PARAFAC Model Research Articles

Angle and mutual coupling estimation in bistatic MIMO radar based on PARAFAC decomposition

In this paper, a PARAFAC decomposition-based algorithm is developed for joint direction-of-departure and direction-of-arrival estimation in the presence of unknown mutual coupling for bistatic multiple-input multiple-output radar. A three-order tensor is formulated which links the estimations of coupled direction matrices to the PARAFAC model. The coupling effects of the direction matrices are compensated by two selective matrices, and the angles are obtained from the estimated direction matrices. Then the mutual coupling coefficients of the transmitter and the receiver are estimated using the subspace method. Unlike existing algorithms, PARAFAC decomposition before decoupling operation results in more accurate angle estimation, which brings better mutual coupling coefficients estimation than the ESPRIT-Like and unitary HOSVD methods. The proposed algorithm does not require spectral peak searching or eigenvalue decomposition of the received signal covariance matrix, and it can achieve automatic pairing of the estimated angles. The identifiability and computation complexity of the presented algorithm are analysed and Cramer–Rao bounds of joint angle and mutual coupling estimation are derived. Numerical experiments verify the effectiveness and improvement of our algorithm.

Characterization of Chromophoric Dissolved Organic Matter (CDOM) in the Bohai Sea and the Yellow Sea Using Excitation-Emission Matrix Spectroscopy (EEMs) and Parallel Factor Analysis (PARAFAC)

Compositions and concentrations of chromophoric dissolved organic matter (CDOM) have been determined in the Bohai Sea (BS) and the Yellow Sea (YS) in the summer and autumn in 2013 by excitation and emission matrix spectroscopy (EEMs) and parallel factor analysis (PARAFAC). The PARAFAC model identified three humic-like components (C1, C2, and C3) and one protein-like component (C4). CDOM exhibited higher fluorescence intensities in the coastal areas in both the summer and autumn. However, its distribution patterns were different in the two seasons. Based on spatial and seasonal distributions of four components, as well as correlations with salinity, chlorophyll a (Chl-a), and apparent oxygen utilization (AOU), the following assignments were made. The C1, C2, and C3 components were mainly dominated by terrestrial inputs and influenced by the primary productivity of phytoplankton in the summer as well. C4 was assigned to terrestrial and autochthonous origins and most likely represented a biologically labile component. Terrestrial inputs were the dominant source of CDOM in the BS and YS. The humification index (HIX) and biological index (BIX) suggested that CDOM in the BS was more stable than that in the YS, which had an increase in autochthonous production, and in the summer, CDOM was less stable with a higher CDOM autochthonous production compared with that in the autumn.

Measuring Polycyclic Aromatic Hydrocarbons in Water using the Simultaneous Absorbance and Fluorescence Excitation-Emission Matrix Method: Advantages of Classical Least Squares Regression Over Parallel Factor Analysis

Many Polycyclic Aromatic Hydrocarbons (PAHs) are carcinogenic and hence of important environmental concern in drinking water sources. Most USEPA PAHs of concern exhibit high fluorescence quantum yields and sub mg/l to μg/l detection limits in neat solutions. However, drinking water sources commonly contain other natural compounds, including humic and fulvic acids and proteins, in carbon concentrations of several mg/l that exhibit significant interfering absorbance and fluorescence backgrounds in the PAH detection region. We investigated the use of the new simultaneous Absorbance and fluorescence Excitation-Emission Matrix (ABEEM) method to measure PAHs in raw water. The ABEEM method corrects fluorescence inner-filter effects to yield quantitative fluorescence spectral information to be largely independent of component concentrations. We compared Classical Least Squares Regression (CLS) and Parallel Factor Analysis (PARAFAC) in terms of identification and detection limits for the PAHs. The CLS library was calibrated with respect to concentration for the individual EPA PAHs dissolved in deionized organic-carbon free water. The same data set was used calibrate a model using the 3-way PARAFAC technique. Validation of the CLS and PARAFAC models was compared with PAHs added to natural raw water of varying organic carbon concentrations. Identification of individual PAHs and mixtures and detection limits were significantly better with CLS. The evaluation is consistent with the fundamental principles CLS vs. PARAFAC and primarily because CLS can be distinctly calibrated with the individual PAH component spectra. We conclude that the identification and detection limits of the ABEEM and CLS analysis can be significant enough for environmental contamination monitoring in drinking water sources; certainly, as a first screening method before the EPA standard solid-phase extraction and liquid or gas-chromatographic analysis methods.

