When the double helical interleaver is used in the TPC coded OFDM system of Fig.
4.3(b) as the channel interleaver, we have the system shown in Fig. 4.6. As mentioned above, the OFDM outputs are deinterleaved before TPC decoding commences and the decoded hard decisions are sent back to the channel estimator via the double helical interleaver for iterative channel estimation.
Numerical examples presented in this section assume that the TPC used is composed of two extended (32, 26, 4) BCH codes which is the same as the extended Hamming code and the code rate is 0.5625. We use the parameter p = 4 and Proakis C five paths channel model [14] with power profile {0.227, 0.460, 0.688, 0.460, 0.227} at delays equals
57
Figure 4.5: Double helical interleaving of (64, 57, 4) × (64, 57, 4) TPC
to {0, T, 2T, 3T, 4T }. Jakes’ model [13] is used to generate independent fading processes associated with each path.
The effectiveness of helical interleaving on the BER performance is demonstrated in Fig. 4.7, assuming fmT = 0.0001, where fm is the maximum doppler shift and T is the sample period. The dashed-curves represent the performance of the system without channel interleaver and the solid curves represent that of the system with the double helical interleaver. The inclusion of a helical interleaver gives a 3 dB performance gain at BER=10−5. The slope of the performance curve is sharper because the interleaver de-correlates the time and frequency correlations of the slow fading process; see Fig. 4.8.
Fig. 4.8 plots the time-frequency channel response without (a) and with (b) the DHI.
It is clear that the channel response associated with different BCH codewords and code bits are less correlated after interleaving.
In general, a higher modelling order provides more accurate channel description at the cost of larger noise-induced variance as more parameters are involved in the estimation process. At high SNRs the perturbation caused by noise is negligible hence
Figure 4.6: A block diagram of a double helical interleaved OFDM system with iterative joint channel estimation and TPC decoding.
higher-order estimates are preferred if complexity is not of high concern. When SNR is low, however, one would rather use a lower-order estimate as the received vector(s) is corrupted. The choice of the channel model order also depends on the channel dynamic and the modelling block size. For slow fading channels or small modelling blocks, a lower-order estimate is a better choice while for other scenarios, one might consider higher-order estimates. Hence, for a particular channel condition (SNR and coherent time/bandwidth) and modelling block size, there always exists an optimal modelling order.
Fig. 4.9 plots the system performance in a fast-fading channel (fmT = 0.001). The system performance for this channel is better than that in the same channel with smaller Doppler-time product fmT = 0.0001. When perfect channel estimation is available, it gives more than 5 dB improvement at BER=10−5 without DHI and more than 4 dB improvement at BER=10−5 with DHI. For such a fast-fading channel, the first-order model is clearly not sufficient to characterize the channel response. A higher-order model is needed and the simulation results indicate that a 3rd-order model yields the best performance.
At high SNRs, modelling error dominates the performance of the channel estimator [15] and the performance curves shown in Fig. 4.9 suggest that the BER performance is bounded at approximately 2 × 10−5 and 5 × 10−5 for the 3rd-order and 4th-order
6 8 10 12 14 16 18 20 22
Figure 4.7: Performance of DHI-permuted system in a fading channel with fmT = 0.0001.
channel estimators when there is no channel interleaving. With the DHI in place, both the performance and the corresponding bounds are improved–from 5 × 10−5 to 3 × 10−6 for the 4th-order channel estimator.
Pilot planning influences the performance of a system using pilot-assisted synchro-nization and channel estimation. The proposed system uses polynomials to model the true CR. Such a model tends to become less reliable when extrapolating beyond the pilot distribution boundary. For a given time-frequency block (or a time or frequency interval) within which pilots are inserted, the estimated CR tends to be less reliable in places close to the block or interval boundary. One way to remedy such a shortcoming is to place pilots in the modelling block boundary. The following equations, (4.16) and (4.17), define two pilot patterns Pa(i) and Pb(i). The former pattern has pilots on its
0
Figure 4.8: (a) Time-frequency channel response with fmT = 0.0001; (b) Channel re-sponse with DHI permutation.
modelling block edges but not the latter.
