This module serves to introduce the learner to the concepts and properties of the Laplace and Fourier transforms and the associated ideas of transfer functions, filters and the impulse response used in engineering systems analysis.
The Laplace Transform: The Laplace Transform of standard functions; the constant function, powers, the exponential function and the sine and cosine functions. The properties of the Laplace transform; the transform of the derivative, the shift properties of the Laplace Transform and the transform of periodic functions. The transform of piecewise linear functions such as the square wave and sawtooth functions.
The Fourier Transform: The definition of the Fourier Transform and the Fourier transforms of common functions. The properties of the Fourier transform; its shift properties, the existence of the inverse transform, the time and frequency duality of the function. The delta function and its role in the transforms of the single sine and cosine functions. The Laplace transform with complex argument and the link to the Fourier transform.
Applications of Transforms: Solution of Ordinary Differential equations and the Laplace and Fourier transforms. The concept of an engineering system and its modelling as a differential equation, with input and output functions and their transforms identified, and the role of the transform domain in this analysis. The concept of a transfer function. The definition and use of the impulse response and its link to the transfer function. The convolutions of functions and the properties of transforms of convolutions. Ordinary Differential equations and Filters.
