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_Signal_FAQ10_Real_&amp;_Complex_Fourier_Series

# _Signal_FAQ10_Real_&amp;_Complex_Fourier_Series - Are...

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Are the real Fourier series using sines and cosines equivalent to the complex Fourier series using complex sinusoids? No. The complex Fourier series based on the complex sinusoid t jk e 0 ϖ can be regarded as the superset. The complex Fourier series includes real Fourier series using ( 29 t k 0 sin ϖ and ( 29 t k 0 cos ϖ as special cases. Basically, no matter what one does to a real Fourier series using ( 29 t k 0 sin ϖ and ( 29 t k 0 cos ϖ , one cannot get, say, a purely imaginary periodic signal. It is like real and complex number. Real number is a special case of complex number, but not the other way round. You cannot get an imaginary number, say, 2 - , by adding or multiplying real numbers. Thus, the correct way to see Foruier series is this. The set of complex sinusoids, , , 0 2 t j e ϖ - t j e 0 1 ϖ - , t j e 0 0 ϖ - t j e 0 1 ϖ , , 0 2 t j e ϖ is our building blocks or atoms for all periodic signals with a fundamental frequency of 0 0 2 f π ϖ = sec rad or 0 f Hz.

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_Signal_FAQ10_Real_&amp;_Complex_Fourier_Series - Are...

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