Аудиториски вежби 2013

4

8. :

{ }

djdF

ttf )()( =

{ } nn

n

djFd

tfjt

)()()( =

9. :

== dftfdtfftftf )()()()()(*)( 212121

{ } )()()(*)( 2121 jFjFtftf =

10. :

== pi

pi

pi

djFjFdjFjFjFjF )]([)]([21)]([][

21)(*)(

21

212121

)()()(*)(21

21211 tftfjFjF =

pi

1. )(tf

t2/2/

2

2sin

)(

EjF =

2. )(tf

t2/2/

2

4

4sin

2)(

=

EjF

3. nF )(tf ,T )(0 jF )(0 tf 0 n= :

>

>

>

11f

12f

13f

11F

12F

13F

1S123f

F

gf

2S

21F

22F

23F

21A

22A

23A

21f

22f

23f

:

=

=

=

+

+=

nnn

TnTtstfTnTtstfnTtstftf

32)(

3)()()()( 00130012011123

)(ts . :

=

=

=

+

+=

nnn

TnTttfTnTttfnTttftf

32)(

3)()()()( 00130012011123

, ( ) )(tf :

=

=

=

++

++=

nnn

TnTt

TnTfTnTtTnTfnTtnTftf

32)

32(

3)

3()()()( 0000130000120011123

)(tT :

58

=

=

==

n

tjn

n

T eTnTtt 01)()( 0

{ }

=

=

=

===

nnn

T nnTnTtt )()(2)()( 0000

pi

{ }=)()( ttu Tm [ ]

=

=

=

n

m

n

tjnm njUTeTtu )(

11)( 00

.

)(11 tf

)(12 tf

)(13 tf

t

t

t

t

)(123 tf

kHzfm 4=

kHzff m 820 ==

sfT m125

21

0 ==

s

NT 67,41

300

===

0T

T

F ( )0(11f ) :

pi

deEeAty tjtjFg

g

=

2

2sin

21)( 1010 0

59

2

2sin

1

2

2sin

=

, :

)()(sin

2)(0

01010 tt

ttfEAtyg

ggF

=

,

. )(11 tf :

=

m

n fnfE

2111

n )(11 tf :

)2

(

)2

(sin2)(

0

0

11

m

g

m

g

gnFn

fn

tt

fn

tt

fEAty

=

,

.

t

t

NT0

gf21

0T

1 12

3

gfN

T2

10= ..

mffT

211

00 ==

60

gm fNf 21

21

= mg Nff = , kHzfm 4= kHzf g 12=

2S F :

11

t

)(2 0nTffA gF

T

21F

2S ( ) gnF fEA 21 . 21F ( )0(11f ) :

=

m

m

deefEAAety tjtjgFtj

pi

0'

0 221)( 10210

)()(sin4)(

0'

0

0'

0210210 ttt

tttEAAffty

m

m

Fmg

=

21F )(11 tf :

;)

2(

)2

(sin4)(

0'

0

0'

02

121

m

m

m

m

nFmgn

fn

ttt

fn

ttt

EAAffty

=

=

m

n fnfE

2111

,1== FAA ,42

mg ff :

mg ffAAA 2232221 4

1

===

.mf

) kHzff mm 721 == kHzfm 153 =

max0 2 ff { }321max ,,max mmm ffff =

61

kHzkHzf 301520 == sfT 3.331

00 == s

TNT

T 1.11300

===

kHzNff g 450 ==

) 21F kHzfm 71 = ( ) :

=

=

nm

m

mg

fn

t

fn

t

fnffftf

)(

)(sin4)(

01

01

011

2121

,1mf 2mf .3mf

:

21

2221 221

mg ffAA == [ ]dBAA 2.772221 ==

20

23

23 21

221

ffffA gmg== [ ]dBA 6.7023 =

2S , miff 20

=

=

nm

m

mg

fn

t

fn

t

fnffftf

)(

)(sin4)(

01

01

011

2121

,

t

t

m

m

1

1sin

12 mfk

,

)(

)(sin

01

01

fn

t

fn

t

m

m

.

