Algebraic signal processing provides a general framework for studying theoretical problems (sampling, transform domain analysis etc.) in the classical signal processing. In this study, the authors extend algebraic representation for the conventional Fourier transform (FT) to the fractional FT (FRFT) domain, from which the algebraic structures for the FRFT on infinite and finite one-dimensional signal models are obtained. They show that FRFTs on the infinite and finite discrete-time (DT) signal models, respectively, are none other than the DTFRFT and the closed-form discrete FRFT. They also derive FRFTs on the infinite and finite discrete-nearest neighbour signal models, and finally they discuss their applications in optical and time–frequency signal processing.

In order to recover weak signal that is submerged in heavy background noise, a weak signal recovery method based on three-dimensional curvelet transform and generalised cross validation (GCV) is presented. Their method includes three stages: firstly, three-dimensional dataset was decomposed into sub-volumes by three-dimensional curvelet transform, and then Graphics Processing Unit (GPU) is used to improve the speed of parallel processing of sub-volumes. Secondly, the GCV criterion and genetic algorithm were used to improve the signal-to-noise ratio (SNR) of the processed signal. Finally, according to different distribution between the effective signal energy and the noise energy, adaptive filter is used to enhance the recovered weak signal. Furthermore, to verify the availability of the method, a wedge of simulation data and 100 groups of three-dimensional seismic data for testing are analysed, the stratigraphic structure information is much clearer in the processed seismic data than in the original data. The results show that the SNR of the recovered data is improved by 3 dB and the band of frequency is increased by 100 Hz. Their method is four to five times faster than a recovery method based on CPU processing.

In some applications, the signals received by an array are a mixture of signals emitted by both far-field and near-field sources. This study develops a new cumulant-based multiple signal classification (MUSIC) algorithm for source localisation using a new structural sparse array for scenarios where both far-field and near-field sources coexist. The key feature of this algorithm is that it utilises fourth-order cumulants to compute the virtual covariance matrix and constructs a new special cumulant matrix to acquire the largest number of virtual sensors and the largest array aperture for a given number of sensors. The authors provide a geometric proof to justify the utilisation of the proposed sparse linear array and compute the effective aperture of the array. The proposed algorithm increases resolution ability, direction of arrival (DOA) and range estimation accuracy, and the number of sources to be localised. Moreover, the new method has the main advantage that it does not use the information of all sensors; so that it provides somewhat low computational complexity while it uses many actual and virtual sensors. Monte Carlo simulations are provided to demonstrate the effectiveness of the proposed method.

This study considers the performance comparison between the matched filter (MF) and the locally optimal (LO) detector when the noise information is not accurate. For a given binary hypothesis testing problem, the authors derive the condition under which the LO detector performs worse than the MF with the assumptions of large number samples and low signal-to-noise ratio. The condition is an inequality, which involves the noise probability density function (pdf) used in the LO detector, the real noise pdf, and the noise variance. Simulations show that the condition is theoretically sound under both known and unknown noise pdf's.

A mainlobe maintenance method based on shrinkage estimator is presented here to promote the adaptive digital beamforming performance when there exists mainlobe jamming (MLJ). First, block matrix preprocessing (BMP) method is applied to suppress the MLJ. Then, the linear combination of estimated covariance matrix and identity matrix is optimised to generate more accurate estimation of the covariance matrix. After that, the improved covariance matrix is utilised to generate the adaptive weights to suppress the sidelobe jamming. Finally, the simulation shows that the proposed method is capable of eliminating peak offset of mainlobe and high sidelobes introduced by BMP and provides robustness against finite data samples effects. Accordingly, it outperforms noise whitening with error compensation, diagonal loading, and robust covariance matrix reconstruction in output SINR.

