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Stochastic Calculus Estimation of the Instantaneous Signal Parameters For the Weighted Array Portion of Stochastic Resonance Array babak vosooghzadeh (Extracted from a previous work in June 2010) massive-wireless-network-found.webnode.com

I- Introduction In this document, stochastic calculus along with Kalman Filtering Markovian State Transition concepts are used to estimate the signal instantaneous mean and instantaneous variance of the weighted (transport path) portions of the Stochastic Resonance Array in Fig. 1. Array path signals are estimated because the only measurement access point is the weighted array output in baseband signal format, and the instantaneous estimations are critical for array weight adjustments. The update procedure for the instantaneous signal calculations are summarized in Fig. 2. In order to justify the use of stochastic calculus and Markovian state machine for the estimation process, a simple Electronic Transport Mechanism is described in Appendix I.

II- Weighted Array Transport Path Signal Parameter Estimation For each array transport path (Fig. 1), the terms  i (t ) and  i2 (t ) represent the drift (instantaneous mean) and diffusion (instantaneous variance), respectively. Each transport path with signal S Ri (t ) will be multiplied by the emphasis or weight factor wi (t ) for i = 1 to M at each update period. The weighted transport paths become yi (t ) for i = 1 to M and the array output becomes yT (t ) 

M

y j 1

j

(t ) + measurement noise. The array output signal yT (t ) becomes the detectable

signal. The multiplication can be performed by a semi-ballistic nanostructure Gilbert cell such as nano FET Gilbert cell transistors. [1,2] The instantaneous mean and variance can be estimated by using the array output measurement and the method of stochastic calculus. The stochastic differential expressions for the transport paths, weighted transport paths i (i = 1 to M) and array output become

S Ri (t )dt   i (t )Z i (t )dt  i (t )dt   i (t )dBi (t )

(1)

 dLi (t ) yi (t )dt   Li (t )dt   Li (t )dBi (t )

 dLwi (t )  f  i .wi (t )i (t )dt  wi (t ) i (t )dBi (t ) M

yT (t )dt  T (t )dt   Lj (t )dB j (t ) j 1

 T (t )dt   T (t )dB(t ) 1/ 2

M   dLT (t ) , where  T (t )    Lj2 (t )  j 1  2 The terms  and  are the drift (instantaneous mean) and diffusion (instantaneous variance) for the respective stochastic process. The term dBi (t ) is the brown motion or Wiener process with variance d 2 Bi (t )  dt . The drift and diffusion coefficients can be estimated by the work of Ait-Sahalia, Jiang, Knight, Stanton, Chapman and Pearson and the relevant procedures are outlined in Ref. [3-5]. If the sampled base band version of the weighted path yi (t ) , i.e. yi (n) were available, then by using numerical





Integration methods such as Trapezoid Law Lwi (n  j )  0.5T yi (n  j )  yi (n  j  1)  Lwi (n  j  2) , the drift and diffusion parameters  Li (n) and  Li2 (n) of the weighted paths could have been estimated by the following -1-

kernel weighted equations:

 L K

 Li (n) 

J 0

 L

wi

K

 ( n)  2 Li

J 0

wi

 L wi (n)  L wi (n  j  1)  (n  j )  L wi (n  j  1) Kern   hS   K  L wi (n)  Lwi (n  j  1)  T  Kern   hS J 0  



(2)

 Lwi (n)  Lwi (n  j  1)  2 (n  j )  Lwi (n  j  1) Kern   hS   K  Lwi (n)  Lwi (n  j  1)  T  Kern   hS J 0  



where K+1 sampling points have been considered . The smoothing parameter hS  1.06 var( Lwi ).( K  1) 1 / 5 and the

Kern( x)  Exp ( x 2 / 2) / 2 is the Gaussian Kernel. By knowing the drift and diffusion coefficients of the weighted paths and the weights wi (n) , the instantaneous noise variance of the paths  i2 (n) and the effective



instantaneous path signal power f  i . i (n)



