Continuous Phase Modulation Signals Research Articles

New Results on the Spectral Analysis of Multi-h CPM Signals

This paper deals with the spectral analysis of multi-h continuous phase modulation (CPM) signals. A new solution to the problem is obtained by identifying an inner hidden Markov model in the CPM signal, which can be accomplished by expressing the modulator as the cascade of a sequential machine (SM), and a bank of PAM modulators. An in-depth study of the resulting Markov chain shows that it is both cyclostationary and reducible, and proper tools are identified to address these peculiarities. As a result, the multi-h CPM spectrum is expressed as a combination of Fourier transforms of portions of a CPM signal, thus allowing an efficient numerical evaluation through fast Fourier transforms (FFTs).

Equalization algorithms in the frequency domain for continuous phase modulations

In this paper, novel equalization algorithms for continuous phase modulations (CPMs) are illustrated. Both conventional (linear and decision-feedback) and turbo equalization techniques are derived using the Laurent decomposition of CPM signals. All of them operate in the frequency domain and process two samples of the received signal per channel symbol. Numerical results show that on one hand, conventional equalization strategies offer good performance for binary partial response signaling over severely frequency-selective wireless channels at a moderate complexity. On the other hand, there is evidence that turbo techniques provide a small energy saving at the price of a substantial computational burden.

Non-stationary signal processing using time-frequency filter banks with applications

We present a new approach to the design of time-frequency (TF) filter banks for non-stationary noisy signals. In this work, we concentrate on multicomponent signals represented by the minimum cross-entropy TF distribution (MCE-TFD). The proposed method is based on an array of time-varying filters, each filter processes one component of the signal according to its specific TF support. The output of the filtering algorithm is a set of the signal components. Two applications of the proposed algorithm are considered. The first application is of interference mitigation of continuous phase modulation (CPM) signals which have been extensively used in recent years, especially for wireless communications. The second application is cancellation of wideband interference in direct-sequence code division multiple access (DS-CDMA) communications, based on the MCE-TFD and the filter banks approach.

Minimal PAM Decompositions of CPM Signals With Separable Phase

This letter proposes a new pulse amplitude modulation decomposition of continuous phase modulation (CPM) signals under the constraint of phase response separability. The separability condition is met by a broad class of CPM signals, including full-response signals. We show that the proposed decomposition has minimal cardinality of N/sub c/=L(M-1), where M is the alphabet size and L is the CPM memory length. The cardinality increases linearly with L, while previous proposed decompositions have cardinalities M/sup L/-M/sup L-1/, which grow exponentially with L.

PAM Decomposition of<tex>$M$</tex>-ary Multi-<tex>$h$</tex>CPM

It is known that any multilevel continuous phase-modulated (CPM) signal with a single modulation index can be exactly represented by a sum of pulse-amplitude modulated (PAM) waveforms. In this paper, we show how multi-h CPM signals can also be represented in this manner. The decomposition is presented in general terms as a function of the alphabet size, modulation indexes, and phase pulse of the CPM scheme. The number of pulses required to exactly construct the signal is shown to increase over that previously given for single-h schemes; this increase is in proportion to the number of modulation indexes. We propose an approximation which significantly reduces the number of signal pulses and which minimizes the mean-squared error for an arbitrary set of modulation indexes. We show that this approximation can have two objectives: 1) to reduce the number of pulses in the same manner as has been proposed for single-h schemes; and/or 2) to reduce the number of multi-h pulses; we also show the conditions where this latter objective is most practical. We compare this minimum mean-squared error approximation with another method which was recently proposed for CPM. We also give numerical results on detection performance which demonstrate the practicality of the proposed approximation.

Frequency-domain equalization for continuous phase modulation

Single-carrier frequency-domain equalization (SC-FDE) is proposed for continuous phase modulation (CPM). Two different discrete representations of the CPM signal are obtained. One is an orthogonal representation obtained from the Gram-Schmidt orthonormalization procedure, and the other is obtained from Laurent's decomposition for binary CPM. The methodology for maintaining phase continuity with a cyclic guard interval is presented. Various optimal and suboptimal reduced-complexity approaches are suggested for SC-FDE of CPM signals. General full- and partial-response CPM waveforms, and specific examples of minimum-shift keying (MSK) and Gaussian MSK (GMSK) with BT=0.3 are investigated. Simulation results show that SC-FDE for CPM is an effective approach for CPM on channels having long impulse responses.

