Extended Single-Sideband Light Intensity Modulation Formats

Abstract—A theoretical analysis of various single-sideband modulation formats used in fiber-optic communication lines with an external electrooptical two-electrode Mach–Zehnder modulator for generating optical signals resistant to dispersive power degradation has been performed. A family of new single-sideband modulation formats is presented, the features of their use, advantages, and disadvantages are considered. A decrease in nonlinear distortions of signals during their transportation through dispersive optical fiber is assumed.

APPENDIX

In fiber-optic communication lines, the stage of transporting light signals through an optical fiber is described in terms of classical electromagnetic waves, i.e., in terms of the strengths of electric and magnetic fields. Registration of the incoming signals with the help of a photodetector is based on the phenomenon of the photoelectric effect and returns us to the photon interpretation and the need to determine the intensity of light. Thus, the optical signal detector is quadratic with respect to the electric field strength of the light wave. Determination of the spectral components of light intensity (9) based on knowledge of the spectral composition of corresponding electromagnetic wave (6) leads to the calculation of series of type (10). Such calculations are usually performed using the Graf summation theorem for Bessel functions. However, the traditional presentation of this theorem, available in well-known monographs [13, 14] devoted to Bessel functions, leads to expressions that are rather difficult to directly apply for the purposes of analyzing the transmission of light signals in fiber-optic communication lines. In this Appendix, we will reformulate and prove Graf’s theorem in terms most suitable for such analytical calculations.

We start with the well-known integral representation of the Bessel function

$${{J}_{n}}({{z}_{3}}) = \frac{1}{{2\pi }}\int\limits_{ - \pi }^\pi {\exp [i({{z}_{3}}\sin \theta - n\theta )]d\theta } .$$

(A.1)

Change of variables $\theta = \psi + \gamma$, $d\theta = d\psi$ leads (A.1) to form

$$\begin{gathered} {{J}_{n}}({{z}_{3}}) = \frac{1}{{2\pi }} \\ \times \,\,\int\limits_{ - \pi - \gamma }^{\pi - \gamma } {\exp [i{{z}_{3}}\sin (\psi + \gamma )]\exp [ - in(\psi + \gamma )]d\psi } . \\ \end{gathered} $$

(A.2)

Next, we multiply both sides of equality ${{z}_{1}}\exp (i\alpha ) + {{z}_{2}}\exp (i\beta )$ $= {{z}_{3}}\exp (i\gamma )$ by $\exp (i\psi )$ and equate the imaginary parts of the resulting expression. Then, we get

$${{z}_{1}}\sin (\alpha + \psi ) + {{z}_{2}}\sin (\beta + \psi ) = {{z}_{3}}\sin (\gamma + \psi ).$$

Substituting this into formula (A.2), we arrive at relation

$$\begin{gathered} \exp (in\gamma ){{J}_{n}}({{z}_{3}}) \\ = \frac{1}{{2\pi }}\int\limits_{ - \pi - \gamma }^{\pi - \gamma } {\exp [i{{z}_{1}}\sin (\alpha + \psi )]} \exp [i{{z}_{2}}\sin (\beta + \psi )]\exp ( - in\psi )d\psi . \\ \end{gathered} $$

(A.3)

Now using the Jacobi–Anger decomposition, we write

$$\begin{gathered} \exp [i{{z}_{1}}\sin (\alpha + \psi )] = \sum\limits_{m = - \infty }^{ + \infty } {{{J}_{m}}({{z}_{1}})\exp [im(\alpha + \psi )]} , \\ \exp [i{{z}_{2}}\sin (\beta + \psi )] = \sum\limits_{k = - \infty }^{ + \infty } {{{J}_{k}}({{z}_{2}})\exp [ik(\beta + \psi )]} \\ \end{gathered} $$

(A.4)

and substitute in (A.3). Then, we get

$$\begin{gathered} \exp (in\gamma ){{J}_{n}}({{z}_{3}}) \\ = \frac{1}{{2\pi }}\int\limits_{ - \pi - \gamma }^{\pi - \gamma } {\sum\limits_{m = - \infty }^{ + \infty } {{{J}_{m}}({{z}_{1}})\exp [im(\alpha + \psi )]\,\sum\limits_{k = - \infty }^{ + \infty } {{{J}_{k}}({{z}_{2}})\exp [ik(\beta + \psi )]} \exp ( - in\psi )d\psi } } \\ = \sum\limits_{m = - \infty }^{ + \infty } {{{J}_{m}}({{z}_{1}})\exp (im\alpha )} \sum\limits_{k = - \infty }^{ + \infty } {{{J}_{k}}({{z}_{2}})\exp (ik\beta )} \times \,\,\frac{1}{{2\pi }}\int\limits_{ - \pi - \gamma }^{\pi - \gamma } {\exp (im\psi )\exp (ik\psi )\exp ( - in\psi )d\psi } . \\ \end{gathered} $$

(A.5)

Change of variables $\psi = \chi - \gamma$, $d\psi = d\chi$ in integral (A.5) gives

$$\begin{gathered} \exp (in\gamma ){{J}_{n}}({{z}_{3}}) = \sum\limits_{m = - \infty }^{ + \infty } {{{J}_{m}}({{z}_{1}})\exp (im\alpha )} \sum\limits_{k = - \infty }^{ + \infty } {{{J}_{k}}({{z}_{2}})\exp (ik\beta )} \\ \times \,\,\frac{1}{{2\pi }}\int\limits_{ - \pi }^\pi {\exp [i(m + k - n)\chi ]\exp [ - i(m + k - n)\gamma ]d\chi } \\ = \sum\limits_{m = - \infty }^{ + \infty } {{{J}_{m}}({{z}_{1}})\exp (im\alpha )} \sum\limits_{k = - \infty }^{ + \infty } {{{J}_{k}}({{z}_{2}})\exp (ik\beta )} \exp [ - i(m + k - n)\gamma ]{{\delta }_{{m + k,n}}}, \\ \end{gathered} $$

(A.6)

where we used equality

$$\frac{1}{{2\pi }}\int\limits_{ - \pi }^\pi {\exp [i(m - n)\varphi ]d\varphi } = {{\delta }_{{mn}}}$$

(A.7)

and introduced Kronecker symbol ${{\delta }_{{mn}}}$ equal to 1 at m = n and 0 otherwise. As a result, we obtain a convenient formula for summing series:

$$\begin{gathered} \exp (in\gamma ){{J}_{n}}({{z}_{3}}) \\ = \sum\limits_{m = - \infty }^{ + \infty } {{{J}_{m}}({{z}_{1}})\exp (im\alpha )} {{J}_{{n - m}}}({{z}_{2}})\exp [i(n - m)\beta ], \\ \end{gathered} $$

(A.8)

where complex numbers ${{z}_{1}}\exp (i\alpha )$, ${{z}_{2}}\exp (i\beta )$, and ${{z}_{3}}\exp (i\gamma )$ are related by relationship ${{z}_{1}}\exp (i\alpha ) +$ ${{z}_{2}}\exp (i\beta )$ $= {{z}_{3}}\exp (i\gamma )$. This statement is the essence of Graf’s theorem. Equivalent statement (10) used in this article to calculate the intensity of the light signal is obtained from (A.8) by replacing index m by –m .

The formulations of Graf’s theorem on the summation of series of Bessel functions, given in well-known textbooks [13, 14], are obtained directly from (A.8) as a special case for $\alpha = 0$.