Exponential Fourier Series based Numerical-1 |How to sketch Complex Frequency (Amp & Phase) Spectrum

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The quadrature and polar forms of the Fourier series are one-sided spectral components, meaning the spectrum can exist for DC and positive frequencies, but on the other hand, the complex exponential Fourier series has two-sided spectral components. The complex exponential Fourier series is a simple form, in which the orthogonal functions are the complex exponential functions.

The main result from the Fourier series analysis is that an arbitrary periodic signal can approximate by summing individual cosine terms with specified amplitudes and phases. This result serves as much of the conceptual and theoretical framework for the field of signal analysis. In practice, the Fourier series is a useful tool for modeling various types of quasi-periodic signals.

An alternative and somewhat more convenient form of this result is obtained by noting that complex exponential functions are directly related to sinusoids and cosines through Euler's identities

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