Serial encoding and iterative detection of continuous phase modulations

Continuous phase modulation (CPM) is known for its attractive spectral properties. Furthermore, CPM has the property that the modulation signal maintains a constant envelope, so CPM is effective for satellite communications. Recently, generalized nonlinear CPM (GNCPM), which can achieve a larger minimum Euclidean distance than ordinary CPM, was introduced. In this paper, we propose a convolutional coded GNCPM to improve the bit error rate (BER) performance of uncoded GNCPM without expanding bandwidth. Combination CPM with convolutional coding causes an increased number of phase trajectory patterns which can cause the bandwidth expansion. Therefore, in our proposed model, to control the bandwidth, we pay attention to the phase trajectory pattern which is already owned by uncoded GNCPM and change the modulation index appropriately, depending on the input symbols, to keep the number of phase trajectory patterns the same as in uncoded GNCPM. First, we consider the most suitable assignment of the modulation index which can increase the Euclidean distances without bandwidth expansion. Next, we theoretically derive the upper bound on the error event probability of the proposed model in additive white Gaussian noise (AWGN). The performance of the proposed convolutional GNCPM, obtained by computer simulation, shows that the proposed model can improve BER performance without bandwidth expansion. © 1998 Scripta Technica. Electron Comm Jpn Pt 1, 81(8): 42–50, 1998

This paper consists of an overview of recent results on combined constant amplitude continuous phase modulation (CPM) and convolutional channel codes. We also compare the best coded CPM systems to trellis-coded amplitude/phase modulations. The ideal additive white Gaussian channel and systems with perfect carrier phase and symbol time synchronization are considered in this paper. Signals which have a constant amplitude at all times are sometimes needed when highly nonlinear amplifiers are used. Memory is introduced into the constant amplitude signals by means of continuous phase at all times, controlled intersymbol interference to further smooth the phase changes and combinations of CPM and convolutional codes. Analytical tools for calculating the power spectrum and upper bounds on the error probability are given. Results for the special case of Continuous Phase Frequency Shift Keying (CPFSK) with up to 256 levels are considered. Some new results on bounds on more advanced coded systems of higher complexity where the coding is performed jointly over several consecutive channel symbols are also given. Current research topics such as trellis-coded modulation for fading channels, and practical implementations and applications are also briefly discussed.

In order to understand the optimal parameter settings of spectrally efficient continuous phase modulation (CPM), we elaborate on our previous work on numerically optimized CPM. An empirical model is developed based on an expression for the effective bandwidth of CPM, an approximate expression for the smallest Euclidean distances of spectrally efficient CPM and observed properties of the numerically optimized phase pulses. We find that the optimized phase pulses, which resulted from a joint optimization of modulation index and phase pulse, share common properties for different alphabet sizes and different spectral efficiencies. These properties are described by three phase pulse parameters, and we show that they attain approximately equal values at the so-called saturation length. The model can predict the minimum Euclidean distance at the saturation length for a CPM system using an even-integer alphabet and jointly optimized modulation index and phase pulse, and it partly extends to less spectrally efficient CPM systems.

In this paper, we propose a concatenated code by combining a Recursive Systematic Convolutional (RSC) code with a Low-Density Parity-Check (LDPC) code. The proposed concatenated code is encoded in parallel by an RSC encoder and an LDPC encoder without interleavers between them. When decoding the proposed concatenated code, the information bits are decoded by two decoding algorithms, and soft information is exchanged between the RSC decoder and the LDPC decoder. This allows for the elimination of wrong codewords output by each decoder. The use of an LDPC code allows us to halt the decoding process when the valid codeword is obtained. We evaluate the error rate performance of the proposed concatenated code by computer simulations. We show that the proposed concatenated code does not have a high error floor, and achieves better BLock Error Rate (BLER) performance than either conventional LDPC codes or conventional turbo codes in the high Eb/No region. We also show that since the proposed concatenated code can correct wrong codewords output by each decoder, it achieves better BLER than the regular LDPC code at the same Bit Error Rate (BER).

A new concept of Bit Geometrical Uniformity (BGU) is introduced to help the analysis and estimation of performance of systems with Iterative Detection/Decoding (ID). It is also shown that the BGU labelings of the signal constellation allows a fast search for best component binary Recursive Systematic Convolutional (RSC) codes of a BICM-ID system using both of overall and in-line interleaving. Best rate-1/2 RSC codes are reported for 4PSK constellation as an example. The new concept and new search method can be directly extended to MPSK and MQAM constellations.

