Research Article Log-Likelihood Ratio Calculation...

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 970126 10 pageshttpdxdoiorg1011552013970126

Research ArticleLog-Likelihood Ratio Calculation for Iterative Decoding onRayleigh Fading Channels Using Padeacute Approximation

Gou Hosoya and Hiroyuki Yashima

Department of Management Science Faculty of Engineering Tokyo University of Science 1ndash3 KagurazakaShinjuku-ku Tokyo 162ndash8601 Japan

Correspondence should be addressed to Gou Hosoya hosoyamskagutusacjp

Received 22 March 2013 Accepted 9 July 2013

Academic Editor D R Sahu

Copyright copy 2013 G Hosoya and H Yashima This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Approximate calculation of channel log-likelihood ratio (LLR) for wireless channels using Pade approximation is presented LLRis used as an input of iterative decoding for powerful error-correcting codes such as low-density parity-check (LDPC) codes orturbo codes Due to the lack of knowledge of the channel state information of a wireless fading channel such as uncorrelated fiatRayleigh fading channels calculations of exact LLR for these channels are quite complicated for a practical implementation Theprevious work an LLR calculation using the Taylor approximation quickly becomes inaccurate as the channel output leaves somederivative pointThis becomes a big problemwhen higher order modulation scheme is employed To overcome this problem a newLLR approximation using Pade approximation which expresses the original function by a rational form of two polynomials withthe same total number of coefficients of the Taylor series and can accelerate the Taylor approximation is devised By applying theproposed approximation to the iterative decoding and the LDPC codes with some modulation schemes we show the effectivenessof the proposed methods by simulation results and analysis based on the density evolution

1 Introduction

In recent years iterative decoding techniques based onmessage passing algorithm such as turbo decoding [1] orbelief-propagation (BP) decoding [2ndash4] have been attractedby their significant performance which attain close to theShannon limit The BP decoding algorithm a well-knowniterative decoding algorithm for LDPC codes [2 3] has beenwidely studied for the binary erasure channel or the additivewhiteGaussian noise (AWGN) channel [2ndash8]The algorithmsfirstly derive channel log-likelihood ratios (LLR) where themessages in the decoder are initialized to these LLR valuesTo exhibit good performance with BP decoding this channelLLR should be obtained with high accuracy but it becomescomplicated for some channel models such as wireless fadingchannel [9]

In this study we focus on a calculation of channel LLRover the uncorrelated flat Rayleigh fading channels where thediscrete-time component transmitted signal is input to aband-limited channel that is 119910

119905= 119903119905119908119905+ 119911119905[10] Here 119910

119905 119903119905

119908119905 and 119911

119905denote a channel output a fading gain a channel

input and an additive white Gaussian noise (AWGN) 119911119905sim

N(0 1205902) with variance 1205902 at time 119905 respectively Hereinafterwe drop the subscript 119905 If 119903 at each received bit positionis known to the receiver we call this case known channelstate information (CSI) If 119903 at each received bit position isunknown to the receiver we call this case unknown CSI For aknown CSI case channel LLR ℓ can be easily calculated usingthe channel outputs 119910 1205902 and 119903 However for an unknownCSI case which is more practical than known CSI and is ourmain interest a calculation of the channel LLR ℓ is rathercomplex due to an integration of 119903

The studies of wireless fading channels with the LDPCcodes or turbo codes were presented in [11ndash20] with severalmodulation schemes [9 10 21] such as binary modulation(binary phase shift keying (BPSK)) or nonbinary modula-tions In [14] for BPSK Hou et al have studied designingirregular LDPC codes [6] using density evolution [7 8] andhave shown that these codes can approach the Shannon limit

2 Journal of Applied Mathematics

But they have used the following simple linear approximationℓex for a calculation of the channel LLR

ℓex =2

1205902sdot 119864 (119903) sdot 119910 (1)

where 119864(119903) denotes the expectation of the channel gain 119903Although the previous approximation is simple and is easy toimplement it is inaccurate that degradation in the decodingperformance compared with the true LLR [19] can be seenYazdani and Ardakani [19 20] have also proposed a linearLLR approximation whose performance is almost identical tothe true LLR

ℓopt = 119910 (2)

where is obtained by maximizing

= argmax120572(1 minus int

infin

minusinfin

log2(1 + exp (minusℓ)) 119891

120572(ℓ) 119889ℓ ) (3)

Here 119891120572(ℓ) is given by

119891120572(ℓ)

=radic2Δ2

radic120587120572120590exp(minus(Δℓ

120572120590)

2

)

times [exp(minus( Δℓ

radic21205721205902)

2

) +Δradic120587ℓ

radic21205721205902sdot erfc(minus Δℓ

radic21205721205902)]

(4)

where Δ = radic1205902(21205902 + 1) and erfc(sdot) denotes the comple-mentary error function [9] However an optimization of using (3) and (4) requires for each channel parameter 120590 sothat it needs large complexity to implement

Recently Asvadi et al [11] have applied Taylor approxima-tion of order 119899 to the true LLR function such that

ℓT119899 =119899

sum

119894=0

119888119894119910119894 (5)

where 119888119894denotes the coefficient of Taylor series of order 119894

They have derived both linear and nonlinear approximationswith small orders For a linear approximation (Taylor seriesof order 119899 = 1) it is given by

ℓT1 = 1198881119910 (6)

For a nonlinear approximation (Taylor series of order 119899 = 3)it is given by

ℓT3 = 1198881119910 + 11988831199103 (7)

From the previous approximations one can obtain accurateLLRwithout optimizing complicated functions such as in [1920]

Tomove our attentions to nonbinarymodulations whichis more practical case the LLR calculations are performedbitwise [15 21] The previous work by Yazdani and Ardakani[20] which is an extension of [19] has devised the LLR

approximation method but it becomes complex to evaluateLLR due to the increment of the number of parameters forthe optimization To fit the true LLR functions the authorsin [11] have modified the approximation functions of Taylorseries of order 3Thismodification is not easy to replicate andis required for each parameter of the channels Moreover it iswell known that the Taylor approximation quickly becomesinaccurate as the variable 119910 leaves the derivative point

To overcome these problems we devise a new LLRapproximation using Pade approximation [22] on the uncor-related flat Rayleigh fading channels with unknown CSIfor BPSK and 8-PAM Pade approximation expresses theoriginal function by rational form of two polynomials withthe same total number of coefficients of the Taylor seriesand it can accelerate the Taylor approximation GenerallyPade approximation is accurate not only at the derivativepoint but also at the wide range of intervals of variables Weshow by simulation results and analysis based on the densityevolution that our method can approximate LLR functionwith high accuracy and can yield almost the same decodingperformance as the true LLR The Pade approximation is ageneralization of the Taylor approximation and the proposedmethod exhibits slightly better performance than themethodusing Taylor approximation [11] Moreover we design irregu-lar LDPC codes based on our LLR approximation function

This paper is organized as follows Section 2 gives thechannelmodel LLR calculationmethod and LDPC codes InSection 3 we briefly review Taylor and Pade approximationsand then we present the proposed LLR calculation by PadeapproximationNumerical results are shown in Section 4 andSection 5 concludes the paper

2 Preliminaries

21 Channel Model and LLR Calculation We here considerthe following discrete-time channel model

119910 = 119903119908 + 119911 (8)

where 119908 isin 120594 and 119910 119911 isin R represent the channel inputoutput and noise respectively and 120594 R denote a set oftransmitted symbols and that of real numbers respectively(As mentioned in Section 1 we drop a time subscript for119910 119903 119908 and 119911) Moreover 119903 ge 0 is the channel gain withan uncorrelated flat Rayleigh distribution by its probabilitydensity function (pdf) 119875(119903) = 2119903119890minus119903

2

and 119911 is the whiteGaussian noise with mean 0 and variance 1205902

Using the bit-interleaved coded modulation (BICM)scheme [10 15 21] for a transmission an information bitsequence is mapped to the codeword (bit sequence) of length119873 by error-correcting codes and then it is partitioned into119873119902 blocks denoted by b

119896= 119887

(1)

119896 119887(2)

119896 119887

(119902)

119896 119896 =

1 2 119873119902 of length 119902 Hereinafter we drop the subscript119896 of both b

119896and 119887(119894)119896

to simplify discussions This block b ismapped to transmit a signal 119908 isin 120594 in a Gray-labeled 119876-arysignal constellation 120594 of size 119876 = 2119902 The signal constellationfor BSPK and 8-PAM is depicted in Figure 1

For the fading channel two cases can be consideredwhichdepends on the knowledge of 119903 at the receiver

Journal of Applied Mathematics 3

BPSK

w

minus1 1

0

b

8PAM

111 110 100 101 001 000 010 011

minus7 minus5 minus3 minus1 1 3 5 7

1

w

b

Figure 1 Signal constellations for BPSKand 8-PAMwithGray label-ing

211 Known CSI For a known CSI case we can use channelfading gain 119903 for each bit position 119894 = 1 2 119902 The channelLLR is given by

ℓ(119894)

CSI = log119875 (119910 | 119887

(119894)(119908) = 0 119903)

119875 (119910 | 119887(119894) (119908) = 1 119903)

= logsum119908isin120594(119894)

0

119875 (119910 | 119908 119903)

sum119908isin120594(119894)

1

119875 (119910 | 119908 119903)

(9)

where 119887(119894)(119908) denotes the 119894th bit of block b which is mappedto 119908 and 120594(119894)

119895is a set of blocks b which satisfies 119887(119894)(119908) = 119895 for

119895 = 0 1 Moreover a base of logarithm takes a natural number119890 and 119875(119910 | 119908 119903) is given by

119875 (119910 | 119908 119903) =1

radic21205871205902sdot exp(minus

(119910 minus 119903119908)2

21205902) (10)

For BPSK (9) is reduced to

ℓ(1)

CSI =2119903119910

1205902 (11)

The calculation of the previous equation is not a difficult task

212 Unknown CSI For an unknown CSI case we cannotuse channel fading gain 119903 for each received bit position Thechannel LLR is given by

