Fundamental limitations of dispersion mitigation filters

Er’el Granot

doi:10.1364/OE.385446

1. Introduction

There are several methods to compensate for Chromatic Dispersion's (CD) distortions in optical communications channels [1]. Since dispersion is a linear effect [2] (unlike nonlinear effects, such as Brillouin scattering, Four-wave mixing, self-phase modulation, etc.) with known parameters (unlike Polarization Modes Dispersion), it can, in principle, be eliminated. However, dispersion remains a challenge since for numerous reasons there are always some residues of dispersion, which affects signal decoding.

The straightforward method to eliminate Dispersion Distortion (DD) is to use low dispersion fibers [3], however, to suppress nonlinear effects, it is advisable to avoid perfect CD compensation [2]. Consequently, these dispersion residues still demand compensation.

Dispersion Compensating Fiber (DCF) and modules [4–12], which are the most ubiquitous methods to mitigate CD, can eliminate the channel's dispersion only for a single wavelength, and since the C-band bandwidth in optical communication is several THz, it is clear that with simple fibers no complete dispersion cancellation is feasible. Moreover, DCFs are relatively costly, and cannot be affective in flexible or affordable networks.

In coherent-detection optical channels (since the field, rather than the intensity, is measured, (see chapter 10 in Ref. [1]), there is no need to compensate dispersion in the optical domain. Instead, it can be eliminated in the electrical or the digital domain [13–15]. It should be noted that recently it was suggested to use linear filters (either digital or analog) [16,17] to compensate dispersion even in direct detection channels, however, in this case, the compensation is partial due to fundamental nonlinearities, which cannot be eliminated completely in direct-detection schemes.

Linear filters are an extremely low-cost solution when done in the digital domain. However, as will be explained below, most generic rigid digital solutions (such as Refs. [13,14]) are inappropriate for compensating weak dispersion channels. In simple modulation scenarios, such as On-Off-Keying (OOK), weak dispersion has a negligible effect, and only large dispersion requires attention. However, modern modulation schemes, such as high order M-ary QAM, are much more sensitive to dispersion, and even weak dispersion's distortion can be detrimental to the channel's operation. Since the amount of information (number of bits) per symbol is proportional to $\def\uppartial{\unicode[Times]{x2202}}\def\uptau{\unicode[Times]{x03C4}}\def\upzeta{\unicode[Times]{x03B6}}\def\uppi{\unicode[Times]{x03C0}}\def\updelta {\unicode[Times]{x03B4}}\def\uppsi {\unicode[Times]{x03C8}}\def\upsigma{\unicode[Times]{x03A3}}\def\upbeta{\unicode[Times]{x03B2}}\def\upomega{\unicode[Times]{x03C9}}{\log _2}M$, but the distance between clusters’ center in the constellation map is proportional to $1/\sqrt M$ (see below, sec.4), then the Signal-to-Noise Ratio (SNR) decreases exponentially with the amount of information per bit. Thus, in high M constellations, even weak dispersion, which in older generation technologies were considered negligible, can prevent data decoding.

It should be stressed that low-dispersion compensation is important not only in high QAM digital networks but in analog channels as well. Analog optical channels are recently used in RF over Fiber (ROF) channels [18]. In these channels, optical fibers replace electrical coaxial cables in transmitting analog data. While digital networks can, in principle, encode data in weak dispersion channel errorlessly, any weak distortion in an analog channel has a permanent impact on the data. Moreover, adaptive filters are useless in analog channels. Therefore, affordable and rigid dispersion mitigation may be essential in analog optical links. Affordability is a key issue here due to the low cost of most ROF channels.

In recent years, adaptive filters have been used to mitigate linear (either optical or electrical) distortions in the communication channel (see, for example, Refs. [15,19–21]). Since these are adaptive filters, they can mitigate dispersion effects as well. In theory, these filters can compensate for any linear distortion in the optical line.

Despite the fact that adaptive filters are common in long distances channels, they are scarce in Passive-Optical Networks (PON). Currently, they are not needed in these networks, however, in next-generation PONs, when data rate increases substantially, affordable solutions will be needed, and therefore rigid filters will be preferred.

Rigid filters, which usually termed Fiber Dispersion(FD) filters [13,14], predict an optimal filter length, and therefore can yield only limited mitigation, i.e., they predict that for a given dispersion the compensating filter has to have a specific (finite) length, beyond which the filter's performances decline. However, adaptive filters demonstrate superior performances, and do not seem to have an optimal length, i.e., the larger the filter the better are the performances (see, for example, Ref. [21]).

Rigid filters, which are based on the least-square method [22], support the fact that the filter should be, ideally, infinitely long; however, this point was not proven. Moreover, there was no attempt to investigate the weak dispersion mitigation regime, and therefore all the universalities and fundamental performances’ boundaries were missed.

Therefore, several questions arise: 1) How can the adaptive filters mitigate very weak dispersion distortion, while the FD ones can compensate only relatively large distortions? 2) How low a dispersion can these filters compensate? 3) How much can these filters improve the fundamental limitations on the channel's distance? 4) What is the relation between the length of the filter and the maximum channel's distance? 5) For a given channel's parameters, what is the minimum required filter's length?

