Editing Fourier series (section)

==Definition==
The Fourier series of a complex-valued {{math|''P''}}-periodic function <math>s(x)</math>, integrable over the interval <math>[0,P]</math> on the real line, is defined as a [[trigonometric series]] of the form
<math display="block">\sum_{n=-\infty}^\infty c_n e^{i 2\pi \tfrac{n}{P} x }, </math>
such that the ''Fourier coefficients'' <math>c_n</math> are complex numbers defined by the integral{{sfn|Folland|1992|pp=18-25}}{{sfn|Hardy|Rogosinski|1999|pp=2-4}}
<math display="block">c_n = \frac{1}{P}\int_0^P s(x)\ e^{-i 2\pi \tfrac{n}{P} x }\,dx.</math>
The series does not necessarily converge (in the [[pointwise convergence|pointwise]] sense) and, even if it does, it is not necessarily equal to <math>s(x)</math>. Only when certain conditions are satisfied (e.g. if <math>s(x)</math> is continuously differentiable) does the Fourier series converge to <math>s(x)</math>, i.e.,
<math display="block">s(x) = \sum_{n=-\infty}^\infty c_n e^{i 2\pi \tfrac{n}{P} x }.</math>
For functions satisfying the [[Dirichlet–Jordan_test|Dirichlet sufficiency conditions]], pointwise convergence holds.{{sfn|Lion|1986}} However, these are not [[Necessity_and_sufficiency|necessary conditions]] and there are many theorems about different types of [[convergence of Fourier series]] (e.g. [[uniform convergence]] or [[mean convergence]]).{{sfn|Edwards|1979|pp=8-9}} The definition naturally extends to the Fourier series of a (periodic) [[Distribution_(mathematics)#Distributions|distribution]] <math>s</math> (also called ''Fourier-Schwartz series'').{{sfn|Edwards|1982|pp=57,67}} Then the Fourier series converges to <math>s(x)</math> in the distribution sense.{{sfn|Schwartz|1966|pp=152-158}}

The process of determining the Fourier coefficients of a given function or signal is called '''''analysis''''', while forming the associated trigonometric series (or its various approximations) is called '''''synthesis'''''.

=== Synthesis ===
A Fourier series can be written in several equivalent forms, shown here as the <math>N^\text{th}</math> [[Series_(mathematics)#Partial_sum_of_a_series|partial sums]] <math>s_N(x)</math> of the Fourier series of <math>s(x)</math>:<ref>
{{Citation
| last = Strang      
| first = Gilbert 
| author-link = Gilbert Strang
| title = Fourier Series And Integrals
| publisher = Wellesley-Cambridge Press
| year = 2008
| edition = 2
| chapter = 4.1
| chapter-url = https://math.mit.edu/~gs/cse/websections/cse41.pdf
| page = 323 (eq 19)
| url = https://math.mit.edu/~gs/cse/
}}
</ref>
[[File:Fourier_series_illustration.svg|right|thumb|400x400px|Fig 1. The top graph shows a non-periodic function <math>s(x)</math> in blue defined only over the red interval from 0 to ''P''. The function can be analyzed over this interval to produce the Fourier series in the bottom graph. The Fourier series is always a periodic function, even if original function <math>s(x)</math> is not.]]

{{Equation box 1 |title=Sine-cosine form
|indent=:|border|border colour=#0073CF|background colour=#F5FFFA|cellpadding=6
|equation={{NumBlk|
|<math>s_N(x) = a_0 + \sum_{n=1}^N \left( 
a_n \cos \left(2 \pi \tfrac{n}{P} x \right) + 
b_n \sin \left(2 \pi \tfrac{n}{P} x  \right) \right)</math> &nbsp; &nbsp;
|{{EquationRef|Eq.1}}}}}}<br/>

{{Equation box 1 |title=Exponential form
|indent=:|border|border colour=#0073CF|background colour=#F5FFFA|cellpadding=6
|equation={{NumBlk|
|<math>s_N(x) = \sum_{n=-N}^N c_n\ e^{i 2\pi \tfrac{n}{P}x}</math> &nbsp; &nbsp;
|{{EquationRef|Eq.2}}}}}}
The harmonics are indexed by an integer, <math>n,</math> which is also the number of cycles the corresponding sinusoids make in interval <math> P</math>. Therefore, the sinusoids have''':'''

* a [[wavelength]] equal to <math>\tfrac{P}{n}</math> in the same units as <math>x</math>.
* a [[frequency]] equal to <math>\tfrac{n}{P}</math> in the reciprocal units of <math>x</math>.

