A New Method to Construct Algorithms for Fast Discrete Fourier Transform of Finite Complex and Real Signals Based on Thesecond Type Parametric Discrete Fourier Transforms

The article develops a simple, efficient and effective method for fast discrete Fourier transform, which allows to calculate Fourier coefficients (bins) independently at positive and negative frequenciesfor finite complex and real signals.The algebraic and matrix forms of the discrete Fourier transform and the structure of its basis - the basis of exponential Fourier functions - are briefly considered.The main section of the article discusses generalizations of the discrete Fourier transform in the form of parametric discrete Fourier transforms.Two types of parametric discrete Fourier transforms have been studied, that have a parameter θ in a frequency variable or a parameterθ in a time variable. The structure and properties of the bases of these transformations being the bases of parametric discrete exponential functions were analyzed and investigated.Based on parametric discrete Fourier transforms of the second type, a new method for constructing algorithms for fast discrete Fourier transforms of complex and real signals has been developed and described in detail.In order to verify the obtained theoretical results, a step-by-step testing of a new method for constructing algorithms for fast discrete Fourier transform of finite complex and real signals was carried out.Testing of a new method for constructing algorithms for fast discrete Fourier transform of finite complex and real signals has fully confirmed the validity of the obtained results. For finite complex signals, the result obtained is (until the corresponding practical problem appears) of a theoretical nature.For finite real signals, the obtained result has theoretical and important practical significance.Since they have a redundant characterdue to the property of Hermitian symmetry of the finite real signalspectra.They can only be calculated at positive or negative frequencies.This allows to reduce the required amount of memory and the number of basic operationsfor finite real signals.