MMSE-Optimal Approximation of Continuous-Phase Modulated Signal as Superposition of Linearly Modulated Pulses

The optimal linear modulation approximation of any M-ary continuous-phase modulated (CPM) signal under the minimum mean-square error (MMSE) criterion is presented in this paper. With the introduction of the MMSE signal component, an M-ary CPM signal is exactly represented as the superposition of a finite number of MMSE incremental pulses, resulting in the novel switched linear modulation CPM signal models. Then, the MMSE incremental pulse is further decomposed into a finite number of MMSE pulse-amplitude modulated (PAM) pulses, so that an M-ary CPM signal is alternatively expressed as the superposition of a finite number of MMSE PAM components, similar to the Laurent representation. Advantageously, these MMSE PAM components are mutually independent for any modulation index. The optimal CPM signal approximation using lower order MMSE incremental pulses, or alternatively, using a small number of MMSE PAM pulses, is also made possible, since the approximation error is minimized in the MMSE sense. Finally, examples of the MMSE-optimal CPM signal approximation and its comparison with the Laurent approximation approach are given using raised-cosine frequency-pulse CPM schemes.

On the blind estimation of the parameters of continuous phase modulated signals

In this paper, a blind estimator of the technical parameters of continuous phase modulated (CPM) signals is proposed. It consists in estimating jointly the modulation index, the symbol period and the frequency offset. It is based on the following observations. First, the inverse of the index is the smallest positive real number a CPM signal should be raised to in order to generate a deterministic harmonic signal; second, the frequencies of the harmonic signal are simply related to the symbol period and the carrier frequency. The practical implementation of this joint estimator is described and the asymptotic behavior of the estimation error is studied. If N is the number of signaling intervals, the estimate of the modulation index is shown to converge to a non-Gaussian distribution at rate 1/N, while the estimate of the frequency offset and the estimate of the symbol period converge at rate 1/N/sup 3/2/. We also investigate the case where the modulation index and the symbol period are available at the receiver side. An estimator of the frequency offset is proposed by adapting the above joint estimator to the latter case. The asymptotic behavior of this estimator is studied and compared with the case where all of the parameters are unknown, so as to evaluate the possible degradation of the performance due to the ignorance of certain technical parameters. Simulations results sustain our theoretical claims.

Reduced complexity receivers for layered space-time CPM

Layered space-time (LST) transmissions employing continuous phase modulations (CPM) are well motivated for both bandwidth- and power-limited multiantenna communications. However, one of the major challenges for LST-CPM is the high complexity it incurs with maximum likelihood (ML) detection. In this paper, we develop reduced complexity LST-CPM receivers. First, we consider single antenna systems. Specifically, we study a reduced complexity Viterbi receiver for binary CPM. Based on this design, we introduce differential encoding for a class of CPM signals and analyze its performance gain both theoretically and with simulations. Second, we focus on multiantenna LST systems with minimum shift-keying (MSK)-type modulations. With group nulling-canceling (NC) and low-complexity linear equalization, we convert a coded multiuser detection problem into an uncoded one with small equalization loss. We also find that the combination of sphere decoding with hard-decision iterative processing is effective in boosting performance with a controllable complexity increase. Both analytical and simulated performance confirm that the novel LST-MSK receiver exhibits markedly improved performance relative to conventional NC detectors with moderate complexity increase.

Blind equalization of turbo trellis-coded partial-response continuous-phase modulation signaling over narrow-band Rician fading channels

In this paper, a blind maximum-likelihood channel estimation algorithm is developed for turbo trellis-coded/continuous-phase modulation (TTC/CPM) signals propagating through additive white Gaussian noise (AWGN) and Rician fading environments. We present CPM for TTC signals, since it provides low spectral occupancy and is suitable for power- and bandwidth-limited channels. Here, the Baum-Welch (BW) algorithm is modified to estimate the channel parameters. We investigate the performance of TTC/CPM for 16-CPFSK over AWGN and Rician channels for different frame sizes, in the case of ideal channel state information (CSI), no CSI, and BW estimated CSI.

