Transform Theory

Abstract

This long and important chapter introduces a mathematical tool called transforms. This is probably the most important mathematical operation used in electrical measurements. It transforms a signal in time space to frequency space, and this is extremely common (and useful) to understand and analyze your measurement signal. The focus in this chapter is the understanding of transforms and it starts with understanding exactly what is meant by the frequency (and later we must understand why the frequency can be a complex number). Transform theory is by most students perceived to be ‘hard’ and the main reason for that is that there appears to be so many different transform expressions; depending on the nature of the (time) signal, it is necessary to use different mathematical expressions, but they all really do the same thing (i.e., transfer a time signal to frequency space). Because there are so many different expressions, this chapter tries to organize them for you (see Table 7.6). Several different transforms are introduced; the Fourier transform, the discrete Fourier transform, the Fast Fourier transform, the Laplace transform, and the z transform, but remember they all do the same thing; they take your signal from time space to frequency space. The main objective of this chapter is to help the reader understand transforms and see how they are related. This chapter also introduces the Bode plot and defines LTI systems (linear and time-invariant system).