GB2440584A

GB2440584A - Bit Metric Calculation for Gray Coded M-QAM using (piecewise) linear equations derived from Max-Log approximations

Info

Publication number: GB2440584A
Application number: GB0615516A
Authority: GB
Inventors: Kassem Benzair
Original assignee: Individual
Current assignee: Individual
Priority date: 2006-08-04
Filing date: 2006-08-04
Publication date: 2008-02-06
Also published as: GB0615516D0

Abstract

Traditional systems use a look-up table to derive bit metrics, which avoids the need to perform calculations but requires large amounts of memory. The invention simplifies the bit metric calculation in receivers for a digital communications system utilizing binary channel coding and higher-order modulation, so that the calculations become feasible and the tables and memory can be done away with. In this invention, the max-log approximation is applied to the optimum bit metric equations for square M-QAM. The approximation offers on-line computation of bit metrics with simple operations. The invention further recommends a Gray mapping for square M-QAM. The constellation mapping further simplifies the bit metric computation, resulting in linear or piecewise linear bit metric equations for 4, 16, 64 and 256-QAM. The approximations are near optimum at high SNR, independent of the channel SNR and simple to implement.

Description

Bit Metric Calculation for M-QAM with Constellation Labeling

I. Introduction

In many digital communication systems, higher-order modulation is used in conjunction with binary channel coding. This ad hoc coded modulation scheme benefits from the spectral efficiency of the higher-order modulation and the decoder simplicity of the binary channel coding. While this ad hoc coded modulation scheme is non-optimum, it offers complexity advantages over trellis-coded modulation schemes while working quite well.

As part of any ad hoc coded modulation scheme, the coded bits are mapped onto a higher-order constellation, such as M-QAM. At the receiver, bit metrics are constructed from the demodulated constellation points. A binary channel decoder uses the bit metrics to produce estimates of the information bits. For many block-coding schemes, such as Reed-Solomon or BCH codes, the bit metrics will be hard metrics (e.g. one-bit). For convolution or "turbo" like coding schemes, the metrics must be soft.

For M-ary modulation, many receiver implementations generate the soft-bit metrics using look-up tables (LUT). In this implementation, the received, quantized constellation point (r[,rQ) indexes a LUT to generate the soft-bit metrics. When the number of bits used to represent ri and rQ are small, the LUT-based approach is attractive. When the number of bits required for ri and rQ becomes large, which will often be the case for higher-order constellations, the LUT can become quite large. As an example, the LUT for 16-QAM with 8-bit ri and rQ values and 3-bit softbit metrics would require roughly 98 kbytes of memory.

In this work, we consider square M-QAM where M=2m. For square M-QAM, we propose a specific Gray mapping for constellation labeling, which provides for independent labeling in :. each signal dimension (I and Q). With this bit labeling, we derive the optimum bit metric * *** equations for 4, 16 and 64-QAM. We further show simple approximations to the optimum bit *:"* metrics, which result from the proposed constellation bit labeling. The bit metric approximations are piecewise linear, independent of the channel signal-to-noise ratio (SNR) and converge to the * *. optimum bit metrics for large SNR. The linear approximations to the bit metrics offer a nice * **. implementation alternative to look-up table approaches for bit metric calculation in digital communications systems using ad hoc coded modulation with square M.-QAM.

II. Ad Hoc Coded Modulation In Figure 1, we show an ad hoc coded modulation scheme using two dimensional amplitude and phase modulation. Here, a binary channel code is used with the coded bits mapped onto a two dimensional M-ary constellation. At the receiver, the received IIQ constellation points are used to generate coded bit metrics, which are used by a binary channel decoder.

Ill. Bit Metric Calculators for M-ary Amplitude and Phase Modulation The bit metric calculator is a key component of the ad hoc coded modulation scheme. For M-ary amplitude and phase modulation, the bit metric calculator computes the log-likelihood ratio (LLR) for each of the coded bits given the received constellation points ri and rQ. For M-ary modulation, the LLR for bit di (i1,2,. . .log2(M)) is defined as: L[di] = log1 P(di = I r! rQ) P(di = 1) [P(di = 01 rI,rQ)P(di = 0) If we denote the transmitted symbol s=I+jQ and we assume that the code bits are transmitted with equal probability (e.g. P(dil)=P(diO)l /2), we may simplify the LLR expression as: P(rI,rQ I s) s,dirl L[di] log P(rl, rQ I sn) s, :di=O Here, the numerator sums all the conditional probabilities of sending symbol s where s has d11. In the denominator, the sum of conditional probabilities is taken over all symbols s such that diO. The simplified bit LLR expression will depend on the constellation bit labeling and the statistics of the received constellation values (ri and rQ).

