4 Applications of Linear Systems

In this chapter, we will consider various applications of linear systems analysis in the biological sciences. Before embarking on specific cases, though, it is worth setting up a basic framework that will allow us to think about what all these cases have in common. As discussed in chapter 3, a linear system converts an input signal into an output signal such that the overall transformation is linear (figure 4.1). In some applications, the linear system is the actual biological entity under study, whose properties are mysterious at the outset, and we seek to understand how it converts its inputs to outputs. In other cases, the linear system represents a measuring instrument whose transform is well known to us, and we need to reconstruct its input signal from recordings of the output signal. In yet other applications, the linear system may be a data analysis tool, and we need to optimize its properties for a certain task we want to perform. For example, the input or output signals may be corrupted by "noise," by which we mean some random process that is not under the control of the experimenter. In this case, we can design a linear filter to optimally suppress the noise in favor of the signal.