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An impulse response function and a transfer function of use for partially coherent systems are presented. Partially coherent systems are those that transform a complex amplitude input to an output intensity and are encountered in many guises including optical systems, radiometers, scatterometers, astronomical radars, and synthetic aperture radar. (In the fully coherent or noncoherent limits, linear analysis and the associated system descriptions are well known in terms of amplitude or modulation transfer functions, respectively.) Properties are derived herein for partially coherent systems that describe performance dependence on system phase and amplitude match to input signal characteristics and to system and signal partial coherence. Such systems are shown to be energy conservative in the event of coherence variations. Output spectral properties that describe the dependence of multiplicative Rayleigh noise (speckle) on input signal coherence, system coherence, and phase match are presented. Unlike linear systems, partially coherent systems require both (1) the impulse response and (2) a quantitative measure of scene/system coherence to adequately describe their performance. The general formulation of this paper is applied to synthetic aperture radar (SAR) imaging systems to derive closed-form results of importance for that discipline. It is shown that the mean output from a SAR for a random distributed scene input is not dependent on system focus, system coherence, or scene coherence, whereas imagery of scene detail is dependent on these same parameters. The speckle spectrum is a function of system coherence, but not a function of scene coherence or system focus.