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سری فوریه بسطی است که هر تابع متناوب را به صورت حاصل جمع تعدادی نامتناهی از توابع نوسانی ساده (سینوسی، کسینوسی یا تابع نمایی مختلط) بیان می‌کند. این تابع به نام ریاضیدان بزرگ فرانسوی، ژوزف فوریه نامگذاری شده‌است. با بسط هر تابع به صورت سری فوریه، مولفه‌های بسامدی آن تابع به دست می‌آید.

پیش گفتار

توابع مورد استفاده در مهندسی و توابع نمایانگر سیگنال‌ها معمولاً توابعی از زمان هستند یا به عبارت دیگر توابعی که در میدان زمان تعریف شده‌اند. برای حل بسیاری از مسائل بهتر است که تابع در دامنه فرکانس تعریف شده باشد؛ زیرا این دامنه ویژگی‌هایی دارد که به راحتی محاسبات می‌انجامد.

فرض کنید که تابعی به شکل زیر تعریف شده‌است:

x=\sum _{k=1}^{N}{A_{k}}cos(\omega _{k}t+\theta _{k})\,\!

که در آن $N$

یک عدد صحیح مثبت،

A_{k}

دامنه،

\omega _{k}

بسامد و

\theta _{k}

فاز توابع کسینوسی می‌باشد. قابل مشاهده است که با در دست داشتن بسامدها

\omega _{1},\omega _{2}\ldots \omega _{N}

، دامنه‌ها

A_{1},A_{2}\ldots A_{N}

و فازها

\theta _{1},\theta _{2}\ldots \theta _{N}

تابع به‌طور کامل قابل تعریف است. توجه شود که بر اساس گفته‌های بالا تابع مستقل از زمان قابل تعریف است.

Common forms

The Fourier series can be represented in different forms. The amplitude-phase form, sine-cosine form, and exponential form are commonly used and are expressed here for a real-valued function $s(x)$

. (See § Complex-valued functions and § Other common notations for alternative forms).

The number of terms summed, $N$

, is a potentially infinite integer. Even so, the series might not converge or exactly equate to

s(x)

at all values of

x

(such as a single-point discontinuity) in the analysis interval. For the well-behaved functions typical of physical processes, equality is customarily assumed, and the Dirichlet conditions provide sufficient conditions.

The integer index, $n$

, is also the number of cycles the

n^{\text{th}}

harmonic makes in the function's period

P

. Therefore:

The $n^{\text{th}}$ harmonic's wavelength is ${\tfrac {P}{n}}$ and in units of $x$ .

The $n^{\text{th}}$ harmonic's frequency is ${\tfrac {n}{P}}$ and in reciprocal units of $x$ .

Fig 1. The top graph shows a non-periodic function s(x) 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 s(x) wasn't.

Amplitude-phase form

The Fourier series in amplitude-phase form is:

Fourier series, amplitude-phase form

$s_{\scriptscriptstyle N}(x)={\frac {A_{0}}{2}}+\sum _{n=1}^{N}A_{n}\cdot \cos \left({\tfrac {2\pi }{P}}nx-\varphi _{n}\right)$

(Eq.1)

Its $n^{\text{th}}$ harmonic is $A_{n}\cdot \cos \left({\tfrac {2\pi }{P}}nx-\varphi _{n}\right)$ .

$A_{n}$ is the $n^{\text{th}}$ harmonic's amplitude and $\varphi _{n}$ is its phase shift.

The fundamental frequency of $s_{\scriptscriptstyle N}(x)$ is the term for when $n$ equals 1, and can be referred to as the $1^{\text{st}}$ harmonic.

${\tfrac {A_{o}}{2}}$ is sometimes called the $0^{\text{th}}$ harmonic or DC component. It is the mean value of $s(x)$ .

Clearly Eq.1 can represent functions that are just a sum of one or more of the harmonic frequencies. The remarkable thing, for those not yet familiar with this concept, is that it can also represent the intermediate frequencies and/or non-sinusoidal functions because of the potentially infinite number of terms ( $N$

).

Fig 2. The blue curve is the cross-correlation of a square wave and a cosine function, as the phase lag of the cosine varies over one cycle. The amplitude and phase lag at the maximum value are the polar coordinates of one harmonic in the Fourier series expansion of the square wave. The corresponding Cartesian coordinates can be determined by evaluating the cross-correlation at just two phase lags separated by 90º.

