Introduction to Introductions to Fourier Series -1

Most Introductions to Fourier Series, Fourier Transform, Fourier Analysis are not written for common ordinary people. Some or a lot of 'high-level' (by common ordinary people standard) mathematical background knowledge is required to understand them. One of the hard parts at the introduction stage is Fourier coefficients. In most Introductions I have tried Fourier coefficients are given without derivation, reasoning or proof, just given. Plus Fourier coefficients do not look like ordinary (simple) coefficients. They are very long and these are functions as well. And Fourier Coefficients are different from Fourier Series, of course. Fourier Coefficients are in integral form while Fourier Series is in series form. Your understanding of Fourier Transform, Laplace Transform (if you try) will be widen and deepened because these are developments of Fourier Series.
 
Wiki's Fourier Series (as of 05-Aug-2011) says in Definition:

"

Fourier's formula for 2π-periodic functions using sines and cosines
 
For a periodic function ƒ(x) that is integrable on [−π, π], the numbers

and

are called the Fourier coefficients of ƒ. One introduces the partial sums of the Fourier series for ƒ, often denoted by

The partial sums for ƒ are trigonometric polynomials. One expects that the functions S_N ƒ approximate the function ƒ, and that the approximation improves as N tends to infinity. The infinite sum

is called the Fourier series of ƒ.

"

This part is not found any more now in Wiki. Probably those who already know Fourier Series can understand this but those who are new to Fourier Series do not understand this. The point is <
a periodic function ƒ(x) is a periodic function>. Or in one word "periodicity". This is crucial and please do not forget this. The above explanation is very general or too general for those who do not know Fourier Series but want to know it.
 
While most Introductions are like this:

Fourier series of ƒ(x) =

and the coefficients are given as

and

Fourier coefficients are not so simple and may have several meanings including very profound ones depending on how how deep you see and understand them. So Fourier coefficients must be explained in some way, not just given.　(Again ƒ(x) must be a periodic function.)


First - what is periodic function ? (we will see this later.)

Second - What is a coefficient? Coefficient is one of the basic math backgrounds.

From <All about Circuits>

"

<https://www.allaboutcircuits.com/technical-articles/the-fourier-coefficients/>

Learn About Fourier Coefficients 

January 05, 2016 by Donald Krambeck 

Never understood Fourier series coefficients? Now you will.

It says


"

$$f\left(t\right)=a_v+\sum _{n=}^{\infty }{a}_n\cos \left(n{\omega }_{0}t\right)+b_n\sin \left(n{\omega }_{0}t\right)$$(1.1) Fourier series representation of a periodic function

Where n is the integer sequence 1,2,3,...

In Eq. 1.1, $$a_v$$, $$a_n$$, and $$b_n$$ are known as Fourier coefficients and can be found from f(t).
The term $${\omega }_0$$ (or $$\frac{2\pi }{T}$$) represents the fundamental frequency of the periodic function f(t).
The integral multiples of $${\omega }_0$$, i.e. $$2{\omega }_0,3{\omega }_0,4{\omega }_0$$ and so on, are known as the harmonic frequencies of f(t). Thus $$n{\omega }_0$$is the nth harmonic term of f(t). 

 

"

Underline made by ACT.

So the Fourier coefficients can be said inter-related with the original periodic function  f(t).

<x> is general while <t> usually stands for time. But sometimes, especially for generalized <x> may be better to use, not necessarily time. But at the introduction stage generalized <x> (not time) is seldom seen.

Basic math background

"
Coefficient
 

From Wikipedia, the free encyclopedia (as of 05-Aug-2011)

In mathematics, a coefficient is a multiplicative factor in some term of an expression (or of a series); it is usually a number, but in any case does not involve any variables of the expression. For instance in

7x² − 3xy + 1.5 + y

the first three terms respectively have the coefficients 7, −3, and 1.5 (in the third term there are no variables, so the coefficient is the term itself; it is called the constant term or constant coefficient of this expression). The final term does not have any explicitly written coefficient, but is usually considered to have coefficient 1, since multiplying by that factor would not change the term. Often coefficients are numbers as in this example, although they could be parameters of the problem, as a, b, and c in

ax² + bx + c

when it is understood that these are not considered as variables.

"

So let's apply this definition to the Fourier Series.

Fourier series of ƒ(x) =

 

an, bn  are coefficients.
ao - no variable attached, so the coefficient is the term itself or the coefficient to 1(one).
In this case (n=0), ƒ(x) = ao /2 (summation part starts with n =1). But why <1/2> ? (1/2 is a coefficient of ao but fixed (constant).


an, bn - multiplicative factors. cos(nx) is multiplied by an and sin(nx) is multiplied by bn.

In another word, an is a scale (or magnitude) factor of cos(nx) and bn is a scale factor (or magnitude) of sin (nx).

n of nx is a multiplicative factor. x (variable) is multiplied by n. The point is n is an integer (. . . . -4, -3, -2, -1, 0, 1, 2, 3, 4 . . . .)

