Trabajo Colaborativo 2 Consolidad

8/18/2019 Trabajo Colaborativo 2 Consolidad

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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIAESCUELA DE CIENCIAS BÁSICAS TECNOLOGÍA E INGENIERÍA

METODOS NUMERICOSTRABAJO COLABORATIVO 2

PRESENTADO POR

ROBINSON ANDRES CASTROJESUS MARINO GOMEZ MUÑOZ CC7505052

MARIANO ANTONIO MONSALVE ROJAS CC 71265546

MAURICIO ANTONIO BOHORQUEZ CUARTASGLORIA ELENA CARDENAS TAMAYO CC 4 7227!"

GRUPO 100401# 4

TUTORJOSE ADEL BARRERA

MEDELL$N % COLOMBIA

2016


2/31

INTRODUCCION

C&' () *+)(,-).,/' + + + *)3) & ) ,*, & +( .&'&., ,+' &8 , +' ,9,.)' & (& :*&:/ , & ;+ < ,.) + () ', ) + + ,&8 :+* , ,+' & + + +=, +'.,+ +( .&' +', & +( / (&8

&*,+' )' & ) + ,)* () ):(,.).,/' + (& .&'.+: & ; '&* ) ,=, ) + (& ,9+*+' + ,:& ++. ).,&'+ 8 .&' , +*)' & + () 9,')(, ) + + + . * & + + + : + )' , +' ,9,.)* ()

,9+*+' + >+**) ,+' ) + + )' +' (& ? & & ' ?*,.& :)*) 9&* )(+.+* ' + *&.&'&., ,+' & +' +( :*&.+ & + 9&* ).,/'@

@


3/31

1. Construir un cuadro co !arati"o d# $as di%#r#ncias #ntr# $os sist# as $in#a$#s & $os sist# as NO $in#a$#s con a$ #nos un#'# !$o. (D#)# s#r ori*ina$+ no s# ad it#n co!ias )a'adas d# int#rn#t,.

-..

/.0.

.2.3. D

E4INICI5N

6.

17. CUADRO CO89ARATIVO1-. SISTE8A LINEAL 1 . SISTE8A

NOLINEAL

15@ E ' :()' +) ,+' & + , )( ) 8 ,'=&( .*)' & ') & < =)*,)3(+ ) ():*, +*) :& +'.,)8 + '& .&' ,+'+ :*& . & +' *+ () =)*,)3(+ @ E +.,*8+ ') +. ).,/' + ,'=&( .*) &() +' + ) ; *+ ) + ') =)*,)3(+ )() :*, +*) :& +'.,)@ E' +( , + ) .)* + ,)'& *+:*+ +' )' *+. ) @

16@• () +. ).,&'+ , (


4/31

, )?.',.) @

1"@ A ()+. ).,&'+)( +3*),.);

*) .+' +'+ + '&

+ : + +'+ :*+ )* ++ ) 9&* )

+ (+ (() )ecuacionesno lineales.

P&* + + :(&8

20.

x2+ xy= 10

21@

y+3 x y2= 57--. 4

OR8A

- .

24@ Q + ) , 9).+ () , ,+' + :*&:,+ ) +25@ A , ,=, ) 9 ;F 9 F 9 ;F26@ H& & +'+, ) 9 ) F ) 9 F@27@ D& *+ () + & ) ) ) .&' ' & + .&'&.+ .& & :*,'.,:,& +

:+*:& ,.,/'@

2!@

2"@ U')+. ).,/' '&(,'+)( +

')+. ).,/' +()9&* ) F 0:)*) '=)(&*

+ .&'&.,& + U@

7. 9RO9

1@ L) +. ).,&'+ (,'+)(+ '& >& & ?'+) + &* +' & & :+*,&* ,+'+' ():*&:,+ ) + + ') .& 3,').,/' (,'+)( + &( .,&'+ ) 3,?' + ')

&( .,/'@

4@ L)+. ).,&'+'& (,'+)(+'& :& ++'


5/31


6/31

45.

