Modified Cramer-Rao Bound Research Articles

OQPSK Synchronization Parameter Estimation Based on Burst Signal Detection

The fast estimation of synchronization parameters plays an extremely important role in the demodulation of burst signals. In order to solve the problem of high computational complexity in the implementation of traditional algorithms, a synchronization parameter (frequency offset, phase offset, and timing error) estimation algorithm based on Offset Quadrature Phase Shift Keying (OQPSK) burst signal detection is proposed in this article. We first use the Data-Aided (DA) method to detect where the burst signal begins by taking the segment correlation between the receiving signals and the known local Unique Word (UW). In the sequel, the above results are adopted directly to estimate the synchronization parameters, which is obviously different from the conventional algorithms. In this way, the complexity of the proposed algorithm is greatly reduced, and it is more convenient for hardware implementation. The simulation results show that the proposed algorithm has high accuracy and can track the Modified Cramer–Rao Bound (MCRB) closely.

A Robust Timing and Phase Offset Estimation Technique for CPM-DSSS-Based Secured Communication Link

Continuous Phase Modulation–Direct Sequence Spread Spectrum (CPM-DSSS) is a promising scheme for a secured Point-to-Point (P2P) communication system having Anti-Jamming (AJ) capabilities. However, like other communication systems, synchronization of CPM-DSSS based systems is required for reliable detection of data. This paper presents a novel algorithm for estimating the timing and phase offset for binary full-response CPM-DSSS based systems. A burst mode transmission is considered here, and within each burst, a training sequence is embedded along with data. Firstly, the entire burst is spread using DSSS and then modulated with CPM, which is later upconverted and passed on to the channel. After down-conversion at the receiver, the proposed algorithm estimates timing and phase offset from the received spread training sequence. Then estimation accuracy of the proposed algorithm is compared with the state-of-the-art. The results indicate that the proposed joint estimator is superior by 9 dB and 10 dB for the estimation of timing and phase offset, respectively, as compared to the existing algorithm. Furthermore, the estimator’s variance for three different frequency pulses is compared with Modified Cramer-Rao Bound (MCRB). The results suggest that the phase offset estimator’s variance approaches MCRB for all Signal-to-Noise Ratios (SNRs). The effectiveness of the proposed algorithm in the multipath fading channel is also demonstrated.

ML-Based Forward Symbol Timing Offset Estimation for Burst Signals

In order to ensure the performance of burst signal demodulation, forward symbol timing offset estimation algorithm is usually used for symbol timing recovery. Aiming at the shortcomings of common forward symbol timing estimation algorithms, a new algorithm based on maximum likelihood estimation joint trigonometric polynomial interpolation is proposed in this article, which is suitable for the data of four samples per symbol and can resist large frequency offset. Hereafter, in order to make it more suitable for engineering practice, optimize it into an improved data-aided (DA) forward symbol timing offset estimation algorithm with multiple characteristics, e.g., moderate frequency offset capture range, insensitive to shaping coefficient, relatively low complexity, excellent estimation performance, flexible algorithm structure and less sample data required for calculation. The simulation results show that the performance of the improved algorithm in this article can approach the modified cramer-rao bound (MCRB) within the frequency offset capture range, under the conditions of low signal to noise ratio (SNR) and small shaping coefficient.

Modified Cramér-Rao Bound for $M$-FSK Signal Parameter Estimation in Cauchy and Gaussian Noise

The Cramer-Rao bound (CRB) provides an efficient standard for evaluating the quality of standard parameter estimators. In this paper, a modified Cramer-Rao bounds (MCRB) for modulation parameter estimations of $\boldsymbol{M}$ -ary frequency-shift-keying ( $\boldsymbol{{M}}$ -FSK) signals is proposed under the condition of the Gaussian and non-Gaussian additive interference. We extend the MCRB to the estimation of a vector of non-random parameters in the presence of nuisance parameters. Moreover, the MCRB is applied to the joint estimation of phase offset, frequency offsets, frequency deviation, and symbol period of $\boldsymbol{M}$ -FSK signal with two important special cases of alpha stable distributions, namely, the Cauchy and the Gaussian. The extensive simulation studies are conducted to contrast the MCRB for the modulation parameter vector in different noise environments.