The method of multiple trellis-coded modulation is applied to continuous phase modulation (CPM). Double trellis-coded quaternary CPM schemes using convolutional codes with an effective rate of 2/4 are considered. Tables of optimum codes and minimum Euclidean distances are given for several frequency impulses and ratios of trellis-branches-per-symbol-interval, which have to be processed by a Viterbi-decoder. The results are clearly superior to single trellis-coded schemes with 1/2-rate codes. These improvements are accomplished by a proper selection of double-channel symbols. This method also offers improved receiver implementation properties. For a further reduction of complexity, a modified decision-feedback-sequence estimation is proposed. >

CPM waveforms which are utilized in both MIL-STD-188-181B (181B) and high performance waveform SATCOM operation undergo appreciable distortion due to the effects of the bandlimiting and hard-limiting satellite transponder. In this application, the traditional LMS equalizer can remove some of the intersymbol interference, but at the higher symbol rates, the modem performance is still far off from the same bit error rate achieved in an environment which only contains additive white Gaussian noise. Observation of the SATCOM channel has shown that the CPM waveform is predictable, distorted in phase during symbol transitions. Large phase transitions incur a larger, proportional amount of distortion when compared to smaller phase transitions. A characterization of the phase error as a function of all the possible clockwise and counterclockwise adjacent phase transitions for the CPM phase constellation illuminates this proportion property of the distortion caused by the intersymbol interference. Once this error is characterized for all the symbols, it can be removed via complex multiplication. This method has been shown to outperform the traditional LMS equalizer in this application especially when considering the noise enhancement inherent in the LMS equalizer implementation. This paper documents the simulation comparison of the LMS equalizer and the new non-linear equalizer approach along with actual radio bit error rate curves.

The detection of convolutional codes over rings suitable for continuous phase modulation (CPM) are presented based on the Rimoldi's Decomposition of CPM. Because the ring convolutional codes are on the same ring as the inner codes of CPM, We can use reduced state sequence detection (RSSD) to analyze and detect the overall encoder. Numerical result shows the detection complexity of this RSSD is significantly reduced with a very small degradation in error performance.

Joint source and channel (JSC) coding using combined trellis coded quantization (TCQ) and continuous phase modulation (CPM) is studied. The channel is assumed to be the additive white Gaussian noise (AWGN) channel. Analytical bounds on the channel distortion for jointly designed TCQ/CPM systems with maximum likelihood sequence detection (MLSD) are developed. For a memoryless uniform source, our upper bounds for the investigated TCQ/CPM systems are shown to be asymptotically tight for increasing channel signal-to-noise ratio (SNR) values. It is concluded that the minimum Euclidean distance of the CPM system alone is not enough to evaluate the performance of the combined TCQ/CPM systems. The number of the error events having the minimum Euclidean distance and the total distortion caused by those error events also affect the asymptotic performance of the investigated systems. This work provides an analysis tool to estimate the performance for a given combined TCQ/CPM system. The analysis method is very general, it may be applied to any trellis based JSC coding schemes.

A new class of recursive convolutional space-time codes (ReC-STC) with an arbitrary number of transmit antennas is designed by adopting several parallel two-state recursive systematic convolutional codes (RSCs). An intercross linear mapping rule that distributes the output bits of an RSC to the different positions of different transmitted signals is also described. The proposed ReC-STC can not only increase the data rate with the number of transmit antennas, but also performs well when it is used in a serially concatenated space-time code (SCSTC). The convergence analysis shows that ReC-STC based SCSTC has a lower decoding threshold than the available SCSTC.

