Unknown Carrier Phase Research Articles

A Refinement to the Viterbi-Viterbi Carrier Phase Estimator and an Extension to the Case With a Wiener Carrier Phase Process

We provide a theoretical foundation for further analysis and optimization of the Mth-power (MP) carrier phase estimator for MPSK modulation. Also known as the Viterbi-Viterbi (VV) estimator, it is commonly used in practice because it leads to low-latency receiver implementations. The MP carrier phase estimator first raises the received noisy signal samples to the Mth-power to remove the unknown phase modulation, and then extracts the unknown carrier phase of the mid-symbol using a weighted sum of these modulation-wiped-off received signal samples over a symmetrical observation window. Our starting point is the single-term, complex exponential expression for a complex sinusoid received in complex, additive, white, Gaussian noise (AWGN), which leads to a great deal of simplicity in dealing with arbitrary powers of the noisy received signal sample when compared with the conventional approach of raising the sum of signal plus noise to higher powers. The single-exponential expression enables us to first optimize the weighting coefficients of the MP carrier phase estimator with respect to the statistics of the AWGN, in a manner much simpler than previous approaches. Then, it enables us to apply the linear minimum mean square error (LMMSE) criterion to optimize the MP estimator with respect to both the statistics of the AWGN and the carrier phase noise that we model here as a Wiener process. Although the LMMSE MP estimator is computationally intensive for online implementation, a much less complex version is suggested that can be efficiently implemented in real time. Extensive simulation results are presented to demonstrate the improved performance of the LMMSE MP estimator over the conventional MP estimator. By using a sufficiently long symmetrical observation window, the LMMSE estimator does not suffer from the block length effect, which leads to much performance gain over the VV/MP estimator especially at high signal-to-noise ratio (SNR) and high phase noise. A phase unwrapping algorithm is also presented for accurate unwrapping of the estimated carrier phase before it is used in data detection. The proposed LMMSE carrier phase estimator is suitable for implementing a coherent receiver at all SNRs.

Bounding the quantum limits of precision for phase estimation with loss and thermal noise

We consider the problem of estimating an unknown but constant carrier phase modulation $\theta$ using a general -- possibly entangled -- $n$-mode optical probe through $n$ independent and identical uses of a lossy bosonic channel with additive thermal noise. We find an upper bound to the quantum Fisher information (QFI) of estimating $\theta$ as a function of $n$, the mean and variance of the total number of photons $N_{\rm S}$ in the $n$-mode probe, the transmissivity $\eta$ and mean thermal photon number per mode ${\bar n}_{\rm B}$ of the bosonic channel. Since the inverse of QFI provides a lower bound to the mean-squared error (MSE) of an unbiased estimator $\tilde{\theta}$ of $\theta$, our upper bound to the QFI provides a lower bound to the MSE. It already has found use in proving fundamental limits of covert sensing, and could find other applications requiring bounding the fundamental limits of sensing an unknown parameter embedded in a correlated field.

Noncoherent Multiple-Symbol Detector of Binary CPFSK in Physical-Layer Network Coding

It is well known that the binary continuous phase frequency shift keying (CPFSK) signal is power efficient with constant envelope, because it allows transmission with inexpensive and energy efficient nonlinear amplifiers. Recently, physical-layer network coding (PNC) using the binary CPFSK is considered as a potentially attractive solution to the dual needs of high throughput and high power efficiency in two-way relay wireless network. However, most recent studies on PNC have assumed that the carrier phases of the two transmitted signals arriving at the relay are ideally aligned. In fact, the carrier phase offset between the two node-signals always exists in practice. In this letter, a noncoherent multiple-symbol detector of binary CPFSK signals with unknown carrier phase offset in physical-layer network coding is proposed, which is based on the log-likelihood ratio test. It is shown by simulation that the algorithm with five symbol observation interval achieves 6.7 dB performance gain, compared with only one symbol observation interval.

