Good Morning
Theory vs. technology <ul><li>Theory always precedes technology </li></ul><ul><ul><li>Physical experiment supports theory ...
Noise Immunity Study Of Hermite Polynomials In Digital Transmission In A Multi-user Environment Debasish Som IAS  (Retired...
Hermite Polynomial:  A little bit of mathematics <ul><li>Hermite Polynomial definition </li></ul><ul><li>H n (x)    (-1) ...
Orthogonal functions <ul><li>∞ </li></ul><ul><li>∫ -∞    n (x)     m (x) dx = δ mn   </li></ul><ul><li>where  δ mn  is t...
Generalized Orthogonal functions <ul><li>A more generalized definition of the orthogonal function is as follows: </li></ul...
Kernel k(x) <ul><li>exp(-x 2 /2) </li></ul>∞ ∫ -∞  H n (x)H m (x)  dx= n! √2π  when n =m 0 when n ≠ m WOW!  Orthogonal !!!...
Modified Hermite Polynomial <ul><li>We have defined Modified Hermite polynomial </li></ul><ul><li>h n (x)  =    exp(-x 2 /...
Modified Hermite Polynomial: Equations they obey <ul><li>Time Domain </li></ul><ul><li>d 2 h n (t)  + (n +  1  -  t 2 )  h...
<ul><li>And they are </li></ul><ul><li>h 0 (x) =  exp (-x 2 /4) </li></ul><ul><li>h 1 (x) =  exp (-x 2 /4) </li></ul><ul><...
Actually what happens???? <ul><li>e x =1+x+x 2 /2!+x 3 /3!+x 4 /4!+…x n /n!+…. ∞ </li></ul><ul><li>h 8 (x)= exp (-x 2 /4) ...
Modified Hermite Polynomial: They Look Like:  Prettily time-limited indeed!!!
Lets now take stock of what we know and what we have <ul><li>Modified Hermite polynomials are orthogonal </li></ul><ul><li...
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...
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! <ul><li>Mostly technical! </li></ul>...
What have  I ( not we )  learnt? <ul><li>Most importantly  I  am dabbling in the peripheral area!!!!! </li></ul><ul><li>MS...
The Generic Lesson <ul><li>Mathematics can break the barrier of the time-limited vs. band-limited issue that has been a ma...
Thank you    for    A Very Very   Patient Hearing to  AN AMATEUR   Good Day!
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Noise Immunity With Hermite Polynomial Presentation Final Presentation

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Noise Immunity With Hermite Polynomial Presentation Final Presentation

