Subspace and wavelet-packet algorithms for de-noising and classifying broadband THz transients

The results from a range of different signal processing schemes used for the further processing of THz transients are contrasted. The performance of different classifiers after adopting these schemes are also discussed.

Classification of lactose and mandelic acid THz spectra using subspace and wavelet-packet algorithms

This work compares classification results of lactose, mandelic acid and dl-mandelic acid, obtained on the basis of their respective THz transients. The performance of three different pre-processing algorithms applied to the time-domain signatures obtained using a THz-transient spectrometer are contrasted by evaluating the classifier performance. A range of amplitudes of zero-mean white Gaussian noise are used to artificially degrade the signal-to-noise ratio of the time-domain signatures to generate the data sets that are presented to the classifier for both learning and validation purposes. This gradual degradation of interferograms by increasing the noise level is equivalent to performing measurements assuming a reduced integration time. Three signal processing algorithms were adopted for the evaluation of the complex insertion loss function of the samples under study; a) standard evaluation by ratioing the sample with the background spectra, b) a subspace identification algorithm and c) a novel wavelet-packet identification procedure. Within class and between class dispersion metrics are adopted for the three data sets. A discrimination metric evaluates how well the three classes can be distinguished within the frequency range 0. 1 - 1.0 THz using the above algorithms.

Subspace system identification framework for the analysis of multimoded propagation of THz-transient signals

We provide a system identification framework for the analysis of THz-transient data. The subspace identification algorithm for both deterministic and stochastic systems is used to model the time-domain responses of structures under broadband excitation. Structures with additional time delays can be modelled within the state-space framework using additional state variables. We compare the numerical stability of the commonly used least-squares ARX models to that of the subspace N4SID algorithm by using examples of fourth-order and eighth-order systems under pulse and chirp excitation conditions. These models correspond to structures having two and four modes simultaneously propagating respectively. We show that chirp excitation combined with the subspace identification algorithm can provide a better identification of the underlying mode dynamics than the ARX model does as the complexity of the system increases. The use of an identified state-space model for mode demixing, upon transformation to a decoupled realization form is illustrated. Applications of state-space models and the N4SID algorithm to THz transient spectroscopy as well as to optical systems are highlighted.

Wiener-System subspace identification for mobile wireless mm-wave networks

We discuss the feasibility of wireless terahertz communications links deployed in a metropolitan area and model the large-scale fading of such channels. The model takes into account reception through direct line of sight, ground and wall reflection, as well as diffraction around a corner. The movement of the receiver is modeled by an autonomous dynamic linear system in state space, whereas the geometric relations involved in the attenuation and multipath propagation of the electric field are described by a static nonlinear mapping. A subspace algorithm in conjunction with polynomial regression is used to identify a single-output Wiener model from time-domain measurements of the field intensity when the receiver motion is simulated using a constant angular speed and an exponentially decaying radius. The identification procedure is validated by using the model to perform q-step ahead predictions. The sensitivity of the algorithm to small-scale fading, detector noise, and atmospheric changes are discussed. The performance of the algorithm is tested in the diffraction zone assuming a range of emitter frequencies (2, 38, 60, 100, 140, and 400 GHz). Extensions of the simulation results to situations where a more complicated trajectory describes the motion of the receiver are also implemented, providing information on the performance of the algorithm under a worst case scenario. Finally, a sensitivity analysis to model parameters for the identified Wiener system is proposed.

Comparison of subspace and ARX models of a waveguide's terahertz transient response after optimal wavelet filtering

A quasi-optical deembedding technique for characterizing waveguides is demonstrated using wide-band time-resolved terahertz spectroscopy. A transfer function representation is adopted for the description of the signal in the input and output port of the waveguides. The time-domain responses were discretized and the waveguide transfer function was obtained through a parametric approach in the z-domain after describing the system with an AutoRegressive with eXogenous input (ARX), as well as with a state-space model. Prior to the identification procedure, filtering was performed in the wavelet domain to minimize both signal distortion, as well as the noise propagating in the ARX and subspace models. The optimal filtering procedure used in the wavelet domain for the recorded time-domain signatures is described in detail. The effect of filtering prior to the identification procedures is elucidated with the aid of pole-zero diagrams. Models derived from measurements of terahertz transients in a precision WR-8 waveguide adjustable short are presented.

