[PDF] The Finite Fourier Transforms
Sn sin(n?x/L) This transform should be used with Dirichlet boundary conditions that specify the value of u at x = 0 and x = L
[PDF] The infinite Fourier transform - Convolution theorem - Parsevals
The infinite Fourier transform - Sine and Cosine transform - Properties - Inversion theorem - Convolution theorem - Parseval's identity - Finite Fourier
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[PDF] Finite Fourier Transform - Unit III Part IV (Integral Transforms)
The finite Fourier sine transform of F (x)0 < x < l is defined as fs (p) = l ? 0 F (x)sin p?x l dx;p ? I Similarly the finite Fourier cosine
[PDF] Chapter 3 - Sine and Cosine Transforms
Transforms with cosine and sine functions as the transform kernels represent an important area of analysis It is based on the so-called half-range
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transform Fourier transform of derivatives convolution The function has finite number of discontinuities in Fourier Sine and Cosine transform
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as the inverse Fourier cosine transform ii) We define F s (u) = 0 t ? = ? f(t) sin ut dt as the Fourier sine transform of f(t)
[PDF] fourier transforms and their applications
When Kernel is sine or cosine or Bessel's function the transformation is called Fourier sine or which is called inverse finite Fourier sine transform
[PDF] transforms and their applications semester – v academic year 2020
FOURIER SINE AND COSINE TRANSFORMS 16 III FINITE FOURIER TRANSFORMS 31 IV Z - TRANSFORM 42 V INVERSE Z TRANSFORMS
[PDF] FOURIER TRANSFORMS
The function has a finite number of maxima and minima Example 4 Show that Fourier sine and cosine transforms of are and respectively
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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 - USM
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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
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 byà 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
Inverse Fourier Transform
B:T; L5Properties of Fourier Transform
. Linear Property F{af(x) + bg(x)} = a F{f(x)} + b F{g(x)} where F is the Fourier transform2. 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
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
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
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
Ô(@ae
ÔASDANA=
Mr4. 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
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
Convolution Theorem
The convolution of two functions f(x) and g(x) is defined asB:T;ÛC:T;
Ls¾tè
±B:P;C:T
FP;@P Fourier Transform Page 2SATHYABAMA UNIVERSITY UNIT III SMT1201 ENGINEERING MATHEMATICS IIIFourier Transform of Convolution of two functions
Fourier Transform of Convolution of f(x) and g(x) is the product of their FourierTransform
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
Inverse Fourier Cosine Transform
B:T; L §6 4Infinite 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
Inverse Fourier Sine Transform
B:T; L §6 4Properties 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
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
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
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
Page 3
SATHYABAMA UNIVERSITY UNIT III SMT1201 ENGINEERING MATHEMATICS IIIWOEsÀo[ 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 44 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 fPutt ax
dta dx is/ 1 2 t adtF fax ft ea f is/ 1 1.2 t af te dta f1. .sF fax Fa a
Problem2Findthe Fouri ersinetransformof 3xe.
Solution:
02sinsF fx fx sxdx
3 3 02sinx x
sF ee sxdx 3 2 023sincos9
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 exSolution:
02sinsF fx fx sxdx
02sinaxa x
sF ee sxdx 2 22s s aByi nverseSinetra nsform,we get
02sinsf xF ssx ds
2 2 02 2sinssx dss a
PROBLEMSPage 5SATHYABAMA UNIVERSITY UNIT III SMT1201 ENGINEERING MATHEMATICS III 2 2 02 sins sxf x dss a
2 2 0 sin 2 s sxf x dss a 2 2 0 sin 2 axs sxe dss aPut1,ax
2 0 sin 2 1 s sxe dssReplace 's' by 'x'
2 0 sin.1 2 s sxdx exProblem4Prove that1cos2C C CF f x ax F s a F s a.
Solution:
02cosc cF s F f x f x sx dx
02cos cos coscF f x ax f x ax sxdx
0 cos cos2 2 a s x a s xf x dx 0 01 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 aSolution:
0 02 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 SProblem6Findax
CF xeandax
SF xe.
Solution:
ax c sdF xe F f xds ax ax c sdF xe F eds 02sinaxde sxdxds
Page 6SATHYABAMA UNIVERSITY UNIT III SMT1201 ENGINEERING MATHEMATICS III 2 222 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 02cosaxde 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
02coscF f x f x sx dx
2 2 0 023 cos 3 cosx x x x
cF e e e sxdx e sxdx2 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 ttThe 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 dtFourier transform of1
2f tLaplace transform of g(t) where g(t) is defined by (1).
Problem 11Find the Fourier sine transform of1
x.Solution:
02sinsF f x f x sx dx
01 2 1sinsF sx dxx x
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