Representation of multidimensional periodic functions by a finite weighted sum of sampled values

Abstract

A closed form analytic representation of multidimensional periodic functions with a finite Fou rier spectrum is obtained as a modified interpolation series with a finite number of terms. The corresponding theorem is formulated, and the multidimensional periodic kernel of the expansion is analyzed. It is shown that such a representation makes it possible to calculate values of the derivatives of an arbitrary order of a peri odic function with a finite spectrum from the samples of this function.

Mathematical model of parametrically controlled matched filters

“…The condition of the expression (21) mean that, at an input signal with a linear law of variation of the frequency modulation steepness, the law of the frequency modulated control signal should be twice as large fig. 2 (Ryzhak, 2003;Cook, 1971;Naydenov, 2014).…”

“…Where, in the case of a periodic increase of each of the frequency filters with frequency i ω the member ( ) (Ryzhak, 2003). As a result, we will obtain a system of equations defining the required characteristics of a manageable coherent filter:…”

“…It is clear from the expressions (17 and 18) that the quadratic element of the order (17) generates the second harmonic of i υ . We ignore the influence of the higher harmonic members in the second member, due to their small values (Ryzhak, 2003). …”

“…Again using the same theorem, but in the time domain we will get delays with deviations, each of which is a function of time (Ryzhak, 2003). In a number of cases, signal processing in a delayed-line filter model is practically unacceptable as large amount of deviations are required one after the other at extremely short intervals.…”