A. Square M-QAM Bit Metric Calculator on A WGN Channel: Consider square M-QAM modulation where M=2m with m coded bits carried per transmitted symbol s=I+jQ. For an additive white Gaussian noise (AWGN) channel, the received signal model is r = s + n with transmitted symbol s = I+jQ and channel noise variable n=nI +j nQ.

Here, ni and nQ are i.i.d. Gaussian random variables with zero mean and variance No/2. With this model, the conditional probability P(rI,rQ s) is: S. P(rI,rQ Is) = * ((ri _j)2 +(rQ_Q)2)] I...

<p≥ K(ri,rQ,No).exp[.(i2 -2rI.i-i-Q2 _2.rQ.Q)] From the conditional probability, we can form the bit LLR and simplify to give

S

exp[_.(J2_2.r1.J+Q2_2.rQ.Q)] L[di]= log s:diI r 1 expLT--.(i2_2.r1.i+Q2 _2.rQ.Q)j -s:dIO No Here, we see the numerator and denominator of the optimum bit LLR involves the log of the sum of exponentials. For the turbo code literature [1] recall a simple a simple relation involving the log of the sum of exponentials given as: 1o[exP(4)] max{A1} This approximation is tight provided one of the arguments Ai is much larger than the other arguments. This tightness in the bit LLR is very closely tied to the signal to noise ratio. When the SNR is large, one of the conditional probabilities P(rJ,rQIs) will be much larger than the other conditional probabilities. Applying approximation to the bit LLR expression, we have that: L[di} max{ (12_2. ri / + -2. rQ Q)} -max{-. (j2 -2 rJ I + -2 rQ. Q)} Since most soft-decision decoders are invariant to input scaling, provided the scaling is constant for all inputs, we may drop the scaling of lfNo. Thus, the bit LLR approximation simplifies to: L[di]max{2rI.J+2.rQQ-12 _Q2}_ma,42.rI 1 +2.rQ.Q-12 Q2} r.dir1 r.dr4) As can be noted, the bit LLR approximation offers reduced complexity versus the computation of the optimum bit LLR. Furthermore, the approximation is not a function of the SNR (e.g. No).

This was a benefit of the max-log approximation found for turbo decoding. In the next sections, we will show that with proper constellation bit labeling, the bit LLR approximation may be simplified to a linear function of rI and/or rQ for square M-QAM.

B. SignedlMagnitude M-QAM Constellation Mapping: For ad hoc coded modulation schemes, the constellation bit map (or labeling) will impact the complexity of the bit metric calculator and the performance of the binary decoder. To improve the performance of the binary decoder, Gray mapping is typically used, whereby the Hamming distance between all adjacent constellation points is one. It can be shown that this constellation mapping reduces the probability of bit error for uncoded transmission. It can be further shown :. that for ad hoc coded modulation, Gray mapping improves the reliability of the bit metrics (e.g. bit LLRs) and thus the performance of the binary decoder. * S**

In this work, we consider square M-QAM and using a special Gray mapping whereby the in-s. . phase and quadrature values are labeled independently. For this constellation Gray mapping, we * " partition the coded bits into even indexed bit and odd indexed bits. The need for equal number of odd and even bits restricts this scheme to square M-QAM with M=2m and m even. Here, the m/2 odd indexed bits determine the transmitted I value and the m/2 even indexed bits determine :. the transmitted Q value. For both land Q, a single bit determines the sign of the constellation point, while the remaining bits determine the magnitude. We call this constellation mapping ** .: signed/magnitude mapping and note that Gray mapping is possible provided the magnitude bits are Gray mapped.