ℓ(119894)

NSI = log119875 (119910 | 119887

(119894)(119908) = 0)

119875 (119910 | 119887(119894) (119908) = 1)

= logsum119909isin120594(119894)

0

119875 (119910 | 119908)

sum119908isin120594(119894)

1

119875 (119910 | 119908)

(12)

where 119875(119910 | 119908) is given by

119875 (119910 | 119908)

= int

infin

0

119875 (119910 | 119908 119903) 119875 (119903) 119889119903

= int

infin

0

1


(119910 minus 119903119908)2

21205902) times 2119903 exp (minus1199032) 119889119903

(13)

For BPSK (12) is given by

ℓ(1)

NSI = logΨ(119910radic2)

Ψ (minus119910radic2) (14)

where = 1205902(1+21205902) andΨ(119909) = 1+radic120587119909 exp(1199092) erfc(minus119909)The calculation of (12) is so complicated that several workshave tried to reduce the computational complexity by approx-imations [11 13 14 19 20]

For the 119876-ary PAM 119875(119910 | 119908) in (13) becomes

119875 (119910 | 119908)

=exp (minus11991022

1)

radic1205873

1

times(erfc(minus119908119910

radic21205901

)timesradic120587119908119910+radic21205901exp(minus

11990821199102

2120590221

))

(15)

where 1= radic1199082 + 21205902 Using (15) the LLR in (12) can

be evaluated The previous equations are so complicated toimplement that several works have tried to reduce the com-putational complexity by approximations

Notice that for the fading channels with some modula-tions the log-sum approximation was used for bitwise linearapproximation This approximation is only efficient for aknown CSI case since an integration of fading factor 119903 fora calculation of LLR is not needed But for an unknown CSIcase an integration of fading factor 119903 is neededMoreover it iseffective only for a high signal-to-noise (SNR) region wherethe sum in (9) is dominated by a single large term

22 LDPC Codes We here consider binary LDPC codesAn LDPC code is represented by the Tanner graph whichconsists of the variable nodes and the check nodes Thesenodes are incident with the edges Let 119889V and 119889119888 denote themaximumnumber of edges incident to the variable nodes andcheck nodes respectively Let 120582(119909) = sum119889V

119894=2120582119894119909119894minus1 and 120588(119909) =

sum119889119888

119894=2120588119894119909119894minus1 be variable node degree distribution and check

node degree distribution where 120582119894and 120588

119894denote fractions

of the number of edges incident to the variable node andcheck node of degrees 119894 in the Tanner graph of the coderespectively An LDPC code is specified by119873 120582(119909) and 120588(119909)The rate of the codes is given by 119877 (= 1 minus 119872119873) where 119872denotes the number of check nodes and is given by 119872 =

119873int1

0120588(119909)119889119909 int

1

0120582(119909)119889119909


An ensemble of LDPC codes [2] is denoted by C119873(120582(119909)120588(119909)) Combined with the BP decoding algorithm LDPCcodes with optimized (120582(119909) 120588(119909)) by density evolution [57 8] can attain high performance which is close to thetheoretical limit (Shannon limit) The iterative threshold 120590⋆of an ensemble of LDPC codes Cinfin(120582(119909) 120588(119909)) is defined asthe maximum standard deviation 120590 on the channel in (8)such that lim

119905rarrinfin119875119890(119905) = 0 where 119875

119890(119905) denotes the message

error probability in iteration 119905 of the BP decoding algorithm119875119890(119905) is calculated recursively by the density evolution [5 7 8]

which keeps track the message error probability of the BPdecoding algorithm from the pdf of channel LLR Iterativethreshold is sometimesmeasured by119864

1198871198730or signal-to-noise

ratio (SNR) such that (1198641198871198730)⋆= 12119877(120590

⋆)2 (for BPSK)

or 212(120590⋆)2 (for 8-PAM) respectively where 119864119887and 119873

0

denote the average energy per information bit and one-sidedpower spectral density of the additive white Gaussian noise(AWGN)

3 LLR Approximation Based onPadeacute Approximation

Before approximating the true LLR function in (12) we brieflyexplain the Taylor approximation and then describe the Padeapproximation

31 Brief Review of Taylor and Pade Approximations Let119891(119910)be the original function and let 119891(119899)(119910) be 119899th derivative of119891(119910) Let [119904

1 1199042] and (119904

1 1199042) be closed and open intervals

between 1199041and 1199042 1199042ge 1199041 respectively

Definition 1 Suppose that 119891(119910) has 119899 derivatives at point1199100isin [1199041 1199042] and 119891(119899+1)(120585) has a derivative at (119904

1 1199042) where

120585 = 1199041+ 120579(1199042minus 1199041) 0 lt 120579 lt 1 Assume that 119891(1)(119910

0)

119891(2)(1199100) 119891

(119899)(1199100) are required to be continuous on [119904

1 1199042]

and 119891(119899+1)(120585) is required to exist on (1199041 1199042) The Taylor

polynomial of order 119899 for 119891(119910) at point 1199100is then defined by

119875119899(119910) = 119891 (119910

0) + 119891(1)(1199100) (119910 minus 119910

0) + sdot sdot sdot

+119891(119899)(1199100)

119899(119910 minus 119910

0)119899

=

119899

sum

119896=0

119891(119896)(1199100)

119896(119910 minus 119910

0)119896

(16)

The remainder term which is a difference between true valueof the function and its Taylor series of polynomial is given by

119891(119899+1)(120585)

(119899 + 1)(119910 minus 119910

0)119899+1

(17)

For some 119899 and 1199100 one can approximate the original function

by 119875119899(119910) in (16) However this function quickly becomes

inaccurate as 119910 leaves 1199100 even though 119899 is large

The approximation in (16) can often be accelerated byrearranging it into a ratio of two series using Pade approxima-tion It generalized the Taylor approximation with the same

total number of coefficients of two series Before describingthe Pade approximation we rewrite (16) as follows

119875119899(119910) =

119899

sum

119896=0

119891(119896)(1199100)

119896(119910 minus 119910

0)119896

=

119899

sum

119896=0

119888119896119910119896

(18)

Definition 2 Suppose that119891(119910) is approximated by theTaylorseries in (18) The Pade approximation of order (1198981015840 1198991015840) 119899 =1198981015840+ 11989910158401198981015840 1198991015840 ge 0 is given by

1198751198991015840

1198981015840 (119910) =

sum1198991015840

119896=0119886119896119910119896

sum1198981015840

119896=0119887119896119910119896 (19)

where 1198860 1198861 119886

1198991015840 and 119887

0 1198871 119887

1198981015840 are determined so that

the coefficients of the terms 1199100 1199101 119910119899 of1198751198991015840

1198981015840(119910) in (19) are

equal to those of 119875119899(119910) in (18)

FromDefinition 2 the polynomialssum1198991015840

119896=0119886119896119910119896sum119898

1015840

119896=0119887119896119910119896

and sum119899119896=0119888119896119910119896 satisfy the equation

1198991015840

sum

119896=0

119886119896119910119896minus (

1198981015840

sum

119896=0

119887119896119910119896) sdot (

119899

sum

119896=0

119888119896119910119896) = 119874(119910

119899+1) (20)

Equation (20) tells that all the coefficients of 1199100 1199101 119910119899

of sum1198991015840

119896=0119886119896119910119896 and (sum119898

1015840

119896=0119887119896119910119896) (sum119899

119896=0119888119896119910119896) in left-hand side of

(20) are equalWe can express these relations by the followingsimultaneous equation for 119894 = 0 1 119899

119894

sum

119895=max(119894minus1198981015840 0)119888119895119887119894minus119895= 119886119894 119894 = 0 1 119899

1015840

0 119894 = 1198991015840+ 1 1198991015840+ 2 119899

(21)

Notice that the terms 119910119899+1 119910119899+2 119910119899+1198981015840

in left-hand sideof (20) are included in 119874(119910119899+1) in right-hand side of theequation These terms are not necessary for the evaluation of1198860 1198861 119886

1198991015840 and 119887

0 1198871 119887

1198981015840

We have already evaluated the coefficients 1198880 1198881 119888

119899in

(18) by Taylor series Moreover we assume that 1198860 1198861 119886

1198991015840

and 1198870 1198871 119887

1198981015840 are normalized so we set 119887

0= 1 This

normalization is valid since the Pade approximation in (19)is of a rational form Substituting 119888

0 1198881 119888

119899and 119887

0=

1 into (20) we can obtain coefficients 1198860 1198861 119886

1198991015840 and

1198870 1198871 119887

1198981015840 Note that the Taylor approximation and the

Pade approximation are equivalent to each other if 1198981015840 =0 Therefore we can see that the Pade approximation is ageneralization of the Taylor series approximation

32 Applying Pade Approximation to LLR Function

321 LLR Calculation for BPSK By applying the Padeapproximation to the true LLR function for BPSK in (14) wecan obtain the approximated function To fit the true LLRfunction we have searched the approximated function forseveral pairs of (1198981015840 1198991015840) and found that Pade approximation of


minus8 minus6 minus4 minus2 0 2 4 6 8

minus30

minus20

minus10

0

10

20

30

LLR

True LLRTaylor3

Pade23Taylor5

y

(a) LLR calculation

0 1 2 3 4 5 6 7 8

Taylor3Pade23Taylor5

100

10minus15

10minus10

10minus5

|Tru

e LLR

appr

oxim

atio

n|

y

(b) Absolute difference

Figure 2 Comparison of true LLR and approximated LLRs (Taylor3 Taylor5 and Pade23) for the uncorrelated flat Rayleigh fading channelwith unknown CSI and 120590 = 06449 From (a) Pade23 gives nearly the same values as the true function for various channel output 119910 andTaylor5 shows inaccurate as |119910| becomes large From (b) accuracy of Pade23 and Taylor5 is good especially for small |119910| But accuracy ofthem is different for large |119910| that is Pade23 shows the best accuracy in three methods but Taylor5 shows the worst accuracy

order (1198981015840 1198991015840) = (2 3) is more accurate than Taylor series oforder 119899 = 3 around 0 in (7) which is previously known as thebest approximated one [11]TheproposedLLR approximationis given as follows

