quaternary phase shift keying (BPSK and QPSK), but is not for higher order QAM, which is evident even from a bare-eye inspection of histograms of LLRs.
23rd Biennial Symposium on Communications
Probability Density Functions of Logarithmic Likelihood Ratios in Rectangular QAM Mustapha Benjillali, Leszek Szczeci´nski, Sonia A¨ıssa INRS-EMT, Montreal, Canada {jillali,leszek,aissa}@emt.inrs.ca
Abstract— Closed-form expressions for the probability density function (PDF) of logarithmic likelihood ratios (LLR) in rectangular quadrature amplitude modulations are derived. Taking advantage of assumed Gray mapping, the problem is solved in one dimension corresponding to the real or imaginary part of the symbol. The results show that sought PDFs are linear combinations of truncated Gaussian functions. This simple result stands in contrast with often assumed Gaussian distribution for the LLRs. Histograms of LLRs obtained via simulations confirm our analysis.
Index terms- Logarithmic Likelihood Ratio, Probability Density Function, QAM, PAM, BICM, Gray mapping. I. I NTRODUCTION Quadrature amplitude modulation (QAM) is widely used in communication systems. When applied in increasingly popular bit interleaved coded modulation (BICM) [1], the calculation of soft bits’ metrics under the form of logarithmic likelihood ratios (LLR) is required [1]. The probabilistic description of LLRs defines then the properties of resulting effective BICM channel. In particular, since LLRs are the input to the soft-input decoder, knowledge of their probability density function (PDF) is required to evaluate the performance of the latter, e.g. [2, 3]. Gaussian modeling of LLR is known to be exact for binary and quaternary phase shift keying (BPSK and QPSK), but is not for higher order QAM, which is evident even from a bare-eye inspection of histograms of LLRs. Despite the importance of such probabilistic description of the LLRs, to the best of our knowledge, no work has gone beyond the simplistic Gaussian assumption. The objective of this paper and its main contribution is, therefore, to present exact expressions for the PDF of LLRs in rectangular M -ary QAM for M = 4, 8, 16, 32, 64. Covering such wide family of modulation is possible thanks to assumed Gray mapping, which allow us to decompose the complex QAM into two pulse amplitude modulations (PAM) corresponding to the real and imaginary parts of the QAM. Assumption of Gray mapping is well justified thanks to its enormous popularity and theoretical justification as the one which maximizes the capacity of the BICM channel [1]. The paper is organized as follows. In Section II, we introduce the system model and notations. The expressions of the bit LLRs are presented in Section III and the PDF forms are derived in Section IV where a comparison between analytical and simulation results is also shown. Conclusions are drawn in Section V.
0-7803-9528-X/06/$20.00 ©2006 IEEE