We study a serially interleaved concatenated code construction, where the outer code is a standard convolutional code, and the inner code is a recursive convolutional code of rate 1. We focus on the ubiquitous inner differential encoder (used, in particular, to resolve phase ambiguities), double differential encoder (used to resolve both phase and frequency ambiguities), and another rate 1 recursive convolutional code of memory 2. We substantiate analytically the rather surprising result, that the error probabilities corresponding to a maximum-likelihood (ML) coherently detected antipodal modulation over the additive white Gaussian noise (AWGN) channel for this construction are advantageous as compared to the stand-alone outer convolutional code. This is in spite of the fact that the inner code is of rate 1. The analysis is based on the tangential sphere upper bound of an ML decoder, incorporating the ensemble weight distribution (WD) of the concatenated code, where the ensemble is generated by all random and uniform interleavers. This surprising result is attributed to the WD thinning observed for the concatenated scheme which shapes the WD of the outer convolutional code to resemble more closely the binomial distribution (typical of a fully random code of the same length and rate). This gain is maintained regardless of a rather dramatic decrease, as demonstrated here, in the minimum distance of the concatenated scheme as compared to the minimum distance of the outer stand-alone convolutional code. The advantage of the examined serially interleaved concatenated code, given in terms of bit and/or block error probability which is decoded by a practical suboptimal decoder, over the optimally decoded standard convolutional code is demonstrated by simulations, and some insights into the performance of the iterative decoding algorithm are also discussed. Though we have investigated only specific constructions of constituent inner (rate 1) and outer codes, we trust, hinging on the rational of the arguments here, that these results extend to many other constituent convolutional outer codes and rate 1 inner recursive convolutional codes.

In this paper, the performance comparison of serially concatenated Recursive systematic convolutional (RSC) code system with Non-systematic convolutional (NSC) code system is conduct. The two models of serially concatenated convolutional code is tested over Additive White Gaussian (AWGN) channel, Binary Phase-Shift Keying (BPSK) modulation and Viterbi decoder using MATLAB simulation. In first model of serially concatenated convolutional code uses Recursive systematic convolutional (RSC) as outer and inner encoder. In second model of serially concatenated convolutional code uses Non-systematic convolutional (NSC) as outer and inner encoder. The outer and inner encoders are concatenated with the help of interleaver. For decoding a non-iterative Viterbi decoding scheme is performed with two Viterbi decoder are concatenated by de-interleaver. The MATLAB simulation results shows that the performance of RSC-RSC concatenated code gives the better bit error rate (BER) than the NSC-NSC concatenated code.

Some properties of digitally phase modulated (DPM) signals are presented. Phase modulation with (overlapping) pulses generated by a digital FIR filter belong to this class, which may be considered to be a practical approximation to continuous phase modulated (CPM) signals. The power spectra of DPM signals are derived analytically. The ability of these signals to operate through an additive white Gaussian noise channel is assessed by calculating their minimum Euclidean distance. Their noise and spectral properties are found to be similar to those of CPM signals. Assuming that a Viterbi decoder is used to resolve symbol interference, the out-of-band power tends to decrease as the pulse duration increases, and the noise immunity is enhanced. At the same time the receiver complexity grows exponentially. Hence, noise immunity and spectrum compactness are achieved at the cost of higher received complexity. Modems for DPM signals are believed to be easier to implement than those for CPM signals. This is because filter design is simple and a residual carrier component can be retained to facilitate carrier regeneration. Furthermore, the accumulated carrier phase does not need to be continuously evaluated in order to perform matched filtering. The analytic results derived are supported by measurements and simulations.

The paper shows that some continuous phase modulation (CPM) schemes have some properties that allow a near optimum reception with very low complexity. The proposed reception method applies simultaneously three techniques for complexity reduction of CPM receivers, which have been previously object of separate analysis with additive white Gaussian noise (AWGN). These techniques are: first-quadrant metrics calculus on a F-dimensional Walsh space, derivation of the other quadrant metrics using a no loss derivation algorithm and finally the use of constrained complexity maximum likelihood sequence detection (MLSD) using the M-algorithm. The cumulative effect on total performance loss is negligible and no more than the sum of the partial losses due to the Walsh space and the M-algorithm, known to be negligible when observing the respective standalone rules for the complexity reduction limits.

Constant envelope and high spectral efficiency are among the key properties which make continuous phase modulation (CPM) a preferred choice for many applications. Numerous methods are available for designing concatenated CPM systems to achieve additional coding gains. In this paper, we propose a new serial concatenated CPM scheme. Our design uses a multi-alphabet binary CPM as an inner code, a modified turbo code as an outer code, and specially designed interleavers. The extrinsic information transfer (EXIT) chart technique has been employed to assist the outer code design. Using soft decision decoding, we show that our design gives noticeable improvements compared with previous concatenated CPM schemes over additive white Gaussian noise (AWGN) channel.