ℓP23 =1198860+ 1198861119910 + 11988621199102+ 11988631199103

1198870+ 1198871119910 + 11988721199102

(22)

where 1198860= 0 119886

2= 0 1198870= 1 and 119887

1= 0

1198861= radic2120587

119887

2=minus35 + 30120587 minus 6120587

2

20 (minus3 + 120587)

1198863= minusradic

120587

2sdot15 minus 30120587 + 8120587

2

30 (minus3 + 120587) radic

(23)

The previous approximation is obtained by the Taylor seriesof order 119899 = 5

ℓT5 = 1198881119910 + 11988831199103+ 11988851199105 (24)

where

1198881= radic2120587

119888

3=radic120587 (minus3 + 120587)

3radic2

1198885=

radic2120587 (35 minus 30120587 + 61205872)

1202radic

(25)

Then (21) becomes

11988801198870= 1198860

11988801198871+ 11988811198870= 1198861

11988801198872+ 11988811198871+ 11988821198870= 1198862

11988811198872+ 11988821198871+ 11988831198870= 1198863

11988821198872+ 11988831198871+ 11988841198870= 0

11988831198872+ 11988841198871+ 11988851198870= 0

(26)

Substituting 1198880 1198881 119888

5in (24) and 119887

0= 1 into (26) we get

1198860 1198861 1198862 1198863 and 119887

1 1198872

We then compare the accuracy of the approximated LLRfunctions Figures 2(a) and 3(a) show LLR values for theuncorrelated flat Rayleigh fading channel with unknown CSIfor 120590 = 06449 and 10264 respectively LLR values in thesefigures are evaluated by the true LLR in (14) Taylor seriesof orders 119899 = 3 and 119899 = 5 (ldquoTaylor3rdquo and ldquoTaylor5rdquo) in(7) and (24) respectively the Pade approximation of order(1198981015840 1198991015840) = (2 3) (ldquoPade23rdquo) in (22) and linear approximation

(ldquoExrdquo) in (1) From these figures Pade approximation isalmost identical to the true LLR for various channel output119910 This may be contributory to the fact that the order 119899 (=1198981015840+ 1198991015840= 5) of Pade23 is larger than that of Taylor3 (119899 = 3)

However we can see that Taylor5 (119899 = 5) is inaccurate as|119910| becomes large Therefore large 119899 is not an answer for anaccurate LLR approximation especially for using the Taylorapproximation

To compare the LLR approximations in detail Figures2(b) and 3(b) show the absolute differences between true LLR


minus15 minus10 minus5 0 5 10 15minus40

minus30

minus20

minus10

0

10

20

30

40LL

R

True LLRTaylor3

Pade23Taylor5

y

(a) LLR calculation

0 5 10 15


100

10minus6

10minus4

10minus2

|Tru

e LLR

appr

oxim

atio

n|

y


Figure 3 Comparison of true LLR and approximated LLRs (Taylor3 Taylor5 and Pade23) for the uncorrelated flat Rayleigh fading channelwith unknown CSI and 120590 = 10264 Pade23 also gives almost the same values as the true function

and approximated LLRs (Taylor3 Taylor5 and Pade23) Weonly show the case of 119910 ge 0 since the true LLR functionis odd symmetric that is 119891(119910) = minus119891(minus119910) From Figures 2and 3 accuracy of Pade23 and Taylor5 are good especially forsmall |119910| But accuracy of them are quite different for large |119910|that is Pade23 shows the best accuracy but Taylor5 shows theworst accuracy

Next we consider analysis based on the density evolutionfor different LLR calculation methods We derive the pdf ofLLR assuming that119908 = +1 is transmitted From (10) we have

119875 (119910 | 119908 = +1 119903) =1


(119910 minus 119903)2

21205902) (27)

Averaging (27) over 119903 by an integration we obtain

119875 (119910 | 119908 = +1)

=radic2

radic120587 (1 + 21205902)sdot exp(minus

1199102

1 + 21205902)

times (120590 sdot exp(minus1199102

2) +119910

2radic2120587

1 + 21205902sdot erfc(minus

119910

radic2))

(28)

Then the density of the LLR function can be expressed in aparametric form [4] For ℓP23 in (22) this is given by

(ℓP23119875 (119910 | 119908 = +1)

ℓ1015840P23) (29)

where ℓ1015840P23 denotes a derivative of the function ℓP23 with119910 Equation (29) is a case of the proposed approximationfunction but one can obtain the pdf of the other LLR

functions For example replacing ℓP23 with ℓ(1)

NSI in (14) wecan obtain the pdf of the true LLR function ℓ(1)NSI

322 LLR Calculation for 8-PAM We demonstrate Padeapproximation for bitwise LLR of 8-PAM constellation withGray labeling on the Rayleigh fading channel without CSI(SNR = 791 (dB) (120590 = 1699)) in Figure 4 The coefficientsof LLR approximation functions of Taylor3 and Pade approx-imation are listed in Table 1 The derivative points 119910

0for each

bit LLR are chosen where these functions take ℓ(119894)NSI = 0 for119894 = 1 2 3 Notice that we can omit the coefficients for thecase 119910

0lt 0 (bits 2 and 3) since these LLR functions are even

functions that is we can derive from119891(119910) = 119891(minus119910) for119910 lt 0The orders of Pade approximation for each bit are differentsince each bit LLR function is distinct

For bit 1 this point is 1199100= 0 For bit 2 these

points are 1199100= plusmn33449 (2 points) So we have two LLR

approximation functions for bit 2 We chose the order pairsof Pade approximation (4 7) and (2 4) (denoted by ldquoPade47rdquoand ldquoPade24rdquo) for 119894 = 1 and 2 respectively The trueLLR function for bit 2 is an even function so we switchtwo LLR approximation functions on 119910 = 0 Since theseapproximation functions intersect on 119910 = 0 the resultingfunction becomes continuous

For bit 3 these points are 1199100= plusmn18848 plusmn69832 (4

points) For 1199100= plusmn18848 and 119910

0= plusmn69832 we use

Pade approximation of orders (4 1) and (3 4) (denoted byldquoPade41rdquo and ldquoPade34rdquo) respectively The true LLR functionfor bit 3 is also an even function so we switch two LLRapproximation functions whose derivative points are 119910

0=

plusmn18848 (bit 3 (a) in Table 1) on the intersection point 119910 = 0But two LLR approximation functions (bit 3 (a) and (b) inTable 1) do not intersect on any 119910 Therefore we searched two


minus10 minus5 0 5 10

minus30

minus20

minus10

0

10

20

30LL

R

True LLRTaylor3Pade47

y

(a) Bit 1


minus12

minus10

minus8

minus6

minus4

minus2

0

2

LLR


y

(b) Bit 2


minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

LLR

True LLRTaylor3Pade41 and pade34

y

(c) Bit 3

4 45 5 55

05

06

07

08

09

1LL

R

True LLRTaylor3

Weighted mean

Switch point at

Switch point aty = 452504

y = 485671

Taylor3 (y0 = 18848)Taylor3 (y0 = 69832)

(d) Bit 3 (detail for Taylor3)

47 475 48 485 49 495 5

066

067

068

069

07

071

072

073

LLR

True LLRPade41 and pade34

Weighted mean

y

Switch point aty = 481638 494772

Pade41 (y0 = 18848)Pade34 (y0 = 69832)

(e) Bit 3 (detail for Pade41 and Pade34)

0 5 10 15minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

LLR

Trule LLRPade45Pade41 and pade34

y

(f) Bit 3 (Pade45)

Figure 4 Comparison of bitwise True LLR and approximated LLRs for the 8-PAM with Gray labeling (SNR = 791 (120590 = 1699))


Table 1 The coefficients of LLR approximation functions for 8-PAM at SNR = 791 (120590 = 1699)

(a) Taylor series of order 119899 = 3

1199100

1198880

1198881

1198882

1198883

bit 1 00 00 12135 00 00241bit 2 33449 23273 minus07706 00205 00006bit 3 (a) 18848 minus07855 00544 02848 minus00491bit 3 (b) 69832 22528 minus02497 minus00181 00011

(b) Pade approximation of numerator

1199100

1198860

1198861

1198862

1198863

1198864

1198865

1198866

1198867

bit 1 00 00 12135 00 00334 00 00025 00 261119864 minus 05

bit 2 33449 33382 minus15496 01836 00039 minus00028 mdash mdash mdashbit 3 (a) 18848 minus11396 06046 mdash mdash mdash mdash mdash mdashbit 3 (b) 69832 56135 minus31331 05963 minus00481 00017 mdash mdash mdash

(c) Pade approximation of denominator

1199100

1198870

1198871

1198872

1198873

1198874

bit 1 00000 10000 00000 00076 00000 00016bit 2 33449 16585 minus03831 00557 mdash mdashbit 3 (a) 18848 14533 minus06204 03220 minus00852 00113bit 3 (b) 69832 13191 minus05502 00496 minus00026 mdash

switch points between the interval of 119910 isin [18848 69832]to minimize the loss of accuracy and we set these pointsfor Taylor approximation on 452504 485671 We take thefunction (bit 3 (a)) for 119910 lt 452504 and take the function (bit3 (b)) for 119910 gt 485671 Between 119910 isin [452504 485671] wetake weightedmean of two approximation functions (bit 3 (a)and (b)) which is shown in Figure 4(d) Likewise we switchLLR functions for Pade approximation on 481638 494772as shown in Figure 4(e) From Figure 4(c) we can easily seeswitch points for Taylor3 but for Pade approximation wecannot see these points explicitly This is because accuracy ofPade approximation is higher than that of Taylor3 for widerange of variables