The coefficients $A_{n}$

and

\varphi _{n}

can be understood and derived in terms of the cross-correlation between

s(x)

and a sinusoid at frequency

{\tfrac {n}{P}}

. For a general frequency

f,

and an analysis interval

[x_{0},x_{0}+P],

the cross-correlation function:

$\mathrm {X} _{f}(\tau )={\tfrac {2}{P}}\int _{x_{0}}^{x_{0}+P}s(x)\cdot \cos \left(2\pi f(x-\tau )\right)\,dx;\quad \tau \in \left[0,{\tfrac {2\pi }{f}}\right]$

(Eq.2)

is essentially a matched filter, with template $\cos(2\pi fx)$

. The maximum of

\mathrm {X} _{f}(\tau )

is a measure of the amplitude

(A)

of frequency

f

in the function

s(x)

, and the value of

\tau

at the maximum determines the phase

(\varphi )

of that frequency. Figure 2 is an example, where

s(x)

is a square wave (not shown), and frequency

f

is the

4^{\text{th}}

harmonic. It is also an example of deriving the maximum from just two samples, instead of searching the entire function. That is made possible by a trigonometric identity:

Equivalence of polar and Cartesian forms

$\cos \left({\tfrac {2\pi }{P}}nx-\varphi _{n}\right)\ \equiv \ \cos(\varphi _{n})\cdot \cos \left({\tfrac {2\pi }{P}}nx\right)+\sin(\varphi _{n})\cdot \sin \left({\tfrac {2\pi }{P}}nx\right)$

(Eq.3)

Substituting this into Eq.2 gives:

{\begin{aligned}\mathrm {X} _{n}(\varphi )&={\tfrac {2}{P}}\int _{P}s(x)\cdot \cos \left({\tfrac {2\pi }{P}}nx-\varphi \right)\,dx;\quad \varphi \in [0,2\pi ]\\&=\cos(\varphi )\cdot \underbrace {{\tfrac {2}{P}}\int _{P}s(x)\cdot \cos \left({\tfrac {2\pi }{P}}nx\right)\,dx} _{\triangleq \ a_{n}}+\sin(\varphi )\cdot \underbrace {{\tfrac {2}{P}}\int _{P}s(x)\cdot \sin \left({\tfrac {2\pi }{P}}nx\right)\,dx} _{\triangleq \ b_{n}}\\&=\cos(\varphi )\cdot a_{n}+\sin(\varphi )\cdot b_{n}\end{aligned}}

which introduces the definitions of

a_{n}

and

b_{n}

. And we note for later reference that

a_{0}

and

b_{0}

can be simplified:

a_{0}={\tfrac {2}{P}}\int _{P}s(x)dx~,\quad b_{0}=0~.

The derivative of

\mathrm {X} _{n}(\varphi )

is zero at the phase of maximum correlation.

\mathrm {X} '_{n}(\varphi _{n})=\sin(\varphi _{n})\cdot a_{n}-\cos(\varphi _{n})\cdot b_{n}=0\quad \longrightarrow \quad \tan(\varphi _{n})={\frac {b_{n}}{a_{n}}}\quad \longrightarrow \quad \varphi _{n}=\arctan(b_{n},a_{n})

And the correlation peak value is:

{\begin{aligned}A_{n}\triangleq \mathrm {X} _{n}(\varphi _{n})\ &=\cos(\varphi _{n})\cdot a_{n}+\sin(\varphi _{n})\cdot b_{n}\\&={\frac {a_{n}}{\sqrt {a_{n}^{2}+b_{n}^{2}}}}\cdot a_{n}+{\frac {b_{n}}{\sqrt {a_{n}^{2}+b_{n}^{2}}}}\cdot b_{n}={\frac {a_{n}^{2}+b_{n}^{2}}{\sqrt {a_{n}^{2}+b_{n}^{2}}}}&={\sqrt {a_{n}^{2}+b_{n}^{2}}}.\end{aligned}}

Therefore $a_{n}$

and

b_{n}

are the Cartesian coordinates of a vector with polar coordinates

A_{n}

and

\varphi _{n}.

Sine-cosine form

Substituting Eq.3 into Eq.1 gives:

{\displaystyle s_{\scriptscriptstyle N}(x)={\frac {A_{0}}{2}}+\sum _{n=1}^{N}\left[A_{n}\cos(\varphi _{n})\cdot \cos \left({\tfrac {2\pi }{P}}nx\right)+A_{n}\sin(\varphi _{n})\cdot \sin \left({\tfrac {2\pi }{P}}nx\right)\right]}

In terms of the readily computed quantities, $a_{n}$

and

b_{n}

, recall that:

\cos(\varphi _{n})=a_{n}/A_{n}

\sin(\varphi _{n})=b_{n}/A_{n}

A_{0}={\sqrt {a_{0}^{2}+b_{0}^{2}}}={\sqrt {a_{0}^{2}}}=a_{0}

Therefore an alternative form of the Fourier series, using the Cartesian coordinates, is the sine-cosine form:

