1. 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 N

2. 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

3. 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.

4. 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

5. 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

6. Performance Comparison10
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

7. 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.

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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.

14. 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% ?

15. 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

16. 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

17. 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.

18. 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.