Representation of Audio Signals

Abstract

In many instances it is more appropriate to describe audio signals as some function of frequency rather than time, since we perceive sounds in terms of their tonal content and their frequency representation often offers a more compact representation. The Fourier Transform is the basic tool that allows us to transform from functions of time like x(t) into corresponding functions of frequency like X(f). In this chapter, we first review some basic math notation and the “Dirac delta function”, since we will make use of its properties in many derivations. We then describe the Fourier Transform and its inverse to see how signals can be translated between their frequency and time domain representations. We also describe summary characteristics of signals and show how they can be calculated from either the time or the frequency-domain information. We discuss the Fourier series, which is a variation of the Fourier Transform that applies to periodic signals. In particular, we show how the Fourier series provides a more parsimonious description of time-limited signals than the full Fourier Transform without any loss of information. We show how we can apply the same insight in the frequency domain to prove the Sampling Theorem, which tells us that band-limited signals can be fully represented using only discrete time samples of the signal.