A Synchronous Look at the Simulink Standard Library

Hybrid systems modeling languages are widely used in the development of embedded systems. Two representatives are Simulink/Stateflow1 that combine Ordinary Differential Equations (ODEs), data-flow and difference equations, hierarchical automata a la StateCharts [14] together with imperative features; and the Modelica language [1]2 based on DAEs with features for modeling discrete components. Ptolemy II3 is another example in which several models of computation are combined [15].The formal verification of hybrid systems has been extensively studied [9]. Yet, we share the viewpoint of Lee and Zheng that hybrid modeling languages are best viewed as programming languages that happen to have a hybrid systems semantics [16, 17]. This raises important questions related to their design, semantics and implementation, to get efficient and reliable simulations as well as provably equivalent embedded target code. While sequential code generation in hybrid modeling tools is routinely used for efficient simulation, it is little used or not used at all to produce target embedded code in critical applications that are submitted to strong safety requirements. This results in a break in the development chain: parts of applications must be rewritten into sequential code and all properties verified on the source model cannot be trusted and have to be reverified on the target code.Sequential code generation from a synchronous language like Lustre [12] has been studied in detail [13]. It can be formalized as a series of source-to-source transformations that eliminate high level constructs like hierarchical automata [11] and the generation of a generic intermediate representation for transition functions, turned itself into C code [6]. This approach, initiated in Lucid Synchrone [18], is implemented in the Scade Suite KCG code generator of Scade 64, used for developing various critical applications.Yet, synchronous languages only manipulate discrete-time signals. Their expressiveness is limited to insure important safety properties like determinacy, execution in bounded time and space, and simple, traceable code generation. Their cyclic execution requires minimal run-time support and does not suffer the complications that accompany numerical solvers of ordinary differential equations. Conversely, a hybrid modeling language allows discrete and continuous time behaviors to interact. But this interaction is not constrained enough, nor specified with adequate precision in tools like Simulink/Stateflow and this occasion semantics pitfalls and bugs [10, 5, 2]. A precise description of all the compilation steps, that is, the actual implemented semantics, is mandatory in safety critical development processes where target code must be trustworthy. Our goal, in short, is to increase the expressiveness of synchronous languages without sacrificing any confidence in their compilers.In previous works, we introduced a novel approach for the design and implementation of a hybrid modeling language that reuses synchronous language principles and an existing compiler infrastructure. We introduced an ideal synchronous semantics based on non standard analysis [5] for a Lustre-like language with ODEs [4], and then extended the kernel language with hierarchical automata [3] and a modular causality analysis [2]. These results form the foundation of Zelus [8]5 and were validated inside the industrial Scade Suite KCG code generator (Release 6.4, 2014) developed at Esterel-Technologies/ANSYS [7].In this talk, I will summarize the ongoing work on Zelus and the way it has been applied to the Scade Suite KCG code generator. In the later, it was possible to reuse the existing infrastructure entirely with little modifications: static typing, the causality and initialization analysis, intermediate languages and the various compiler optimizations. The proposed language extension is conservative in that regular synchronous functions are compiled as before --- the same synchronous code is used both for simulation and for execution on target platforms. It also shows the versatility of the KCG infrastructure based on a successive rewriting. The precise definition of all compilation steps, built on the proven compiler infrastructure of a synchronous language avoids the rewriting of control software and may also increase the confidence in simulation results.This work results from a collaboration with Albert Benveniste, Benoit Caillaud (INRIA, Rennes), Timothy Bourke (INRIA, Paris-Rocquencourt), Jean-Louis Colaco, Bruno Pagano and Cedric Pasteur (Esterel-Technologies/ANSYS).

In this paper, the problem of sampling a continuous-time generalized almost-cyclostationary (GACS) signal is addressed. The class of such nonstationary signals includes, as a special case, the almost-cyclostationary (ACS) signals. ACS signals filtered by some linear time-variant channels are further examples. It is shown that the discrete-time signal constituted by the samples of a GACS signal is a discrete-time ACS signal. Thus, the nonstationarity kind of a continuous-time GACS signal cannot be deducted from that of the discrete-time signal of its samples. However, in the paper it is shown how, starting from the sampled signal, the GACS or ACS nature of the continuous-time signal can be conjectured, provided that analysis parameters such as sampling period, padding factor, and data-record length are properly chosen.

Chapter 9 - Discrete-Time Signals and Systems

Chapter 1 - Continuous-Time Signals

Zelus is a new programming language for modeling systems that mix discrete logical time and continuous time behaviors. From a user's perspective, its main originality is to extend an existing Lustre-like synchronous language with Ordinary Differential Equations (ODEs). The extension is conservative: any synchronous program expressed as data-flow equations and hierarchical automata can be composed arbitrarily with ODEs in the same source code. A dedicated type system and causality analysis ensure that all discrete changes are aligned with zero-crossing events so that no side effects or discontinuities occur during integration. Programs are statically scheduled and translated into sequential code that, by construction, runs in bounded time and space. Compilation is effected by source-to-source translation into a small synchronous subset which is processed by a standard synchronous compiler architecture. The resultant code is paired with an off-the-shelf numeric solver.We show that it is possible to build a modeler for explicit hybrid systems a la Simulink/Stateflow on top of an existing synchronous language, using it both as a semantic basis and as a target for code generation.