As cos(x) and sin(x) are period functions so n is considered as a frequency multiplicative factor.

<x> here is very general. More explicitly

As cos($$\frac{2\mathrm{\pi fn}}{}$$x) and sin($$\frac{2\mathrm{\pi fn}}{}$$x)  or 

$$\cos ({\omega }_{0}nt)\sin ({\omega }_0)$$

$$are\; period\; functions.$$$$\frac{2\mathrm{\pi fn\; and}}{}$$$${\omega }_{0}n\; can\; be\; regarded\; as\; as\; coefficient\; of\; variable\; <x>,\; <t>.$$

Also <n> (integer)  is considered as a angular frequency (multiplicative) factor of $$\frac{2\mathrm{\pi ,}}{}$$$${\omega }_0$$


Please note $${\mathrm{\omega \; =}}^{}$$$${\mathrm{2\pi f\; and}}^{}$$

$${\mathrm{f\; =\; frequency:\; how\; many\; times\; (number)\; /\; second.}}^{}$$

$${\mathrm{t\; =\; time:\; how\; long\; or\; how\; many\; (number)\; seconds}}^{}$$

$${\mathrm{\omega 0nt\; =}}^{}$$$${\mathrm{2\pi f\; x\; n\; x\; t}}^{}$$

$${\mathrm{2\pi \; x}}^{}$$$${\mathrm{<how\; many\; times\; (number)\; /\; second>}}^{}$$$${\mathrm{x \; n \; x}}^{}$$ <$${\mathrm{how\; many\; (number)\; seconds>}}^{}$$

$${\mathrm{therefore}}^{}$$

$${\mathrm{The\; unit\; second\; disappearedand\; the\; only\; numbers\; remain.}}^{}$$

$${=}^{}$$$${\mathrm{2\pi \; x<how\; many\; times\; (number\; (1))}}^{}$$$${\mathrm{>x \; n \; x}}^{}$$ <$${\mathrm{how\; many\; (number\; (2))>}}^{}$$


$${\mathrm{This\; may\; be\; the\; key\; to\; Fourier\; Transform.\; Time\; (t)\; and\; Frequency\; (f,}}^{}$$$${\mathrm{Frequency.}}^{}$$

Back to coefficient. Coefficient is

1) scale (or magnitude) factor
2) multiplicative factor

Some other writings use

3) weighing  (factor)

4) magnitude - this is used when explaining Fourier series, amplitude-phase form

5) then, amplitude (more engineering)


But Fourier coefficients are very long like above and have functions (the original periodic function f(x) and trigonometric functions (which are are typical periodic functions) and these two are multiplied first then integrated.
The last two 4) magnitude and 5) amplitude - we will return to this later when we see the exponential form of Fourier Series. Also this form shows trigonometric function (featuring Magnitude and Phase Difference, which is the principal thing (concept) to understand Fourier Series.
Coefficient itself is not difficult to understand as above - no explanation is required. But the concept is important and people use this concept unconsciously or consciously (good practice for your brain for analysis and synthesis). Actually Fourier Series itself is the work of analysis and synthesis.


Another form of  Fourier Series.

(wiki - recent)

Fourier series, amplitude-phase form


An is a coefficient.

This form shows trigonometric function (featuring Amplitude (Magnitude) and Phase Difference, which is the principal thing (concept) to understand Fourier Series.
Coefficient itself is not difficult to understand as above - no explanation is required. But the concept is important and people use this concept unconsciously or consciously (good practice for your brain for analysis and synthesis). Actually Fourier Series itself is the work of analysis and synthesis.


-------

Basic and useful things to know - continued

Basic property of sine and cosine functions

From:
https://www.math.purdue.edu/academic/files/courses/2014fall/MA16021/FourierSeries%28nopauses%29.pdf

Intuitively, periodic functions have repetitive behavior. A periodic function can be defined on an interval, then copied and pasted so that it repeats itself.

Examples

sin x and cos x are periodic with period 2π.

If L is a fixed number, then sin (2πx/L ) and cos(2πx/L ) have period L. Sine and cosine are the most basic periodic functions!

sin x (odd) cos x (even)

The product of two odd functions is even: x sin x is even 

The product of two even functions is even: cos x²  cos x is even

The product of an even function and an odd function is odd: sin x cos x is odd

Multiplication must follow the trigonometric rules.

To find a Fourier series, it is sufficient to calculate the integrals that give the coefficients an, and bn and plug them in to the big series formula, equation (2.1) above.

Typically, f(x) will be piecewise defined.

Big advantage that Fourier series have over Taylor series: the function f(x) can have discontinuities!
Useful identities for Fourier series: as n is an integer, then

sin(nπ) = 0

e.g. sin(1π ) = sin(2π) = sin(3π) = sin(20π) = 0

                      n
cos(nπ ) = (-1)   =

1 when n is even
-1 when n is odd

e.g.

cos(0π) = cos(2π) = cos(3π) = 1;
cos(1π) = cos(3π) = cos(5π) = -1

These are not so difficult to understand either. But some more Trigonometric formulae (including transformation formulae) are required, which will be shown when necessary.