46.

47@ G

R

A

F

I

C

A

4!@ R+:*+ +' ).,/' +' +( :()'& .)* + ,)'& + ') ( '+) *+. )@ E ) 9 '.,/' +: + + + .*,3,* 9 F 3

4"@

50@ U') 9 '.,/'. ) *< ,.)+ ')9 '.,/' +'() 9&* )

f ( x)= a x

51@ S *)9,.)+ (() )')

:)*


7/31

,' +* +..,&'+ +' &'()

&( .,&'++

5!@

x2− 2 x− 8=

5"@

( x+2 ) ( x− 4 )

60@

pues x =− 2

61@

-..

S

OLUCI5N

/.65@Ti#n# so$uci:n ;nica66@

67@

6!@ T,+'+ =)*,)&( .,&'+

6"@ S&( .,/'

',.)70@ N +*&9,', & +

&( .,&'+71@ N +*&

,'9,', & +&( .,&'+

72. N& ,+'+&( .,/'


8/31


9/31

5 y+4 x=− 1 → 4 x=− 5 y− 1 → x= −5 y− 14

2 y+ x= 0 →x=− 2 y

92.− 5 y− 1

4=− 2 y

93. −5 y− 1=− 8 y

94. −5 y+8 y= 1

95. 3 y= 1

96. y=13

97.

98.

99. x=− 2 y

x= 25 −

110.

111.

112.

113.

114.

115.

116.

117.

118.

119.


10/31

100. x=− 2 . 13

101. x=−23

102.

103.

104.

120.

121.

1--.123.

1-/.


11/31

1-0.

1- . -. So$ucion# #$ si*ui#nt# #'#rcicio uti$i>ando $os 8=todo d# #$i inaci:nd# Gauss+ Gauss?@ord n & GaussS#id#$. Co !ar# $os r#su$tados & a*a un!# u# o an $isis.

1-2. 0.1 X 1 +7.0 X 2− 0.3 X 3 =− 19.30

1-3. 3.0 X 1 +0.1 X 2− 0.2 X 3 = 7.85

1-6. 0.3 X 1 +0.2 X 2− 10.0 X 3 = 71.40

1 7. Uti$i>ar un F 7.771

1 1.

1 2@ T*)' 9&* ) & () ) *,- + )'+*) + .)(&') )

1 @0.1 7.0 − 0.33.0 − 0.1 −0.20.3 − 0.2 −10 |− 19.307.8571.40

1 4@

1 5@ f 2,1 = 3.0 −3.0

0.1 ∗0.1 = 0

1 6@ f 2,2 =− 1.0 −3.00.1∗7.0 =− 210,1

1 7@ f 2,3=− 0.2 −3.00.1∗(−0.3 )= 8.8

1 !@ f 2,4 = 7.9 −3.00.1 ∗(−19.30 )= 586.9

1 "@0.1 7.0 − 0.30 − 210.1 8.8

0.3 − 0.2 − 10|− 19.30586.971.40140@

141@ f 3,1 = 3.0 −

3.0

0.1∗0.1 = 0


12/31


13/31

15"@

160@ E. ).,/' =)*,)3(+ X 3

161@− 19077

1910 + X

3 =133861

1910 X

3=−133861

19077 = X

3 =−7.0

162@

16 @ E. ).,/' =)*,)3(+ X 2

164@− 2101

10+ X 2=

− 445

X 3 +11737

20− 2101

10∗ X 2

165@

166@− 44

5∗(− 13386119077 )+(1173720 ) X 2= − 11778538154 X 2=− 3.0

167@ E. ).,/' =)*,)3(+ X 1

16!@1

10+ X 1=− 7∗ X 2+

310∗ X 3−

19310

110∗ X 2

16"@ − 7∗−11778538154

−3

10∗−133861

19077− 193

10 X 1= 2.0

170@

171@

172@ R+ : + )