Continuous-time or analog signals are processed using analog devices such as amplifiers, filters, etc. It is impossible to process signals multiplexed from various sources using a single hardware system in the analog domain. On the other hand, digital signals can be processed using both special-purpose hardware and software systems. Worldwide use of Internet, mobile communications, etc. demands all kinds of data such as video, audio, graphics, etc. In order to receive this information on a single device, computer, for instance, it is impossible to use analog signals and techniques. In order to be able to design and implement digitally based systems, it is absolutely necessary to have an understanding of digital signals and systems. Digital signals are discrete in time and amplitude. However, we will assume discrete-time signals to have a continuum of amplitude in order to be able to analyze such signals and systems mathematically. In this chapter we will describe typical discrete-time signals mathematically and then use them to describe and analyze linear time-invariant discrete-time systems. To help the readers understand the mathematical details, we will work out examples followed by MATLAB-based examples. Since digital signals are obtained from analog sources, we will also discuss the conversion of continuous-time signals into digital signals using analog-to-digital converters (ADC).

A discrete time signal is one that has a value only for a finite or infinite number of time instants whereas a continuous time signal has a value for every (real) time instant. For example, consider: The sound pressure wave of a speech signal The electrical that is the output of a transducer that is used to mesdsure the cound pressure wave of a speech signal The sequence of numbers that is obtained from connecting an analg to digital converter (ADC) to the electrical signal sound pressure wave of a speech signal KeywordsSpeech SignalAnalog SignalAnalog InputLinear Predictive CodeDiscrete Time SignalThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Explicit hybrid systems modelers like Simulink/Stateflow allow for programming both discrete- and continuous-time behaviors with complex interactions between them. A key issue in their compilation is the static detection of algebraic or causality loops. Such loops can cause simulations to deadlock and prevent the generation of statically scheduled code.This paper addresses this issue for a hybrid modeling language that combines synchronous data-flow equations with Ordinary Differential Equations (ODEs). We introduce the operator last(x) for the left-limit of a signal x. This operator is used to break causality loops and permits a uniform treatment of discrete and continuous state variables. The semantics relies on non-standard analysis, defining an execution as a sequence of infinitesimally small steps. A signal is deemed causally correct when it can be computed sequentially and only changes infinitesimally outside of announced discrete events like zero-crossings. The causality analysis takes the form of a type system that expresses dependences between signals. In well-typed programs, signals are provably continuous during integration provided that imported external functions are also continuous.The effectiveness of this system is illustrated with several examples written in Zelus, a Lustre-like synchronous language extended with hierarchical automata and ODEs.

Basics of Multirate Digital Signal Processing

Gabor's expansion and the Zak transform for continuous-time and discrete-time signals

This paper proposes an adaptive digital filtering method which improves continuous-time performance. To take account of continuous-time performance, we make an assumption that the adaptive filter can exploit the continuous-time error signal, and define the integral of the square error signal as a performance index. Using lifting, the performance index can be equivalently reduced to a summation of discrete-time signals. Therefore it is shown that the recursive formula for the filtering problem with the performance index is given by the well-known RLS algorithm. Next, a more implementable case is considered where the output of the unknown system is approximated by the oversampled discrete-time signal and the summation of the squared discrete-time signals is defined as an approximate performance index. It is shown that the approximate performance index can be directly applied to the first result. Finally, some numerical examples are given to illustrate the effectiveness of the proposed method.

When discretizing continuous-time systems or signals, one is often interested in preserving a property termed covariance-invariance. In this paper a technique is outlined for synthesizing discrete-time systems and signals which are covariance-invariant with corresponding continuous-time systems and signals. Applications of the technique to process simulation, minimum mean-squared error estimation, and digital filter synthesis are outlined, with example designs presented for covariance-invariant Butterworth and Chebychev digital filters. Based on the frequency response of these designs it is argued that the method of covariance-invariance is superior to the methods of impulse-invariance and bilinear-z as a response matching design technique for the synthesis of digital filters. This superiority is especially apparent at sampling rates that are marginal with respect to filter critical frequencies.

General Information Product Theory in Economics Science

Central limit theorem, modern extreme value theory and the theory that the discrete orthogonal frequency division multiplexing signals converge weakly to a Gaussian random process are generally employed to derive the best approximation of peak-to-average power ratio distributions of discrete-time and continuous-time signals. In this paper, we arrive at a simple, rigorously justified, and accurate expression of the upper bound of the probability density function of the PAPR in context of orthogonal frequency division multiplexing systems. Since this bound has a complex expression not convenient to deal with in practice, we show also the demonstration of an accurate estimation of the proposed bound.

In this paper, we discuss a concept of the mixed Fourier transformation, when signals are transformed to the time-frequency domain, where the difference between the time and frequency is disappeared. Both cases of continuous-time and discrete-time signals are considered, and properties of the mixed transformations are described. The concept of mixed Fourier transform includes the fractional power of the discrete Fourier transform (DFT) which is referred to as a discrete version of the fractional Fourier transform. Mixed Fourier transformations can be used for calculation of roots of the Fourier and identity transformations. Examples of such root transforms for signal processing and image encryption are given.