Algorithm for detecting frequencies in multi‐sine signals by inductive generation

Prony's method is an approximation approach to decomposing a function into sum of exponents and thus is applicable to unknown frequencies estimation of signals. The concrete algorithms for estimating pure sine wave, triple tone, and quadruple tone have already been derived and presented. This paper aims at deriving the estimating algorithm for multi‐sine signals which consist of unknown sine waves. The new method of generating an algebraic algorithm for detecting unknown frequencies in the signals is derived by mathematical induction. The crux of the generation is dependent on the integer matrices induction. A handy method for generating the matrices is shown as well. Algorithms for the triple tone, quadruple tone, and the higher‐order tone are generated and verified. As a result, they are shown to be identical to the ordinary algorithms. Subjects on the application of the induced algorithm to practical frequency detection are discussed. The algorithm has both instantaneity in the time domain and higher resolution in the frequency domain, that is, the signal analysis by the algorithm can be performed without constraint of the uncertainty principle. An iterative solution for algebraic equation is dominant for calculation in the algorithm. Techniques for detecting frequencies in a multi‐sine of unknown order are also discussed. © 2007 Wiley Periodicals, Inc. Electr Eng Jpn, 160(3): 27– 38, 2007; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/eej.20404