Fourier series, integral theorem, and transforms: a review

Abstract

A continuous-time system may be described by differential, difference or algebraic equations. In other applications, the independent variable might be a spatial dimension, rather than time. It is often the case that when examining either signals or the behaviour of a system, the use of harmonic analysis and transform methods provides valuable information. An obvious example is provided if we consider vibrations — which might be positively unwelcome. In a mechanical system, vibrations can be potentially or actually destructive. The component parts have their own natural frequencies of vibration, and internal or external ‘noise’ might produce oscillations of an unacceptably large amplitude. This is called ‘resonance’, and if the source is identified then corrective measures can be considered, such as redesigning the damping characteristics of an affected component or (if possible) removing the cause of the noise. Similar factors are important in the design of the speakers and amplifiers of an electrical system.