Notice that for bit 3 it is not necessary to considerweighted mean of two approximation functions if we usePade approximation of higher order pairs In this case wefound that Pade approximation of orders (4 5) at point1199100= plusmn argmax

119910ge0ℓ(3)

NSI provides almost the same accuracyas the combination of (3 4) and (4 1) which is shown inFigure 4(f) (denoted by ldquoPade45rdquo) But the accuracy of Padeapproximation of (4 5) for large |119910| is not so good comparedwith the combination of Pade approximations (3 4) and(4 1) Also note that the previous work in [11] have modifiedTaylor3 functions to fit the true LLR for 119894 = 2 and 3 But thisis not easy to replicate so we only show the original form ofTaylor3

4 Numerical Results and Discussion

In order to compare the proposed LLR approximations weuse LDPC codes with BP decoding and show results bysimulations and analysis based on the density evolution Weonly show the case where CSI is unknown to the receiver

Table 2 Comparison between the iterative thresholds for LDPCcode ensembles using different LLR calculation methods

(a) Cinfin(1199092 1199095) LDPC code ensemble

True LLR Taylor3 Pade23120590⋆ 0644755 0644754 0644755(1198641198871198730)⋆ [dB] 3810759 3810772 3810759

(b) Cinfin(1199093 11990915) LDPC code ensemble


(c) Cinfin(1199092 1199093) LDPC code ensemble


41 Results for BPSK By using the density evolution [5 7 8]Table 2 shows iterative thresholds of Cinfin(1199092 1199095) Cinfin(119909311990915) and Cinfin(1199092 1199093) LDPC code ensembles whose rates 119877

are respectively 119877 = 05 075 and 025 for different LLRcalculation methods Evaluating preciously the thresholds ofthree methods are almost identical especially for true LLRand Pade23 but there exists a little gap between Taylor3 andthe other methods

Table 3 shows degree distribution profiles and their iter-ative thresholds for (a) threshold optimized and (b) rateoptimized irregular LDPC codes These profiles and theircorresponding thresholds are evaluated based on the trueLLR Taylor3 and Pade23 respectively From Table 3(a) for


Table 3 Degree distribution profiles and their iterative thresholdsfor the uncorrelated flat Rayleigh fading channel with unknown CSIby the density evolution These profiles are (a) threshold optimizedand (b) rate optimized codes The Shannon limit for rate half codeis 120590 = 07436 (119864

1198871198730= 2572 dB)

(a) Threshold optimized

True LLR Taylor3 Pade231205822

0200284000 0200283000 02002840001205823

0228588000 0228591000 02285880001205827

0067795000 0067777920 00677944871205828

0210231000 0210246680 021023161312058230

0292602000 0292601400 02926018501205889

1000000000 1000000000 1000000000Rate 0500000 0500000 0500000120590⋆ 07232423 07232385 07232422(1198641198871198730)⋆ [dB] 27068537 27069429 27068549

(b) Rate optimized


0202860 0202758 02028601205823

0214366 0214369 02143661205827

0157257 0157244 01572581205828

0112911 0112922 011285912058230

0312606 0312607 03126071205889

1000000 1000000 1000000Rate 04946847 04946836 04946845120590⋆ 0740167 0740167 0740167(1198641198871198730)⋆ [dB] 2613406 2613406 2613406

fixed rate the threshold of Pade23 is almost the same as thatof true LLR and is slightly better than that of Taylor3 FromTable 3(b) for fixed threshold the rate of the code by Pade23is almost the same as that by true LLR and is slightly higherthan that by Taylor3 From these thresholds they are close tothe Shannon limit that is for rate half code it is 120590 = 07436(119864119887119873

0= 2572 [dB])

42 Results for 8-PAM Figure 5 shows bit error rate (BER) of(120582(119909) 120588(119909)) = (119909

2 1199093) LDPC codes with 119873 = 15000 for the

uncorrelated flat Rayleigh fading channel with 8-PAM Thenumber of transmitted codewords is 5 times 105 BERs of trueLLR and two Pade approximations (combination of orders (41) and (3 4) and orders (4 5)) are almost the same and thatof Taylor3 is slightly higher than those of three methods

Table 4 shows iterative thresholds of Cinfin(1199092 1199093) LDPCcode ensemble for different LLR calculation methods with8-PAM The threshold of Pade approximation is almostidentical to that of true LLR (inferior to 002 dB) and it isbetter than that of Taylor3 Notice that threshold of Taylor3in [11] was SNR = 786 (dB) and it is better than that of Padeapproximation But the method in [11] performed severalmodifications to fit the true functions for the LLR functionsof bits 2 and 3 it is not easy to replicate the approximationsPade approximation is better than linear approximation in[20] (SNR = 788 (dB)) The threshold of Pade approximation

78 8 82 84 86 88 9 92 94

BER

SNR (dB)

True LLRTaylor3

Pade41 and 34Pade45

10e minus 006

10e minus 005

10e minus 004

10e minus 003

10e minus 002

10e minus 001

10e + 000

Figure 5 Comparison of BER of (1199092 1199093) LDPC codes with 119873 =15000 for the uncorrelated flat Rayleigh fading channel with 8-PAM

Table 4 Comparison between the iterative threshold of Cinfin(1199092 1199093)LDPC code ensembles using different LLR calculation methods on8-PAM

True LLR Taylor3 Pade41 amp Pade34SNR⋆ 785 805 787

of orders (4 5) for bit 3 LLR function is SNR= 788 (dB)whichis the same as the linear approximation

5 Conclusion

In this paper we apply the Pade approximation which isa generalization of the Taylor approximation to the LLRfunction on the uncorrelated flat Rayleigh fading channelswith unknown CSI Using Pade approximation we canaccelerate the accuracy of the LLR function that it is accuratenot only around the derivative point but also at the otherintervals of the variable From the simulation results andanalysis based on the density evolution our method canyield almost the same decoding performance as the trueLLR function and is slightly better than the conventionalapproximation method (Taylor approximation of order 119899 =3) Moreover we derive some irregular LDPC code profileswhose iterative thresholds are close to the Shannon limit

To apply Pade approximation other modulations (eg16-QAM) or other channels (Rician fading) are remained forfurther works

Acknowledgments

The authors are grateful for anonymous reviewers for theirthorough and conscientious reviewing which improves thequality of this paper One of the authors Gou Hosoya wouldlike to thank Dr M Kobayashi at the Shonan Institute of


Technology and Dr H Yagi at the University of Electro-Communications for their valuable comments and discus-sions This research is partly supported by JSPS KAKENHIGrant nos 25820166 and 25420386

References

[1] C Berrou A Glavieux and P Thitimajshima ldquoNear shannonlimit error-correcting coding and decoding turbo-codesrdquo inProceedings of the IEEE International Conference on Commu-nications (ICC rsquo93) pp 1064ndash1070 Geneva Switzerland May1993

[2] R G Gallager Low-Density Parity-Check Codes MIT Press1963

[3] D J C MacKay ldquoGood error-correcting codes based on verysparse matricesrdquo IEEE Transactions on InformationTheory vol45 no 2 pp 399ndash431 1999

[4] T J Richardson and R L Urbanke Modern Coding TheoryCambridge University Press Cambridge UK 2008

[5] S Y Chung On the construction of some capacity-approachingcoding schemes [PhD thesis] MIT Cambridge Mass USA2000

[6] M G Luby M Mitzenmacher M A Shokrollahi and DA Spielman ldquoImproved low-density parity-check codes usingirregular graphsrdquo IEEE Transactions on InformationTheory vol47 no 2 pp 585ndash598 2001

[7] T J Richardson and R L Urbanke ldquoThe capacity of low-density parity-check codes under message-passing decodingrdquoIEEE Transactions on InformationTheory vol 47 no 2 pp 599ndash618 2001

[8] T J Richardson M A Shokrollahi and R L UrbankeldquoDesign of capacity-approaching irregular low-density parity-check codesrdquo IEEE Transactions on Information Theory vol 47no 2 pp 619ndash637 2001

[9] J G Proakis and M Salehi Digital Communications McGraw-Hill 5th edition 2005

[10] A Guillen i Fabregas A Martinez and G Caire ldquoBit-inter-leaved coded modulationrdquo Foundations and Trends in Commu-nications and InformationTheory vol 5 no 1-2 pp 1ndash153 2008

[11] R Asvadi A H Banihashemi M Ahmadian-Attari and HSaeedi ldquoLLR approximation for wireless channels based onTaylor series and its application to BICM with LDPC codesrdquoIEEE Transactions on Communications vol 60 no 5 pp 1226ndash1236 2012

[12] E K Hall and S GWilson ldquoDesign and analysis of turbo codeson Rayleigh fading channelsrdquo IEEE Journal on Selected Areas inCommunications vol 16 no 2 pp 160ndash174 1998

[13] G Hosoya M Hasegawa and H Yashima ldquoLLR calculationfor iterative decoding on fading channels using Pade approx-imationrdquo in Proceedings of the International Conference onWireless Communications and Signal Processing (WCSP rsquo12)CTS 1569651991 pp 1ndash6 Huanshan China October 2012

[14] J Hou P H Siegel and L B Milstein ldquoPerformance analysisand code optimization of low density parity-check codes onRayleigh fading channelsrdquo IEEE Journal on Selected Areas inCommunications vol 19 no 5 pp 924ndash934 2001

[15] J Hou P H Siegel L B Milstein and H D Pfister ldquoCapacity-approaching bandwidth-efficient coded modulation schemesbased on low-density parity-check codesrdquo IEEE Transactions onInformation Theory vol 49 no 9 pp 2141ndash2155 2003

[16] N Inaba H Yashima T Kuroda and T Tsubouchi ldquoErrorperformances of convolutional coded modulation using themaximum likelihood metric and an approximated metric overRayleigh fading channelrdquo IEICE Transactions on Fundamentalsof Electronics vol J78-A no 10 pp 1397ndash1399 1995 (Japanese)

