Design of Fixed-Point Processing Based LDPC Codes Using EXIT Charts

University of Southampton School of Electronics and Computer Science Communications Research Group

Xin Zuo, Robert G. Maunder and Lajos Hanzo

Presented by Robert G. Maunder Email: xz3g08,rm,lh @ecs.soton.ac.uk { }

San Francisco, USA September - 2011 Contents

Contents

J Introduction Low-Density Parity-Check Codes • Decoder and Decoding Algorithm • Fixed-Point Representation • EXIT Charts • J Simulation Results

J Conclusions

Xin Zuo, Robert G. Maunder & Lajos Hanzo Communications Research Group - ECS 2/26 References References

[1] R. G. Gallager, “Low density parity check codes,” IRE Transactions on Information Theory, vol. IT, no. 5, pp. 21–28, 1962.

[2] R. Tanner, “A recursive approach to low complexity codes,” IEEE Transactions on Information Theory, vol. 27, pp. 533–547, Sep. 1981.

[3] J. Hagenauer, E. Offer, and L. Papke, “Iterative decoding of binary block and convolutional codes,” IEEE Transactions on Information Theory, vol. 47, pp. 429–445, Aug. 2002.

[4] F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, “Factor graphs and the sum-product algorithm,” IEEE transactions on Information Theory, vol. 47, pp. 498–519, Feb. 2001.

[5] J. Chen, A. Dholakia, E. Eleftherious, M. P. C. Fossorier, and X. Hu, “Reduced-complexity decoding of ldpc codes,” IEEE Transactions on Communications, vol. 53, no. 8, pp. 1288–1299, 2005.

Xin Zuo, Robert G. Maunder & Lajos Hanzo Communications Research Group - ECS 3/26 References

[6] G. Montorsi and S. Benedetto, “Design of ﬁxed-point iterative decoders for concatenated codes with interleavers,” IEEE Journal on Selected Areas in Communications, vol. 19, pp. 871–882, May 2001.

[7] S. ten Brink, “Convergence behavior of iteratively decoded parallel concatenated codes,” IEEE Transactions on Communications, vol. 49, no. 10, pp. 1727–1737, 2001.

Xin Zuo, Robert G. Maunder & Lajos Hanzo Communications Research Group - ECS 4/26 Introduction - LDPC Codes Introduction

Low-Density Parity-Check (LDPC) codes. • – LDPC codes [1] belong to a class of linear block codes, hence is deﬁned by a Parity-Check Matrix (PCM);   111010    101001  010101 – Alternatively by a corresponding Tanner graph [2], on which the decoding process is based.

˜a ˜e variable node c c ˜re ˜ra

p˜ap˜e check node Figure 1: An example of a Tanner graph. Xin Zuo, Robert G. Maunder & Lajos Hanzo Communications Research Group - ECS 5/26 Introduction - Decoder and Decoding Algorithm

Turbo-like decoder: a serial concatenation of two soft-input soft-output decoders, • Check Node Decoder (CND) and Variable Node Decoder (VND).

u c BPSK x G Mod

AWGN p˜e rã cã π 1 − Soft y CNDpã r˜e VND c˜e π Demod c˜p

Figure 2: The schematic of encoder and decoder of LDPC codes.

– The a priori, extrinsic and a posteriori messages are Log-Likelihood Ratios (LLRs).

Xin Zuo, Robert G. Maunder & Lajos Hanzo Communications Research Group - ECS 6/26 Introduction - Decoder and Decoding Algorithm

Iterative Decoding Algorithm. • a˜

b˜ c˜ a˜ b˜ c˜

(a) (b) Figure 3: Basic operation units in (a) VND and (b) CND. – VND: c˜ =a ˜ + ˜b (1)

– CND: replace + by boxplus operator [3] in Eq (1). c˜ =a ˜ ˜b = sign(˜a)sign(˜b) min( a˜ , ˜b ) | | | | a˜+˜b a˜ ˜b + log 1 + e−| | log 1 + e−| − | (2) − sign(˜a)sign(˜b) min( a˜ , ˜b ). (3) ≈ | | | | Xin Zuo, Robert G. Maunder & Lajos Hanzo Communications Research Group - ECS 7/26 Introduction - Decoder and Decoding Algorithm

Eq (2) is referred as the Log Sum-Product Algorithm (SPA) [4]. ∗ The approximated version in Eq (3) is referred as the Min-Sum algorithm ∗ (MSA) [5]. x˜ Correction function log 1 + e− is implemented by a Look-Up Table (LUT). ∗

a x˜L

a a e a a e x˜L 1 ... x˜2 x˜1 x˜L ... x˜2 x˜1 −

(a) (b) Figure 4: Decoding processings of a (a) VND and (b) CND with high degrees.