A neurofuzzy network knowledge extraction and extended gram-Schmidt algorithm for model subspace decomposition

This paper introduces a new neurofuzzy model construction and parameter estimation algorithm from observed finite data sets, based on a Takagi and Sugeno (T-S) inference mechanism and a new extended Gram-Schmidt orthogonal decomposition algorithm, for the modeling of a priori unknown dynamical systems in the form of a set of fuzzy rules. The first contribution of the paper is the introduction of a one to one mapping between a fuzzy rule-base and a model matrix feature subspace using the T-S inference mechanism. This link enables the numerical properties associated with a rule-based matrix subspace, the relationships amongst these matrix subspaces, and the correlation between the output vector and a rule-base matrix subspace, to be investigated and extracted as rule-based knowledge to enhance model transparency. The matrix subspace spanned by a fuzzy rule is initially derived as the input regression matrix multiplied by a weighting matrix that consists of the corresponding fuzzy membership functions over the training data set. Model transparency is explored by the derivation of an equivalence between an A-optimality experimental design criterion of the weighting matrix and the average model output sensitivity to the fuzzy rule, so that rule-bases can be effectively measured by their identifiability via the A-optimality experimental design criterion. The A-optimality experimental design criterion of the weighting matrices of fuzzy rules is used to construct an initial model rule-base. An extended Gram-Schmidt algorithm is then developed to estimate the parameter vector for each rule. This new algorithm decomposes the model rule-bases via an orthogonal subspace decomposition approach, so as to enhance model transparency with the capability of interpreting the derived rule-base energy level. This new approach is computationally simpler than the conventional Gram-Schmidt algorithm for resolving high dimensional regression problems, whereby it is computationally desirable to decompose complex models into a few submodels rather than a single model with large number of input variables and the associated curse of dimensionality problem. Numerical examples are included to demonstrate the effectiveness of the proposed new algorithm.

Robust neurofuzzy rule base knowledge extraction and estimation using subspace decomposition combined with regularization and D-optimality

A new robust neurofuzzy model construction algorithm has been introduced for the modeling of a priori unknown dynamical systems from observed finite data sets in the form of a set of fuzzy rules. Based on a Takagi-Sugeno (T-S) inference mechanism a one to one mapping between a fuzzy rule base and a model matrix feature subspace is established. This link enables rule based knowledge to be extracted from matrix subspace to enhance model transparency. In order to achieve maximized model robustness and sparsity, a new robust extended Gram-Schmidt (G-S) method has been introduced via two effective and complementary approaches of regularization and D-optimality experimental design. Model rule bases are decomposed into orthogonal subspaces, so as to enhance model transparency with the capability of interpreting the derived rule base energy level. A locally regularized orthogonal least squares algorithm, combined with a D-optimality used for subspace based rule selection, has been extended for fuzzy rule regularization and subspace based information extraction. By using a weighting for the D-optimality cost function, the entire model construction procedure becomes automatic. Numerical examples are included to demonstrate the effectiveness of the proposed new algorithm.

Subspace-based blind adaptive multiuser detection for multirate DS/CDMA signals

The existing dual-rate blind linear detectors, which operate at either the low-rate (LR) or the high-rate (HR) mode, are not strictly blind at the HR mode and lack theoretical analysis. This paper proposes the subspace-based LR and HR blind linear detectors, i.e., bad decorrelating detectors (BDD) and blind MMSE detectors (BMMSED), for synchronous DS/CDMA systems. To detect an LR data bit at the HR mode, an effective weighting strategy is proposed. The theoretical analyses on the performance of the proposed detectors are carried out. It has been proved that the bit-error-rate of the LR-BDD is superior to that of the HR-BDD and the near-far resistance of the LR blind linear detectors outperforms that of its HR counterparts. The extension to asynchronous systems is also described. Simulation results show that the adaptive dual-rate BMMSED outperform the corresponding non-blind dual-rate decorrelators proposed by Saquib, Yates and Mandayam (see Wireless Personal Communications, vol. 9, p.197-216, 1998).