Fourier series, sine-cosine form

$s_{\scriptscriptstyle N}(x)={\frac {a_{0}}{2}}+\sum _{n=1}^{N}\left(a_{n}\cos \left({\tfrac {2\pi }{P}}nx\right)+b_{n}\sin \left({\tfrac {2\pi }{P}}nx\right)\right)$

(Eq.4)

Exponential form

Another applicable identity is Euler's formula:

{\begin{aligned}\cos \left({\tfrac {2\pi }{P}}nx-\varphi _{n}\right)&{}\equiv {\tfrac {1}{2}}e^{i\left(2\pi nx/P-\varphi _{n}\right)}+{\tfrac {1}{2}}e^{-i\left(2\pi nx/P-\varphi _{n}\right)}\\[6pt]&=\left({\tfrac {1}{2}}e^{-i\varphi _{n}}\right)\cdot e^{i2\pi (+n)x/P}+\left({\tfrac {1}{2}}e^{-i\varphi _{n}}\right)^{*}\cdot e^{i2\pi (-n)x/P}\end{aligned}}

(Note: the ∗ denotes complex conjugation.)

Therefore, with definitions:

c_{n}\triangleq \left\{{\begin{array}{lll}A_{0}/2&=a_{0}/2,\quad &n=0\\{\tfrac {A_{n}}{2}}e^{-i\varphi _{n}}&={\tfrac {1}{2}}(a_{n}-ib_{n}),\quad &n>0\\c_{|n|}^{*},\quad &&n<0\end{array}}\right\}={\tfrac {1}{P}}\int _{P}s(x)\cdot e^{-i2\pi nx/P}\,dx,

the final result is:

Fourier series, exponential form

$s_{_{N}}(x)=\sum _{n=-N}^{N}c_{n}\cdot e^{i2\pi nx/P}$

(Eq.5)

This is the customary form for generalizing to § Complex-valued functions. Negative values of $n$

correspond to negative frequency (explained in Fourier transform § Use of complex sinusoids to represent real sinusoids).

نمایش‌های مختلف سری فوریه

نمایش مثلثاتی

اگر $f:\mathbb {R} \rightarrow \mathbb {C}$

یک تابع متناوب با دوره تناوب

T

باشد (یا به عبارتی: ‎

f(t+T)=f(t)

‎) آنگاه این تابع را می‌توان به صورت زیر نوشت:

f(t)={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }[a_{n}\cos(\omega _{n}t)+b_{n}\sin(\omega _{n}t)]\,\!

که در آن $\omega _{n}$

هارمونیک nام سری فوریه با رادیان بوده و ضرایب

a_{n}

،

a_{0}

و

b_{n}

را می‌توان از فرمول‌های اویلر بدست آورد.
فوریه بر این باور بود که هرگونه تابع متناوب را می‌توان به صورت جمعی از توابع سینوسی نوشت. این مطلب درست نیست. شرایط لازم برای هر تابع متناوب برای اینکه به صورت سری فوریه نوشته شود به صورت زیر است:

تابع در هر دورهٔ تناوبی انتگرال پذیر باشد:

\int _{a}^{a+T}{\left\vert f(x)\right\vert }dx<\infty

تابع فقط شمار محدودی بیشینه و کمینه داشته باشد.
تابع فقط شمار محدودی ناپیوستگی داشته باشد.

نمایش مختلط

سری فوریه می‌تواند به صورت زیر نیز نوشته شود:

f(t)=\sum _{n=-\infty }^{+\infty }c_{n}e^{i\omega _{n}t}\,\!

و در اینجا:

c_{n}={\frac {1}{T}}\int _{t_{1}}^{t_{2}}f(t)e^{-i\omega _{n}t}\,dt\,\!

این رابطه با کمک فرمول اویلر قابل گسترش به صورت زیر است:

\sum _{n=-\infty }^{+\infty }(c_{n}cos({w_{n}}t)+ic_{n}sin({w_{n}}t))\,\!

اگر این رابطه را به‌طور مستقیم با نمایش مثلثی مقایسه کنیم مشاهده می‌شود که $c_{n}$

به طریق زیر نیز قابل محاسبه است:

c_{n}={\frac {1}{2}}(a_{n}-ib_{n})\,\!

c_{-n}={\frac {1}{2}}(a_{n}+ib_{n})\,\!

نمایش کسینوس-با-فاز

نمایش زیر که در واقع شکل ویژه‌ای از نمایش مثلثی می‌باشد، نمایش کسینوس-با-فاز نام دارد. از این نمایش در رسم طیف خطی (به انگلیسی: line spectra) استفاده می‌شود.

x={a_{0}+}\sum _{k=1}^{N}{A_{k}}cos({\omega _{k}}t+\theta _{k})\,\!