Pa(i) = Fig. 4.11 compares the BER performance of the above two pilot patterns. The former pilot pattern results in 0.5 and 4 dB gains at BER=10−4 when 3rd-order and 4th-order models are used. To validate our claim that pilot pattern does affect the system performance and to examine the relation between model order and channel dynamic, we plots CRs of the two pilot patterns of the 3rd-order channel estimate (i.e., the one using a degree-3 polynomial channel model) and the true CR (marked by solid stars) in Fig. 4.12. Fig. 4.13 is similar to Fig. 4.12 except that 4th-order model is used. The square and cross markers on both figures represent CR estimates using two different pilot patterns. It can be seen that high modelling order tends to incur larger estimation errors at positions close to the edge. Using a pilot pattern with pilots in the boundary positions does help reducing the estimation error.
Finally, in Fig. 4.14 we plot the BER performance when the receiver employs joint iterative channel estimation and TPC decoding. A 3rd-order channel estimator is used
0
Figure 4.9: Time-frequency response of the Proakis C five-path fading channel with fmT = 0.001.
and both two- and four-iteration detection are considered. The performance gain with respect to the no-iteration receiver is about 1.0 dB at BER = 10−4 when two-iteration is used and is increased by another 0.3 dB with a four-iteration receiver. As expected, iterative channel estimation does enhances the system performance, bringing about per-formance closer to that of the ideal receiver with perfect channel estimate.
6 8 10 12 14 16 18 20 1E-5
1E-4 1E-3 0.01 0.1
Bits error rate
Average Eb/N0 (dB)
(32,26,4)x(32,26,4) Helical interleaving Perfect CE Order = 1 Order = 2 Order = 3 Order = 4 No interleaving
Perfect CE Order = 3 Order = 4
Figure 4.10: BER Performance comparison for the channel shown in the above figure.
0 2 4 6 8 10 12 14 16 18 20 1E-6
1E-5 1E-4 1E-3 0.01 0.1
Bits error rate
Average Eb/N0 (dB) (32,26,4)x(32,26,4)
Perfect CE Pilot added as Pa,i
Order = 3 Order = 4 Pilot added as Pb,i
Order = 3 Order = 4
Figure 4.11: Performance curve of different pilot insert position in fmT = 0.001 fading channel.
0 10 20
Figure 4.12: Estimated CR using a third-order channel model using pilot patterns (4.16) and (4.17); the true CR (solid star markers) is included for comparison; fmT = 0.001.
0 10 20
Figure 4.13: Estimated CR using a fourth-order channel model using pilot patterns (4.16) and (4.17). The true CR is marked by solid stars; fmT = 0.001.
0 2 4 6 8 10 12 14 16 1E-5
1E-4 1E-3 0.01 0.1
Bits error rate
Average Eb/N0 (dB) (32,26,4)x(32,26,4)
Perfect CE
Order = 3 channel estimation No iteration
Iteration 2 Iteration 4
Figure 4.14: BER performance curves of the receiver with iterative joint channel esti-mation and TPC decoding in a fading channel with fmT = 0.001.
Chapter 5
Parallel Concatenated Product Codes
We extend the concept of Pyndiah’s block turbo codes and propose a new class of codes based on product codes called parallel concatenated product codes whose structure is similar to that of turbo codes [9], replacing the constituent convolutional codes in a turbo code by product codes. This new class of codes provides more flexible choices of code rates and component codes. When simple systematic product codes are used as component codes, the corresponding decoding complexity remains relatively low and the achievable performance outperforms that of a turbo product code with comparable rate at high SNR.
The interleaver we use is the Fibonacci interleaver. Besides its simplicity in imple-mentation, we prove that such an interleaver guarantees that the resulting PCPC has a minimum distance larger than that of its constituent product codes if a proper code length is selected.
5.1 Encoder
The structure of parallel concatenated product codes (PCPC) is shown in Fig. 5.1 in which “TPC encoder” is the encoder of a product code given in Fig. 1. A codeword
TPC encoder
TPC encoder
π
Figure 5.1: The structure of the PCPC encoder.
consists of the product codeword of the upper branch and the parity part of the lower branch output. The only interleaver considered here is the Fibonacci interleaver.