2 30 mfn

fn

=

.

) kHzf 150 =

00

1fT = 1 2 3.

62

11f

12f

13f

1S 2S

21f

22f

23f

1

3

t

0T

3

21

:

gf

21

40

= gf2

11541

=

kHzkHzf g 30152 == .

,1mf ,2mf ,3mf :

21

2221 221

mg ffAA == [ ]dBAA 7.802221 ==

23

23 221

mg ffA = [ ]dBA 15.7423 =

19. . :

1)

2) | |

( ) .a t

Tp t e a const

= =

3)

00

0

0 1)sin()(T

ftf

tfth ==

pi

pi

2)sin()(

=

tftf

tum

m

m pi

pi ( )T t max

. ( ), ( ), ( ), ( )x t y t X j Y j .

63

:

1( ) ( ) ( ) ( ) ( ) ( )Tn n

u t t p t p t nT d p t nT

= =

= = =

,

.

:

12 2

2

( ) ( ) ( ) , ( ) ( ) , ( ) ( )( )

j tT T

n

a

TU j j P j j n P j p t e dta

T

=

= = = =+

1( ) ( ) ( ) ( ) ( ) o n n

U j P j n P jn n

= =

= =

:

1( ) ( ) ( ) ( ) ( )m mn

x t u t u t u t p t nT

=

= =

1 11 1( ) ( ) ( ) ( ) ( ( ))

2 2

1 1( ) ( ) ( ) ( ) ( ( ))2 2

m m

m o o m

n n

X j U j U j U jv U j v dv

U jv P n n v dv P n U j n

pi pi

pi pi

= =

= = =

=

n=0 ( )mU j . ( )mU j

n . n

2 2

2

( )( ) ( )

o

o

a

TP na

nT

=

+ n.

64

( )x t (3) , 2o mf f= :

2

22(0)

( )

aTTP

a a

T

= = :

65

:

2 2

2sin sin1 1( ) m m

m

m m m m

f t f ty t faf f t af f t

pi pi

pi pi

= =

20. :

1)

tftf

tU m0

0 )sin()(pi

pi= ( )T t

1of T=

2) | |

( ) .a t

Tp t e a const

= =

3) sin( ) oo

f th t f tpi

pi=

( ), ( )x t X j a ( ) ( ) ( )m

t y t u t = 0t = .

: :

=

=

===

n

m

n

mTm nTtnTunTttuttutu )()()()()()()(1

=

=

===

n

m

n

m nTtpnTunTtnTutptutptx )()()()(*)()(*)()( 1 :

11 1( ) ( ) ( ) ( ) ( ( ))

2 2

1 1( ) ( ) ( ( ))2

m T m T

m m

n n

U j U j j U jv j v dv

U jv n v dv U j nT

pi pi

pi

= =

= = =

=

1( )U j ( )mU j .

1( )( ) ( ) ( ) ( ( ))m

n

P jX j P j U j U j nT

=

= =

66

( )X j

( )mU j , ( )P j , 2 2

2

( )( )

a

TP ja

T

=

+.

( )X j . ( )p t

| |( ) .

a t

Tp t e a const

= =

:

1 1( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )m mn n

x t u t p t p u t d p u nT t nT d u nT p t nT

= =

= = = =

2

2 2

2

21 1 1 1( )

2 ( )

o

o

j t

o o

a

Ty t e daf T fT

pi

=

+ .

0t =

2 2 2

2 22 2 0 0

2

0

21 1 1 2 2 1 2 1(0)

2 2( ) 1 1

2 2 12 2

2

o o o

oo o o

o

oo o

a

T a TTy d d da af f a f T aT TT T a a

T farctg arctg tg a ff a f a a tg

pi pi pi

pipi pi pipipi pi

= = = =

+ + +

= = = = =

NF ( )r

h t

: 1( ) ;( ) 2

okH j P j

=

21. . 1( )mu t 2 ( )mu t

: 0

sin( )( ) om

f tU t f tpi

pi=

67

. .

NF gf . gf .

, , ', '', ', ''A B C C D D .