17 @

174@ X 1= 2.0 X 2=− 3.0 X 3=− 7.0

175@

[email protected] 7.0 − 0.33.0 − 0.1 −0.20.3 − 0. −10.0 |−

19.307.85

71.40

177@ f 2,1= 3.0 −

3.0

1.0∗1.0 = 0


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17!@ f 2,2=− 0.1 −3.01.0∗7.0 =− 210.1

17"@ f 2,3=− 0.2 −3.01.0∗(−0.3 )= 8.8

1!0@ f 2,4 = 7.9 −3.01.0∗(−19.30 )= 58.69

1!1@

[email protected] 70.0 − 0.30 − 210.1 − 8.8

0.3 − 0.2 − 10.0|− 19.30586.971.40

1! @

1!4@ f 3,1 = 3.0 −3.01.0∗1.0 = 0

1!5@ f 3,2 =− 2.0 −3.01.0∗70.0 =− 21.2

1!6@ f

3,3=− 10.0 − 3.0

1.0∗(− 0.3 )=− 9.1

1!7@ f 3,4 = 71.4 −3.01.0∗(−19.3 )= 129.3

1!!@

1!"@0.1 70.0 − 0.30 − 210.1 − 8.80 − 21.2 − 9.1|−

19.30586.9129.3

1"0@

1"1@ f 2,2=−− 210.1− 210.1 = 1

1"2@ f 2,3 =− 0.00.01.0∗1.0 =− 0.0

1" @ f 3,2 =− 21.2 − − 21.2− 210.1 ∗(−210.1 )= 0


15/31

1"4@ f 3,3 =− 9.1 − − 21.2− 210.1 ∗(8.8 )=− 10.0

1"5@

0.1 70.0 −0.3

0 1 −0.00 0 −10.0 |− 19.30

− 2.870.1

1"6@

1"7@ f 2,3 =− 0.00.01.0∗1.0 =− 0.0

1"!@ f 4,2=− 2.8 −− 0.01.0∗(−7.0 )=− 3.1

1""@ f 3,3 =− 10.0 /(10.0 )= 1

200@ f 4,3=− 2.8 −− 0.01.0∗(− 7.0 )=− 3.1

201@

202@

0.1 70.0 −0.30 1 00 0 1 |

− 19.30− 3.1− 70.1

20 @

204@

205@ f 2,1=− 70.0 −− 3.01.0∗(0.0 )= 70

206@ f 1,2 = 70.0 −

70.0

1.0(0.0 )= 0

207@ f 1,4 =− 193.0− 3.01.0

(−7.0 )=− 214.1

20!@

20"@1 0 00 1 00 0 1|− 214,0− 3.1− 70.1


16/31


17/31

231.

17¿

− 2 − 3 = 508 − 5 +¿− 21− 2 = 200 ¿ − 5

5 +¿22 = 30 ¿

232.

233.

1 − 0,11 −0,17 |29,41 |− 5 21 2 |200 |− 5 −5 22 |30 |

234.

235.

1 − 0,11 −0,17 |29,41 |0 26 3 |205 |

− 5 −5 22 |30 |

236.

237.

1 −0,11 − 0,17 |29,41 |0 26 3 |205 |0 0 27 |35 |

238.

1 −

0,11 −

0,17 |29,41 |0 1 0,11 | 7,8 |0 0 27 |35 |

239.

1 −0,11 − 0,17 |29,41 |0 1 0,11 | 7,8 |0 0 1 |1,29 |

240.

241.

*+*. G#$SS-S '

243.

