148

Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

Embed Size (px)

Citation preview

Page 1: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

Page 2: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• ! " # $%& $%&$''

• (%&$')#*

• (

+

Page 3: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• %&$', *

$'#

*

,V I+ +

,V I− −

-

Page 4: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• .

• +*

%&$'%&$'%&$'

#

+* &

Page 5: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• %&$'+* $'

/* #$'0* #

1

Page 6: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• % )

• ( # ) #

2

Page 7: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• " #

3

Page 8: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• ( # 4567 #) 8 + 8

9

Page 9: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• %&$'*

$' #

:

Page 10: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(+)

58 6

# #

• . +

1 2 2

1 2 2

V A V BI

I CV DI

= +

= +2

1

2 0

VV I = 2

1

2 0

VI V =

;

Page 11: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• !"

$'

%&$'" $'"

$'<

Page 12: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• "

• "

+

Page 13: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• <

-

Page 14: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• % )

• # • #

• ( )

Page 15: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• !

56 #

1

Page 16: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• %&" +*

2

Page 17: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• =

• (

3

Page 18: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

#$

• > # ?

• 6 @ ? ?

• (

• # )

9

Page 19: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• " # )

8

• " #

A A

#

:

Page 20: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• (

) )

#

1 11 1 12 2

2 21 1 22 2

V S V S V

V S V S V

- + +

- + +

= +

= +

+;

Page 21: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

8

+

• ( # )

2V -2V +

• ( # )

+

Page 22: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• %&(

+*

1V +2V +

1V −2V −

++

Page 23: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• %

• ( B

ijS

0,k

i

j k j

VV

V

-

+

= ¹

jV+

+-

Page 24: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

&

• &

• ( 8 +

• 4/7 #

+

Page 25: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

+1

Page 26: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• .) 4/7

4"7 #4"7 #

• <

)&&)

+2

Page 27: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• % )

+3

Page 28: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• ( # )

+9

Page 29: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• # /

#

• " )

• / #

• 58 +)

+:

Page 30: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• 8 +)

( )012n n nV V Z I+ = +

• ( # )

( )012n n nV V Z I- = -

-;

Page 31: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

-

Page 32: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• " #C="

? )

#

&&& &&&

• ( )

&

• > [ ][ ]( )' *1

Re2

P V Iav =

-+

Page 33: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

--

Page 34: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• " )

?A #

• =

[ ]n sI

• =

2*Re 0 Re 0m mn n n nnI Z I I Z= = =

-

Page 35: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• ?

A #

• ()

• % )

2 2* *Re 0m mn n n nm m n nn m mmI Z I I Z I I Z I Z+ + + =

-1

Page 36: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

-2

Page 37: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• =) #

• " ) #

-3

Page 38: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

-9

Page 39: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

-:

Page 40: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

;

Page 41: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

Page 42: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

+

Page 43: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

-

Page 44: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• $'C

#

• % 56 #)• % 56 #)

Page 45: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• (+ 4/78

4567 #

• " )

1

Page 46: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

$'C

#

• ( #)

2

Page 47: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

3

Page 48: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

9

Page 49: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

'(&

• *

.* D*

%&%&

• ( ) AE;

+AE

:

Page 50: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• %&$'D* $'.*

1;

Page 51: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• .*

#

CZ

CD

ZC

AZ cba

111 =−=−=

BD

YB

AY

BY cba

111 −=−==

1

Page 52: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• . #

#

21122211 )1)(1( ssssA

+−+= 211222110 )1)(1( ssssZ

B−++

=21

21122211

2)1)(1(

sssss

A+−+

=21

211222110

2)1)(1(

sssssZ

B−++

=

210

21122211

2)1)(1(

sZssss

C−−−

=21

21122211

2)1)(1(

sssss

D++−

=

1+

Page 53: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• .?

/)/ /

• (

) )

• ( A #) <

<

1-

Page 54: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

) * " '" !'

!+ ,

6 &C "F

1

Page 55: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

• ()

?

