Noise Immunity With Hermite Polynomial Presentation Final Presentation

Good Good Morni ngMorni ng

Theory vs. technologyTheory vs. technology Th e o r y a lw a y s p r e c e d e s

t e c h n o lo g y– P h y s ic a l e x p e r im e n t s u p p o r t s

t h e o r y o r n e g a t e s it:E x a m p le

’ E in s t e in s “ G e n e r a l Th e o r y G e n e r a l Th e o r y o f R e la t iv it y ” o f R e la t iv it y ”

’ F e y n m a n s ( Q ED Q u a n t u m ( Q ED Q u a n t u m )E le c t r o D y n a m ic s )E le c t r o D y n a m ic s

N a n o t e c h n o lo g yN a n o t e c h n o lo g y ( Ag a in’ 1 9 6 0 F e y n m a n s t a lk o n

“ ” Th e r e is P le n t y o f R o o m Th e r e is P le n t y o f R o o m A t Th e B o t t o m A t Th e B o t t o m ” w a s t h e

f ir s t t a lk o n n a n o t e c h n o lo g y a n d Te c h n o lo g y c a u g h t u p !!!)f o u r d e c a d e s la t e r

’ S h a n n o n s “ A M a t h e m a t ic a l A M a t h e m a t ic a l Th e o r y O f C o m m u n ic a t io n Th e o r y O f C o m m u n ic a t io n ”

p a p e r t h a t g a v e b ir t h t o In f o rm a t io n t h e o r y

– Th e r e S h o u ld B e N o P r e s u m p t io n Th a t ; Th is P a p e r Is In An y O f Th a t C a t e g o r y

– - (1 0Ac t u a lly It Is In Th e N a n o c a t e g o r y -9 ) C o m p a r e d To Th e E x a m p le s Ab o v e

– B u t it t a lk s o f a t h e o r y t h a t is d o a b le

N o is e Im m u n it y S t u d y O f H e rm it e P o ly n o m ia ls In D ig it a l Tr a n s m is s io n

- In A M u lt i u s e rE n v ir o n m e n t

D e b a s is h S o m IAS ( )R e t ir e d

, P r e s id e n t F e e d b a c k Ve n t u r e s P v tLt d&

M a n a g in g D ir e c t o r B e n g a l F e e d b a c k Ve n t u r e s P v t Lt d

Hermite Polynomial: A little bit of mathematics

Hermite Polynomial definition

Hn(x) ∆ (-1)x exp(x2/2) d n[exp(-x 2/2)]

= dxn

Generating Function of Hermite polynomial∞

(e x p t x -t 2 /2 ) = ∑a n ( )x t n ; a n ( ) = x Hn(x)/n! n=0

H0(x) = 1H1(x) = xH2(x) = x2-1H3(x) = x3-3xH4(x) = x4-6x2+3

O r t h o g o n a l f u n c t io n s

∞

∫-∞ φn(x) φ m(x) dx = δmn where δmn is the Kronecker's Delta=> δmn=1; m=n

=0; m≠n U n f o r t u n a t e ly H e rm it e

p o ly n o m ia l is n o t a s t r a ig h t f o r w a r d O r t h o g o n a l!!!!!F u n c t io n

Standard

definition

G e n e r a liz e d O r t h o g o n a lf u n c t io n s

A more generalized definition of the orthogonal function is as follows:

∞

∫-∞ φn(x) φ m(x) K(x) dx = δmn where δmn is the Kronecker's Delta=> δmn=1; m=n

=0; m≠n

( ) => K x k e r n e l

General

definition

Kernel k(x)

exp(-x2/2)

∞

∫-∞ Hn(x)Hm(x) dx=! 2 n √ π

=w h e n n m0 w h e n n ≠ m

! WOW! WOW !!!!!O r t h o g o n a l !!!!!O r t h o g o n a l

D e f in e M o d if ie d H e rm it e p o ly n o m ia lh n ( )= x (-e x p x 2 /4 ) ( )H n x

√ ( ! 2 )n √ π ∞∫ -∞ h n ( ) x h m ( ) x d x = δm n where δmn is the Kronecker's Delta.

Modified Hermite Polynomial

We h a v e d e f in e d M o d if ie d H e rm it e p o ly n o m ia l

h n ( ) = x (-e x p x 2 /4 ) H n ( )x ;

w h e r e

√ ( ! 2 )n √ n

∞

∫ -∞ h n ( ) x h m ( ) x d x = δm n where δmn is the Kronecker's Delta

And the G O O D N EWS is IT IS

n o t o n ly O R TH O G O N AL b u t it isO R TH O N O RMAL !!