Tracking fluorescent components of dissolved organic matter from soils in large-scale irrigated area.

Combination of fluorescence excitation-emission matrix spectroscopy with parallel factor analysis (PARAFAC) and principal component analysis (PCA) was engaged to track fluorescent components of dissolved organic matter (DOM) extracted from soils, to seek potential factors, and to reveal their correlations with physico-chemical properties of soils. Soil samples at different depths were collected in Hetao irrigated area of Inner Mongolia, China. Five fluorescent components (C1 to C5) were identified by PARAFAC modeling of DOM extracted from the soil samples. C1 was referred as fulvic-like fluorescent component, by which DOM was dominated in the whole soil samples. C2 was associated with salinity and agriculture, which was similar to marine humic-like fluorescent component. C3 was assigned as traditional humic-like fluorescent component. The three components were of the terrestrial origin. C4 was involved in tryptophan-like fluorescent component, which was autochthonous productions of biological degradation. C5 might be a polycyclic aromatic hydrocarbon contaminant, which could be relative to anthropogenic sources of pesticides. The C1, C2, and C3 were the potential factors of characterizing DOM fractions using PCA on fluorescent components and physico-chemical parameters. Moreover, DOM might restrained by exchangeable sodium percentage, and its formation and decomposition might be influenced by soil moisture.

Europium binding to humic substances extracted from deep underground sedimentary groundwater studied by time-resolved laser fluorescence spectroscopy

ABSTRACTHumic substances (HSs) are ubiquitous in various environments including deep underground and play an important role in the speciation and mobility of radionuclides. The binding of Eu3+, a chemical homologue of trivalent actinide ions, to HSs isolated from sedimentary groundwater at −250 m below the surface was studied by time-resolved laser fluorescence spectroscopy combined with parallel factor analysis (PARAFAC) as a function of pH and salt concentration. PARAFAC modeling reveals the presence of multiple factors that correspond to different Eu3+ species. These factors resemble those observed for Eu3+ binding to HSs from surface environments; however, detailed comparison shows that there are some particularities in Eu3+ binding to the deep groundwater HSs. The distribution coefficients (Kd) of Eu3+ binding to the HSs calculated from the PARAFAC modeling exhibits a rather strong salt effect. At 0.01 M NaClO4, the Kd values are relatively large and comparable to those to the surface HSs; they are decreaed at 0.1 M NaClO4 by more than an order of the magnitude. The Kd values are larger for the humic acid fraction of the deep underground HSs than the fulvic acid fraction over the entire range of pH and salt concentration investigated in this study.

Compositional analysis of trade flows structure

Statistical analysis of trade flows structure can significantly help to reveal or to confirm important macroeconomic phenomena. Because of relative character of these multivariate observations, application of standard multivariate methods directly to raw data can lead to meaningless results, affected by trade sizes of different countries. As a way out, it is proposed to employ the logratio methodology that is able to capture interesting features through logratios between compositional parts. Particularly, the perturbation operation together with clr coefficients for coordinate representation of compositions seem to be easy to handle and to interpret for the purpose. Popular exploratory tools, principal component analysis and PARAFAC modeling of threeway data, resulting from a long-term study of the export/import structure, are applied in the compositional context for data from OECD and WIOD databases. The results show that the logratio methodology enables to reveal interesting features of world trade flows and thus provides a preferable alternative to existing exploratory tools.

Humic-like substances from urban waste as auxiliaries for photo-Fenton treatment: a fluorescence EEM-PARAFAC study.

In this work, analysis of excitation-emission-matrices (EEM) has been employed to gain further insight into the characterization of humic like substances (HLS) obtained from urban wastes (soluble bio-organic substances, SBOs). In particular, complexation of these substances with iron and changes along a photo-Fenton process have been studied. Recorded EEMs were decomposed by using parallel factor analysis (PARAFAC). Three fluorescent components were identified by PARAFAC modeling of the entire set of SBO solutions studied. The EEM peak locations (λex/λem) of these components were 310-330 nm/400-420 nm (C1), 340-360 nm/450-500 nm (C2), and 285 nm/335-380 nm (C3). Slight variations of the maximum position of each component with the solution pH were observed. The interaction of SBO with Fe(iii) was characterized by determining the stability constants of the components with Fe(iii) at different pH values, which were in the order of magnitude of the ones reported for humic substances and reached their highest values at pH = 5. Photochemical experiments employing SBO and Fe(iii), with and without H2O2, showed pH-dependent trends for the evolution of the modeled components, which exhibited a strong correlation with the efficiency reported for the photo-Fenton processes in the presence of SBO at different pH values.

