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As a novice in signal processing, I have been going through Signals & Systems by Oppenheim to try and understand how continuous time periodic signals are represented by Fourier series coefficients.

The book defines a pair of equations which represent the Fourier series of a periodic continuous time signal.

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I cannot understand when one should use the synthesis equation (3.38) and when one should use the analysis equation (3.39) to determine the set of Fourier series coefficients.

Can someone give examples/explain as to when I should use one equation over another? Why is this so?

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  • $\begingroup$ Eq. (3.39) is the definition of the Fourier coefficients, so if you have a function $x(t)$ and you want to know its Fourier series coefficients, then that's what you have to do. I'm not sure I see the problem though. $\endgroup$ – Matt L. Dec 8 '18 at 10:33
  • $\begingroup$ I agree. $(3.39)$ is used in $(3.38)$; perhaps a bit of an oversight to miss the $a_k$ at the beginning of $(3.39)$? $\endgroup$ – Marcus Müller Dec 8 '18 at 11:10
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When you are given a continuous-time periodic signal $x(t)$ and you want to find out the corresponding CTFS (continuous-time Fourier series) coefficients $a_k$ associated with $x(t)$, then you use the analysis equation; i.e, analyse $x(t)$ to find out $a_k$ use

$$ \boxed{ a_k = \frac{1}{T} \int_{0}^{T} x(t) e^{-j \frac{2 \pi}{T} k t} dt }\tag{1}$$

On the other hand, when you are given a set of CTFS coefficients $a_k$ and you want to obtain the corresponding continuous-time periodic signal $x(t)$, then you use the synthesis equation; i.e., sum up exponentials weighted by $a_k$ and create $x(t)$ as in

$$ \boxed{ x(t) = \sum_{k=-\infty}^{\infty} a_k e^{j \frac{2\pi}{T} k t } } \tag{2}$$

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