17¿

− 2 − 3 = 508 − 5 +¿− 21− 2 = 200 ¿ − 5

5 +¿22 = 30 ¿

244. X

1=

500 +2 X 2 +3 X 317


18/31

245. X 2=200 +5 X 1+2 X 3

21

246. X 3=

30 +5 X 1 +5 X 222

247. X 2= 0 Y X 3= 0

248. X 10 = 500

17= 29,41

249. X 2200 +5 (29,41 )

21= 16,52

250. X 30 = 30 +5 (29,41 )+5 (16,52 )

22 = 11,80

251. X 11=

500 +2 (16,52 )+3 (11,80 )17

= 33,73

252. X 21= 200 +5 (33,43 )+2 (11,80 )

21

= 18,60

253. X 31= 30 +5 (33,43 )+5 (18,60 )

22= 13,18

254. [ X 11 − x10 = |33,43 − 29,41 |]= 4,02

255. [ X 21 − x20 = |18,60 − 16,52 |]= 2,08

256. [ X 31 − x30= |13,18 − 11,80 |]= 1,38

257. X 12= 500 +2 (18,60 )+3 (13,18 )= 33,92

17

258. X 22= 200 +5 (33,92 )+2 (13,18 )= 18,85

21


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259. X 32= 3 +5 (33,92 )+5 (18,85 )= 13,35

22

260. | X 12 − X 1

1|= | 33,92 − 33,43 |= 0,4

261.

262. X 13= 500 +2 (18,85 )+3 (13,35 )= 33,98

17

263. X 23= 200 +5 (33,98 )+2 (13,35 )= 18,88

21

264. X 33= 30 +5 (33,98 )+5 (18,88 )= 13,37

22

265. [ X 21 − x31 = |33,98 − 33,92 |]= 0,06

266. X 41 = 500 +2 (18,88 )+3 (13,37 )= 33,99

17

267. X 34 = 200 +5 (33,99 )+2 (13,37 )= 18,89

21

268. X 34 = 30 +5 (33,99 )+5 (18,89 )= 13,38

22

269. | X 41 − X 3

1|=|33,99 − 33,98 |= 0,01

270. X 1= ¿33,99 X 2= ¿18,89 X 3= ¿13,38

*/0.

*/*. G#$SS

273.17

274.-2

275.-3

276.500

277.f1/17

278.-5

279.21

280.-2

281.200

282.

283. 284. 285. 286. 287.


20/31

-5 -5 22 30*11.

289.1

290.-

0,11764706

291.-

0,17647059

292.29,4117

647293.

294.-5

295.21

296.-2

297.200

298.f2+5f1

299.-5

300.-5

301.22

302.30

303.f3+5f1

304.305. 306. 307. 308.

309. 310. 311. 312. 313.

314.1

315.-

0,11764706

316.-

0,17647059

317.29,4117

647318.

319.0

320.20,4117

647

321.-

2,88235294

322.347,058

824

323.f2/20,4

11

324.0

325.-

5,58823529

326.21,1176

471

327.177,058

824328.

329.330.

331. 332. 333. 334.

335.1

336.-

0,11764706

337.-

0,17647059

338.29,4117

647339.

340.0

341.1

342.-

0,14121037

343.17,0028

818344.

345.0

346.-

5,588235

347.21,1176

471

348.177,058

824

349.f3+5,5

88f 2


21/31

29350.

351. 352. 353. 354.

355. 356. 357. 358. 359.

360.1

361.-

0,11764706

362.-

0,17647059

363.29,4117

647364.

365.0

366.1

367.-

0,14121037

368.17,0028

818369.

370.0

371.-

2,5072E-09

372.20,3285

303

373.272,074

928

374.f3/20,3

28375.

376. 377. 378. 379.

380. 381. 382. 383. 384.

385.1

386.

-0,11764706

387.

-0,17647059

388.29,4117

647

389.

390.0

391.1

392.-

0,14121037

393.17,0028

818

394.

395.0

396.-

1,2333E-10

397.1

398.13,3838

956

399.

400@

+ 'etermine el 2olinomio de nterpolación de agrange para la siguiente ta la.

