An Alternative Derivation of the Method of Stationary Phase for Multivariate Signals

The method of stationary phase (msp) is a commonly established technique to asymptotically determine integrals (such as Fourier integrals) with strongly oscillating integrands. Basically, the method states that an integral over a strongly oscillating integrand only delivers contributions at those (“stationary”) points or regions, where the integrand does not change, i.e., where the phase derivative of its oscillation vanishes. In the engineering literature, various approaches for the derivation exist for univariate (1-D) signals, which, however, find their limitations for multivariate (multidimensional) signals, which cannot be factorized. On the other hand, from the field of applied mathematics, there are derivations, formulating and solving the problem in a rigorous and sophisticated way. This paper tries to close what the author interprets as a gap between these two worlds, aiming to substitute intuitive insights by some basic Dirac impulse modeling concepts, which are familiar to signal processing engineers. We first start with the general multivariate (multidimensional) case, introducing vectorial time and frequency representations. Here, we replace 1-D Taylor-series-based quadratic phase signals by multivariate signals with phase histories, being given by more general quadratic forms. The exponential, containing the quadratic form, is then related to an
n
-variate Dirac impulse, which provides the sifting property needed to establish the basic msp property that the integration over the entire integration range yields only one value of the integrand, evaluated in the point of stationary phase. This sifting capacity in connection with (time limiting) window functions in the integrand migrates time limitations of the signals to frequency limitations of the spectra-a multiplicative time window is transformed in a multiplicative frequency window of the same form, rather than into a convolution of the Fourier transforms. In this respect, this paper provides a closed-form comprehensive treatment of the subject.