2

could be retrieved. After the measurement of array output, a posteriori

estimates of the weighted transport paths shall be used for the mentioned drift and diffusion calculations. The Milstein method [6] for estimating Lwi (n) (or yi (n) ) from Lwi (n  1) (or yi (n  1) ) is indicated below:



 (n) Bi2 (n  1)  T Lwi (n)  Lwi (n  1)   Li (n)T   Li (n).Bi (n  1)  0.5 Li (n) Li  Lwi (n  1)  f  i .wi (n)i (n)T  wi (n) Li (n).Bi (n  1)



 0.5wi (n) i (n)(wi (n) i (n)  wi (n) i (n)) Bi2 (n  1)  T



(3)



The Brownian increments Bi (n  1) are nearly independent with variance (Bi ) 2  T . During the (n-1) period, the

~ (n) , the diffusion ~ 2 (n) , the Brownian increment a priori estimates for the next period (n) which include the drift  i i

~ ~ Bi (n  1) , and the multiplication reduction factor f  i can be determined by using either simple predictors at the start of the operation or by using the pseudo random noise Markovian state models [7-8] (see Appendix I). The method for estimating the base band signals of the transport paths by measuring the array output is presented here. After making the measurement from the output of the array yT (n) or equivalently LT (n) due to yT (t ) 

dLT (t ) dt

, the

a posteriori estimates for the above terms can be determined. For this purpose, the Kalman filtering method [9] will be used. For the Kalman state equation, we regroup Lwi (n) , as follows

Lwi (n)  Lwi (n  1)  ui (n  1)  ei (n  1) for i =1 to M

(4)

Where Lwi (n) and Lwi (n  1) are the current and previous Kalman process states, respectively. The term ui (n  1) is the driving function and ei (n  1) is the process noise with variance Qi (n)  ei2 (n  1) .

~ ~ ui (n  1)  f  i .wi (n)~i (n)T  wi (n)~i (n).Bi (n  1) ~  0.5w (n)~ (n)(w (n)~ (n)  w (n)~ (n)) B 2 (n  1)  T i

i

i

i

i

i



i

-2-



(5)

Nano Antenna Port

Nano Cavity

Nano Cavity

Nano Cavity ASRA Pre Amplifier

S1 (t )

S i (t )

S M (t )

S Ri (t )

S R1 (t )

S RM (t ) ooo

ooo

w1 (t )

wi (t )

y1 (t )

wM (t )

yi (t )

y M (t )

yT (t ) Side Demodulator/ Sampler

LNA

ooo

Injecting Signal Generators and Modulators

yT (n)

Analog Source Separation Module

Weight & injecting signal parameter update for maximizing SR Quality Measure

Demodulator & Sampling Stages

Digital Signal Source Separation / Interference Reduction Module

Stochastic Resonance Quality Measure (Mutual Information, Normalized Correlation, etc…)

sˆD (n) Signal Quality

Other Post Sampling Modules Reference or Wireless Network identification Sequence

Fig. 1- Adaptive Stochastic Resonance Array

~ ~ ei (n  1)  [ f  i .wi (n) i (t )  f  i .wi (n)~i (n)]T  wi (n)[ i (n).Bi (n  1)  ~i (n).Bi (n  1)]



 0.5wi (n) i (n)(wi (n) i (n)  wi (n) i (n)) Bi2 (n  1)  T ~  0.5w (n)~ (n)(w (n)~(n)  w (n)~ (n)) B 2 (n  1)  T i

i

i

i

i

i



i



(6)



For the Kalman measurement equation, the array output will be used, as follows: M

LT (n)   Lwj (n)  v L (n) where vL (n) is the zero mean measurement noise with variance R(n)  v L2 (n) j 1

-3-

By using the a posteriori estimates Lˆ wi (n  1) of the (n-1) cycle, we obtain the following a priori estimates:

~ Lwi (n)  Lˆ wi (n  1)  ui (n  1) for i = 1 to M, ~ ~ e~i  Lwi (n)  Lwi (n) is the a priori estimate error with variance Pi (n)  ~ ei 2 (n)

(7)

eˆi  Lwi (n)  Lˆ wi (n) is the a posteriori estimate error with variance Pˆi (n)  eˆi2 (n) ~ The index n in the variance expressions Q (n) , R(n) , P (n) and Pˆ (n) implies that the variance estimates were obtained i

i

i

by using all of the available data up to the nth period.

