Human gait analysis has been widely used in medical and health fields. It is essential to extract spatio-temporal gait features (e.g., single support duration, step length, and toe angle) by partitioning the gait phase and estimating the footprint position/orientation in such fields. Therefore, we propose a method to partition the gait phase given a foot position sequence using mutually constrained piecewise linear approximation with dynamic programming, which not only represents normal gait well but also pathological gait without training data. We also propose a method to detect footprints by accumulating toe edges on the floor plane during stance phases, which enables us to detect footprints more clearly than a conventional method. Finally, we extract four spatial/temporal gait parameters for accuracy evaluation: single support duration, double support duration, toe angle, and step length. We conducted experiments to validate the proposed method using two types of gait patterns, that is, healthy and mimicked hemiplegic gait, from 10 subjects. We confirmed that the proposed method could estimate the spatial/temporal gait parameters more accurately than a conventional skeleton-based method regardless of the gait pattern.

In this letter, a piecewise linear (PWL) sigmoid function approximation based on the statistical distribution probability of the neurons' values in each layer is proposed to improve the network recognition accuracy with only addition circuit. The sigmoid function is first divided into three fixed regions, and then according to the neurons' values distribution probability, the curve in each region is segmented into sub-regions to reduce the approximation error and improve the recognition accuracy. Experiments performed on Xilinx's FPGA-XC7A200T for MNIST and CIFAR-10 datasets show that the proposed method achieves 97.45% recognition accuracy in DNN, 98.42% in CNN on MNIST and 72.22% on CIFAR-10, up to 0.84%, 0.57% and 2.01% higher than other approximation methods with only addition circuit.

In this study we investigate the synchronization of relaxation oscillators having individual differences by using non-periodic signal injection. When a common non-periodic signal is injected into the relaxation oscillators, the oscillators exhibit synchronization phenomena. Such synchronization phenomena can be classified as injection locking. We also consider the relation between the synchronization state and the individual difference. Further, we pay attention to the effect of the fluctuation range of the non-periodic injected signal. When the fluctuation range is wide, we confirm that the synchronization range increases if the individual difference is small.

Pulse Pairs (PPs) generated by Distance Measure Equipment (DME) cause severe interference on L-band Digital Aeronautical Communication System type 1 (L-DACS1) which is based on Orthogonal Frequency Division Multiplexing (OFDM). In this paper, a novel and practical PP mitigation approach is proposed. Different from previous work, it adopts only time domain methods to mitigate interference, so it will not affect the subsequent signal processing in frequency domain. At the receiver side, the proposed approach can precisely reconstruct the deformed PPs (DPPs) which are often overlapped and have various parameters. Firstly, a filter bank and a correlation scheme are jointly used to detect non-overlapped DPPs, also a weighted average scheme is used to automatically measure the waveform of DPP. Secondly, based on the measured waveform, sparse estimation is used to estimate the precise positions of DPPs. Finally, the parameters of each DPP are estimated by a non-linear estimator. The key point of this step is, a piecewise linear model is used to approximate the non-linear carrier frequency of each DPP. Numerical simulations show that comparing with existing work, the proposed approach is more robust, closer to interference free environment and its Bit Error Rate is reduced by about 10dB.

This paper studies a privacy-preserving decision tree learning protocol (PPDT) for vertically partitioned datasets. In vertically partitioned datasets, a single class (target) attribute is shared by both parities or carefully treated by either party in existing studies. The proposed scheme allows both parties to have independent class attributes in a secure way and to combine multiple class attributes in arbitrary boolean function, which gives parties some flexibility in data-mining. Our proposed PPDT protocol reduces the CPU-intensive computation of logarithms by approximating with a piecewise linear function defined by light-weight fundamental operations of addition and constant multiplication so that information gain for attributes can be evaluated in a secure function evaluation scheme. Using the UCI Machine Learning dataset and a synthesized dataset, the proposed protocol is evaluated in terms of its accuracy and the sizes of trees*.

In this letter, we investigate the performance of cooperative decode-and-forward multiple-input multiple-output relaying system using orthogonal space-time block codes with piecewise-linear (PL) receiver over correlated Nakagami-m fading channels for integer values of m. We derive the closed-form expression for the exact bit error rates of binary phase shift keying signals. The analytical expression is validated through numerical results. It is shown that the performance of PL receiver outperforms that of conventional maximal ratio combining receiver.