:

1 21 1( ) ( ) ( ) ( ) ( )

2 2A m mn n o mT

u t u t t nT u t t nT f f

= =

= + = =

21 2

1 1( ) ( ) ( ) oo oTjnjn t jn t

A m mn n

u t u t e u t e eT T

= =

= +

1 21 1( ) ( ( )) ( 1) ( ( ))nA m o m o

n n

U j U j n U j nT T

= =

= +

2 cos( ) sin( ) cos( ) ( 1)oTjn jn ne e n j n n pi pi pi pi = = = =

68

: 1

2 gTN f= . :

1 1 1 22 2 2 g m oo m g

T f f fN Nf f f= = = = =

( )AU j . :

( )Bu t :

' ''( ) ( ) ( ) ( ) ( ) ( )

2C B C Bn nT

u t u t t nT u t u t t nT

= =

= =

=

=

n

BC njUTjU ))((1)( 0'

=

=

==

n

jnB

n

Tntj

BC enjUTenjUTjUpi

pi

))((1))((1)( 02

0"

0

70

22. =1,68Mbps. 30 . , , ,

n . max n.

(-5,5)(V). .

:

.

of n N , : of -

n -

N -

6

31.68 10 7 [ ]

2 2 4 10 30o o mC C bitif n N C n nf N f N primerok

< < = = =

.

U , : UUq

= ( , )2 2U U

, q .

7 10( )2 2 128; ( , ) ( 5,5) 10( ) 78.1252 2 128

n U U Vq U V U mV= = = = = = =

:

22( ) 1 ( )

12 12NqU UP

q

= = ( 1R = ,

).

: 2sP s= , 2s

: 22

2 2 2 3 3 2 2

2

1 1 1 1( ) [( ) ( ) ]2 2 12 12

U

U

U U Us s p s ds s ds q U

U U s

= = = = =

/ :

71

2

2 2 22

2

112 128 16384 10log 42.144[ ]1

12

s

Nq

UP q q dBUPq

= = = = =

23. :

(1) , (2) q=8 , (3) .

x(t) [ , ]

2 2A A

.

NqP . / . .

x(t) 4 kHz?

:

1

q

Nq Nqii

P P=

=

72

1 2 22 2 3

223

3

2 2

1 1 1

1( ) ( ) ( ) ( ) ( ) ( ) ( )3

( )1( ) 2( ) ( ) ,3 2 12

( ) ( )( ) ( )12 12

iiqi

qii

ii qii

qi

xxx

xx

Nqi qi qi qi qi qix

x xxx

i iNqi qi qi i

q q q

Nq Nqi qi qi Nqi i i

P x x p x dx p x x x p x dx p x x x

x xP p x p x x x

x xP P p x x p x x P

+

+ +

= = =

= = =

= = =

= = =

2( )

12x

=

22

2

( )A

SA

P x p x dx

= ( )p x [ , ]2 2A A

1( )p xA

= .

3 3 22

3

2

1 1 2,

3 3 2 12

A

SA

x A A AP xA A q

= = = =

2 2

22 2

2

12 12, 8 64 2 8 3[ ]( ) 1

12 12

ns s

Nq Nq

A AP P bitiq q q n ld

x AP P primerokq

= = = = = = = =

:

32 2 4 10 3 1 24b o mf f n N f n N kbps= = = = ( 1N = , 4mf kHz= )

24. 256 , N 4kHz.

( ) ( ) n bn

U t a x t nT

=

= ,

n

a 0 1, bT ( )x t :

1 ,2( ) 0.5

0 ,2

tx t s

t

= = >

:

73

1 ,2( ) 0.5

0 ,2

th t s

t

= = >

N ,

. 0

max

2E

, 1 max2

E

. maxE .

