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184 views19 pagesUnit 6 RK Jain
Uploaded by
Basanti DasCopyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
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/ 19
Chapter 9
fourier Series, Fourier Integrals and
Fourier Transforms
al Introduction
sachapter7 (section 7.2.4), we have defined the orthogonal and orthonormal functions. For example,
te tancions cos mx and sin mx are orthogonal over the interval [-1, zt]. The orthonormal set of
jomtions corresponding t0 cos mx are given by (Example 7.7)
_1_ cosx cos
pn" ve
‘Animportant application discussed in section 7.2.4 was the series expansion of (suitable) functions
intems of a complete set of orthogonal or orthonormal functions. The expansion of a continuous
fission f(x), having continuous derivatives over the interval [-1, 1] in terms of Legendre poly-
nomials was discussed in section 7.2.6. This series was called Fourier-Legendre series. The Fourier-
Bessel series was discussed in section 7.4.2. A series expansion in terms of the trigonometric functions
cosmeand sin mx is called a Fourier series. Many functions including some discontinuous periodic
functions can be expanded in a Fourier series. Therefore, Fourier series solution method is a powerful
tol in solving some ordinary and partial differential equations.
cos mx
92 Fourier Series
Let (490), 6) (2), (a). - .} be an orthogonal set of functions, orthogonal with respect 10 a weight
nection W(x), on an interval a, b}. Let f (x) be a continuous function defined on the same interval
(0.5, Then, f(x) can be expanded in an infinite series of the form (sce section 7.2.4)
F(X) = codo(x) + erPils) + 2G20) #--- 9.1)
The coefficients cj, i= 0, 1, 2, «are given by
= (f Fo) WO8: cons| / oe(a)) IPs F= 0,1, 20--- a 2 Engineering Mathe
* ;
f wong? Cds
where He corr
of orthogonal finetions
Bray )
Consider now, the set
ie
{i
o(4¢) sn (2) | (2)
+ weight f .
hich thogonat on the interval [= fF] wills respect 0 the weight Funetion WO) 1 yy
Which are orthogonal on the interval [=
functions have the following propertics
: . q
i vos (Fas =f sin( 4)
Jai 1
" dx = 0, men,
[fom (282) oo (SF) ) 0
at
ees ams) dy = 0, for all mand 1 (96)
Er t
v mtx
J, cos? (2) a= f, me aa = 2)
where m and n are integers.
Now, let f(x) be a periodie function of period 2/ defined on [= 1, 1], that is ((x + 21) = f(x) and assume
that it can be expanded in an orthogonal series in terms of the trigonometric functions. We shall
discuss later in this section, the conditions under which such an expansion is possible. Let the series
be written as
FI 7
[a os (4) +a, cvs (24) te | + [* sin (#) + by sin (4) + |
+E [+ cos (4) +b, sin (*)} (08)
ie coefficients ay, ay, a3, .. «bi, by... can be determined by using the orthogonal properties of
the trigonometric functions gi i
interval (=, ], we obtain EA O-4) t0 (9.7). Integrating Eq, (9.8) term by term on te
t te
if fordr=2 [deg Sq [' (nm. y
Ei pe n+ Z| ay [, cos (Blars by j sin (SE) = Ido
ot
since cos (nzx/)
and sin
(axl) are orthogonal with Fespect to W(x) = 1, on [= 1, {J Therefor"
7
a
i f S(x)dx,
Now, multiply both si .
i
Len TW ett iS OF Ea, (08) by cog (mmx/I) and int tei"
rn legrate term by term on the rourter Series, Fourier Imegrals and Fourier Transforms 9.3
ay f! max . :
=a | cs| —Jdx+ Dla, cos ( M7 wee
i 7 wat [Gn J S08 | TJ cos =
y mmx
by [cos an ( 22E
f, (= ) sin (2 ax).
je ontogonal properties given in Eqs. (9.4) to (9.7), we get
g the
wi ;
f Feoyeos (Jax ata
EF 7
an =4 j F(x) cos (M2) 4
set TJ dx.
iging bot sides of Eq, (.8) by sin (mi) and integrating term by term on the interval
; 1) we obtain
y = 1
{nom (™E)e= $f 130 SFE) te [om fan SFE) oo (SE) ex
bp [sin 222) sin( ax
1 t T r
tng he othogonal properties given in Eqs. (9.4) to (9.7), we get
1
j f(x) sin (AE ax = [Bp
-1
1
Terefore, bn = + j f(x) sin (7 as.
a1
ltcan be observed that the expressions for ay and aq, can be combined as a single expression. It is to
obtin this simplicity of notation that ao/2 is used in Eq. (9.8). This does not mean that the value of
can be obtained after evaluating a, and setting n = 0 in this expression.