Modified Hermite Polynomial:Equations they obey

Time Domain

d 2h n(t) + (n + 1 - t 2) hn(t) = 0 dt2 2 4

dh n(t) + t hn(t) = nhn-1(t)

dt 2

hn+1(t) = t hn(t) –dh n(t) 2 dt

R e c u r s iv e in!!!!!n a t u r e

F r e q u e n c yD o m a in. .H n ( ) +1 6 f π 2 ( +n ½ -4 π 2 f 2 ) H n ( ) = 0 f

.8 j π 2 ( ) + ( ) = 4 f H n f jH n f π nH -1n ( ) f

.H +1n ( ) = f j H n ( ) - 2f j π f H n ( )f

A ls o R e c u r s iv e in!!!!!n a t u r e

An d R e c u r s iv e f u n c t io n s c a n b e g e n e r a t e d b y D S P t e c h n iq u e s

An d t h e y a r eh 0 ( )x = (-e x p x 2 /4 )h 1( )x = (-e x p x 2 /4 )h 2 ( )x = (-e x p x 2 /4 ) (x 2 -1 )h 3 ( )x = (-e x p x 2 /4 ) (x 3 -3 )xh 4 ( )x = (-e x p x 2 /4 ) (x 4 -6 x 2 +3 )h 5 ( )x = (-e x p x 2 /4 ) (5 x 4 -3 0 x 2 +15 )h 6 ( )x = (-e x p x 2 /4 )

(x 6 -1 5 x 4 +4 5 x 2 -1 5 )h 7 ( )x = (-e x p x 2 /4 )

(x 7 -2 1 x 5 +10 5 x 3 -1 0 5 )xh 8 ( )x = (-e x p x 2 /4 )

(x 8 -2 8 x 6 +2 10 x 4 -4 2 0 x 2 +10 5 )

Modified Hermite Polynomial:We got our generating equations !!!!

Ac t u a lly w h a t? ? ? ?h a p p e n s

ex=1+x+x2/2!+x3/3!+x4/4!+…xn/n!+…. ∞

h8(x)= exp (-x2/4) (x8-28x6+210x4-420x2+105)⇒ As x increases or decreases h8(x) -> 0⇒ h8(x) is time-limited if x=time

Modified Hermite Polynomial:They Look Like: Prettily time-limited indeed!!!

Normalised Hermite Polynomial series

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

time

h1(t)h2(t)h3(t)h4(t)h5(t)h6(t)h7(t)h8(t)

Lets now take stock of what we know and what we have

Modified Hermite polynomials are orthogonal We can generate them by DSP techniques They are time limited They are reasonably bandlimited

(98% of energy in finite bandwidth)

So now question……????> ?C a n w e u s e t h e m in c o m m u n ic a t io n> ?Ar e t h e y n o is e im m u n e

> S o w e d e s ig n a n d s im u la t e a s m a ll s y s t e m a n d s e e w h a t

h a p p e n s

The systemThe system

h n ( ): t M o d if ie d H e rm it ep o ly n o m ia ls

p n s : n ; 9 U s e r d a t a To t a l u s e r s y s t e m

User 1pns0

User 2pns1

User 3pns2 ∑

h 1(t)

h 2(t)

h8(t)

User 9Pns8

∑

n(t)Additive White

Gaussian Noise

h 0(t)

∫ pns’0

h 1(t)

∫

h 2(t)

∫

h 8(t)

∫

pns’1

pns’2

pns’8

h 0(t)

Transmitter ReceiverChannel

Integrator and Dump

: Th e r e s u lt s N o N o is e n o e r r o r s i f s a m p lin g is r ig h t

User#1: Tx data and RX Output data

-4

-3

-2

-1

0

1

2

3

4

pns0

pns'0(t)

-4

-3

-2

-1

0

1

2

3

4

5

pns1

pns'1(t)


-8

-6

-4

-2

0

2

4

6

pns2pns'2(t)


-20

-15

-10

-5

0

5

10

15

pns3

pns'3(t)


-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

pns4

pns'4(t)


-1

-0.5

0

0.5

1

1.5

pns5

pns'5(t)


-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

pns6

pns'6(t)


-4

-3

-2

-1

0

1

2

3

pns7pns'7(t)


-4

-3

-2

-1

0

1

2

3

pns8pns'8(t)


Received signal with noise

-8

-6

-4

-2

0

2

4

6

8

10

12

14


: : Th e r e s u lt s S N R 15 d B Ag a in n o e r r o r s if s a m p lin g is r ig h t


-4

-3

-2

-1

0

1

2

3

4

pns0

pns'0(t)

-4

-3

-2

-1

0

1

2

3

4

5

pns1

pns'1(t)


-8

-6

-4

-2

0

2

4

6

pns2pns'2(t)


-20

-15

-10

-5

0

5

10

15

pns3

pns'3(t)


-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

pns4

pns'4(t)


-1

-0.5

0

0.5

1

1.5

pns5

pns'5(t)


-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

pns6

pns'6(t)


-4

-3

-2

-1

0

1

2

3

pns7pns'7(t)


-4

-3

-2

-1

0

1

2

3

pns8pns'8(t)



-8

-6

-4

-2

0

2

4

6

8

10

12

14


: : Th e r e s u lt s S N R -4 .8 d B S t i l l n o e r r o r s if s a m p lin g is r ig h t


-4

-3

-2

-1

0

1

2

3

4

pns0

pns'0(t)