These series can represent functions that are just a sum of one or more frequencies in the [[harmonic spectrum]].  In the limit <math>N\to\infty</math>, a trigonometric series can also represent the intermediate frequencies or non-sinusoidal functions because of the infinite number of terms.

=== Analysis ===
The coefficients can be given/assumed, such as a music synthesizer or time samples of a waveform.  In the latter case, the exponential form of Fourier series synthesizes a [[discrete-time Fourier transform]] where variable <math>x</math> represents frequency instead of time. In general, the coefficients are determined by ''analysis'' of a given function <math>s(x)</math> whose [[domain of definition]] is  an interval of length <math>P</math>.{{efn-ua
|Typically <math>[-P/2, P/2]</math> or <math>[0,P]</math>. Some authors define <math>P \triangleq 2 \pi</math> because it simplifies the arguments of the sinusoid functions, at the expense of generality.}}{{sfn|Stade|2005|p=6}}

{{Equation box 1|title=Fourier coefficients
|indent=:|border|border colour=#0073CF|background colour=#F5FFFA|cellpadding=6
|equation={{NumBlk|
|<math>
\begin{align} 
&a_0 = \frac{1}{P}\int_P s(x) \,dx&\\
&a_n = \frac{2}{P}\int_P s(x) \cos \left( 2\pi \tfrac{n}{P} x \right) \,dx,\ &\textrm{for}~n\geq 1\\
&b_n = \frac{2}{P}\int_P s(x) \sin \left( 2\pi \tfrac{n}{P} x \right) dx,\ &\text{for}~n\geq 1 \\
\end{align}
</math> &nbsp; &nbsp;
|{{EquationRef|Eq.3}}}}}}

The <math>\tfrac{2}{P}</math> scale factor follows from substituting {{EquationNote|Eq.1}} into {{EquationNote|Eq.3}} and utilizing the [[Orthogonal_functions#Trigonometric_functions|orthogonality of the trigonometric system]].<ref>{{cite web | last=Zygmund | first=Antoni |author-link1=Antoni Zygmund | title=Trigonometrical series | website=EUDML | year=1935 | url=https://eudml.org/doc/219339 |page=6 | access-date=2024-12-14}}</ref> The equivalence of {{EquationNote|Eq.1}} and {{EquationNote|Eq.2}} follows from [[Euler%27s_formula#Relationship_to_trigonometry|Euler's formula]]
<math display="block">
\cos x = \frac{e^{ix} + e^{-ix}}{2}, \quad \sin x = \frac{e^{ix} - e^{-ix}}{2i},</math>
resulting in:
{{Equation box 1|title=Exponential form coefficients
|indent=:|border|border colour=#0073CF|background colour=#F5FFFA|cellpadding=6
|equation=
<math>c_n = \begin{cases}
\tfrac{1}{2}(a_n -i b_n) & \text{if } n > 0,\\
a_n & \text{if } n = 0,\\
\tfrac{1}{2}(a_{-n} + i b_{-n}) & \text{if } n < 0,\\
\end{cases}</math>}}
with <math>c_{0}</math> being the [[Mean_of_a_function|mean value]] of <math>s</math> on the interval <math>P</math>.{{sfn|Folland|1992|pp=21}} Conversely:
{{Equation box 1|title=Inverse relationships
|indent=:|border|border colour=#0073CF|background colour=#F5FFFA|cellpadding=6
|equation=<math>\begin{aligned} 
a_0 &= c_0 &\\
a_n &= c_n+c_{-n} \qquad &\textrm{for}~ n > 0 \\
b_n &= i(c_n-c_{-n}) \qquad &\textrm{for}~ n > 0 \end{aligned}</math>
}}
==== Example ====
[[File:sawtooth pi.svg|thumb|right|400px|Plot of the [[sawtooth wave]], a periodic continuation of the linear function <math>s(x)=x/\pi</math> on the interval <math>(-\pi,\pi]</math>]]

[[File:Periodic identity function.gif|thumb|right|400px|Animated plot of the first five successive partial Fourier series]]

