Transcript Document

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
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
• 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
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
• Avalanche Process Multiplication Layer
First impact ionization
Injected electron
p
n
de
dh
de
dh
de
x
W
Eie
de 
qE
Dead space for electron
Eie and Eih are the
average ionization
threshold energies
Eih
dh 
qE
Dead space for hole
Electric field E
Hole
Electron
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 -
–
and
are the intermediate quantities and
• Physical Interpretation of Renewal Relations parent electron
first impact ionization
f e (m, t; x |  )
p
0
+V

f e (m, t  ; x   )


ve
f h (m, t  ; x   ) f e (m, t  ; x   )
vh
ve
x
x+
n
W
• Numerical Computation Recursive Equations
f e (m, t ; x), f h (m, t ; x)
for some fixed z, and
suitable range of t & x
Recursive Equations
z-transform over m
Fe ( z , t ; x), Fh ( z , t ; x )
Initialize data
Fe
( 0)
, Fh
( 0)
Update data
Compute relative change
Fe( n 1)
Err (e)  max Fe( n 1)  Fe( n )
Fh( n )  Fh( n 1)
Err (h)  max Fh( n 1)  Fh( n )
Fe
(n)

No
If
max Err (e), Err (h)
<tolerance level
Gubner and
Hayat method
f G ,T (m, t )
FG ,T ( z, t ) 
z Fe ( z , t ;0)
(for fixed z and hence
m)
Final result
Fe
(n)
 Fe
Fh( n )  Fh
Yes
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
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
Statement of Cramér's Theorem -
Corollary of Cramér's Theorem -
Deduction from Corollary -
Statement of Gärtner-Ellis Theorem -
Bit-Error Probability Computation
Literature Review
• 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
• Salient Points of the Theory –
– Define
• Gk = gain of kth primary electron
• M = no. of primary electrons in signaling interval
• N = thermal noise response of receiver with variance σ2
– Assume {Gk} to be i.i.d. This gives the receiver statistic as:
– Consider the hypotheses:
– Let γ be the decision threshold, define:
– 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/c0)λ0 = (1/c1)λ1, c0 and c1 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.
Result of Letaief and Sadowsky -
– 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 11(D) = 1 if D< γ, zero otherwise; and 10(D) = 1 if D> γ,
zero otherwise.
• W(•) is the importance-sampling weighting function, it was
chosen as:
• Hayat and B. Choi –
– modified the work of Sadowsky and Letaief to include deadspace 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
Generalized Theory
• 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,Tb] and assume current information bit
as ‘1’. Let Gi, Ti be gain and impulse-response duration due to ith
primary electron and τi be the time of impact. Let indicator function
for set A be defined as 1A(x) = 1, x an element of A, zero
otherwise.
Defining
, gives photocurrent as
In case of bit ‘0’, it is assumed Gi = 0 for all i.
– The photocurrent function can be approximated by dividing [0,Tb]
into N equal slots and assuming in each slot all photons get
absorbed at same time t = τi = i(Tb/N). Let ni be no. of absorptions in
ith slot, Gij and Tij be gain and duration for jth absorption in ith slot,
then
– The receiver’s output at index ‘k’ is then
– We define
to get
– We examine R0 and evaluate mgf of A0,j(τi) 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
ni can be assumed to be a Poisson variable with parameter (λ/N), so
that mgf of R0 can be found out as:
– Likewise, mgf of Rk, Rk(μ) can be found as
– With respect to [0,Tb], R0 conveys information about current bit, R1
conveys information about previous bit, in general, Rk conveys
information about kth previous bit, provided the bit-stream entirely
consists of ‘1’s. For general bit-stream, the receiver output is:
Bit-Error Probability Estimation
• For the sake of simplicity, we consider R0, R1, R2 only so that
receiver output Yλ becomes:
• Decision threshold γ is defined as:
which simplifies to
• Bit-error probability is then given by:
• Means and variances for Yλ when current bit is ‘1’ and ‘0’ are:
• The following relations help in calculating the means and variances:
• Large Deviations Theory
– {Yλ} 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
where (•)+ is a function defined as (x)+ = x, x>0, zero else.
– The rate functions I0(x) and I1(x) can be computed as:
– Assuming Hypothesis 1 of Gärtner-Ellis Theorem is true, we have
– Letting
and
gives
– This gives an approximate expression for bit-error probability:
• Gaussian Approximation
– For Gaussian approximation, we assume Yλ ~ N(ρ0,σ02)
when current bit is ‘0’ and Yλ ~ N(ρ1,σ12) when current bit is
‘1’.
– Then, bit-error probability is given as:
Numerical Results
• 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/Tb = 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.
Plot of λ vs. Bit-Error Probability Pb
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.
QUESTIONS ??