Lusari Ask Bibliou pana

7/24/2019 Lusari Ask Bibliou pana

1/44

Ver.


2/44

t

f(t) = etu(t)

12

t

f(t 1) = e(t1)u(t 1)

12

t

f(t + 1) =e(t+1)u(t + 1)

12

t

f(3t) =e3tu(3t) = e3tu(t)

12

t

f(3 t) = e(3t)u(3 t)

12

u(at+b)

a >0

a


3/44

t

u(t)

t

u(2 t)

t

u(3t + 2)

t

2u(3 t)

t

u(t 1)

t

u(2

t)

t

u(t)


4/44


5/44

f3(t) = cos 3t (t 2) = cos 6 (t 2) =(t 2)

f4(t) =u(t)u(2 t)(t 3) = 0

f5(t) = sin 3t(u(t 1) u(t 2))

f6(t) =e

3|t|u(t) =e3tu(t)

t

f6(t)

12

t

f1(t)

f2(t)

f3(t)

t

f5(t)

12

12

0

(t 3)sin (t 2) dt= sin (3 2) = sin .

0

t

3 1

(t3 + t2 1) dt=

0

1

3(t 3)

(t3 + t2 1) dt=

=3

0

(t

3)(t3 + t2

1) dt= 3(33 + 32

1) = 105.


6/44

0

e3t(r

t) dr= e3t

0

(r

t) dr= e3tu(t).

0

(t3 + 2t2 20)(t 2)dt= (1)1 ddt

(t3 + 2t2 20)t=2

=(3t2 + 4t)t=2

=20.

0

(t3)sin (t4) dt= (1)2 d2

dt2sin (t 4)

t=3

=2 sin (t 4)t=3

= 2 sin .

0

(t)sin2t

2t

dt= (

1)1

d

dt

sin2t

2t

t=0

=

4t cos2t 2sin2t

4t2

t=0

=0

0

.

L Hopital

2cos2t 2t sin2t 2cos2t4t

t=0

= sin2t

2

t=0

= 0.

f(t) = 10 sin 5t u(t)

I=

t2t1

f2(t) dt=

t2t1

100 sin2 5t u2(t) dt= 100

t20

1 cos 10t2

dt=

50 t2

0dt 50 t2

0cos 10tdt= 50t2 5010sin 10t|t20 =

50t2 5

sin 10t2.

f(t)

limt2

I= limt2

50t2 5

sin 10t2

=.

limt2

1

t2I= lim

t2

1

t2 50t2 5

sin 10t2

= 50 limt2

5sin10t2t2

= 50


7/44

I= t2t1

f2(t) dt= t2t1

100e10t cos2 5tdt

= 100

t2t1

e10t

2 dt + 100

t2t1

e10t cos10t

2 dt=

50

10 e10t|t2t1+ 50e10t

10cos10t + 10 sin10t100 + 1002

t2t1

=

5

e10t2

sin 10t2 cos10t2

1 + 2 1

e10t1

sin 10t1 cos10t1

1 + 2 1

.

limt2t1

I=

e10t1

limt2t1

1

t2 t1 I= limt2=t1=t1

2tI=

limt e10t

2t =

f(t) = 10e5tu(t)

f(t)

f(t)

Wf =

f

2(t) dt

Wg =

g2(t) dt=

a2f2(bt) dt= a2

f2(bt) dt.

bt= udt = du

b

b >0

Wg = a2

f2(u)du

b

=a2

b

f2(u) du= a2

b

Wf

b


8/44

Pg = limT

12T TT

g2(t) dt= limT

12T TT

a2f2(bt) dt= a2 limT

12T TT

f2(bt) dt.

bt= udt = du

b

Pg = a2 limT

1

2T

bTbT

f2(u)du

b =a2 lim

bT

1

2bT

bTbT

f2(u) du= a2Pf.

W1+2

f1(t)

f2(t)

W1+2=

t2t1

|f1(t) + f2(t)|2 dt= t2t1

(f1(t) + f2(t))(f1(t) + f2(t)) dt=

t2t1

[f1(t)f1 (t) + f1(t)f

2 (t) + f2(t)f

1 (t) + f2(t)f

2 (t)] dt=

t1t2f1(t)f

1 (t) dt +

t2t1

f1(t)f2 (t) dt +

t2t1

f2(t)f1 (t) dt +

t2t1

f2(t)f2 (t) dt

f1(t)

f2(t)

f1(t), f2(t)= t2t1

f1(t)f2 (t) dt= 0

f1(t), f2(t)

=

t2

t1

f1(t)f2 (t) dt

= t2

t1

f1 (t)f2(t)dt= 0.

W1+2=

t2t1

f1(t)f1 (t) dt +

t2t1

f2(t)f2 (t) dt=

t2t1

|f1(t)|2 dt + t2t1

|f2(t)|2 dt= W1+ W2.

P1+2= 1

t

W1+2= 1

t

W1+ 1

t

W2= P1+ P2


9/44

e(t)

f(t)

f(t) =

n ann(t)

e(t) =f(t) n ann(t)

a1= 1

I=

t2t1

(f(t) a11(t))1(t) dt= 0.