Sequence detection for MPSK/MQAM with adaptive phase tracking

In [P.Y. Kam, P. Sinha, A Viterbi-type algorithm for efficient estimation of MPSK sequences over the Gaussian channel with unknown carrier phase, IEEE Trans. Commun. COM-43 (1995) 2429–2433], we developed a Viterbi algorithm for efficient detection of M -ary phase-shift keyed (MPSK) sequences received over the additive, white, Gaussian noise (AWGN) channel with an unknown carrier phase. By assuming the carrier phase is constant over some L symbol intervals and using the sequence detection metric in [P.Y. Kam, Maximum-likelihood digital data sequence estimation over the Gaussian channel with unknown carrier phase, IEEE Trans. Commun. COM-35 (1987) 764–767] as the decoding metric, it approaches the performance of coherent detection, provided the observation interval L for forming the decoding metric is sufficiently long. However, this observation interval L is fixed, and chosen based on prior knowledge of the carrier phase characteristics. Thus, the metric is nonadaptive, and cannot be optimized when a priori statistical knowledge of the carrier phase is not available. The work here renders it adaptive by developing a recursive metric that is adapted on-line based on the received signals, without prior knowledge of the carrier phase characteristics. We first derive a metric for the maximum likelihood sequence detector for a general M -ary quadrature amplitude modulation (MQAM) sequence received in the presence of AWGN and an unknown, slowly- and randomly-varying carrier phase. To reduce the receiver complexity, a practical, suboptimal metric is obtained. A recursive form of the metric using adaptive reference estimation as in [P.Y. Kam, K.H. Chua, X. Yu, Adaptive symbol-by-symbol reception of MPSK on the Gaussian channel with unknown carrier phase characteristics, IEEE Trans. Commun. COM-46 (1998) 1275–1279], is proposed. Comparison between the proposed adaptive, recursive metric and those in the literature is provided. Simulation results using a random-walk carrier phase model show the superior performance of the adaptive sequence detector over the original nonadaptive sequence detector of [P.Y. Kam, P. Sinha, A Viterbi-type algorithm for efficient estimation of MPSK sequences over the Gaussian channel with unknown carrier phase, IEEE Trans. Commun., COM-43 (1995) 2429–2433].

Non-coherent detectors for quadrature-multiplexed continuous phase modulation signals

Here, non-coherent detectors for quadrature-multiplexed continuous phase modulation (QM-CPM) signals are developed (Fonseka et al. , 2008). QM-CPM signals are more spectrally efficient than CPM signals, and multiamplitude QM-CPM signals can be constructed that outperform QAM. The non-coherent detectors view a QM-CPM signal as having composite states that represent the information modulated both onto the in-phase and quadrature-phase carriers. Viterbi decoding is used to jointly detect the quadrature-multiplexed signal components. Optimal sequence-based quadrature-matched filter (QMF) detectors are presented for use with QM-CPM detection and a QM-MSK detector is developed in detail. Differential phase and envelope (DPE) detectors are also presented which require less computation and are able to outperform the QMF-type detectors when the unknown carrier phase varies (phase jitter, phase hits etc.) within the sequence length used in the sequence-based QMF detectors. In the absence of these phase fluctuation effects, as the sequence length used in the QMF detectors increases, the performance of the QMF detectors approaches the performance of optimal non-coherent detection. The DPE-type detectors perform within about 2.5 dB of optimal coherent detection.

Sequence Design for Data-Aided Estimation of Synchronization Parameters

In this letter, we propose a systematic technique for optimum data-sequence design for data-aided (DA) estimation of synchronization parameters. It is assumed that the system operates in burst mode, transmitting a linearly modulated information signal over an additive white Gaussian noise channel. The method is based on the minimization of the corresponding Crameacuter-Rao lower bound subject to a power constraint and is applied in four practical receiver synchronization contexts: estimation of the symbol timing with known carrier frequency offset and either known or unknown carrier phase, joint estimation of the symbol timing and carrier phase with known carrier frequency offset, and joint estimation of the carrier phase and carrier frequency offset with known symbol timing

Semi-blind algorithms for automatic classification of digital modulation schemes

The problem of automatic classification of digital communication modulation schemes is considered in this work. Firstly, the maximum likelihood (ML) classifier for classifying phase-amplitude modulated schemes in coherent environment is presented. It is well known that the ML classifier requires the knowledge of the signal-to-noise ratio (SNR) and has a higher computational complexity. To relax the first requirement, we introduce a novel idea to estimate the SNR and this gives rise to a novel estimated ML (EsML) classifier. After which, in an attempt to reduce the computational complexity of the EML and EsML classifiers, we propose a simplified minimum distance (MD) classifier. The performance of these classifiers are compared against each other's under the ideal channel condition as well as under a channel condition with an unknown carrier phase offset. In the second part of the paper, we adapt a closed form blind source separation (BSS) algorithm for rectifying the carrier phase offset prior to the actual classification procedures.