محاسبه ضرایب فوریه

نمایش مثلثی

نمایش مثلثی بالا را در نظر بگیرید. همان‌طور که گفته شد $T$

دوره تناوب و

{\omega _{n}}

هارمونی nام تابع می‌باشد. در تبدیل فوریه سه ضریب

a_{n}

و

b_{n}

و ضریب ثابت

a_{0}

مطرح است. ضریب‌ها با استفاده از روابط زیر قابل محاسبه هستند.

a_{0}={\frac {1}{T}}\int _{0}^{T}f(x)dx\,\!

a_{n}={\frac {2}{T}}\int _{0}^{T}f(x)Cos(n{\omega }x)dx,n=1,2,\ldots \,\!

b_{n}={\frac {2}{T}}\int _{0}^{T}f(x)Sin(n{\omega }x)dx,n=1,2,\ldots \,\!

بازه [ $\pi ,\pi$

-] یا در کل بازه‌هایی که طول آنها

2\pi

است از مهمترین بازه‌هایی است که درمحاسبه ضرایب استفاده می‌شود. بدین ترتیب

p=2\pi

پس ضرایب عبارتند از:

a_{0}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(x)dx\,\!

a_{n}={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(x)Cos(nx)dx\,\!

b_{n}={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(x)Sin(nx)dx\,\!

همگرایی

چهار مجموع جزئی اول سری فوریه برای یک موج مربعی

در کاربردهای مهندسی، به‌طور کلی فرض می‌شود که سری‌های فوریه تقریباً در همه جا همگرا شوند (استثنائاتی در ناپیوستگی‌های گسسته وجود دارد) زیرا عملکردهایی که در مهندسی مشاهده می‌شوند رفتار بهتری نسبت به توابعی دارند که ریاضیدانان می‌توانند به عنوان نمونه‌های متضاد این فرض ارائه دهند. به‌طور خاص، اگر $s$

پیوسته باشد و مشتق

s(x)

(که ممکن است در همه جا وجود نداشته باشد) مربع انتگرال دار است، پس سری‌های فوریه به‌طور کامل و یکنواخت به

s(x)

همگرا می‌شوند.

چهار جمع جزئی (سری فوریه) با طول ۱ ، ۲ ، ۳ و ۴ جمله ای، که نشان می‌دهد چگونه تقریب با یک موج مربعی با افزایش تعداد جمله‌ها بهبود می‌یابد.
چهار مجموع جزئی (سری فوریه) با طول ۱ ، ۲ ، ۳ و ۴ جمله ای، که نشان می‌دهد چگونه تقریب با یک موج دندانه اره ای با افزایش تعداد جمله‌ها بهبود می‌یابد.
نمونه ای از همگرایی به یک تابع نسبتاً دلخواه. به شکل‌گیری "طنین" (پدیده گیبس) در انتقال به بخش‌های عمودی توجه کنید.

Extensions

Fourier series on a square

We can also define the Fourier series for functions of two variables $x$

and

y

in the square

[-\pi ,\pi ]\times [-\pi ,\pi ]

:

{\begin{aligned}f(x,y)&=\sum _{j,k\in \mathbb {Z} }c_{j,k}e^{ijx}e^{iky},\\[5pt]c_{j,k}&={\frac {1}{4\pi ^{2}}}\int _{-\pi }^{\pi }\int _{-\pi }^{\pi }f(x,y)e^{-ijx}e^{-iky}\,dx\,dy.\end{aligned}}

Aside from being useful for solving partial differential equations such as the heat equation, one notable application of Fourier series on the square is in image compression. In particular, the jpeg image compression standard uses the two-dimensional discrete cosine transform, a discrete form of the Fourier cosine transform, which uses only cosine as the basis function.

For two-dimensional arrays with a staggered appearance, half of the Fourier series coefficients disappear, due to additional symmetry.

Fourier series of Bravais-lattice-periodic-function

A three-dimensional Bravais lattice is defined as the set of vectors of the form:

\mathbf {R} =n_{1}\mathbf {a} _{1}+n_{2}\mathbf {a} _{2}+n_{3}\mathbf {a} _{3}

where

n_{i}

are integers and

\mathbf {a} _{i}

are three linearly independent vectors. Assuming we have some function,

f(\mathbf {r} )

, such that it obeys the condition of periodicity for any Bravais lattice vector

\mathbf {R}

,

f(\mathbf {r} )=f(\mathbf {R} +\mathbf {r} )

, we could make a Fourier series of it. This kind of function can be, for example, the effective potential that one electron "feels" inside a periodic crystal. It is useful to make the Fourier series of the potential when applying Bloch's theorem. First, we may write any arbitrary position vector

\mathbf {r}

in the coordinate-system of the lattice:

\mathbf {r} =x_{1}{\frac {\mathbf {a} _{1}}{a_{1}}}+x_{2}{\frac {\mathbf {a} _{2}}{a_{2}}}+x_{3}{\frac {\mathbf {a} _{3}}{a_{3}}},

where

a_{i}\triangleq |\mathbf {a} _{i}|,

meaning that

a_{i}

is defined to be the magnitude of

\mathbf {a} _{i}

, so

{\hat {\mathbf {a} _{i}}}={\frac {\mathbf {a} _{i}}{a_{i}}}

is the unit vector directed along

\mathbf {a} _{i}

.

Thus we can define a new function,

g(x_{1},x_{2},x_{3})\triangleq f(\mathbf {r} )=f\left(x_{1}{\frac {\mathbf {a} _{1}}{a_{1}}}+x_{2}{\frac {\mathbf {a} _{2}}{a_{2}}}+x_{3}{\frac {\mathbf {a} _{3}}{a_{3}}}\right).

This new function,

g(x_{1},x_{2},x_{3})

, is now a function of three-variables, each of which has periodicity

a_{1}

,

a_{2}

, and

a_{3}

respectively:

g(x_{1},x_{2},x_{3})=g(x_{1}+a_{1},x_{2},x_{3})=g(x_{1},x_{2}+a_{2},x_{3})=g(x_{1},x_{2},x_{3}+a_{3}).

This enables us to build up a set of Fourier coefficients, each being indexed by three independent integers

m_{1},m_{2},m_{3}

. In what follows, we use function notation to denote these coefficients, where previously we used subscripts. If we write a series for

g

on the interval

\left[0,a_{1}\right]

for

x_{1}

, we can define the following:

h^{\mathrm {one} }(m_{1},x_{2},x_{3})\triangleq {\frac {1}{a_{1}}}\int _{0}^{a_{1}}g(x_{1},x_{2},x_{3})\cdot e^{-i2\pi {\frac {m_{1}}{a_{1}}}x_{1}}\,dx_{1}

And then we can write:

g(x_{1},x_{2},x_{3})=\sum _{m_{1}=-\infty }^{\infty }h^{\mathrm {one} }(m_{1},x_{2},x_{3})\cdot e^{i2\pi {\frac {m_{1}}{a_{1}}}x_{1}}

Further defining:

{\begin{aligned}h^{\mathrm {two} }(m_{1},m_{2},x_{3})&\triangleq {\frac {1}{a_{2}}}\int _{0}^{a_{2}}h^{\mathrm {one} }(m_{1},x_{2},x_{3})\cdot e^{-i2\pi {\frac {m_{2}}{a_{2}}}x_{2}}\,dx_{2}\\[12pt]&={\frac {1}{a_{2}}}\int _{0}^{a_{2}}dx_{2}{\frac {1}{a_{1}}}\int _{0}^{a_{1}}dx_{1}g(x_{1},x_{2},x_{3})\cdot e^{-i2\pi \left({\frac {m_{1}}{a_{1}}}x_{1}+{\frac {m_{2}}{a_{2}}}x_{2}\right)}\end{aligned}}

We can write

g

once again as:

g(x_{1},x_{2},x_{3})=\sum _{m_{1}=-\infty }^{\infty }\sum _{m_{2}=-\infty }^{\infty }h^{\mathrm {two} }(m_{1},m_{2},x_{3})\cdot e^{i2\pi {\frac {m_{1}}{a_{1}}}x_{1}}\cdot e^{i2\pi {\frac {m_{2}}{a_{2}}}x_{2}}

Finally applying the same for the third coordinate, we define:

{\begin{aligned}h^{\mathrm {three} }(m_{1},m_{2},m_{3})&\triangleq {\frac {1}{a_{3}}}\int _{0}^{a_{3}}h^{\mathrm {two} }(m_{1},m_{2},x_{3})\cdot e^{-i2\pi {\frac {m_{3}}{a_{3}}}x_{3}}\,dx_{3}\\[12pt]&={\frac {1}{a_{3}}}\int _{0}^{a_{3}}dx_{3}{\frac {1}{a_{2}}}\int _{0}^{a_{2}}dx_{2}{\frac {1}{a_{1}}}\int _{0}^{a_{1}}dx_{1}g(x_{1},x_{2},x_{3})\cdot e^{-i2\pi \left({\frac {m_{1}}{a_{1}}}x_{1}+{\frac {m_{2}}{a_{2}}}x_{2}+{\frac {m_{3}}{a_{3}}}x_{3}\right)}\end{aligned}}

