It has long been known that the pupil dilates as a consequence of attentional effort. But the function that relates attentional input to pupillary output has never been the subject of quantitative analysis. We present a system analysis of the pupillary response to attentional input. Attentional input is modeled as a string ofattentional pulses. We show that the system is linear; the effects of input pulses on the pupillary response are additive. The impulse response has essentially a gamma distribution with two free parameters. These parameters are estimated; they are fairly constant over tasks and subjects. The paper presents a method of estimating the string of attentional input pulses, given some average pupillary output. The method involves the technique of deconvolution; it can be implemented with a public-domain software package, Pupil.