In Table 1, we give the synthesis equations for Gray, signed/magnitude constellation mapping with 4, 16,64 and 256-QAM. Here, D is a constellation spacing parameter, which determines the tinsmitted energy per symbol. It should be noted that other Gray mappings are possible, but slightly more complicated bit LLRs will result Modulation I Constellation Labeling Q Constellation Labeling 4-QAM I-2(dl-1)D Q=2(d2-1)D 16-QAM ID d3=0 ID d4=0 I =2(d1--1)* Q=2(d2-1) ________ _________3D__d3=l _________ 3D d4=l 64-QAM D d3d5 =00 D d4d6 =00 3D d3d501 3D d4d6=0l I = 2(dl -1). 0= 2(d2 -1) 5D d3d5=11 5D d4d6=11 7D d3d5=10 7D d4d6=10 256-QAM D d3d5d7=000 D d4d5d8=000 3D d3dSd7=00l 3D d4d5d8=001 5D d3d5d7=011 5D d4d5d8=011 7D d3d5d7=010 7D d4d5d8=010 I=2(d1-1) Q=2(d2-1) 9D d3d5d7=ll0 9D d4d5d8= 110 1W d3d5d7=111 lID d4d5d8=11l 13D d3d5d7=l01 13D d4d5d8=l01 15D d3d5d7=l00 ISD d4d5d8=100 Table 1: Constellation labeling equations for M-QAM with independent, sigWmagnitude mapping Independent I/Q labeling also the bit LLRs to simplified. Since a given code bit di only impacts on of the signal dimensions (I or Q), LLR computation for that bit need not consider the orthogonal signal dimension in the case of i.i.d. noise. Assuming even indexed bits label! and odd indexedbits label Q, we havethatthebitLLRs may bewrittenas: *::::* P(rIjI) L[di] = log I:d,I i=1,3 5 P(rIII) * *. 1:di=O F(rQIQ) _ J Q:d11 A * * * L4M1J -iOi 1h. ? P(rQLQ) *. : LQth=O * I. where P(r s) = -j. exp[-. (r -)2] -4/No?r No C. 4-QAM Bit LLRc on A WGN Channel From the analysis above, we now construct the bit LLRs for 4-QAM (e.g. QPSK) modulation.

Using the synthesis equations given in Table 1, it can be shown that the optimum bit LLRs are: L[dll = k(No) rI L[d2) = k(No) pQ Here, k(No) is a constant function of the SNR. Since most binary decoders are invariant to input scaling (provided it is constant for all inputs), we may ignore the scaling and construct the bit LLRsas: L[dl] rI L[d2]=rQ D. 16-QAM Bit LLRs on A WGN Channel Next, we construct the bit LLRs for 16-QAM. Using the bit labeling in Table 1, we may construct the bit LLRs for the odd indexed bits using the rI values. By symmetly, the even indexed bits will follow, replacing rI with rQ.

Sign-bit LLR (Lid 11) The optimum bit LLR for the in-phase sign bit dl is given by: e I.(D2_2.rI.D) +exp L.(9D2_6.rJ.D)l L[d1]=lool [No i LN0 J 1 exp[T_.(D2 +6.rI.D)l L LNO J [No J : * For simplification, we apply the log-max approximation and remove the scaling by the SNR to * * a yield an approximation to the bit LLR given as: * **.

I

L[dl}max{2.rI-D,6r1 _9D}-max{-2r1 -D,-6.rI-9D} *. *.* This expression may be further simplified to yield the piecewise linear bit LLR. expression: 2rJ+2D rJ<-ZD a:::': L[d1] ri -2D =r! =2D 2-rI-2D rI>2D * S I * *1 In the approximate bit LRR, we see that the function is a piecewise linear function of the received value ri We note that the slope of the linear function is one for ri values near zero and two for rI values away from zero. We speculate that the bit LLR approximation could be further simplified using a simple, linear expression rather than the two-piece linear expression. Taking the slope of the approximation near zero, we approximate the bit LLR expression using the linear function: L[dlJ=rI In Figure 2, we show the exact and approximate bit LLRs as a function of rI assuming an Es/No of 8 dB (D=1). As can be seen, the exact and max-log approximation are very tight over the range of rL The maximum error is roughly 15%, and the sign of the LLR approximations is never in error. For the linear approximation, the error becomes large for large rI, due to the incorrect slope. However, for these points, the LLRs would convey a large confidence in a one and zero, making the effects of an incorrect LLR magnitude negligible in the decoding. * * * *** *** * * * *** S * S* * S * * S. S... * S S S. * S. *

S S * *0

Magnitude-bit LLR Following the sign-bit LLR, we next consider the magnitude bit d3. Here, the optimum bit LLR is: exp.L-.(9D2_6.rI.D) +ex -.(9D2+6.rI.D) L(d3]=log L J_L J L exP[.(D2 _2.rI.D)j+exP[-.(D2 +2.rI.D)] Applying the log-max approximation to L[d.3] and simplifying, we have that L[d3] max{6. ri-9D,6 ri -9D} -max{-2* -D,2 ri -D} Upon further inspection, the approximate bit LLR. can be further simplified to yield: L[d3] ri I -2D Thus, we see that L[d31 can also be approximated using a linear function of rI.

