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Computing Bit-Error Probability for Avalanche Photodiode Receivers by Large Deviations Theory Abhik Kumar Das Indian Institute of Technology, Kanpur under.

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Presentation on theme: "Computing Bit-Error Probability for Avalanche Photodiode Receivers by Large Deviations Theory Abhik Kumar Das Indian Institute of Technology, Kanpur under."— Presentation transcript:

1 Computing Bit-Error Probability for Avalanche Photodiode Receivers by Large Deviations Theory Abhik Kumar Das Indian Institute of Technology, Kanpur under the guidance of Majeed M. Hayat and P. Sun UNM, Albuquerque

2 OUTLINE APD: Introduction APD: Gain and Build-up Time Relation –Joint PDF of Gain & Build-up Time – Renewal Relations –Numerical Computation Large Deviations Theory –Cramér's Theorem –Gärtner-Ellis Theorem

3 APD: Bit-Error Probability Computation –Literature Review Sadowsky and Letaief Hayat and B. Choi –Generalized Theory –Bit-Error Probability Estimation Large Deviations/Asymptotic Analysis Gaussian Approximation Numerical Results Conclusion

4 APD : Introduction APD - –used in high-speed optical receivers –offers high opto-electronic gain Functioning - –photon-impacts produce primary carriers –primary carriers move due to electric field to produce secondary carriers by avalanche process in multiplication layer –these carriers constitute the photo-current Impulse-response of APD stochastic in nature with both random gain and impulse duration/build-up time

5 Avalanche Process - Injected electron First impact ionization W x dede pn dede dede dhdh dhdh E ie and E ih are the average ionization threshold energies Electric field E Hole Dead space for electronDead space for hole Electron Multiplication Layer

6 Gain and Build-up Time Relation Joint PDF of gain G and build-up time T - –defined as m = no. of electron-hole pairs produced t = time before completion of avalanche build-up –Hayat and P. Sun proposed a method to compute it from coupled renewal relations mentioned below -

7 – and are the intermediate quantities and +V+V Physical Interpretation of Renewal Relations - first impact ionization W p parent electron 0 x x +  n

8 Recursive Equations z-transform over m Initialize data Update data Compute relative change for some fixed z, and suitable range of t & x If

9 Joint distribution function (PDF) of G and T for homo-junction GaAs APD with 160 nm – multiplication layer and average gain Joint density function of G and T for the same APD. Large peaks have been truncated to show details

10 Large Deviations Theory Theory of ‘rare’ events - –concerned with the behavior of ‘tails’ of probability distributions –Law of Large Numbers – special case of the theory Cramér's Theorem - –concerned with i.i.d. random variable sequence –most basic theorem of the theory Gärtner-Ellis Theorem - –generalizes Cramér's Theorem –can be applied to independent, but not necessarily identical random variable sequence

11 Statement of Cramér's Theorem -

12 Corollary of Cramér's Theorem - Deduction from Corollary -

13 Statement of Gärtner-Ellis Theorem -

14 Bit-Error Probability Computation Sadowsky and Letaief - –gave an asymptotic analysis method based on large deviations theory for error probability estimation –formulated an efficient Monte-Carlo estimation method based on importance sampling –assumptions made in the theory: dead-space effect neglected APD functioning considered to be instantaneous OOK-type modulated optical signal direct detection integrate-and-dump receiver Literature Review

15 Salient Points of the Theory – –Define G k = gain of kth primary electron M = no. of primary electrons in signaling interval N = thermal noise response of receiver with variance σ 2 –Assume {G k } to be i.i.d. This gives the receiver statistic as: –Consider the hypotheses: –Let γ be the decision threshold, define:

16 –Bit-error probability is given as: –Asymptotic Analysis – Define,, The function is steep if. The mgf for G given by McIntyre was used by Sadowsky and Letaief, which doesn’t include dead-space effect. Let γ = (1/c 0 )λ 0 = (1/c 1 )λ 1, c 0 and c 1 positive constants. Asymptotic refers to γ being large. The estimate for bit-error probability can be obtained from a result proven by Sadowsky and Letaief.