Innovations-based MAP estimation with application to phase synchronization for serially concatenated CPM

A novel adaptive soft-input soft-output (A-SISO) module is developed for maximum a posteriori (MAP) symbol detection with parameter uncertainty. In contrast to the existing A-SISO algorithms that use linear prediction, the parameter estimation in the proposed structure is operated in a more general least-squares sense. Based on this scheme, a family of fixed-interval A-SISO (FI-A-SISO) algorithms is utilized to implement blind iterative phase synchronization for serially concatenated continuous phase modulation (SCCPM). For SCCPM systems, the forward-only FI-A-SISO algorithms are shown to be much more robust than the forward-backward FI-A-SISO algorithms in acquiring and tracking a time-varying carrier phase. The reason has to do with rotational invariance of CPM signals. This result can be extended to any rotationally invariant convolutional coded system with an unknown carrier phase.

On the probability density function of the GNSS ambiguity residuals

Integer GNSS ambiguity resolution involves estimation and validation of the unknown integer carrier phase ambiguities. A problem then is that the classical theory of linear estimation does not apply to the integer GPS model, and hence rigorous validation is not possible when use is made of the classical results. As with the classical theory, a first step for being able to validate the integer GPS model is to make use of the residuals and their probabilistic properties. The residuals quantify the inconsistency between data and model, while their probabilistic properties can be used to measure the significance of the inconsistency. Existing validation methods are often based on incorrect assumptions with respect to the probabilistic properties of the parameters involved. In this contribution we will present and evaluate the joint probability density function (PDF) of the multivariate integer GPS carrier phase ambiguity residuals. The residuals and their properties depend on the integer estimation principle used. Since it is known that the integer least-squares estimator is the optimal choice from the class of admissible integer estimators, we will only focus on the PDF of the ambiguity residuals for this estimator. Unfortunately the PDF cannot be evaluated exactly. It will therefore be shown how to obtain a good approximation. The evaluation will be completed by some examples.

Iterative demodulation and decoding of frequency-hopped PSK in partial-band jamming

This paper considers the problem of developing and utilizing side information in a frequency-hopped communication system using phase-shift keying (PSK) and operating in an environment with partial-band jamming. Two aspects of side information are studied. The first deals with estimating the unknown random carrier phase that varies from hop to hop. The second aspect is the detection of jamming signals. We use a serially concatenated convolutional code structure with differential M-ary PSK as the inner code. The iterative receiver uses an expanded trellis in the inner decoder to resolve the phase ambiguity and is augmented by a ratio-threshold test for detecting jammer energy. Performance is compared for different dwell interval lengths and both log-APP and max-log-APP decoding algorithms. This paper also considers the effect of different thresholds on the false alarm and detection probabilities of the ratio-threshold test.

True Cramer–Rao Bound for Timing Recovery From a Bandlimited Linearly Modulated Waveform With Unknown Carrier Phase and Frequency

This paper derives the Cramer-Rao bound (CRB) related to the estimation of the time delay of a linearly modulated bandpass signal with unknown carrier phase and frequency. We consider the following two scenarios: joint estimation of the time delay, the carrier phase, and the carrier frequency; and joint estimation of the time delay and the carrier frequency irrespective of the carrier phase. The transmit pulse is a bandlimited square-root Nyquist pulse. For each scenario, the transmitted symbols constitute either an a priori known training sequence or an unknown random data sequence. In spite of the presence of random data symbols and/or a random carrier phase, we obtain a relatively simple expression of the CRB, from which the effect of the constellation and the transmit pulse are easily derived. We show that the penalty resulting from estimating the time delay irrespective of the carrier phase decreases with increasing observation interval. However, the penalty, caused by not knowing the data symbols a priori, cannot be reduced by increasing the observation interval. Comparison of the true CRB to existing symbol synchronizer performance reveals that decision-directed timing recovery is close to optimum for moderate-to-large signal-to-noise ratios.

An Integrated Navigation System Using GPS Carrier Phase for Real-Time Airborne/Synthetic Aperture Radar (SAR)

A Synthetic Aperture Radar (SAR) requires accu- rate measurement of the motion of the imaging plat- form to produce well-focused images with minimal absolute position error. The motion measurement (MoMeas) system consists of a inertial measurement unit (IMU) and a P-code GPS receiver that outputs corrected ephemeris, L1 & L2 pseudoranges, and L1 & L2 carrier phase measurements. The unknown initial carrier phase biases to the GPS satellites are modeled as states in an extended Kalman filter and the resulting integrated navigation solution has po- sition errors that change slowly with time. Position error drifts less than 1- cm/sec have been measured from the SAR imagery for various length apertures.