I=

t2t1

f(t)1(t) dt a1 t2t1

21(t) dt=

0

4

sin t dt 2

4

sin t dt a1 2

0

16

2sin2 t dt=

= 4

cos t|0 + cos t|2

16a1

2

20

1 cos2t2

dt=

16

16a1

2

20

1

2dt

20

cos2t

2 dt

=

16

16a1

.

I = 0 a1 = 1 e(t)

1(t)

I=

t2t1

(f(t) a11(t))1(t) dta1=1

=

t2t1

(f(t) 1(t))1(t) dt= t2t1

e(t)1(t) dt= 0.

f(t) =a00(t) + a11(t) + a22(t)

an=f(t), n(t)

a0=f(t), 0(t)=

f(t)1(t) dt=

10

sin tdt= 1

cos t

10

= 2

a1= f(t), 1(t)=

f(t)1(t) dt=

12

0sin tdt

11

2

sin tdt=

1

cos t|

1

2

0

cos t

|11

2=1

(0

1

(

1

0)) = 0.


10/44

a2 = f(t), 2(t)=

f(t)2(t) dt=

14

0sin tdt

34

1

4

sin tdt +

13

4

sin tdt=

1

cos t| 140 cos t|

34

1

4

+ cos t|134

=2

2 1 .

Wf

Wf =

|f(t)|2 dt= 1

0sin2 tdt=

10

1 sin2t2

dt=1

2

Wf

Wf =

n= |an|

2

=

4

2 + 0 +

4

2 (

2 1)

2

=

8

2 (

2 2)

2 = 1 W

f

Wf=.0503585%

sin5

(t ) d=sin5t

t

sin 5( 3) 3 d =

sin5u

u du= 2

0

sin5u

u du= 2

2 =,

sin 5(t3)t3 (t 3)

sin 5( 3) 3

sin 5(t )t d

( 3) sin 5(t )t d=

sin5(t 3)t 3

f(t) g(t) =

f()g(t ) d =

e3u()e5(t)u(t ) d =

e5t

0

e2u(t

) d.


11/44

t 0

e5t

0e2u(t ) d =e5t

t

0e2d =e5t

e2

2

t0

=e5te2t 1

2 =

e3t e5t2

.

f(t) g(t) = e3t e5t

2 u(t).

f(t) =A

tu(t) 2 t 12u t 12) + (t 1)u(t 1)

f(t) g(t) =

f(t )g() d=B 2

1f(t ) d=

AB

21

(t )u(t ) 2

t 12

u

t 1

2

)+

(t

1)u(t

1) d=

AB

21

(t )u(t ) d 2 2

1

t 1

2

u

t 1

2

d+ 2

1(t 1 )u(t 1 ) d

.

I(t)

f(t)

g(t) =AB I(t)

2It 1

2+ I(t

1) .

I(t) =

21

(t )u(t ) d = t1t2

u() d=

t1

u() d t2

u() d=

t 12

r(t

1)

t 2

2

r(t

2).


12/44

f(t) g(t) =AB t 12

r(t 1) t 22

r(t 2) 2t 322

r

t 32t 52

2 r

t 52

+

+t 2

2 r(t 2) t 3

2 r(t 3)

=

AB

(t 1)2

2 u(t 1)

t 3

2

2u

t 3

2

+

+

t 5

2

2u

t 5

2

(t 3)

2

2 u(t 3)

=

f(t) g(t) =

(t1)2

2 , 1

t

32

12(t2)24 , 32 t 52(t3)2

2 , 5

2 t30,

3

2 5

2 t

f(t) g(t)

1

8

14

f(t), g(t)= 0 (1, 3) f(t) g(t)= 0 (2, 6) f(t) g(t)|t=2= 0

f(t) g(t)|t=3

f(t) = A(t 1)[u(t 1) u(t 2)] +A(t+3)[u(t 2) u(t 3)] g(t) =B [u(t 1) u(t 3)]

f(t) g(t)|t=3=

f()g(3 ) d=

A( 1)[u( 1) u( 2)]B[u(3 1) u(3 3)] d

+

A(+ 3)[u( 2) u( 3)]B[u(3 1) u(3 3)] d=

AB 2

1

(

1)[u(2

)

u(

)] d+ 3

2

(

+ 3)[u(2

)

u(

)] d=


13/44

Fourier

AB

21

( 1)(1 0) d+ 3

2(+ 3)(0 0) d

= AB

21

( 1) d=

AB2

.

Fourier

F n

f(t)

f(t) =

+n=

Fnejnt, Fn= 1

T T

0f(t)ejnt dt,

An= 2|Fn|, n = arctan

j(Fn Fn)Fn+ Fn

.

Fn

f(t )

Fn= 1

T

T0

f(t )ejnt dt,

t=ut = u+

dt= du

Fn= 1

T

T

f(u)ejn(u+) du= ejn

1

T

T

f(u)ejnudu= ejnFn.

