*PhD student Tom White explains one of the key mathematical tools for understanding complex sound fields.*

**1. The shape of sound**

*amplitude*and

*frequency*. The amplitude dictates the strength of the pressure wave and the frequency gives the number of full oscillations completed in one second. For a given point in space, the oscillating pressure looks like the following when measured against time:

$$ \begin{equation} g(t)=A \cos(2\pi f t) \label{eq:eq1} \end{equation}$$

*pressure field*.

When the peaks of the waves meet, the pressure field is amplified, and where the the peaks meet the troughs the pressure cancels out. We call this process of overlapping waves to create a non wave-like pressure field

*superposition*. Sound recorded by a microphone is much more likely resemble the pressure field on the right hand side of Figure 2, than the wave in Figure 1. However, for us to look at the complex pressure field it it difficult to identify its component frequencies. In fact, not only can certain sound fields be created by the superposition of waves, but

**any**sound field can be produced by a unique combination of waves.

**2. Fourier Transform**

*Fourier Transform*is used to convert the time-pressure field into a frequency dependent function. The transform was first proposed by a French mathematician,

**Joseph Fourier**, in the 19th century. It converts a time-domain pressure field as discussed above, into a frequency-domain function, which is dependent of the frequency \(f\). The transform is given by the equation

$$\begin{equation} G(f)=\int_\infty^\infty g(t) \exp\{2\pi i f t\}\, dt, \label{eq:eq2}\end{equation}$$

where \(\exp\{-\}\) is the exponential operator, and \( i = \sqrt{-1}\).

In Figure 3 the first graph shows the resulting field of three waves in superposition, with frequencies \(f = 1/2\pi\), \(f = 1\) and \(f = 2\), given by

$$\begin{equation} g(t)=2\cos(t)+\cos(2\pi t)+\cos(4 \pi t). \label{eq:eq3}\end{equation}$$

**3. Applications of the Fourier transform**

The Fourier transform is not limited to use in sound processing and analysis: it can be applied to any system where there is known to be an oscillating energy source which can operate at a continuous spectrum of frequencies. For example, the Fourier transform is used in a wide range of spectroscopy techniques such as Nuclear Magnetic Resonance (NMR) spectroscopy which uses the Fourier transform to analyse the magnetic fields of atomic nuclei.