سری فوریه

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Fourier2.jpg
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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[ویرایش]

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:

که در آن یک عدد صحیح مثبت، دامنه، بسامد و فاز توابع کسینوسی می‌باشد. قابل مشاهده است که با در دست داشتن بسامدها ، دامنه‌ها و فازها تابع به‌طور کامل قابل تعریف است. توجه شود که بر اساس گفته‌های بالا تابع مستقل از زمان قابل تعریف است.

Common forms[ویرایش]

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

The number of terms summed, , is a potentially infinite integer. Even so, the series might not converge or exactly equate to at all values of (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, , is also the number of cycles the harmonic makes in the function's period .[A] Therefore:

  • The harmonic's wavelength is and in units of .
  • The harmonic's frequency is and in reciprocal units of .
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[ویرایش]

The Fourier series in amplitude-phase form is:

Fourier series, amplitude-phase form

 

 

 

 

(Eq.1)

  • Its harmonic is .
  • is the harmonic's amplitude and is its phase shift.
  • The fundamental frequency of is the term for when equals 1, and can be referred to as the harmonic.
  • is sometimes called the harmonic or DC component. It is the mean value of .

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 ().

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 and can be understood and derived in terms of the cross-correlation between and a sinusoid at frequency . For a general frequency and an analysis interval the cross-correlation function:

 

 

 

 

(Eq.2)

is essentially a matched filter, with template .[B] The maximum of is a measure of the amplitude of frequency in the function , and the value of at the maximum determines the phase of that frequency. Figure 2 is an example, where is a square wave (not shown), and frequency is the 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

 

 

 

 

(Eq.3)

Substituting this into Eq.2 gives:

which introduces the definitions of and .[۱]  And we note for later reference that and can be simplified:
The derivative of is zero at the phase of maximum correlation.
And the correlation peak value is:

Therefore and are the Cartesian coordinates of a vector with polar coordinates and

Sine-cosine form[ویرایش]

Substituting Eq.3 into Eq.1 gives:

In terms of the readily computed quantities, and , recall that:

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

Fourier series, sine-cosine form

 

 

 

 

(Eq.4)

Exponential form[ویرایش]

Another applicable identity is Euler's formula:

(Note: the ∗ denotes complex conjugation.)

Therefore, with definitions:

the final result is:

Fourier series, exponential form

 

 

 

 

(Eq.5)

This is the customary form for generalizing to § Complex-valued functions. Negative values of correspond to negative frequency (explained in Fourier transform § Use of complex sinusoids to represent real sinusoids).

نمایش‌های مختلف سری فوریه[ویرایش]

نمایش مثلثاتی[ویرایش]

اگر یک تابع متناوب با دوره تناوب باشد (یا به عبارتی: ‎‎) آنگاه این تابع را می‌توان به صورت زیر نوشت:

که در آن هارمونیک nام سری فوریه با رادیان بوده و ضرایب ، و 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:

  1. تابع در هر دورهٔ تناوبی انتگرال پذیر باشد:
  1. تابع فقط شمار محدودی بیشینه و کمینه داشته باشد.
  2. تابع فقط شمار محدودی ناپیوستگی داشته باشد.

نمایش مختلط[ویرایش]

سری فوریه می‌تواند به صورت زیر نیز نوشته شود:

و در اینجا:

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

اگر این رابطه را به‌طور مستقیم با نمایش مثلثی مقایسه کنیم مشاهده می‌شود که به طریق زیر نیز قابل محاسبه است:

نمایش کسینوس-با-فاز[ویرایش]

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.

محاسبه ضرایب فوریه[ویرایش]

نمایش مثلثی[ویرایش]

نمایش مثلثی بالا را در نظر بگیرید. همان‌طور که گفته شد دوره تناوب و هارمونی nام تابع می‌باشد. در تبدیل فوریه سه ضریب و و ضریب ثابت مطرح است. ضریب‌ها با استفاده از روابط زیر قابل محاسبه هستند.