Autocorrelation Function of Full-Response CPM Signals and Its Application to Synchronization

Two types of the autocorrelation functions of the full-response continuous phase modulation (CPM) signal are investigated in the paper, the autocorrelation functions are proven to be periodic functions when the modulation index of the CPM signal is h=1. Then the autocorrelation functions are expanded in the form of Fourier series, and two synchronization schemes are proposed to estimate the synchronization parameters from the coefficients of the Fourier series, including the carrier frequency offset, the carrier phase and the timing offset. The performances of the synchronization algorithms are shown by simulations and compared with the modified Cramer-Rao bounds (MCRB) and other synchronization schemes. The simulation results show that the first type of the proposed scheme has better synchronization performances than the available schemes.

Joint Blind Parameter Estimation of Non-cooperative High-Order Modulated PCMA Signals

A joint blind parameter estimation algorithm based on minimum channel stability function aimed at the non-cooperative high-order modulated paired carrier multiple access (PCMA) signals is proposed. The method, which uses hierarchical search to estimate time delay, amplitude and frequency offset and the estimation of phase offset, including finite ambiguity, is presented simultaneously based on the derivation of the channel stability function. In this work, the structure of hierarchical iterative processing is used to enhance the performance of the algorithm, and the improved algorithm is used to reduce complexity. Compared with existing data-aided algorithms, this algorithm does not require a priori information. Therefore, it has significant advantage in solving the problem of blind parameter estimation of non-cooperative high-order modulated PCMA signals. Simulation results show the performance of the proposed algorithm is similar to the modified Cramer-Rao bound (MCRB) when the signal-to-noise ratio is larger than 16 dB. The simulation results also verify the practicality of the proposed algorithm.

On the Estimation of Modulation Index for Binary Full Response CPM Signals

This letter considers a full response binary continuous phase modulation (CPM) scheme with a single modulation index. Two estimators are proposed for estimating the modulation index of the underlying CPM scheme. The first estimator is based on the method of moments (MoM) that makes use of the auto-correlation function of the received signal whereas the second estimator is the best linear unbiased estimator (BLUE), which is based on an approximate linear model. A closed-form expression of the modified Cramer-Rao bound (MCRB) is also derived. The MoM estimator is developed only for the rectangular phase shaping functions, while the MCRB expression and the BLUE estimator are applicable to signals with arbitrary phase shaping functions. Numerical performance analysis in terms of mean-squared error and bit-error rate shows that the method of moments estimator performs better at low signal-to-noise ratio (SNR), whereas the performance of the BLUE is close to MCRB at high SNR.

Non-data aided joint estimation of symbol timing offset and carrier frequency offset for OFDM/OQAM systems

This paper deals with the problem of blind synchronization for orthogonal frequency division multiplexing based on offset quadrature amplitude (OFDM/OQAM) systems without the prior information of the channel response and transmitting signals power. A novel non-data aided (NDA) joint estimation method of symbol timing offset (STO) and carrier frequency offset (CFO) for OFDM/OQAM is conceived in multipath fading channels condition. The multipath fading channels are equivalent to multiple parallel sub-channels with flat Rayleigh fading characteristics firstly, and then STO and CFO are estimated by the second-order cyclic-cumulant with different cycle frequency values and time delay values. Moreover, the modified Cramer-Rao Bound (MCRB) of NDA joint STO and CFO estimation is presented. The simulation results demonstrate that the proposed method is effective and feasible over frequency selective fading channels.