II. S YSTEM MODEL We consider the following baseband system model. Let c(k) be the sequence of bits to be transmitted, for time k = −∞, . . . , +∞. The bits are grouped into codewords cQAM (n) = [cBQAM (n), . . . , c1 (n)] of length BQAM , transformed into symbols sQAM (n) = MQAM [cQAM (n)], and transmitted over additive white Gaussian noise (AWGN) channel. The received signal rQAM (n) = sQAM (n) + ηQAM (n) is corrupted by the complex noise ηQAM (n) with variance given by N0 = 1/γ. With Gray mapping, each QAM symbol may be treated as a superposition of independently modulated real and imaginary parts [4], each being a PAM symbol. Thus, in the following we analyze 2B -ary PAM, which may correspond to the real or imaginary part of the symbol. By combining PAM constellations, we can get different rectangular QAM constellations (e.g. 32-QAM = 8-PAM × 4-PAM ...). We keep the introduced notations but we take away the sub-indexing note “QAM” to refer to the signals and operations in the PAM context. Then s(n) belongs to S = {a0 , . . . , aM −1 } where am = (2m + 1 − M )∆ and ∆ denotes half the minimum distance between the constellation symbols. To alleviate the notation we abandon the time index n, which should not lead to any confusion as all considerations are static with respect to n due to the memoryless nature of the modulation and the channel. At the receiver, LLR for the k-th bit in codeword c (k = 1, . . . , B) is obtained as [5] |r−M[b]|2 exp − k b∈C1 N0 Pr{ck = 1|r} = ln λB,k (r) = ln |r−M[b]|2 Pr{ck = 0|r} − k exp b∈C0
N0
γ[ min |r − M[b]|2 − min |r − M[b]|2 ], b∈C0k
b∈C1k
(1)
where Cxk is the set of codewords b = [bB , . . . , b1 ] with the k-th bit equal to x ∈ {0, 1} and (1) is obtained using the known max-log approximation: ln ( i exp(−Xi )) ≈ −mini (Xi ) [6]. III. D ERIVATION OF LLR’ S EXPRESSIONS The LLR in (1) can now be simplified to λB,k (r) = γ[(r − sˆk0 )2 − (r − sˆk1 )2 ]
= 2γ · r[ˆ sk1 − sˆk0 ] + γ[(ˆ sk0 )2 − (ˆ sk1 )2 ],
(2)
sˆkx
is the symbol with the k-th labelling bit equal to x, where closest to the received signal r, i.e.
283
sˆkx = M[arg min |r − M[b]|2 ]. k b∈Cx
(3)
23rd Biennial Symposium on Communications
20
In the following, we provide the explicit expressions of the LLRs when B = 1, 2, 3, using for normalization purpose the 1 . coefficient β = 4γ∆ Case B = 1: Given that k ≡ 1 in this case, for every received r we have sˆ10 = −∆ and sˆ11 = +∆. Hence, using (2), the LLR expression is given by 1 λ1,1 (r) = r. β
λ2,1 (r) =
λ2,2 (r) =
+ β1 r − 2 − β r − −1r β2 −βr +
λ2,k (r) γ∆2
5
0
−5
−10
−15
−20 −4
2∆ β 2∆ β
if r ≤ 0,
2∆ β
if r ≤ −2∆, if −2∆ ≤ r ≤ 2∆,
−3
0 −∆
0 +∆
1 +3∆
LSB, k = 1
1 −3∆
1 −∆
0 +∆
0 +3∆
MSB, k = 2
3
4
0 0 1 1 −7∆ −5∆ −3∆ −∆
0 0 1 1 MiSB, k = 2 +∆ +3∆ +5∆ +7∆
1 1 1 1 −7∆ −5∆ −3∆ −∆
0 0 0 0 MSB, k = 3 +∆ +3∆ +5∆ +7∆
sˆ40
r ≤ −2∆
−3∆
−∆
−3∆
∆
−2∆ ≤ r ≤ 0
−3∆
−∆
−∆
∆
0 ≤ r ≤ 2∆
3∆
∆
−∆
∆
r ≥ 2∆
3∆
∆
−∆
3∆
λ3,3 (r) =
TABLE I S YMBOLS CLOSEST TO r IN THE 4-PAM CASE (B = 2), CF. (3).
Case B = 3: Gray mapping for 8-PAM with the corresponding borders of the decision regions on sˆk1 and sˆk0 is presented in Fig. 3. Table II describes the decision regions for the three bit positions and subintervals of r. LLRs of LSB (k = 1), middle (significant) bit (MiSB), i.e. k = 2 and MSB (k = 3) are provided respectively in (7), (8) and (9) and their plots shown in Fig. 4.
− β1 r − + 1 r + β 1 − βr + 1 r β
+
− 2 −βr − − β1 r − − 2 r − β λ3,2 (r) = 2 + βr − + β1 r − + 2 r − β − β4 r − 3 − β r − 2 − β r −
MSB sˆ41
2
0 0 1 1 LSB, k = 1 +∆ +3∆ +5∆ +7∆
λ3,1 (r) =
Fig. 1. Bit mapping and decision regions for LSB and MSB, case of four symbols in the real dimension (B = 2).
sˆ20
1
Fig. 3. Bit mapping and decision regions for LSB, MiSB and MSB, case of eight symbols in the real dimension (B = 3).
