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We propose a parametric waveform design approach for improved detection of extended targets embedded in uncorrelated signal-dependent clutter and noise, whose spectral densities are assumed to be known. Unlike canonical waveform design approaches, the transmit waveform is represented as a weighted linear combination of discrete prolate spheroidal sequences. In the optimization problem, the probability of detection is maximized with respect to the weighting factors of the associated discrete prolate spheroidal sequences under the transmit energy constraint. The weighting factors, which are resolved using a numerical method, lead directly to the desired transmit waveform in the time domain. In comparison to the canonical waveform design approaches, the extra step for time sequence synthesis is avoided and the loss in probability of detection produced therein is remedied. Simulation results demonstrate the improvement in the probability of detection for the proposed approach. However, the improvement comes at the cost of higher computational complexity.