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Fourier Series

Represent periodic signals in terms of cosines and sines.

Periods

A period signal repeats its pattern at some period \(T\).

Fourier series of period signal can be used to analyze the signal in another domain.

If \(f(t)\) is a function with period T, then

\[ f(t+nT) = f(t) \quad \forall n \]
Function Period
\(\cos \theta, \sin \theta\) \(2 \pi\)
\(\cos (n\theta), \sin (n\theta)\) \(\dfrac{2\pi}{n}\)

Fourier Series

of a function \(f(x)\) of period \(2\pi\) in the interval \([-\pi, +\pi]\), is defined as

\[ f(x) = \frac{a_0}{ \textcolor{orange}{2} } + \sum\limits_{n=1}^\infty a_n \cos(nx) + \sum\limits_{n=1}^{\infty} b_n \sin (nx) \]

It is always continuous.

Whenever possible, we have to make it into regular summation, ie from \(1 \to \infty\).

Fourier Constants

\[ \begin{aligned} a_0 &= \frac{1}{\pi} \int\limits_{-\pi}^{\pi} f(x) \cdot dx \\ a_n &= \frac{1}{\pi} \int\limits_{-\pi}^{\pi} f(x) \textcolor{orange}{\cos(nx)} \cdot dx \\ b_n &= \frac{1}{\pi} \int\limits_{-\pi}^{\pi} f(x) \textcolor{orange}{\sin(nx)} \cdot dx \end{aligned} \]

Sum of Functions

\[ FS(g_1 \pm g_2) = FS(g_1) \pm FS(g_2) \]

Dirchelet Condition

Even though the function that the FS represents may be discontinuous, the FS itself will be continuous

\[ \text{FS} \stackrel{\text{converges}}{\longrightarrow} \begin{cases} f(a) &, a = \text{Continuous Point} \\ \dfrac{f(a^-) + f(a^+)}{2} &, a = \text{Discontinuous Point} \end{cases} \]

Even/Odd Functions

Even Odd
\(f(-x)\) \(f(x)\) \(-f(x)\)
Fourier Series \(\dfrac{a_0}{2} + \sum\limits_{n=1}^\infty a_n \cos(nx)\) \(\sum\limits_{n=1}^\infty b_n \sin(nx)\)
\(a_0\) \(\dfrac{2}{\pi} \int\limits_0^\pi f(x) dx\) 0
\(a_n\) \(\dfrac{2}{\pi} \int\limits_0^\pi f(x) \cos(nx) dx\) 0
\(b_n\) 0 \(\dfrac{2}{\pi} \int\limits_0^\pi f(x) \sin(nx) dx\)

This is because \(\int f(x) dx = 0\) when \(f(x)\) is even

Note

Consider

\[ \begin{aligned} f(x) &= \begin{cases} g_1(x), & (-a, 0) \\ g_2(x), & (0, a) \end{cases}\\ \implies f(x) &= \begin{cases} \text{Even}, & g_1(-x) = +g_2(x) \\ \text{Odd}, & g_1(-x) = -g_2(x) \end{cases} \end{aligned} \]

Grahphically

We can also plot the points for \(x = \{-\pi, 0, - \pi\}\). Connect the points.

Function Type Symmetric about
Even Y axis
Odd Origin

Sine/Cosine Series

Special types of series, where we represent the fourier series in terms of \(\sin\) alone or \(\cos\) alone in half interval \((0,\pi)\)

Sine/Cosine series may be asked for an odd/even function.

Cosine Series Sine Series
Fourier Series \(\dfrac{a_0}{2} + \sum\limits_{n=1}^\infty a_n \cos(nx)\) \(\sum\limits_{n=1}^\infty b_n \sin(nx)\)
\(a_0\) \(\dfrac{2}{\pi} \int\limits_0^\pi f(x) dx\) 0
\(a_n\) \(\dfrac{2}{\pi} \int\limits_0^\pi f(x) \cos(nx) dx\) 0
\(b_n\) 0 \(\dfrac{2}{\pi} \int\limits_0^\pi f(x) \sin(nx) dx\)

Arbitrary Interval

Fourier series of \(f(x)\) of period \(2l\) defined in the interval \((-l, l), l \in R\) is

\[ f(x) = \frac{ a_0 }{ \textcolor{orange}{2} } + \sum_{n=1}^\infty a_n \cos \left( \frac{n \textcolor{hotpink}{\pi} x}{\textcolor{hotpink}{l}} \right) + \sum_{n=1}^\infty b_n \sin \left( \frac{n \textcolor{hotpink}{\pi} x}{\textcolor{hotpink}{l}} \right) \]
\[ \begin{aligned} a_0 &= \frac{1}{\textcolor{hotpink}{l}} \int\limits_{-l}^l f(x) dx \\ a_n &= \frac{1}{\textcolor{hotpink}{l}} \int\limits_{-l}^l f(x) \cos \left( \frac{n \textcolor{hotpink}{\pi} x}{\textcolor{hotpink}{l}} \right) dx \\ b_n &= \frac{1}{\textcolor{hotpink}{l}} \int\limits_{-l}^l f(x) \sin \left(\frac{n \textcolor{hotpink}{\pi} x}{\textcolor{hotpink}{l}} \right) dx \end{aligned} \]

Changes in Interval

From To For
\((-\pi, \pi)\) \((-l, l)\) FS
\((0, \pi)\) \((0, l)\) CS and SS
\(\cos(nx)\) \(\cos \left( \frac{n \pi x}{l} \right)\) FS, CS, SS
\(\sin(nx)\) \(\sin \left( \frac{n \pi x}{l} \right)\) FS, CS, SS
\(\frac{1}{\pi} \int_{-\pi}^\pi\) \(\frac{1}{l} \int_{-l}^l\) FS, CS, SS
\(\frac{2}{\pi} \int_0^\pi\) \(\frac{2}{l} \int_0^l\) FS, CS, SS

Bernoulli’s Integration Chain Rule

\[ \int(uv) dx = u v_1 - u' v_2 - u'' v_3 - \ldots - u^{(n-1)} v_n \]
Term Meaning
\(u, v\) Given Functions
\(u', u'', \dots\) Derivatives
\(v_1, v_2, v_3, \dots\) Integrals
Last Updated: 2024-05-12 ; Contributors: AhmedThahir

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