Supplementary Proof for “Exact and Approximate Construction of Digital Phase Modulations by Superposition of AMP” by P. A. Laurent

In this letter, we supplement the derivation in Laurent's paper by mathematically proving that combining binary continuous phase modulation (CPM) signals in different bit durations results in the decomposed CPM expression consisting of a set of continuous amplitude-modulated pulse functions.

Fractional Multi-bit Differential Detection Technique for Continuous Phase Modulation

A new low-complexity differential detection technique, fractional multi-bit differential detection (FMDD), is proposed in order to improve the performance of continuous phase modulation (CPM) signals such as Gaussian minimum shift keying (GMSK) and Gaussian frequency shift keying (GFSK). In comparison to conventional one-bit differential detected (1DD) GFSK, the FMDD-employed GFSK provides a signal-to-noise ratio advantage of up to 1.8 dB in an AWGN channel. Thus, the bit-error rate performance of the proposed FMDD is brought close to that of an ideal coherent detection while avoiding the implementation complexity associated with the carrier recovery. In the adjacent channel interference environment, FMDD achieves an even larger SNR advantage compared to 1DD.

Non Data-Aided Estimation of the Modulation Index of Continuous Phase Modulations

In this paper, a new non data-aided estimator of the modulation index of continuous phase modulated (CPM) signals is proposed. It is based on the observation that the inverse of the index is the smallest positive real number a CPM signal should be raised to in order to generate a sinusoid of period 2T, where T is the symbol period. The asymptotic behavior of the estimator is studied. If N is the sample size, the estimation error is shown to converge to a non-Gaussian distribution at a rate of 1/N. Simulation results sustain the conclusions of the theoretical asymptotic analysis.

Blind and semi-blind equalization of CPM signals with the EMV algorithm

We apply the expectation-maximization Viterbi algorithm (EMVA) introduced in a previous paper to the blind or semi-blind maximum-likelihood equalization of continuous-phase modulated (CPM) signals transmitted over a noisy linear finite impulse response (FIR) channel. The EMVA is a computationally efficient technique for simultaneously performing system identification and signal detection whenever the transmission system admits a hidden Markov model (HMM) description. The convergence properties of the EMVA are examined and a method for monitoring the EMVA convergence rate online is presented. It is shown that the order of the FIR channel can be estimated by applying the order-incrementing (OI) and fixed-order (FO) methods proposed previously. A Cramer-Rao bound is derived for the channel impulse response, and it is shown via simulations that the EMVA approaches this bound as the SNR increases. Finally, an EMVA-based technique is described for estimating the error covariance matrix for the system parameters for the case of long observation blocks.

Digital audio broadcasting in the fm band based on continuous phase modulation

A method for broadcasting digital audio signals simultaneously with existing analog frequency modulation radio (88-108 MHz) in adjacent channels is presented. The digital transmission is based on continuous phase modulation (CPM) and a proper reduced-state sequence estimator. With the proposed method, the power level and the symbol rate of the transmitter signal is determined in a manner that the interference the CPM signal poses for the analog FM signal in adjacent channels remains below a level according to the radio frequency emission mask defined by international rules. Due to the multipath propagation of the transmitted signal, the transmission behavior of the radio channel is determined by high dispersion up to 85 /spl mu/s. With the selected bit rate, the receiver has to cope with a channel memory of up to 17 bits. Since Viterbi detection is not feasible due to the number of channel states, detection is performed by a reduced-state sequence estimator that is able to eliminate the complete channel interference by decision feedback. Simulation results show that the detector almost achieves the detection quality of the optimum receiver. CPM achieves data rates of up to 200 kb/s inside a 200 kHz FM channel, which is sufficient for transmission of digital compressed audio signals at compact disc quality. The encouraging results of field tests will be published in another paper.