401@

402@


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40 @

P3 ( x)= f ( x0) ( x− x1) ( x− x2 )( x− x3 )( x0 − x1 )( x0− x2)( x0− x3 )

+f ( x1 ) ( x− x0)( x− x2)( x− x3)( x1− x0)( x1− x2 )( x1 − x3 )

+f ( x2) ( x− x0) ( x− x1)( x− x3)( x2− x0 )( x2− x1)( x2− x3)

+

404@405@

406@

P3 ( x)=− 2( x− 3 ) ( x− 5 ) ( x− 7 )(1 − 3 ) (1− 7 ) (1− 7 )+1

( x− 1 ) ( x− 5) ( x− 7 )(3− 1 ) (3− 5) (3− 7 )+2

( x− 1) ( x− 3 ) ( x− 7 )(5− 1) (5− 3 ) (5 − 5 )− 3

( x− 1 ) ( x− 3 )( x− 5 )(7− 1)(7− 3 )(7 − 5)

407@

40!@

P3 ( x)=− 2( x− 3 ) ( x− 5 ) ( x− 7 )

− 48 +1( x− 1 ) ( x− 5) ( x− 7 )

16 +2( x− 1) ( x− 3 ) ( x− 7 )

− 16 − 3( x− 1 ) ( x− 3)( x− 5)

48

40"@

P3 ( x)=− 2( x− 3 ) ( x− 5 ) ( x− 7 )

24 +1 ( x− 1 ) ( x− 5) ( x− 7 )

16 +2 ( x− 1) ( x− 3 ) ( x− 7 )

8 − 3 ( x− 1 ) ( x− 3)( x− 5)

16

410@

411@

P3 ( x)=( 116 ( x− 1)( x− 5))[ ( x− 7 )−( x− 3 )]+[18 ( x− 3 )( x− 7 )][( x−5

)3 −( x− 1 )]412@

41 @

x− 7− x+3

P3 ( x)=( 116 ( x2 − 6 x− 5 ))[¿]+[18 ( x2 − 10 x+21 )][( x− 5 )3 −( x− 1 )]

414@ P3 ( x)=( 1

16 ( x2

− 6 x− 5 ))[− 4 ]+[1

8 ( x2

− 10 x+21 )][( x− 5 )

3 −( x− 1)]415@ P3 ( x)=( 116 ( x2 − 6 x− 5 ))[− 4 ]+[18 ( x2− 10 x+21 )][ x− 5− 3 z+33 ]416@ P3 ( x)=( 116 ( x2 − 6 x− 5 ))[− 4 ]+[18 ( x2− 10 x+21 )][− 2 x− 23 ]417@ P3 ( x)=(

− 14 ( x

2

− 6 x+5 ))+[18 ( x

2

− 10 x+21 )][ x+1

3 ]− 2


23/31

41!@ P3 ( x)=− (14 ( x2− 6 x+5 ))+[14 ( x2− 10 x+21 )][ x+13 ]

41"@ P3 ( x)=−

( x2

4− 3 x

2+ 5

4)+

[ x2

4− 5 x

2+ 21

4 ][ x+1

3 ]420@ P3 ( x)=− ( x24 − 3 x2 + 54)+[ x

3 + x2

12− 5 x

2+5 x6

+ 21 x+2112 ]

421@ P3 ( x)=− x2

4+3 x

2− 5

4− x

3

12− x

2

12+ 5 x

2

6+5 x

6− 21 x

12− 21

12

422@ P3 ( x)= x3

12 + x2

2 +7 x12 − 3

42 @

424@

425@

426@

427@

42!@42"@

+34.

5 'etermine el 2olinomio de nterpolación $sando la nterpolación de 'i6erencias'i7ididas de )e8ton, e interpole en el punto 9 : 3

4 1@

4 2@

4 @

4 4@ P3 ( x)= b0 +b1 ( x− x0)+b 2( x− x0) ( x− x1)+b3 ( x− x0 )( x− x1)( x− x2 )

4 5@

4 6@ b0 = f ( x0 )= 1430

4 7@


24/31


25/31

454@ ( x4 ! x3 ! x2 ! x1)=f ( x4 ! x3 ! x2)− f ( x3 ! x2 ! x1)

x4 − x1= 9− 49− 4− 6 =

− 40− 10 = 4

455@ P3 ( x)= 1430 +522 ( x− 7 )+69 ( x− 7 ) ( x− 6 )+4 ( x− 7 )( x− 6 )( x− 4 )

