Computing & Simulations
7.2. Expressions for a passive optical link
7.2.2. SNR expressions
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0.001 0.010
0.100 1.000
10.000
Received Photo-current [mA]
Noise power in 1 Ohm [dBm/Hz] TOTAL for RIN#1 Thermal Shot RIN#1
(a)
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0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 Optical Losses [dB]
Noise power in 1 Ohm [dBm/Hz]
TOTAL for RIN#1 Thermal Shot RIN#1
(b)
Figure 7.3.: Noise power components and sum for a lossy optical link as a function of (a) the received photo-current; (b) the optical budget
𝑆𝑁 𝑅1 = 𝐼02/𝐶𝐹
(𝑅𝐼𝑁 ·𝐼02+ 2𝑒·𝐼0 +𝐼𝑡ℎ2 ) Δ𝑓 (7.29) Since a linear relation exists between the link losses 𝐿 and the average detected photo- current 𝐼0, the asymptotic behavior of the logarithmic SNR can be expressed as a function of the link losses.
Three different cases are considered whether the SNR is dominated by the RIN, the shot noise or the thermal noise of the receiver:
The RIN term
𝑆𝑁 𝑅1 = 1/𝐶𝐹
𝑅𝐼𝑁 ·Δ𝑓 𝑆𝑁 𝑅1 𝑑𝐵 = +𝛼 (7.30)
The shot noise
𝑆𝑁 𝑅1 = 𝐼0·1/𝐶𝐹
2𝑒·Δ𝑓 𝑆𝑁 𝑅1 𝑑𝐵 =−10 log(𝐿) +𝛽 (7.31) The thermal noise
𝑆𝑁 𝑅1 = 𝐼02·1/𝐶𝐹
𝐼𝑡ℎ2 ·Δ𝑓 𝑆𝑁 𝑅1 𝑑𝐵 =−20 log(𝐿) +𝛾 (7.32) 7.2.3. SNR behavior versus the optical budget
As a direct consequence of the noise power components, when reusing the noise power terms exposed in Fig. 7.3b, for plotting the SNR versus the optical budget (Fig. 7.4a, the SNR behavior (let’s just consider the 100%-curve) is directly linked: from 0 to 8dB, the SNR is almost constant, while from 10 to 16dB, the SNR drops proportionally to the
optical losses: namely by 1dB per 1dB of optical loss. Finally for 18dB and beyond, the SNR degrades by 2dB for 1dB of optical loss.
Taking into account the remarks of §3.3.1 on page 61, and the chart in Fig. 3.21 we can expect correct EVM figures at the transmitter side for optical budgets up to ≈30dB
—when assuming a 30dB SNR to be sufficient for a 3.84MHz wide channel using a simple QPSK modulation scheme.
Impact of the CF onto the SNR
In Fig. 7.3b and 7.4 we considered a Crest Factor (CF) of 10dB, which corresponds to an typical W-CDMA downlink carrier.
Yet for a dummy QPSK signal with a CF of 7dB, according to Eq. 7.28 the SNR improves by 3dB. While when using two W-CDMA carriers, the SNR will degraded by 3dB. Finally when characterizing the SNR by a sine-signal (CF of 3dB), and keeping the reference bandwidth of 3.84MHz of our case, then the SNR curves will apparently improve by 7dB!
Impact of the BW onto the SNR
In Fig. 7.4 the considered bandwidth values 3.84MHz corresponding to the channel width of a W-CDMA carrier. Yet when increasing the bandwidth to 10 or 20MHz (higher channel bandwidths of the recent 4G LTE radio standard), then the SNRs curves in Fig. 7.4a drop by:
10·log (10 or 20MHz/3.84MHz) = 4.2 or 7.2dB (7.33) Impact of the OMI onto the SNR
In Fig. 7.4, the SNR is plotted for several peak OMI since in IM-DD-schemes the individual OMI of a carrier can be reduced in order to operate in the linear part of laser diode. . . or to avoid over-driving the laser.
This black curve assumed an ideal peak OMI of 100%. For instance passing to a peak OMI of 55% causes the SNR to drop by:
20·log (actual peak OMI %/100%) = 5.2dB (7.34) Thus of course the highest possible OMI is targeted. . .
