Engling Yeo University of California,

Berkeley 1

Architectures and Implementations of

Low-Density Parity Check

Decoding Algorithms

Engling Yeo, Borivoje Nikolić, and Venkat Anantharam

Department of Electrical Engineering and Computer SciencesUniversity of California, Berkeley, CA 94720, USA

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0 1 2 3 4 5 6

10-4

10-3

10-2

10-1

100

SNRB

ER

SNR vs. BER for rate 1/2 codes

IterativeCode

Conv. Code ML decoding

Uncoded

CapacityBound

Background: Iterative Codes

C. Berrou and A. Glavieux, "Near Optimum Error Correcting Coding And Decoding: Turbo-Codes," IEEE Trans. Comms., Vol.44, No.10, Oct 1996.

S. Chung; G.D. Forney, T.J. Richardson, and R. Urbanke, “On the design of low-density parity-check codes within 0.0045 dB of the Shannon limit,” IEEE Communications Letters, vol.5, (no.2), IEEE, Feb. 2001. pp.58-60.

Year Rate ½ Code SNR Required for BER < 10-5

1948 SHANNON 0dB

1967 (255,123) BCH 5.4dB

1977 Convolutional Code 4.5dB

1993 Iterative Turbo Code 0.7dB

2001 Iterative LDPC Code 0.0045dB4

dB

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Outline

LDPC Codes Soft Decoding of LDPC Codes Parallel vs. Serial Architectures Platforms Methods for Code Construction

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Low Density Parity Check Codes (LDPC)

• LDPC representation by bi-partite graph.

• Decoding by message computation and relay along edges

• Iteratively improved estimates of log-likelihood ratios

• Example code:

- R. G. Gallager, IRE Trans. Info. Theory, Vol. 8(1962) p. 21

4096 information bits 4608 variable nodes , 512 check

nodes512 parity bits

CHECK NODES

VARIABLE NODES

)0(

)1(ln

n

n

p

p

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Outline

LDPC Codes Soft Decoding of LDPC Codes Parallel vs. Serial Architectures Platforms Methods for Code Construction

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Qnm

Message from Variable n to Check m

R1n

R2n

R 3n

1 2 3m

n

CHECK NODES

VARIABLE NODE

mnnm

nmn

nnm RR

p

pQ

)('')0(

)1(ln

)0(

)1(ln

n

n

p

pDecoder input:

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n1 2 3

Message from Check m to Variable n

0);()2

1tanh(log)( 1

xxxx

Q 1m

Q 2m

Q3m

mR

nm

CHECK NODE

VARIABLE NODE

)sgn(sgn ')(')('

'1

mnmn

nmnmmNn

mnmn QQQQR

• Signed magnitude representation

• MSB represents parity information

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Hardware for Computation of Rmn

b : Wordlength of messages

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Outline

LDPC Codes Soft Decoding of LDPC Codes Parallel vs. Serial Architectures Platforms Methods for Code Construction

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Parallel Architecture of LDPC decoders

Throughput Efficiency

Power Efficiency Complex

Interconnect

A. Blanksby and C. J. Howland, “A 220mW 1-Gbit/s 1024-Bit Rate-1/2 Low Density Parity Check Code Decoder,” Proc IEEE CICC, Las Vegas, NV, USA, pp. 293-6, May 2001.

PEVC,1

SoftInput1

SoftOutput1

...

PECV,1

PEVC,2 PEVC,3 PEVC,4

PECV,2 PECV,M...

PEVC,N

SoftInput2

SoftOutput2

SoftInput3

SoftOutput3

SoftInput4

SoftOutput4

SoftOutputN-1

SoftInputN

SoftOutputN

SoftInputN-1

PEVC,N-1

PECVCheck-to-Variable

Processing Element

PEVCVariable-to-Check

Processing Element

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Serial Architecture of LDPC decoders

High logic density Memory requirement grows linearly with number of

edges G. Al-Rawi, J. Cioffi, and M. Horowitz, “Optimizing the mapping of low-density parity check codes on parallel decoding architectures,” Proc. IEEE ITCC, Las Vegas, NV, USA, pp.578-86, Apr 2001.

Processing Element for BothClasses of MessageComputation

PE

Memory

SoftInputs

CrossbarSwitch

SoftOutputs.

.

.

.

.

