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Using the Fourier method, we have derived a three-dimensional, fully analytic model of a shielded magnetoresistive or giant magnetoresistive head for perpendicular replay. The head may include side shields. The field and the spectral response function are expressed in closed form. Here, we use the model to show the effect of varying the sensor-shield spacings and the air-bearing-surface-underlayer separation on the field and response of the sensor at high areal density.