An

An= 2|Fn|= 2|ejnFn|= 2|ejn||Fn|= 2|Fn| An= An

n

n = arctan

j(Fn Fn)Fn+ F

n

=arctan

j(ejnFn ejnFn)

ejnFn+ ejnFn

n=arctan

j(ej2n

Fn Fn)e2jnFn+ Fn

.

g(t)

g(t) = 3 cos(52t)+5cos(62t)

Fourier

g(t) =a0

2

+

n=1

ancos nt +

n=1

bnsin nt


14/44

= 2

g(t)

bn= 0n N

an= 0,

n

N

{5, 6

}

a5= 3

a6= 5.

An=

a2n+ b2n=

a2n=|an|,nN

An=

3,

n= 5

5,

n= 6

0,

n=arctanbnan

= 0,nN.

g(t) = 3 cos 10t +5cos12t

g(t)

g(t) = 4 sin(1 1t) + 10 sin(3 1t)

Fourier

= 1

g(t)

an= 0nN

bn= 0,n N {1, 3}, b1= 4 b3= 10

An=

a2

n+ b2

n=

b2

n=|bn|,nN

An=

4,

n= 1

10,

n= 30,

n=arctanbnan

=

2 ,

n= 1, 3

0,

g(t) = 4sin t+ 10 sin3t

fe(t) = 12 [f(t) + f(t)]fe(t) = 12 [f(t) + f((t))] = 12 [f(t) + f(t)] =fe(t)

fo(t) = 1

2 [f(t) f(t)] fo(t) = 12 [f(t) f((t))] = 12 [f(t) f(t)] =fo(t)

ue(t) = 12 [u(t) + u(t)] = 12 ,tR

uo(t) = 12 [u(t) u(t)] =

12 ,

t 0

f(t) = cos 3t

fe(t) = cos 3t

fo(t) = 0


15/44

Fourier

n

An

3 cos 10t+ 5cos12t

n

n

2

2

3 cos 10t+ 5cos12t

n

An

4 sin t+ 10 sin 3t

n

n

2

2

4 sin t+ 10 sin 3t

fT(t) =

A sin t, 0t0, t2

T = 2

= 2T

= 1

a0= 2

T

T0

fT(t) dt= A

0

sin t dt=2A

an= 2

T T

0

fT(t)cos ntdt=A

0

sin t cos ntdt


16/44

0

sin mt cos ntdt=

0, m + n

2mm2n2

, m + n

,

an=

0, n

2A

1n21

, n

bn= 2

T

T0

fT(t)sin ntdt=A

0

sin t sin ntdt

0sin mt sin ntdt =

0,

m=n

2

,

m= n,

bn= 0, n= 1A

2

, n= 1

f(t) =A

+

A

2 sin t A

cos2t

1 3 +cos4t

3 5 + . . .

An

n

An= 2|Fn|= 212 (anjbn)

=

A

, n= 0A2 , n= 1A

(n1)(n+1) , n= 2, N {1}

n=arctanbnan

=

2 , n= 1, n= 2, N {1}

0, n

T = 3

0, T2

fT2

(t) =A

u(t) u

t 1

2

2A(t 1)

u

t 1

2

u(t 1)

bn= 0,nN dc

a02

= 2

T

T2

0fT

2

(t) dt= 2A

T

12

0dt +

4A

T

11

2

(t 1) dt= AT

+4A

T

t2

2 t

1

1

2

=3A

2T =

A

2

an=2

T

T2

0fT

2

(t)cos ntdt=2A

T

12

0cos ntdt +

4A

T

11

2

(t 1)cos ntdt= . . .=

2A

n

sinn

2

2

n0

sinn

6

+ cosn

6


17/44

Fourier

sin n2

n

sin n2 = 1

n

4 + 1

sin n2 =1

n

4 + 3

f(t) =A

22A

n=1

1

2n 1cos n 2

(2n 1)0 sin(2n 1)

6 + cos

(2n 1)6

cos(2n 1)0t

An=|an|= 2A

1

n

cosn6 3nsin n6

n= 0

cosn

6 3

nsin

n

6

T = 2

0, T2

fT2

(t) =Aet

5 [u(t) u(t )] dc

an

n N bn

bn=4

T

T2

0fT

2

(t)sin n0t dt= 2A

0

et

5 sin ntdt=2A

et

5

15sin nt + n cos nt

1

25 + n2

0

= 50A

(1 + 25n2) (e

5

n cos n n) = 50An

(1 + 25n2)

1 (1)n

e

5

n

f(t) =50A

n=1

(1 (1)ne5 ) n1 + 25n2

sin nt,

An=|bn|= 50A

(1 (1)ne5 n1 + 25n2

)

n= 0,nN.

