Bridging the Gap Between

Parallel and Serial Concatenated Codes

Naveen ChandranCMG plc

Richmond, VA

Matthew C. Valenti (presenter)Lane Dept. of Comp. Sci. & Elect. Eng.

West Virginia University

This work was supported by the Office of Naval Researchunder grant N00014-00-0655

Overview Review of Concatenated Convolutional

Codes Parallel (PCCC) vs. serial (SCCC) concatenation

PCCC’s are a special case of SCCC’s In other words, SCCC’s are a generalization of

PCCC’s. It is possible to modify a SCCC encoder to

make it produce a PCCC. Illustrative proof Implications

A new class of hybrid concatenated codes Simulation results

Turbo Codes Key features:

Concatenated Convolutional Codes. PCCC: Parallel Concatenated Convolutional Codes. SCCC: Serial Concatenated Convolutional Codes.

Nonuniform interleaving. Recursive encoding.

RSC: Recursive Systematic Convolutional Codes. For PCCC both encoders are RSC. For SCCC at least the inner encoder is recursive.

Iterative decoding algorithm. MAP/APP based.

“SISO” Soft-Input, Soft-Output Log-MAP: In logarithmic domain.

PCCC’s Features of parallel concatenated

convolutional codes (PCCC’s): Both encoders are RSC. Performance close to capacity limit for BER

down to about 10-5 or 10-6. BER flooring effect at high SNR.

RSCEncoder #1

RSCEncoder #2

NonuniformInterleaver

Input

ParityOutput

Systematic Output

ix

SCCC’s Features of serially concatenated

convolutional codes (SCCC’s): Inner encoder must be recursive.

Could even be just a differential encoder. Outer encoder can be recursive or nonrecursive. Performance not as good as PCCC’s at low SNR. However, performance is better than PCCC’s at

high SNR because the BER floor is much lower.

Outer Encoder

Inner Encoder

NonuniformInterleaver

Input OutputOptional

Puncturing

Performance Comparison AWGN channel Parameters:

Rate = ⅓ Frame size = 512

bits K=5 RSC encoders Spread interleaver Log-MAP decoder

0 0.5 1 1.5 2 2.5 3 3.5

10-6

10-4

10-2

100

Eb / No in dB

BE

R

SCCC

PCCC

Key Observation PCCC’s are actually a subclass of

SCCC’s PCCC’s are, in fact, a particular type of

SCCC. Equivalently, SCCC’s are a generalization of

PCCC’s. Thus, a PCCC can be encoded by a SCCC

encoder. However, this requires a restriction to be

placed on the SCCC.

PCCC Encoding Using a SCCC Encoder

Requirements for the SCCC encoder: Encoder restriction

Both inner and outer encoder are RSC. Interleaver restriction

Interleaver must output all of the outer encoder’s systematic bits before it outputs any of its parity bits.

Puncturing restriction The “double parity” bits must be punctured.

Outer Encoder

Inner Encoder

NonuniformInterleaver

Input OutputOptional

Puncturing

An Alternative Representation

Because of the interleaver restriction and the fact that both encoders are systematic:

Outputs constitute a rate ¼ SCCC.

Outputs constitute a rate ⅓ PCCC.

RSCEncoder

#1

s

p

RSCEncoder

#2

uu

p1~p1

~u ~ ~u p 1

~ ~u p 1

p p2s

2p

equivalentinterleavers

~ ~u p ps 1 2

~ ~u p p ps p 1 2 2

alternately puncture for rate ⅓ SCCC

Equivalent PCCC Encoder If is not transmitted, then the

encoder can be expressed as a PCCC encoder

p2p

RSCEncoder

#1

s

RSCEncoder

#2

u s

p

u

p1~p1

~u

~ ~u p ps 1 2

~up2

s

Only difference with standard PCCCis that this part is interleaved

Is It Really a PCCC?

0 0.5 1 1.5 2 2.5 3 3.5

10-6

10-4

10-2

Conventional PCCC

PCCC from SCCC codec

Es / No in dB

BE

R

Parameters: Rate = ⅓ Frame size = 512 bits K=5 RSC encoders Log-MAP decoder

No apparent performance loss due to using the interleaver restriction.

SCCC Performance Loss Due to Interleaver

Restriction?

0 0.5 1 1.5 2 2.5 310

-8

10-6

10-4

10-2

100

Eb / No in dB

BE

R

Conventional SCCC

SCCC with interleaver structuring

Parameters: Rate = ⅓ Frame size = 512 bits K=5 RSC encoders Log-MAP decoder

No apparent performance loss due to using the interleaver restriction.

Implications Because a PCCC code may be encoded

(decoded) by a SCCC encoder (decoder), IC designers should focus on SCCC codecs.

Note however that the SCCC decoder is 1.5 times more complex than the equivalent PCCC decoder.

An incremental redundancy approach can be taken in ARQ data transmissions.

First send the rate ⅓ PCCC. If necessary, send the extra parity to create a rate ¼

SCCC. Y. Wu and M.C. Valenti, “An ARQ technique using

related parallel and serial concatenated convolutional codes,” in Proc. IEEE Int. Conf. on Commun. (ICC), (New Orleans, LA), June 2000.

Hybrid Turbo Codes If we delete all the double parity, we get a

rate ⅓ PCCC code. i.e. maintain field p2

s but drop field p2p

p2p is 100% punctured (p2

s is 0% punctured)

The rate ⅓ SCCC code is created by puncturing alternate parity bits at inner encoder’s output i.e. maintain exactly half of both fields p2

s and p2

p

p2p is 50% punctured (p2

s is 50% punctured)

What if instead we puncture p2p by some

ratio between 50% and 100% ?

Performance of Hybrid Codes

0 0.5 1 1.5 2 2.5 3 3.5

10-8

10-6

10-4

10-2

Hybrid Code B (87.5% puncturing)

Hybrid Code A (75% puncturing)

Conventional PCCC and PCCC from SCCC codec

Conventional SCCC andSCCC with Interleaver Structuring

Eb / No in dB

BE

R

Rate = ⅓ Frame size = 512 bits K=5 RSC encoders Log-MAP decoder

Observations Results for larger frame sizes (1K, 2K,

4K, and 8K) are given in the paper. In general,

A double parity puncturing ratio close to 100% gives performance close to PCCC.

A double parity puncturing ratio close to 50% gives performance close to SCCC.

A double parity puncturing ratio of about 80% gives performance halfway between PCCC & SCCC

Relationship to Divsalar and Pollara’s Hybrid Codes Hybrid codes have been previously

proposed by D. Divsalar and F. Pollara, “Hybrid concatenated codes and iterative

decoding,” JPL TDA Progress Report, April 1997.

Our hybrid codes are different Only 1 interleaver and 2 encoders. Similar performance, but at less complexity.

Conclusion An SCCC encoder can be used to encode a

PCCC. This result was used to develop a new class of

hybrid concatenated codes with performance between that of SCCC and PCCC codes.

The decision to use PCCC or SCCC codes no longer needs to be “black and white”; rather a middle ground (shades of “gray”) exists that can give the system designer more flexibility.

Formal guidelines for designing hybrid codes are needed

Gaussian density evolution may be helpful.