ISSN: 2041-6792

Clinical Investigation

Harmonic Analysis

 The traditional way to transform a signal U i.e., to change amplitude-frequency characteristic (AFC) of U in some desired manner, is: 1) use Fourier transform to obtain the AFC of the signal; 2) apply some mechanism to recalculate the row coefficients; and 3) use inverse Fourier transform to obtain the transformed signal. Also, the traditional way to calculate values of a function U + i*V, analytical inside a unit circle, from given values on the unit circle, so that new values resemble values of that function on some other concentric circle, is: 1) use Fourier transform to obtain row coefficients; 2) recalculate the coefficients to reflect that the new values are on the circle with radius ‘r’ and starting argument ‘ψ’; and 3) use reverse Fourier transform to obtain the desired values. The traditional way to transform a signal also has a well-known obstacle. The obstacle is the increasing relative cost of calculations that occurs with an increasing number of sampling points N. If N is a power of 2 the cost is N*log2 (N), but in other cases the cost is higher. A worst-case scenario is the absence of small prime dividers, which results in costs proportional to N^2. I intend to prove here that one circulant matrix operator can do both transforms: transform the AFC, and transform from analytical function values on the unit circle to the values on the concentric circle with a different radius and starting point argument. In both transforms, only U is used as an input vector, while the result of the transform is a complex vector.   

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