I have a quick question about the relationship between the complex Fourier coefficient,[tex]\alpha_n[/tex] and the real Fourier coefficients, [tex]a_n[/tex] and [tex]b_n[/tex].(adsbygoogle = window.adsbygoogle || []).push({});

Given a real-valued function, I could just find the real coefficients and plug them into the relation below, right?

Fourier Coefficients for periodic functions of period 2a.

Complex Form:

[tex]\alpha_n = \frac{1}{2a}\int_{-a}^{a} f\left(t\right)e^{\frac{-jn\pi t}{a}dt[/tex]

Real Form:

[tex]a_0 = \frac{1}{a}\int_{-a}^{a} f\left(t\right)dt[/tex]

[tex]a_n = \frac{1}{a}\int_{-a}^{a} f\left(t\right) cos\left(\frac{n\pi t}{a}\right)dt[/tex]

[tex]b_n = \frac{1}{a}\int_{-a}^{a} f\left(t\right) sin\left(\frac{n\pi t}{a}\right)dt [/tex]

Relation

[tex]\alpha_n = \left\{

\begin{array}{lr}

\frac{1}{2}\left(a_n + jb_n\right) & : n < 0\\ \\

\frac{1}{2}a_0 & : n = 0\\ \\

\frac{1}{2}\left(a_n - jb_n\right) & : n > 0

\end{array}

\right.[/tex]

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# Homework Help: Fourier Coefficient Relations

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