Different from conventional phased-array providing only angle-dependent beampattern, frequency diverse array (FDA) produces angle-range-dependent and time-variant transmit beampattern. Existing investigations show that FDA offers improved performance in interference suppression and target localization, but the time-variant beampattern will bring interferences to subsequent matched filtering. More seriously, the range-dependent signal phase may be canceled out in the filtering process. In fact, traditional single-channel receiver does not fully exploit the multicarrier feature in FDA signals. In this paper, we propose a multichannel matched filtering structure with considering the time-variance property for receiving pulsed-FDA signals. A coherent pulsed-FDA radar signal model to deal with the angle-range-dependent and time-variance problem is devised under additive colored Gaussian noise scenarios, followed by the corresponding waveform design principle. Moreover, closed-form expressions of the output signal-to-interference-plus-noise ratio and Cramér-Rao bounds for angle and range are derived. The proposed receiver design approach and corresponding theoretical performance derivations are verified by extensive numerical results.