Complex-valued Signals Research Articles

Ultrahigh-speed, phase-sensitive full-field interferometric confocal microscopy for quantitative microscale physiology.

We developed ultra-high-speed, phase-sensitive, full-field reflection interferometric confocal microscopy (FFICM) for the quantitative characterization of in vivo microscale biological motions and flows. We demonstrated 2D frame rates in excess of 1 kHz and pixel throughput rates up to 125 MHz. These fast FFICM frame rates were enabled by the use of a low spatial coherence, high-power laser source. Specifically, we used a dense vertical cavity surface emitting laser (VCSEL) array that synthesized low spatial coherence light through a large number of narrowband, mutually-incoherent emitters. Off-axis interferometry enabled single-shot acquisition of the complex-valued interferometric signal. We characterized the system performance (~2 μm lateral resolution, ~8 μm axial gating depth) with a well-known target. We also demonstrated the use of this highly parallelized confocal microscopy platform for visualization and quantification of cilia-driven surface flows and cilia beat frequency in an important animal model (Xenopus embryos) with >1 kHz frame rate. Such frame rates are needed to see large changes in local flow velocity over small distance (high shear flow), in this case, local flow around a single ciliated cell. More generally, our results are an important demonstration of low-spatial coherence, high-power lasers in high-performance, quantitative biomedical imaging.

Complex variational mode decomposition for signal processing applications

Complex-valued signals occur in many areas of science and engineering and are thus of fundamental interest. The complex variational mode decomposition (CVMD) is proposed as a natural and a generic extension of the original VMD algorithm for the analysis of complex-valued data in this work. Moreover, the equivalent filter bank structure of the CVMD in the presence of white noise, and the effects of initialization of center frequency on the filter bank property are both investigated via numerical experiments. Benefiting from the advantages of CVMD algorithm, its bi-directional Hilbert time-frequency spectrum is developed as well, in which the positive and negative frequency components are formulated on the positive and negative frequency planes separately. Several applications in the real-world complex-valued signals support the analysis.

High-Accuracy Compressed Sensing Decoder Based on Adaptive <inline-formula> <tex-math notation="LaTeX">$(\ell_{0},\ell_{1})$</tex-math> </inline-formula> Complex Approximate Message Passing: Cross-layer Design

Compressed sensing (CS) allows a signal that is sparse in certain domain to be acquired and reconstructed accurately with only a small number of samples. In this paper, we propose an adaptive $(\ell_{0},\ell_{1})$ complex approximate message passing (CAMP) algorithm and its hardware implementation for complex-valued sparse signal recovery. Compared with the existing CAMP algorithm which solves $\ell_{1}$ -regularized least squares problems, our proposed algorithm adaptively switches between $\ell_{0}$ and $\ell_{1}$ -regularized least squares and therefore significantly outperforms the original CAMP. We implement the architecture in a medium-sized field-programmable gate array (FPGA) chip. For a sparse stepped frequency waveform radar application, we perform experiments on the simulated data and the data collected by a real radar system. According to the result, the decoder design achieves 7.2 dB improvement over the conventional CAMP architecture with less than 27.4% extra hardware cost.

Incremental augmented complex adaptive IIR algorithm for training widely linear ARMA model

In this paper, we propose a distributed adaptive learning algorithm to train the coefficients of a widely linear autoregressive moving average model by measurements collected by the nodes of a network. We assume that each node uses the augmented complex adaptive infinite impulse response (ACA-IIR) filter as the learning rule, and nodes interact with each other under an incremental mode of cooperation. To derive the proposed algorithm, called the incremental ACAIIR (IACA-IIR), we firstly formulate the distributed adaptive learning problem as an unconstrained minimization problem. Then, we apply stochastic gradient optimization argument to solve it and derive the proposed algorithm. We further find the step size range where the stability of the proposed algorithm is guaranteed. We also introduce a reduced-complexity version of the IACA-IIR algorithm. Since the proposed algorithm relies on the augmented complex statistics, it can be used to model both types of complex-valued signals (proper and improper signals). To evaluate the performance of the proposed algorithm, we use both synthetic and real-world complex signals in our simulations. The results exhibit superior performance of the proposed algorithm over the non-cooperative ACA-IIR algorithm.

Optimum time delay estimation for complex-valued stationary signals

The problem of finding the time delay between complex-valued sequences received at two spatially separated sensors is addressed. Considering white signal and noise processes, three delay estimation algorithms based on the minimum mean square error criterion are devised. Their asymptotic mean and mean square error expressions are derived which show that all are unbiased estimators with performance approaching the Cramér–Rao lower bound. Numerical results are included to validate the theoretical development and contrast the three techniques.