Optical Characteristics of Dissolved Organic Matter from Two Different Full Mixed Reservoirs in Winter Based on UV-vis and EEMs

The differences in the resource and characteristics of DOM between Zhoucun Reservoir and Jinpen Reservoir were studied by fluorescence ultraviolet-visible (UV-vis) and excitation-emission matrix spectra combined with parallel analysis (EEMs-PARAFAC). The results showed that three fluorescent components were identified by PARAFAC model in Zhoucun Reservoir and Jinpen Reservoir, including long wave humus-like component (C1: 350/460 nm), visible fulvic-like (C2: 335/410 nm) and protein-like (C3: 260,285/360 nm). However, the fluorescence intensity and the relative proportions of DOM exhibited significant difference (P<0.01) in two reservoirs. Moreover, the concentration of protein-like component in Zhoucun Reservoir was higher than that in Jinpen Reservoir, while the concentration of humus-like component presented the opposite trend. Based on the comparison of fluorescence index, biological index, humification index and the freshness index, the DOM of Zhoucun Reservoir where the land-use type was cultivated, livestock and residential, the internal pollution of sediments was serious, indicating a strong autochthonous component and aquatic bacterial origin, whereas the DOM of Jinpen Reservoir dominated by forest system had a higher terrigenous contribution in winter. From all the results, hydrological conditions of the reservoir and the characteristics of coastal ecological environment were important factors to influence the DOM sources and characteristics.

A novel algorithm for resolution of three-component mixtures of fluorophores by fluorescence quenching

The article presents a new algorithm for resolving the fluorescence spectra of three-component systems. The proposed method refers to a technique already known in the literature combining self-modeling based on singular value decomposition with numerical optimization of the error function derived from the linear Stern–Volmer equation (SM-SVD-SV). Previously, this method was successfully used for resolving the fluorescence spectra of two-component systems. A formal extension of mathematical concepts employed by the SM-SVD-SV method to the case of three-component mixtures of fluorophores, even though possible, leads to expressions which because of its complexity are ineffective in the optimization process and thus have to be simplified. To this purpose, the suitably developed two-component difference fluorescence spectra are used to a broad extent. This way the search for the fluorescence spectra of pure components performed in the 3D space spanned by three significant eigenvectors is reduced to much easier and efficient searches in the 2D space. Further reduction of the number of the optimized parameters is achieved by a neat application of the rank annihilation factor analysis (RAFA) providing convenient estimates for the searched values of the Stern–Volmer constants of all three fluorophores. The effectiveness of the developed technique was tested on model spectra to which a homoscedastic noise was added. The real data set including the quenched fluorescence spectra of a three-component mixture of 9,10-dicyanoanthracene, 10-chloro-9-cyanoanthracene and 9-cyanoanthracene in methanol was also analyzed. The obtained results compared with those yielded by PARAFAC and direct hard modeling allow to consider the proposed algorithm as an useful tool that facilitates resolution and refinement of the spectra of pure components derived from suitably quenched mixtures of three fluorescing species.

Light Absorption and Excitation-Emission Fluorescence of Urban Organic Aerosol Components and Their Relationship to Chemical Structure.

The present study used a combination of solvent and solid-phase extractions to fractionate organic compounds with different polarities from total suspended particulates in Nagoya, Japan, and their optical characteristics were obtained on the basis of their UV-visible absorption spectra and excitation-emission matrices (EEMs). The relationship between their optical characteristics and chemical structures was investigated based on high-resolution aerosol mass spectra (HR-AMS spectra), soft ionization mass spectra and Fourier transform infrared (FT-IR) spectra. The major light-absorption organics were less polar organic fractions, which tended to have higher mass absorption efficiencies (MAEs) and lower wavelength dependent Ångström exponents (Å) than the more polar organic fractions. Correlation analyses indicate that organic compounds with O and N atoms may contribute largely to the total light absorption and fluorescence of the organic aerosol components. The extracts from the aerosol samples were further characterized by a classification of the EEM profiles using a PARAFAC model. Different fluorescence components in the aerosol organic EEMs were associated with specific AMS ions and with different functional groups from the FT-IR analysis. These results may be useful to determine and further classify the chromophores in atmospheric organic aerosols using EEM spectroscopy.