Pn

Pn=a20

4 +

1

2

n

k=1

(a2k+ b2k)


18/44

n

An

14

12

34

A = 1

n

n

2

2

n

An

A = 1

n

n

2

2

n

An

14

12

34

A = 1

n

n

2

2


19/44

Fourier

P

P= WT

= 1T

T2T

2

|fT(t)|2 dt

fT(t)

P = 2

T2

0

4t2

T2 dt=

1

3

P2 =

a20

4 +

1

2

2

k=1(a

2

k+ b

2

k)

ak = 0,k N

bn= 22

T

T2

0

2t

T sin ntdt= 2

cos n

n =

2(1)nn

b2n= 4

2n2

P2= 1

2(b21+ b

22) =

1

2 4

42+

1

42= 5

22

P2P

=5

22

13

= 15

22 = 75.9%

An= 2|Fn|= 2

1

|2n|ejarctan n

2

= 1

|n|

n= arctanj(Fn Fn)

Fn+ Fn=arctan

j2|n|

ejarctan

n

2 ejarctan n2

12|n|

ejarctan

n

2 + ejarctan n

2

= arctanj

j tan

arctan

n

2

=arctann

2

|F3|2 + |F3|2|F4|2 + |F4|2 =

2|F23 |2|F24 |

=191

16

=16

9


20/44

n

An

18

14

38

1

2

n

n

2

2

f(t) = 2 cos 1t cos 2t= cos(1 2)t + cos(1+ 2)t

F() = FT {f(t)}=FT {cos(1 2)t} + FT {cos(1+ 2)t}

[(+ (1 2)) + ( (1 2))] + [(+ (1+ 2)) + ( (1+ 2))]

f(t) = 2 cos 1t cos 1t

F() =2 1

2FT {cos 1t} F T {cos 2t}

1

[(+ 1) + ( 1)] [(+ 2) + ( 2)]

[(+ 1+ 2) + (+ 1 2) + ( 1+ 2) + ( 1 2)]

f(t) =g(t)cos2 1t= g(t)

1 + cos 21t

2

=

1

2g(t) + g(t)

cos21t

2

F() =1

2FT {g(t)} + 1

2FT {g(t)} 1

2FT {cos21t}

1

2G() +

1

4G() [(+ 21) + ( 21)]

1

2

G() +1

4

[G(+ 21) + G(

21)]


21/44

Fourier

f(t) =g(t)cos2 1t= g(t)cos 1t cos 1t

F() = 1

2FT {g(t)} F T {cos 1t cos 1t}

F() =1

4G() [(+ 21) + 2() + ( 21)]

1

4 (G(+ 21) + 2G() + G( 21))

f(t) = g(at+b)

h(t) = g(at)

h(t)

Fourier

H() =FT {g(at)}= 1|a|G

a

f(t) =g(at + b) =h

t + ba

F()

F() =FT

h

t + ba

= e+j

b

a H() = e+j

a b

|a| G

a

sin 0t= cos

0t 2

FT {sin 0t}= FT

cos

0t 2

=

1

0ej

0

2 F

0

,

F() =FT {cos t}

FT {sin 0t}= 10

ej

0

2

0+ 1

+

0 1

=

1

0e

j 0

2 [0(+ 0) + 0( 0)] =

ej0

0

2 (+ 0) + ej

0

0

2 ( 0)

=

cos

2+j sin

2

(+ 0) +

cos

2j sin

2

( 0)

=

j [(+ 0)

(

0)]


22/44

FT {cos 0t cos 0t}= 12

FT {cos 0t} F T {cos 0t}=1

2[(+ 0) + ( 0)] [(+ 0) + ( 0)] =

2[(+ 20) + () + () + ( 20)] =

2[(+ 20) + 2() + ( 20)]

F()

2

0

0

F() = 2(+ 0) + 2( 0) Fourier (t)

FT {t(t)}= Sinc

2

FT {2(t)}= 2Sinc

2

FTSinc

t= 22()

FT1 {2()}=

Sinc

t

f(t) = FT1 {F()}=FT1 {2(+ 0)} + FT1 {2( 0)}ej0t FT1 {2()} + ej0t FT1 {2()}

ej0t + ej0t

Sinc

t

=

2

cos 0t Sinc

t

f(t) = FT1{F()}= 12

+

F()ejt d= 1

2

0+0

ejt d+ 1

2

0+0

ejt d

1

2

1

jt

ejt

0+0

+ ejt0+0

=

1

2jt

ej(0)t ej(0+)t + ej(0+)t ej(0)t

=

1

2jt(2j sin(0+ )t 2j sin(0 )t) =

1

t(sin 0t cos t + sin t cos 0t sin 0t cos t + sin t cos 0t) = 2

tcos 0t sin t =

2

cos 0tsin t

t

=2

cos 0t Sinc

t


23/44

Fourier

F() =FT {ej(t)}=FT {cos (t) +j sin (t)}=F() =FT {cos (t)} +j FT {sin (t)}

FT {ej(t)}=FT {(ej(t))}= F()F() =FT {cos (t)} j FT {sin (t)}

2 FT {cos (t)}= F() + F() FT {cos (t)}= F() + F()

2

2j FT {sin (t)}= F() F() FT {sin (t)}= F() F()