:

: ( ) ( ) , {0,1} n b nn

U t a x t nT a

=

= ( )x t (). :

( ) ( )* ( )y t x t h t=

:

=

=

n

bn nTtyatU )()(

( ) ( ) ( )y t h x t d

=

1. t <

2. 0t < <

74

3. t <

4. t >

=

=

n

bn nTtyatU )()( . 101 .

t yt y

= = . max (0)E y = = ,

:

(0) (0)2 ( ) 2 22 2 4

34 4

b

b

y yy T y t t

T t

> > = >

= =

b , b =

nNf

m

b

= 21 25.312

1=

=

nfN mb .31=N

75

25. N 4kHz ,

. [ ; ]2 2U U

0, 2U V= .

1 W . 3E V= 0.5 s = , . NF

0.825gf MHz= . . N: ) ) . max .

1 0.6062v g

sf = =

:

max

12 8 125oo

f f kHz T sf = = = = ; TTN

=

.

bT TTn nN

= =

.

2

2

2

( )12 2

1 1 , 0.212

Nq n

Nq

U U UP Uq

UP W U Vq

= = =

= < =

7.57=q , q 2n 664 2q = =

6n ldq= =

76

) bT , .

, 18.83 18bb

TT N NnT

= = = =

) 101 .

:

maxmax2 22 4

y ty y t

> = <

( ) 0.4282

VbT t s

= + =

, 0.5 s = bT < ,

0.5bT s= . 41.67 41N N= =

: { })()()( = txtxEtp HH

:

{ })()()( = tytyEty HH : , 1.106V s = + =

:

max

V

y E

=

:

max

max

y t ty yy

= =

77

26. 0 .2N .

: ).()( APAP =

: ..

:

2

2

2

21)( N

n

N

enp pi

=

:

)()()( tntsty +=

t

)(ts

A

A

t

)(tn

)"0/"(yp )"1/"(yp "0" ".1"

)(2

1)"0/"( 02)(

2

2

ypeyp NAy

N

==

+

pi

)(2

1)"1/"( 12)(

2

2

ypeyp NAy

N

==

pi

78

A A

)"0/"(yp )"1/"(yp

y

, ,0=pU .

:

)."1/"()"1(")"0/"()"0(" PPPPPe +=

+=p

p

U

Ue dyypdyypP )"1/"(2

1)"0/"(21

, :

+

===

N

N

p A

u

NN

Ay

NUe duedyedyypP

pipi

20

2)(

22

2

22

12

1)"0/"(

.

2212

21

2

2

==

NA

u

e

AerfcdueP

N

pi

:

.

221

log12

=

Nb

derfc

MM

MP

:

.2Ad =

27. ./50 sbRb = , . ,5.0 V .2.0 V .

: , .

:

79

.2.0 VN =

:

( ) 31045.577.121

2.025.0

21

221

==

=

= erfcerfcAerfcP

Ne

,ebe PRN = :

.7.31045.550

1113 sPRN

tebe

p =

=

==

28. ,1MHzf g = .5.0 VA = : ./02.0)( HzWSS NN ==

: :

.04.02)(2 WSfdffSg

g

f

fNgNN ===

, :

.2.0 VN =

:

( ) 31045.577.121

2.025.0

21

221

==

=

= erfcerfcAerfcP

Ne

29. .4.02 WN = (SNR) dB10 "1" ,2.0)"1(" =P .

:

"0" :

.8.0)"1("1)"0(" == PP

80

:

.)(8.0)(2.0 222 AAAPs =+=

SNR :

.10log10log10 21010 dBPSNR

N

s==

1022

2 ==

NN

s AP

.2.0 VA =

,

Bayes:

,)"1(")"0("

)()(

0

1

PP

UpUp

p

p=

)(1 yp )(0 yp "1" ".0"

2

2

2)(

0 21)( N

Ay

N

eyp pi

+

=

2

2

2)(

1 21)( N

Ay

N

eyp pi

=

A A

)"0/"(yp )"1/"(yp

y

, :

2

22

2

)()(

42.08.0

N

pp AUAU

e

+

==

:

81

.138.04ln2

2

VA

U Np ==

:

)."1/""0(")"1(")"0/""1(")"0(" PPPPPe += ( ) ( )

+

+=+=p

N

p

N

p

p

U Ay

NU

Ay

N

U

Ue dyedyedyypdyypP

2

2

2

2

22

212.0

218.0)"1/"(2.0)"0/"(8.0

pipi

.1066.622

12.022

18.0 4=

+

+=

N

p

N

pe

UAerfcUAerfcP

30.

