Performance analysis of phase modulated RoF based two and three tone transmission against intermodulation distortion over a dispersive link

Abstract

Analytical model for distribution of multi-tone radio frequencies signal is developed in the present paper. Based on the analytical model, signal to noise distortion ratio (SNDR) has been investigated for different modulation indices at dispersion of 400 and 800 ps² including the effects of intermodulation distortion and noise. It is found that SNDR decreases by 17 dB as dispersion increases from 400 to 800 ps². The modulation index should be kept small to minimise the unwanted modulation nonlinearities. Further system dynamic range has also been investigated at the considered dispersion values. On comparison, it has been found that dynamic range reduces by 7.754 dB in 1-Hz bandwidth for a three tone transmission system compared to a two tone transmission system. Simulation is also carried out for studying output spectrum of signals using Optsim and verifies the results of analytical analysis. There is trade-off between the output power and system dynamic range which should be considered in the system design.

Appendix

Beating takes place in photodetector between optical components. Different orders of electrical harmonic components are generated due to beating. We find the kth harmonic component, ${ \exp }\left({jk\omega_{m} t} \right),$ with an angular frequency $(k_{1} \omega_{1} + k_{2} \omega_{2} + k_{3} \omega_{3})$ can be expressed as:

$$\begin{aligned} i_{{k_{1},k_{2},k_{3}}} \left(t \right) & = \rho \mathop \sum \limits_{p = - \infty}^{\infty} \mathop \sum \limits_{q = - \infty}^{\infty} \mathop \sum \limits_{r = - \infty}^{\infty} \left\{j^{{p + q + r + k_{1} + k_{2} + k_{3}}} J_{{p + k_{1}}} J_{{q + k_{2}}} J_{{r + k_{3}}} \exp \left[{{\text{j}}\left({q + k_{2}} \right){\upvarphi}} \right] \right. \\ & \quad \left. \times \exp \left[{{\text{j}}\left(2 \right)\left({r + k_{3}} \right){\upvarphi}} \right]{ \exp }\left({\text{j}}\left[\left(p + k_{1}\right)\omega_{1} + (q + k_{2})\omega_{2}\right.\right.\right.\\ &\quad\left.\left.\left.+\,(r + k_{3})\omega_{3} \right]{\text{t}} \right)\exp \left(j\frac{1}{2}\beta_{2} L\left[(p + k_{1})\omega_{1} + \left({q + k_{2}} \right)\omega_{2} + {\left(r + k_{3}\right)\omega_{3}} \right]^{2} \right)\right\} \\ & \quad \times \left\{j^{- p - q - r} J_{p} J_{q} J_{r} \exp \left({- jq{\upvarphi}} \right)\exp \left[{- j\left(2 \right)r{\upvarphi}} \right]\right.\\ &\left.\quad\times \exp \left[- j\left({p\omega_{1} + q\omega_{2} + r\omega_{3}} \right)t\right]\exp \left({- j\frac{1}{2}\beta_{2} L\left({p\omega_{1} + q\omega_{2} + r\omega_{3}} \right)^{2}} \right) \right\} \end{aligned}$$

(17)

After simplification, we get

$$\begin{aligned} i_{{k_{1},k_{2},k_{3}}} &= \rho j^{{k_{1} + k_{2} + k_{3}}} \exp \left({jk_{2} \varphi} \right)\exp \left[{j\left(2 \right)k_{3} \varphi} \right]\exp \left[j\left(k_{1} \omega_{1} + k_{2} \omega_{2}\right.\right.\\ &\quad\left.\left.+\,k_{3} \omega_{3} \right)t \right]\exp \left[{- j\frac{1}{2}\beta_{2} L\left({k_{1} \omega_{1} + k_{2} \omega_{2} k_{3} \omega_{3}} \right)^{2}} \right]{\text{S}}_{1} \cdot {\text{S}}_{2} \cdot {\text{S}}_{3} \end{aligned}$$

