# Calculation of Receiver Sensitivities in (Orthogonal) Subcarrier Multiplexing Microwave-Optical Links

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Electro-Optical System

## 3. Mathematical Model

_{F}represents the quality factor of the digital communication, N states the number of electrical subchannels, J

_{n}(x) stands for the n

^{th}order Bessel function of the first kind, m represents the rms OMI per subchannel (m = πV

_{AC}/2V

_{π}where V

_{π}is the half-wave voltage of the MZMs and V

_{AC}is the peak voltage of a tone in theory but represents the rms voltage per subchannel at the electrical input of the MZMs in practice [10]), and, finally, D

_{IMD}represents the distortion associated with IMD.

_{i}− Ω

_{j}), third order intermodulation (Ω = Ω

_{i}± 2Ω

_{j}), and a different case of third order intermodulation that is called triple beat (Ω = Ω

_{i}± Ω

_{j}± Ω

_{k}), where Ω

_{i}, Ω

_{j}, and Ω

_{k}are any three arbitrary frequencies [15]. For systems consisting of many subchannels, triple beat is the dominant third order intermodulation [15], but for the presented case which consists of a low number of subchannels, the described subdivision of the third order IMD is necessary. For each subchannel in a given frequency plan, there can be a different number of interfering intermodulation products of each kind. These values are denoted as N

_{CSO}for the second order intermodulation, N

_{IM}

_{3}for the third order intermodulation, and N

_{CTB}for the triple beat. In the case under analysis, with the MZMs biased at quadrature, the received photocurrent is free of second order intermodulation [10]. For that reason, only N

_{IM}

_{3}and N

_{CTB}are analysed, and are reported in Table 1 for the two frequency plans under analysis. Finally, from [10], D

_{IMD}is expressed as

_{IMD}can be different for every subchannel, which translates into different optical sensitivities for each subchannel.

## 4. Results

^{10}− 1 bits were generated in each baseband source. Each PRBS was delayed by a random integer number of bits to ensure a degree of decorrelation between subchannels. In the baseband receivers, the digital signals were sampled in the middle of the bit period before performing a binary decision. It is important to note that the BER was calculated emulating a real system, in contrast with other approaches that estimate BER according to statistical distributions of noise. For different values of OMI, the attenuation of the VOA in the receiver was increased until a BER of 3.8 × 10

^{−3}was achieved, which is the typical BER threshold for hard-decision forward error correction (HD-FEC) codes with a 7% overhead. With this procedure, the optical receiver sensitivities were obtained as a function of OMI. The simulated results, obtained with the VPI software, are illustrated in Figure 2, where they are also compared with the theoretical calculations obtained with the mathematical model presented in Section 3. Note that the results are shown as a function of overall rms OMI with respect to V

_{π}:

_{rms}is the rms voltage per subchannel and V

_{RMS}is the overall rms voltage.

_{%}≤ 35%). When nonlinearities are dominant, especially triple-beat for Subchannel 2, the error can be higher. This effect occurs because the Gaussian approximation is more accurate to model the ASE noise than the IMD. In reality, the effect of IMD is also related to the phase alignment of subchannels and can vary for different sampling points. Apart from that, the IMD count in this particular case is small, and Gaussian approximations tend to be more accurate when a higher number of intermodulation products take place. However, and despite the inaccuracies of the model, it can be observed that the optimum OMI that gives rise to the best achievable optical sensitivity for each subchannel is accurately predicted by the mathematical model (M

_{%}≈ 30%).

_{%}≤ 35%), the error is higher than in the previous case, ≤4 dB. However, similarly to the previous case, the optimum OMI for each subchannel is predicted with a negligible error (M

_{%}≈ 25% for Subchannel 2 and M

_{%}≈ 35% for Subchannels 1 and 3).

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Direct Detection (DD) electro-optical scheme compatible with subcarrier multiplexing (SCM)/single side band (SSB) and orthogonal subcarrier multiplexing (OSCM)/SSB transmission based on optical IQ modulator and pre-amplified optical receiver; (

**b**) example of transmitted optical spectrum for the case of SCM/SSB consisting of three 2.7 Gbaud quadrature phase shift keying (QPSK) subchannels; (

**c**) example of transmitted optical spectrum for the case of OSCM/SSB consisting of three orthogonally overlapping 2.7 Gbaud QPSK subchannels.

**Figure 2.**Simulated and theoretical best achievable optical receiver sensitivities for (

**a**) the SCM/SSB frequency plan consisting of three 2.7 Gbaud QPSK subchannels and (

**b**) the OSCM/SSB frequency plan consisting of three orthogonally overlapping 2.7 Gbaud QPSK subchannels. The values employed in the simulation and the theoretical calculations are: N = 3, F = 5 dB, v = 193.4 THz (1550 nm), Q

_{F}= 2.67 (BER = 3.8 × 10

^{−3}), B = 2.7 GHz, and intermodulation product count according to Table 1.

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**MDPI and ACS Style**

Gutiérrez, F.A.; Martin, E.P.; Perry, P.; Ellis, A.D.; Barry, L.P.
Calculation of Receiver Sensitivities in (Orthogonal) Subcarrier Multiplexing Microwave-Optical Links. *Appl. Sci.* **2017**, *7*, 184.
https://doi.org/10.3390/app7020184

**AMA Style**

Gutiérrez FA, Martin EP, Perry P, Ellis AD, Barry LP.
Calculation of Receiver Sensitivities in (Orthogonal) Subcarrier Multiplexing Microwave-Optical Links. *Applied Sciences*. 2017; 7(2):184.
https://doi.org/10.3390/app7020184

**Chicago/Turabian Style**

Gutiérrez, Fernando A., Eamonn P. Martin, Philip Perry, Andrew D. Ellis, and Liam P. Barry.
2017. "Calculation of Receiver Sensitivities in (Orthogonal) Subcarrier Multiplexing Microwave-Optical Links" *Applied Sciences* 7, no. 2: 184.
https://doi.org/10.3390/app7020184