Probability Density Functions of Logarithmic Likelihood Ratios in Rectangular QAM

7/29/2019 Probability Density Functions of Logarithmic Likelihood Ratios in Rectangular QAM

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Probability Density Functions of Logarithmic

Likelihood Ratios in Rectangular QAMMustapha Benjillali, Leszek Szczecinski, Sonia Assa

INRS-EMT, Montreal, Canada{jillali,leszek,aissa}@emt.inrs.ca

Abstract Closed-form expressions for the probability densityfunction (PDF) of logarithmic likelihood ratios (LLR) in rec-tangular quadrature amplitude modulations are derived. Takingadvantage of assumed Gray mapping, the problem is solved inone dimension corresponding to the real or imaginary part of thesymbol. The results show that sought PDFs are linear combinationsof truncated Gaussian functions. This simple result stands incontrast 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. INTRODUCTION

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-aryQAM for M = 4, 8, 16, 32, 64. Covering such wide familyof 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.

I I . SYSTEM 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 sym-bols sQAM(n) = MQAM[cQAM(n)], and transmitted overadditive white Gaussian noise (AWGN) channel. The received

signal rQAM(n) = sQAM(n) + QAM(n) is corrupted by the

complex noise QAM(n) with variance given by N0 = 1/.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 realor imaginary part of the symbol. By combining PAM constel-

lations, we can get different rectangular QAM constellations

(e.g. 32-QAM = 8-PAM 4-PAM ...). We keep the introducednotations 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, . . . , aM1} where am = (2m + 1 M) and denotes half the minimum distance between theconstellation symbols.

To alleviate the notation we abandon the time index n, whichshould not lead to any confusion as all considerations are static

with respect to n due to the memoryless nature of the modulationand the channel.

At the receiver, LLR for the k-th bit in codeword c (k =1, . . . , B) is obtained as [5]

B,k(r) = lnPr{ck = 1|r}Pr{ck = 0|r} = ln

bCk1 exp

|rM[b]|2

N0

bCk0 exp |rM[b]|2

N0

[min

bCk0|r M[b]|2 min

bCk1|r M[b]|2], (1)

where Ckx is the set of codewords b = [bB, . . . , b1] with thek-th bit equal to x {0, 1} and (1) is obtained using the knownmax-log approximation: ln (

i

exp(Xi))

mini(Xi) [6].

I I I . DERIVATION OF LLRS EXPRESSIONS

The LLR in (1) can now be simplified to

B,k(r) = [(r sk0)2 (r sk1)2]= 2 r[sk1 sk0 ] + [(sk0)2 (sk1)2], (2)

where skx is the symbol with the k-th labelling bit equal to x,closest to the received signal r, i.e.

skx = M[arg minbCk

x

|r M[b]|2]. (3)

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In the following, we provide the explicit expressions of the

LLRs when B = 1, 2, 3, using for normalization purpose thecoefficient = 14 .

Case B = 1:Given that k 1 in this case, for every received r we haves10 = and s11 = +. Hence, using (2), the LLR expressionis given by

1,1(r) =1

r. (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 sk1 and sk0 is given in Table I. Accordingly,

the LLR expressions for the LSB and MSB are respectively

given by

2,1(r) =

1r 2

if r 0,

+ 1r 2

if r 0. (5)

2,2(r) =

2 r 2 if r 2, 1r if2 r 2,

2r + 2

if r 2.

(6)

A normalized representation of these functions ((5) and (6)) is

shown in Fig. 2.

1 1 0 0

1 0 0 13

3

+

+

+3

+3

LSB, k = 1

MSB, k = 2

Fig. 1. Bit mapping and decision regions for LSB and MSB, case of foursymbols in the real dimension (B = 2).

r LSB MSB

s21

s20

s41

s40

r 2 3 3

2 r 0 3

0 r 2 3

r 2 3 3

TABLE I

SYMBOLS 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 sk1 and sk0 is presented in Fig. 3. Table II

describes the decision regions for the three bit positions and sub-

intervals of r. LLRs of LSB (k = 1), middle (significant) bit(MiSB), i.e. k = 2 and MSB (k = 3) are provided respectivelyin (7), (8) and (9) and their plots shown in Fig. 4.

4 3 2 1 0 1 2 3 420

15

10

5

0

5

10

15

20

MSBLSB

2,

k

(r)

2

r

Fig. 2. LLR as a function ofr for LSB and MSB in the case of four symbolsin the real dimension (B = 2) with = 5dB.

7

7

7

5

5

5

3

3

3

+

+

+

+3

+3

+3

+5

+5

+5

+7

+7

+7LSB, k = 1

MiSB, k = 2

MSB, k = 30000

0

0000

000

11

1

11

111

1111

Fig. 3. Bit mapping and decision regions for LSB, MiSB and MSB, case ofeight symbols in the real dimension (B = 3).

3,1(r) =

1r 6

if r 4,

+ 1r + 2

if4 r 0,

1r + 2

if 0 r 4,

+ 1r 6

if r 4.

(7)

3,2(r) =

2r

10

if r

6,

1r 4

if6 r 2,

2r 6

if2 r 0,

+ 2r 6

if 0 r 2,

+ 1r 4

if 2 r 6,

+ 2r 10

if r 6.

(8)

3,3(r) =

4r 12

if r 6,

3r 6

if6 r 4,

2r 2

if4 r 2,

1r if2 r 2,

2r + 2

if 2 r 4,

3 r +

6 if 4 r 6,

4r + 12

if r 6.