In this study, a simple but effective spoofing detection method using a global positioning system (GPS) directional antenna is proposed which exploits the difference between the estimated direction-of-arrival (DOA) and the measured DOA from a GPS almanac and ephemeris data. The receiving signal's DOA is estimated by using a single antenna power measurement-based complex extended Kalman filter (EKF) which is a complex valued state space based estimation technique. Furthermore, an adaptive logic is applied to the complex EKF to reduce the effect of the measurement disturbance. To maintain the validity of the proposed algorithm, it is assumed that the spoofer is aware of the target's location, but that its DOA is not perfectly the same as that of the authentic GPS signal. In addition, the orientation of the directional antenna is known by using an attitude and heading reference system that is attached to the antenna, and its antenna radiation pattern is also known. The proposed detection method is evaluated using a theoretical analysis and simulations. It is finally confirmed that the proposed algorithm can detect a spoofing signal according to the different direction angles of the spoofing signal, and especially those with low DOAs.

This study presents the shrinkage-based sequential source localisation and range estimation algorithms. The shrinkage factor is found using the variance of the estimate in the existing shrinkage algorithm. However, the variance of the estimate is difficult to calculate when the form of the estimate is complex. To circumvent this problem, the authors propose a shrinkage algorithm that employs the Cramér–Rao lower bound (CRLB) instead of the variance for the maximum likelihood (ML) estimate. The variance of the ML estimate and CRLB were found to be similar in simulation results. Furthermore, Stein's unbiased risk estimator and Ledoit–Wolf methods are used to determine the shrinkage factor. The resulting estimation accuracy of the proposed shrinkage-based sequential source localisation and range estimation methods was similar with that of the existing shrinkage algorithm.

A novel machine learning method named extended support vector data description with negative examples (ESVDD-neg) is developed to classify the fast Fourier transform-magnitude feature of complex high-resolution range profile (HRRP), motivated by the problem of radar automatic target recognition. The proposed method not only inherits the close non-linear boundary advantage of support vector data description with negative examples model but also incorporates a new learning paradigm named learning using privileged information into the model. It leads to the appealing application with no assumptions regarding the distribution of data and needs less training samples and prior information. Besides, the second order central moment is selected as privileged information for better recognition performance, weakening the effect of translation sensitivity, and the normalisation contributes to eliminating the amplitude sensitivity. Hence, there will be a remarkable improvement of recognition accuracy not only with small training dataset but also under the condition of low signal-to-noise ratio. Numerical experiments based on two publicly UCI datasets and HRRPs of four aircrafts demonstrate the feasibility and superiority of the proposed method. The noise robust ESVDD-neg is ideal for HRRP-based radar target recognition.

The current popular dictionary learning algorithms for sparse representation of signals are K-means Singular Value Decomposition (K-SVD) and K-SVD-extended. Only rank-1 approximation is used to update one atom at a time and it is unable to cope with large dictionary efficiently. In order to tackle these two problems, this study proposes M-Principal Component Analysis-N (M-PCA-N), which is an algorithm for dictionary learning and sparse representation. First, M-Principal Component Analysis (M-PCA) utilised information from the top M ranks of SVD decomposition to update M atoms at a time. Then, in order to further utilise the information from remaining ranks, M-PCA-N is proposed on the basis of M-PCA, by transforming information from the following N non-principal ranks onto the top M principal ranks. The mathematic formula indicates that M-PCA may be seen as a generalisation of K-SVD. Experimental results on the BBC Sound Effects Library show that M-PCA-N not only lowers the MSE between original signal and approximation signal in audio signal sparse representation, but also obtains higher audio signal classification precision than K-SVD.

Multiple-input multiple-output (MIMO) radars have attracted much attention for their superior ability to enhance a system's performance. In this study, the authors’ goal was the study of the spatial multiplexing gain of MIMO radars with widely separated antennas (WS-MIMO), which the authors showed that is equal to the number of unambiguously detectable targets. They obtained this number from two different aspects: first, by defining the ambiguity function of a WS-MIMO radar in the case of multiple targets, suitable for such purpose; Second, by modelling the MIMO radar system with a MIMO wireless channel. They showed that a MIMO radar is indeed a MIMO wireless system communicating the information about the existence of the targets. By such modelling, they could easily make the relation between dual concepts of MIMO radar and MIMO communication, one of which is the multiplexing gain.