• .

G"! C

11

Page 56: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

H H

• .$

'

12

Page 57: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

• (

– #

– 5– 5

– (

– B

13

Page 58: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

• ! *

?

• 5)

? ?

19

Page 59: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

• (

1:

Page 60: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

-

• (? *

#

• . #

2;

Page 61: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

• < #CIBJ

/;

• . CIBJ • . CIBJ

# K@

? *

%&

2

Page 62: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

• L *

#A

'A

2+

Page 63: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

• < #

/;$

/;'

• .• .

/

• .

2-

Page 64: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

• <

• .

l01

R jXZ

Z+=

• . 5 #%& A

• " %&

2

Page 65: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

• %&.$',

$' #

21

Page 66: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

• %&.$',

$' #

22

Page 67: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

oq max 0 01 maxZ Z Z r= ⋅ ⋅ oq min 0 01 minZ Z Z r= ⋅ ⋅

0qZ

• #6

31*B-1

1;?

23

Page 68: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

• .

>L >C

• .L 1;H+C

75 351.5 0.7

50L

jZ j

−= = −

• .L 1;H+C

+M1;$ %&'

• . ?

50 25 35.34oqZ = × = Ω 50 100 70.7oqZ = × = Ω

29

Page 69: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

. /$

• <*

• + • +

• % <* )

< ?

%&

2:

Page 70: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

• *

$' I

$' I

3;

Page 71: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

• % I )N;

• % )

0

11/( )

Z jXjB R jX

= ++ +

3

• C ? J5

0 1/( )L L

Z jXjB R jX

= ++ +

0 0

0

( )

(1 )L L L

L L L

B XR X Z R Z

X BX BZ R X

− = −− = −

Page 72: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

• " J ?

+ ??

? 5

3+

• N;

? 8

2 20 02 2

/L L L L L

L L

X R Z R X Z RB

R X

± + −=

+

0 01 L

L L

X Z ZX

B R BR= + −

Page 73: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• " C<O/;

0

1 1( )L L

jBZ R j X X

= ++ +

0 0

0

( )

( )L L

L L

BZ X X Z R

X X BZ R

+ = −+ =

"

3-

0( )L LX X BZ R+ =

0

0

0

( )

( ) /

L L L

L L

X R Z R X

Z R RB

Z

= ± − −

−= ±

Page 74: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

%& *

$' I

$' I 3

Page 75: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

• <*

E E

• ( )

31

Page 76: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

• "

• #6<*

C

/ E+;;*B;;;;/<E+;;*B;;;;

?1;;=A

32

Page 77: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

• /< $ A'IB#

<* %&

• .

/<$ A'

"

33

Page 78: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

• . )5)0< $ A'

• ) 5$" ?'

• .+

• " "

39

Page 79: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

• % )

6

• % 6)

$"

'

• 6 $+*B')

5$;& IB;&+')$;& IB;&1'6$*B&+'

3:

Page 80: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

0 0

0.3 0.95B B

jB j j C j C pFZ Z

ωω

= = = =

XZ001.2 38.2

XZjX j j L jXZ L nHω

ω= = = =

9;

Page 81: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

• " #$'" $'

<

! "#$%&

'

(

!

LY

LZ

)*%

9

Page 82: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

9+

Page 83: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

%$0

$'.*

9-

$'.*

&

$'C

?

$'

Page 84: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

#$&1 /$

• ! # && && $PQ' PQ'

• " ? ,=A

9

Page 85: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

%&"*

$'"

$'"

91

$'"

Page 86: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• +B

– !

• % *)

(

92

• % *)

– " "

";I

Page 87: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

– . R

– .

(

93

• % *)

– " /

/;I!

– . R!