------

Basic and useful things to know - further continued

The next step is then what Fourier coefficients are.

If you a novice in math like me, understanding step by spec (from one step to the next step) may be a good way to proceed.

Now we are familiar with the following equations:

Fourier series of ƒ(x) =

and the coefficients are given as 

and

But what do these Fourier coefficients mean? The formulae are an integral of the original function  f(t) which multiplied by a sinusoidal function cos(nx) or sin(nx).
To understand Fourier coefficients an and bn or what an and bn are as well as where an and bn come from we need some more basic knowledge of trigonometry including trigonometric integration of

f(x) sin(nx) dx

<x> is variable and can be abstract - just number but continuous with no unit, or you can think it location on straight line or horizontal axis. If you use more practically time (t)

f(t) sin(nt) dt

Some < Integration Rules> which will be used (or may not be used)

Multiplication by constant. c - constant

∫cf(x) dx    =  c∫f(x) dx

Trigonometry (x in radians)

∫cos(x) dx   =   sin(x) + C

∫sin(x) dx   =   -cos(x) + C

Integration by parts

∫u v dx = u∫v dx − ∫u' (∫v dx) dx

• u is the function u(x)
• v is the function v(x)
• u' is the derivative of the function u(x)

List of integrals of trigonometric functions (wiki)

Here x is not a function f(x) but a variable. However these are suggestive.

-----

The point or a difficult or strange part is that the coefficient equation contains the original function f(x), ( which we are required to find ?).

To find f(x) we need to know an and bn but an and bn equations have f(x) (in integral) plus f(x) can be regarded as coefficient, or coefficient function, of trigonometric functions. Very complicated or we seem to have been put into a magic box, but this seems absurd, too. (Is it absurd because if we find an or bn then we find f(x) before calculating it by summation.



? )

However, again as far as I checked samples of  f(x) shown in Introductory articles sample functions are rather simple periodical functions with their visually seen graphs. So f(x) is almost already known. Though not so straightforward and a bit difficult but it is possible to calculate an and bn by using a computer sin, cos calculator, hoping meaningfully.


But,  <It is absurd because if we find an or bn then we find f(x) before calculating it by summation.>
is an absurd statement. f(x) is known in time domain. We try to find an equivalent but in a Series form having cosine and sine functions which have <time(t)> and cycle (frequency) variable as a part. This is a kind of transform rather than equation. Again please do not forget f(x) is a periodic function and cosine and sine functions are periodic as well.

How to get them. Some a bit forward introduction writings show Derivation of an and bn equation.

For instance

1) https://lpsa.swarthmore.edu/Fourier/Series/DerFS.html#Finda_n  (this is a very good article)

How do we find a_n?

I cannot copy and past here so please go to the above site.

It says,

"
Without justification we multiply both sides by $\cos \left(m{\omega }_{0}t\right)$
$"\; some\; re-arrangement\; of\; the\; equation\; and\; then\; use$
$"$$the\; trig(onometric)\; identity\; cos(a)cos(b)=½(cos(a+b)+cos(a-b))$

$"$
This is the point as this is a transformation from Product form to Summation form.

and then use the property of integration of sine and cosine of nπ.

Another one

2) https://planetmath.org/derivationoffouriercoefficients1 (this is a very good article, too)

Derivation of Fourier Coefficients

A very similar way to the above (actually the same thing)


"
Now, in order to find $a_k$, we multiply both sides of (2) by $\cos \left({\omega }_{k}t\right)$ and we arrive at
"

and then


"
By using orthogonality relationships or by literally evaluating the above integrals, we get the following
"

<Orthogonality relationships> roughly means the property of integration of sine and cosine of nπ.

Please refer to the above articles or some other <Derivation of Fourier Coefficients>, which is not a waste of time or rather crucial to get one step further understanding of Fourier Series, Fourier Transform, Fourier Analysis.

-------

Noe as I indicated we will check <Fourier series, amplitude-phase form>

(wiki)

Fourier series, amplitude-phase form




This is also given but we can now more guessing.  An is a coefficient (mutiplicative factor, magnitude, amplitude).

2πnx
------
   P

is now familiar. P is Period (time for one cycle, usually). 1/P is frequency - how many cycles per unit time (usually second). Calculation wise

                         Number of cycles
Frequency =    ----------------------   
                          one (1 ) sec


Reciprocal

                                                                       
        1                              1                           one (1) sec               
--------------  =    ---------------------   =   -----------------------   =   
Frequency          Number of cycles         Number of cycles        
                          ----------------------
                           one (1) sec


  Number
-----------------    x  sec   =   P (Period), usually, (1/100 )x sec = 0.01 sec, (1/1,000) sec = 0.001 sec, etc
One (1) Cycle


           


ACT