Fourier Series

Fourier Series¶

f (x) = \frac{a_{0}}{2} + \sum_{n = 1}^{\infty} (a_{n} \cos \frac{n π}{p} x + b_{n} \sin \frac{n π}{p} x)

\begin{array}{r} a_{0} = \frac{1}{p} \int_{P} f (x) d x \\ a_{n} = \frac{1}{p} \int_{P} f (x) \cos \frac{n π}{p} x d x \\ b_{n} = \frac{1}{p} \int_{P} f (x) \sin \frac{n π}{p} x d x \end{array}

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{π}{p} = \frac{2 π}{2 p} = ω_{0}

Notice that the close form of $a_{n}$ is often undefined on $0$ .

\begin{array}{r} lim_{n \to 0} a_{n} \to \pm \infty \end{array}

Fourier Cosine and Sine Series¶

$f (x)$ is even => Fourier cosine series
$f (x)$ 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{aligned} f (x) & = \frac{a_{0}}{2} + \sum_{n = 1}^{\infty} a_{n} \cos \frac{n π}{p} x \end{aligned}

in which,

\begin{aligned} a_{0} & = \frac{1}{p} \int_{- p}^{p} f (x) d x = \frac{2}{p} \int_{0}^{p} f (x) d x \\ a_{n} & = \frac{1}{p} \int_{- p}^{p} f (x) \cos \frac{n π}{p} x d x = \frac{2}{p} \int_{0}^{p} f (x) \cos \frac{n π}{p} x d x \end{aligned}

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

Fourier Sine Series¶

\begin{array}{r} f (x) = \sum_{n = 1}^{\infty} b_{n} \sin \frac{n π}{p} x \end{array}

in which,

\begin{array}{r} b_{n} = \frac{1}{p} \int_{- p}^{p} f (x) \sin \frac{n π}{p} x d x = \frac{2}{p} \int_{0}^{p} f (x) \sin \frac{n π}{p} x d x \end{array}

notice : 1. $f (x)$ is odd. ( $[f (x) \sin α x]$ 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} (a_{n} \cos \frac{n π}{p} x + b_{n} \sin \frac{n π}{p} x)

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

Phase Angle Form¶

Definition

\begin{array}{r} f (x) = \frac{a_{0}}{2} + \sum_{n = 1}^{\infty} c_{n} \cos (n ω_{0} x + δ_{n}) \end{array}

in which

\begin{aligned} ω_{0} & = \frac{π}{p} \\ c_{n} & = \sqrt{a_{n}^{2} + b_{n}^{2}} \\ δ_{n} & = \tan^{- 1} (\frac{- b_{n}}{a_{n}}) \end{aligned}

The phase angle form is aka harmonic form. $c_{n}$ is the $n$ th harmonic amplitude, $δ_{n}$ is the $n$ th phase angle of $f (x)$ , and the term $\cos (n ω_{0} x + δ_{n})$ is the $n$ th harmonic of $f (x)$ . ( $n$ 階諧波)

Amplitude Spectrum¶

Graph of the polar points $(θ, r) = [(n ω_{0}, \frac{c_{n}}{2}) \cup (0, \frac{a_{0}}{2})]$ in which $n ω_{0}$ is the frequency and $\frac{c_{n}}{2}$ is the amplitude.

Complex Fourier Series¶

With Euler Formula

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

, we can rewrite Fourier Series expansion as

\begin{aligned} f (x) & = d_{0} + \sum_{n = 1}^{\infty} d_{n} e^{i n ω_{0} x} + \sum_{n = 1}^{\infty} \overset{―}{d_{n}} e^{- i n ω_{0} x} \\ = d_{0} + \sum_{n = - \infty, n \neq 0}^{\infty} d_{n} e^{i n ω_{0} x} \end{aligned}

in which,

\begin{array}{r} d_{n} = \frac{1}{2} (a_{n} - i b_{n}) = \frac{1}{2 p} \int_{P} f (t) e^{- i n ω_{0} t} d t \end{array}

notice that $p$ is also the half of period here, and that

\begin{array}{r} \overset{―}{d_{n}} = d_{- n} \end{array}

Amplitude Spectrum¶

Graph of the polar points $(θ, r) = (n ω_{0}, | d_{n} |)$ , in which $n ω_{0}$ is the frequency and $| d_{n} |$ is the amplitude.