[17] N Iviyani and J Weber ldquoAnalysis of random regular LDPCcodes on Rayleigh fading channelsrdquo in Proceedings of the 27thSymposium on Information Theory in the Benelux Noordwijkthe Netherlands June 2006

[18] J Lin andWWuPerformanceAnalysis of LDPCCodes onRicianFading Channels Higher Education Press and Springer 2006

[19] R Yazdani and M Ardakani ldquoLinear LLR approximation foriterative decoding on wireless channelsrdquo IEEE Transactions onCommunications vol 57 no 11 pp 3278ndash3287 2009

[20] R Yazdani andMArdakani ldquoEfficient LLR calculation for non-binarymodulations over fading channelsrdquo IEEETransactions onCommunications vol 59 no 5 pp 1236ndash1241 2011

[21] G Caire G Taricco and E Biglieri ldquoBit-interleaved codedmodulationrdquo IEEE Transactions on InformationTheory vol 44no 3 pp 927ndash946 1998

[22] C Brezinski History of Continued Fractions and Pade Approxi-mants vol 12 Springer Berlin Germany 1991

Submit your manuscripts athttpwwwhindawicom

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


But they have used the following simple linear approximationℓex for a calculation of the channel LLR

ℓex =2

1205902sdot 119864 (119903) sdot 119910 (1)

where 119864(119903) denotes the expectation of the channel gain 119903Although the previous approximation is simple and is easy toimplement it is inaccurate that degradation in the decodingperformance compared with the true LLR [19] can be seenYazdani and Ardakani [19 20] have also proposed a linearLLR approximation whose performance is almost identical tothe true LLR

ℓopt = 119910 (2)

where is obtained by maximizing

= argmax120572(1 minus int

infin

minusinfin

log2(1 + exp (minusℓ)) 119891

120572(ℓ) 119889ℓ ) (3)

Here 119891120572(ℓ) is given by

119891120572(ℓ)

=radic2Δ2

radic120587120572120590exp(minus(Δℓ

120572120590)

2

)

times [exp(minus( Δℓ

radic21205721205902)

2

) +Δradic120587ℓ

radic21205721205902sdot erfc(minus Δℓ

radic21205721205902)]

(4)

where Δ = radic1205902(21205902 + 1) and erfc(sdot) denotes the comple-mentary error function [9] However an optimization of using (3) and (4) requires for each channel parameter 120590 sothat it needs large complexity to implement

Recently Asvadi et al [11] have applied Taylor approxima-tion of order 119899 to the true LLR function such that

ℓT119899 =119899

sum

119894=0

119888119894119910119894 (5)

where 119888119894denotes the coefficient of Taylor series of order 119894

They have derived both linear and nonlinear approximationswith small orders For a linear approximation (Taylor seriesof order 119899 = 1) it is given by

ℓT1 = 1198881119910 (6)

For a nonlinear approximation (Taylor series of order 119899 = 3)it is given by

ℓT3 = 1198881119910 + 11988831199103 (7)

From the previous approximations one can obtain accurateLLRwithout optimizing complicated functions such as in [1920]

Tomove our attentions to nonbinarymodulations whichis more practical case the LLR calculations are performedbitwise [15 21] The previous work by Yazdani and Ardakani[20] which is an extension of [19] has devised the LLR

approximation method but it becomes complex to evaluateLLR due to the increment of the number of parameters forthe optimization To fit the true LLR functions the authorsin [11] have modified the approximation functions of Taylorseries of order 3Thismodification is not easy to replicate andis required for each parameter of the channels Moreover it iswell known that the Taylor approximation quickly becomesinaccurate as the variable 119910 leaves the derivative point

To overcome these problems we devise a new LLRapproximation using Pade approximation [22] on the uncor-related flat Rayleigh fading channels with unknown CSIfor BPSK and 8-PAM Pade approximation expresses theoriginal function by rational form of two polynomials withthe same total number of coefficients of the Taylor seriesand it can accelerate the Taylor approximation GenerallyPade approximation is accurate not only at the derivativepoint but also at the wide range of intervals of variables Weshow by simulation results and analysis based on the densityevolution that our method can approximate LLR functionwith high accuracy and can yield almost the same decodingperformance as the true LLR The Pade approximation is ageneralization of the Taylor approximation and the proposedmethod exhibits slightly better performance than themethodusing Taylor approximation [11] Moreover we design irregu-lar LDPC codes based on our LLR approximation function

This paper is organized as follows Section 2 gives thechannelmodel LLR calculationmethod and LDPC codes InSection 3 we briefly review Taylor and Pade approximationsand then we present the proposed LLR calculation by PadeapproximationNumerical results are shown in Section 4 andSection 5 concludes the paper

2 Preliminaries

21 Channel Model and LLR Calculation We here considerthe following discrete-time channel model

119910 = 119903119908 + 119911 (8)

where 119908 isin 120594 and 119910 119911 isin R represent the channel inputoutput and noise respectively and 120594 R denote a set oftransmitted symbols and that of real numbers respectively(As mentioned in Section 1 we drop a time subscript for119910 119903 119908 and 119911) Moreover 119903 ge 0 is the channel gain withan uncorrelated flat Rayleigh distribution by its probabilitydensity function (pdf) 119875(119903) = 2119903119890minus119903

2

and 119911 is the whiteGaussian noise with mean 0 and variance 1205902

Using the bit-interleaved coded modulation (BICM)scheme [10 15 21] for a transmission an information bitsequence is mapped to the codeword (bit sequence) of length119873 by error-correcting codes and then it is partitioned into119873119902 blocks denoted by b

119896= 119887

(1)

119896 119887(2)

119896 119887

(119902)

119896 119896 =

1 2 119873119902 of length 119902 Hereinafter we drop the subscript119896 of both b

119896and 119887(119894)119896

to simplify discussions This block b ismapped to transmit a signal 119908 isin 120594 in a Gray-labeled 119876-arysignal constellation 120594 of size 119876 = 2119902 The signal constellationfor BSPK and 8-PAM is depicted in Figure 1

For the fading channel two cases can be consideredwhichdepends on the knowledge of 119903 at the receiver


BPSK

w

minus1 1

0

b

8PAM

111 110 100 101 001 000 010 011


1

w

b



ℓ(119894)

CSI = log119875 (119910 | 119887

(119894)(119908) = 0 119903)

119875 (119910 | 119887(119894) (119908) = 1 119903)

= logsum119908isin120594(119894)

0

119875 (119910 | 119908 119903)

sum119908isin120594(119894)

1

119875 (119910 | 119908 119903)

(9)




119875 (119910 | 119908 119903) =1


(119910 minus 119903119908)2

21205902) (10)


ℓ(1)

CSI =2119903119910

1205902 (11)



ℓ(119894)

NSI = log119875 (119910 | 119887

(119894)(119908) = 0)

119875 (119910 | 119887(119894) (119908) = 1)

= logsum119909isin120594(119894)

0

119875 (119910 | 119908)

sum119908isin120594(119894)

1

119875 (119910 | 119908)

(12)

where 119875(119910 | 119908) is given by

119875 (119910 | 119908)

= int

infin

0

119875 (119910 | 119908 119903) 119875 (119903) 119889119903

= int

infin

0

1


(119910 minus 119903119908)2

21205902) times 2119903 exp (minus1199032) 119889119903

(13)


ℓ(1)





119875 (119910 | 119908)

=exp (minus11991022

1)

radic1205873

1


radic21205901


11990821199102

2120590221

))

(15)





119894=2120582119894119909119894minus1 and 120588(119909) =

sum119889119888





119873int1

0120588(119909)119889119909 int

1

0120582(119909)119889119909



119905rarrinfin119875119890(119905) = 0 where 119875






⋆)2 (for BPSK)


0





1 1199042] and (119904




1 1199042) where


0)

119891(2)(1199100) 119891


1 1199042]



119875119899(119910) = 119891 (119910

0) + 119891(1)(1199100) (119910 minus 119910

0) + sdot sdot sdot

+119891(119899)(1199100)

119899(119910 minus 119910

0)119899

=

119899

sum

119896=0

119891(119896)(1199100)

119896(119910 minus 119910

0)119896

(16)


119891(119899+1)(120585)

(119899 + 1)(119910 minus 119910

0)119899+1

(17)






119875119899(119910) =

119899

sum

119896=0

119891(119896)(1199100)

119896(119910 minus 119910

0)119896

=

119899

sum

119896=0

119888119896119910119896

(18)


1198751198991015840

1198981015840 (119910) =

sum1198991015840

119896=0119886119896119910119896

sum1198981015840

119896=0119887119896119910119896 (19)

where 1198860 1198861 119886

1198991015840 and 119887

0 1198871 119887



1198981015840(119910) in (19) are



119896=0119886119896119910119896sum119898

1015840

119896=0119887119896119910119896


1198991015840

sum

119896=0

119886119896119910119896minus (

1198981015840

sum

119896=0

119887119896119910119896) sdot (

119899

sum

119896=0

119888119896119910119896) = 119874(119910

119899+1) (20)


of sum1198991015840

119896=0119886119896119910119896 and (sum119898

1015840

119896=0119887119896119910119896) (sum119899



119894

sum

119895=max(119894minus1198981015840 0)119888119895119887119894minus119895= 119886119894 119894 = 0 1 119899

1015840

0 119894 = 1198991015840+ 1 1198991015840+ 2 119899

(21)



1198991015840 and 119887

0 1198871 119887

1198981015840


119899in


1198991015840

and 1198870 1198871 119887


0= 1 This


0 1198881 119888

119899and 119887

0=


1198991015840 and

1198870 1198871 119887







minus30

minus20

minus10

0

10

20

30

LLR

True LLRTaylor3

Pade23Taylor5

y

(a) LLR calculation

0 1 2 3 4 5 6 7 8


100

10minus15

10minus10

10minus5

|Tru

e LLR

appr

oxim

atio

n|

y




ℓP23 =1198860+ 1198861119910 + 11988621199102+ 11988631199103

1198870+ 1198871119910 + 11988721199102

(22)

where 1198860= 0 119886

2= 0 1198870= 1 and 119887

1= 0

1198861= radic2120587

119887

2=minus35 + 30120587 minus 6120587

2

20 (minus3 + 120587)

