Average performance bounds based on uniform interleaving are considered for a serial concatenation consisting of an outer code, an interleaver and an inner coded modulation system. The exact input–output weight spectrum is derived for the case when only the outer code has the uniform error property. Thereby, an upper bound to the bit error probability under maximum likelihood detection is obtained. However, numerous examples for serially concatenated continuous phase modulation show that neglecting the non-uniform error property of the inner system gives an accurate and less complex approximation. Other inner coded modulations may behave differently.

The orthonormal basis for the space spanned by a given signal set can be chosen in many different ways. However, when the basis is truncated to fewer dimensions, the quality of the approximated signals differs, depending on the choice of the original basis. We study two energy-related quality measures and show that the optimal lower-dimensional approximation is given by the principal components (PC) method, which is also a simple and efficient alternative to Gram-Schmidt techniques. In addition, we derive and bound the average decrease in squared Euclidean distances over one symbol interval caused by the PC method. This measure is relevant in serially concatenated continuous phase modulation, where a manifold of signal pairs contributes to the bit-error rate for low-to-medium signal-to-noise ratios. By a numerical evaluation, we find that for this measure, the decrease is lower than that of a previous method by J. Huber and W. Liu (see IEEE J. Select. Areas Commun., vol.7, p.1437-49, 1989). Finally, we compare the minimum squared Euclidean distance for error events, which is relevant for uncoded CPM systems. Here, the loss with the PC method is generally larger than with Huber and Liu's method, although examples of the opposite exist.

A convergence analysis of serially concatenated continuous phase modulation (SCCPM) with iterative decoding is presented, based on extrinsic information density evolution. Earlier conclusions from union bounds to maximum likelihood decoding that could not be verified by simulations can now be explained. This includes the effect of more complex outer codes and of inner CPM systems with input weight 1 error events. Due to their trivial feed-forward polynomial, all CPM systems provide good initial decoder convergence. Still the role of the CPM state complexity is not well explained. Finally, bit error rate performance per iteration is estimated from the diagrams, showing good coherence with simulation results.

Serially concatenated and interleaved continuous phase modulation (CPM) with iterative decoding is investigated. An a posteriori probability (APP) algorithm for CPM is developed based on the classic APP algorithm for channel codes. The system is analyzed through upper bounds on the average bit error probability. For coded and interleaved minimum shift keying, the weight spectrum is computed, resulting in a transfer function bound. This is cumbersome for a general CPM system; instead, only the most significant error events contributing to the weight spectrum are identified. Simulations show that, firstly, these events give a satisfactory view of system performance when equal outer codes are used, and secondly, that remarkably good performance can be obtained for some simple systems. Finally, power spectral densities and bandwidths are computed, allowing for a bandwidth/performanee comparison of different combinations.

Serially concatenated continuous phase modulation (SCCPM) with iterative decoding is analyzed and simulated. Earlier results on outer rate 1/2 convolutional codes combined with inner binary and quaternary CPM systems are extended to rate 2/3 and 3/4 codes with low-complexity octal and hexadecimal CPM systems, having the 2RC and 3RC frequency pulses. Analysis and simulations show that-among the evaluated systems-the octal 2RC systems provide the highest power/bandwidth efficiency. As an example, rate 2/3-coded octal 2RC with modulation index 1/4 provides a 5.7 dB gain over uncoded minimum shift keying at a bit error rate of 10/sup -3/ and an input delay 4096 bits, without any bandwidth expansion.