Xin Zuo, Robert G. Maunder & Lajos Hanzo Communications Research Group - ECS 8/26 Introduction - Decoder and Decoding Algorithm

– For a VN connected to L 1 CNs (L 2) in the Tanner graph, the extrinsic − ≥ message can be computed using forward-backward algorithm.  ˜b if i = 1  2 e  x˜i = f˜i 1 + ˜bi+1 if 1 < i < L , (4)  −  f˜L 1 if i = L −  a  x˜1 if i = 1 f˜i = , (5) a ˜  x˜i + fi 1 if i > 1 −  a  x˜L if i = L ˜bi = . (6) a ˜  x˜i + bi+1 if i < L – Likewise, the decoding in a CN connected to L VNs (L 2) can be obtained by ≥ replacing + by .

Xin Zuo, Robert G. Maunder & Lajos Hanzo Communications Research Group - ECS 9/26 Introduction - FP Representation

Fixed-Point (FP) representation. • – Notation FP (x, y, z). – Two’s complement [6]: from left to right, y integer bits including one sign bit (MSB) at the most left, the imaginary decimal point and z fraction bits. – Clipping is a usual method to control overﬂow. x represents the number of integer bits after clipped. – An example.

yyyy.zz 0100.10 + 4.50 clipped xxx.zz 011.11 + 3.75

Xin Zuo, Robert G. Maunder & Lajos Hanzo Communications Research Group - ECS 10/26 Introduction - FP Representation

– When the decoder adopts FP representation, z decides the number of entries in LUT.

0.8 correction function z = 1 0.7 z = 2 z = 3 0.6

) 0.5 ˜ x − e 0.4

log(1 + 0.3

0.2

0.1

0 0 1 2 3 4 5 6 x˜

x˜ Figure 5: Correction function log 1 + e− and its approximation by LUT.

Xin Zuo, Robert G. Maunder & Lajos Hanzo Communications Research Group - ECS 11/26 Introduction - EXIT Charts

EXtrinsic Information Transfer (EXIT) charts. • – EXIT charts [7] characterise the iteratively exchanged extrinsic information between the VND and CND. Mutual Information (MI) I(x˜; x) is employed to quantify the reliability of information. – Histogram-based method [7, Eq (19)] of evaluating MI can accurately quantify the degradation imposed by sub-optimal decoding algorithms, which compares the histogram of LLR sequence x˜ with the content of the bit sequence x. – EXIT charts can offer insights into performance degradation of iterative decoders. – By the aid of EXIT bands, more accurate prediction can be allowed, even when a short code is employed.

Xin Zuo, Robert G. Maunder & Lajos Hanzo Communications Research Group - ECS 12/26 Introduction - EXIT Charts

– An example of ﬂoating point EXIT charts and the bands.

1 1

0.8 0.8 ) ) r r

, 0.6 , 0.6 e e ˜ ˜ r r ), I( ), I( p p , , a a ˜ ˜ p 0.4 p 0.4 I( I(

code length = 500 code length = 5000

coding rate = 0.5, d =3 coding rate = 0.5, d =3 0.2 v 0.2 v Eb/N0 = 3dB Eb/N0 = 3dB

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

I(p˜e,p), I(r˜a,r) I(p˜e,p), I(r˜a,r)

(a) (b) Figure 6: EXIT charts of two half-rate regular LDPC codes having code length of (a) 500 and (b) 5000 bits.