1198863= minusradic

120587

2sdot15 minus 30120587 + 8120587

2

30 (minus3 + 120587) radic

(23)


ℓT5 = 1198881119910 + 11988831199103+ 11988851199105 (24)

where

1198881= radic2120587

119888

3=radic120587 (minus3 + 120587)

3radic2

1198885=

radic2120587 (35 minus 30120587 + 61205872)

1202radic

(25)

Then (21) becomes

11988801198870= 1198860

11988801198871+ 11988811198870= 1198861

11988801198872+ 11988811198871+ 11988821198870= 1198862

11988811198872+ 11988821198871+ 11988831198870= 1198863

11988821198872+ 11988831198871+ 11988841198870= 0

11988831198872+ 11988841198871+ 11988851198870= 0

(26)

Substituting 1198880 1198881 119888

5in (24) and 119887


1198860 1198861 1198862 1198863 and 119887

1 1198872







minus30

minus20

minus10

0

10

20

30

40LL

R

True LLRTaylor3

Pade23Taylor5

y

(a) LLR calculation

0 5 10 15


100

10minus6

10minus4

10minus2

|Tru

e LLR

appr

oxim

atio

n|

y





119875 (119910 | 119908 = +1 119903) =1


(119910 minus 119903)2

21205902) (27)


119875 (119910 | 119908 = +1)

=radic2


1199102

1 + 21205902)


2) +119910

2radic2120587


119910

radic2))

(28)


(ℓP23119875 (119910 | 119908 = +1)

ℓ1015840P23) (29)





0for each











0=




minus30

minus20

minus10

0

10

20

30LL

R


y

(a) Bit 1


minus12

minus10

minus8

minus6

minus4

minus2

0

2

LLR


y

(b) Bit 2


minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

LLR


y

(c) Bit 3

4 45 5 55

05

06

07

08

09

1LL

R

True LLRTaylor3

Weighted mean

Switch point at


y = 485671

Taylor3 (y0 = 18848)Taylor3 (y0 = 69832)


47 475 48 485 49 495 5

066

067

068

069

07

071

072

073

LLR


Weighted mean

y


Pade41 (y0 = 18848)Pade34 (y0 = 69832)


0 5 10 15minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

LLR


y

(f) Bit 3 (Pade45)





1199100

1198880

1198881

1198882

1198883



1199100

1198860

1198861

1198862

1198863

1198864

1198865

1198866

1198867

bit 1 00 00 12135 00 00334 00 00025 00 261119864 minus 05



1199100

1198870

1198871

1198872

1198873

1198874




119910ge0ℓ(3)
















1198871198730= 2572 dB)



0200284000 0200283000 02002840001205823

0228588000 0228591000 02285880001205827

0067795000 0067777920 00677944871205828

0210231000 0210246680 021023161312058230

0292602000 0292601400 02926018501205889

1000000000 1000000000 1000000000Rate 0500000 0500000 0500000120590⋆ 07232423 07232385 07232422(1198641198871198730)⋆ [dB] 27068537 27069429 27068549

(b) Rate optimized


0202860 0202758 02028601205823

0214366 0214369 02143661205827

0157257 0157244 01572581205828

0112911 0112922 011285912058230

0312606 0312607 03126071205889

1000000 1000000 1000000Rate 04946847 04946836 04946845120590⋆ 0740167 0740167 0740167(1198641198871198730)⋆ [dB] 2613406 2613406 2613406


0= 2572 [dB])





78 8 82 84 86 88 9 92 94

BER

SNR (dB)

True LLRTaylor3

Pade41 and 34Pade45

10e minus 006

10e minus 005

10e minus 004

10e minus 003

10e minus 002

10e minus 001

10e + 000





5 Conclusion



Acknowledgments




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










BPSK

w

minus1 1

0

b

8PAM

111 110 100 101 001 000 010 011


1

w

b



ℓ(119894)

CSI = log119875 (119910 | 119887

(119894)(119908) = 0 119903)

119875 (119910 | 119887(119894) (119908) = 1 119903)

= logsum119908isin120594(119894)

0

119875 (119910 | 119908 119903)

sum119908isin120594(119894)

1

119875 (119910 | 119908 119903)

(9)




119875 (119910 | 119908 119903) =1


(119910 minus 119903119908)2

21205902) (10)


ℓ(1)

CSI =2119903119910

1205902 (11)



ℓ(119894)

NSI = log119875 (119910 | 119887

(119894)(119908) = 0)

119875 (119910 | 119887(119894) (119908) = 1)

= logsum119909isin120594(119894)

0

119875 (119910 | 119908)

sum119908isin120594(119894)

1

119875 (119910 | 119908)

(12)

where 119875(119910 | 119908) is given by

119875 (119910 | 119908)

= int

infin

0

119875 (119910 | 119908 119903) 119875 (119903) 119889119903

= int

infin

0

1


(119910 minus 119903119908)2

21205902) times 2119903 exp (minus1199032) 119889119903

(13)


ℓ(1)





119875 (119910 | 119908)

=exp (minus11991022

1)

radic1205873

1


radic21205901


11990821199102

2120590221

))

(15)





119894=2120582119894119909119894minus1 and 120588(119909) =

sum119889119888





119873int1

0120588(119909)119889119909 int

1

0120582(119909)119889119909



119905rarrinfin119875119890(119905) = 0 where 119875






⋆)2 (for BPSK)


0





1 1199042] and (119904




1 1199042) where


0)

119891(2)(1199100) 119891


1 1199042]



119875119899(119910) = 119891 (119910

0) + 119891(1)(1199100) (119910 minus 119910

0) + sdot sdot sdot

+119891(119899)(1199100)

119899(119910 minus 119910

0)119899

=

119899

sum

119896=0

119891(119896)(1199100)

119896(119910 minus 119910

0)119896

(16)


119891(119899+1)(120585)

(119899 + 1)(119910 minus 119910

0)119899+1

(17)






119875119899(119910) =

119899

sum

119896=0

119891(119896)(1199100)

119896(119910 minus 119910

0)119896

=

119899

sum

119896=0

119888119896119910119896

(18)


1198751198991015840

1198981015840 (119910) =

sum1198991015840

119896=0119886119896119910119896

sum1198981015840

119896=0119887119896119910119896 (19)

where 1198860 1198861 119886

1198991015840 and 119887

0 1198871 119887



1198981015840(119910) in (19) are



119896=0119886119896119910119896sum119898

1015840

119896=0119887119896119910119896


1198991015840

sum

119896=0

119886119896119910119896minus (

1198981015840

sum

119896=0

119887119896119910119896) sdot (

119899

sum

119896=0

119888119896119910119896) = 119874(119910

119899+1) (20)


of sum1198991015840

119896=0119886119896119910119896 and (sum119898

1015840

119896=0119887119896119910119896) (sum119899



119894

sum

119895=max(119894minus1198981015840 0)119888119895119887119894minus119895= 119886119894 119894 = 0 1 119899

1015840

0 119894 = 1198991015840+ 1 1198991015840+ 2 119899

(21)



1198991015840 and 119887

0 1198871 119887

1198981015840


119899in


1198991015840

and 1198870 1198871 119887


0= 1 This


0 1198881 119888

119899and 119887

0=


1198991015840 and

1198870 1198871 119887







minus30

minus20

minus10

0

10

20

30

LLR

True LLRTaylor3

Pade23Taylor5

y

(a) LLR calculation

0 1 2 3 4 5 6 7 8


100

10minus15

10minus10

10minus5

|Tru

e LLR

appr

oxim

atio

n|

y




ℓP23 =1198860+ 1198861119910 + 11988621199102+ 11988631199103

1198870+ 1198871119910 + 11988721199102

(22)

where 1198860= 0 119886

2= 0 1198870= 1 and 119887

1= 0

1198861= radic2120587

119887

2=minus35 + 30120587 minus 6120587

2

20 (minus3 + 120587)

1198863= minusradic

120587

2sdot15 minus 30120587 + 8120587

2

30 (minus3 + 120587) radic

(23)


ℓT5 = 1198881119910 + 11988831199103+ 11988851199105 (24)

where

1198881= radic2120587

119888

3=radic120587 (minus3 + 120587)

3radic2

1198885=

radic2120587 (35 minus 30120587 + 61205872)

1202radic

(25)

Then (21) becomes

11988801198870= 1198860

11988801198871+ 11988811198870= 1198861

11988801198872+ 11988811198871+ 11988821198870= 1198862

11988811198872+ 11988821198871+ 11988831198870= 1198863

11988821198872+ 11988831198871+ 11988841198870= 0

11988831198872+ 11988841198871+ 11988851198870= 0

(26)

Substituting 1198880 1198881 119888

5in (24) and 119887


1198860 1198861 1198862 1198863 and 119887

1 1198872







minus30

minus20

minus10

0

10

20

30

40LL

R

True LLRTaylor3

Pade23Taylor5

y

(a) LLR calculation

0 5 10 15


100

10minus6

10minus4

10minus2

|Tru

e LLR

appr

oxim

atio

n|

y





119875 (119910 | 119908 = +1 119903) =1


(119910 minus 119903)2

21205902) (27)


119875 (119910 | 119908 = +1)

=radic2


1199102

1 + 21205902)


2) +119910

2radic2120587


119910

radic2))

(28)


(ℓP23119875 (119910 | 119908 = +1)

ℓ1015840P23) (29)





0for each











0=




minus30

minus20

minus10

0

10

20

30LL

R


y

(a) Bit 1


minus12

minus10

minus8

minus6

minus4

minus2

0

2

LLR


y

(b) Bit 2


minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

LLR


y

(c) Bit 3

4 45 5 55

05

06

07

08

09

1LL

R

True LLRTaylor3

Weighted mean

Switch point at


y = 485671

Taylor3 (y0 = 18848)Taylor3 (y0 = 69832)


