Bridging the Gap Between Parallel and Serial Concatenated Codes Naveen Chandran CMG 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 Research under 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. Systematic Output Input RSC Encoder #1 Parity Output RSC Encoder #2 Nonuniform Interleaver

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. Input Output Outer Encoder Nonuniform Interleaver Inner Encoder Optional Puncturing

Performance Comparison 10 AWGN channel Parameters: Rate = ⅓ Frame size = 512 bits K=5 RSC encoders Spread interleaver Log-MAP decoder SCCC -2 10 -4 10 BER PCCC -6 10 0.5 1 1.5 2 2.5 3 3.5 Eb / No in dB

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. Input Output Outer Encoder Nonuniform Interleaver Inner Encoder Optional 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. RSC Encoder #1 RSC Encoder #2 equivalent interleavers alternately puncture for rate ⅓ SCCC

Equivalent PCCC Encoder If is not transmitted, then the encoder can be expressed as a PCCC encoder RSC Encoder #1 Only difference with standard PCCC is that this part is interleaved RSC Encoder #2

Is It Really a PCCC? Parameters: Rate = ⅓ Frame size = 512 bits K=5 RSC encoders Log-MAP decoder No apparent performance loss due to using the interleaver restriction. -2 10 -4 10 Conventional PCCC PCCC from SCCC codec BER -6 10 0.5 1 1.5 2 2.5 3 3.5 Es / No in dB

SCCC Performance Loss Due to Interleaver Restriction? 10 Parameters: Rate = ⅓ Frame size = 512 bits K=5 RSC encoders Log-MAP decoder No apparent performance loss due to using the interleaver restriction. -2 10 Conventional SCCC BER -4 10 -6 10 SCCC with interleaver structuring -8 10 0.5 1 1.5 2 2.5 3 Eb / No in dB

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 p2s but drop field p2p p2p is 100% punctured (p2s 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 p2s and p2p p2p is 50% punctured (p2s is 50% punctured) What if instead we puncture p2p by some ratio between 50% and 100% ?

Performance of Hybrid Codes Rate = ⅓ Frame size = 512 bits K=5 RSC encoders Log-MAP decoder Hybrid Code A (75% puncturing) -2 10 -4 10 Conventional SCCC and SCCC with Interleaver BER Structuring Conventional PCCC and PCCC from SCCC codec -6 10 Hybrid Code B (87.5% puncturing) -8 10 0.5 1 1.5 2 2.5 3 3.5 Eb / No in dB

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.