2j

f(t)R f(t) =f(t)F() =F()

f1(t) +jf2(t) = 1

0

F()ejt d

(f1(t) +jf2(t)) =f1(t) jf2(t) =

1

0

F()ejt d

=

1

0

F()ejt d

2f1(t) = 1

0F()ejt d+

1

0F()ejtd

=ud =du

f1(t) = 1

2

0

F()ejt d+

0

F(u)ejut(du)

=

1

2

0

F()ejt d+

0

F()ejutd

=

1

2

F()ejt d=

FT1

{F()

}= f(t)


24/44

2jf2(t) =

1

0 F()ejt

d 1

0 F

()ejt

d

=ud =du

f2(t) = 1

2j

0

F()ejt d

0F(u)ejut(du)

=

1

2j

0

F()ejt d 0

F()ejut d

=

1

2j

F()sgn()ejt d= 1

jFT1 {F()sgn()}=

1

jFT1 {F()} F T 1 {sgn()}=1

j f(t) FT1 {sgn()}

FT {sgn(t)}= 2j

FT

2

jt

= 2 sgn() =2 sgn FT1 {sgn }= 1

jt

f2(t) = 1

jf(t)

1

jt

=

1

f() 1

t d

FT {f(at)ej0t}= 12

FT {f(at)}FT{ej0t}= 12

1

|a|F

a

2(0) = 1|a|F

0

a

FT1

1

(j + 1)(j + 2)

= (et e2t)u(t)

Laplace

LT {(2et

+ t)u(t)}=LT {2et

u(t)} + LT {tu(t)}= 2 LT {et

u(t)} + 1s2 =

2s+1 +

1s2


25/44

Laplace

LT {u(t 2)}= e2s LT {u(t)}= e2s

s

f(t) =etu(1

t)

Laplace

Laplace

f(t)u(t)

f(t)u(t) =etu(1 t)u(t) =et(u(t) u(t 1))

LT {f(t)u(t)}= LT {etu(t)} L T {e1e(t1)u(t 1)}= 1s

s=s+1

e1 LT {e(t1)u(t 1)}=

1

s + 1 e1es LT {etu(t)}= 1

s + 1 e(s+1)

1

s

s=s+1

= 1

s + 1e

(s+1)

s + 1 =

1 e(s+1)s + 1

.

LT {f(t)u(t)}= LT {et(u(t) u(t 1))}=LT {(u(t) u(t 1))}|s=s+1=

1

s e

s

s

s=s+1

=

1 e(s+1)s + 1

.

(t 1)u(t) =tu(t) u(t)

LT {(t 1)u(t)}= LT {tu(t)} L T {u(t)}= 1s2

1s

=1 s

s2 =

es LT {tu(t)}= 1 ss2

es.

x(t) = tu(t) LT {x(t)}= X(s) = 1

s2

y(t) =atu(at) =x(at)

a >0

Laplace

Y(s) = 1

aX

s

a=

a

s2

z(t) =y

t b

a

= x

a

t b

a

= x(at b) = (at b)u(at b).

Z(s) =eb

asY(s) =

aeb

as

s2 .

Laplace

(at b)u(t)

LT {(at

b)u(t)

}= a

LT {tu(t)

} L T {bu(t)

}=

a

s2

b

s

=as b

s2


26/44

LT

dydt

= s LT {y(t)} y(0).

LT {et(tu(t))}=LT {tu(t)}|s=s+1= 1

s2

s=s+1

= 1

(s + 1)2,

LT

d

dt(tetu(t))

= s LTtetu(t) tetu(t)

t=0

= s

(s + 1)2.

x(t) =

0

u(t ) d= tu(t) u(t)

X(s) =LT {tu(t) u(t)}=LT {tu(t)}LT{u(t)}= 1s2

1

s =

1

s3

x(t)

x(t) =u(t) u(t 0.5) T = 1 Laplace

X(s) =LT {u(t)} L T {u(t 0.5)}= 1s e

0.5s

s =

1 e0.5ss

X(s) = X(s) 1

1 eTs = 1 e0.5ss(1 es) .

X(s) = 1

s2+4s+4= 1

(s+2)2 x(t) =LT1{X(s)}=LT1

1

(s+2)2= te2tu(t)

X(s) = s3+3s+2(s+1)3

= s3+3s2+3s+1

(s+1)3 + 3s

2+1(s+1)3

= 1 + 3s2+1

(s+1)3

3s2+1(s+1)3

= As+1+

B(s+1)2

+ C(s+1)3

=. . .= 3s+1 +

6(s+1)2

+ 2(s+1)3

X(s) =1 + 3s + 1

+ 6

(s + 1)2+

2(s + 1)3

x(t) =(t) 3etu(t) + 6tetu(t) 2 t2

2etu(t)

(t) (t2 6t + 3)etu(t).

