Fundamentals of Electronics 2

Continuous-time Signals and Systems

Pierre Muret

• Publisher: Wiley-ISTE
• Year: 2018
• ISBN:
  9781119489115
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Power Resources
• Index: Technology & Engineering

This book presents a synthesis of Electronics through keynotes which are substantiated in three volumes. The first one comprises four chapters devoted to elementary devices, i.e. diodes, bipolar transistors and related devices, field effect transistors and amplifiers. In each of one, device physics, non linear and linearized models, and applications are studied. The second volume is devoted to systems in the continuous time regime and contains two chapters: one describes different approaches to the transfer function concept and applications, and the following deals with the quadripole properties, filtering and filter synthesis. The third volume presents the various aspects of sampling systems and quantized level systems in the two last chapters.

1
Continuous-time Systems: General Properties, Feedback, Stability, Oscillators

The linear and stationary systems that concern us here deliver output signal y(t) when input signal x(t) is applied to them, solution to a real and linear ordinary differential equation, where t represents the time variable:

which can also be seen as a linear application:

Function exp(α t), with real or complex α, is of special importance since it is the specific function of the system’s differential equation, which means that if x(t) = exp(α t), the output signal is also proportional to exp(α t). It is this fundamental property that warrants the approaches discussed in the following sections 1.1 and 1.2. Another method, based on the state-space form, also applicable to nonlinear systems, is presented in sections 1.4.5 and 1.5.

1.1. Representation of continuous time signals

These signals are real electrical quantities and thus measurable functions of time variable t, which itself is a continuous variable. They are also referred to as analog signals. An additional representation is formed by the frequency spectrum.

1.1.1. Sinusoidal signals

In general, any real sinusoidal signal of angular frequency ω₁ and constant frequency f₁ (ω₁ = 2π f₁) is written as y(t) = A cos(ω₁t + φ₁), once a time and phase origin has been selected. But in complex numbers, this can also be written as:

Both exponential terms with imaginary exponent appear with the same A coefficient and are always complex conjugates, two conditions that are required if y(t) is real. The two vectors corresponding to images on the complex plane rotate in opposite directions; thus, frequency −f₁ is consistently found at the same time as frequency f₁.

Figure 1.1. Representation of a sinusoidal signal on the complex plane

The spectral or frequency representation is thus formed simply by two lines of amplitude A/2 at frequencies f₁ and −f₁, and phase lines φ₁ and −φ₁ at these same frequencies.

Figure 1.2. Spectrum of a sinusoidal signal (amplitude solid line, phase dotted)

Indeed, sinusoidal signals of the same frequency form a two-dimensional vector space for which a basis is provided by exp[jω₁t] and exp[−jω₁t] (cos[ω₁t] and sin[ω₁t] form another basis). Thus, we can write:

with

and

where c₁ and c₋₁ are complex con...

Table of contents

• Cover
• Table of Contents
• Title
• Copyright
• Preface
• Introduction
• 1 Continuous-time Systems: General Properties, Feedback, Stability, Oscillators
• 2 Continuous-time Linear Systems: Quadripoles, Filtering and Filter Synthesis
• Appendix: Notions of Distribution and Operating Properties
• Bibliography
• Index
• End User License Agreement