This study introduces a new method for speech signal encryption and compression in a single step. The combined compression/encryption procedures are accomplished using compressive sensing (CS). The contourlet transform is used to increase the sparsity of the signal required by CS. Due to its randomness properties and very high sensitivity to initial conditions, the chaotic system is used to generate the sensing matrix of CS. This largely increases the key size of encryption to when logistic map is used. A spectral segmental signal-to-noise ratio of −36.813 dB is obtained as a measure of encryption strength. The quality of reconstructed speech is given by means of signal-to-noise ratio (SNR), and perceptual evaluation speech quality (PESQ). For 60% compression ratio the proposed method gives 48.203 dB SNR and 4.437 PESQ for voiced speech segments. However, for continuous speech (voiced and unvoiced), it gives 41.097 dB SNR and 4.321 PESQ.

A non-linear multi-sensory adaptive noise canceller (ANC, MsANC) with both multi-primary and multi-reference channels is proposed for foetal electrocardiogram (FECG) extraction. The primary channels are connected by a linear combiner (LC) whose output serves as a primary signal for the whole MsANC. A finite impulse response (FIR) filter or a non-linear filter [a Volterra filter, or an FIR filter plus a functional link artificial neural network (FLANN), or an FIR filter plus a generalised FLANN] is placed in each reference channel to approximate the linear and non-linear mappings between the chest maternal ECG (ANC reference signal) and the abdominal ECG (ANC primary signal). The LC connecting the primary channels is updated by a constrained recursive least square algorithm, while the linear and non-linear filters placed in reference channels are updated in a least mean square sense. Two real datasets derived from the Physionet non-invasive FECG database as well as the DaISy database are used to show the effectiveness of the proposed MsANC. Experimental results have revealed that the proposed MsANC results in considerable performance improvement as the number of primary channels is increased in comparison with existing ANCs with a single primary channel.

This study proposes a new representation of discrete Non-linear AutoRegressive with eXogenous inputs (NARX) model by developing its coefficients associated to the input, the output, the crossed product, the exogenous product and the autoregressive product on five independent Laguerre orthonormal bases. The resulting model, entitled NARX-Laguerre, ensures a significant parameter number reduction with respect to the NARX model. However, this reduction is still subject to an optimal choice of the Laguerre poles defining the five Laguerre bases. Therefore, the authors propose to use the genetic algorithm to optimise the NARX-Laguerre poles, based on the minimisation of the normalised mean square error. The performances of the resulting NARX-Laguerre model and the proposed optimisation algorithm are validated by numerical simulations and tested on the benchmark Continuous Stirred Tank Reactor.

The long time for collecting the data and a considerable amount of data are important technical challenges in microwave imaging for the detection of breast cancer. From the other point of view, compressive sensing (CS) is an interesting representation and analysis of sparse signals. In this study, a new imaging method for monostatic ultra-wideband microwave imaging of breast cancer using CS is presented. Instead of using all of the conventional radar returned signals, a few received signals, by random choosing the antenna, are sufficient for obtaining reliable images even at high noise levels. Using simulations done, it is shown that sparser images are obtained comparing to the delay-and-sum beamforming technique using only a few received signals.

This study focuses on the design methods for uniform linear array (ULA) based directional antenna arrays by optimising the radiation characteristics of elements. To improve the performance of direction-of-arrival (DOA) estimation in a predetermined objective spatial sector which includes all the potential directions of incidence, Cramér–Rao bound based optimisation models are established by utilising the least squares fitting technique. Besides, a modified simulated annealing (SA) algorithm with the iteration of parameters is proposed, aiming to solve the optimisation problems when the classic SA is invalid. Compared with the corresponding conventional ULA, an optimised array can obtain higher accuracy of DOA estimation in the objective spatial sector with little fluctuation. Additionally, the optimised design of radiation characteristics can also suppress the ambiguities, and remains effective for the arrays with different aperture. Simulation results verify the effectiveness of the proposed methods and the superiority of the optimised arrays.