Page 88: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• #% /<E2;*B9;)*$&&' 1;+,=A&

• (

(

99

• ( "

• IB +

Page 89: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• . $+'

• C! ., E$;&32*;&;21'SE;&;S+E$;&-+1*;&;21'SE;&+2;S

(

9:

E;&+2;S

• +) &E&;IB& 3+E&;*B& 3

• % ) )&&? RB& 3IB& 3

Page 90: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• <

• " &&

RB& 3)E$;&-1*;&+1'SE;&;:1S

• " &&

(

:;

• " &&

B& 3)+E$;&+1I;&1+'SE;& ;1S

• %&" #$'" $'

< $'

<

Page 91: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

LY

LZ

#

+

+#

,& ,&0.110λ 0.260λ

-../0/&/&

/

&

-../0/&/&

/

&

0.095λ 0.405λ

:

Page 92: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

:+

Page 93: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

:-

%$

C ?

Page 94: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• . )EH"EI

!

00

0

( ) tan( ) tan

L L

L L

R jX jZ dZ Z

Z j R jX dββ

+ +=+ +

(

:

2

2 20

20 02 2

0 0

1

(1 tan )where

( tan )

tan ( tan )( tan )[ ( tan ) ]

L

L L

L L L

L L

Y G jBZ

R dG

R X Z d

R d Z X d X Z dB

Z R X Z d

ββ

β β ββ

= + =

+=+ +

− − +=+ +

Page 95: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

• , E0;EH/;)

2 2 20 0 0 0

2 2

( ) tan 2 tan ( ) 0L L L L LZ R Z d X Z d R Z R Xβ β− − + − − =

± − +

• (C< E/;)#$%!H+;

• +

:1

2 20 0

00

[( ) ] /tan , for L L L L

LL

X R Z R X Zd R Z

R Zβ

± − += ≠

Page 96: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

1

0 0

1

1tan for - 0

2 2 2

1tan for - 0

2 2 2

L L

L L

X XZ Zd

X XZ Z

πλ

ππ

− ≥ =

+ − <

• . ? ) E*

:2

0 02 2 2Z Zπ

1 10

0 0

1 1tan tan

2 2Sl B B

Y Yλ π π− −

= = −

Page 97: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

1 10 0 01 1tan tan

2 2S

l Y YB Bλ π π

− − = − =

:3

Page 98: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

"1&1 /$

• ( B B

• . %&

• %&6$'" "$'( $'( +

:9

Page 99: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

LY

1y

'1

y

2y

'2y

1b

'1b

2b

'2b

d

0Y0Y 2jb 1jb 1

d

0Y0Y '2

jb'1

jb 1

::

2l

0Y 0Y

--2-- --2--

2l

0Y 0Y

--2-- --2--

Page 100: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

$'

;;

&

$'?*

&

Page 101: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

;

Page 102: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

• . $@') $@' IB

• ! )• ! )+ $@'+ $+@'IB

• . +$+@') )

;+

Page 103: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

• #6*

/<E2;*

B9;1;&.

&&SH9 &&&SH9 &

?+,=A&

• "

;-

60 801.2 1.6 0.3 0.4

50L L

jZ j Y j

−= = − = +

Page 104: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

• # IB

SH9&!

)

• .&&

;

'2 21.0 3.38; 1.0 3.38y j y j= − = +

'2 23.38; 3.38b j b j= = −

' '1 1 2 20.358 ; 0.204 ; 0.298 ; 0.146l l l lλ λ λ λ= = = =

Page 105: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

• ! SH9

&.

;1

'1 11.8 0.4 1.3; 0.3 0.4 0.1b b= − = = − = −

Page 106: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

LY1y

'1

y

'2y

1b

'1b

'2b

;2

1y

2y2b

Page 107: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

"#

;3

"#

$'"

*

Page 108: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

$'.

*

&

$'C

;9

$'C

?

$'

Page 109: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

• "

• . $ 5'

%&)

0 E, < IB$5<I5' 0< E, < IB5<0 E, < IB$5<I5' 0< E, < IB5<

. +)

• )CT0+UE0; , <

;:

1 02 0

0 1

( ) where tan

( )L L

L L

G j B B Y tY Y t d

Y jt G jB jBβ+ + += =

+ + +$?'

