Optical Networks: A Practical Perspective - Part 80 pptx

Efficient routing and scheduling algorithms for optical networks.. Bandwidth allocation algorithms on tree-shaped all-optical networks with wavelength converters.. IEEE Network: Special

760 RECEIVER NOISE STATISTICS

An optical amplifier introduces spontaneous emission noise to the signal in addition

to providing gain Consider a system with an optical preamplifier shown in Figure 4.7 The electric field at the input to the receiver may be written as

Eft) ~/-2P cos(2rCfct 4- cb) 4- N(t)

Here, P is the signal power, fc is the carrier frequency, and (I) is a random phase uniformly distributed in [0, 2zr] N(t) represents the amplifier spontaneous emission

noise For our purposes, we will assume that this is a zero-mean Gaussian noise

process with autocorrelation RN(r)

The received power is given by

P(t) - E2(t) - 2P cos2(27Cfct + ~) + 2x/~-fiN(t)cos(27rfct + ~) + N2(t)

The mean power is

To calculate the autocovariance, note that since N(t) is a Gaussian process,

E[N2(t)N2(t + r)] n2(0) + 2R2(r)

using the moment formula (H.1) Using this fact, the autocovariance of P(.) can be calculated to be

p2

The corresponding spectral density is given by

S p ( f ) L p ( r ) e -i27rfr dr

o o

= 2 S N ( f ) 9 S N ( f ) 4- 2 P [ S N ( f - fc) + S N ( f + fc)]

p2

The 9 denotes the convolution operator, where f (x) 9 g(x) - f-~oo f (u)g(x - u)du

After photodetection, the last term in (I.5) and (I.6) can be omitted because the

2fc components will be filtered out

In order to derive the noise powers, we return to (I.3) and substitute for E[P(.)]

and Lp(.) from (I.4) and (I.6), respectively, to obtain

L i f t ) e ~ [ P + RN(0)]8(r) + 7-~2[4PRN(r)cos(2zrfcr)] + ~212R2(r)]

1.2 Amplifier Noise 761

We also have

S i ( f ) eT-~[P + RN(0)] + 72~,22P[SN(f fc) + SN(f + fc)]

+ 7"~212SN(f) * S N ( f ) ] (I.7) The first term on the right-hand side represents the shot noise terms due to the signal and the amplifier noise The second term represents the signal-spontaneous beat noise, and the last term is the spontaneous-spontaneous beat noise Note that

we have so far assumed that the amplifier noise is Gaussian but with an arbitrary spectral shape SN ( f ) In practice, it is appropriate to assume that the amplifier noise

is centered at f~ and is white over an optical bandwidth Bo < 2f~, with

Pn(G-1)

S N ( f ) = O, 2

Bo

I f + fcl <_ T

otherwise

Here, P~ is given by nsphfc, where nsp is the spontaneous emission factor Corre- spondingly, we have

RN(O) f_~c

O0

S N ( f ) d f - P n ( G - 1)Bo

The spectral density of the photocurrent SI(f) from (I.7) is plotted in Figure 1.1, assuming the preceding value for S N ( f ) Note that, as before, the shot noise is white, but the signal-spontaneous beat noise spectrum has a rectangular shape, and the spontaneous-spontaneous beat noise a triangular shape Moreover, the incident optical power P is given by G Pi, where Pi is the input power to the amplifier

Shot

/

Sponta

Signal-spontaneous

Power spectral density

~ I ) ] B o

2

29t Pn(G- 9

I

B e Bo/2

e~R[ GPi + Pn( G-1)Bo]

B o Frequency

Figure 1.1 Photocurrent spectral density

762 R E C E I V E R N O I S E STATISTICS

Note that the photocurrent is passed through a low pass filter with bandwidth

Be The noise power at the output of the filter is given by

fBe

cr 2 m S I ( f ) d f m O.shot _~_ Crsig_spont q_ O.spont_spont, 2 2 2

Be

where

2 _ 2eTg[GPi + P n ( G - 1)Bo]Be

O'shot

2 47p~2GPi Pn(G - 1)Be,

O'sig_spon t

and

2 ,/pv2 ]2 O'spont_spon t [ P n ( G - 1) ( 2 B o - Be)Be

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