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A comprehensive theory of matched illumination waveforms for both deterministic and stochastic extended targets is presented. Design of matched waveforms based on maximization of both signal-to-noise ratio (SNR) and mutual information (MI) is considered. In addition the problem of matched waveform design in signal-dependent interference is extensively addressed. New results include SNR-based waveform design for stochastic targets, SNR-based design for a known target in signal-dependent interference, and MI-based design in signal-dependent interference. Finally we relate MI and SNR in the context of waveform design for stochastic targets.