This paper proposes an adaptive algorithm for identifying unknown systems containing nonlinear amplitude characteristics. Usually, the nonlinearity is so small as to be negligible. However, in low cost systems, such as acoustic echo canceller using a small loudspeaker, the nonlinearity deteriorates the performance of the identification. Several methods preventing the deterioration, polynomial or Volterra series approximations, have been hence proposed and studied. However, the conventional methods require high processing cost. In this paper, we propose a method approximating the nonlinear characteristics with a piecewise linear curve and show using computer simulations that the performance can be extremely improved. The proposed method can also reduce the processing cost to only about twice that of the linear adaptive filter system.

In this paper, we propose a variational method to derive the coefficients of piecewise-linear (PWL) models able to accurately approximate nonlinear functions, which are vector fields of autonomous dynamical systems described by continuous-time state-space models dependent on parameters. Such dynamical systems admit limit cycles, and the supercritical Hopf bifurcation normal form is chosen as an example of a system to be approximated. The robustness of the approximations is checked, with a view to circuit implementations.

Piecewise linear (PWL) circuit modules operating on sigma-delta (ΣΔ) modulated signals and nonlinear signal processors built of these modules are proposed. The proposed module library includes absolute circuits, min/max selectors and negative resistances. Their output signal-to-noise ratio is higher than 50dB when their oversampling ratio is 28. A nonlinear filter and a stochastic resonator are presented as applications of the PWL modules to ΣΔ domain signal processing. The filter is structured with 37% of logic gates consumed by an equivalent filter with a 5-bit parallel signal form.

In this paper, a novel type of neural networks called grey neural network (GNN) is proposed and applied to improve short term load forecasting (STLF) performance. This work is motivated by the following observations: First, the forecasting performance of neural network is affected by the randomness in STLF data. That is, poor performance results from large randomness and vice versa. Second, the grey first-order accumulated generating operation (1-AGO) is reported having randomness reduction property. By the observations, the GNN is proposed and expected to have better STLF performance. The GNN consists of grey 1-AGO, the piecewise linear neural network (PLNN), and grey first-order inverse accumulated generating operation (1-IAGO). Given a set of STLF data, the data is first converted by grey 1-AGO and then is put into the PLNN to perform forecasting. Finally, the predicted load of GNN is obtained through grey 1-IAGO. For comparison, the original STLF data is also put into the PLNN itself. With identical training conditions, the simulation results indicate that with various network structures the GNN, as expected, outperforms the PLNN itself in terms of mean squared error.

In this paper, we consider a switching system modeled by a discrete-time flow model. By simulation, it is shown that a lot of border-collision bifurcations occur since the system is piecewise linear. By using its characteristics, we classify its dynamics into modes, and we define blocks and a kind of Poincare map based on the modes. We calculate occurrence conditions of each block and all the Poincare points by computer-assisted analysis. We consider two bifurcation phenomena, and we show that a Poincare point hits a boundary of the occurrence conditions of a block. So, both bifurcations are indeed border-collision bifurcations.

In this article, we propose a square wave generator whose switching threshold values are switched by external inputs. This circuit is designed to simulate the synchronized luminescence of coupled fireflies. We investigate the behavior of the solutions in two coupled oscillators. The dynamics are demonstrated by a linear autonomous equation piecewisely, therefore, a one-dimensional return map is derived. We also prove the existence of stable in-phase synchronization in the coupled oscillator by using the return map, and we show the existence of regions of periodic solutions within a parameter space. Some theoretical results are confirmed by laboratory measurements.

Periodic solutions of slow-fast systems called "canards," "ducks," or "lost solutions" are examined in a second order autonomous system. A characteristic feature of the canard is that the solution very slowly moves along the negative resistance of the slow curve. This feature comes from that the solution moves on or very close to a curve which is called slow manifolds or "rivers." To say reversely, solutions which move very close to the river are canards, this gives a heuristic definition of the canard. In this paper, the generation mechanism of the canard is examined using a piecewise linear system in which the cubic function is replaced by piecewise linear functions with three or four segments. Firstly, we pick out the rough characteristic feature of the vector field of the original nonlinear system with the cubic function. Then, using a piecewise linear model with three segments, it is shown that the slow manifold corresponding to the less eigenvalue of two positive real ones is the river in the segment which has the negative resistance. However, it is also shown that canards are never generated in the three segments piecewise linear model because of the existence of the "nodal type" invariant manifolds in the segment. In order to generate the canard, we propose a four segments piecewise linear model in which the property of the equilibrium point is an unstable focus.

We analyze dynamics of a simple hysteresis network (ab. SHN) which has only two parameters. We classify the periodic orbits and clarify the number of attractors and their domain of attraction. The SHN is a piecewise linear system, and therefore we can calculate the trajectory using exact solutions. We clarify the bifurcation sets on which equilibrium attractors bifurcate to the periodic orbits. We also give a sufficient condition for stability of the periodic orbits, and the stability is verified by laboratory experiment. The results of this paper may contribute to the development of an efficient multi functional artificial neural network.

