical Engineering at the University of Pittsburgh in 1975. He has been with the Department of Electronics Engineering at Universidad T´ecnica Federico Santa Mar´ıa since 1975, where he is the head of the telecommunications area. His current interests include RF channel modeling, digital communications, microwave system design and RF measurement.