In Figures 4 and 5, we show the exact and approximate bit LLRs as a function of rI assuming an Es/No of 8 dB (D1). As can be seen, the exact and max-log approximation are very tight over the range of rL The maximum error is roughly 15%, however the sign of the LLR is never in error. Note, the error equation has a discontinuity at rP2D due to zero in the optimum bit LLR expression. a * * * *.* * * **** * ** * * S * S. S... * . S S. * *. a * S S S 5,

Based on this analysis, we can approximate the bit LLRs for 16-QAM using the signed/magnitude constellation labeling of Table 1 using: Sign-bit LLR Magnitude-bit LLR 2r1+2D rI<-2D I L[dl] rI -2D = rI = 2D L[d3] = ri I -2D 2r1-2D rI>2D _____________________ -2rQ+2D rQ<-2D Q L[d2] = rQ -2D = rQ = 2D L[d4] rQ I -2D 2rQ-2D rQ>2D _______________ Table 2:Max-log approximation to bit LLRs for 1 6-Q,4M Sign-bit LLR Magnitude-bit LLR I L[dl] = rI L[d3] = rI I -2D Q L[d2] = rQ L[d41 rQ I -2D Table 3: Linear approximation to max-log LLR approximations for 16QAM E. 64-QAMBitLLRS onAWGN In Figure 6, we plot the optimum LLRs for the in-phase coded bits for 64-QAM using signed/magnitude constellation mapping from Table I with an Es/No of 12 dB. As can be seen, the LLRs are approximately piecewise linear. Applying the log-max approximation to the optimum LLRs and linearizing as for the 16-QAM case, Table 4 shows the approximate, linearized bit ILR values for 64-QAM In Figure 7, we plot the optimum LLRs and the linear approximations for an Es/No of 12 dB. a. * . a **s *... a, S... * .. * S S * .. *55 * S a a. . a. * * . S * aa

F. 256-QAM Bit Metrics on A WGN As with 4, 16 and 64-QAM, the bit LLRS for 256-QAM using the constellation mapping in Table I also may be approximated using piecewise linear functions. The process is the same as above.

First, we detennine the optimum LLRs. Next, we apply the max-log approximation and linearize as possible. Here, we omit the computation of the approximate and linear approximate bit LLRs.

1V. Performance and Implementation Issues For the optimum bit LLR, knowledge of the noise variance (No) and the constellation spacing parameter (D) are required. In the max-log and linearized max-log approximation, knowledge of the constellation spacing parameter D is required, but no knowledge of the noise variance (No) is required. The independence of the LLR computation and the noise variance No was also observed for Turbo decoding using the log-max approximation [xx].

Knowledge of the constellation spacing parameter is required for the bit LLR approximations.

This information can be achieved by 1.) estimation of D or 2.) applying gain control to fix the rI and iQ values to a desired D value. In the estimation method, D is estimated from the received rI and rQ values and used in the bit LLR expressions of Tables 2 or 3. In the gain control method, a desired value of D is chosen and an automatic gain control (AGC) circuit is used prior to the bit metric calculator.

For most digital implementations, ri and rQ will be quantized with 4-10 bits of precision. The soft bit metrics (e.g. bit LLRs) need only 3-4 bits of precision for most soft-input decoding algorithms. Therefore, the bit LLRs must be quantization perfonned prior to the encoding. In Figure 8, we show the bit meiric calculator and typical supporting circuitry using gain control. * S * *.* I... * S S... * ** * S * * .. *.. * S S S. * S. S * I * * S. Gi

