Optimal radar waveform design for target detection has been addressed in prior research literature under various assumptions regarding noise and clutter. A common model of the radar scene in work of this kind is a linear time-invariant (LTI) operator with additive Gaussian noise that acts on the transmitted signal to produce the received signal. This model is intrinsically ill-suited to dynamic scenes or moving radar platforms because it cannot account for Doppler. This paper introduces scene models based on Hilbert-Schmidt class (HS) operators on the space of finite-energy signals. This category of models generalizes the LTI category in the sense that every LTI operator is also a HS operator, but the HS class includes operators that account for frequency shifts as well as time shifts and are thus suitable for modeling radar scenes involving Doppler. Every HS operator is uniquely expressible as a superposition of elementary time and frequency shift operators, thus providing a convenient interpretation of a scene in terms of these physically meaningful operations on the transmitted signal. Application of this perspective to waveform design for target detection in noise and to optimal receiver processing for a given waveform for target detection in clutter and noise are demonstrated.