Can fluorescence spectrometry be used as a surrogate for predicting the dissolved organic nitrogen and its bioavailable portion in wastewater effluents?

This study investigated the applicability of fluorescence excitation–emission matrices coupled with parallel factor analysis (EEM–PARAFAC) for predicting dissolved organic nitrogen (DON) and bioavailable DON (ABDON) in wastewater effluents. A four component PARAFAC model was developed using 40 EEMs of effluent samples from four full-scale wastewater treatment plants. The DON showed a poor correlation with the PARAFAC components (−0.117 < r < 0.308, p = 0.053–0.471), while ABDON correlated significantly with protein-like components C2 (r = 0.511, p < 0.05) and C3 (r = 0.576, p < 0.05, Pearson test). Further, a multivariate linear regression model having a good predictive capability for ABDON with mean absolute error of 12.0% was developed. Results demonstrate that the application of fluorescence spectroscopy for predicting effluent DON may have limitations, however, it could provide an alternative tool for predicting effluent ABDON.

Normalization techniques for PARAFAC modeling of urine metabolomic data

Introduction One of the body fluids often used in metabolomics studies is urine. The concentrations of metabolites in urine are affected by hydration status of an individual, resulting in dilution differences. This requires therefore normalization of the data to correct for such differences. Two normalization techniques are commonly applied to urine samples prior to their further statistical analysis. First, AUC normalization aims to normalize a group of signals with peaks by standardizing the area under the curve (AUC) within a sample to the median, mean or any other proper representation of the amount of dilution. The second approach uses specific end-product metabolites such as creatinine and all intensities within a sample are expressed relative to the creatinine intensity.

Contrasting dissolved organic matter quality in groundwater in Holocene and Pleistocene aquifers and implications for influencing arsenic mobility

The discontinuous nature of elevated arsenic (As) in drinking water wells of West Bengal and other regions in the Bengal Basin has led to increased interest in the role that sediment-derived organic matter may play in enhancing reductive dissolution and As mobilization. Higher As concentrations have been observed in groundwater in reduced Holocene (grey) aquifers when compared to oxidized Pleistocene (orange) aquifers. In order to evaluate if the differences in the chemical character of dissolved organic matter (DOM) are present in groundwater in the Holocene and the Pleistocene aquifers that may influence dissolved As concentrations, shallow groundwater and surface water samples were collected from four study sites in Murshidabad district, West Bengal, India, and analyzed for water chemistry parameters and characteristics of DOM. For wells known to typically contain high As concentrations (in Holocene sediments) in Beldanga (10–4622 μg/L, at 35–45 m depth) and Hariharpara (5–695 μg/L, at 6–37, depth) sites, as well as wells characterized by low As concentrations (Pleistocene sediiments) in Nabagram (0–16 μg/L, at 20–45 m depth) and Kandi (5–50 μg/L, at 20–55 m depth), detailed DOM characterization was carried out using fluorescence spectroscopy and parallel factor analysis (PARAFAC). Results from statistical analysis of a variety of optical (absorbance and fluorescence) DOM properties revealed that the DOM in groundwater in the Holocene aquifer had high humification index (HIX) and low freshness index (β:α) values, whereas groundwater in the Pleistocene aquifer comprised more labile and microbial DOM sources. Consistent with the more labile nature of DOM in groundwater in the Pleistocene aquifer, two ratios 1) humic-like to protein-like components (humic:protein) and 2) terrestrially-derived to microbially-derived components (terr:microb) obtained from a four-component PARAFAC model were 1.9 and 2.9 times greater, respectively, in groundwater in the Holocene aquifer than in that of the Pleistocene aquifer, which suggests that the absence of humic-like DOM may be an important limitation to As mobility.