1 −3∆
sˆ21
0
r ∆
(6)
if r ≥ 2∆.
LSB
−1
0 0 1 1 −7∆ −5∆ −3∆ −∆
A normalized representation of these functions ((5) and (6)) is shown in Fig. 2.
r
−2
Fig. 2. LLR as a function of r for LSB and MSB in the case of four symbols in the real dimension (B = 2) with γ = −5dB.
(5)
if r ≥ 0.
2∆ β
10
(4)
Case B = 2: The mapping of least significant bit (LSB) and most significant bit (MSB) is presented in Fig. 1 and the correspondence between the observation r and sˆk1 and sˆk0 is given in Table I. Accordingly, the LLR expressions for the LSB and MSB are respectively given by − β1 r −
MSB LSB 15
− β1 r − 2 r β − β3 r − 4 r β
+ + +
6∆ β 2∆ β 2∆ β 6∆ β
if r ≤ −4∆, if −4∆ ≤ r ≤ 0, if 0 ≤ r ≤ 4∆, if r ≥ 4∆.
10∆ β 4∆ β 6∆ β 6∆ β 4∆ β 10∆ β
if r ≤ −6∆,
12∆ β 6∆ β 2∆ β
if r ≤ −6∆,
2∆ β 6∆ β 12∆ β
(7)
if −6∆ ≤ r ≤ −2∆, if −2∆ ≤ r ≤ 0, if 0 ≤ r ≤ 2∆,
(8)
if 2∆ ≤ r ≤ 6∆, if r ≥ 6∆. if −6∆ ≤ r ≤ −4∆, if −4∆ ≤ r ≤ −2∆, if −2∆ ≤ r ≤ 2∆,
(9)
if 2∆ ≤ r ≤ 4∆, if 4∆ ≤ r ≤ 6∆, if r ≥ 6∆.
IV. P ROBABILITY DENSITY FUNCTIONS Now, for each of the three cases presented in the previous section, our aim is to derive an expression of the conditional PDF of the LLR d PB,k (λ|s), (10) pB,k (λ|s) = dλ
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23rd Biennial Symposium on Communications
LSB
MiSB
MSB
Analytical Simulated
1
sˆ21
sˆ20
sˆ41
sˆ40
sˆ61
sˆ60
r ≤ −6∆
−7∆
−5∆
−7∆
−3∆
−7∆
+∆
−6∆ ≤ r ≤ −4∆
−7∆
−5∆
−5∆
−3∆
−5∆
+∆
−4∆ ≤ r ≤ −2∆
−∆
−3∆
−5∆
−3∆
−3∆
+∆
−2∆ ≤ r ≤ 0
−∆
−3∆
−5∆
−∆
−∆
+∆
0 ≤ r ≤ +2∆
+∆
+3∆
+5∆
+∆
−∆
+∆
+2∆ ≤ r ≤ +4∆
+∆
+3∆
+5∆
+3∆
−∆
+3∆
+4∆ ≤ r ≤ +6∆
+7∆
+5∆
+5∆
+3∆
−∆
+5∆
r ≥ +6∆
+7∆
+5∆
+7∆
+3∆
−∆
+7∆
0.9
p2,k (λ|s3 )
r
0.8 0.7
LSB
0.6 0.5 0.4 0.3 0.2
MSB
0.1 0 −80
−60
−40
−20
0
λ/(γ∆2 )
20
40
60
Fig. 5. The PDF of LSB and MSB conditioned on the transmission of s3 = +∆ in the case of B = 2; γ = −5dB.
TABLE II S YMBOLS CLOSEST TO r IN THE 8-PAM CASE (B = 3), CF. (3).