An Efficient POCS Interpolation Method in the Frequency-Space Domain

Sampling irregularity in observed seismic data may cause a significant complexity increase in subsequent processing. Seismic data interpolation helps in removing this sampling irregularity, for which purpose complex-valued curvelet transform is used, but it is time-consuming because of the huge size of observed data. In order to improve efficiency as well as keep interpolation accuracy, I first extract principal frequency components using forward Fourier transform. The size of the principal frequency-space domain data is at least halved compared with that of the original time-space domain data because the complex-valued components of the representation of a real-valued signal (i.e., a complex-valued signal with zero as its imaginary component) exhibit conjugate symmetry in the frequency domain. Then, the projection onto convex projection (POCS) method is used to interpolate frequency-space data based on complex-valued curvelet transform. Finally, interpolated seismic data in the time-space domain can be obtained using inverse Fourier transform. Synthetic data and field data examples show that the efficiency can be improved more than two times and the performance is slightly better in the frequency-space domain compared with the POCS method directly performed in the time-space domain, which demonstrates the validity of the proposed method.

Phaseless computational imaging with a radiating metasurface.

Computational imaging modalities support a simplification of the active architectures required in an imaging system and these approaches have been validated across the electromagnetic spectrum. Recent implementations have utilized pseudo-orthogonal radiation patterns to illuminate an object of interest-notably, frequency-diverse metasurfaces have been exploited as fast and low-cost alternative to conventional coherent imaging systems. However, accurately measuring the complex-valued signals in the frequency domain can be burdensome, particularly for sub-centimeter wavelengths. Here, computational imaging is studied under the relaxed constraint of intensity-only measurements. A novel 3D imaging system is conceived based on 'phaseless' and compressed measurements, with benefits from recent advances in the field of phase retrieval. In this paper, the methodology associated with this novel principle is described, studied, and experimentally demonstrated in the microwave range. A comparison of the estimated images from both complex valued and phaseless measurements are presented, verifying the fidelity of phaseless computational imaging.

Quaternion Wigner–Ville distribution associated with the linear canonical transforms

The quaternion linear canonical transform (QLCT), a generalization of the classical 2D Fourier transform, has gained much popularity in recent years because of its applications in many areas, including color image and signal processing. There are relationship between Wigner distribution and ambiguity function. But, these relations are only suitable for complex-valued signals, and have not been investigated in quaternion linear canonical transforms. The purpose of this paper is to propose an equivalent relationship for the quaternion Wigner distribution and quaternion ambiguity function in the QLCT setting. First, we propose the 2D quaternion Wigner distribution (QLWD) and quaternion ambiguity function associated with the QLCT. Next, the relationship between these two novel concepts are derived. Moreover, the connection with the corresponding analytic signal are investigated. Examples with bandpass analytic signal illustrate the features of the proposed distributions. Finally a novel algorithm for the detection of quaternion-valued linear frequency-modulated signal is presented by using the proposed QLWD.

Analysis on LUT based digital predistortion using direct learning architecture for linearizing power amplifiers

In modern wireless communication system, power amplifier (PA) is an important component which is expected to be operated at the region of high power efficiency, but in this region, PA is inherently nonlinear. Thus, the linearization of high power efficient PA is necessary. In this paper, direct learning architecture (DLA) and indirect learning architecture (ILA) are firstly compared. It shows that DLA is more robust than ILA. Then a baseband digital predistortion (DPD) method with DLA is proposed for power amplifier linearization based on combined look-up tables (LUT) and memory polynomial (MP) model. The main innovation is that a LUT-based approach is proposed to calculate directly the complex-valued predistorted signal. Moreover, some interpolation techniques are introduced to reduce the LUT size. The proposed DPDs are validated experimentally. Additionally, the influences of some important parameters in experimental setup, such as the number of bits of analog-to-digital converter (ADC) and the instrument bandwidth, are analyzed.

Using sparsity information for iterative phase retrieval in x-ray propagation imaging.

For iterative phase retrieval algorithms in near field x-ray propagation imaging experiments with a single distance measurement, it is indispensable to have a strong constraint based on a priori information about the specimen; for example, information about the specimen's support. Recently, Loock and Plonka proposed to use the a priori information that the exit wave is sparsely represented in a certain directional representation system, a so-called shearlet system. In this work, we extend this approach to complex-valued signals by applying the new shearlet constraint to amplitude and phase separately. Further, we demonstrate its applicability to experimental data.

Joint algebraic coded modulation and lattice‐reduction‐aided preequalisation

Lattice-reduction-aided (LRA) preequalisation for the multiple-input/multiple-output broadcast channel has most often been considered for the uncoded case so far. However, recent advantages in the closely related field of integer-forcing equalisation, where the cancellation of the multiuser interference and the channel coding are combined, create doubt to this separated point of view. In this Letter, the philosophy of matching both the channel code and the complex-valued signal constellation to the same finite-field arithmetic is proposed. The consequences on the factorisation task present in LRA preequalisation are discussed and covered by numerical performance evaluations based on non-binary low-density parity-check codes.

