Fourier Cosine Transforms • Examples on the Use of Some Operational Rules Processing • Image Compression by the Discrete Local Sine Transform (DLS)
equation, it can be very helpful to use a finite Fourier transform In particular, we it is best to use the finite cosine transform Examples of the Sine Transform
Sn Sin( nπ L x) Fourier cosine transform Cn = 2 L ∫ L 0
theorem - Parseval's identity - Finite Fourier sine and cosine transform Unit III FOURIER Properties of Fourier Sine and Cosine Transforms 1 Linear Property
associated with second order partial differential equations on the semi-infinite inter- val x > 0 Because property 11 43d for the Fourier sine transform utilizes the
The examples of a kernel are i) When k (s, x) INFINITE FOURIER TRANSFORMS Consider, FOURIER COSINE AND SINE TRANSFORMS 1) We define F
24 nov 2014 · To introduce the sine and cosine transforms and use them to solve an infinite- diffusion problem To define the Fourier and inverse Fourier
are called the Inverse Fourier cosine inverse Fourier sine transform of Fc(s) Let X Y be two discrete r v , f(xi, yi) is the joint p d f of X and Y then (X + Y) is
FOURIER TRANSFORMATIONS1 A W JACOBSON 1 Introduction, The finite sine transformation and the finite cosine transformation of F(x) with respect to x
The function has a finite number of maxima and minima 3 has only a finite Example 4 Show that Fourier sine and cosine transforms of are and respectively
A periodic function of time f(t) of period T can be represented by a Fourier transform F(n) = 1 T ZT/2 ?T/2 f(t)e?jn?0tdt f(t) = X? n=?? F(n)e+jn?0t where F(n) is the corresponding Fourier series where F(n) =1 2 (Cn? iSn) F(?n) =1 2 (Cn+ iSn) and F(0) = C0
The Finite Fourier Transforms When solving a PDE on a nite interval 0
The Finite Fourier Transform and the Fast Fourier Transform Algorithm 1 Introduction: Fourier Series Early in the Nineteenth Century Fourier in studying sound and oscillatory motion conceived of the idea of representing periodic functions by their coefficients in an expansion as a sum of sines and cosines rather than their values
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Page 1
SATHYABAMA UNIVERSITY UNIT III SMT1201 ENGINEERING MATHEMATICS III The infinite Fourier transform - Sine and Cosine transform - Properties - Inversion theorem - Convolution theorem - Parseval's identity - Finite Fourier sine and cosine transform.
FOURIER TRANSFORMS
Complex Fourier Transform (Infinite)
-LULv-wise continuous in each finite partial interval then the complex Fourier transform of f(x) is defined by
à L(:O; L5 Inverse Fourier Transform
B:T; L5 Properties of Fourier Transform
. Linear Property F{af(x) + bg(x)} = a F{f(x)} + b F{g(x)} where F is the Fourier transform 2. Shifting Theorem
If à L(:O;PDAJà F=;= LAÜaeÔ(:O;
3. Change of Scale property
à L(:O;PDAJà L5 Ô(@ae
ÔASDANA=
Mr 4. Modulation Theorem
à L(:O;PDAJà L5 6>(:O F=; E(:O E=;? Convolution Theorem
The convolution of two functions f(x) and g(x) is defined as B:T;ÛC:T;
Ls ¾tè
±B:P;C:T
FP;@P Fourier Transform Page 2SATHYABAMA UNIVERSITY UNIT III SMT1201 ENGINEERING MATHEMATICS III Fourier Transform of Convolution of two functions
Fourier Transform of Convolution of f(x) and g(x) is the product of their Fourier Transform
B:T;ÛC:T;
Infinite Fourier Cosine Transform
àÖ L §6 4 Inverse Fourier Cosine Transform
B:T; L §6 4 Infinite Fourier Sine Transform
àae L §6 4 Inverse Fourier Sine Transform
B:T; L §6 4 Properties of Fourier Sine and Cosine Transforms
1 Linear Property
àÖ<=B:T;
G>C:T;=
L=àÖ G>àÖ àae<=B:T;
G>C:T;=
L=àae G>àae Page 3
SATHYABAMA UNIVERSITY UNIT III SMT1201 ENGINEERING MATHEMATICS III WOEsÀo[ Identity
ìB:T;
2 Modulation Property
5 Identities
If FC (s) and Gc(s) are the Fourier cosine transforms and Fs(s) and Gs(s) are the transforms of f(x) and g(x) respectively then i) ii) iii) 4 4 4 Page 4SATHYABAMA UNIVERSITY UNIT III SMT1201 ENGINEERING MATHEMATICS III
Problem1Ift heFour iertransformoff xisF sthen,wha tisFouri ertr ansformof f ax? Solution:
Fouriertransf ormoff xis
is1 2 xF sF fx fx edx f is1 2 xF fax fax edx f Putt ax
dta dx is/ 1 2 t adtF fax ft ea f is/ 1 1.2 t af te dta f 1. .sF fax Fa a
Problem2Findthe Fouri ersinetransformof 3xe.