  1. 1. Good Morning
  2. 2. Theory vs. technology <ul><li>Theory always precedes technology </li></ul><ul><ul><li>Physical experiment supports theory or negates it </li></ul></ul><ul><ul><ul><li>Example: </li></ul></ul></ul><ul><ul><ul><ul><ul><li>Einstein’s “ General Theory of Relativity” </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>Feynman’s QED (Quantum Electro Dynamics) </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>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!!!) </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>Shannon’s “ A Mathematical Theory Of Communication ” paper that gave birth to Information theory </li></ul></ul></ul></ul></ul><ul><ul><li>There Should Be No Presumption That This Paper Is In Any Of That Category; </li></ul></ul><ul><ul><li>Actually It Is In The Nano-category (10 -9 ) Compared To The Examples Above </li></ul></ul><ul><ul><li>But it talks of a theory that is doable </li></ul></ul>
  3. 3. 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
  4. 4. Hermite Polynomial: A little bit of mathematics <ul><li>Hermite Polynomial definition </li></ul><ul><li>H n (x)  (-1) x exp(x 2 /2) d n [exp(-x 2 /2)] </li></ul><ul><li>= dx n </li></ul><ul><li>Generating Function of Hermite polynomial </li></ul><ul><li>∞ </li></ul><ul><li>exp(t x -t 2 /2) = ∑a n (x)t n ; a n (x) = H n (x)/n! n=0 </li></ul>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
  5. 5. Orthogonal functions <ul><li>∞ </li></ul><ul><li>∫ -∞  n (x)  m (x) dx = δ mn </li></ul><ul><li>where δ mn is the Kronecker's Delta=> δ mn =1; m=n </li></ul><ul><li> =0; m ≠n </li></ul>Unfortunately Hermite polynomial is not a straight forward Orthogonal Function!!!!! Standard definition
  6. 6. Generalized Orthogonal functions <ul><li>A more generalized definition of the orthogonal function is as follows: </li></ul><ul><li>∞ </li></ul><ul><li>∫ -∞  n (x)  m (x) K(x) dx = δ mn </li></ul><ul><li>where δ mn is the Kronecker's Delta=> δ mn =1; m=n </li></ul><ul><li> =0; m ≠n </li></ul>K(x) => kernel General definition
  7. 7. Kernel k(x) <ul><li>exp(-x 2 /2) </li></ul>∞ ∫ -∞ 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.
  8. 8. Modified Hermite Polynomial <ul><li>We have defined Modified Hermite polynomial </li></ul><ul><li>h n (x) = exp(-x 2 /4) H n (x) ; where </li></ul><ul><li> √ (n!√2n) </li></ul><ul><li>∞ </li></ul><ul><li>∫ -∞ h n (x) h m (x) dx = δ mn </li></ul><ul><li>where δ mn is the Kronecker's Delta </li></ul><ul><li>And the GOOD NEWS is IT IS </li></ul><ul><li>not only ORTHOGONAL but it is ORTHONORMAL !! </li></ul>
  9. 9. Modified Hermite Polynomial: Equations they obey <ul><li>Time Domain </li></ul><ul><li>d 2 h n (t) + (n + 1 - t 2 ) h n (t) = 0 dt 2 2 4 </li></ul><ul><li>dh n (t) + t h n (t) = nh n-1 (t) </li></ul><ul><li>dt 2 </li></ul><ul><li>h n+1 (t) = t h n (t) – dh n (t) 2 dt </li></ul><ul><li>Recursive in nature!!!!! </li></ul><ul><li>Frequency Domain </li></ul><ul><li>.. </li></ul><ul><li>H n (f) +16 π 2 (n + ½ -4 π 2 f 2 ) H n (f) = 0 </li></ul><ul><li> . </li></ul><ul><li>j8 π 2 f Hn(f) +jHn(f) = 4 π n H n-1 (f) </li></ul><ul><li>. </li></ul><ul><li>H n+1 (f) = j H n (f) - j2 π f H n (f) </li></ul><ul><li>Also Recursive in nature!!!!! </li></ul>And Recursive functions can be generated by DSP techniques
  10. 10. <ul><li>And they are </li></ul><ul><li>h 0 (x) = exp (-x 2 /4) </li></ul><ul><li>h 1 (x) = exp (-x 2 /4) </li></ul><ul><li>h 2 (x) = exp (-x 2 /4) (x 2 -1) </li></ul><ul><li>h 3 (x) = exp (-x 2 /4) (x 3 -3x) </li></ul><ul><li>h 4 (x) = exp (-x 2 /4) (x 4 -6x 2 +3) </li></ul><ul><li>h 5 (x) = exp (-x 2 /4) (5x 4 -30x 2 +15) </li></ul><ul><li>h 6 (x) = exp (-x 2 /4) (x 6 -15x 4 +45x 2 -15) </li></ul><ul><li>h 7 (x) = exp (-x 2 /4) (x 7 -21x 5 +105x 3 -105x) </li></ul><ul><li>h 8 (x) = exp (-x 2 /4) (x 8 -28x 6 +210x 4 -420x 2 +105) </li></ul>Modified Hermite Polynomial: We got our generating equations !!!!
  11. 11. Actually what happens???? <ul><li>e x =1+x+x 2 /2!+x 3 /3!+x 4 /4!+…x n /n!+…. ∞ </li></ul><ul><li>h 8 (x)= exp (-x 2 /4) (x 8 -28x 6 +210x 4 -420x 2 +105) </li></ul><ul><li>As x increases or decreases h 8 (x) -> 0 </li></ul><ul><li>h 8 (x) is time-limited if x=time </li></ul>
  12. 12. Modified Hermite Polynomial: They Look Like: Prettily time-limited indeed!!!
  13. 13. Lets now take stock of what we know and what we have <ul><li>Modified Hermite polynomials are orthogonal </li></ul><ul><li>We can generate them by DSP techniques </li></ul><ul><li>They are time limited </li></ul><ul><li>They are reasonably bandlimited </li></ul><ul><ul><ul><ul><li>(98% of energy in finite bandwidth) </li></ul></ul></ul></ul><ul><li>So now question……???? </li></ul><ul><ul><li>Can we use them in communication? </li></ul></ul><ul><ul><li>Are they noise immune? </li></ul></ul><ul><li>So we design and simulate a small system and see what happens </li></ul>
  14. 14. 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
  15. 15. The results: No Noise no errors if sampling is right
  16. 16. The results: SNR: 15 dB Again no errors if sampling is right
  17. 17. The results: SNR: -4.8 dB Still no errors if sampling is right
  18. 18. The results: SNR: -10.8 dB We are still sustaining with very low(1) error
  19. 19. The results: SNR: -18.6 dB Errors have started (3 errors)
  20. 20. The results: SNR: -24 dB Oops! Eight errors!!!
  21. 21. What Is Yet To Be Done, What Could Have Been Done And What Has NOT Been Done! <ul><li>Mostly technical! </li></ul><ul><ul><li>Synchronization of clocks have been assumed! </li></ul></ul><ul><ul><li>Real experimental measurements have not been done! </li></ul></ul><ul><ul><li>BER have not been calculated </li></ul></ul><ul><ul><li>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! </li></ul></ul><ul><ul><li>Why it will be only applicable for UWB and not for other spectrum areas have not been explored! </li></ul></ul><ul><li>Mathematical question: </li></ul><ul><ul><li>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 ? </li></ul></ul>
  22. 22. What have I ( not we ) learnt? <ul><li>Most importantly I am dabbling in the peripheral area!!!!! </li></ul><ul><li>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! </li></ul><ul><li>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! </li></ul>
  23. 23. The Generic Lesson <ul><li>Mathematics can break the barrier of the time-limited vs. band-limited issue that has been a major challenge to communication engineers </li></ul><ul><li>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 </li></ul><ul><li>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 </li></ul>
  24. 24. Thank you for A Very Very Patient Hearing to AN AMATEUR Good Day!

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