Musicians have been using filters for thousands of years to shape the sounds of their art in various ways. For example, the evolution of the physical dimensions of the violin constitutes an evolution in filter design. The choice of wood, the shape of the cutouts, the geometry of the bridge, and everything that affects resonance all have a bearing on how the violin body filters the signal induced at the bridge by the vibrating strings. Once a sound is airborne there is yet more filtering performed by the listening environment, by the pinnae of the ear, and by idiosyncrasies of the hearing process.

What is a Filter?

Any medium through which the music signal passes, whatever its form, can be regarded as a filter. However, we do not usually think of something as a filter unless it can modify the sound in some way. For example, speaker wire is not considered a filter, but the speaker is (unfortunately). The different vowel sounds in speech are produced primarily by changing the shape of the mouth cavity, which changes the resonances and hence the filtering characteristics of the vocal tract. The tone control circuit in an ordinary car radio is a filter, as are the bass, midrange, and treble boosts in a stereo preamplifier. Graphic equalizers, reverberators, echo devices, phase shifters, and speaker crossover networks are further examples of useful filters in audio. There are also examples of undesirable filtering, such as the uneven reinforcement of certain frequencies in a room with ``bad acoustics.'' A well-known signal processing wizard is said to have remarked, ``When you think about it, everything is a filter.''

A digital filter is just a filter that operates on digital signals, such as sound represented inside a computer. It is a computation which takes one sequence of numbers (the input signal) and produces a new sequence of numbers (the filtered output signal). The filters mentioned in the previous paragraph are not digital only because they operate on signals that are not digital. It is important to realize that a digital filter can do anything that a real-world filter can do. That is, all the filters alluded to above can be simulated to an arbitrary degree of precision digitally. Thus, a digital filter is only a formula for going from one digital signal to another. It may exist as an equation on paper, as a small loop in a computer subroutine, or as a handful of integrated circuit chips properly interconnected.

Why learn about filters?

Computer musicians nearly always use digital filters in every piece of music they create. Without digital reverberation, for example, it is difficult to get rich, full-bodied sound from the computer. However, reverberation is only a surface scratch on the capabilities of digital filters. A digital filter can arbitrarily shape the spectrum of a sound. Yet very few musicians are prepared to design the filter they need, even when they know exactly what they want in the way of a spectral modification. A goal of this book is to assist sound designers by listing the concepts and tools necessary for doing custom filter designs.

There is plenty of software available for designing digital filters [10,8,22]. In light of this available code, it is plausible to imagine that only basic programming skills are required to use digital filters. This is perhaps true for simple applications, but knowledge of how digital filters work will help at every phase of using such software.

Also, you must understand a program before you can modify it or extract pieces of it. Even in standard applications, effective use of a filter design program requires an understanding of the design parameters, which in turn requires some understanding of filter theory. Perhaps most important for composers who design their own sounds, a vast range of imaginative filtering possibilities is available to those who understand how filters affect sounds. In my practical experience, intimate knowledge of filter theory has proved to be a very valuable tool in the design of musical instruments. Typically, a simple yet unusual filter is needed rather than one of the classical designs obtainable using published software.

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The Simplest Lowpass Filter
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