Three-Frequency Quaternion Fourier Transform

Abstract

Vadim Sovetov

As is known, the spectra of real analog signals are calculated using the Fourier Transform (FT). The FT calculated as a definite integral over time over the duration of the signal. The kernel of the transformation is the exponential to the power of a complex number -iωt, where i – imaginary unit of a complex number, ω – circular frequency, t – time. The exponential to the power of a complex number can be represented by Euler's formula as a sum of harmonic functions: exp{-iωt} = cos(ωt) - isin(ωt). Integrating a signal over time allows us to calculate what portion of the signal's energy is concentrated at certain frequencies. Therefore, FT gives us a representation of the analog signal in the form of harmonic components with the corresponding amplitude and phase. The phase of harmonics is determined by the amplitudes of the cosine and sine values on the orthogonal coordinate axes of the complex plane, i.e., by the angle of the corresponding vector. The length of the vector is calculated using the Pythagorean theorem from the value of the amplitudes of the cosine and sine.

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