LT 1sn= tn1(n1)! u(t)


27/44

t, , s

X(s) = 1e4s

3s3+2s2 = 1

s2(3s+2) e4s

s2(3s+2)

1

s2(3s + 2) =

A

s+

B

s2+

C

(3s + 2)=

(3A + C)s2 + (2A + 3B)s + 2B

s2(3s + 2) =3

4s+

1

s2+

9

4(3s + 2)

t, , s

f(t)

f(t) =g(t) + k(t)

g(t)

Fourier

f(t)

g(t)

FT {f(t)}= F() =FR() +jFI()

FT {g(t)}= G() =GR() +jGI()

FT {f(t)}=FT {g(t) + k(t)}= G() + k= GR() + k+jGI() =FR() +jFI()

FR() =GR() + k

FI() =GI()

g(t)

Hilbert

GR() = 1

GI()

d

GI() =1

GR()

d

FR() = 1

GI()

d+ k= 1

FI()

d+ k,

FI() =

1

GR()

d=

1

FR() k

d.


28/44

Hilbert

R() = 1

I()

d= 1

0

I()

d+1

0

I()

d

=d=d

R() =1

0

I()+

d+ 1

0

I()

d

I()

R() =1

0

I()

+ d+

1

0

I()

d= 1

0

I()

I()

+

d=

2

0

I()

2 2d

Hilbert

I() =1

R()

d=1

0

R()

d1

0

R()

d

=d=d

I() = 1

0

R()+

d 1

0

R()

d

R()

I() =1

0

R()

+ , d1

0

R()

d=1

0

R()

+ +

R()

d=2

0

R()

2 2d

Hilbert

I() =1

R()

d=1

() 1

d=1

X() =R() +jI() =()

j

1

=() + 1

j

=

FT {u(t)

}


29/44

t, , s

Paley-Wiener

|ln A()|1 + 2

d 0

0

> 0: A()< M ea ln A()< ln M a

A(0)< 1ln A()< 0

ln A()> a ln M | ln A()|> a ln M.

| ln A()|1 + 2

d=

0

| ln A()|1 + 2

d+

0

| ln A()|1 + 2

d= +

0

| ln A()|1 + 2

d > 0

a

1 + 2d

0

ln M

1 + 2d =

a

2ln(2 + 1)

0

ln Marctan |0 =.

A(0)1

A()

1

> 1: A()< 1ln A()< 0.

ln A()> a ln M | ln A()|> a ln M, > 1

| ln A()|1 + 2

d=

1

| ln A()|1 + 2

d+

1

| ln A()|1 + 2

d= +

1

| ln A()|1 + 2

d >

1

a

1 + 2d

1

ln M

1 + 2d =

a

2ln(2 + 1)

1 ln Marctan |1 =.


30/44

()

a() =

ln A()

()

Hilbert

a() =a(0) 2

()

(2 2)d

()

1(22)

a() =a(0) 22

0

()

(2 2)d= a(0) 220

c

1

(2 2)d=

a(0) 220

1

22 ln 1

2

2

c

=a(0) +0

ln 1

2

2c

A() =ea() =ea(0)

0

ln

122c=ea(0)e

ln

122c

0

=A(0)eln

22c

0

Sinc

FT {Sa(t)}=FTSinc t= 2(),

Sa(t)

[, ] rad/sec

4

3

2

2

3

4

t

Sa(t) = sintt

14

14

12

34

2()

Sa50t

[50, 50] rad/s 502 , 502

=

25

, 25

Hz

fs = 2fm

fs = 50 Hz


31/44

t, , s

Sa2(t)

Fourier

[ , + ] = [2, 2] Sa2 50t [100, 100] rad/s 1002, 1002 = 50, 50 Hz fs = 2fm fs = 100 Hz

Sa50t+ Sa25t

[50, 50] rad/s 502 , 502 = 25, 25 Hz

Sa25t

Sa50t

Sa25t

Fourier

fs = 2fm

fs= 50

Hz

Fourier X()

x(t) = e3|t|

X() = 69+2

|| > c

x(t)

c

c

W =

|x(t)|2 dt= 2

0|x(t)|2 dt= 2

0

e6t dt=26

e6t

0 =

1

3

Wc = 1

2

cc

|X()|2 dt= 2 12

c0

|X()|2 dt= 1

c0

36

(9 + 2)2dt =

36

2 32 (2 + 32)+ 1

2 33arctan

3

c0

= 2

c

2c + 32

+1

3arctan

c3

WcW

t

e3|t|

12

c c

X()

23


32/44

x(t)

xs(t)

p(t)

2Ts Ts Ts 2Ts t

p(t)1

WcW =

2

c2c+3

2 + 13arctan

c3

13 =

2

c3

c3

2 + 1 + arctan

c

3

WcW

c

c Wc/W

rad/s

rad/s

rad/s

p(t)

1

Ts

Fourier

p(t) =

n=

Pnejnst,

s = 2Ts

Pn=

1

Ts Ts2Ts

2

p(t)ejnst

dt=

1

Ts 2

2

1

ejnst

dt=

1

TsSinc

n

Ts

FourierXs()

xs(t) =x(t)p(t)

Xs() =

xs(t)ejt dt=

x(t)p(t)ejt dt=

x(t)ejt

n=

Pnejnst dt=

n=

Pn

x(t)ejtejnst dt=

n=

Pn

x(t)ej(ns)t dt=

n=

PnX( ns) =

1

Ts

n=

Sincn

TsX( ns)


