engineering math 4,ch 10, fourier analysis

7/31/2019 ENGINEERING Math 4,Ch 10, Fourier Analysis

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(ENGINEERING MATHEMATICS 4, CH-10, SEC-

10.1,10.2,10.3,10.4,10.5,10.9,10.10, & 10.11)

B I O - M E D I C A L - E N G I N E E R I N G - G U I D E B Y M O H - S I K A N D A R - L O D H I Page 1-OF-Monday, August 1

2012

2010

Bio-medical-engineering

mathematics

By Mohammad Sikandar-khan-Lodhi

[FOURIER-ANALYSIS: CH-10]


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FOURIER-SERIES:(10.1)

PERIODIC-FUNCTIONS:Any function f(x) have some positive no (p) and its defined for all

real (x) .

[ * +, - ,

-.]

fimilar periodic function are sine & cosine functions.

(f=c=constant) => its also a periodic-function, because its

satisfied equation (A) for all (p).

GRAPH:

those function whose are not periodic are followed

Point

(a)

Point (b)

There is any

interval of

length (p) b/w

point (a) & (b)

f(x)

x-axes


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(i-e)

[ () .]

TRIGONOMETRIC-SYSTEM:Our problem in the first few sections of this chapter will be the

representation ofvarious functions of period ( ) in termsof the simple functions.

(i-e)


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And so on

Above figure show the cosine and sine function having the

period ( ) From (0 2[pi]).TRIGONOMETRIC-SERIES:There is a series which aries by above equation (D)

(i-e)

Where , a0,a1,a2,,an.; & b0,b1,b2,,bn .; all are real constant this

series is called as trigonometric-series we may also write

above series which was given in equation (E) as ,


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Also the an & bnare Coefficient of series where the set of

function in eq (D) above is called as Trigonometric-System.

Key point we see that each terms of series in equation (E)

above has the period ( 2) hence if the series of eq(E) convergesthen its sum will be a function of period (2);Key-point--> trigonometric series in equation (E) is also called

fourier-series of f .

_____________

(10.2) start here

FOURIER-SERIES:STATEMENTS;

The fourier series aries from the partical-task for representing a

periodic function [f(x)=f(x+p)] in terms of Cosine and Sine function.

The Fourier-series are often also called as trigonometric-

series(Eq-[E] in section 10.1), whose coefficient are find from f(x) by

using certain formula Euler-Formula (below).

Now, we first drive the Euler formulas which are most use full for

finding the coefficient of fourier or trigonometric series.


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EULER-FORMULAS-FOR-FOURIER-COEFFICIENTS:Let, f(x) is a periodic function of period (2), which can berepresented by trigonometric-series(or Fourier-Series).

(i-e)

We assume that this series converges & has f(x) as its Sum.

Key we want to find the coefficients *a0, an & bn of the above

fourier-series in equation (E).

Where * n=1,2,3,4,..,inifinity+.

FOR-a0 : (Eular-formula for a0):

Let , integrated on both sides in eq(E) from * ] we get,


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(above)(Eq-CE) Its the equation of a0(the coefficient of fourier-

series in equation[E]).

FOR an=m: FOR am :When (m=n),

[m= any fixed +ve No]

_________________________________


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Formulas:

__________________________________

We multiply eq(E) by cos(mx) & then integrated from ( ),where [m=any fixed +ve No].


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CONSIDER :


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Now placing the value of eq(Zi) & eq (Zii) in eq(G1)then

eq(G1)becomes;


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When (n=m):

FOR bn=m OR bm :


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*m=n=1,2,3,,+

_____________

Rough-work:

When (n=m):


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__________________

LIST OF EULAR-FORMULA OF FOURIER COEFFICIENT:

Where:

[ , -.]Where (an,bn & a0 ) are real-Integer-No,also these number(an,bn

& a0 ) are called fourier-Coefficients of f(x).


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And above eq(E) is called as fourier series or trigonometrical-

series.

_________________(FINISHED-HERE)__________________

EXAMPLE#1) SQUARE-WAVE:

Find the fourier coefficients of periodic-function f(x) in below

function which is given in eq (A) and sketch the graph of f(x)?

Solution:

REQUIRED:

A). Sketch the graph of f(x) of eq(A)=?;

B). find fourier coefficient of f(x) .A). Sketch the graph of f(x) in eq(A):

Graph :


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B). find fourier coefficient of f(x) .1)FOR a0 :By using Eular formula:

2

-2

+k

(amplitude)

-k

(Amplitude)

x-axis


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___________

Rough work:


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______________

FOR bn:By using Eular formula:

___________

Rough work:


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CONSIDER ():In general ()

FOR FOURIER SERIES-OF-f(x):


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FORMULA:

FOR PARTIAL SUM OF ABOVE FOURIER SERIES ARE:

GRAPHICAL-REPRESENTATION-OF-ABOVE-FOURIER-SERIES INEQUATION (V):


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k

-k

-

x

(2)=P=(-

For

./

k

-k

-

x

(2)=P=(-

For k

-k

-

x

(2)=P=(-

For


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Key these above figure show that fourier series is convergent

and has sum f(x) we notice that partial-sum cause to be

converge and has the sum f(x).

f(x)

k

f(x)


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In above example we noted that at (x=0) & (x=) & (x=)the f(x) is discontinutes & the value of partial sum is zero, this

means that the series of partial-sum is convergent and has the

sum is f(x) [as represented in above graph].

Let, f(x) is the sum of series then Assume we set ( ),

(10.3) Start-Here.

FUNCTION-OF-ANY-PERIOD[ ]:The function considered so far had period ( ) ,but in mostapplication the most periodic function will have other periods

rather than period( ),We can converted those function whose have ( ) in

the transition from functions of ( ) basically astretch of scale on the axis;

The periodic function f(x) of period (p=2L) has a fourierseries.


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With the fourier-coefficients of f(x) given by the euler-formulas.

That is,

[]=V

[]=V

-VV

V

f(x) (v) instead of(x);

[ ,-]


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PROOF (DERIVATION):Now, we want to drive the formula of

x=-L

x=L

x

V

f(x) is a

function of

(v) so, f(x) =

g(v)

* +

[]

[-]

P=

2L

x=-LP=

2L

x=L

x

f(x)


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Solution:

We want to set the scale, we set ,

Thus, regarded as a function of (v) which we call g(v);

Key-point) we derive this above formulas of f(x) , ao , an , & bn

from the section (10.2) formulas of f(x)1 , ao1 , an1 , & bn1 ;

Formula of section (10.2);

(i-e)


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FOR g(v):We know that, [:. g(v) = f(x)1 ].

So,

So, by replacing (v) insteated of (x) , so f(x)1 becomes as,


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STEP2:o FORf(x):

Where: [ g(v)=f(x) ]

Placing the above value in eq(G), then above eq (G) becomes,

FOR aO , an & bn :as we know that,

[ 0 1 0 1 * + ]Consider above eq (a) , (b) & (c) :


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[ 0 1 ,-

,- ,

01 ,-,- , ./ ,-

; ]


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____________________________________________________


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90

270

360=2

pi

1

-1

0

Cos(2x)


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engineering math 4,ch 10, fourier analysis

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