A basic understanding of linear systems is fundamental to understanding the nature of sound and hearing, including sound sources such as speech and music, sound propagation and mixing, and sound analysis in the inner ear. An understanding of linear systems is also a base on which to build an understanding of important nonlinear aspects of hearing.

In this chapter, we attempt to bring the novice and the expert to a common level of shared understanding about linear systems. The terminology and understanding developed here will support the material in subsequent chapters.

The concepts that we discuss include filters, circuits, impulse responses, frequency responses, convolution, transfer functions (including magnitude, phase, and delay), poles and zeros, transforms, time and frequency domains, eigenfunctions, sinusoids, and complex exponentials. The typical electrical engineering undergraduate education covers all these concepts, but too often some of the important connections between them are neglected. The typical bioengineering or hearing science education may only touch on half of the concepts, sometimes omitting Laplace and Z transforms and the important concept of poles and zeros that we need for our cochlear models. Depending on the level at which one wants to work in human or machine hearing, a deeper understanding of this area can be very helpful.

Linear systems are often encountered in the study of electrical, mechanical, acoustic, and even quantum systems, as well as in the processing of nonphysical data sequences such as stock prices. In a modern engineering or computer science curriculum, linear systems will likely be taught using discrete-time data sequences, processed by digital filters in computer programs. A more traditional approach, at least in electrical engineering (EE), is to teach linear systems using electrical circuits. This approach connects more directly with the continuous-time nature of sound waves and sound processing in the mechanical structures of the ear, so that's where we start.