Tensor LRR and sparse coding-based subspace clustering

Subspace clustering groups a set of samples from a union of several linear subspaces into clusters, so that the samples in the same cluster are drawn from the same linear subspace. In the majority of the existing work on subspace clustering, clusters are built based on feature information, while sample correlations in their original spatial structure are simply ignored. Besides, original high-dimensional feature vector contains noisy/redundant information, and the time complexity grows exponentially with the number of dimensions. To address these issues, we propose a tensor low-rank representation (TLRR) and sparse coding-based (TLRRSC) subspace clustering method by simultaneously considering feature information and spatial structures. TLRR seeks the lowest rank representation over original spatial structures along all spatial directions. Sparse coding learns a dictionary along feature spaces, so that each sample can be represented by a few atoms of the learned dictionary. The affinity matrix used for spectral clustering is built from the joint similarities in both spatial and feature spaces. TLRRSC can well capture the global structure and inherent feature information of data, and provide a robust subspace segmentation from corrupted data. Experimental results on both synthetic and real-world data sets show that TLRRSC outperforms several established state-of-the-art methods.

Pulse shaping using state space algorithms

We discuss the use of pulse shaping for optimal excitation of samples in time-domain THz spectroscopy. Pulse shaping can be performed in a 4f optical system to specifications from state space models of the system's dynamics. Subspace algorithms may be used for the identification of the state space models.

State-space model identification of a wireless THz network

We model the large scale fading of wireless THz communications links deployed in a metropolitan area taking into account reception through direct line of sight, ground or wall reflection and diffraction. The movement of the receiver in the three dimensions is modelled by an autonomous dynamic linear system in state-space whereas the geometric relations involved in the attenuation and multi-path propagation of the electric field are described by a static non-linear mapping. A subspace algorithm in conjunction with polynomial regression is used to identify a Wiener model from time-domain measurements of the field intensity.

A novel blind channel estimation method for MIMO system

Channel estimation method is a key issue in MIMO system. In recent years, a lot of papers on subspace(SS)-based blind channel estimation have been published, and in this paper, combining SS method with a space-time coding scheme, we proposed a novel blind channel estimation method in MIMO system. Simulation result demonstrates the effectiveness of this method.

MIMO Wiener model identification for large scale fading of wireless mobile communications links

The large scale fading of wireless mobile communications links is modelled assuming the mobile receiver motion is described by a dynamic linear system in state-space. The geometric relations involved in the attenuation and multi-path propagation of the electric field are described by a static non-linear mapping. A Wiener system subspace identification algorithm in conjunction with polynomial regression is used to identify a model from time-domain estimates of the field intensity assuming a multitude of emitters and an antenna array at the receiver end.

Backward elimination model construction for regression and classification using leave-one-out criteria

A fundamental principle in practical nonlinear data modeling is the parsimonious principle of constructing the minimal model that explains the training data well. Leave-one-out (LOO) cross validation is often used to estimate generalization errors by choosing amongst different network architectures (M. Stone, "Cross validatory choice and assessment of statistical predictions", J. R. Stast. Soc., Ser. B, 36, pp. 117-147, 1974). Based upon the minimization of LOO criteria of either the mean squares of LOO errors or the LOO misclassification rate respectively, we present two backward elimination algorithms as model post-processing procedures for regression and classification problems. The proposed backward elimination procedures exploit an orthogonalization procedure to enable the orthogonality between the subspace as spanned by the pruned model and the deleted regressor. Subsequently, it is shown that the LOO criteria used in both algorithms can be calculated via some analytic recursive formula, as derived in this contribution, without actually splitting the estimation data set so as to reduce computational expense. Compared to most other model construction methods, the proposed algorithms are advantageous in several aspects; (i) There are no tuning parameters to be optimized through an extra validation data set; (ii) The procedure is fully automatic without an additional stopping criteria; and (iii) The model structure selection is directly based on model generalization performance. The illustrative examples on regression and classification are used to demonstrate that the proposed algorithms are viable post-processing methods to prune a model to gain extra sparsity and improved generalization.