LED communications with linear complexity compensation of dynamic nonlinear distortion

An in-depth study of the second order Volterra kernel of light emitting diodes (LEDs) is provided, with a special focus on compensation of dynamic nonlinear distortion generated by the LED when used as a communications signal transmitter. It is shown that from the factorization of the kernel in the frequency domain follows a simplified time-domain model for LED nonlinearity, consisting of three operations: two linear filtrations and squaring. Next, using the pth inverse theory, the corresponding block structures of nonlinear pre- and postdistorters are derived. As it turns out, they exhibit linear computational complexity. It is shown that the model coefficients defining the kernel may be estimated up to the Nyquist frequency. Numerical results indicate that pre- and postdistortion achieve the same performance with respect to receiver signal to noise power ratio (SNR) and have a considerable advantage over linear equalizers. However, in a practical scenario, the predistortion increases the peak to average power ratio of the transmitted signal (unless it is not high already), possibly leading to performance inferior to postdistortion. Finally, it is shown that the simplified equalizer based on the LED model has roughly identical performance to the regular, quadratic Volterra equalizer.

A new method for the accurate measurement of higher-order frequency response functions of nonlinear structural systems

Higher-order frequency response functions (FRFs) are important to the analysis and identification of structural nonlinearities. Though much research effort has been devoted recently to their potential applications, practical issues concerning the difficulty and accuracy of higher-order FRF measurement have not been rigorously assessed to date. This paper presents a new method for the accurate measurement of higher-order FRFs. The method is developed based on sinusoidal input, which is ideal for exciting a nonlinear structure into desired regimes with flexible control, and the correlation technique, which is a novel signal processing method capable of extracting accurate frequency components present in general nonlinear responses. The correlation technique adopted is a major improvement over Fourier transform based existing methods since it eliminates leakage and aliasing errors altogether and proves to be extremely robust in the presence of measurement noise. Extensive numerical case studies have been carried out to critically assess the capability and accuracy of the proposed method and the results achieved are indeed very promising. Interesting nonlinear behavior such as frequency shift and jump have been observed in first-, second- and third-order FRFs, as well as solitary islands which have been identified over which higher-order FRFs virtually do not change as input force amplitude varies. Higher-order FRFs over such solitary islands are essentially their theoretical counterparts of Volterra transfer functions which can be measured with very low input force and can be profitably employed for the identification of physical parameters of structural nonlinearities. Subsequently, a nonlinear parameter identification method has also been developed using measured higher-order FRFs and results are presented and discussed.

Nonlinear properties of medial entorhinal cortex neurons reveal frequency selectivity during multi-sinusoidal stimulation

The neurons in layer II of the medial entorhinal cortex are part of the grid cell network involved in the representation of space. Many of these neurons are likely to be stellate cells with specific oscillatory and firing properties important for their function. A fundamental understanding of the nonlinear basis of these oscillatory properties is critical for the development of theories of grid cell firing. In order to evaluate the behavior of stellate neurons, measurements of their quadratic responses were used to estimate a second order Volterra kernel. This paper uses an operator theory, termed quadratic sinusoidal analysis (QSA), which quantitatively determines that the quadratic response accounts for a major part of the nonlinearity observed at membrane potential levels characteristic of normal synaptic events. Practically, neurons were probed with multi-sinusoidal stimulations to determine a Hermitian operator that captures the quadratic function in the frequency domain. We have shown that the frequency content of the stimulation plays an important role in the characteristics of the nonlinear response, which can distort the linear response as well. Stimulations with enhanced low frequency amplitudes evoked a different nonlinear response than broadband profiles. The nonlinear analysis was also applied to spike frequencies and it was shown that the nonlinear response of subthreshold membrane potential at resonance frequencies near the threshold is similar to the nonlinear response of spike trains.

Spectral analysis for nonstationary and nonlinear systems: a discrete-time-model-based approach

A new frequency-domain analysis framework for nonlinear time-varying systems is introduced based on parametric time-varying nonlinear autoregressive with exogenous input models. It is shown how the time-varying effects can be mapped to the generalized frequency response functions (FRFs) to track nonlinear features in frequency, such as intermodulation and energy transfer effects. A new mapping to the nonlinear output FRF is also introduced. A simulated example and the application to intracranial electroencephalogram data are used to illustrate the theoretical results.