We write

g

as:

g(x_{1},x_{2},x_{3})=\sum _{m_{1}=-\infty }^{\infty }\sum _{m_{2}=-\infty }^{\infty }\sum _{m_{3}=-\infty }^{\infty }h^{\mathrm {three} }(m_{1},m_{2},m_{3})\cdot e^{i2\pi {\frac {m_{1}}{a_{1}}}x_{1}}\cdot e^{i2\pi {\frac {m_{2}}{a_{2}}}x_{2}}\cdot e^{i2\pi {\frac {m_{3}}{a_{3}}}x_{3}}

Re-arranging:

g(x_{1},x_{2},x_{3})=\sum _{m_{1},m_{2},m_{3}\in \mathbb {Z} }h^{\mathrm {three} }(m_{1},m_{2},m_{3})\cdot e^{i2\pi \left({\frac {m_{1}}{a_{1}}}x_{1}+{\frac {m_{2}}{a_{2}}}x_{2}+{\frac {m_{3}}{a_{3}}}x_{3}\right)}.

Now, every reciprocal lattice vector can be written (but does not mean that it is the only way of writing) as

\mathbf {G} =m_{1}\mathbf {g} _{1}+m_{2}\mathbf {g} _{2}+m_{3}\mathbf {g} _{3}

, where

m_{i}

are integers and

\mathbf {g} _{i}

are reciprocal lattice vectors to satisfy

\mathbf {g_{i}} \cdot \mathbf {a_{j}} =2\pi \delta _{ij}

(

\delta _{ij}=1

for

i=j

, and

\delta _{ij}=0

for

i\neq j

). Then for any arbitrary reciprocal lattice vector

\mathbf {G}

and arbitrary position vector

\mathbf {r}

in the original Bravais lattice space, their scalar product is:

\mathbf {G} \cdot \mathbf {r} =\left(m_{1}\mathbf {g} _{1}+m_{2}\mathbf {g} _{2}+m_{3}\mathbf {g} _{3}\right)\cdot \left(x_{1}{\frac {\mathbf {a} _{1}}{a_{1}}}+x_{2}{\frac {\mathbf {a} _{2}}{a_{2}}}+x_{3}{\frac {\mathbf {a} _{3}}{a_{3}}}\right)=2\pi \left(x_{1}{\frac {m_{1}}{a_{1}}}+x_{2}{\frac {m_{2}}{a_{2}}}+x_{3}{\frac {m_{3}}{a_{3}}}\right).

So it is clear that in our expansion of

g(x_{1},x_{2},x_{3})=f(\mathbf {r} )

, the sum is actually over reciprocal lattice vectors:

f(\mathbf {r} )=\sum _{\mathbf {G} }h(\mathbf {G} )\cdot e^{i\mathbf {G} \cdot \mathbf {r} },

where

h(\mathbf {G} )={\frac {1}{a_{3}}}\int _{0}^{a_{3}}dx_{3}\,{\frac {1}{a_{2}}}\int _{0}^{a_{2}}dx_{2}\,{\frac {1}{a_{1}}}\int _{0}^{a_{1}}dx_{1}\,f\left(x_{1}{\frac {\mathbf {a} _{1}}{a_{1}}}+x_{2}{\frac {\mathbf {a} _{2}}{a_{2}}}+x_{3}{\frac {\mathbf {a} _{3}}{a_{3}}}\right)\cdot e^{-i\mathbf {G} \cdot \mathbf {r} }.

Assuming

\mathbf {r} =(x,y,z)=x_{1}{\frac {\mathbf {a} _{1}}{a_{1}}}+x_{2}{\frac {\mathbf {a} _{2}}{a_{2}}}+x_{3}{\frac {\mathbf {a} _{3}}{a_{3}}},

we can solve this system of three linear equations for

x

,

y

, and

z

in terms of

x_{1}

,

x_{2}

and

x_{3}

in order to calculate the volume element in the original cartesian coordinate system. Once we have

x

,

y

, and

z

in terms of

x_{1}

,

x_{2}

and

x_{3}

, we can calculate the Jacobian determinant:

{\begin{vmatrix}{\dfrac {\partial x_{1}}{\partial x}}&{\dfrac {\partial x_{1}}{\partial y}}&{\dfrac {\partial x_{1}}{\partial z}}\\[12pt]{\dfrac {\partial x_{2}}{\partial x}}&{\dfrac {\partial x_{2}}{\partial y}}&{\dfrac {\partial x_{2}}{\partial z}}\\[12pt]{\dfrac {\partial x_{3}}{\partial x}}&{\dfrac {\partial x_{3}}{\partial y}}&{\dfrac {\partial x_{3}}{\partial z}}\end{vmatrix}}

which after some calculation and applying some non-trivial cross-product identities can be shown to be equal to:

{\frac {a_{1}a_{2}a_{3}}{\mathbf {a} _{1}\cdot (\mathbf {a} _{2}\times \mathbf {a} _{3})}}

(it may be advantageous for the sake of simplifying calculations, to work in such a Cartesian coordinate system, in which it just so happens that

\mathbf {a} _{1}

is parallel to the x axis,

\mathbf {a} _{2}

lies in the xy-plane, and

\mathbf {a} _{3}

has components of all three axes). The denominator is exactly the volume of the primitive unit cell which is enclosed by the three primitive-vectors

\mathbf {a} _{1}

,

\mathbf {a} _{2}

and

\mathbf {a} _{3}

. In particular, we now know that

dx_{1}\,dx_{2}\,dx_{3}={\frac {a_{1}a_{2}a_{3}}{\mathbf {a} _{1}\cdot (\mathbf {a} _{2}\times \mathbf {a} _{3})}}\cdot dx\,dy\,dz.

We can write now

h(\mathbf {G} )

as an integral with the traditional coordinate system over the volume of the primitive cell, instead of with the

x_{1}

,

x_{2}

and

x_{3}

variables:

h(\mathbf {G} )={\frac {1}{\mathbf {a} _{1}\cdot (\mathbf {a} _{2}\times \mathbf {a} _{3})}}\int _{C}d\mathbf {r} f(\mathbf {r} )\cdot e^{-i\mathbf {G} \cdot \mathbf {r} }

writing

d\mathbf {r}

for the volume element

dx\,dy\,dz

; and where

C

is the primitive unit cell, thus,

\mathbf {a} _{1}\cdot (\mathbf {a} _{2}\times \mathbf {a} _{3})

is the volume of the primitive unit cell.

Hilbert space interpretation

In the language of Hilbert spaces, the set of functions $\left\{e_{n}=e^{inx}:n\in \mathbb {Z} \right\}$

is an orthonormal basis for the space

L^{2}([-\pi ,\pi ])

of square-integrable functions on

[-\pi ,\pi ]

. This space is actually a Hilbert space with an inner product given for any two elements

f

and

g

by:

\langle f,\,g\rangle \;\triangleq \;{\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(x)g^{*}(x)\,dx,

where

g^{*}(x)

is the complex conjugate of

g(x).

The basic Fourier series result for Hilbert spaces can be written as

f=\sum _{n=-\infty }^{\infty }\langle f,e_{n}\rangle \,e_{n}.

Sines and cosines form an orthonormal set, as illustrated above. The integral of sine, cosine and their product is zero (green and red areas are equal, and cancel out) when

m

,

n

or the functions are different, and π only if

m

and

n

are equal, and the function used is the same.

This corresponds exactly to the complex exponential formulation given above. The version with sines and cosines is also justified with the Hilbert space interpretation. Indeed, the sines and cosines form an orthogonal set:

\int _{-\pi }^{\pi }\cos(mx)\,\cos(nx)\,dx={\frac {1}{2}}\int _{-\pi }^{\pi }\cos((n-m)x)+\cos((n+m)x)\,dx=\pi \delta _{mn},\quad m,n\geq 1,

\int _{-\pi }^{\pi }\sin(mx)\,\sin(nx)\,dx={\frac {1}{2}}\int _{-\pi }^{\pi }\cos((n-m)x)-\cos((n+m)x)\,dx=\pi \delta _{mn},\quad m,n\geq 1

(where δ_mn is the Kronecker delta), and

\int _{-\pi }^{\pi }\cos(mx)\,\sin(nx)\,dx={\frac {1}{2}}\int _{-\pi }^{\pi }\sin((n+m)x)+\sin((n-m)x)\,dx=0;

furthermore, the sines and cosines are orthogonal to the constant function

1

. An orthonormal basis for

L^{2}([-\pi ,\pi ])

consisting of real functions is formed by the functions

1

and

{\sqrt {2}}\cos(nx)

,

{\sqrt {2}}\sin(nx)

with n= 1,2,.... The density of their span is a consequence of the Stone–Weierstrass theorem, but follows also from the properties of classical kernels like the Fejér kernel.

Table of common Fourier series

Some common pairs of periodic functions and their Fourier Series coefficients are shown in the table below.

$s(x)$ designates a periodic function defined on $0<x\leq P$ .
$a_{0},a_{n},b_{n}$ designate the Fourier Series coefficients (sine-cosine form) of the periodic function $s(x)$ .