Analysis of Incremental Augmented Affine Projection Algorithm for Distributed Estimation of Complex-Valued Signals

In this paper the aim is to solve the problem of distributed estimation in an incremental network when the measurements taken by the nodes follow a widely linear model. The proposed algorithm, which we refer to as incremental augmented affine projection algorithm (incAAPA), utilizes the full second order statistical information in the complex domain. Moreover, it exploits the spatio-temporal diversity to improve the estimation performance. We derive steady-state performance metric of the incAAPA in terms of mean-square deviation. We further derive sufficient conditions to ensure mean-square convergence. Our analysis illustrates that the proposed algorithm is able to process both second-order circular (proper) and non-circular (improper) signals. The validity of the theoretical results and the good performance of the proposed algorithm are demonstrated by several computer simulations.

A Novel and Fast Algorithm for Solving Permutation in Convolutive BSS, Based on Real and Imaginary Decomposition

In this paper, a new fast method for solving the permutation problem in convolutive BSS is presented. Typically, by transferring signals to the frequency domain, the convolutive BSS problem is converted to an instantaneous BSS, and deconvolution takes place in each frequency bin. However, another major problem arises which is permutation ambiguity in the frequency domain. Solving the permutation ambiguity for N sources in frequency domain needs N! comparisons between adjacent frequency bins. This drastically increases the overall computational complexity of the convolutive BSS. In our new approach, the complex-valued signals are decomposed into real and imaginary parts in each frequency bin. We show that the ideal mixing matrix has to possess a simple and symmetric structure. Accordingly, the structure can be exploited for solving the permutation ambiguity in frequency domain. Although separation in subband is accomplished by the FastICA algorithm, the proposed method requires modification of the separation algorithm, and a new structure is imposed on the mixing matrix. After that signals are separated by means of the FastICA, the permutation correction takes place only by N comparisons, decreasing the computational complexity. Comparing to five competitive methods, we experimentally demonstrate that permutation ambiguity is resolved accurately by this very fast approach while substantially decreasing the order of calculations. In terms of the separation performance and signal quality, the proposed method is superior to four of the compared methods and almost similar to the best of them.

Compact design for two-dimensional electronic spectroscopy

We present a passively phase-stabilized two-dimensional electronic spectroscopy (2DES) with a compact size, and the ease of implementation and maintenance. Our design relies on a mask beam-splitter with four holes to form non-collinear box geometry, and a homebuilt stacked retroreflector, which introduces the phase-locked pulse sequence, remedying the instability of commonly used translation stages. The minimized size of the setup suppresses the influences of optical path-length fluctuations during measurements, improving the phase stability and precise timing of pulse sequences. In our 2DES, only few conventional optical components are used, which make this sophisticated instrumentation convenient to establish and particularly easy to conduct alignment. In data analysis, the self-referencing spectral interferometry (SRSI) method is first introduced to extract the complex-valued signal from spectral interferometry in 2DES. The alternative algorithm achieves the improvement of the signal-to-noise ratio (SNR) and considerable reduction of data acquisition time. The new setup is suitable over a tunable range of spectroscopic wavelength, from ultraviolet (UV) to the near-infrared (NIR) regime, and for ultra-broadband bandwidth, few-cycle laser pulses.

Smoothing localized directional cyclic autocorrelation and application in oil-film instability analysis

In this paper, smoothing localized directional cyclic autocorrelation (SLDCA) is proposed as a novel time–frequency distribution to analyze the vibration signal of rotor-bearing system in oil-film instability. Based on the cyclostationarity of real-valued signal, directional cyclic autocorrelation (DCA) is defined for the complex-valued signal. In order to suppress the cross-term interferences, the DCA is localized and filtered with two-dimensional time–frequency window which allows the smoothing kernel function to adapt to the signal time–frequency-varying characteristics. And then the localized DCA is smoothed by the localized optimal radially Gaussian kernel and cascaded to produce the SLDCA with high time–frequency resolution and less susceptibility to cross-term interferences. The application of SLDCA in the oil-film instability analysis verifies that the SLDCA can not only precisely detect the instantaneous frequency information in the time–frequency distribution with high resolution and robustness to the noises and cross-terms, but also provide the phase coupling information of rotor instantaneous planar motion by which the directivity and shape of the planar motion can be determined.