بازه [-] یا در کل بازه‌هایی که طول آنها است از مهمترین بازه‌هایی است که درمحاسبه ضرایب استفاده می‌شود. بدین ترتیب پس ضرایب عبارتند از:

همگرایی[ویرایش]

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 پیوسته باشد و مشتق (که ممکن است در همه جا وجود نداشته باشد) مربع انتگرال دار است، پس سری‌های فوریه به‌طور کامل و یکنواخت به همگرا می‌شوند.[۲]

Extensions[ویرایش]

Fourier series on a square[ویرایش]

We can also define the Fourier series for functions of two variables and in the square :

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.[۳]

Fourier series of Bravais-lattice-periodic-function[ویرایش]

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

where are integers and are three linearly independent vectors. Assuming we have some function, , such that it obeys the condition of periodicity for any Bravais lattice vector , , 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 in the coordinate-system of the lattice:
where meaning that is defined to be the magnitude of , so is the unit vector directed along .

Thus we can define a new function,

This new function, , is now a function of three-variables, each of which has periodicity , , and respectively:
This enables us to build up a set of Fourier coefficients, each being indexed by three independent integers . In what follows, we use function notation to denote these coefficients, where previously we used subscripts. If we write a series for on the interval for , we can define the following:
And then we can write:
Further defining:
We can write once again as:
Finally applying the same for the third coordinate, we define:
We write as:
Re-arranging:
Now, every reciprocal lattice vector can be written (but does not mean that it is the only way of writing) as , where are integers and are reciprocal lattice vectors to satisfy ( for , and for ). Then for any arbitrary reciprocal lattice vector and arbitrary position vector in the original Bravais lattice space, their scalar product is:
So it is clear that in our expansion of , the sum is actually over reciprocal lattice vectors:
where
Assuming
we can solve this system of three linear equations for , , and in terms of , and in order to calculate the volume element in the original cartesian coordinate system. Once we have , , and in terms of , and , we can calculate the Jacobian determinant:
which after some calculation and applying some non-trivial cross-product identities can be shown to be equal to:
(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 is parallel to the x axis, lies in the xy-plane, and 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 , and . In particular, we now know that
We can write now as an integral with the traditional coordinate system over the volume of the primitive cell, instead of with the , and variables:
writing for the volume element ; and where is the primitive unit cell, thus, is the volume of the primitive unit cell.

Hilbert space interpretation[ویرایش]

In the language of Hilbert spaces, the set of functions is an orthonormal basis for the space of square-integrable functions on . This space is actually a Hilbert space with an inner product given for any two elements and by:

where is the complex conjugate of

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

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 , or the functions are different, and π only if and 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:

(where δmn is the Kronecker delta), and
furthermore, the sines and cosines are orthogonal to the constant function . An orthonormal basis for consisting of real functions is formed by the functions and , 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[ویرایش]

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

  • designates a periodic function defined on .
  • designate the Fourier Series coefficients (sine-cosine form) of the periodic function .
Time domain

Plot Frequency domain (sine-cosine form)

Remarks Reference
PlotRectifiedSineSignal.svg
Full-wave rectified sine [۴]: p. 193 
PlotHalfRectifiedSineSignal.svg
Half-wave rectified sine [۴]: p. 193 
PlotRectangleSignal.svg
PlotSawtooth1Signal.svg
[۴]: p. 192 
PlotSawtooth2Signal.svg
[۴]: p. 192 
PlotParabolaSignal.svg
[۴]: p. 193 

جستارهای وابسته[ویرایش]

منابع[ویرایش]

  1. 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.
  2. Tolstov, Georgi P. (1976). Fourier Series. Courier-Dover. ISBN 0-486-63317-9.
  3. Vanishing of Half the Fourier Coefficients in Staggered Arrays
  4. ۴٫۰ ۴٫۱ ۴٫۲ ۴٫۳ ۴٫۴ Papula, Lothar (2009). Mathematische Formelsammlung: für Ingenieure und Naturwissenschaftler [Mathematical Functions for Engineers and Physicists] (به آلمانی). Vieweg+Teubner Verlag. ISBN 978-3834807571.

کتاب‌شناسی[ویرایش]

  • 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.


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