Page 110: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

222 0 1

0 2 2

2 220 1

0 2 2 2 20

( )10

4 ( )11 1

2 (1 )

LL L

LL

Y B t B ttG G Y

t t

t Y B t B ttG Y

t Y t

− −+− + =

− −+= ± − +

$?+'

$?-'

• "& )??

*

;

2 20 12 2 2

0

4 ( )0 1

(1 )Lt Y B t B t

Y t− −≤ ≤

+

Page 111: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

#$R ? ? ') ?$+'

20

0 2 2

10

sinL

YtG Y

t dβ+≤ ≤ =

?$+'

• .+ ?$'

2 2 20 0

1

(1 ) L LL

Y t G Y G tB B

t

± + −= − +

Page 112: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

(

• 5+ E2 2 2

0 0 0(1 )L L L

L

Y Y G t G t G Y

G t

± + − +

• .*

• . *

+

LG t

10

0

1tan

2l B

Yλ π−

=

10 01tan

2l Y

Bλ π− = −

Page 113: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

) " " '" !'

!+ ,

6 &C "F

-

Page 114: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

.. "C

• % )

B

?

• " • "

B

• <

#

Page 115: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

1

%$0> *

Page 116: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

.. "C

"*".

2 1 21 2 1 3

2 1 2

2 121 1 12 2

2 1 2 1

, , ,

2 21 , 1

L

L

Z Z Z ZZ Z Z Z

Z ZT T

Z Z Z Z

− −Γ = Γ = −Γ Γ =+ +

= + Γ = = + Γ =+ +

2

2 1 2 1

2 2 41 12 21 3 12 21 3 2

212 21 3

1 22 31

j j

j

j

Z Z Z Z

T T e T T e

T T ee

θ θ

θ

θ

− −

+ +

Γ = Γ + Γ + Γ Γ +

Γ= Γ +− Γ Γ

21 3

21 31

j

j

ee

θ

θ

Γ + ΓΓ =+ Γ Γ

131 <<ΓΓθje 2

31−Γ+Γ=Γ

Page 117: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

.

B

• VA• VA

3

Page 118: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

.

1 0 1, , ,n n L NZ Z Z Z Z Z+− − −Γ = Γ = Γ =

9

• /

• )

1 0 10

1 0 1

, , ,n n L Nn N

n n L N

Z Z Z Z Z ZZ Z Z Z Z Z

+

+

− − −Γ = Γ = Γ =+ + +

2 4 20 1 2( ) j j jN

Ne e eθ θ θθ − − −Γ = Γ + Γ + Γ + + Γ

0 , 1 1, 2 2, etc.N N N− −Γ = Γ Γ = Γ Γ = Γ

Page 119: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• ()

)

( 1) ( 1)0 1( ) [ ] [ ] jN jN jN j N j Ne e e e eθ θ θ θ θθ − − − − −Γ = Γ + + Γ + + +

( 1) / 2 ( )j jN e eθ θ−

−Γ +

/ 2NΓ

0 1( ) 2 [ cos cos( 2)

1

jNe N Nθθ θ θ−Γ = Γ + + Γ − +

:

/ 2

0 1

( 1) / 2

1], for N even,

2( ) 2 [ cos cos( 2)

1cos ], for N odd,

2

N

jN

N

e N Nθθ θ θ

θ

+ Γ

Γ = Γ + + Γ − +

+ Γ

Page 120: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

.

• .

A

?$W') X ?$W') X

$'

• .%

#

+;

Page 121: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

& & /$

• .

?#

• . ) *

) *

+

) *

YZ$W'Y ; ;

• " X$W'

2( ) (1 )j NA e θθ −Γ = +2( )

2 cos

Nj j

NN

A e e e

A

θ θ θθ

θ

− −Γ = +

=

Page 122: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• YZ$W'YE; WEDH+)$ YZ$W'YH' '

E;WEDH+ E)+)[ )(*

• .\E DH+ ESH

• 5 ;8 W;0

0

(0) 2N L

L

Z ZA

Z Z−Γ = =+

0

0

2 N L

L

Z ZA

Z Z− −=

+N

++

2 2

0

( ) (1 ) ,

!where

( )! !