47 475 48 485 49 495 5

066

067

068

069

07

071

072

073

LLR


Weighted mean

y


Pade41 (y0 = 18848)Pade34 (y0 = 69832)


0 5 10 15minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

LLR


y

(f) Bit 3 (Pade45)





1199100

1198880

1198881

1198882

1198883



1199100

1198860

1198861

1198862

1198863

1198864

1198865

1198866

1198867

bit 1 00 00 12135 00 00334 00 00025 00 261119864 minus 05



1199100

1198870

1198871

1198872

1198873

1198874




119910ge0ℓ(3)
















1198871198730= 2572 dB)



0200284000 0200283000 02002840001205823

0228588000 0228591000 02285880001205827

0067795000 0067777920 00677944871205828

0210231000 0210246680 021023161312058230

0292602000 0292601400 02926018501205889

1000000000 1000000000 1000000000Rate 0500000 0500000 0500000120590⋆ 07232423 07232385 07232422(1198641198871198730)⋆ [dB] 27068537 27069429 27068549

(b) Rate optimized


0202860 0202758 02028601205823

0214366 0214369 02143661205827

0157257 0157244 01572581205828

0112911 0112922 011285912058230

0312606 0312607 03126071205889

1000000 1000000 1000000Rate 04946847 04946836 04946845120590⋆ 0740167 0740167 0740167(1198641198871198730)⋆ [dB] 2613406 2613406 2613406


0= 2572 [dB])





78 8 82 84 86 88 9 92 94

BER

SNR (dB)

True LLRTaylor3

Pade41 and 34Pade45

10e minus 006

10e minus 005

10e minus 004

10e minus 003

10e minus 002

10e minus 001

10e + 000





5 Conclusion



Acknowledgments




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119905rarrinfin119875119890(119905) = 0 where 119875






⋆)2 (for BPSK)


0





1 1199042] and (119904




1 1199042) where


0)

119891(2)(1199100) 119891


1 1199042]



119875119899(119910) = 119891 (119910

0) + 119891(1)(1199100) (119910 minus 119910

0) + sdot sdot sdot

+119891(119899)(1199100)

119899(119910 minus 119910

0)119899

=

119899

sum

119896=0

119891(119896)(1199100)

119896(119910 minus 119910

0)119896

(16)


119891(119899+1)(120585)

(119899 + 1)(119910 minus 119910

0)119899+1

(17)






119875119899(119910) =

119899

sum

119896=0

119891(119896)(1199100)

119896(119910 minus 119910

0)119896

=

119899

sum

119896=0

119888119896119910119896

(18)


1198751198991015840

1198981015840 (119910) =

sum1198991015840

119896=0119886119896119910119896

sum1198981015840

119896=0119887119896119910119896 (19)

where 1198860 1198861 119886

1198991015840 and 119887

0 1198871 119887



1198981015840(119910) in (19) are



119896=0119886119896119910119896sum119898

1015840

119896=0119887119896119910119896


1198991015840

sum

119896=0

119886119896119910119896minus (

1198981015840

sum

119896=0

119887119896119910119896) sdot (

119899

sum

119896=0

119888119896119910119896) = 119874(119910

119899+1) (20)


of sum1198991015840

119896=0119886119896119910119896 and (sum119898

1015840

119896=0119887119896119910119896) (sum119899



119894

sum

119895=max(119894minus1198981015840 0)119888119895119887119894minus119895= 119886119894 119894 = 0 1 119899

1015840

0 119894 = 1198991015840+ 1 1198991015840+ 2 119899

(21)



1198991015840 and 119887

0 1198871 119887

1198981015840


119899in


1198991015840

and 1198870 1198871 119887


0= 1 This


0 1198881 119888

119899and 119887

0=


1198991015840 and

1198870 1198871 119887







minus30

minus20

minus10

0

10

20

30

LLR

True LLRTaylor3

Pade23Taylor5

y

(a) LLR calculation

0 1 2 3 4 5 6 7 8


100

10minus15

10minus10

10minus5

|Tru

e LLR

appr

oxim

atio

n|

y




ℓP23 =1198860+ 1198861119910 + 11988621199102+ 11988631199103

1198870+ 1198871119910 + 11988721199102

(22)

where 1198860= 0 119886

2= 0 1198870= 1 and 119887

1= 0

1198861= radic2120587

119887

2=minus35 + 30120587 minus 6120587

2

20 (minus3 + 120587)

1198863= minusradic

120587

2sdot15 minus 30120587 + 8120587

2

30 (minus3 + 120587) radic

(23)


ℓT5 = 1198881119910 + 11988831199103+ 11988851199105 (24)

where

1198881= radic2120587

119888

3=radic120587 (minus3 + 120587)

3radic2

1198885=

radic2120587 (35 minus 30120587 + 61205872)

1202radic

(25)

Then (21) becomes

11988801198870= 1198860

11988801198871+ 11988811198870= 1198861

11988801198872+ 11988811198871+ 11988821198870= 1198862

11988811198872+ 11988821198871+ 11988831198870= 1198863

11988821198872+ 11988831198871+ 11988841198870= 0

11988831198872+ 11988841198871+ 11988851198870= 0

(26)

Substituting 1198880 1198881 119888

5in (24) and 119887


1198860 1198861 1198862 1198863 and 119887

1 1198872







minus30

minus20

minus10

0

10

20

30

40LL

R

True LLRTaylor3

Pade23Taylor5

y

(a) LLR calculation

0 5 10 15


100

10minus6

10minus4

10minus2

|Tru

e LLR

appr

oxim

atio

n|

y





119875 (119910 | 119908 = +1 119903) =1


(119910 minus 119903)2

21205902) (27)


119875 (119910 | 119908 = +1)

=radic2


1199102

1 + 21205902)


2) +119910

2radic2120587


119910

radic2))

(28)


(ℓP23119875 (119910 | 119908 = +1)

ℓ1015840P23) (29)





0for each











0=




minus30

minus20

minus10

0

10

20

30LL

R


y

(a) Bit 1


minus12

minus10

minus8

minus6

minus4

minus2

0

2

LLR


y

(b) Bit 2


minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

LLR


y

(c) Bit 3

4 45 5 55

05

06

07

08

09

1LL

R

True LLRTaylor3

Weighted mean

Switch point at


y = 485671

Taylor3 (y0 = 18848)Taylor3 (y0 = 69832)


47 475 48 485 49 495 5

066

067

068

069

07

071

072

073

LLR


Weighted mean

y


Pade41 (y0 = 18848)Pade34 (y0 = 69832)


0 5 10 15minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

LLR


y

(f) Bit 3 (Pade45)





1199100

1198880

1198881

1198882

1198883



1199100

1198860

1198861

1198862

1198863

1198864

1198865

1198866

1198867

bit 1 00 00 12135 00 00334 00 00025 00 261119864 minus 05



1199100

1198870

1198871

1198872

1198873

1198874




119910ge0ℓ(3)
















1198871198730= 2572 dB)



0200284000 0200283000 02002840001205823

0228588000 0228591000 02285880001205827

0067795000 0067777920 00677944871205828

0210231000 0210246680 021023161312058230

0292602000 0292601400 02926018501205889

1000000000 1000000000 1000000000Rate 0500000 0500000 0500000120590⋆ 07232423 07232385 07232422(1198641198871198730)⋆ [dB] 27068537 27069429 27068549

(b) Rate optimized


0202860 0202758 02028601205823

0214366 0214369 02143661205827

0157257 0157244 01572581205828

0112911 0112922 011285912058230

0312606 0312607 03126071205889

1000000 1000000 1000000Rate 04946847 04946836 04946845120590⋆ 0740167 0740167 0740167(1198641198871198730)⋆ [dB] 2613406 2613406 2613406


0= 2572 [dB])





78 8 82 84 86 88 9 92 94

BER

SNR (dB)

True LLRTaylor3

Pade41 and 34Pade45

10e minus 006

10e minus 005

10e minus 004

10e minus 003

10e minus 002

10e minus 001

10e + 000





5 Conclusion



Acknowledgments




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











minus30

minus20

minus10

0

10

20

30

LLR

True LLRTaylor3

Pade23Taylor5

y

(a) LLR calculation

0 1 2 3 4 5 6 7 8


100

10minus15

10minus10

10minus5

|Tru

e LLR

appr

oxim

atio

n|

y




ℓP23 =1198860+ 1198861119910 + 11988621199102+ 11988631199103

1198870+ 1198871119910 + 11988721199102

(22)

where 1198860= 0 119886

2= 0 1198870= 1 and 119887

1= 0

1198861= radic2120587

119887

2=minus35 + 30120587 minus 6120587

2

20 (minus3 + 120587)

1198863= minusradic

120587

2sdot15 minus 30120587 + 8120587

2

30 (minus3 + 120587) radic

(23)


ℓT5 = 1198881119910 + 11988831199103+ 11988851199105 (24)

where

1198881= radic2120587

119888

3=radic120587 (minus3 + 120587)

3radic2

1198885=

radic2120587 (35 minus 30120587 + 61205872)

1202radic

(25)

Then (21) becomes

11988801198870= 1198860

11988801198871+ 11988811198870= 1198861

11988801198872+ 11988811198871+ 11988821198870= 1198862

11988811198872+ 11988821198871+ 11988831198870= 1198863

11988821198872+ 11988831198871+ 11988841198870= 0

11988831198872+ 11988841198871+ 11988851198870= 0

(26)

Substituting 1198880 1198881 119888

5in (24) and 119887


1198860 1198861 1198862 1198863 and 119887

1 1198872







minus30

minus20

minus10

0

10

20

30

40LL

R

True LLRTaylor3

Pade23Taylor5

y

(a) LLR calculation

0 5 10 15


100

10minus6

10minus4

10minus2

|Tru

e LLR

appr

oxim

atio

n|

y





119875 (119910 | 119908 = +1 119903) =1


(119910 minus 119903)2

21205902) (27)