The orthogonal series for f (x) given in Eq. (9.8) is called the Fourcer series. The coefficients ao,
4g, are called the Fourier coefficients on (~ 1, 1]. The expressions for the coefficients
1
f fix)dx, (9.9)
1
a,=4 f F(x) 0s (=) dx, (9.10)
ot
1
Da f fo sin( "ax (9.1)
x
“Called the Buler formulas.
pli cod of the function is 27, that is f (x) is defined on (-7, 7]. then the Euler formulas are
led as net "pads ©.)
wa),
; 1" f(a) cos nx de. ny
oneal,
elf" poo sinardx. ny
bn =
i is continuous or piecewise ¢
tegrals, we have that iff (2) is ONtinuoy,
finitic efinite integrals,
From the definition of definit 4
on the integrals given in Eqs. (9.9),
vara finite number of finite jumps) ten the integ 1015.14
(continuous except fo e ea
exist and f (x) can bé expanded as a Fourier Serie sie tuncon
i - i eriodic ful
Example 9.1 Find the Fourier series expansion of the ee
, cos (nil) are even functions on (~ I, 1], since
FOR = 0" = (xP = = f(y),
F(X) = 008 (~ natxIN) = cos (nztx/l) = f (2).
-f£@), -Isxs
xsl 9.16)
equ v
yy. sin (nat/l) are odd functions on (— 1), since
FRE CaP = CAP! eet oP epee)
fx) = sin (— naxil) = = sin (neil) =~ F(x).
gapis of even functions | x |, x? and a typical cosine like function are given in Figs. 9.3. 2, b.c.
Ll x 4018)
0 x oO}
(@) (b) ©
Fig, 9.3, Graphs of even functions.
Guaphs of odd functions x, x° and a typical sine like function are given in Figs. 9.4. a, b, ¢
|.
- iO * / 0 x
a f@)
@ ) =
Fig. 9.4. Graphs of odd functions.
tra
"/0) isan even function on [- I, I], then we have
f
J jonas=2 | finds. on
1 0 9.8 Engineering Mathematics
Le} then we have
IE F(x) is an ok function on [-
v= 0. /
fod on
om the definition
7 x results can be easily proved from U
a: function) (even function) = even function
(even functio' 5
cen function) (odd function) = odd function
(even fune
(odd funetion) (odd function) = even function
od
finition, we have that
Therefore, from the definition, we have t
si II) is odd,
if f(x) is even then f (x) cos (nzx/l) is even and f (x) sin (aztx/l) is 0
it 1) is even,
if f (x) is odd then f(x) cos (nz-x/l) is odd and f (x) sin (n7tx 2 ’
Hence, if f(x) is an even function on [- J, J. then we have the following Fourier series
ey (2
O05) aHeos (oa
F(a) = B+ F ay 08 | oy
nx
where 23f f(x)dx and a, = 2f Fea) eos (7 Jax. oxo
The Fourier series of an odd function on the interval (- /, 1] is given by
F(x) = & by sin way
where Ff F(x) sin (a ax. 02)
The series given in Eq. (9.19) is called the Fourier cosine series and the series given in Eq, (9.21)is
called the Fourier sine series.
Consider Example 9.1. The function f (x)
series.
The function f (x) = x, -1<$x <1 is an even function and we obtain a cosine series.
2+x, -2 +26 1)" ay
at both the end points, x = / and ~ /.
Remark 2
Let the sum upto j terms of the Fourier series be denoted by
ME) +0, sin") 212300.
that is, the approximations Tepresent
Then, partial sums S; give successive approximations to f (x),
F (2) closer and closer as j increases.
Example 9.5 Using the results of Examples 9.2, 9.3 and 9.4 prove the following
i) eed 7
O rtatat reget
Ptyts Gi) Seat
10s 2263 2h
ii) (ee, 1
(iii) 1 zo
Solution In Example 9.2, the Fourier series expansion of
fa)= {a “M (1-x), OKIS%
16. fayelebo bers
i 18. f(x) = sinh (x), — 1 <8 <1.
17. fw =l+lah =x, -2 i) cos nn.
the complex form of the Fourier series ig
clots
her .
y= Sinha 1) ns
PO IEE Be (Lott),
ssyto convert the complex exponental form of the Fourier series to a real trigonometric form,
ise
Wehave
0 20) ty = oy + Cm bn =i (eye),
ana ateas of engineering, the complex form of Fourier series is used. Since f (x) is defined
eb Nand fx+ 20) =f (3), the Sundamental period Tot J (2) is T= 21. We define @ = 2n/T =
vias the Jundamental angular frequency, In erms of «, we can write the ‘ourier series as
L)= Bs Fla, cos (nx) + by sin (nox) (9.31)
2" ast I}
y F(x) = Z cyeinor, (9.32)
Tepotof the points (1, | cy |) where «ois the fund:
lamer
ceffcients defined in Eq. (
9-30), is called the frequenc
Sumple99 Find the frequency spectrum of the pet
ntal angular frequency and cy
are the Fourier
-Y'spectrum of f (x).
odie pulse defined by
vl, -lsx