Adaptive Complex-Valued Independent Component Analysis Based on Second-Order Statistics

This paper proposes a two-stage fast convergence adaptive complex-valued independent component analysis based on second-order statistics of complex-valued source signals. The first stage constructs a cost function by extending the real-valued whiten cost function to a complex-valued domain and optimizes the cost function using a complex-valued gradient. The second stage uses the restriction that the pseudocovariance matrix of the separated signal is a diagonal matrix to construct the cost function and the geodesic method is used to optimize the cost function. Compared with other adaptive complex-valued independent component analysis, the proposed method shows a faster convergence rate and smaller error. Computer simulations were performed on synthesized signals and communications signals. The simulation results demonstrate the validity of the proposed algorithm.

Underdetermined blind modal identification of structures by earthquake and ambient vibration measurements via sparse component analysis

Sparse component analysis (SCA) approach was adopted to handle underdetermined blind modal identification of structures, where the number of sensors is less than the number of active modes. To exploit the sparsity of structural responses in time–frequency domain, Short Time Fourier Transform (STFT) was used in this study. The proposed SCA-based approach has two main stages: modal matrix estimation and modal displacement estimation. In the first stage, hierarchical clustering algorithm was used to estimate the modal matrix. The clustering algorithm was preceded by a preprocessing step to select the points in time–frequency domain that only one mode makes contribution in the responses. These points were fed to the clustering algorithm as an input. Performing this analysis enhanced the modal matrix estimation accuracy and reduced the computational cost while conducting clustering analysis. Having estimated mixing matrix, the complex-valued modal responses in the transformed domain were recovered via Smoothed zero-norm (SL-0) algorithm. In a broad sense, using the SL-0 algorithm permits researchers to use any kind of transform in seeking sparsity, regardless of obtaining real-valued or complex-valued signals in transformed domain. Natural frequencies and damping ratios were extracted from the recovered modal responses. Performance of the proposed method was investigated using a synthetic example and a benchmark structure with earthquake and ambient excitation, respectively.

Improved velocimetry in optical coherence tomography using Bayesian analysis.

OCT is a popular cross-sectional microscale imaging modality in medicine and biology. While structural imaging using OCT is a mature technology in many respects, flow and motion estimation using OCT remains an intense area of research. In particular, there is keen interest in maximizing information extraction from the complex-valued OCT signal. Here, we introduce a Bayesian framework into the data workflow in OCT-based velocimetry. We demonstrate that using prior information in this Bayesian framework can significantly improve velocity estimate precision in a correlation-based, model-based framework for Doppler and transverse velocimetry. We show results in calibrated flow phantoms as well as in vivo in a Drosophila melanogaster (fruit fly) heart. Thus, our work improves upon the current approaches in terms of improved information extraction from the complex-valued OCT signal.

Statistical analysis of motion contrast in optical coherence tomography angiography.

Optical coherence tomography angiography (Angio-OCT), mainly based on the temporal dynamics of OCT scattering signals, has found a range of potential applications in clinical and scientific research. Based on the model of random phasor sums, temporal statistics of the complex-valued OCT signals are mathematically described. Statistical distributions of the amplitude differential and complex differential Angio-OCT signals are derived. The theories are validated through the flow phantom and live animal experiments. Using the model developed, the origin of the motion contrast in Angio-OCT is mathematically explained, and the implications in the improvement of motion contrast are further discussed, including threshold determination and its residual classification error, averaging method, and scanning protocol. The proposed mathematical model of Angio-OCT signals can aid in the optimal design of the system and associated algorithms.

Improper Complex-Valued Bhattacharyya Distance

Motivated by application of complex-valued signal processing techniques in statistical pattern recognition, classification, and Gaussian mixture (GM) modeling, this paper derives analytical expressions for computing the Bhattacharyya coefficient/distance (BC/BD) between two improper complex-valued Gaussian distributions. The BC/BD is one of the most widely used statistical measures for evaluating class separability in classification problems, feature extraction in pattern recognition, and for GM reduction (GMR) purposes. The BC provides an upper bound on the Bayes error, which is commonly known as the best criterion to evaluate feature sets. Although the computation of the BC/BD between real-valued signals is a well-known result, it has not yet been extended to the case of improper complex-valued Gaussian densities. This paper addresses this gap. We analyze the role of the pseudocovariance matrix, which characterizes the noncircularity of the signal, and show that it carries critical second-order statistical information for computing the BC/BD. We derive upper and lower bounds on the BD in terms of the eigenvalues of the covariance and pseudocovariance matrices of the underlying densities. The theoretical bounds are then used to introduce the concept of β -dominance in the context of statistical distance measures. The BC is pseudometric, since it fails to satisfy the triangle inequality. Using the Matusita distance (a full-metric variant of the BC), we propose an intuitively pleasing indirect distance measure for comparing two general GMs. Finally, we investigate the application of the proposed BC/BD measures for GMR purposes and develop two BC-based GMR algorithms.