Time domain $s(x)$	Frequency domain (sine-cosine form) ${\begin{aligned}&a_{0}\\&a_{n}\quad {\text{for }}n\geq 1\\&b_{n}\quad {\text{for }}n\geq 1\end{aligned}}$	Remarks
$s(x)=A\left\|\sin \left({\frac {2\pi }{P}}x\right)\right\|\quad {\text{for }}0\leq x<P$	${\begin{aligned}a_{0}=&{\frac {4A}{\pi }}\\a_{n}=&{\begin{cases}{\frac {-4A}{\pi }}{\frac {1}{n^{2}-1}}&\quad n{\text{ even}}\\0&\quad n{\text{ odd}}\end{cases}}\\b_{n}=&0\\\end{aligned}}$	Full-wave rectified sine
$s(x)={\begin{cases}A\sin \left({\frac {2\pi }{P}}x\right)&\quad {\text{for }}0\leq x<P/2\\0&\quad {\text{for }}P/2\leq x<P\\\end{cases}}$	${\begin{aligned}a_{0}=&{\frac {2A}{\pi }}\\a_{n}=&{\begin{cases}{\frac {-2A}{\pi }}{\frac {1}{n^{2}-1}}&\quad n{\text{ even}}\\0&\quad n{\text{ odd}}\end{cases}}\\b_{n}=&{\begin{cases}{\frac {A}{2}}&\quad n=1\\0&\quad n>1\end{cases}}\\\end{aligned}}$	Half-wave rectified sine
$s(x)={\begin{cases}A&\quad {\text{for }}0\leq x<D\cdot P\\0&\quad {\text{for }}D\cdot P\leq x<P\\\end{cases}}$	${\begin{aligned}a_{0}=&2AD\\a_{n}=&{\frac {A}{n\pi }}\sin \left(2\pi nD\right)\\b_{n}=&{\frac {2A}{n\pi }}\left(\sin \left(\pi nD\right)\right)^{2}\\\end{aligned}}$	$0\leq D\leq 1$
$s(x)={\frac {Ax}{P}}\quad {\text{for }}0\leq x<P$	${\begin{aligned}a_{0}=&A\\a_{n}=&0\\b_{n}=&{\frac {-A}{n\pi }}\\\end{aligned}}$
$s(x)=A-{\frac {Ax}{P}}\quad {\text{for }}0\leq x<P$	${\begin{aligned}a_{0}=&A\\a_{n}=&0\\b_{n}=&{\frac {A}{n\pi }}\\\end{aligned}}$
$s(x)={\frac {4A}{P^{2}}}\left(x-{\frac {P}{2}}\right)^{2}\quad {\text{for }}0\leq x<P$	${\begin{aligned}a_{0}=&{\frac {2A}{3}}\\a_{n}=&{\frac {4A}{\pi ^{2}n^{2}}}\\b_{n}=&0\\\end{aligned}}$

جستارهای وابسته

منابع

↑ Dorf, Richard C.; Tallarida, Ronald J. (1993). Pocket Book of Electrical Engineering Formulas (1st ed.). Boca Raton,FL: CRC Press. pp. 171–174. ISBN 0849344735.
↑ Tolstov, Georgi P. (1976). Fourier Series. Courier-Dover. ISBN 0-486-63317-9.
↑ Vanishing of Half the Fourier Coefficients in Staggered Arrays
↑ Papula, Lothar (2009). Mathematische Formelsammlung: für Ingenieure und Naturwissenschaftler [Mathematical Functions for Engineers and Physicists] (به آلمانی). Vieweg+Teubner Verlag. ISBN 978-3834807571.

کتاب‌شناسی

Yitzhak Katznelson, An introduction to harmonic analysis, Second corrected edition. Dover Publications, Inc. , New York, 1976. ISBN 0-486-63331-4
Felix Klein, Development of mathematics in the 19th century. Mathsci Press Brookline, Mass, 1979. Translated by M. Ackerman from Vorlesungen uber die Entwicklung der Matematik im 19 Jahrhundert, Springer, Berlin, 1928.
Walter Rudin, Principles of mathematical analysis, Third edition. McGraw-Hill, Inc. , New York, 1976. ISBN 0-07-054235-X
Kamen, Edward W.; Heck, Bonnie S. (2007). Signals And Systems. Prentice Hall. ISBN 0-13-168737-9.

[3] Dorf, Richard C.; Tallarida, Ronald J. (1993). Pocket Book of Electrical Engineering Formulas (1st ed.). Boca Raton,FL: CRC Press. pp. 171–174. ISBN 0849344735.

[5] Tolstov, Georgi P. (1976). Fourier Series. Courier-Dover. ISBN 0-486-63317-9.

[6] Vanishing of Half the Fourier Coefficients in Staggered Arrays

[Papula-7] Papula, Lothar (2009). Mathematische Formelsammlung: für Ingenieure und Naturwissenschaftler [Mathematical Functions for Engineers and Physicists] (به آلمانی). Vieweg+Teubner Verlag. ISBN 978-3834807571.