.

.2N

: ) :

bA bA

)(0 yp

y

)(1 yp

"0" "1" :

2

2

2)(

0 21)"0/"()( N

bAy

N

eypyp pi

+

==

2

2

2)(

1 21)"1/"()( N

bAy

N

eypyp pi

==

:

+

=

0

2)(

2

2

21 dyeP N

bAy

Ne

pi

82

.

221

=

N

be

AerfcP

:

( ) 222 5.05.0 bbbbs AAAP =+=

:

.

221

221

2

2

=

=

b

N

be

SNRerfcAerfcP

) .

uA

)(0 yp

y

)(1 yp

"0" "1" :

2

2

20 2

1)"0/"()( Ny

N

eypyp pi

==

2

2

2)(

1 21)"1/"()( N

uAy

N

eypyp pi

==

:

( ).

2221

21

21

21

21 2 2

2

2 22

2

2

=+=

N

u

AAy

NA

y

Ne

AerfcdyedyeP

u

N

u

u

N

pipi

( ) ,2

5.005.02

2 uu

Us

AAP =+=

:

83

=

=

421

821

2

2 SNRerfcAerfcP

N

u

e

:

N

b

N

u AA 222

=

bu AA 2=

:

( ).22

22

22

22b

sbbuU

s PAAA

P ====

31. - . 5=m l . :

) ; ) .

A "0" ".1" .2

:

i) ;8.02

=

N

RA

ii) ;22

=

N

RA

iii) ;32

=

N

RA

: RA .

:

) :

)(1 tu

)(2 tn )(1 tnm )(tnm

,l ,l ,l ,l

)(1 tn

)(tum>> > >A RA A RA RA

,l

:

84

.10 ][][ ldBlNpR AeAA

==

,

.

, ,

:

.

2

1

22N

m

iNN mim ==

=

:

=

N

Re

m

AerfcP

m 221

:

N

RA2

0.8 2 3

N

R

m

A2

0.358 0.89 1.34

5eP 0.306 0.103 0.029

) :

)(1 tu )(2 tu )(1 tum )(tum

,l ,l ,l ,l

:

.

221)1(

==

N

Re

AerfcPp

.5=m , :

=

=

5,3,1

)( ),,(k

e

m

e kmPP

85

),( kmPe k .m :

,)1(),( kmkkme ppCkmP = .)!(!

!kmk

mC km

=

5 : 052341 )1(

55)1(

35)1(

15

ppppppPe

+

+

=

)(meP :

N

RA2

0.8 2 3

)1(eP 0.129 2.339 310 1.1 510

)5(eP 0.387 1.1585 210 5.5 510

,

.

32. , V 3,0= 1 )( jfHT ).( jfH L )( jfH R .

.02.0)(HzWSfS NN ==

. .

1f1f

)( jfHT1

12 f12 f

)( jfH L1

1f1f

)( jfH R1

)( jfHT )( jfH L )( jfH R

)(tn

kT

)0(y)(tx

TP

86

: )()( ttx =

1)( =jfX )()()()()( jfHjfHjfHjfXjfY RLT=

1f1f

)( jfY1

)(ty . :

=

=

ii iTtyAtU )()( iA

.A 0)( =kTy 0k ,

2 1fnT = n .

:

.04.021)()( 122

1

1

WfSdfSdffHfS Nf

fNRN ====

).0(yAi

102 2)()()0( fdfjfYdfejfYy

t

ftj===

=

pi

0, 12 fA , 1, 12 fA . ,

.. .0=pU :

0013.02

321

2)0(

21

2=

=

= erfcAyerfcPe

.A

= dfjfHjfXTAP T

s

T22

2

)()(

=

dfjfHjfXTPA

T

sT

22 )()(.

)0(y A :

87

=

dffHfSdfjfHjfX

dfjfYTPerfcP

RNT

sT

e222

2

)()()()(

)(

21

Аудиториски вежби 2013

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