456@ P3 ( x)= 1430 +522 x− 3654 +69 ( x2 − 13 x+42 )+4 ( x2− 13 x+42 )( x− 4 )

457@ P3 ( x)= 1430 +522 x− 3654 +69 x2− 897 x+2898 +4 ( x3− 13 x2+42 x− 4 x2 +52 x− 168 )

45!@

45"@

P3 ( x)= 1430 +522 x− 3654 +69 x2

− 897 x+2898 +4 x3

− 52 x2

+168 x− 16 x2

+208 x− 672

460@ P3 ( x)= 4 x3+ x2+ x+2

461@ P3 (2 )= 4 (2 )3 +(2 )2+2+2

462@ P3 (2 )= 40

46 @ P

3(3 )= 4 (3 )3 +(3 )2 +3 +2

464@

465@ P3 (3 )= 122

466@

; 2ara la siguiente ta la o tenga el 2olinomio de nterpolación de di6erencias 6initasde )e8ton e nterpole en el punto 9 : -0+


26/31

4!4@4!5@4!6@ P3 ( x)= b0 +b1 ( x− x0 )+b 2 ( x− x0 ) ( x− x1 )+b3 ( x− x1 )( x− x2)

4!7@

4!!@ b0 = f ( x0 )=− 44!"@4"0@4"1@ P*, +*) ,9+*+'.,) ,=, , )4"2@

4" @ b1= ( x1 ! x0 )=f ( x1 ! x0 ) x1− x0

=

− 329

−(− 4 )

− 23

−(− 1 )=

4913

= 43

4"4@ ( x2 ! x1 )=f ( x2 ! x1) x2− x1

=

− 83 −(

− 329 )

− 13

−( − 23

)=

8913

= 83

4"5@

4"6@ ( x3 ! x2 )= f ( x3 ! x2) x3− x2

=− 2 + 8

313

=

23913

= 2

4"7@4"!@ S+ ' ) ,9+*+'.,) ,=, , )4""@

500@ b2= ( x2 ! x1 ! x0 )=f ( x2 !x1 )− f ( x1 ! x0 )

x2− x0=

83

− 43

− 13

+1=

4323

= 2

501@

502@ ( x3 ! x2 ! x1 )= f ( x3 ! x2 )− f ( x2 ! x1 )

x3− x1=

2 − 83

0 +23

=

− 2323

=− 1

50 @504@ T+*.+*) ,9+*+'.,)505@506@

507@ b3 = ( x3 ! x2 ! x1 ! x0 )=f ( x3 ! x2 ! x1 )− f ( x2 ! x1 ! x0)

x3− x0= − 1 − 2

0 +1 =− 3

1=− 3

50!@

50"@ P3 ( x)=− 4+43 x( x+1 )+2 ( x+1 )( x+

23)− 3 ( x+1)( x+

23 )( x+

13 )


27/31

510@

511@ P3 ( x)=− 4+43

x+ 43

+2 ( x2+ 53

x+ 23

)− 3 ( x2+ x+ 29

)( x+1 )

512@

51 @ P3 ( x)=− 4+43 x+

43 +2 x

2

+103 x+

43 − 3 ( x

3

+ x2

+29 x+ x

2

+ x+29 )

514@

515@ P3 ( x)=− 4+43

x+ 43

+2 x2 + 103

x+43

− 3 ( x3 +2 x2+ 119

x+ 29

)

516@

517@ P3 ( x)=− 4+43

x+ 43

+2 x2 + 103

x+43

− 3 x3− 6 x2− 113

x− 23

51!@

51"@ P3 ( x)=− 3 x3

− 4 x2

+ x− 2520@

521@ x=− 1415

P3(− 1415 )=− 3(− 1415 )3

− 4(− 1415 )2

+(− 1415 )− 2522@52 @

524@ x=− 1415

P3(− 1415 )= − 1492375525@526@0-2. 2. Dados $os !untos (?/.0+ 7.2,+ (? .-+ -. ,+ (?1./+ .3,+ (7.3+ 0.7,+ (-.0+

0.0,+ (/.1+ 0. , d#t#r in# $os !o$ino ios d# *rado / & 0. Gra%icar !arad#t#r inar $a cur"a s a!roHi ada.

