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# Noise Immunity With Hermite Polynomial Presentation Final Presentation

## on Dec 03, 2009

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## Noise Immunity With Hermite Polynomial Presentation Final PresentationPresentation Transcript

• Good Morning
• Theory vs. technology
• Theory always precedes technology
• Physical experiment supports theory or negates it
• Example:
• Einstein’s “ General Theory of Relativity”
• Feynman’s QED (Quantum Electro Dynamics)
• Nanotechnology (Again Feynman’s 1960 talk on “” There is Plenty of Room At The Bottom ” was the first talk on nanotechnology and Technology caught up four decades later!!!)
• Shannon’s “ A Mathematical Theory Of Communication ” paper that gave birth to Information theory
• There Should Be No Presumption That This Paper Is In Any Of That Category;
• Actually It Is In The Nano-category (10 -9 ) Compared To The Examples Above
• But it talks of a theory that is doable
• Noise Immunity Study Of Hermite Polynomials In Digital Transmission In A Multi-user Environment Debasish Som IAS (Retired) President, Feedback Ventures Pvt Ltd & Managing Director Bengal Feedback Ventures Pvt Ltd
• Hermite Polynomial: A little bit of mathematics
• Hermite Polynomial definition
• H n (x)  (-1) x exp(x 2 /2) d n [exp(-x 2 /2)]
• = dx n
• Generating Function of Hermite polynomial
• exp(t x -t 2 /2) = ∑a n (x)t n ; a n (x) = H n (x)/n! n=0
H 0 (x) = 1 H 1 (x) = x H 2 (x) = x 2 -1 H 3 (x) = x 3 -3x H 4 (x) = x 4 -6x 2 +3
• Orthogonal functions
• ∫ -∞  n (x)  m (x) dx = δ mn
• where δ mn is the Kronecker's Delta=> δ mn =1; m=n
• =0; m ≠n
Unfortunately Hermite polynomial is not a straight forward Orthogonal Function!!!!! Standard definition
• Generalized Orthogonal functions
• A more generalized definition of the orthogonal function is as follows:
• ∫ -∞  n (x)  m (x) K(x) dx = δ mn
• where δ mn is the Kronecker's Delta=> δ mn =1; m=n
• =0; m ≠n
K(x) => kernel General definition
• Kernel k(x)
• exp(-x 2 /2)
∞ ∫ -∞ H n (x)H m (x) dx= n! √2π when n =m 0 when n ≠ m WOW! Orthogonal !!!!! Define Modified Hermite polynomial h n (x)= exp(-x 2 /4) Hn(x) √ ( n!√2π) ∞ ∫ - ∞ h n (x) h m (x) dx = δmn where δmn is the Kronecker's Delta.
• Modified Hermite Polynomial
• We have defined Modified Hermite polynomial
• h n (x) = exp(-x 2 /4) H n (x) ; where
• √ (n!√2n)
• ∫ -∞ h n (x) h m (x) dx = δ mn
• where δ mn is the Kronecker's Delta
• And the GOOD NEWS is IT IS
• not only ORTHOGONAL but it is ORTHONORMAL !!
• Modified Hermite Polynomial: Equations they obey
• Time Domain
• d 2 h n (t) + (n + 1 - t 2 ) h n (t) = 0 dt 2 2 4
• dh n (t) + t h n (t) = nh n-1 (t)
• dt 2
• h n+1 (t) = t h n (t) – dh n (t) 2 dt
• Recursive in nature!!!!!
• Frequency Domain
• ..
• H n (f) +16 π 2 (n + ½ -4 π 2 f 2 ) H n (f) = 0
• .
• j8 π 2 f Hn(f) +jHn(f) = 4 π n H n-1 (f)
• .
• H n+1 (f) = j H n (f) - j2 π f H n (f)
• Also Recursive in nature!!!!!
And Recursive functions can be generated by DSP techniques
• And they are
• h 0 (x) = exp (-x 2 /4)
• h 1 (x) = exp (-x 2 /4)
• h 2 (x) = exp (-x 2 /4) (x 2 -1)
• h 3 (x) = exp (-x 2 /4) (x 3 -3x)
• h 4 (x) = exp (-x 2 /4) (x 4 -6x 2 +3)
• h 5 (x) = exp (-x 2 /4) (5x 4 -30x 2 +15)
• h 6 (x) = exp (-x 2 /4) (x 6 -15x 4 +45x 2 -15)
• h 7 (x) = exp (-x 2 /4) (x 7 -21x 5 +105x 3 -105x)
• h 8 (x) = exp (-x 2 /4) (x 8 -28x 6 +210x 4 -420x 2 +105)
Modified Hermite Polynomial: We got our generating equations !!!!
• Actually what happens????
• e x =1+x+x 2 /2!+x 3 /3!+x 4 /4!+…x n /n!+…. ∞
• h 8 (x)= exp (-x 2 /4) (x 8 -28x 6 +210x 4 -420x 2 +105)
• As x increases or decreases h 8 (x) -> 0
• h 8 (x) is time-limited if x=time
• Modified Hermite Polynomial: They Look Like: Prettily time-limited indeed!!!
• 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……????
• Can we use them in communication?
• Are they noise immune?
• So we design and simulate a small system and see what happens
• The system h n (t): Modified Hermite polynomials pns n: User data; Total 9 user system User 1 pns 0 User 2 pns 1 User 3 pns 2 ∑ h 1 (t) h 2 (t) h 8 (t) User 9 Pns 8 ∑ 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 Receiver Channel Integrator and Dump
• The results: No Noise no errors if sampling is right
• The results: SNR: 15 dB Again no errors if sampling is right
• The results: SNR: -4.8 dB Still no errors if sampling is right
• The results: SNR: -10.8 dB We are still sustaining with very low(1) error
• The results: SNR: -18.6 dB Errors have started (3 errors)
• The results: SNR: -24 dB Oops! Eight errors!!!
• What Is Yet To Be Done, What Could Have Been Done And What Has NOT Been Done!
• Mostly technical!
• 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!
• Mathematical question:
• 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 ?
• What have I ( not we ) learnt?
• 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: Physics , Mathematics , Communication Theory 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
• Thank you for A Very Very Patient Hearing to AN AMATEUR Good Day!