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Noise Immunity With Hermite Polynomial Presentation Final Presentation
 

Noise Immunity With Hermite Polynomial Presentation Final Presentation

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    Noise Immunity With Hermite Polynomial Presentation Final Presentation Noise Immunity With Hermite Polynomial Presentation Final Presentation Presentation 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!