17 Result of Letaief and Sadowsky -

18 –Monte-Carlo Estimation – A sequence D (l) = (M (l),G (l),N (l) ), l = 1,2,…,L, is generated according to their twisted distributions. The error probability is estimated using: where 1 1 (D) = 1 if D γ, zero otherwise. W() is the importance-sampling weighting function, it was chosen as:

19 Hayat and B. Choi – –modified the work of Sadowsky and Letaief to include dead- space effect –other assumptions were held intact –same approach was adopted, only dead-space generalized mgf for G was used in place of the one given by McIntyre –similar approach was also used for Monte-Carlo estimation

20 The generalized theory, takes into account dead-space effect and doesn’t assume instantaneous functioning of APD. The theory uses the model for APD as proposed by Hayat; impulse-response of APD is considered to be a random-duration rectangular function. Salient points of the Theory- –Consider the time interval [0,T b ] and assume current information bit as ‘1’. Let G i, T i be gain and impulse-response duration due to ith primary electron and τ i be the time of impact. Let indicator function for set A be defined as 1 A (x) = 1, x an element of A, zero otherwise. Defining, gives photocurrent as In case of bit ‘0’, it is assumed G i = 0 for all i. Generalized Theory

21 –The photocurrent function can be approximated by dividing [0,T b ] into N equal slots and assuming in each slot all photons get absorbed at same time t = τ i = i(T b /N). Let n i be no. of absorptions in ith slot, G ij and T ij be gain and duration for jth absorption in ith slot, then –The receiver’s output at index ‘k’ is then –We define to get

22 – We examine R 0 and evaluate mgf of A 0,j ( τ i ) as – I is a boolean function function with I(x) = 1, if x is true, zero otherwise. Total no. of photons is a Poisson variable with parameter, say λ, then n i can be assumed to be a Poisson variable with parameter (λ/N), so that mgf of R 0 can be found out as:

23 –Likewise, mgf of R k, R k (μ) can be found as –With respect to [0,T b ], R 0 conveys information about current bit, R 1 conveys information about previous bit, in general, R k conveys information about kth previous bit, provided the bit-stream entirely consists of ‘1’s. For general bit-stream, the receiver output is:

24 For the sake of simplicity, we consider R 0, R 1, R 2 only so that receiver output Y λ becomes: Decision threshold γ is defined as: Bit-Error Probability Estimation which simplifies to

25 Bit-error probability is then given by: Means and variances for Y λ when current bit is ‘1’ and ‘0’ are:

26 The following relations help in calculating the means and variances:

27 Large Deviations Theory –{ Y λ } can be seen as an infinite sequence of random variables w.r.t. λ. We define which on simplification gives –Defining corresponding to cases of current bit being ‘0’ and ‘1’ gives

28 where () + is a function defined as (x) + = x, x>0, zero else. – The rate functions I 0 (x) and I 1 (x) can be computed as: – Assuming Hypothesis 1 of Gärtner-Ellis Theorem is true, we have

29 – Letting and gives – This gives an approximate expression for bit-error probability:

30 Gaussian Approximation –For Gaussian approximation, we assume Y λ ~ N(ρ 0,σ 0 2 ) when current bit is ‘0’ and Y λ ~ N(ρ 1,σ 1 2 ) when current bit is ‘1’. –Then, bit-error probability is given as:

31 Computation of bit-error probability was carried out for InP APD receiver with 100-nm multiplication layer and average gain The optical-link speed was 40 Gbps (i.e. 1/T b = 40 Gbps), the time interval was divided into N = 1000 equal slots and value of decision threshold was γ = A plot for λ (average no. of photons in the time interval) vs. bit-error probability was made, with λ ranging from 1000 to Numerical Results

32 Plot of λ vs. Bit-Error Probability P b

33 Conclusion The generalized theory – –takes into account dead space effect –assumes the functioning of APD to be non-instantaneous The use of asymptotic-analysis techniques and other approximation methods are extended to a wider class of APDs. Large Deviations give a better estimate of error probability compared to Gaussian approximation.


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