Spatiotemporal variations in the abundance and composition of bulk and chromophoric dissolved organic matter in seasonally hypoxia-influenced Green Bay, Lake Michigan, USA

Green Bay, Lake Michigan, USA, is the largest freshwater estuary in the Laurentian Great Lakes and receives disproportional terrestrial inputs as a result of a high watershed to bay surface area ratio. While seasonal hypoxia and the formation of “dead zones” in Green Bay have received increasing attention, there are no systematic studies on the dynamics of dissolved organic matter (DOM) and its linkage to the development of hypoxia. During summer 2014, bulk dissolved organic carbon (DOC) analysis, UV–vis spectroscopy, and fluorescence excitation-emission matrices (EEMs) coupled with PARAFAC analysis were used to quantify the abundance, composition and source of DOM and their spatiotemporal variations in Green Bay, Lake Michigan. Concentrations of DOC ranged from 202 to 571μM-C (average=361±73μM-C) in June and from 279 to 610μM-C (average=349±64μM-C) in August. In both months, absorption coefficient at 254nm (a254) was strongly correlated to bulk DOC and was most abundant in the Fox River, attesting a dominant terrestrial input. Non-chromophoric DOC comprised, on average, ~32% of bulk DOC in June with higher terrestrial DOM and ~47% in August with higher aquagenic DOM, indicating that autochthonous and more degraded DOM is of lower optical activity. PARAFAC modeling on EEM data resulted in four major fluorescent DOM components, including two terrestrial humic-like, one aquagenic humic-like, and one protein-like component. Variations in the abundance of DOM components further supported changes in DOM sources. Mixing behavior of DOM components also indicated that while bulk DOM behaved quasi-conservatively, significant compositional changes occurred during transport from the Fox River to the open bay.

Use of fluorescence EEM to monitor the removal of emerging contaminants in full scale wastewater treatment plants

This study investigated the applicability of different techniques for fluorescence excitation/emission matrices data interpretations, including peak-picking method, fluorescence regional integration and PARAFAC modelling, to act as surrogates in predicting emerging trace organic compounds (ETOrCs) removal during conventional wastewater treatments that usually comprise primary and secondary treatments. Results showed that fluorescence indexes developed using alternative methodologies but indicative of a same dissolved organic matter component resulted in similar predictions of the removal of the target compounds. The peak index defined by the excitation/emission wavelength positions (λex/λem) 225/290nm and related to aromatic proteins and tyrosine-like fluorescence was determined to be a particularly suitable surrogate for monitoring ETOrCs that had very high removal rates (average removal >70%) (i.e., triclosan, caffeine and ibuprofen). The peak index defined by λex/λem=245/440nm and the PARAFAC component with wavelength of the maxima λex/λem=245, 350/450, both identified as humic-like fluorescence, were found remarkably well correlated with ETOrCs such as atenolol, naproxen and gemfibrozil that were moderately removed (51–70% average removal). Finally, the PARAFAC component with wavelength of the maxima λex/λem=<240, 315/380 identified as microbial humic-like fluorescence was the only index correlated with the removal of the antibiotic trimethoprim (average removal 68%).

Changing Dynamics of Dissolved Organic Matter Fluorescence in the Northern Gulf of Mexico Following the Deepwater Horizon Oil Spill.

The characteristics of fluorescent components of dissolved organic matter (DOM) were examined using excitation emission matrix (EEM) fluorescence spectroscopy combined with parallel-factor analysis (PARAFAC) for seawater samples obtained from the northern Gulf of Mexico (NGoM) before, during, and after the 2010 Deepwater Horizon (DwH) oil spill. An EEMs PARAFAC modeling of samples collected within 16 km of the wellhead during the oil spill in May 2010, which included one typical subsurface sample with a PAH concentration of 1.09 μg/L, identified two humic-like and two previously reported oil-like components. Compared to prespill levels, however, there were order-of-magnitude higher fluorescence intensities associated with these components that are consistent with an oil-spill source. The spectral decomposition of the EEMs data using individual and combined data sets from coastal and offshore waters impacted by the DwH spill further revealed the changing nature of fluorescent DOM composition. Although the PAHs concentrations were at prespill conditions after the spill in 2012 and 2013 near the DwH site, the variable and anomalous levels of fluorescence intensities and DOC concentrations three years after the spill suggest the potential long-term persistence of the oil in the DOC pool in the NGoM.

The impact of land use on riparian soil dissolved organic matter and on streamwater quality on Chongming Island, China