Consider a sawtooth function:
<math display="block">s(x) = s(x + 2\pi k)  = \frac{x}{\pi}, \quad \mathrm{for } -\pi < x < \pi,\text{ and } k \in \mathbb{Z}.</math>
In this case, the Fourier coefficients are given by
<math display="block">\begin{align}
a_0 &= 0.\\
a_n & = \frac{1}{\pi}\int_{-\pi}^{\pi}s(x) \cos(nx)\,dx = 0, \quad n \ge 1. \\
b_n & = \frac{1}{\pi}\int_{-\pi}^{\pi}s(x) \sin(nx)\, dx\\
&= -\frac{2}{\pi n}\cos(n\pi) + \frac{2}{\pi^2 n^2}\sin(n\pi)\\
&= \frac{2\,(-1)^{n+1}}{\pi n}, \quad n \ge 1.\end{align}</math>
It can be shown that the Fourier series converges to <math>s(x)</math> at every point <math>x</math> where <math>s</math> is differentiable, and therefore:
<math display="block">\begin{align}
s(x) &= a_0 + \sum_{n=1}^\infty \left[a_n\cos\left(nx\right)+b_n sin\left(nx\right)\right] \\[4pt]
&=\frac{2}{\pi}\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} \sin(nx), \quad \mathrm{for}\ (x-\pi)\ \text{is not a multiple of}\ 2\pi.
\end{align}</math>
When <math>x=\pi</math>, the Fourier series converges to 0, which is the half-sum of the left- and right-limit of <math>s</math> at <math>x=\pi</math>. This is a particular instance of the [[Convergence of Fourier series#Convergence at a given point|Dirichlet theorem]] for Fourier series.

This example leads to a solution of the [[Basel problem]].

===Amplitude-phase form===
If the function <math>s(x)</math> is real-valued then the Fourier series can also be represented as{{sfn|Stade|2005|pp=59-64}}
{{Equation box 1 |title=Amplitude-phase form
|indent=:|border|border colour=#0073CF|background colour=#F5FFFA|cellpadding=6
|equation={{NumBlk|
|<math>s_N(x)=A_0 + \sum_{n=1}^N A_n \cos\left( 2\pi \tfrac{ n }{ P } x - \varphi_n\right)</math> &nbsp; &nbsp;
|{{EquationRef|Eq.4}}}}}}
where <math>A_{n}</math> is the [[amplitude]] and <math>\varphi_{n}</math> is the [[phase shift]] of the <math>n^{th}</math> harmonic.

The equivalence of {{EquationNote|Eq.4}} and {{EquationNote|Eq.1}} follows from the [[List_of_trigonometric_identities#Angle_sum_and_difference_identities|trigonometric identity]]:
<math display="block">\cos\left(2\pi \tfrac{n}{P}x-\varphi_n\right) = \cos(\varphi_n)\cos\left(2\pi \tfrac{n}{P} x\right) 
+  \sin(\varphi_n)\sin\left(2\pi \tfrac{n}{P} x\right),</math>
which implies<ref name="Kassam">
{{cite web
| url = https://www.seas.upenn.edu/~kassam/tcom370/n99_2B.pdf
| title = Fourier Series (Part II)
| last = Kassam
| first = Saleem A.
| date = 2004
| access-date = 2024-12-11
| quote = The phase relationships are important because they correspond to having different amounts of "time shifts" or "delays" for each of the sinusoidal waveforms relative to a zero-phase waveform.}}
</ref>
<math display="block>a_n = A_n \cos(\varphi_n)\quad \text{and}\quad b_n = A_n \sin(\varphi_n)</math>
[[File:Correlation_function.svg|right|thumb|300px|Fig 2. The blue curve is the cross-correlation of a square wave and a cosine template, as the phase lag of the template varies over one cycle. The amplitude and phase at the maximum value are the polar coordinates of one harmonic in the Fourier series expansion of the square wave. The corresponding rectangular coordinates can be determined by evaluating the correlation at just two samples separated by 90°.]]
are the [[rectangular coordinates]] of a vector with [[Polar coordinate system|polar coordinates]] <math>A_n</math> and <math>\varphi_n</math> given by
<math display="block">A_n = \sqrt{a_n^2 + b_n^2}\quad \text{and}\quad \varphi_n = \operatorname{Arg}(c_n) = \operatorname{atan2}(b_n, a_n)</math> where <math>\operatorname{Arg}(c_n)</math> is the [[Argument_(complex_analysis)|argument]] of <math>c_{n}</math>.