.

Memory

Memory

PE

PE

PE

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Outline

LDPC Codes Soft Decoding of LDPC Codes Parallel vs. Serial Architectures Platforms Methods for Code Construction

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Decoding with Software Approach

General purpose microprocessors and Digital Signal Processors (DSP)

Limited number of Processing Elements (ALUs) Serial Architecture Few hundreds of kbps throughput Design, simulate, and perform comparative analysis of LDPC

codes Low throughput applications with fast time to market

element

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Decoding with Hardware Approach

Parallel architecture Power and throughput efficiency

FPGA Parallel adders and table lookups Need to fit PEs and routing onto single

FPGA die Existing implementations with serial

architecture limited to 56Mbps throughput

[M. M. Mansour and N. R. Shanbhag, “Memory-efficient turbo decoder architectures for LDPC codes,” Proc. IEEE SIPS 2002, San Diego, CA, Oct. 2002.]

[T. Zhang and Keshab Parhi, “A 56Mbps (3,6)-Regular FPGA LDPC Decoder,” Proc. IEEE SIPS 2002, San Diego, CA, Oct. 2002.]

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Decoding with Hardware Approach

Custom ASIC Parallel implementation demonstrated with

1Gbps throughput[A.J. Blanksby and C.J. Howland, “A 690-mW 1-Gb/s 1024-b, rate-1/2 low-density parity-check code decoder,” IEEE Journal of Solid-State Circuits, vol.37, (no.3), (Proceedings of the IEEE 2001 Custom Integrated Circuits Conference, San Diego, CA, USA, 6-9 May 2001.) IEEE, March 2002. p.404-12. ]

Routing congestion Logic density is 50% Design not scalable to codes with larger

block sizes

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Solving Routing Congestion in Hardware

Serial architecture with groups of parallel optimized processing elements

Full utilization of pipelined hardware with alternating blocks E.g. 128x parallelism in commercial IP (FlarionTM) Further memory reduction through staggered decoding

schedule[E. Yeo, P. Pakzad, B. Nikolic, and V. Anantharam, "High throughput low-density parity-check architectures," Proc. IEEE Globecom2001, San Antonio, TX, pp.3019-24, Nov 2001. ]

Bank ofMemories

Variable-to-Check PEs

CrossbarSwitch

SoftInputs

Bank ofMemories

Check-to-Variable PEs

SoftOutputs

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Platform vs. Throughput Summary

103 104 105 106 107 108 109

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Outline

LDPC Codes Soft Decoding of LDPC Codes Parallel vs. Serial Architectures Platforms Methods for Code Construction

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Density Evolution

Density Evolution Very good codes (< 0.0045dB from theoretical

bound)

Large variable edge degree (~ 100)

Large block size (107)

Cayley and Ramanujan Graphs Unstructured interconnects

Algebraic Constructions Cyclic or quasi-cyclic properties

Use of shift registers

Parallel implementation has to address sparse code / interconnect issue.

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Summary

Difficulties with routing or memory requirement

Parallel architectures are optimal for power/ throughput efficiency

Different platforms (microprocessor/FPGA/ASIC) offers possibilities for various applications

Methods for code construction need to consider implementability

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END

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Low Density Parity Check Codes (LDPC)

• LDPC representation by bi-partite graph.

• Non-zero entries in each

• Row m represent the set of bits that are connected to check m, (m) = {n: Hm,n = 1}

• Column n represent the set of checks that are connected to bit n. (n) = {m: Hm,n = 1} is

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Sparse Graph

Limited spatial locality between input edges

Rearrangement of nodes has limited effect in improving spatial locality

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...

PEVC,1PECV,1

PEVC,2

PEVC,3

PEVC,4

Memory

PECV,2

PECV,3

PECV,4

Memory...

Memory

...

LDPC Decoding Algorithms

Pipeline stall delays

Throughput hardly increased

Limited spatial locality

• Sparse graph

Hardware Pipelining of serial architecture

STA

LL!!

Traditional DSP Algorithms

• e.g. FFT, Digital Filters

Throughput increases

High spatial locality

...

ButterflyUnit

Memory

Memory...

Memory

...

ButterflyUnit

ButterflyUnit

ButterflyUnit

...

ButterflyUnit

ButterflyUnit

ButterflyUnit

ButterflyUnit