Space-time code design with continuous phase modulation

This paper addresses the space-time code design for Rayleigh-fading channels using continuous phase modulation (CPM). General code construction is desirable due to the nonlinearity and inherent memory in CPM signals which make hand design or computer search computationally impractical. Several sufficient conditions for full rank are identified for CPM signaling schemes and for linear representations of CPM. Simulation results verify the resulting performance.

A BEM-based detector for CPM signals transmitted over frequency-flat fading channels

In this paper, the expectation-maximization (EM) algorithm for maximum a posteriori (MAP) estimation of a random vector is applied to the problem of symbol detection for continuous phase modulation signals transmitted over time-selective Rayleigh-fading channels. This results in a soft-in-soft-out detection algorithm suitable for iterative detection/decoding schemes. Simulation results show that the error performance provided by the proposed solution is very close to that of a MAP detector endowed with an ideal knowledge of the channel state both in uncoded and coded transmissions.

The PAM decomposition of CPM signals with integer modulation index

This article presents the solution for decomposing continuous-phase modulated (CPM) signals with integer modulation index into pulse-amplitude modulated components. The notion of main complex pulse is also introduced. A simplified demodulator for CPM signals with integer modulation index is proposed as an application example and simulation results using a quaternary 2 raised cosine (RC) scheme are given.

New lower bounds on symbol error probability for MLSD of CPM signals on AWGN channel

Symbol error probability bounds for maximum likelihood sequence detection (MLSD) of continuous phase modulation (CPM) signals are studied. The calculation of the upper bound is based on the transfer function technique, which has been generalised. A new method for constructing a lower bound for CPM systems is proposed. From numerical comparisons, it can be seen that the proposed algorithm can substantially improve the lower bound compared to previous approaches by considering the entire set of transmitted data sequences and not only the worst case. This generalised algorithm may be applied to any system that can be described as a finite-state machine.

Iterative demodulation and decoding of high‐order CPM signals in Rician fading channels

AbstractSpace communications require high data rate information transmission at low bit error rate (BER). In this paper, we present a pilot symbol‐assisted, iterative demodulation and decoding system for high‐order continuous phase modulated (CPM) signals. We consider CPM signals because they are especially suitable for space communications due to their high spectral efficiency and constant envelope property.Coherent demodulation is adopted because of its superior BER performance over non‐coherent demodulation like differential detection. To track the time‐varying channel state information (CSI) caused by the Doppler effect in space communications, pilot symbols are inserted periodically in the data stream to serve as references for channel estimation. Due to the inherent memory in continuous phase modulation, those pilot symbols are set to be data dependent so that the CPM phase trellis can be forced to some pre‐defined reference state at certain time epochs. In addition, pilot symbols are used as parity check symbols in demodulating each pilot frame to help reduce decoding errors. Powerful forward error correcting codes (convolutional codes) are then combined with the phase trellis of the CPM signal to form a structure similar to serial concatenated convolutional codes so that the BER performance can be significantly improved by applying iterative processing. Compared with other joint detection and decoding systems, the proposed system has high power‐bandwidth efficiency and low computational complexity by using pilot symbols to separate channel estimation from demodulation, which is even more significant in high‐order modulation.To further improve the receiver performance, information is exchanged between the CPM demodulator, the convolutional decoder, and the channel estimator. First, the channel estimator uses the information from CPM demodulator to construct an estimate of the phase trajectory of the transmitted signal and filters all received samples on this estimated phase trajectory to obtain a more accurate estimation of the CSI. Then the CPM demodulator combines this newly estimated CSI with the extrinsic information from the convolutional decoder to generate more reliable soft decisions, which is to be used as input to the convolutional decoder. This process can be carried out iteratively to improve the BER performance. Although there may be some errors in reconstructing the phase trajectory which will cause an increase in channel estimation errors, the number of errors will decrease at high signal‐to‐noise ratio and the accuracy of the estimated CSI will improve due to the increased sampling rate.At last, it should be pointed out that although the proposed system is designed for CPM signals, it applies in general to other signals modulated with memory such as differential phase‐shift keying. We have already obtained similar results for differential QPSK in Rician fading channels. Copyright © 2002 John Wiley & Sons, Ltd.