Xin Zuo, Robert G. Maunder & Lajos Hanzo Communications Research Group - ECS 13/26 Introduction - EXIT Charts – Generations of EXIT functions for VND and CND, both of which operate on FP variables. u c u cBPSK x G G Mod

a a I(r˜ ; r) Generate r I(p˜ ; p) Generate p r Repeat π Repeat LLRs AWGN LLRs

a a Q r˜ p˜ Q a c˜ Soft y VND CND e Q Demod p˜ e r˜ 1 1 − − Q Q e e I(r˜ ; r) Measure I(p˜ ; p) Measure MI MI

(a) VND (b) CND Figure 7: schematics used to depict the generations of EXIT functions for the (a) VND, (b) CND, where indicates clipping , represents conversion from ﬂoating point values × 1 Q to FP values and − is the inverse. Q Xin Zuo, Robert G. Maunder & Lajos Hanzo Communications Research Group - ECS 14/26 Introduction - Novel Contributions

Novel contributions of this paper: • – We introduce the use of EXIT charts to determine the minimum Operand Width (OW), namely the value of x, y and z, of a LDPC decoder. This approach overcomes the time-consuming problem when bit-error ratio simulations are used to achieve comprehensive investigation, and it offers insights into the speciﬁc causes of the performance degradation encountered. – By the use of EXIT charts, a FP scheme having an overall OW of 6 bits is proposed.

Xin Zuo, Robert G. Maunder & Lajos Hanzo Communications Research Group - ECS 15/26 Simulation Results Simulation Results

First x and z are separately considered, then y is decided by combined • consideration of x and z. – Simulation parameters. Table 1: Parameters used for FP EXIT chart simulations of LDPC codes.

clipped integer OW x 1, 2, 3, 4, 5, 6 bits fraction OW z 0, 1, 2, 3, 4, 5 bits integer OW y 3, 4, 5, bits inﬁnite value 32 bits ∞ code length 500 bits

CN degree dc 4, 8, 16, 32

VN degree dv 2, 4, 8, 16 benchmarker ﬂoating point Log-SPA and MSA

Xin Zuo, Robert G. Maunder & Lajos Hanzo Communications Research Group - ECS 16/26 Simulation Results

Determining fraction OW z. • 1 1

0.8 0.8

dv =2, 4, 8, 16 from bottom to top 0.6 0.6 ) ) r p , , e a ˜ ˜ r p I( I( 0.4 0.4 dc =4, 8, 16, 32 from right to left FP( , ,0) FP( , ,0) dc =8 FP(∞,∞,1) FP(∞,∞,1) FP(∞,∞,2) FP(∞,∞,2) ∞ ∞ 0.2 FP(∞,∞,3) 0.2 FP( , ,3) FP(∞,∞,4) FP(∞,∞,4) ∞ ∞ dv =4 FP(∞,∞,5) FP( , ,5) ∞ ∞ Log-SPA/MSA∞ ∞ Log-SPA MSA 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 I(r˜a,r) I(p˜e,p)

(a) VND (b) CND Figure 8: The EXIT functions for FP implementations of LDPC codes employing various fraction OWs z, as well as VN and CN degrees, for communication over an AWGN

channel having an Eb/N0 of 3 dB.

Xin Zuo, Robert G. Maunder & Lajos Hanzo Communications Research Group - ECS 17/26 Simulation Results

Determining clipped integer OW x. • 1 1

0.8 0.8

0.6 0.6 dv =2, 4, 8, 16 ) ) from bottom to top r p , , e a ˜ ˜ r p I( I( 0.4 0.4 dc =4, 8, 16, 32 from right to left FP(1, , ) FP(1, , ) FP(2,∞, ∞) FP(2,∞, ∞) ∞ ∞ ∞ ∞ FP(3, , ) 0.2 FP(3, , ) dv =4 0.2 FP(4,∞, ∞) FP(4,∞, ∞) ∞ ∞ ∞ ∞ FP(5, , ) FP(5, , ) FP(6,∞, ∞) dc =8 FP(6,∞, ∞) ∞ ∞ ∞ ∞ Log-SPA Log-SPA/MSA MSA 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 a I(r˜ ,r) I(p˜e,p)

(a) VND (b) CND Figure 9: The EXIT functions for FP implementations of LDPC codes employing various clipped integer OWs x, as well as VN and CN degrees, for communication over an

AWGN channel having an Eb/N0 of 3 dB.

Xin Zuo, Robert G. Maunder & Lajos Hanzo Communications Research Group - ECS 18/26 Simulation Results

Decide fraction OW z. • – CND: the degradation imposed by a fraction OW z 2 is less than that imposed ≥ by MSA, and z = 3 approaches Log-SPA. – VND: z 1 is acceptable, and z = 2 approaches Log-SPA. ≥ – To achieve better trade-off between complexity and performance, and simplify the decoder’s architecture, z is selected as 2.