(9)

IV. PROBABILITY 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

pB,k(|s) =d

dPB,k(|s), (10)

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r LSB MiSB MSB

s21

s20

s41

s40

s61

s60

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

TABLE II

SYMBOLS CLOSEST TO r IN THE 8-PAM CASE (B = 3), CF. (3).

8 6 4 2 0 2 4 6 860

40

20

0

20

40

60LSBMiSBMSB

3,

k

(r)

2

r

Fig. 4. LLR as a function ofr for LSB, MiSB and MSB in the case of eightsymbols per real dimension (B = 3), with = 5dB.

as a derivative of the cumulative distribution function (CDF) for

each variable

PB,k(|s) = Pr{B,k(r) |s} = Pr{r I|s}, (11)where I = {r : B,k(r) } is the interval (or unionof intervals) in which B,k(r) . The latter may be easilyobtained from equations (4)-(9) (or Fig. 2 and Fig. 4).

According to our system model (i.e., r N(s, 12)), we canwrite

PB,k(|s) = 1/

rI

exp|r s|2 dr, (12)

and change the variable r in the integration with its in-verse expression 1B,k() in the sub-intervals of I from (4)-(9). Though straightforward, the mathematical derivations are

lengthy. Hence, in what follows, we only present the final resultsfor the three cases ofB.

Case B = 1:Applying (10) and considering (12) and (4), we obtain

p1,1(|s) =1

4

exp| s|2 , (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

80 60 40 20 0 20 40 60

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

AnalyticalSimulated

LSB

MSB

/(2)

p2,

k(|

s3

)

Fig. 5. The PDF of LSB and MSB conditioned on the transmission ofs3 = +in the case of B = 2 ; = 5dB.

120 100 80 60 40 20 0 20 40 600

0.1

0.2

0.3

0.4

0.5

0.6

AnalyticalSimulated

LSB

MSB

/(2)

p2,

k(|

s4

)

Fig. 6. The PDF of LSB and MSB conditioned on the transmission ofs4 =+3 in the case of B = 2 ; = 5dB.

LSB and MSB is respectively given by (14) and (15) which

demonstrate that each distribution is a piecewise Gaussian.

p2,1

(|s) =

14

exp

| + 2 s|2

+ exp | + 2 + s|2 if 2 ,0 if 2.

(14)

p2,2(|s) =

18

exp|2 + + s|2

if 2

,

14

exp| + s|2 if2

2

,

18

exp|2 + s|2

if 2

.

(15)

Note that the PDFs defined in (15) for the MSBs are sym-

metric, i.e. p2,2(|s) = p2,2(| s). This is not the case forthe LSBs (14).

Figures 5 and 6 show the comparison between the histograms

of the LLRs, obtained from simulated data, and the analyticalformulas when the PDF is conditioned on the transmission of

s = and s = 3 respectively, and considering = 5dB. Itis 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

= 14 .


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p3,1(|s) =

u{1,1} exp| + 6 + u s|2

if 2

,

u{1,1}

exp

| + 6 + u s|2

+ exp

| 2 + u s|2

if 2

2

,

0 if 2.

(16)

p3,2(|s) =

2

u{1,1} exp

|

2 + 5 + u s|2

if 2

,

u{1,1} exp

| + 4 + u s|2

if2

2

,

2

u{1,1} exp|

2 + 3 + u s|2

if6

6

,0 if 6

.

(17)

40 20 0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

AnalyticalSimulated

/(2)

p3,1

(|

s7

)

Fig. 7. The PDF of LSB conditioned on the transmission ofs7 = +5 inthe case ofB = 3 ; = 5dB.

p3,3(|s) =

4 exp

|4 3 + s|2

if 12

,

3 exp

|3 2 + s|2

if 6

12

,

2 exp

|2 + s|2

if 2

6

,

exp

| + s|2

if 2

2

,

2 exp

|2 + + s|2 if6 2 ,3 exp

|3 + 2 + s|2

if12

6

,

4 exp

|4 + 3 + s|2

if 12

.

(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 ofB.

V. CONCLUSION

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.

60 40 20 0 20 40 60 80 100 120 1400

0.5

1

1.5

2

AnalyticalSimulated

/(2)

p3,2

(|

s4

)

Fig. 8. The PDF of MiSB conditioned on the transmission ofs4 = inthe case ofB = 3 ; = 5dB.

150 100 50 0 50 100 150 200 250 300 3500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 AnalyticalSimulated

/(2)

p3,3

(|

s1

)

Fig. 9. The PDF of MSB conditioned on the transmission ofs1 = 7 inthe case ofB = 3 ; = 5dB.

REFERENCES

[1] G.Caire, G.Taricco, and E. Biglieri, Bit-interleaved coded modulation,IEEE Transactions on Information Theory, vol. 44, no. 3, pp. 927946,May 1998.

[2] A. G. Fabregas, A. Martinez, and G. Caire, Error probability of bit-interleaved coded modulation using the Gaussian approximation, in Con-

ference 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 Transactionson Communications, vol. 52, no. 2, pp. 228235, Feb. 2004.

[4] K. Hyun and D. Yoon, Bit metric generation for Gray coded QAM signals,IEE Proc.-Commun, no. 6, pp. 11341138, December 2005.

[5] G.Caire, G.Taricco, and E. Biglieri, Capacity of bit-interleaved channels,IEE Electronics Letters, vol. 32, no. 12, pp. 10601061, June 1996.

[6] A. J. Viterbi, An intuitive justification and a simplified implementation ofthe MAP decoder for convolutional codes, IEEE Journal of Selected Areasin Communication, no. 2, pp. 260264, 1998.

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Probability Density Functions of Logarithmic Likelihood Ratios in Rectangular QAM