Nj N N jn

nn

Nn

A e A C e

NC

N n n

θ θθ − −

=

Γ = + =

=−

2 2 4 20 1 2

0

( )N

N jn j j jNn N

n

A C e e e eθ θ θ θθ − − − −

=

Γ = =Γ + Γ + Γ + + Γ

Page 123: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• Z

• "Z ) ]+$*

'H$I')

Nn nACΓ =

1 1

1

1 0

1ln

2

ln 2 2 2(2 ) 2 ln

n n nn

n n n

N N N N Nn L L

Z Z ZZ Z Z

Z Z Z ZAC C C

+ +

+

− −+

−Γ =+

−Γ = =

E

+-

1 0

0 0

ln 2 2 2(2 ) 2 lnN N N N Nn L Ln n n n

n L

Z Z Z ZAC C C

Z Z Z Z− −+ −Γ = =

+ E E

Page 124: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

& & /$

LLL

LN

LNnn

Z

N

Z

N

ZZ

N

ZZ

ZZ

N

ZZ

ZZn

N

ZZ

=

=

=∴

=+

001

01

ln1

expln0

expln1

exp

ln20

exp

ln2

exp

+

LL

N

N

k

N

NL

N

n

kn

LN

LN

LN

LN

LN

ZZZk

N

ZZZZk

N

ZZ

ZZ

N

ZZ

N

Z

ZZ

ZZ

ZZZ

ZZ

=

=

=

=

+

=

=

=

=

+=

+0

001

0

001

000

000

012

ln2

2exp;ln

2exp

ln21

ln20

exp

ln21

expln20

expln21

exp

Page 125: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• .

1/1

0

2 | | cos

1cos

2 | |

2( )2 4

N Nm m

N

mm

m m

A

A

f fff f

θ

θ

θπ

Γ =

Γ=

−∆ = = −

2( )

2 cos

Nj j

NN

A e e e

A

θ θ θθ

θ

− −Γ = +

=

6

+1

0 0

1/14 1

2 cos2 | |

N

m

f f

A

π

π− Γ= −

θπ

ππλββθ 0

00

0 224

24

ff

ff

f

v

vf

l p

p

=====

6

Page 126: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

& & /$

• #6 *

-;;1;

^ 5!

XE;&XE;&

• "

• E1H+

• ^ 5! E9 ^

+2

Page 127: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

& & /$

( )

NN

OhmZZ

N

ZZZZk

N

ZZ LN

LN

n

kn 9.556ln

161

exp50ln20

exp;ln2

exp

21

001

0

001

=

=

=

=

=

+

+3

( ) ( )

( ) OhmZZk

N

ZZ

OhmZZk

ZZOhmZZk

ZZ

LN

k

LN

kLN

k

4.1716ln1615

exp50ln2

exp

4.1716ln1611

exp50ln2

exp;5.876ln165

exp50ln2

exp

0

3

004

0

003

0

002

=

=

=

=

=

==

=

=

=

==

Page 128: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

+9

C ?

#< E

1;_ ; E;;_

Page 129: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

!/12/ & /$

• ? )

5! #

• V @

+:

Page 130: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

!/12/ & /$

!/12/ &2 &

• .

1

22

( ) ,

( ) 2 1,

T x x

T x x

=

= −

-;

• = *

33

4 24

( ) 4 3 ,

( ) 8 8 1.

T x x x

T x x x

= −

= − +

1 2( ) 2 ( ) ( )n n nT x xT x T x− −= −

Page 131: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

-

. )$'

Page 132: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

!/12/ & /$

( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) θθθθ

θθθθθθ

θθ

cos3cos43cos

1cos22cos

cos1cos

10cos

3

22

1

0

−=×=

−=×=

=×==×=

T

T

T

T

-+

( ) ( ) ( ) ( ) θθθθ cos3cos43cos3 −=×=T

" $W'

Page 133: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• >

– % *` `)Y)$'Y` ±?

