A Biproportional Construction Algorithm for Correctly Calculating Fourier Series of Aperiodic Non-Sinusoidal Signal

The Fourier series (FS) applies to a periodic non-sinusoidal function satisfying the Dirichlet conditions, whereas the being-processed function in practical applications is usually an aperiodic non-sinusoidal signal. When is aperiodic, its calculated FS is not correct, which is still a challenging problem. To overcome the problem, we derive a direct calculation algorithm, a constant iteration algorithm, and an optimal iteration algorithm. The direct calculation algorithm correctly calculates its Fourier coefficients (FCs) when is periodic and satisfies the Dirichlet conditions. Both the constant iteration algorithm and the optimal iteration algorithm provide an idea of determining the states of . From the idea, we obtain an algorithm for determining the states of based on the optimal iteration algorithm. In the algorithm, the variable iteration step is introduced; thus, we present an algorithm for determining the states of based on the variable iteration step. The presented algorithm accurately determines the states of . On the basis of these algorithms, we build a biproportional construction theory. The theory consists of a first and a second proportional construction theory. The former correctly calculates the FCs of at the present sampling time