Solution:
0 2sinsF fx fx sxdx
3 3 0 2sinx x
sF ee sxdx 3 2 0 23sincos9
xesxs xs @2 22 2sinsi ncos9 ax axs ee bxdx abx bbx s ab Problem3Findthe Fourie rsinetransformofaxf xe ,0a. Hencededuce that 2 0 sin 1 2 x xdx ex Solution:
0 2sinsF fx fx sxdx
0 2sinaxa x
sF ee sxdx 2 22s s a Byi nverseSinetra nsform,we get
0 2sinsf xF ssx ds
2 2 0 2 2sinssx dss a
PROBLEMSPage 5SATHYABAMA UNIVERSITY UNIT III SMT1201 ENGINEERING MATHEMATICS III 2 2 0 2 sins sxf x dss a
2 2 0 sin 2 s sxf x dss a 2 2 0 sin 2 axs sxe dss a Put1,ax
2 0 sin 2 1 s sxe dss Replace 's' by 'x'
2 0 sin.1 2 s sxdx ex Problem4Prove that1cos2C C CF f x ax F s a F s a.
Solution:
0 2cosc cF s F f x f x sx dx
0 2cos cos coscF f x ax f x ax sxdx
0 cos cos2 2 a s x a s xf x dx 0 0 1 2 1 2cos cos2 2f x s a xdx f x s a xdx
1.2c cF s a F s a
Problem5Find the Fourier cosine transform ofcos , 0 0, x x af xx a Solution:
0 0 2 2cos cos cos
a cF f x f x sxdx x sxdx 0 cos 1 cos 12 2 as x s xdx 0 sin 1 sin 11 1 12 as x s x s s sin 1 sin 11 1 12 s a s a s s , provided1, 1.S S Problem6Findax
CF xeandax
SF xe.
Solution:
ax c sdF xe F f xds ax ax c sdF xe F eds 0 2sinaxde sxdxds
Page 6SATHYABAMA UNIVERSITY UNIT III SMT1201 ENGINEERING MATHEMATICS III 2 2 22 22 2
2 2.d s a s
ds s as a ax ax s c s cd dF xe F e F xf x F f xds ds 0 2cosaxde sxdxds
22 22 2
2 2 2.d a as
ds s as a Problem7IfF sis the Fourier transform off x, then prove that the Fourier transform of axe f xisF s a. Solution:
is1 2 xF s F f x f x e dx is1 2 iax iax xF e f x e f x e dx 1 2 i a s xe f x dx .F s a Problem8Find the Fourier cosine transform of23x xe e. Solution:
Let23x xf x e e
0 2coscF f x f x sx dx
2 2 0 0 23 cos 3 cosx x x x
cF e e e sxdx e sxdx 2 22 2 3.4 1s s
Problem9State convolution theorem.
Solution:
If F(s) and G(s) are Fourier transform off xandg xrespectively, Then the Fourier transform of the convolutions ofandf x g xis the product of their Fourier transforms. i.e.*F f x g x F f x F g x Problem 10Derive the relation between Fourier transform and Laplace transform. Solution:
Consider, 010 , 0
xte g t tf tt The Fourier transform off xis given by
is1 2 tF f t f t e dt is1 2 xt te g t e dt Page 7SATHYABAMA UNIVERSITY UNIT III SMT1201 ENGINEERING MATHEMATICS III is1 2 x te g t dt 1 2 pte g t dt wherep xis 0 1 2 stL g t L f t e f t dt Fourier transform of1
2f tLaplace transform of g(t) where g(t) is defined by (1).
Problem 11Find the Fourier sine transform of1
x. Solution:
0 2sinsF f x f x sx dx
0 1 2 1sinsF sx dxx x
Letsxquotesdbs_dbs11.pdfusesText_17
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