– % YYN)Y)$'YN

– % YYN)Y)$'Y #

• EW YYO

--

• EW YYO

• .#

)

(cos ) cosnT nθ θ=

1

1

( ) cos( cos ) for | | 1,

( ) cosh( cosh ) for | | 1.n

n

T x n x x

T x n x x

= <

= >

Page 134: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

!/12/ & /$

• ! $D* $%

• % )

( )( )xθcos=

• . $#'

-

( )( )m

xθcos

=

220

πθθπθ =+−= mm

Page 135: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• . )

1cos cos( ) cos (sec cos ) cos coscos cosn m

m m

T n nθ θθ θ θθ θ

− = =

(sec cos ) sec cos ,T θ θ θ θ=

-1

1

22

33

44

2

(sec cos ) sec cos ,

(sec cos ) sec (1 2cos ) 1,

(sec cos ) sec (cos3 3cos ) 3sec cos ,

(sec cos ) sec (cos 4 4cos 2 3)

4sec (cos 2 1) 1.

m m

m m

m m m

m m

m

T

T

T

T

θ θ θ θθ θ θ θθ θ θ θ θ θ θθ θ θ θ θ

θ θ

=

= + −

= + −

= + +

− + +

Page 136: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

6.

0 1( ) 2 [ cos cos( 2)

cos( 2 ) ]

(sec cos )

jN

N

jN

e N N

N n

Ae T

θ

θ

θ θ θθ

θ θ

Γ = Γ + + Γ − ++Γ − +

=

-2

• <W E;)

(sec cos )jNN mAe Tθ θ θ−=

0

0

0

0

(0) (sec )

1(sec )

LN m

L

L

L N m

Z ZAT

Z Z

Z ZA

Z Z T

θ

θ

−Γ = =+−=+

Page 137: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• (#

Z

• *$+,

0 0

0 0 0

1 1 1(sec ) ln

2L L L

N mL m L m

Z Z Z Z ZT

A Z Z Z Z Zθ − −= =

+ Γ + Γ

E

-3

1

1 0

0

1

0

cosh( cosh (sec ))

1 1sec cosh cosh

1 1cosh cosh ln

2

m

Lm

m L

L

m

N

Z ZN Z Z

ZN Z

θ

θ

=

−= Γ +

Γ

E

Page 138: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

• W )

!/12/ "$

6 *

-;;1;

0

2 4 mff

θπ

∆ = −

-9

-;;1;

XE;&&

Page 139: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

-:

C ?

#

Page 140: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

;

&$'.

&

$' &

Page 141: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

# &$'

G &$'C

&

Page 142: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

.5*%

• $A

' 5! a

• ! *#

5! a

• =#

a

+

Page 143: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

.5*% C

C<

$b;

-

$b;

?

'&$'> -&

$'" -&$'> &

$'" &

Page 144: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

.5*%

• %&

C5*

%

• X $b' • X $b'

Page 145: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

.5*%

• A

%&$'$#'

π∞ 111

1

RCdd

mm

πωωωω

Γ∆=

Γ=

Γ ∆

∞ 1ln

1ln

1ln

0

Page 146: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

2

( 5*% &$'

&$' A A

&

Page 147: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

.5*%

• % C) 5! $cb'

#

• . • .

A 5! E; >

?

• C )?

$cb HX' = L

L

3

Page 148: Lecture Slides 3 - iitg.ac.in · ( 1) ( 1) ( ) { 0 [ ] 1 } Gq = e-jNq jNq +- jNq j N- q - j N- q + ( 1)/2 ( ) j N e e q - q G - + G N /2 ( ) 2 [0 cos 1 cos( 2) 1 G q = e-jNq G Nq+

C

• 6&&>A ). /011)! )

+;;1

• -: .?d V=F

• &&& • &&&

• 1 12

/ , 3

9