0-3.0-6.0 7.0 1.

3. 9$ant## & so$ucion# dos #'#rcicios so)r# $a t# tica d# Trans%or adadiscr#ta d# 4ouri#r.

0 -.

0 .

)@ x[n ]= {1,2,1,0 }

0 /.

5 5@ # = 0 X $ [0 ]= ∑ x[n]= 1+2+1+0= 4


28/31

5 6@ # = 1 X $ [1 ]= % x[n]exp (− j2 & 4 )= 1 +2exp (− j& 2 )+exp (− j& )0− j2

5 7@ # = 2 X $ [2]= % x[n ]exp

(− j 2 & 2

4 )= 1+2exp (− j& )+exp (− j2 & )= 0

0 3. # = 3 X $ [3]= % x[n ]exp (− j 2 & 34 )= 1+2exp (− j3 & 2 )+exp (− j 3 & )= j 2

5 "@ P&* (& )' & () D T + x[n ]es X $ [# ]= {4,− j 2.0 . j2 } para# = 0,1,2,3

540@

541@542@

3@

x[n ]= ' [n]⇒ ~ x= ∑# = (

(

' [n− #) ]

54 @

544@ a# =1

) ∑n= ⟨ ) ⟩~ x[n ]. e

− j n# 2 & ) = 1

) para 0 * n * ) − 1

545@546@ x1[n]= ' [n]⇒ X 1 (+ )= 1 X 1 (+ )≠ X 2 (+ )

547@54!@ x2 [n]= ' [n− ) ]⇒ X 2 (+ )= e

j+)

54"@550@

551@ S/(& >); + )')(,-)* +' + 1=2 ) ; +' + & =)(&*+

X 1 (+ )= X 2(+ )⇒ noi,portael -

552@

55 @ X (# )= a # =1

) ∑# = 0 ) − 1

x[n]. e− jn# 2 &

) x[n ]= ∑

# = 0

) −1 ~ X [# ]. e jn# 2 &

)

554@


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555@ 0 * n* ) − 1 1 *# * ) − 1 556@557@ T*)' 9&* ) ) D, .*+ ) + & *,+* 55!@

55"@ x[n ]= ∑# = 0

) − 1

X [# ]. e jn# 2 & ) ∨¿[∑# = 0

) − 1

X ∗[# ]. e− jn# 2 & )

]560@561@562@56 @564@565@566@567@56!@56"@570@571@572@57 @574@575@576@577@57!@57"@5!0@5!1@5!2@5! @5!4@5!5@5!6@5!7@5!!@

5!"@5"0@5"1@5"2@5" @5"4@5"5@5"6@5"7@5"!@5""@

600@601@


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7-. CONCLUSIONES7 .7/.

605@ C&' +( :*+ +' + *)3) & .&()3&*) ,=& + .& :*+' ,+*&' (& .&'.+: &+ () ', ) II +( . * & + M? & & N ?*,.& @ A( '& + (& .&'.+: &

+ ,) & + *+().,&')' .&' (& ? & & , +*) ,=& + +(, ,').,/' ; (),' +*:&().,&'+ + :&(,'& ,& 8 :&* + + & ,=& + ,+'+ +( .&'&., ,+' &'+.+ )*,& :)*) ,(,-)*(& +' .) & *+)(+ + () =, ) :*&9+ ,&')(@

606@607@60!@60"@610@611@612@61 @614@615@616@617@61!@61"@620@621@622@62 @624@625@626@627@62!@62"@6 0@6 1@6 2@6 @6 4@

6 5@6 6@6 7@6 !@6 "@640@641@642@64 @644@645@

646@ /2. BIBLIOGRA4IA


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/3.64"@650@651.

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