[Part 1 discusses pressure waves and sound transmission. Part 2 covers sound intensity, power and pressure level. Part 3 looks at adding sounds together. Part 4 discusses the "inverse square law" for sound. Part 5 begins a look at sound interactions, including refraction, absorption and reflection. Part 6 continues a look at sound interactions with a discussion of sound interference, standing waves, diffraction and scattering. Part 7 looks at the analysis of sound waves using time and frequency domain representations.]

1.7 Analysing spectra
Because the spectrum of a sound is an important part of the way we perceive it there is often a need to look at the spectrum of a real signal. The way this is achieved is to use a bank of filters, as shown in Figure 1.51.

Figure 1.51 A filter bank for analysing spectra.

1.7.1 Filters and filter types
A filter is a device which separates out a portion of the frequency spectrum of a sound signal from the total, this is shown in Figure 1.52.

Figure 1.52 The effect of different filter types.

There are four basic types of filters, which are classified in terms of their effect as a function of signal frequency. This is known as the filters' frequency response. These basic types of filter are as follows:

• Low-pass: the filter only passes frequencies below a frequency known as the filter's cut-off frequency.
• High-pass: the filter only passes frequencies above the cut-off frequency.
• Band-pass: the filter passes a range of frequencies between two cut-off frequencies. The frequency range between the cut-off frequencies is known as the filter's bandwidth.
• Band-reject: the filter rejects a range of frequencies between two cut-off frequencies.

The effects of these four types of filter are shown in Figure 1.52 and one can see that a bank of band-pass filters are the most appropriate for analysing the spectrum of a sound wave. Note that although practical filters have specified cut-off frequencies which are determined by their design, they do not cut off instantly as the frequency changes. Instead they take a finite frequency range to attenuate the signal.

The effect of a filter on a spectrum is multiplicative in that the output spectrum after filtering is the product of the filter's frequency response with the input signal's spectrum. Thus we can easily determine the effect of a given filter on a signal's spectrum. This is a useful technique which can be applied to the analysis of musical instruments, as we shall see later, by treating some of their characteristics as a form of filtering. In fact, filtering can be carried out using mechanical, acoustical and electrical means, and many instruments perform some form of filtering on the sounds they generate (see Chapter 4).