A novel method for image thresholding using interval type-2 fuzzy set and Bat algorithm

In this paper, we propose a novel image thresholding method based on the interval type-2 fuzzy set (IT2FS). The interval type-2 fuzzy membership function (IT2FMF) is generated from a bag of type-1 fuzzy membership functions (T1FMFs) chosen adaptively based on the image characteristics for a given problem. An evolutionary algorithm called Bat algorithm is used to enhance the computational efficiency of the proposed thresholding technique. It is expected that the IT2FS based threshold technique will be better than that of the methods based on type-1 fuzzy set due to superior uncertainty handling capacity of the former technique. This fact is experimentally verified using benchmark dataset. The performance and robustness of the proposed method under different noise corruptions are measured statistically on the dataset by modified Cramer–Rao bound. The segmentation performance of the proposed method is compared experimentally with that of the state-of-the-art methods based on fuzzy and non-fuzzy frameworks. It is observed that the proposed method can achieve a higher segmentation accuracy in comparison to other state-of-the-art methods when they are benchmarked against the Modified Cramer–Rao Bound. Also, on average the misclassification error (ME) of the proposed method in segmentation is found to be minimum in comparison to the state-of-the-art methods.

Modified Cramer–Rao bound of parameter estimation for single‐channel mixtures of adjacent‐frequency signals

Present research on the bounds of parameter estimation performance is focused solely on single-source signals; thus, no benchmark for such bounds of mixed signals is currently available. For the case of a single-channel mixture of adjacent-frequency digitally modulated source signals, the modified Cramer–Rao bounds (MCRBs) of parameter estimation for frequency offsets, carrier phases, and time delays have been derived. A detailed comparison among the MCRBs of jointly estimating different parameters set is then conducted. Moreover, comparisons between maximum likelihood estimators and MCRBs, true Cramer–Rao bounds and MCRBs have been conducted, through theoretical analysis and numerical simulation.

MCRB for timing, phase and frequency estimation in presence of self-phase modulation for low-rate optical communication

In this paper we derive the theoretical lower bound, namely Modified Cramer-Rao bound (MCRB) for symbol timing, phase and frequency offset in presence of nonlinear self-phase modulation (SPM) in a dispersion compensated long-haul coherent fiber link. The system model considers multiple span of fiber each associated with optical amplifier. Dual polarization multilevel quadrature amplitude modulation is opted for data transmission to support the data rate lower than 10 Gigabaud. We find that SPM induces underdamped oscillation on the MCRB bounds depending on the pulse shapes (symmetric and asymmetric) utilized. In presence of realistic low-pass filter at the receiver front end, the MCRB degrades significantly due to SPM. We also show the effect of SPM on symbol error rate degradation. Simulation is carried out with symmetric return-to-zero pulse with duty cycles of 33, 67 % and self-generated asymmetric pulse to verify the theoretical results.

Lower bounds on the estimation performance in low complexity quantize-and-forward cooperative systems

Cooperative communication can effectively mitigate the effects of multipath propagation fading by using relay channels to provide spatial diversity. A relaying scheme suitable for half-duplex devices is the quantize-and-forward (QF) protocol, in which the information received from the source is quantized at the relay before being forwarded to the destination. In this contribution, the Cramer-Rao bound (CRB) is obtained for the case where all channel parameters in a QF system are estimated at the destination. The CRB is a lower bound (LB) on the mean square estimation error (MSEE) of an unbiased estimate and can thus be used to benchmark practical estimation algorithms. Additionally, the modified Cramer-Rao bound (MCRB) is also presented, which is a looser but computationally less complex bound. An importance sampling technique is developed to speed up the computation of the MCRBs, and the MSEE performance of a practical estimation algorithm is compared with the (M)CRBs. We point out that the parameters of the source-destination and relay-destination channels can be accurately estimated but that inevitably the source-relay channel estimate is poor when the instantaneous SNR on the relay-destination channel is low; however, in this case, the decoder performance is not affected by the inaccurate source-relay channel estimate.

