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Probing waveform synthesis and receive filter design play crucial roles in achievable performance for many active sensing applications, including radar, sonar, medical imaging, and communications (channel estimation and spread spectrum). A flexible receive filter design approach can be used to compensate for missing features of the probing waveforms, at the costs of lower signal-to-noise ratio (SNR) and higher computational complexity. A well-synthesized waveform, meaning one with good auto- and cross-correlation properties, can reduce computational burden at the receiver and improve performance. In this article, we will highlight the interplay between waveform synthesis and receiver design. We will provide a tutorial review of recent novel, cyclic approaches to single and multiple waveform designs. Both aperiodic and periodic correlations will be considered. We show that by making use of fast Fourier transforms (FFTs), we can now efficiently design sequences that were previously impossible to synthesize. Furthermore, we will provide an overview of some advanced techniques for receiver design, including data-independent instrumental variables (IV) filters and a data-adaptive iterative approach. We will show how these designs can significantly outperform conventional techniques in various active sensing applications.