In response to a recent paper, this correspondence points out the difficulty of applying the concept of the energy distribution in time and frequency of deterministic signals to random signals and finite-time-averaged power spectra. For deterministic signals, an appropriately defined finite-time spectrum will indicate the frequency band within which the signal energy is concentrated at a particular time. However, it does not appear meaningful to apply such concepts to finite-time averages of random processes since taking sample averages would typically destroy any time-frequency distribution of the energy perhaps found in an individual sample of the process.

The Fourier representation of signals and its relation to the signal structure in time and frequency, and more generally the inherent properties of phase-modulated signals, have received considerable attention in the past. These topics have led to such seemingly unrelated studies as the representation of a signal in a combined time-frequency plane, "instantaneous power spectra," and the ambiguity function and its transform relations. It is shown in this paper that the studies can be unified by the introduction of the concept of the complex energy density function of a signal. The function is an extension and combination of the one-dimensional energy density functions in time and frequency, the energy density spectrum|\Psi(f)|^{2}, and energy density waveform|\psi (t)|^{2}. On the basis of the complex energy density function, the significance of complicated-appearing transform relations is readily understood. The new concept also conveys a good insight into the internal structure of phase-modulated signals.

Conventional resolution theory is based on the assumption of constant range rate, and it neglects distortions of the modulation function as well as effects of the antenna pattern. Among the generalizations of the theory reported in the literature, some include the antenna pattern and others retain higher-order range derivatives in both carrier term and modulation function. The first part of this paper is tutorial and discusses the significance of the various generalizations of the ambiguity function. It is shown that an important class of problems can be accommodated when the conventional theory is extended by permitting higher-order range derivatives in the carrier term. Target resolvability on the basis of motion then depends only on the relative phase shifts introduced over the signal duration. In the second part, the ambiguity function is formally extended to arbitrary target motion. The modified theory retains the simplicity of the conventional approach and allows utilization of well-known results.