Decide Clipped integer OW x. • – CND: a clipped integer OW x 3 is approaching Log-SPA. ≥ – VND: x 3 approaches Log-SPA, but droops happen at low a priori mutual ≥ information region when x = 1, 2. – The minimum acceptable z is selected as 3.

Xin Zuo, Robert G. Maunder & Lajos Hanzo Communications Research Group - ECS 19/26 Simulation Results

Determining integer OW x, based on the decisions of y and z. • 1 1

0.8 0.8

dv =2, 4, 8, 16 0.6 from bottom to top 0.6 ) ) r p , , e a ˜ ˜ r p I( I( 0.4 0.4 dc =4, 8, 16, 32 dc =8 from right to left

FP(3,3,2) FP(3,3,2) 0.2 FP(3,4,2) 0.2 FP(3,4,2) FP(3,5,2) d =4 FP(3,5,2) Log-SPA/MSA v Log-SPA MSA 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 I(r˜a,r) I(p˜e,p)

(a) VND (b) CND Figure 10: The EXIT functions for FP implementations of LDPC codes employing various integer OWs y, as well as VN and CN degrees, for communication over an AWGN

channel having an Eb/N0 of 3 dB.

Xin Zuo, Robert G. Maunder & Lajos Hanzo Communications Research Group - ECS 20/26 Simulation Results

Decide integer OW y. • – x = 3 and z = 2 are ﬁxed, and y is considered from the range of 3, 4, 5. – For both CND and VND, y = 4 is sufﬁcient. FP(3, 4, 2) achieves the most desirable trade-off.

Xin Zuo, Robert G. Maunder & Lajos Hanzo Communications Research Group - ECS 21/26 Simulation Results

Explanation of the unusual ‘droop’ occured on VND EXIT functions. • – Two conditions: a ) integer OW clipped into too low value x = 1, 2, b ) low a priori mutual information region I(˜ra; r) < 0.1. – Consequences: Condition a ): Clipping makes the shape of histogram of channel output c˜a ∗ nearly two ‘spikes’ located at the upper and lower saturations limits. Condition b ): The generated LLRs ˜ra contains little information, hence most ∗ of them provide wrong indications of the content of the binary bits, which makes the two spikes less focused and more overlapped in the neighborhood of 0. According to the histogram based MI mearsuring method, the resultant ∗ histogram conveys less MI even than the dual-‘spike’ like histogram.

Xin Zuo, Robert G. Maunder & Lajos Hanzo Communications Research Group - ECS 22/26 Simulation Results

Various BER simulation results of FP based LDPC decoder as z or x varies, where • the adopted LDPC code is half-rate regular and has code length of 500 bits and dv of 3. The BER results proved our observations based on EXIT functions in Figure 8

and Figure 9. 100 z =0 z =1 1 z =2 10− z =3 z =4 z =5 2 10− z =6 z =7 Log-SPA z increases 3 MSA 10− BER

4 10−

5 10−

6 10− 0 1 2 3 4 5 6

Eb/N0 Figure 11: BER plots of LDPC codes using different fraction OWs.

Xin Zuo, Robert G. Maunder & Lajos Hanzo Communications Research Group - ECS 23/26 Simulation Results

100

1 10−

2 10−

3 10− x increases BER x=1 x=2 4 10− x=3 x=4 x=5 10 5 x=6 − x=7 Log-SPA MSA 6 10− 0 1 2 3 4 5 6

Eb/N0 Figure 12: BER plots of LDPC codes using different clipped integer OWs.

Xin Zuo, Robert G. Maunder & Lajos Hanzo Communications Research Group - ECS 24/26 Conclusions

Conclusions

The use of EXIT charts is prosed to investigate the performance degradation • caused by limited OWs used in LDPC decoders.

Compared to BER simulations, EXIT charts overcome the time-consuming • drawback, thus using EXIT charts, it is able to fully investigate the effects caused by limited OWs and others.

EXIT charts are able to predict the convergence behaviour of turbo-like decoders, • and provide extra insight into performance degradation.

Based on investigation using EXIT charts, a scheme FP(3, 4, 2) is proposed. •

Xin Zuo, Robert G. Maunder & Lajos Hanzo Communications Research Group - ECS 25/26 Thank you!

Thank you !

Xin Zuo, Robert G. Maunder & Lajos Hanzo Communications Research Group - ECS 26/26