A. Performance and Implementation Issues for 16-QAM To test the validity of the bit LLR approximations from Tables 2 and 3, we simulated the ad hoc coded modulation scheme. Here, the binary code is a rate 4/5 convolutional code and 16-QAM modulation is used with the constellation labeling from Table 1. In Figure 9, we show the bit error rate (BER) performance after Viterbi decoding using the different bit LLRs. Here, the bit LLRs and the Viterbi decoding used floating-point arithmetic. As can be seen, the approximate bit LLRs result in little to no degradation in the decoder performance for the values of Es/No considered. * . * *** **.. * * **** * * * S S * SI S.., * S S S. * S. S * I* * ** tO

Based on Figure 7, we can use the bit metric values in Table 3 without significant performance degradation. Based on these equations, the implementation involves two subtractions and two ABS operations to generate four soft-bit metrics. For comparison, we contrast this implementation with a LUT-based approach. First, the LUT-baseci approach would require the same AGC circuit to fix the constellation spacing parameter prior to the bit metric calculator.

B. Gain control and estimation error In real-world digital communications systems, signal fading and power fluctions will prevent perfect knowledge of the constellation spacing parameter (or perfect gain control prior to the bit metric calculator). For LLR expression which rely on knowledge of the constellation spacing paramter, these fiuctions will result in performance degradations. As an example consider the 16-QAM linear LLR approximations. Here, an incorrect estimate of D (or gain control error) would result in errors in the L[d31 and L[4], while not affecting L[I] and L[2]. For 4-QAM, the constellation spacing parameter is not needed.

Gain control and estimation error will be more severe for higher-order constellations and fading channels. For 4 and I 6-QAM, the effects may be minimum. Gain control and estimation error will also cause problems when quantization of the bit LLRs is performed. For any ad hoc coded modulation scheme, the effects of gain control and estimation error should be studied in order to characterize performance degradations.

References: [1] S. S. Pietrobon and S. A. Barbulescu, "A Simplification of the modified Bahi decoding algorithm for systematic convolutional codes," Tnt Symposium on Information Theoiy and Its Application, Syndey, Australia, pp. 1073-1077, Nov. 1994. * I* * * * .a. * a S. * * .. * S. I. S * .* * *q I' (2-

Claims

CLAIM

1. TITLE OF INVENTION:

BIT METRIC CALCULATION FOR SQUARE M-QAM

2. Description

The purpose of this invention is to simplify the bit metric calculation in the receiver for a digital communications system utilizing binary channel coding and higher-order modulation. If the computational complexity of the bit metric calculation can be reduced, on-line computation is preferred over traditional look-up table approaches due to the reduction in required memory. In this invention, the max-log approximation is applied to the optimum bit metric equations for square M-QAM. The approximation offers on-line computation of bit metrics with simple operations. The invention further recommends a Gray mapping for square M-QAM. The constellation mapping further simplifies the bit metric computation, resulting in linear or piecewise linear bit metric equations for 4, 16, 64 and 256-QAM. The approximations are near optimum at high SNR, independent of the channel SNR and simple to implement.

3. Claim: What is new or different? The use of a linear circuit, rather than a look-up table to compute bit metrics for an ad hoc coded modulation scheme using M-QAM is novel. It is well know that such a linear circuit can be used for 4-QAM, however this work extends the linear bit metric to 16, 64 and 256-QAM. The linear circuit approach is different from traditional approaches using a look-up table to store pre-computed bit metrics.

4. Claim: Advantages over past practices The linear bit-metric calculation has very simple digital implementation. It does not require a large memory to store look-up table (LUT) values. Unlike the LUT approach, the bit metrics computed by the linear method are invariant to the signal-to-noise ratio of the received signal.

5. Other writings: publications, patents, products that relate to this invention Look-up table based bit metric calculators are well documented and patented. In [I], a simplified bit metric calculator is discussed for 16-PSK. Qualcomm and others use full LUT based approaches as in [2]. The max-log approximation is very heavily used in the Turbo coding literature. It was first proposed for this application in [3].

[1] A. Abbaszadeh, B. Currie, et. al "Simplified Bit Metric Calculator for 16-PSK" US Patent submission, L-3 Communications.

[2] Qualcomm Data Sheet, "QI 875: Pragmatic Trellis Decoder Technical Data Sheet".

[3] S. S. Pietrobon and S. A. Barbulescu, "A Simplification of the modified BahI decoding algorithm for systematic convolutional codes," Int. Symposium on Information Theory and Its Application, Syndey, Australia, pp. 1073-1077, Nov. 1994.