-4

-3

-2

-1

0

1

2

3

4

5

pns1

pns'1(t)


-8

-6

-4

-2

0

2

4

6

8

pns2pns'2(t)


-20

-15

-10

-5

0

5

10

15

pns3

pns'3(t)


-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

pns4

pns'4(t)


-1

-0.5

0

0.5

1

1.5

pns5

pns'5(t)


-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

pns6

pns'6(t)


-4

-3

-2

-1

0

1

2

3

pns7pns'7(t)


-4

-3

-2

-1

0

1

2

3

pns8pns'8(t)



-8

-6

-4

-2

0

2

4

6

8

10

12

14


: : Th e r e s u lt s S N R -1 0 .8 d B We a r e s t i l l (1 ) s u s t a in in g w it h v e r y lo w e r r o r


-4

-3

-2

-1

0

1

2

3

4

pns0

pns'0(t)

-4

-3

-2

-1

0

1

2

3

4

5

pns1

pns'1(t)


-6

-4

-2

0

2

4

6

8

pns2pns'2(t)


-20

-15

-10

-5

0

5

10

15

pns3

pns'3(t)


-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

pns4

pns'4(t)


-1

-0.5

0

0.5

1

1.5

pns5

pns'5(t)


-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

pns6

pns'6(t)


-4

-3

-2

-1

0

1

2

3

pns7pns'7(t)


-4

-3

-2

-1

0

1

2

3

pns8pns'8(t)



-10

-5

0

5

10

15


: : Th e r e s u lt s S N R -1 8 .6 d B E r r o r s (3 )h a v e s t a r t e d e r r o r s


-4

-3

-2

-1

0

1

2

3

4

5

6

7

pns0

pns'0(t)

-4

-3

-2

-1

0

1

2

3

4

5

pns1

pns'1(t)


-8

-6

-4

-2

0

2

4

6

8

pns2pns'2(t)


-25

-20

-15

-10

-5

0

5

10

15

pns3

pns'3(t)


-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

pns4

pns'4(t)


-1

-0.5

0

0.5

1

1.5

pns5

pns'5(t)


-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

pns6

pns'6(t)


-6

-5

-4

-3

-2

-1

0

1

2

3

4

pns7pns'7(t)


-4

-3

-2

-1

0

1

2

3

pns8pns'8(t)



-10

-5

0

5

10

15

20


: : Th e r e s u lt s S N R -2 4 d B ! O o p s !!!E ig h t e r r o r s


-8

-6

-4

-2

0

2

4

6

8

pns0

pns'0(t)

-6

-4

-2

0

2

4

6

8

pns1

pns'1(t)


-10

-8

-6

-4

-2

0

2

4

6

8

10

pns2pns'2(t)


-20

-15

-10

-5

0

5

10

15

20

pns3

pns'3(t)


-1

-0.5

0

0.5

1

1.5

pns4

pns'4(t)


-1

-0.5

0

0.5

1

1.5

2

pns5

pns'5(t)


-1

-0.5

0

0.5

1

1.5

pns6

pns'6(t)


-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

pns7pns'7(t)


-5

-4

-3

-2

-1

0

1

2

3

4

5

6

pns8pns'8(t)



-15

-10

-5

0

5

10

15

20


Wh a t Is Ye t , To B e D o n e Wh a t C o u ld H a v e B e e n D o n e

An d Wh a t H a s N O T !B e e n D o n e

!M o s t ly t e c h n ic a l– Synchronization of clocks have been assumed!– Real experimental measurements have not been done!– BER have not been calculated– Hardware design for generating Hermite Polynomials have not

been explored- can it be really done with DSP- I believe it can be done but that’s not enough!

– Why it will be only applicable for UWB and not for other spectrum areas have not been explored!

:M a t h e m a t ic a l q u e s t io n– Is the kernel which is Gaussian is contributing to the excellent

simultaneous time-limiting and band-limiting attributes (after all Gaussian pulse is our favorite for that!!) and not much contribution from the Hermite polynomial ?

Wh a t h a v e I ( n o t w e ) ?le a r n t

Most importantly I am dabbling in the peripheral area!!!!!

MS Excel is a powerful tool: I am told Matlab is better: but I don’t know Matlab: another sign of the extent of peripheral dabbling!

Last but not the least: P h y s ic s , M a t h e m a t ic s , C o m m u n ic a t io nTh e o r y all are so intricately related that we need to have a better and intensive understanding of everything to innovate!

The Generic Lesson

Mathematics can break the barrier of the time-limited vs. band-limited issue that has been a major challenge to communication engineers

There is no end to the theoretical research and we must address that instead of focusing only on protocol issues in the area of Digital Communication &Communication Theory

Hermite polynomials are still an enigma and full of surprises: probably there are more such functions in mathematics: which can have realm practical applications specially when spectrum is becoming one of the major resource issues

Th a n ky o u

for A Very Very

Patient Hearingto

ANAMATEU R G o o d

!D a y

Technology

Noise Immunity With Hermite Polynomial Presentation Final Presentation