The resolution of closely adjacent spectral lines

Abstract. The problem is that of determining the parameters in a trigonometric polynomial when it is observed with added stationary noise. The frequencies, in particular, must be determined and the situation especially considered is that where these are close together. A similar problem arises if an angular frequency is close to zero or π. The method of estimation is the maximization of the regression sum of squares as a function of the unknown frequencies. In the asymptotic theory, the closely adjacent frequencies are separated by an amount that is of the order T^‐1, where T is the length of the series. Simulations show that this asymptotic treatment gives a better approximation in cases where the separation is of this magnitude than that obtained by treating the frequencies as fixed.