Quadratic sinusoidal analysis of voltage clamped neurons

Nonlinear biophysical properties of individual neurons are known to play a major role in the nervous system, especially those active at subthreshold membrane potentials that integrate synaptic inputs during action potential initiation. Previous electrophysiological studies have made use of a piecewise linear characterization of voltage clamped neurons, which consists of a sequence of linear admittances computed at different voltage levels. In this paper, a fundamentally new theory is developed in two stages. First, analytical equations are derived for a multi-sinusoidal voltage clamp of a Hodgkin-Huxley type model to reveal the quadratic response at the ionic channel level. Second, the resulting behavior is generalized to a novel Hermitian neural operator, which uses an algebraic formulation capturing the entire quadratic behavior of a voltage clamped neuron. In addition, this operator can also be used for a nonlinear identification analysis directly applicable to experimental measurements. In this case, a Hermitian matrix of interactions is built with paired frequency probing measurements performed at specific harmonic and interactive output frequencies. More importantly, eigenanalysis of the neural operator provides a concise signature of the voltage dependent conductances determined by their particular distribution on the dendritic and somatic membrane regions of neurons. Finally, the theory is concretely illustrated by an analysis of an experimentally measured vestibular neuron, providing a remarkably compact description of the quadratic responses involved in the nonlinear processing underlying the control of eye position during head rotation, namely the neural integrator.

Stimulus-invariant processing and spectrotemporal reverse correlation in primary auditory cortex

The spectrotemporal receptive field (STRF) provides a versatile and integrated, spectral and temporal, functional characterization of single cells in primary auditory cortex (AI). In this paper, we explore the origin of, and relationship between, different ways of measuring and analyzing an STRF. We demonstrate that STRFs measured using a spectrotemporally diverse array of broadband stimuli-such as dynamic ripples, spectrotemporally white noise, and temporally orthogonal ripple combinations (TORCs)-are very similar, confirming earlier findings that the STRF is a robust linear descriptor of the cell. We also present a new deterministic analysis framework that employs the Fourier series to describe the spectrotemporal modulations contained in the stimuli and responses. Additional insights into the STRF measurements, including the nature and interpretation of measurement errors, is presented using the Fourier transform, coupled to singular-value decomposition (SVD), and variability analyses including bootstrap. The results promote the utility of the STRF as a core functional descriptor of neurons in AI.

Parallel cascade identification and its application to protein family prediction

Parallel cascade identification is a method for modeling dynamic systems with possibly high order nonlinearities and lengthy memory, given only input/output data for the system gathered in an experiment. While the method was originally proposed for nonlinear system identification, two recent papers have illustrated its utility for protein family prediction. One strength of this approach is the capability of training effective parallel cascade classifiers from very little training data. Indeed, when the amount of training exemplars is limited, and when distinctions between a small number of categories suffice, parallel cascade identification can outperform some state-of-the-art techniques. Moreover, the unusual approach taken by this method enables it to be effectively combined with other techniques to significantly improve accuracy. In this paper, parallel cascade identification is first reviewed, and its use in a variety of different fields is surveyed. Then protein family prediction via this method is considered in detail, and some particularly useful applications are pointed out.

Volterra series modeling of spatial light modulators

We present a multiple-input, single-output, weakly nonlinear model of spatial light modulators by use of a second-order Volterra series and describe an experimental method to measure the nonlinear transfer functions by means of sinusoidal perturbation and synchronous detection with a lock-in amplifier. We also present an application of this method to a liquid-crystal light valve.

Nonlinear analysis of biological systems using short M-sequences and sparse-stimulation techniques

The m-sequence pseudorandom signal has been shown to be a more effective probing signal than traditional Gaussian white noise for studying nonlinear biological systems using cross-correlation techniques. The effectiveness is evidenced by the high signal-to-noise (S/N) ratio and speed of data acquisition. However, the "anomalies" that occur in the estimations of the cross-correlations represent an obstacle that prevents m-sequences from being more widely used for studying nonlinear systems. The sparse-stimulation method for measuring system kernels can help alleviate estimation errors caused by anomalies. In this paper, a "padded sparse-stimulation" method is evaluated, a modification of the "inserted sparse-stimulation" technique introduced by Sutter, along with a short m-sequence as a probing signal. Computer simulations show that both the "padded" and "inserted" methods can effectively eliminate the anomalies in the calculation of the second-order kernel, even when short m-sequences were used (length of 1023 for a binary m-sequence, and 728 for a ternary m-sequence). Preliminary experimental data from neuromagnetic studies of the human visual system are also presented, demonstrating that the system kernels can be measured with high signal-to-noise (S/N) ratios using short m-sequences.

