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Fourier Series

Fourier Series

\[ f(x)=\frac{a_0}{2}+\sum_{n=1}^{\infty}{\bigg(a_n\cos{\frac{n\pi}{p}x}+b_n\sin{\frac{n\pi}{p}x}\bigg)} \]
\[ \begin{align} \\ a_0 = \frac{1}{p}\ \int_{P}{f(x)\ dx} \\\\ a_n = \frac{1}{p}\int_{P}{f(x)\cos{\frac{n\pi}{p}x}\ dx}\\\\ b_n = \frac{1}{p}\int_{P}{f(x)\sin{\frac{n\pi}{p}x}\ dx}\\\\ \end{align} \]

in which \(a_0,\ a_n,\ b_n\) are called as Fourier coefficients, \(P\) is any of full period of \(f(x)\).

Let \(p\) always be a half of period, then we can understand that

\[\frac{\pi}{p}=\frac{2\pi}{2p}=\omega_0\]

Notice that the close form of \(a_n\) is often undefined on \(0\).

\[ \begin{align} \lim_{n\to 0}a_n\to\pm \infty \end{align} \]

Fourier Cosine and Sine Series

  1. \(f(x)\) is even => Fourier cosine series
  2. \(f(x)\) is odd \(\,\) => Fourier sine series


  • Tips

  • cosine & sine series : interval is change into \([-L,\ L]\), set \(p=L\)

  • Fourier series : interval \([-p,p]\) is change into \([0,L]\), set \(p=L/2\)

  • \(p\) is always a half of period.

Fourier Cosine Series

\[ \begin{align} f(x)&=\frac{a_0}{2}+\sum_{n=1}^{\infty}{a_n\cos{\frac{n\pi}{p}x}} \\\\ \end{align} \]

in which,

\[ \begin{align} a_0 &= \frac{1}{p}\int_{-p}^{p}{f(x)\ dx}= \frac{\mathbf{2}}{p}\int_{\mathbf{0}}^p{f(x)\ dx}\\\\ a_n&=\frac{1}{p}\int_{-p}^{p}{f(x)\ \cos{\frac{n\pi}{p}}x\ dx}=\frac{\mathbf{2}}{p}\int_{\mathbf{0}}^p{f(x)\cos{\frac{n\pi}{p}x}\ dx} \end{align} \]

notice : 1. \(f(x)\) is even. 2. Take \([-p, p]\) for \(P\). 3. Half range extension (compare to Fourier series).

Fourier Sine Series

\[ \begin{align} f(x)=\sum_{n=1}^{\infty}{b_n\sin{\frac{n\pi}{p}x}} \\\\ \end{align} \]

in which,

\[ \begin{align} b_n=\frac{1}{p}\int_{-p}^p{f(x)\sin{\frac{n\pi}{p}x}\ dx} =\frac{\mathbf{2}}{p}\int_{\mathbf{0}}^p{f(x)\sin{\frac{n\pi}{p}x}\ dx} \end{align} \]

notice : 1. \(f(x)\) is odd. (\(\big[f(x) \sin\alpha x\big]\) will thus be even) 2. Take \([-p,p]\) for \(P\) 3. Half range extension (compare to Fourier series).

Gibbs Phenomenon

\[ f(x)=\frac{a_0}{2}+\sum_{n=1}^{N}{\bigg(a_n\cos{\frac{n\pi}{p}x}+b_n\sin{\frac{n\pi}{p}x}\bigg)} \]

There will be "overshooting" near discontinuities when \(N\) isn't infinity.

While \(N \to \infty\), "overshooting" will be more and more narrow.

And according to convergence theorem, for all \(a\) in the domain of \(f(x)\), we have

\[ \begin{align} \lim_{n\to \infty}f(a)=\frac{1}{2}\big[f(a_-)+f(a_+)\big] \end{align} \]

Phase Angle Form

  • Definition
\[ \begin{align} f(x)=\frac{a_0}{2}+\sum_{n=1}^{\infty}{c_n\ \cos{(n\omega_0 x+\delta_n)}} \end{align} \]

in which

\[ \begin{align} \omega_0 &= \frac{\pi}{p}\\\\ c_n&=\sqrt{a_n^2+b_n^2}\\\\ \delta_n&=\tan^{-1}{(\frac{-b_n}{a_n})} \end{align} \]

The phase angle form is aka harmonic form. \(c_n\) is the \(n\)th harmonic amplitude, \(\delta_n\) is the \(n\)th phase angle of \(f(x)\), and the term \(\cos{(n\omega_0 x+\delta_n)}\) is the \(n\)th harmonic of \(f(x)\). (\(n\)階諧波)

Amplitude Spectrum

Graph of the polar points \((\theta, r)= \bigg[(n\omega_0 , \frac{c_n}{2}) \ \cup\ (0, \frac{a_0}{2})\bigg]\) in which \(n\omega_0\) is the frequency and \(\frac{c_n}{2}\) is the amplitude.

Complex Fourier Series

With Euler Formula

\[e^{ix}=\cos{(x)}+i\ \sin{(x)}\]

, we can rewrite Fourier Series expansion as

\[ \begin{align} f(x)&= d_0 + \sum_{n=1}^{\infty}{d_ne^{in\omega_0 x}}+ \sum_{n=1}^{\infty}{\overline{d_n}e^{-in\omega_0 x}} \\\\ &=d_0+\sum_{n=-\infty,\ n\ne0}^{\infty}{d_ne^{in\omega_0 x}} \end{align} \]

in which,

\[ \begin{align} d_n=\frac{1}{2}(a_n-ib_n)=\frac{1}{2p}\int_{P}{f(t)e^{-in\omega_0 t}dt} \end{align} \]

notice that \(p\) is also the half of period here, and that

\[ \begin{align} \overline{d_n}=d_{-n} \end{align} \]

Amplitude Spectrum

Graph of the polar points \((\theta, r)=(n\omega_0 , |d_n|)\), in which \(n\omega_0\) is the frequency and \(|d_n|\) is the amplitude.