30 35 40 45 50 55 60 65 70
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 Optical loss [dB]
SNR [dB/3.84MHz]
100%
85%
70%
55%
40%
25%
(a) (b)
Figure 7.4.: SNR of a lossy optical link for several peak OMIs and signal bandwidth of (a) 3.84MHz and (b) 20MHz. For both CF=10dB, other parameters as for Fig. 7.3b
Non-Linear Distortion (NLD) noise
. . . yet the clipping at the transmitter sides causes non-linear distortions and can be considered useless thus as an additional noise component. According to [108, 109] these NLD can be modeled in terms of noise source on the receiver as:
𝑃𝑁 𝐿𝐷 =
√︃2
𝜋 ·𝐼02·𝜇5·exp
(︃
− 1 2·𝜇2
)︃
in𝐴2 (7.35)
where N is number of multiplexed carriers, and𝜇 is the total Root-Mean-Square (RMS) modulation index (and furthermore assuming identical modulation index for each channel):
𝜇=√
𝑁 · 𝑂𝑀 𝐼𝑝𝑒𝑎𝑘
√
𝐶𝐹 =√
𝑁 ·𝑂𝑀 𝐼𝑅𝑀 𝑆 (7.36)
Contrary to the contributions of RIN, of the shot and thermal noises,𝑃𝑁 𝐿𝐷 does not depend upon the considered channel bandwidth. NB: according to [110] this NLD-modeling is not included in simulations tools likeVPItransmissionMaker.
When assuming the NLD to be the dominating noise, then the SNR expression can be simplified to:
𝑆𝑁 𝑅1 =
√︂𝜋
2 ·𝜇−3·exp
(︃
+ 1
2·𝜇2
)︃
𝑆𝑁 𝑅1 𝑑𝐵 =−30·log (𝜇) + 5
ln 10·𝜇2 +𝜀 (7.37) Fig. 7.5 illustrates Eq. 7.37 for a single carrier as a function of the RMS OMI, and as a function the peak OMI assuming a signal with a 10dB Crest Factor. The NLDs’
contributions to the SNR term are negligible for RMS OMIs up to 20% (≡63% peak OMI for a 10dB-crest-factor signal).
Since in this thesis, we will target —in some cases— applications requiring a SNR of 50dB over important optical budgets, the temptation of increasing the OMI in order to
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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
OMI (RMS or PEAK(CF=10dB))
SNR [dB]
SNR for NLD = f (OMI rms) SNR for NLD = f (OMI peak) target SNR #1 target SNR #2
Figure 7.5.: SNR when considering only NLD as function of the OMIs (rms and peak) not taking into account any losses
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0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 Optical loss [dB]
Noise power [dBm] for a single channel
w/ NLD for pk OMI 100%
w/ NLD for pk OMI 85%
w/ NLD for pk OMI 80%
w/ NLD for pk OMI 75%
w/ NLD for pk OMI 65%
w/o NLD
(a)
-155 -150 -145 -140 -135 -130 -125 -120 -115 -110 -105 -100 -95 -90
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 Optical loss [dB]
Noise power [dBm] for a single channel
w/ NLD for pk OMI 100%
w/ NLD for pk OMI 85%
w/ NLD for pk OMI 80%
w/ NLD for pk OMI 75%
w/ NLD for pk OMI 65%
w/o NLD
(b)
Figure 7.6.: Noise power with and without the NLD component for several peak OMIs (10dB- crest-factor signal) and for channel bandwidths of (a) 3.84MHz, and (b) 20MHz
maximize the SNR exists. Yet given the NLD, the OMI should only be increased to its tolerable maximum.
In order to assess the impact of the NLD contribution to the SNR (Eq. 7.38), Fig. 7.6 shows, for several peak OMIs, whether the NLD distortions are prevalent over sum of the RIN, shot, and thermal noises.
𝑆𝑁 𝑅 = 𝐼02·𝑂𝑀 𝐼𝑝𝑒𝑎𝑘2 /𝐶𝐹
(𝜎𝑅𝐼𝑁2 +𝜎2𝑠ℎ𝑜𝑡+𝜎𝑡ℎ𝑒𝑟𝑚.2 ) +𝑃𝑁 𝐿𝐷 (7.38) Finally
For a 3.84MHz bandwidth, Fig. 7.7a shows that when taking into account the NLD noise term, the highest optical budget is no longer necessarily reached by the highest OMI.
Indeed, for an SNR ≥50dB, the optimum peak OMI (assuming a crest factor of 10dB) ranges from 40 to 65%. For the upper-bound OMI, the SNR is limited by the NLD component while the limit of the lower-bound OMI is mainly set by the thermal noise component. Yet if the targeted SNR is below 40dB, then there is no point in considering the OMI, and over-modulation is no longer an issue.
Similar conclusions can be made when considering a 20MHz channel.
Introducing the NLD issue decreases the maximum SNR.
25 30 35 40 45 50 55 60 65
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 Optical Budget [dB]
SNR [dB] for a single channel
100%
85%
80%
75%
65%
55%
40%
(a)
25 30 35 40 45 50 55 60 65
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 Optical Budget [dB]
SNR [dB] for a single channel
100%
85%
80%
75%
65%
55%
40%
(b)
Figure 7.7.: SNR for several peak OMIs, an optical launch power of +7.02dBm, channel band- widths of (a) 3.84MHz, and (b) 20MHz