~ Pi (n)  Pˆi (n  1)  Qi (n  1) is the error variance projection ~ Pi (n) K i ( n)  M is the Kalman gain ~  Pj (n)  R(n  1)

(8)

j 1

After receiving the demodulated and sampled output of the array YT (n) (or equivalently LT (n) ), the a posteriori estimates of the weighted path Lˆwi (n) become M ~ ~ Lˆ wi (n)  Lwi (n)  K i (n)( LT (n)   Lwj (n)) , i = 1 to M

(9)

j 1

~  Lwi (n) 

Pˆi (n  1)  Qi (n  1) M

 ( Pˆ (n  1)  Q (n  1))  R(n  1) j 1

i

M ~ ( LT (n)   Lwj (n)) j 1

i

~ Pˆi (n)  (1  K i (n)) Pi (n) for the updated variance of the a posteriori estimate error of the weighted paths. By the knowledge of a posteriori weighted path estimates Lˆ wi (n) , the estimated drift ˆ Li (n) and diffusion ˆ Li2 (n) can be determined by the weighted kernel method of equation (2). Since the weights wi (n) are known, the a posteriori estimates for drift ˆ i (n) and the product of drift and the multiplication factor fˆ i ˆ i (n) can be easily determined by simple division of ˆ Li (n) and ˆ i (n) by the weights wi (n) . Then, the process noise estimates can be retrieved by the difference between the a priori and a posteriori weighted path estimates.

~ eˆi (n  1)  Lˆ wi (n)  Lwi (n) , which can be used to update the process noise variance Qi (n) .

(10)

The a posteriori estimates of the Brownian increments Bˆ i (n  1) can be retrieved by using the process noise estimates in quadratic format, as follows:

~ ~ eˆi (n  1)  [ fˆ i .wi (n)ˆ i (t )  f  i .wi (n)~i (n)]T  wi (n)[ˆ i (n).Bˆ i (n  1)  ~i (n).Bi (n  1)]

 

 0.5wi (n)ˆ i (n)(wi (n)ˆ i (n)  wi (n)ˆ i (n)) Bˆ i2 (n  1)  T ~  0.5wi (n)~i (n)(wi (n)~i(n)  wi (n)~i (n)) Bi2 (n  1)  T

 

(11)

The mentioned a posteriori estimates and the a priori estimates will be used to update the drift and diffusion predictors and the pseudo noise state model. The latest estimate of the measurement noise vˆL (n) becomes: M

vˆ L (n)  LT (n)   Lˆ wj (n)

(12)

j 1

The measurement variance R(n) of the array output will be updated by using the measurement noise estimate at the nth cycle. The block diagram for the update procedure is shown in Fig 2. -4-

During the (n-1) operation cycle, calculate the (n) cycle path a priori estimates for

~ ( n) , drift  i ~

diffusion  i (n) , Brownian increment

~ ~ Bi (n  1) and multiplication reduction factor f  i by

using the (n-1) cycle posteriori estimates, simple predictors, random noise observable Markovian state machine, paths i=1 to M

During the (n-1) operation cycle, estimate the Kalman Filter driving function path weights

wi (n) for the (n) operation cycle,

ui (n  1) , the

i = 1 to M

During the (n-1) operation cycle, calculate the weighted paths a posteriori estimates

~ Lwi (n) for the nth cycle by using the driving function and the a posteriori estimates Lˆ (n  1) wi

During the (n) operation cycle, by using the output of the array, calculate the Kalman filter

ˆ (n) for the weighted paths and update the Kalman gain and the a posteriori estimates L wi filter parameters

ˆ (n) for the weighted paths, the (n) cycle a posteriori By using the a posteriori estimates L wi weighted paths drift