Analytical Simulated
p2,k (λ|s4 )
0.6
0.5
60
0.4
LSB MiSB MSB
0.3
40
λ3,k (r) γ∆2
LSB
0.2
20
MSB 0.1
0 0 −120
−20
−80
−60
−40
−20
0
λ/(γ∆2 )
20
40
60
Fig. 6. The PDF of LSB and MSB conditioned on the transmission of s4 = +3∆ in the case of B = 2; γ = −5dB.
−40
−60 −8
−100
−6
−4
−2
0
r ∆
2
4
6
8
Fig. 4. LLR as a function of r for LSB, MiSB and MSB in the case of eight symbols per real dimension (B = 3), with γ = −5dB.
as a derivative of the cumulative distribution function (CDF) for each variable PB,k (λ|s) = Pr{λB,k (r) ≤ λ|s} = Pr{r ∈ Iλ |s},
(11)
where Iλ = {r : λB,k (r) ≤ λ} is the interval (or union of intervals) in which λB,k (r) ≤ λ. The latter may be easily obtained from equations (4)-(9) (or Fig. 2 and Fig. 4). 1 )), we can According to our system model (i.e., r ∼ N (s, 2γ write
1 (12) PB,k (λ|s) = exp −γ|r − s|2 dr, π/γ r∈Iλ and change the variable r in the integration with its inverse expression λ−1 B,k (λ) in the sub-intervals of Iλ from (4)(9). Though straightforward, the mathematical derivations are lengthy. Hence, in what follows, we only present the final results for the three cases of B. Case B = 1: Applying (10) and considering (12) and (4), we obtain p1,1 (λ|s) =
1 √ exp −γ|βλ − s|2 , 4∆ γπ
(13)
which is exactly a Gaussian PDF in this case. Case B = 2: Similarly, it is easy to show in this case that the PDF for
LSB and MSB is respectively given by (14) and (15) which demonstrate that each distribution is a piecewise Gaussian. 1 2 4∆√γπ exp −γ|βλ + 2∆ − s| p2,1 (λ|s) = + exp −γ|βλ + 2∆ + s|2 if λ ≥ − 2∆ , β 2∆ 0 if λ ≤ − β . 1 √ exp −γ| β2 λ + ∆ + s|2 8∆ γπ 1 2 p2,2 (λ|s) = 4∆√ γπ exp −γ|βλ + s| √ β 1 2 exp −γ| λ − ∆ + s| 8∆ γπ 2
(14) if λ ≥
2∆ , β
if − 2∆ ≤λ≤ β if λ ≤ − 2∆ . β
(15) Note that the PDFs defined in (15) for the MSBs are symmetric, i.e. p2,2 (λ|s) = p2,2 (−λ| − s). This is not the case for the LSBs (14). Figures 5 and 6 show the comparison between the histograms of the LLRs, obtained from simulated data, and the analytical formulas when the PDF is conditioned on the transmission of s = ∆ and s = 3∆ respectively, and considering γ = −5dB. It is clear that the PDFs are not Gaussian and the match is perfect between the analytical and simulated results. Case B = 3: Using the same derivations as for the previous cases, we obtain the piecewise Gaussian PDFs for LSB, MiSB and MSB, shown respectively in (16), (17) and (18) where we use the notation 1 α = 4∆√ γπ .
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2∆ , β
23rd Biennial Symposium on Communications
−γ|βλ + 6∆ + u · s|2 α u∈{−1,1} exp
2 p3,1 (λ|s) = α u∈{−1,1} exp −γ|βλ + 6∆ + u · s| + exp −γ|βλ − 2∆ + u · s|2 0 β α 2 , if λ ≥ 2∆ 2 u∈{−1,1} exp −γ| 2 λ + 5∆ + u · s| β 2∆ 2 α if − exp −γ|βλ + 4∆ + u · s| ≤ λ ≤ 2∆ , u∈{−1,1} β β p3,2 (λ|s) = β α 6∆ 6∆ 2 2 if − β ≤ λ ≤ β , u∈{−1,1} exp −γ| 2 λ + 3∆ + u · s| . 0 if λ ≤ − 6∆ β
2∆ , β
if λ ≥ if
− 2∆ β
if λ ≤
≤λ≤
2∆ , β
(17)
Analytical Simulated
3
(16)
− 2∆ . β
Analytical Simulated
p3,2 (λ|s4 )
p3,1 (λ|s7 )
2 2.5
1.5
2
1.5 1 1 0.5 0.5
0 −40
0 −20
0
20
40
λ/(γ∆2 )
60
80
100
−60
Fig. 7. The PDF of LSB conditioned on the transmission of s7 = +5∆ in the case of B = 3; γ = −5dB.