The identification of nonlinear biological systems: Volterra kernel approaches

Representation, identification, and modeling are investigated for nonlinear biomedical systems. We begin by considering the conditions under which a nonlinear system can be represented or accurately approximated by a Volterra series (or functional expansion). Next, we examine system identification through estimating the kernels in a Volterra functional expansion approximation for the system. A recent kernel estimation technique that has proved to be effective in a number of biomedical applications is investigated as to running time and demonstrated on both clean and noisy data records, then it is used to illustrate identification of cascades of alternating dynamic linear and static nonlinear systems, both single-input single-output and multivariable cascades. During the presentation, we critically examine some interesting biological applications of kernel estimation techniques.

The identification of nonlinear biological systems: Volterra kernel approaches

Representation, identification, and modeling are investigated for nonlinear biomedical systems. We begin by considering the conditions under which a nonlinear system can be represented or accurately approximated by a Volterra series (or functional expansion). Next, we examine system identification through estimating the kernels in a Volterra functional expansion approximation for the system. A recent kernel estimation technique that has proved to be effective in a number of biomedical applications is investigated as to running time and demonstrated on both clean and noisy data records, then it is used to illustrate identification of cascades of alternating dynamic linear and static nonlinear systems, both single-input single-output and multivariable cascades. During the presentation, we critically examine some interesting biological applications of kernel estimation techniques.

Wiener and Volterra analyses applied to the auditory system

The application of a particular branch of non-linear system analysis, the functional series expansion or integral method, to the auditory system is reviewed. Both the Volterra and Wiener approach are discussed and an extension of the Wiener method from its traditional white-noise stimulus approach to that of Poisson distributed clicks is presented. This type of analysis has been applied to compound and single-unit responses from the auditory nerve, cochlear nucleus, auditory midbrain and medial geniculate body. Most studies have estimated only first-order Wiener kernels but in recent years second-order Wiener and Volterra kernels have been estimated, particularly with reference to dynamic non-linearities. A particular form of second-order analysis, the Spectro Temporal Receptive Field, offers an alternative to first-order cross-correlation when phase-lock is absent. The correlation method has revealed that neural synchronization is less affected by intensity changes and damage to the hair cells than is neural firing rate. Although the presence of the static cochlear non-linearity could be demonstrated on the basis of the intensity dependence of the first-order Wiener kernel, the identification of the exact form of the nonlinearity of the peripheral auditory system on basis of higher-order Wiener kernels has so far been inconclusive. However, successes of the method can be found in the description of the dynamic non-linearities and non-linear neural interactions.

Interpretation of functional series expansions

While much research has been devoted to the implementation and application of Volterra and Wiener functional series expansions in the identification and characterization of biological systems, little effort has been focused on the fundamental problem of interpreting the resulting kernels. This paper describes the application of the series to the components of a known model of the human pupil control system. As more complicated elements are put together, insight into kernel interpretation is built up incrementally until the total system is identified. Practical limitations and methods are also discussed.

Practical identification of functional expansions of nonlinear systems submitted to non-Gaussian inputs

Time-domain identification of nonlinear systems represented by functional expansions is considered. A general framework is defined for the analysis of three identification methods: the widely used cross-correlation method, Korenberg's method, and a suboptimal least-squares method based on a stochastic approximation algorithm. First, the major characteristics of the underlying estimation problem are pointed out. Then, the identification methods are interpreted as approximations to an optimal estimator, which helps gain insight into their internal functioning and to the investigation of their connections and differences. Examination of results previously published and of the simulations reported in this article indicate that stochastic approximation is an interesting alternative to other existing methods. Identification of a biological system stimulated by a non-Gaussian input confirms the practicality of this approach.