This paper describes sensor calibration and signal analysis techniques applicable to the method of acoustic emission (AE) and ultrasonic testing. They are particularly useful for obtaining absolute measurements of AE wave amplitude and shape, which can be used to constrain the physics and mechanics of the AE source. We illustrate how to perform calibration tests on a thick plate and how to implement two different mechanical calibration sources: ball impact and glass capillary fracture. In this way, the instrument response function can be estimated from theory, without the need for a reference transducer. We demonstrate the methodology by comparing calibration results for four different piezoelectric acoustic emission sensors: Physical Acoustics (PAC) PAC R15, PAC NANO30, DigitalWave B1025, and the Glaser-type conical sensor. From the results of these tests, sensor aperture effects are quantified and the accuracy of calibration source models is verified. Finally, this paper describes how the effects of the sensor can be modeled using an autoregressive-moving average (ARMA) model, and how this technique can be used to effectively remove sensor-induced distortion so that a displacement time history can be retrieved from recorded signals.