33/44

t, , s

Xs()

xs(t) = x(t)n=n=

(t

kTs)

H()

0 0

X()

100 1000 0

Xs()

0

2a

Xf()

2a= 20

2a > 20

2a < 20

Parseval

=

|x(t) y(t)|2 dt= 12

|X() Y()|2 dt=1

2

|X() Y()|2 d+

|X() Y()|2 d+

|X() Y()|2 d

Y() = 0

||>

= 1

2

|X()

|2 d+

|X()

Y()

|2 d+

+

|X()

|2 d


34/44

100 1000 0

Xs() H()

100 100

Xf,1()

100 1000 0

Xs() H()

100 100

Xf,2()

100 1000 0

Xs() H()

100 100

Xf,3()

X() =Y()

||<

h(t) =LT1{H(s)}=LT1Y(s)X(s)

y(t)

x(t) =(t)

d2y

dt2 + 2

dy

dt + y(t) =x(t)

s2Y(s)

sy(0)

dy

dtt=0

+ 2(sY(s)

y(0)) + Y(s) =X(s)


35/44

(s2 + 2s + 1)Y(s) =X(s)

Y(s)

X(s) =H(s) =

1

(s + 1)2

h(t) =LT1{H(s)}= tetu(t)

x(t) =u(t)X(s) =LT {x(t)}=LT {u(t)}= 1s

Y(s) = 1

(s + 1)2X(s) =

1

s(s + 1)2

1

s(s + 1)2 =

A

s +

B

s + 1+

C

(s + 1)2 =. . .=

1

s 1

s + 1 1

(s + 1)2

y(t) = LT1{Y(s)}=LT1

1

s 1

s + 1 1

(s + 1)2

=

LT1

1

s

LT1

1

s + 1

LT1

1

(s + 1)2

=

u(t) etu(t) tetu(t) = (1 et tet)u(t)

yss(t)

steady state

yss(t) = limt y(t) = limt(1 et

tet

)u(t) =u(t)

yt(t)

transient

yt(t) =y(t) yss(t) =(1 + t)etu(t)

y(t) =

t(x() x( t0)) d

Y() = 1

jX() + X(0)() 1

jX()ejt0 X(0)ej0t0() =

1j

(1 ejt0)X()

Y()

X()=H() =

1 ejt0j

h(t)

x(t) =(t)

h(t) =

t

[() ( t0)] d = t

() d t

( t0) d=

u(t)

u(t

t0) = t0 t

t0

2


36/44

h(t) = FT1

{H()}=FT11 e

jt0

j

=FT1ejt0

j FT1e

jt0

j

=1

2sgn t 1

2sgn(t t0)

t0 t

12sgn(t)

12

1

2

t0 t

12sgn(t)

12

12

+ t0 t

h(t) =u(t)

u(t

t0)

12

12

H() =H4()H3() (1 + H2()) H1(),

H1() = FT

d

dt

sin 0t

2t

= j

02

FT

sin 0t

0t

= j

02

FT

Sinc0

t

=

j02

020() =

j

2 20(),

H2() =ej 2

0

,

H3() = FT

sin30t

t

=

30

FT

sin30t

30t

=

30

FT

Sinc

30

t

=

30

30

60() =

60(),


37/44

H4() = FT {u(t)}= () + 1j

.

H() =

() +

1

j

60()

1 + e

j 20

j

2 20()

6020 = 20

f(t)(t t0) =f(t0)

H() =()

1 + ej 2

0j

2 20() +

1

j

1 + e

j 20

j

2 20() =

()1 + ej2

00j0

2

20(0) +1

2

20() +1

2

ej 2

0

20() =

1

220() +

1

2ej 2

0

20()

h(t) = FT1{H()}= 12

202

Sinc

0

t

+ Sinc

0

t 2

0

=

02

Sinc

0

t

+ Sinc0

t 2

x(t) =x1(t) + x2(t) = sin 20t + cos0t

2

FT {x(t)}=X() =X1() + X2() =j [(+ 20) (+ 20)] +

(+

02

) + (+0

2 )

.

Y() =X()H() =X1()H() + X2()H().

X1()H() = 0

X2()H() =

2+ 0

2+

0

2 1 + ej 20 20() =

2

1 + e

j 20

0

2

20

02

+0

2

+

1 + ej 2

0

0

2

20

02

+0

2

=

2

(1 + ej) 1

+

02

+ (1 + ej ) 1

0

2

=

2

(1 1)

+

02

+ (1 1)

0

2

= 0.


