Introduction of Continuous Time Signals

ANALYSIS OF CONTINUOUS TIME SIGNALS

INTRODUCTION

A signal is said to be a continuous-time signal if it is available at all Instants of time. The analysis of a signal is far more convenient in the frequency domain. There are three important cases of transformation methods available for continuous-time systems. They are

1. Fourier series

2. Fourier Transform

3. Laplace Transform

Out of these three methods, the Fourier series is applicable only to periodic signals. ie, signals which repeat periodically over - ∞ < t < ∞.

Not all periodic signal can be represented by Fourier series. In this chapter, we discuss the conditions to be satisfied for a periodic signal to be represented by Fourier series, different types of Fourier series representations, conversion from one type to other and analysis of symmetry present in the waveforms are also discussed and their utilization of simplifying the computations is illustrated.

REPRESENTATION OF FOURIER SERIES

The Fourier analysis is also sometimes called the harmonic analysis Fourier series is applicable only for periodic signals. It cannot be applied to non-periodic signals. A periodic signal is one which repeats itself at regular intervals of time, i.e. periodically over -∞ to ∞.

The representation of signals over a certain interval of time in terms of the linear combination of orthogonal functions is called Fouries series. Three important classes of Fourier series methods are available. They are

i. Trigonometric form

ii. Cosine form

iii. Exponential form.

If the orthogonal functions are trigonometric functions, then it is called trigonometric Fourier series

If the orthogonal function are exponential function, then it is called exponential Fourier series.

Existense of Fourier Series

For the Fourier series to exist for a periodic signal. It must satisfy certain conditions. The conditions under which a periodic signal can be represented by a Fourier series are known as Dirichlet's conditions. They are as follows.

i. The function x(t) must be a single valued function.

ii. The function x(t) has only a finite number of maxima and minima.

iii. The function x(t) has a finite number of discontinuities.

iv. The function x(t) is absolutely integrable over one period, that is