119875 (119910 | 119908 = +1)

=radic2


1199102

1 + 21205902)


2) +119910

2radic2120587


119910

radic2))

(28)


(ℓP23119875 (119910 | 119908 = +1)

ℓ1015840P23) (29)





0for each











0=




minus30

minus20

minus10

0

10

20

30LL

R


y

(a) Bit 1


minus12

minus10

minus8

minus6

minus4

minus2

0

2

LLR


y

(b) Bit 2


minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

LLR


y

(c) Bit 3

4 45 5 55

05

06

07

08

09

1LL

R

True LLRTaylor3

Weighted mean

Switch point at


y = 485671

Taylor3 (y0 = 18848)Taylor3 (y0 = 69832)


47 475 48 485 49 495 5

066

067

068

069

07

071

072

073

LLR


Weighted mean

y


Pade41 (y0 = 18848)Pade34 (y0 = 69832)


0 5 10 15minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

LLR


y

(f) Bit 3 (Pade45)





1199100

1198880

1198881

1198882

1198883



1199100

1198860

1198861

1198862

1198863

1198864

1198865

1198866

1198867

bit 1 00 00 12135 00 00334 00 00025 00 261119864 minus 05



1199100

1198870

1198871

1198872

1198873

1198874




119910ge0ℓ(3)
















1198871198730= 2572 dB)



0200284000 0200283000 02002840001205823

0228588000 0228591000 02285880001205827

0067795000 0067777920 00677944871205828

0210231000 0210246680 021023161312058230

0292602000 0292601400 02926018501205889

1000000000 1000000000 1000000000Rate 0500000 0500000 0500000120590⋆ 07232423 07232385 07232422(1198641198871198730)⋆ [dB] 27068537 27069429 27068549

(b) Rate optimized


0202860 0202758 02028601205823

0214366 0214369 02143661205827

0157257 0157244 01572581205828

0112911 0112922 011285912058230

0312606 0312607 03126071205889

1000000 1000000 1000000Rate 04946847 04946836 04946845120590⋆ 0740167 0740167 0740167(1198641198871198730)⋆ [dB] 2613406 2613406 2613406


0= 2572 [dB])





78 8 82 84 86 88 9 92 94

BER

SNR (dB)

True LLRTaylor3

Pade41 and 34Pade45

10e minus 006

10e minus 005

10e minus 004

10e minus 003

10e minus 002

10e minus 001

10e + 000





5 Conclusion



Acknowledgments




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











minus30

minus20

minus10

0

10

20

30

40LL

R

True LLRTaylor3

Pade23Taylor5

y

(a) LLR calculation

0 5 10 15


100

10minus6

10minus4

10minus2

|Tru

e LLR

appr

oxim

atio

n|

y





119875 (119910 | 119908 = +1 119903) =1


(119910 minus 119903)2

21205902) (27)


119875 (119910 | 119908 = +1)

=radic2


1199102

1 + 21205902)


2) +119910

2radic2120587


119910

radic2))

(28)


(ℓP23119875 (119910 | 119908 = +1)

ℓ1015840P23) (29)





0for each











0=




minus30

minus20

minus10

0

10

20

30LL

R


y

(a) Bit 1


minus12

minus10

minus8

minus6

minus4

minus2

0

2

LLR


y

(b) Bit 2


minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

LLR


y

(c) Bit 3

4 45 5 55

05

06

07

08

09

1LL

R

True LLRTaylor3

Weighted mean

Switch point at


y = 485671

Taylor3 (y0 = 18848)Taylor3 (y0 = 69832)


47 475 48 485 49 495 5

066

067

068

069

07

071

072

073

LLR


Weighted mean

y


Pade41 (y0 = 18848)Pade34 (y0 = 69832)


0 5 10 15minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

LLR


y

(f) Bit 3 (Pade45)





1199100

1198880

1198881

1198882

1198883



1199100

1198860

1198861

1198862

1198863

1198864

1198865

1198866

1198867

bit 1 00 00 12135 00 00334 00 00025 00 261119864 minus 05



1199100

1198870

1198871

1198872

1198873

1198874




119910ge0ℓ(3)
















1198871198730= 2572 dB)



0200284000 0200283000 02002840001205823

0228588000 0228591000 02285880001205827

0067795000 0067777920 00677944871205828

0210231000 0210246680 021023161312058230

0292602000 0292601400 02926018501205889

1000000000 1000000000 1000000000Rate 0500000 0500000 0500000120590⋆ 07232423 07232385 07232422(1198641198871198730)⋆ [dB] 27068537 27069429 27068549

(b) Rate optimized


0202860 0202758 02028601205823

0214366 0214369 02143661205827

0157257 0157244 01572581205828

0112911 0112922 011285912058230

0312606 0312607 03126071205889

1000000 1000000 1000000Rate 04946847 04946836 04946845120590⋆ 0740167 0740167 0740167(1198641198871198730)⋆ [dB] 2613406 2613406 2613406


0= 2572 [dB])





78 8 82 84 86 88 9 92 94

BER

SNR (dB)

True LLRTaylor3

Pade41 and 34Pade45

10e minus 006

10e minus 005

10e minus 004

10e minus 003

10e minus 002

10e minus 001

10e + 000





5 Conclusion



Acknowledgments




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











minus30

minus20

minus10

0

10

20

30LL

R


y

(a) Bit 1


minus12

minus10

minus8

minus6

minus4

minus2

0

2

LLR


y

(b) Bit 2


minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

LLR


y

(c) Bit 3

4 45 5 55

05

06

07

08

09

1LL

R

True LLRTaylor3

Weighted mean

Switch point at


y = 485671

Taylor3 (y0 = 18848)Taylor3 (y0 = 69832)


47 475 48 485 49 495 5

066

067

068

069

07

071

072

073

LLR


Weighted mean

y


Pade41 (y0 = 18848)Pade34 (y0 = 69832)


0 5 10 15minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

1

LLR


y

(f) Bit 3 (Pade45)





1199100

1198880

1198881

1198882

1198883



1199100

1198860

1198861

1198862

1198863

1198864

1198865

1198866

1198867

bit 1 00 00 12135 00 00334 00 00025 00 261119864 minus 05



1199100

1198870

1198871

1198872

1198873

1198874




119910ge0ℓ(3)
















1198871198730= 2572 dB)



0200284000 0200283000 02002840001205823

0228588000 0228591000 02285880001205827

0067795000 0067777920 00677944871205828

0210231000 0210246680 021023161312058230

0292602000 0292601400 02926018501205889

1000000000 1000000000 1000000000Rate 0500000 0500000 0500000120590⋆ 07232423 07232385 07232422(1198641198871198730)⋆ [dB] 27068537 27069429 27068549

(b) Rate optimized


0202860 0202758 02028601205823

0214366 0214369 02143661205827

0157257 0157244 01572581205828

0112911 0112922 011285912058230

0312606 0312607 03126071205889

1000000 1000000 1000000Rate 04946847 04946836 04946845120590⋆ 0740167 0740167 0740167(1198641198871198730)⋆ [dB] 2613406 2613406 2613406


0= 2572 [dB])





78 8 82 84 86 88 9 92 94

BER

SNR (dB)

True LLRTaylor3

Pade41 and 34Pade45

10e minus 006

10e minus 005

10e minus 004

10e minus 003

10e minus 002

10e minus 001

10e + 000





5 Conclusion



Acknowledgments




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1199100

1198880

1198881

1198882

1198883



1199100

1198860

1198861

1198862

1198863

1198864

1198865

1198866

1198867

bit 1 00 00 12135 00 00334 00 00025 00 261119864 minus 05



1199100

1198870

1198871

1198872

1198873

1198874




119910ge0ℓ(3)
















1198871198730= 2572 dB)



0200284000 0200283000 02002840001205823

0228588000 0228591000 02285880001205827

0067795000 0067777920 00677944871205828

0210231000 0210246680 021023161312058230

0292602000 0292601400 02926018501205889

1000000000 1000000000 1000000000Rate 0500000 0500000 0500000120590⋆ 07232423 07232385 07232422(1198641198871198730)⋆ [dB] 27068537 27069429 27068549

(b) Rate optimized


0202860 0202758 02028601205823

0214366 0214369 02143661205827

0157257 0157244 01572581205828

0112911 0112922 011285912058230

0312606 0312607 03126071205889

1000000 1000000 1000000Rate 04946847 04946836 04946845120590⋆ 0740167 0740167 0740167(1198641198871198730)⋆ [dB] 2613406 2613406 2613406


0= 2572 [dB])





78 8 82 84 86 88 9 92 94

BER

SNR (dB)

True LLRTaylor3

Pade41 and 34Pade45

10e minus 006

10e minus 005

10e minus 004

10e minus 003

10e minus 002

10e minus 001

10e + 000





5 Conclusion



Acknowledgments




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198871198730= 2572 dB)



0200284000 0200283000 02002840001205823

0228588000 0228591000 02285880001205827

0067795000 0067777920 00677944871205828

0210231000 0210246680 021023161312058230

0292602000 0292601400 02926018501205889

1000000000 1000000000 1000000000Rate 0500000 0500000 0500000120590⋆ 07232423 07232385 07232422(1198641198871198730)⋆ [dB] 27068537 27069429 27068549

(b) Rate optimized


0202860 0202758 02028601205823

0214366 0214369 02143661205827

0157257 0157244 01572581205828

0112911 0112922 011285912058230

0312606 0312607 03126071205889

1000000 1000000 1000000Rate 04946847 04946836 04946845120590⋆ 0740167 0740167 0740167(1198641198871198730)⋆ [dB] 2613406 2613406 2613406


0= 2572 [dB])





78 8 82 84 86 88 9 92 94

BER

SNR (dB)

True LLRTaylor3

Pade41 and 34Pade45

10e minus 006

10e minus 005

10e minus 004

10e minus 003

10e minus 002

10e minus 001

10e + 000





5 Conclusion



Acknowledgments




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Log-Likelihood Ratio Calculation...

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