(1) Theory: Many systems are, or may be approximated as a linear, shift-invariant system. When such systems are applied to a sine wave input, the output will be exactly the same sine wave (i.e., same frequency, orientation, speed, etc.) except for a possible phase shift and scaling of amplitude. Because of this property, it is useful to describe input signals (e.g., sensory stimuli) as sums of sine waves (the "Fourier transform"). Then, the response of the system may be predicted by the separate response of the system to the stimulus' sine wave components. Thus, we summarize system performance by the phase shift and response scaling as a function of frequency (the "Modulation transfer function").

(2) Physiological application: Sensory neurons are generally described by reference to their receptive field. In many cases, the operation of the neuron may be summarized as a linear receptive field, i.e., a linear weighting of different locations in the input (possibly followed by a threshold nonlinearity). One can then analyze the operation of a collection of neurons sharing identical receptive fields except located at different positions (e.g., locations in the visual field). The group treated as a whole constitutes a linear, shift-invariant system producing as output a "neural image". Linear systems theory, as applied to the receptive field weighting function, provides a concise summary of the function of this class of neurons.

(3) Analogous psychophysical application: The results of behavioral experiments in spatial vision (using such methods as masking, adaptation and subthreshold summation) may be modeled as resulting from "spatial frequency channels". Each channel is analogous to a neural image; channels differ in their modulation transfer functions, and channel outputs are integrated nonlinearly (e.g., by a logical "or" rather than by linear summation).

(4) Broader application: Any time you analyze continuous data of one or more variables, the techniques of Fourier Analysis will likely be helpful. Obvious examples include any physiological measurements gathered over time, including fMRI, EEG or MEG.

(5) Principle: When analyzing a sensory mechanism, it is helpful to start with the simplest possible model: a linear, shift-invariant system. Thus, one starts by measuring the frequency response of the system, and by testing whether the properties of linearity and shift-invariance do indeed hold. When violations occur, only then do you consider additional, nonlinear stages, or nonlinear pooling over multiple, linear mechanisms.