For a fuzzy control of manipulated variable so as to match a required output of a plant, tuning of fuzzy rules are necessary. For its purpose, various methods to tune their rules automatically have been proposed. In these method, some of them necessitate much time for its tuning, and the others are lacking in the generalization capability. In the fuzzy control by the steepest descent method, a use of piecewise linear membership functions (MSFs) has been proposed. In this algorithm, MSFs of the premise for each fuzzy rule are tuned having no relation to the other rules. Besides, only the MSFs corresponding to the given input and output data for the learning can be tuned efficiently. Comparing with the conventional triangular form and the Gaussian distribution of MSFs, an expansion of the expressiveness is indicated. As a result, for constructing the inference rules, the training cycles can be reduced in number and the generalization capability to express the behavior of a plant is expansible. An effectiveness of this algorithm is illustrated with an example of a parallel parking of an autonomous mobile robot.

This paper proposes a tool to analyze complicated phenomena from a simple hysteresis network. The simple hysteresis network is described by a piecewise liner ordinal differential equation and has only two parameters: self feedback and DC team. Then this simple system exhibits various kinds of attractors: stable equilibria, periodic orbits, tori and chaos. In order to perform the numerical analysis, we derive return map and propose a fast calculation algorithm for the return map and its Lyapunov exponents based on the exact solutions. Using this algorithm, we have clarified chaos generation and related bifurcation phenomena. Also, we give theoretical formula that give fundamental bifurcation set.

A parallel blower system is quite familiar in hydraulic machine systems and quite often employed in many process industries. It is dynamically dual to the fundamental functional element of digital computer, that is, the flip-flop circuit, which was extensively studied by Moser. Although the global dynamic behaviour of such systems has significant bearing upon system operation, no substantial study reports have hitherto been presented. Extensive research concern has primarily been concentrated upon the local stability of the equilibrium point. In the paper, a piecewise linear model is used to analytically and numerically investigate its manifold global dynamic behaviour. To do this, first the Poincar map is defined as a composition boundary map, each of which is defined as the point transformation from the entry point to the end point of any trajectory on some boundary planes. It was already shown that, in some parameter region, the system exhibits the so-called chaotic states. The chaos generating process is investigated using the above Poincar map and it is shown that the map contains the contracting, stretching and folding operations which, as we often see in many cases particularly in horse shoe map, produce the chaotic states. Considering the one dimensional motions of the orbits by such Poincar map, that is, the stretching and folding operations, a one dimensional approximation of the Poincar map is introduced to more closely and exactly study manifold bifurcation processes and some illustrative bifurcation diagrams in relation to system parameters are presented. Thus it is shown how many types of bifurcations like the Hopf, period doubling, saddle node, and homoclinic bifurcations come to exist in the system.

In this paper we introduce the Piecewise Linear Radial Basis Function Model (PWL-RBFM), a new nonlinear model that uses the well known RBF framework to build a PWL functional approximation by combining an l1 norm with a linear RBF function. A smooth generalization of the PWL-RBF is proposed: it is obtained by substituting the modulus function with the logistic function. These models are applied to several time series prediction tasks.

The purpose of our research is to get further improvement in the performance of order statistic filters. The basic idea found in our research is the use of a robust median estimator to obtain median differential order information which the classes of order statistic filter required in order to sort the input signal in the filter window. In order to give the motivation for using a median estimator in the classes of order statistic filters, we derive theorems characterizing the median filters and prove them theoretically using the characteristic that the order statistic filter has the performance for a monotonic signal equivalent with the FIR linear filter. As an application of median operation, we propose and investigate the Median Differential Order Statistic Filter to reduce impulsive noise as well as Gaussian noise and regard it as a subclass of the Order Statistic Filter. Moreover, we introduce the piecewise linear function in the Median Differential Order Statistic Filter to improve performance in terms of edge preservation. We call it the Piecewise Linear Median Differential Order Statistic Filter. The effectiveness of proposed filters is verified theoretically by computing the output Mean Square Error of the filters in parts of edge signals, impulsive noise, small amplitude noise and their combination. Computer simulations also show that the proposed filter can improve the performance in both noise (small-amplitude Gaussian noise and impulsive noise) reduction and edge preservation for one-dimensional signals.

This article proposes a four dimensional autonomous hyperchaos generator whose nonlinear element is only one diode. The circuit is analyzed by regarding the diode as an ideal switch. Hence we can derive the two dimensional return map rigorously and its Lyapunov exponents confirm the hyperchaos generation. Also, a novel mathematical basis for the simplification to the ideal switch is given.