ˆ Li (n)

and diffusion ˆ Li (n) terms can be determined by the methods 2

of stochastic calculus, such as the weighted Gaussian kernel method

Estimate the (n) cycle a posteriori drift and the Brownian increments

ˆ i (n) , diffusion ˆ i2 (n) , multiplication factor fˆ i ,

Bˆ i (n  1) for paths

i = 1 to M , update the array output

measurement noise R(n) and weighted path estimate error

Qi (n) statistics, and update the

pseudo noise Markovian state machine

Fig. 2- Update procedure for the instantaneous signal calculations

Appendix I- Electronic Transport Mechanism Boltzmann transport equation [10-13] is used for examining the receiver transport and noise mechanism due to its simple structure. However, for the instantaneous noise power analysis and its pseudo randomness, a simpler method that is based on stochastic calculus will be employed.

  dx   1  v (k )   k  (k ) , (I.1) dt      where x is the electron position, v is the electron drift velocity, k is the wave vector,  (k ) is the energy vector,        Eff ( x, t )   Int ( x, t )   Ext ( x, t ) (I.2)   where the internal field  Int ( x , t ) is generated by internal forces and density profiles     d dk   q Eff  Fr , dt dt (I.3)







where  is the momentum,  Eff is the effective electric field, Fr is the random impulse force on electron due to -5-

scattering and it is given by

    Fr   ui  (t  t i )  {Fr }  Fr0 , i

(I.4)





where {Fr } is the momentum driven drag force and Fr0 is the zero mean fluctuating force.

       (I.5) {Fr }   uW (k , k  u )du   (0)me v and   dB Fr0   ( Fr ) t dt   where Bt is a zero mean Wiener process,  2 ( Fr ) is the variance of the random scattering force on the electron,  (k ) is    the scattering rate for the electron wave vector (k ) , and W (k , k ) is the transition rate satisfying      (k )   W (k , k )dk  ,  (0)  1 /  f , where f is the average time between collisions Therefore,

  dB  (I.6) Fr   (0)me v   ( Fr ) t dt   (k )t is the probability that a jump in momentum will occur in a small time interval t and if a scattering event has      occurred at time t i , k i  k (t i ) and k i  ui  k (t i ) , then the probability distribution function for the amplitude of the    W (k i , k i  u i )  jump would be  k (u i )    (k i ) Note that for a conduction path, there is a correspondence between the scatterings and a Markovian state transition probability model. The momentum jumps in a given time frame indicate transitions into new states. For semi-ballistic transport, the number of Markovian states is limited, because the scattering rate is lower with respect to non-ballistic transport and depending on the relaxation time period, there is a tendency for electron distribution to return to equilibrium. Even though there is a weak dependency between the signal power and the noise power transitions, it is customary to model one state machine for a range of average signal power to noise power ratio. Let q : {Sp1 , Sp2 ,..., Sp L } be the L Markovian states for a noise analysis of a simple transport path for a known range of signal to noise power ratio and let {1 ( N2 ),  2 ( N2 ), ... ,  V ( N2 )} be the V possible noise power symbols which are normalized with respect to the average noise power  N2 . The Markovian state machine [7,8] is modelled by optimizing the observable symbol probability distributions,

b j (k )  Pr{ k ( 2 ) at time period n | q(n)  Sp j } for j = 1 to L and k = 1 to V and the state transition distributions aij (k )  Pr{q(n  1)  Sp j | q(n)  Spi } for i, j = 1 to L. In practice, the Markovian state machine is constructed by indirect noise power measurement during each measurement sampling period. Therefore, the Markovian transitions are modelled discretely with the same time frame as the measurement sampling period. If the measurement sampling period is  , the probability of electron scattering in time period  is specified below,

Pr{(t i  t i 1 )  }  1  exp{

ti 1 

ti 1



 (k (t ))dt } .