An example of determining the parameter <math>\varphi_n </math> for one value of <math>n</math> is shown in Figure 2. It is the value of <math>\varphi </math> at the maximum correlation between <math>s(x)</math> and a cosine ''template,'' <math>\cos(2\pi \tfrac{n}{P} x -  \varphi)</math>. The blue graph is the [[Cross-correlation|cross-correlation function]], also known as a [[matched filter]]:

:<math>\begin{align}
\Chi(\varphi) &= \int_{P} s(x) \cdot \cos\left( 2\pi \tfrac{n}{P} x -\varphi \right)\, dx\quad \varphi \in \left[ 0, 2\pi \right]\\
&=\cos(\varphi) 
\underbrace{\int_{P} s(x) \cdot \cos\left( 2\pi \tfrac{n}{P} x\right) dx}_{X(0)}
+
\sin(\varphi) \underbrace{\int_{P} s(x) \cdot \sin\left( 2\pi \tfrac{n}{P} x\right) dx}_{ X(\pi/2) } 
\end{align}</math>

Fortunately, it is not necessary to evaluate this entire function, because its derivative is zero at the maximum:
<math display="block">X'(\varphi) = \sin(\varphi)\cdot X(0) - \cos(\varphi)\cdot X(\pi/2) = 0,  \quad \textrm{at}\ \varphi = \varphi_n.</math> Hence
<math display="block">\varphi_n  \equiv  \arctan(b_n/a_n) = \arctan(X(\pi/2)/X(0)).  
</math>
===Common notations===
The notation <math>c_n</math> is inadequate for discussing the Fourier coefficients of several different functions. Therefore, it is customarily replaced by a modified form of the function (<math>s,</math> in this case), such as <math>\widehat{s}(n)</math> or <math>S[n],</math> and functional notation often replaces subscripting''':'''

:<math>\begin{align}
s(x) &= \sum_{n=-\infty}^\infty \widehat{s}(n)\cdot e^{i 2\pi \tfrac{n}{P} x} &&  \scriptstyle \text{common mathematics notation} \\
&= \sum_{n=-\infty}^\infty S[n]\cdot e^{i 2\pi \tfrac{n}{P} x} &&  \scriptstyle \text{common engineering notation}
\end{align}</math>

In engineering, particularly when the variable <math>x</math> represents time, the coefficient sequence is called a [[frequency domain]] representation. Square brackets are often used to emphasize that the domain of this function is a discrete set of frequencies.

Another commonly used frequency domain representation uses the Fourier series coefficients to [[Modulation|modulate]] a [[Dirac comb]]:
:<math>S(f) \ \triangleq \ \sum_{n=-\infty}^\infty S[n]\cdot \delta \left(f-\frac{n}{P}\right),</math>

where <math>f</math> represents a continuous frequency domain. When variable <math>x</math> has units of seconds, <math>f</math> has units of [[hertz]]. The "teeth" of the comb are spaced at multiples (i.e. [[harmonics]]) of <math>\tfrac{1}{P}</math>, which is called the [[fundamental frequency]]. <math>s(x)</math> can be recovered from this representation by an [[Fourier inversion theorem|inverse Fourier transform]]:

:<math>\begin{align}
\mathcal{F}^{-1}\{S(f)\} &= \int_{-\infty}^\infty \left( \sum_{n=-\infty}^\infty S[n]\cdot \delta \left(f-\frac{n}{P}\right)\right) e^{i 2 \pi f x}\,df, \\[6pt]
&= \sum_{n=-\infty}^\infty S[n]\cdot \int_{-\infty}^\infty \delta\left(f-\frac{n}{P}\right) e^{i 2 \pi f x}\,df, \\[6pt]
&= \sum_{n=-\infty}^\infty S[n]\cdot e^{i 2\pi \tfrac{n}{P} x} \ \ \triangleq \ s(x).
\end{align}</math>

The constructed function <math>S(f)</math> is therefore commonly referred to as a '''Fourier transform''', even though the Fourier integral of a periodic function is not convergent at the harmonic frequencies.{{efn-ua|
Since the integral defining the Fourier transform of a periodic function is not convergent, it is necessary to view the periodic function and its transform as [[Distribution (mathematics)|distributions]]. In this sense <math>\mathcal{F} \{ e^{i 2\pi \tfrac{n}{P} x} \}</math> is a [[Dirac delta function]], which is an example of a distribution.
}}