New design of pilot patterns for joint semi-blind estimation of CFO and channel for OFDM systems

The greatest threat to the OFDM system is the loss of orthogonality between subcarriers due to the carrier frequency offset (CFO) which results in significant performance degradation. In addition to this last problem was the effect of propagation over multipath fading channel. Hence, both the channel and the CFO need to be estimated and corrected at the receiver for subsequent symbol detection.A new joint semi-blind CFO and channel estimation method for OFDM systems operating in multipath fading channels is investigated in this paper. The proposed algorithms are based on virtual subcarriers (VSC) in one hand and on a new design of pilot patterns in OFDM blocks in other hand.We propose to use Newton–Raphson algorithm for the estimation of CFO that has a faster convergence and a lower complexity compared to similar estimators that not employed this algorithm.Finally, we derive the Modified Cramer–Rao bounds (MCRB) for semi-blind CFO and channel estimation to evaluate the performance bounds of the proposed methods for commonly encountered scenarios.The simulation results are given to verify the validity of the proposed methods.

Simple timing recovery algorithm for binary CPFSK based on phase unwrapping

A reduced-complexity timing recovery method for binary continuous-phase frequency shift keying (CPFSK) is presented. The proposed algorithm is based on phase unwrapping and is easy to implement. Theoretical analysis and computer simulation both show that the new algorithm is able to perform close to the theoretical limit given by the modified Cramer–Rao bound (MCRB).

MCRB for Timing and Phase Offset for Low-Rate Optical Communication with Self-Phase Modulation

We derive the modified Cramer-Rao bound (MCRB) for symbol timing and phase offset estimation in the presence of nonlinear self-phase modulation (SPM) in a dispersion compensated long-haul optical fiber link with coherent detection at data rates below 10 Gigabaud. In the presence of a low-pass filter at the receiver front-end, we find that SPM degrades the MCRB. Moreover, depending on the pulse shape, SPM induces underdamped oscillation on the bounds.

A Null-Subcarrier-Aided Reference Symbol Mapping Scheme for 3GPP LTE Downlink in High-Mobility Scenarios

In this paper, a new reference symbol (RS) mapping scheme for the Third-Generation Partnership Project (3GPP) long-term evolution (LTE) and LTE-Advanced (LTE-A) downlink is proposed to improve channel estimation performance in high-mobility communication scenarios. The proposed scheme employs null subcarriers to guard RSs, which helps mitigate the effect of intercarrier interference (ICI) on subcarriers carrying RSs. Additionally, the proposed scheme allows the ICI gain parameters to be estimated via a simple frequency-domain estimator. Modified Cramer–Rao bound (MCRB) expressions are derived for the proposed scheme, as well as for the conventional RS mapping scheme defined in the 3GPP LTE and LTE-A standards to compare their performance at high mobile user speeds. These bounds, together with mean square errors obtained from simulations, reveal superior performance achieved by the proposed scheme in high-mobility scenarios. Additionally, at high mobile user speeds, the proposed scheme offers significant bit-error-rate (BER) performance improvement over the standard RS mapping.

Frequency-Domain AoA Estimation and Beamforming with Wideband Hybrid Arrays

High speed long range millimetre-wave (mm-wave) links can be achieved by using wideband hybrid antenna arrays of sub-arrays. Due to the array architecture difference, conventional wideband angle-of-arrival (AoA) estimation and beamforming techniques are not applicable to such wideband hybrid arrays. Targeted at point-to-point line-of-sight wireless transmission in the 70/80 GHz E bands, a unified frequency-domain AoA estimation and beamforming algorithm suitable for large scale wideband hybrid arrays of both interleaved and side-by-side sub-arrays is proposed in this paper. The AoA estimation performance is analyzed by deriving a recursive modified Cramer-Rao bound (MCRB). The effect of mutual coupling among antenna elements on the estimation performance is also considered for the hybrid array of side-by-side sub-arrays. The analytical results can be used to determine system parameters according to required system specifications. Simulation results show that the proposed AoA estimation algorithm is robust against practical impairments, and the frequency dependency of the array pattern is significantly reduced after digital beamforming. Simulated mean square errors of AoA estimation are also compared with the analytical bounds, showing that the derived recursive MCRB provides a meaningful indication to the AoA estimation performance.