(I.7)

The Boltzmann Transport Equation (BTE) is given by the expression,

        q        df ( x , k )    v (k ). X f ( x, k )   Eff ( x, t ). k f ( x, k )   f ( x, k )W (k , k )dk    (k ) f ( x, k ) I.8) dt  3       2 1  1/ 2  [ ij (k ) f ( x , k )]   k .[ E{Fr } f ( x , k )]  i , j 1 k i k j         where f ( x , k ) is the electron distribution function, q is the electronic charge and  ij (k )   ui u j W (k i , k i  ui )dui The current longitudinal noise auto covariance function can be retrieved as the transient solution of BTE, subject to the -6-

following special initial condition,

  f ( x, k , t )

t 0

      (v (k )  v  X ) f SS ( x, k ) ,

(I.9)

        where f SS ( x , k ) is the steady state solution of the electron distribution function and  v  X   v (k ) f SS ( x, k )dk . Typically, the Fermi distribution is used for the steady state electron distribution,

   f SS ( x, k )  [exp( (k )   f ) / K BT  1]1 , where K B is the celebrated Boltzmann constant and T is the Temperature.

The current noise auto covariance function becomes

      K J ( x, t )  q 2  v (k ) f ( x, k , t )dk , t  0    J ( x, w)   K J ( x, )e jw d is the current density noise power spectral density with the average noise power of   J (x,0) .

For ballistic transport, the average current noise power at room temperature is given by 8K BTq 2 .TTR h , where TTR is the ballistic net transmission coefficient. For semi-ballistic transport, there is a partial contribution of the non-ballistic current noise power density of

4 K B T .L where  2f  1 2 (0) is the square of the average time between 2 2 (1  w  f )nq e A

collisions (relaxation time), A is the cross sectional area of the conduction path, L is the length of the conduction path and  e is the electron mobility. The average current noise power for non-ballistic transport path is

4 K B T .L nq e A

For the instantaneous noise power analysis, the force equation is transformed into stochastic differential equation,

      dB    d dk dv (k )   me  q Eff  Fr  q Eff   (0)me v   ( Fr ) t dt dt dt dt     me dv  (q Eff   (0)me v )dt   ( Fr ) dBt 

(I.10)



For the weak dependence of the noise velocity v N on the external electric field  Ext (Vbias, f MIX (S D (t ), S IN (t )) and the





mean fluctuation force, and the relatively strong dependence of the noise velocity to the internal electric field  Int ( x , t ) , the coefficients  N 1 ,  N 2 and  N 3 are used in the following approximation,

      q N 1 Int ( x , t )  q N 2  Ext   N 3  (0)me v N  ( Fr )  dv N  dt  dBt me me   ( Fr )   dv N   (v N )dt  dBt me

(I.11)

The instantaneous noise power  N2 (t ) in the transport path direction with noise velocity v N is approximated by the following expression,  N2 (t )  C N q 2 n 2 v N2 , where C N is a constant, q is electronic charge and n is the electronic density By using stochastic calculus,

   d N2  d N2  ( Fr )  d 2 N2  2 ( Fr )  d    (v N )  0.5 . (I.12)  dt    dBt 2 me  me2  dv N  dv N  dv N   2     ( Fr )  2 2 2 2  ( Fr ) 2 2 2 2  2C N q n v N  (v N )  C N q n  dt  2C N q n v N .  dBt   ( N )dt   ( N )dBt 2 me  me    The pseudo randomness of the instantaneous noise power  N2 (t ) is evident from the drift contribution  ( N2 ) , and 2 N

therefore, Markovian state machine can be constructed by limited states due to semi-ballistic transport and relatively  lower variance of the random scattering  2 ( Fr ) . -7-

The Markovian state machine parameters, i.e. observable symbol probability distributions, b j (k ) and the state transition distributions aij (k ) are related to the time evolutionary Fokker Planck probability distribution P( N2 , t ) ,

P( N2 , t )  2 2 2   2 [  ( N ) P( N , t )]  0.5 [ ( N2 ) P( N2 , t )] 2 2 t  N  N

(I.13)

The Milstein method [6] for estimating  N2 (n) from  N2 (n  1) is expressed below:

 N2 (n)   N2 (n  1)   ( N2 , n)T   ( N2 , n).Bt (n  1)  0.5 ( N2 , n) ( N2 , n) Bt2 (n  1)  T 

(I.14)

where the derivative is performed with respect to time.

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