Fast computation algorithm for multiple symboldetection of uncodedMPSK sequences over an AWGN channel

A fast computation algorithm is presented for multiple symbol detection of MPSK sequences over an AWGN channel with unknown carrier phase. The central idea is to define a decision statistic for each symbol based on the information of the past decided symbol sequence and to establish a simple relation between successive symbol decision statistics. Simulation results for uncoded 4PSK and 8PSK show its good performance.

Burst demodulation of QAM with unknown carrierphase

A new algorithm for the burst demodulation of QAM with unknown carrier phase is presented. This algorithm utilises sequence detection and the Viterbi algorithm, and does not require estimation of the carrier frequency and phase. Its applications can be widely found in broadband wireless access and other systems that use burst transmission.

Adaptive symbol-by-symbol reception of MPSK on the Gaussian channel with unknown carrier phase characteristics

An adaptive version of the optimum, decision-aided, symbol-by-symbol receiver obtained in our previous works for M-ary phase shift keying over the Gaussian channel with unknown carrier phase characteristics is considered. It uses a first-order adaptive filter for carrier reference tracking, and the filter gain is adapted on-line based on the observed signal samples using a least-mean-square-error criterion. The phase tracking performance of the filter in the steady-state tracking mode is analyzed using a linearized model. The phase tracking error variance is minimized due to the risk function used in designing the filter gain adaptation algorithm. The bit-error probability for quadrature phase shift-keying is obtained, both analytically and via computer simulations.

Efficient estimation of continuous phase modulation with unknown carrier phase

An efficient algorithm is presented for the estimation of continuous phase modulation (CPM) sequences over the Gaussian channel with unknown carrier phase. It operates on the excess phase trellis like the Viterbi algorithm, except that it is not a maximum-likelihood (ML) estimator. Its decision metric for survivor selection at each node is chosen so that it achieves the node error event probability of coherent ML estimation in the limit as the carrier phase remains constant over a long interval, so that the observation interval for forming the metric can be made large. It is shown for the case of minimum-shift-keying (MSK) that the bit-error probability (BEP) of perfectly coherent ML estimation is attained.

Data detection of MPSK in the presence of rapidly changing carrier phase

We present a novel method for data detection of M-ary phase shift keying (MPSK) in the presence of unknown carrier phase and additive white Gaussian noise. Here, the unknown carrier phase is rapidly changing and random. That is, the unknown phase changes from symbol to symbol in a nondeterministic manner. The rate of phase change is such that phase change over two symbol intervals is negligible, but it is not necessarily negligible over a longer interval. Our proposed system outperforms traditional differential phase shift keying (DPSK) by 1.5 dB at error probabilities of 10/sup -3/ and 10/sup -4/. We also compare the performance of our scheme with that of multiple symbol differential detection (MSDD). Except under very slow phase change conditions, our scheme outperforms MSDD. Our decoder is well suited to mobile communication systems. Here, receivers usually move with different speeds at different times.

Data detection of MPSK in presence ofunknown carrier phase at low complexity

Two novel methods for performing maximum likelihood (ML) data detection in the presence of an unknown carrier phase over a block of N symbols are introduced. The first method corresponds to an averaging over the range of phase and the second method corresponds to a limited search over the range of phase.

A Viterbi-type algorithm for efficient estimation of M-PSK sequences over the Gaussian channel with unknown carrier phase

A Viterbi-type algorithm for the efficient estimation of a convolutionally/trellis coded, or uncoded M-PSK sequence over the Gaussian channel with unknown carrier phase is presented. It performs coherent detection of M-PSK sequences with no explicit recovery of coherent references. The metric computations involve only linear operations. Theoretical and simulated performance results are presented for various carrier phase processes. The algorithm is called the suboptimum ML estimator (SMLE). Its BER in the presence of a time invariant carrier phase is obtained analytically. The simulation results for the BER with time-invariant as well as time-varying carrier phase processes allow a comparison of the performance of the SMLE with other sequence estimation algorithms, and enable the applicability of the SMLE to burst mode TDMA communications to be ascertained. >

Approximate maximum likelihood coherent detectionof M -ary PSK signalsusing Viterbi algorithm

Reduced complexity coherent detection based on maximum likelihood sequence estimation using the Viterbi algorithm is proposed for the reception of uncoded M-ary PSK signals with unknown carrier phase. The computer-simulated bit error rate performance of 4PSK is reported.