38/44

h1(t) = 10e

tu(t)

H1(s) = 10

s+1

H1(s)

sp =1

H1() = H1(s)|s=j = 10

1 +j.

y1(t)

x1(t) = (t)

h(t)

x2(t) =u(t)

Y2() =X2()H() = 10

1 +j

() +

1

j

= 10() +

10

(1 +j)j =

10() + 10j

101 +j

y2(t) =10u(t) 10etu(t) = 10(1 et)u(t).

x(t) H1()

H2()

y(t)

t

y1(t) =etu(t)

t

y2(t) = 10(1 et)u(t)

s

Y(s) =H1(s)(X(s) + H2(s)Y(s))


39/44

Y(s)X(s) =H(s) = H1(s)1 H1(s)H2(s) =

10

1+s1 10k1+s

= 10s + 1 10k .

H(s)

sp = 10k 1

k 0.1

H()

H() = H(s)|s=j = 10

1 10k+j .

y1(t)

h(t) =FT1{H()}=FT1{H()}= 10e(110k)tu(t),

y2(t)

y2(t) = FT1{H()X2()}=FT1

10

1 10k+j1

j

=

10

1 10kFT1

1

j 1

1 10k+j

= 10

1 10k (u(t) e(110k)tu(t)) =

10

1

10k

(1 e(110k)t)u(t).

d2ydt2

+ 3dydt

+ 2y(t) = 1 + t

y(0) = 0 dydt

t=0

= 1

Laplace

s2Y(s) sy(0) dydt

t=0

+ 3sY(s) 3y(0) + 2Y(s) =1s

+ 1

s2

(s2 + 3s + 2)Y(s) 1 = s + 1s2

(s + 1)(s + 2)Y(s) = s + 1

s2 + 1

Y(s) = 1

s2(s + 2)+

1

(s + 1)(s + 2)

1

s2(s + 2)+

1

(s + 1)(s + 2)=

A

s +

B

s2+

C

s + 2+

D

s + 1+

E

s + 2=...

=14

1

s+

1

2

1

s2+

1

4

1

s + 2+

1

s + 1 1

s + 2


40/44

x(t) H1()

H2()

y(t)

t

y1(t) = 10e11tu(t)

y2(t) = 1011

(1 e11t)u(t)

k = 1

t

2.5 1025 102

7.5 102103

y1(t) = 10e9tu(t)

y2(t) = 109

(e9t 1)u(t)

k = 1

y(t) = LT1{Y(s)}=14

u(t) +1

2tu(t) +

1

4e2tu(t) + etu(t) e2tu(t) =

1

4(2t 1) +

et 3

4e2t

u(t)

d2y

dt2 + 2n

dy

dt + 2ny(t) =

2nx(t)

s2Y(s) + 2nsY(s) + 2nY(s) =

2nX(s)

Y(s)

X(s)=

2ns2 + 2ns + 2n

422n 42n= 42n(2 1)

2 1> 02 >1


41/44

y(t) =x(t) h1(t) h2(t)Y() =X()H1()H2(),

X() =FT {(t 1)}= ej ,

H1() =

FT {etu(t)

}=

1

1 +j

, H2() =

FT {e2tu(t)

}=

1

2 +j

.

Y() =ej 1

1 +j

1

2 +j

Wo

Parseval

W

= 1

2

ej

(1 +j)(2 +j)

2

d= 2 1

2

0

1

(1 + 2)(4 + 2)d =

1

1

3

0

1

1 + 2d 1

1

3

0

1

4 + 2d =

1

3

arctan 1

2arctan

2

0 =1

3

2 0 1

2

2 0

=

1

12.

[0, 10]

W10= 1

2

1010

fracej (1 +j)(2 +j)2 d= 2 12

100

1

(1 + 2)(4 + 2)d =

1

3

arctan 1

2arctan

2

10

0

= 1

3

arctan 10 1

2arctan5

W10W

=1

3

arctan 10 12arctan 5

1

12

= 4

arctan 10 1

2arctan 5

= 99.88%.

W

y(t) =x(t) h1(t) h2(t) =

( 1)e(t)u(t ) d

h2(t) =

t

(

1)e(t) d

h2(t)


42/44

t


43/44

h(t)

h(t) =e3|t| sin5t= e3|t| sin5tu(t) + e3|t| sin5tu(t) =e3t sin5tu(t) + e3t sin5tu(t).

f(t) =e3t sin5tu(t)

h(t) =f(t) f(t)

f(t)

Laplace

LT {f(t)}= F(s) = 5s2 + 6s + 34

Fourier

FT {f(t)}= F() = F(s)|s=j = 534 2 +j6

FT {f(t)}= F() FT {f(t)}= F()

H() =F() F() = 534 2 +j

5

34 2 j

Y() =X1() X2()y(t) = 12

x1(t)x2(t),

x1(t) =FT1

1

a +j

= eatu(t)

x2(t) =FT1

6

9 + 2

= e3|t|.

y(t) = 1

2eatu(t)e3|t| =

1

2eate3tu(t) =

1

2e(a+3)tu(t).

y(t) =x(t)cos2(103t) = x(t)

2 +

x(t) cos(2 103t)2

Y() = FT {y(t)}= 12

X() +1

2

1

2X() [(+ 2 103t) + ( 2 103t)]

=1

2X() +

1

4[X(+ 2000) + X( 2000)].


44/44

2000 2000

Y()

Date post:	23-Feb-2018
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