In this paper, the problem of joint transmit waveform and receive filter design for dual-function radar-communication (DFRC) systems is studied. The considered system model involves a multiple antenna base station (BS) of a cellular system serving multiple single antenna users on the downlink. Furthermore, the BS simultaneously introduces sensing capabilities in the form of point-like target detection from the reflected return signals in a signal-dependent interference environment. A novel framework based on constrained optimization problems is proposed for the joint design of the transmit waveform and the radar receive filter such that different constraints related to the power amplifiers and the radar waveform are satisfied. In contrast to the existing approaches in the DFRC systems' literature, the proposed approach does not require the knowledge of a predetermined radar beampattern in order to optimize the performance of the radar part through its approximation. Instead, a beampattern is generated by maximizing the radar receive signal-to-interference ratio (SINR) thus, enabling a more flexible design. Moreover, the radar receive filter processing and its optimization is considered for the first time on DFRC systems, enabling the effective exploitation of the available degrees of freedom in the radar receive array. Efficient algorithmic solutions with guaranteed convergence are developed for the defined constrained nonconvex optimization problems. The effectiveness of the proposed solutions is verified via numerical results.