Riparian wetland controls the transfer of terrestrial dissolved organic matter (DOM) to surface water bodies. However, the effects of land use on riparian soil DOM quality and its contribution to aquatic environment are largely unknown. In this study, the amount and composition of water-extracted soil organic matter (WSOM) in riparian wetlands were determined to evaluate the effect of land uses on spatial patterns of WSOM and streamwater quality on Chongming Island, China. The fluorescent properties of WSOM and fluvial DOM were analyzed using EEM spectra-combining PARAFAC model and accurate MS and MS/MS identification. Our findings showed no differences in the riparian WSOC contents between land use types (agricultural land, natural wetland, commercial land and industrial land). However, the fluorescent WSOM and its humic-like (Comp.1 and Comp.2) and microbial degradation (Comp.4) components significantly varied under different land uses (P < 0.05). Overall, the fluorescent WSOM quantities and its components (Comp.1, Comp.2 and Comp.4) were present at markedly lower concentrations for agricultural land use relative to the other three land uses. The same distribution pattern was observed for carbonyl compounds and fatty acids in the riparian WSOM molecules (P < 0.05), but the distribution patterns of the lipids were different between the four land uses (P < 0.05). Industrial land could result in the input of more organic matter into the riparian wetland. Our results showed that fluvial Comp.1 and Comp.2 were significantly correlated with WSOM Comp.2 and WSOM Comp.4 (P < 0.05). We also observed that the fluvial trophic status was significantly higher when the fluvial DOM components increased (P < 0.05). These results indicated that land uses can alter the composition of riparian WSOM, reshape fluvial DOM compositions and significantly affect fluvial water quality.

Drivers of fluorescent dissolved organic matter in the global epipelagic ocean

AbstractFluorescent dissolved organic matter (FDOM) in open surface waters (&lt; 200 m) of the Atlantic, Pacific, and Indian oceans was analysed by excitation‐emission matrix (EEM) spectroscopy and parallel factor analysis (PARAFAC). A four‐component PARAFAC model was fit to the EEMs, which included two humic‐ (C1 and C2) and two amino acid‐like (C3 and C4) components previously identified in ocean waters. Generalized‐additive models (GAMs) were used to explore the environmental factors that drive the global distribution of these PARAFAC components. The explained variance for the humic‐like components was substantially larger (&gt; 70%) than for the amino acid‐like components (&lt; 35%). The environmental variables exhibiting the largest effect on the global distribution of C1 and C2 were apparent oxygen utilisation followed by chlorophyll a. Positive non‐linear relationships between both predictor variables and the two humic‐like PARAFAC components suggest that their distribution are biologically controlled. Compared with the dark ocean (&gt; 200 m), the relationships of C1 and C2 with AOU indicate a higher C1/AOU and C2/AOU ratios of the humic‐like substances in the dark ocean than in the surface ocean where a net effect of photobleaching is also detected. C3 (tryptophan‐like) and C4 (tyrosine‐like) variability was mostly dictated by salinity (S), by means of positive non‐linear relationships, suggesting a primary physical control of their distributions at the global surface ocean scale that could be related to the changing evaporation‐precipitation regime. Remarkably, bacterial biomass (BB) only contributed to explain a minor part of the variability of C1 and C4.

Fluorescence excitation–emission matrix spectroscopy as a tool for determining quality of sparkling wines

Browning in sparkling wines was assessed by the use of excitation–emission fluorescence spectroscopy combined with PARAllel FACtor analysis (PARAFAC). Four different cava sparkling wines were monitored during an accelerated browning process and subsequently storage. Fluorescence changes observed during the accelerated browning process were monitored and compared with other conventional parameters: absorbance at 420nm (A420) and the content of 5-hydroxymethyl-2-furfural (5-HMF). A high similarity of the spectral profiles for all sparkling wines analyzed was observed, being explained by a four component PARAFAC model. A high correlation between the third PARAFAC factor (465/530nm) and the commonly used non-enzymatic browning indicators was observed. The fourth PARAFAC factor (280/380nm) gives us also information about the browning process following a first order kinetic reaction. Hence, excitation–emission fluorescence spectroscopy, together with PARAFAC, provides a faster alternative for browning monitoring to conventional methods, as well as useful key indicators for quality control.

Closed-Form Semi-Blind Receiver For MIMO Relay Systems Using Double Khatri–Rao Space-Time Coding

In this letter, we consider a one-way two-hop AF relaying scheme employing two independent Khatri-Rao space-time (KRST) codings at the source and relay nodes. The signals received at destination form a fourth-order tensor whose dimensions correspond to four signal diversities, and which satisfies a nested PARAFAC model. Exploiting this nested structure, we derive two matrix unfoldings expressed in terms of two Khatri-Rao products which are used to propose a closed-form semi-blind receiver allowing to jointly estimate the information symbols and the individual channels. A numerical analysis shows that this new receiver achieves a substantial computational complexity reduction over an iterative (ALS-based) semi-blind receiver, especially in presence of a great number of source and/or relay antennas.