Lecture IVStatistical Models in Optical Communications

DIRECT DETECTION• Gaussian approximation for single-shot link performance• Receiver thermal noise• Single-shot OOK receiver modelling -2nd order statistics at

sampler output • Symbol-Noise Ratios and BER for ISI-free OOK in the Poisson-

Gauss regime

(ch. 11 – part 2 “Notes”)

Detection in shot + gaussian noise

( )rz t

ArbitraryShape

Receivefilter

SufficientStatisticExtractor

Slicer

Binary transmission system

ReceivedSignal

We consider the performance of a receiver which operates in the single-shot mode,or equivalently, we ignore the effect of intersymbol interference upon transmitting trains of pulses.

The second term of the last equation is automatically gaussian, but we have seen the central theorem result whereby a filtered Poisson process such as the first term of the last equation, tends to gaussian in the limit of large rate. Hence, we end up with the sum of two gaussian variables, which is gaussian itself.

Gaussian approximation for single-shot link performance (I)

signal separation - defined as the difference of the two means of the received sufficient statistics, conditioned on the two respective transmission hypotheses:

Gaussian approximation for single-shot link performance (II)

To express the BER as a function of the threshold we require the likelihood functions:

the optimal ML threshold:

Gaussian approximation for single-shot link performance (III)

Gaussian approximation for single-shot link performance (IV)

It turns out that there exists a better choice of threshold, albeit not optimal within the rules of the gaussian approximation, but better approximating the true threshold of the mixed Poisson-GaussThis modified value of threshold is the one making the two conditional error probabilities equal,(rather than making the likelihood functions equal as in the ML solution). Hence, we choose the threshold so as to make the integrated tails of the two likelihoods of equal area (rather than rendering heights of the likelihood functions equal), a condition which may written as

Gaussian approximation for single-shot link performance (V)

Gaussian approximation for single-shot link performance (VI)

Gaussian approximation for single-shot link performance (VII)

Gaussian approximation for single-shot link performance (VIII)

Gaussian approximation for single-shot link performance (IX)

Gaussian approximation for single-shot link performance (X)

Gaussian approximation for single-shot link performance (XI)

Gaussian approximation for single-shot link performance (XII)

Receiver thermal noise (I)

• <<drawing>>

Receiver thermal noise (II)

Receiver thermal noise (III)

Receiver thermal noise (IV)

Receiver thermal noise (V)

Receiver thermal noise (VI)

The spot noise figure

Receiver thermal noise (VII)

Receiver thermal noise (VIII)

Noisless Noiseless

Freq. dependent white

Noisy

+

Receiver thermal noise (IX)

Receiver thermal noise (X)

where

Single-shot OOK receiver modelling -2nd order statistics at sampler output (I)

Single-shot OOK receiver modelling -2nd order statistics at sampler output (II)

Single-shot OOK receiver modelling -2nd order statistics at sampler output (III)

Single-shot OOK receiver modelling -2nd order statistics at sampler output (IV)

Single-shot OOK receiver modelling -2nd order statistics at sampler output (V)

Use Cambell’s theorem

Single-shot OOK receiver modelling -2nd order statistics at sampler output (VI)

Single-shot OOK receiver modelling -2nd order statistics at sampler output (VII)

Single-shot OOK receiver modelling -2nd order statistics at sampler output (VIII)

Single-shot OOK receiver modelling -2nd order statistics at sampler output (IX)

( )

Dark current

Single-shot OOK receiver modelling -2nd order statistics at sampler output (X)

The other approach is to calculate the shot fluctuations associated with the dark current, using Campbell’s theorem:

Single-shot OOK receiver modelling -2nd order statistics at sampler output (XI)

It is worthwhile representing the dark current as an effective constant intensity optical source,which would yield the same average current as the level of the dark current. In terms of mean arrival rate,

Single-shot OOK receiver modelling -2nd order statistics at sampler output (XII)

Single-shot OOK receiver modelling -2nd order statistics at sampler output (XIII)

Single-shot OOK receiver modelling -2nd order statistics at sampler output (XIV)

Single-shot OOK receiver modelling -2nd order statistics at sampler output (XV)

Symbol-Noise Ratios and BER for ISI-free OOK in the Poisson-Gauss regime (I)

Symbol-Noise Ratios and BER for ISI-free OOK in the Poisson-Gauss regime (II)

Symbol-Noise Ratios and BER for ISI-free OOK in the Poisson-Gauss regime (III)

Symbol-Noise Ratios and BER for ISI-free OOK in the Poisson-Gauss regime (IV)

(repeated from previous slide for easy see)

Symbol-Noise Ratios and BER for ISI-free OOK in the Poisson-Gauss regime (V)

Symbol-Noise Ratios and BER for ISI-free OOK in the Poisson-Gauss regime (VI)

Symbol-Noise Ratios and BER for ISI-free OOK in the Poisson-Gauss regime (VII)

Symbol-Noise Ratios and BER for ISI-free OOK in the Poisson-Gauss regime (VIII)

where the s-factor is given by the harmonic mean of the two symbol-noise ratios, which add as resistors in parallel to generate the overall snr ratio, which determines the probability of error.

Symbol-Noise Ratios and BER for ISI-free OOK in the Poisson-Gauss regime (IX)

Symbol-Noise Ratios and BER for ISI-free OOK in the Poisson-Gauss regime (X)

shot noise limited

Symbol-Noise Ratios and BER for ISI-free OOK in the Poisson-Gauss regime (XI)

…shot noise limited – continued…

Symbol-Noise Ratios and BER for ISI-free OOK in the Poisson-Gauss regime (XII)

thermal noise limited

which seems correct at least as far as the linear dependence on optical power (as well as responsivity and mean APDgain) and the inverse dependence on the noise equivalent current is concerned (the noise is constant but the signalseparation in the current or voltage domains grows linearly with the optical power).

Symbol-Noise Ratios and BER for ISI-free OOK in the Poisson-Gauss regime (XIII)…thermal noise limited – continued…

Direct detection link with intersymbol interference (I)

The total received optical field

Direct detection link with intersymbol interference (II)

Direct detection link with intersymbol interference (III)

Direct detection link with intersymbol interference (IV)

Direct detection link with intersymbol interference (V)

Direct detection link with intersymbol interference (VI)

Direct detection link with intersymbol interference (VII)

Direct detection link with intersymbol interference (VIII)

Direct detection link with intersymbol interference (IX)

Direct detection link with intersymbol interference (X)

Math Detour: Cyclostationary processes, and PAM modulation (I)

Math Detour: Cyclostationary processes, and PAM modulation (II)

Math Detour: Cyclostationary processes, and PAM modulation (III)

Math Detour: Cyclostationary processes, and PAM modulation (IV)

Math Detour: Cyclostationary processes, and PAM modulation (V)

Time-average:

Math Detour: Cyclostationary processes, and PAM modulation (VI)

Math Detour: Cyclostationary processes, and PAM modulation (VII)

Proof: