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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 the guidance of Majeed M. Hayat and P. Sun UNM, Albuquerque

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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

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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

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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

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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

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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 -

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

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

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Joint distribution function (PDF) of G and T for homo-junction GaAs APD with 160 nm – multiplication layer and average gain 10.46 Joint density function of G and T for the same APD. Large peaks have been truncated to show details

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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

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Statement of Cramér's Theorem -

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Corollary of Cramér's Theorem - Deduction from Corollary -

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Statement of Gärtner-Ellis Theorem -

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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

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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:

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–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.

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Result of Letaief and Sadowsky -

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–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:

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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

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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

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–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

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– 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:

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–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:

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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

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

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The following relations help in calculating the means and variances:

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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

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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

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

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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:

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Computation of bit-error probability was carried out for InP APD receiver with 100-nm multiplication layer and average gain 10.699. 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 = 5.325. A plot for λ (average no. of photons in the time interval) vs. bit-error probability was made, with λ ranging from 1000 to 3000. Numerical Results

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Plot of λ vs. Bit-Error Probability P b

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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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