(18)

where

$${\text{S}}_{1} = \mathop \sum \limits_{p = - \infty}^{\infty} J_{{p + k_{1}}} J_{p} \exp \left[{{\text{j}\upbeta}_{2} {\text{L}}\left({k_{1} + p} \right)\left({k_{1} \omega_{1}^{2} + k_{2} \omega_{1} \omega_{2} + \omega_{1} k_{3} \omega_{3}} \right)} \right]$$

(19)

$${\text{S}}_{2} = \mathop \sum \limits_{q = - \infty}^{\infty} J_{{q + k_{2}}} J_{q} \exp \left[{{\text{j}\upbeta}_{2} {\text{L}}\left({k_{2} + q} \right)\left({k_{2} \omega_{2}^{2} + k_{1} \omega_{1} \omega_{2} + k_{3} \omega_{3} \omega_{2}} \right)} \right]$$

(20)

$${\text{S}}_{3} = \mathop \sum \limits_{r = - \infty}^{\infty} J_{{r + k_{3}}} J_{r} \exp \left[{{\text{j}\upbeta}_{2} {\text{L}}\left({k_{3} + r} \right)\left({k_{3} \omega_{3}^{2} + k_{2} \omega_{2} \omega_{3} + k_{1} \omega_{1} \omega_{3}} \right)} \right]$$

(21)

By using the addition theorem for the Bessel function, ${\text{S}}_{1}$ can be simplified as:

$${\text{S}}_{1} = j^{{{\text{k}}_{1}}} J_{{k_{1}}} 2m\sin \frac{{\theta_{1}}}{2} {\text{e}}^{{{\text{jk}}_{1} \frac{{\theta_{1}}}{2}}}$$

(22)

Similarly, ${\text{S}}_{2}$ and ${\text{S}}_{3}$ can be simplified as:

$${\text{S}}_{2} = j^{{{\text{k}}_{2}}} J_{{k_{2}}} 2m\sin \frac{{\theta_{2}}}{2} {\text{e}}^{{{\text{jk}}_{2} \frac{{\theta_{2}}}{2}}}$$

(23)

$${\text{S}}_{3} = j^{{k_{3}}} J_{{k_{3}}} 2m\sin \frac{{\theta_{3}}}{2} {\text{e}}^{{{\text{j}}k_{3} \frac{{\theta_{3}}}{2}}}$$

(24)

Thus,

$$\begin{aligned} i_{{k_{1},k_{2},k_{3}}} & = \rho \left({- 1} \right)^{{k_{1} + k_{2} + k_{3}}} \exp \left({jk_{2} \varphi} \right)\exp \left[{j\left(2 \right)k_{3} \varphi} \right]\exp \left[{j\left({k_{1} \omega_{1} + k_{2} \omega_{2} + k_{3} \omega_{3}} \right)t} \right] \\ & \quad \times\,J_{{k_{1}}} \left(2m\sin \frac{{\theta_{1}}}{2}\right) \times J_{{k_{2}}} \left({2m\sin \frac{{\theta_{2}}}{2}} \right) \times J_{{k_{3}}} \left({2m\sin \frac{{\theta_{3}}}{2}} \right) \\ \end{aligned}$$

(25)

and the two tone signal with RF frequencies $\omega_{1} \,and\, \omega_{2}$ is represented as:

$$i_{{k_{1},k_{2}}} = \rho \left({- 1} \right)^{{k_{1} + k_{2}}} \exp \left({\begin{array}{*{20}c} {jk_{2} \varphi} \\ \end{array}} \right)J_{{k_{1}}} \left({2m\sin \frac{{\theta_{1}}}{2}} \right) \times J_{{k_{2}}} \left({2m\sin \frac{{\theta_{2}}}{2}} \right)\exp \left[{j\left({k_{1} \omega_{1} + k_{2} \omega_{2}} \right)t} \right]$$

(26)