The Bruns approach was incorporated in resulting in a more computationally balanced algorithm. This method was extended in to calculate the Fourier coefficients of both the even and odd components of a periodic function. Similar algorithms were studied by Wintner and Sadasiv for the calculation of the Fourier coefficients of even periodic functions. It was discovered by Bruns at the beginning of the last century. The arithmetic Fourier transform (AFT) offers a convenient method, based on the construction of weighted averages, to calculate the Fourier coefficients of a periodic function. The frequency response of the system is obtained by evaluating the transform on the unit circle. The system is assumed to be stable and the transform is convergent for, where. We consider a causal, linear, time-invariant system with an infinite impulse response. A general algorithm is developed to compute the inverse transform on the unit circle, and an error estimate is given for the truncated series representation. A generalized Möbius transform is presented.
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