Design and performance analysis of non-data-aided carrier phase estimators for amplitude and phase shift keying signals

This study studies the feedforward (FF) non-data-aided (NDA) carrier phase estimation of the amplitude and phase shift keying (APSK) signals. The true Cramer–Rao bounds (CRBs) for NDA phase estimation of APSK signals are derived and evaluated numerically using Gauss–Hermite quadrature. The jitter variance of the FF Viterbi–Viterbi (V&V) algorithm is analysed assuming the absence of data pattern noise. It is proved that, when the design parameter μ=2, the jitter variance is able to converge asymptotically to the modified CRB (MCRB) at high signal-to-noise ratios (SNRs). For practical application, the parameter μ is also optimised for 16/32/64-APSK signals at different SNRs. The analytical results of the jitter variance are verified by Monte-Carlo evaluations. It is shown that, for 32/64-APSK signals, the plain V&V algorithm cannot approach the CRBs due to the data pattern noise. A modified V&V algorithm based on the constellation partition and a linear combination of the sub-estimators is proposed to eliminate the divergence. Simulation results show that the jitter variance of the proposed estimator is very close to the CRB at the SNRs of interest and converges to the MCRB at high SNRs.

Lower Bounds to the Performance of Bit Synchronization for Bandwidth Efficient Pulse-Shaping

Bandwidth efficient signaling requires using pulse-shaping that inherently has intersymbol interference (ISI) except when the timing is perfect. Several lower bounds to the mean-square-error (MSE) of non-data-aided bit-synchronizers are derived for this case. The optimal lower bound is derived using the maximum likelihood (ML) criterion for a sequence of binary pulse amplitude modulated pulses in the presence of ISI and Gaussian noise. This lower bound is used as a benchmark to evaluate the performance of other synchronizers in a practical scenario. It is shown that a previous lower bound based on the ISI-free ML synchronizer cannot be used to lower bound the MSE of bit-synchronizers. A detection theory bound (DTB) (also called Ziv-Zakai bound) is applied to the symbol timing recovery problem in the presence of ISI and it is shown that this bound is a tight lower bound on the MSE of the ML synchronizer. A simple lower bound on this DTB is derived and it is shown that the simple bound is almost as tight as the well known modified Cramer-Rao bound (MCRB) at moderate values of SNR, while it does not suffer from the shortcomings of the MCRB at small values of SNR.

Security Analysis on Add-SS Watermarking with GSM

Watermarking security has emerged as the domain of extensive research in recent years. Attack to the security of watermarking is to estimate the secret keys used in watermarking communications. The security level of watermarking system can be evaluated by Cramer-Rao bound (CRB) for secret keys estimation. As a widely used method, the security issue of add spread spectrum (Add-SS) based watermarking has drawn great attention. The previous works on watermarking security were mainly based on the assumption that host signal was Gaussian distributed and ignored the impacts of non-Gaussian characteristics of nature images. With incorporation of the Gaussian scale mixture (GSM) model for host signals, this paper presents a theoretical analysis on the security of Add-SS watermarking system. By giving the CRB and MCRB (modified Cramer-Rao bound) for the estimation of secret carriers under KMA (known message attack) and WOA (watermarked only attack), this paper also reveals the factors that may influent the Add-SS watermarking security such as the times of observation, the length of secret carriers, the embedding energy, and the distribution of embedded messages. The results obtained in this paper will be helpful for designing the new secure and robust watermarking system.