The theoretically optimal approach to multitarget detection, tracking, and identification is a suitable generalization of the recursive Bayes nonlinear filter. However, this optimal filter is so computationally challenging that it must usually be approximated. We report on a novel approximation of a multi-target non-linear filtering based on the spectral compression (SPECC) non-linear filter implementation of Stein-Winter probability hypothesis densities (PHDs). In its current implementation, SPECC is a two-dimensional, four-state, FFT-based filter that is Bayes-Closed. It replaces a log-posterior or log-likelihood with an approximate log-posterior or log-likelihood, that is a truncation of a Fourier basis. This approximation is based on the minimization of the least-squares error of the log-densities. The ultimate operational utility of our approach depends on its computational efficiency and robustness when compared with similar approaches. Another novel aspect of the proposed algorithm is the propagation of a first-order statistical moment of the multitarget system. This moment, the probability hypothesis density (PHD) is a density function on single-target state space which is uniquely defined by the following property: its integral in any region of state space is the expected number of targets in that region. It is the expected value of the point process of the random track set (i.e., the density function whose integral in any region of state space is the actual number of targets in the region). The adequacy, and the accuracy of the algorithm when applied to simulated and real scenarios involving ground targets are demonstrated.