−40
−20
0
20
40
60
λ/(γ∆2 )
80
100
120
140
Fig. 8. The PDF of MiSB conditioned on the transmission of s4 = −∆ in the case of B = 3; γ = −5dB. 0.9
Analytical Simulated
β α 2 exp −γ| λ − 3∆ + s| 4 4 α exp −γ| β λ − 2∆ + s|2 3 3 α exp −γ| β λ − ∆ + s|2 2 2 p3,3 (λ|s) = α exp −γ|βλ + s|2 β α 2 2 exp −γ| 2 λ + ∆ + s| α β 2 exp −γ| λ + 2∆ + s| 3 3 α exp −γ| β λ + 3∆ + s|2 4 4
p3,3 (λ|s1 )
0.8 0.7
if λ ≥
12∆ β
0.6
,
0.5
if
6∆ β
≤λ≤
12∆ β
if
2∆ β
≤λ≤
6∆ β
if − 2∆ β ≤λ≤
,
0.4 0.3
,
0.2
2∆ β ,
0.1
if − 6∆ ≤ λ ≤ − 2∆ , β β if
− 12∆ β
if λ ≤
≤λ≤
− 12∆ β
− 6∆ β
0 −150
−100
−50
0
50
100
150
λ/(γ∆2 )
200
250
300
350
, Fig. 9. The PDF of MSB conditioned on the transmission of s = −7∆ in 1 the case of B = 3; γ = −5dB.
.
(18) The remark about symmetry made in the previous section still holds for MSB in equation (18). Similar to the previous case, the simulated histograms confirm again our analytical expressions, cf. Fig. 7, Fig. 8 and Fig. 9. Due to lack of space we show just three examples of the PDFs for LSB, MiSB and MSB conditioned on different transmitted symbols. It is clear that the PDF cannot be well approximated as a Gaussian which is also confirmed for higher values of B.
V. C ONCLUSION In this paper, we presented the closed-form expressions for the probability density functions (PDF) of the logarithmic likelihood ratios in rectangular QAM. Our results show that this PDF is piecewise Gaussian and simulation results confirmed our formulas. The new expressions that we advanced provide a tool necessary for the analysis of Bit-Interleaved Coded Modulation (BICM) transmissions.
R EFERENCES [1] G.Caire, G.Taricco, and E. Biglieri, “Bit-interleaved coded modulation,” IEEE Transactions on Information Theory, vol. 44, no. 3, pp. 927–946, May 1998. [2] A. G. Fabregas, A. Martinez, and G. Caire, “Error probability of bitinterleaved coded modulation using the Gaussian approximation,” in Conference on Information Sciences and Systems, 2004. [3] A. Abedi and A. K. Khandani, “An analytical method for approximate performance evaluation of binary linear block codes,” IEEE Transactions on Communications, vol. 52, no. 2, pp. 228–235, Feb. 2004. [4] K. Hyun and D. Yoon, “Bit metric generation for Gray coded QAM signals,” IEE Proc.-Commun, no. 6, pp. 1134–1138, December 2005. [5] G.Caire, G.Taricco, and E. Biglieri, “Capacity of bit-interleaved channels,” IEE Electronics Letters, vol. 32, no. 12, pp. 1060–1061, June 1996. [6] A. J. Viterbi, “An intuitive justification and a simplified implementation of the MAP decoder for convolutional codes,” IEEE Journal of Selected Areas in Communication, no. 2, pp. 260–264, 1998.
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