These are all fundamental questions, which are related to the limits of the channel's performances.

In this paper, we will address these questions, and show theoretically that it is possible to mitigate arbitrarily residual weak DD. We will implement and investigate these filters performances in M-ary QAM modulation.

2. General theory and the problem with FD filters

Dispersion is governed by the Schrödinger equation [1,2]

(1)$$i\frac{{\uppartial E({\uptau ,\upzeta } )}}{{\uppartial \upzeta }} = \frac{1}{2}\frac{{{\uppartial ^2}E({\uptau ,\upzeta } )}}{{\uppartial {\uptau ^2}}}$$

where $E({\uptau ,\upzeta } )$ is the electromagnetic (EM) pulse envelope, $\upzeta \equiv {\upbeta _2}z/{T^2} = {\upbeta _2}z{B^2}$ is the dimensionless distance, z is the length of the dispersive medium (the fiber), ${\upbeta _2}$ is the dispersion coefficient of the medium, T is the period of a symbol (i.e., $B = {T^{ - 1}}$ is the symbol rate), $\uptau = ({t - z/v} )/T = ({t - z/v} )B$, where t is the real time, i.e., $\uptau$ is the dimensionless time in a frame of reference that travels at the speed of light in the medium ($v$).

If the transmitted EM field is known at the medium's (the fiber) entrance, $E({\uptau ,\upzeta = 0} )$, then the field at its exit (after a distance $\upzeta$) is [1]

(2)$$E({\uptau ,\upzeta } )= \int\limits_{ - \infty }^\infty {K({\uptau - \uptau^{\prime},\upzeta } )} E({\uptau^{\prime},0} )d\uptau ^{\prime}$$

where the kernel is

(3)$$K({\uptau - \uptau^{\prime},\upzeta } )= {({ - 2\uppi i\upzeta } )^{ - 1/2}}\exp \left( { - i\frac{{{{({\uptau - \uptau^{\prime}} )}^2}}}{{2\upzeta }}} \right).$$

Therefore, the dispersion effect can be completely compensated by applying a similar convolution but with $\upzeta$ replaced by $- \upzeta$ in the kernel, namely

(4)$${K^\ast }({\uptau - \uptau^{\prime},\upzeta } )= K({\uptau - \uptau^{\prime}, - \upzeta } )= {({2\uppi i\upzeta } )^{ - 1/2}}\exp \left( {i\frac{{{{({\uptau - \uptau^{\prime}} )}^2}}}{{2\upzeta }}} \right).$$

Since the signal is discrete, the field is measured at the symbols’ center, i.e., at $\uptau = n$, $E({n,0} )$, and therefore, the straightforward approach to compensate the signal is to discretize and to truncate the kernel to the form

(5)$$\bar{K}({n,\upzeta } )= \left\{ {\begin{array}{cc} {{{({2\uppi i\upzeta } )}^{ - 1/2}}\exp \left( {i\frac{{{n^2}}}{{2\upzeta }}} \right)}&{ - N \le n \le N}\\ 0&{else} \end{array}} \right.$$

where $N = \lfloor{\uppi \upzeta } \rfloor = \lfloor{\uppi {\upbeta_2}z/{T^2}} \rfloor$ was determined by the Nyquist criterion to avoid aliasing. The brackets $\lfloor{}\;\rfloor$ stand for the rounding operation. Note, that N should be added to the kernel's index to maintain causality. Therefore, the larger the dispersion, the longer is the required filter. Equation (5) is the FD filter [13,14], which, for a given dispersion ($\upzeta$), has to have a finite length N.

When $\uppi |{{\upbeta_2}} |z/{T^2}\;<\;1$ (or $\upzeta\;<\;{\uppi ^{ - 1}}$) there is only one tap in the filter leading to incomplete dispersion compensation for fiber length smaller than $z\;<\;{T^2}/\uppi |{{\upbeta_2}} |$. In the binary protocols (such as OOK), this incomplete dispersion compensation does not pose a severe problem since the eye-opening is larger than zero [23]. However, if the protocol is based on high M QAM, then clearly this kind of filter can be used only by increasing the sampling frequency (i.e., decreasing T), but this solution (usually termed: over sampling) requires faster electronics, which increases the complexity and system's cost.

Filters, which work in the frequency domain do not mitigate the problem, because they also require faster electronics and substantially increase the system's complexity.

Adaptive filters, however, solve these problems. They can mitigate weak dispersion and their performance improves with the length of the fiber. So a question arises: How do they do that?

3. Filter designed for specific pulses

To understand how adaptive filters solve these problems we take advantage of the fact, that the EM sequence can be written as a superposition of given pulses $f({\uptau ,0} )$, namely

(6)$$E({\uptau ,0} )= {E_0}\sum\limits_{n ={-} \infty }^\infty {{a_n}f({\uptau - n,0} )} .$$

where ${a_n}$ is the amplitude of the nth symbol in the sequence, and $\uptau = ({t - z/v} )/T$ is again the normalized time in the moving frame of reference.

The pulse can have any shape, provided that initially, it has discrete delta characteristics, i.e.,

(7)$$f({n,0} )= \updelta (n )$$

(for example, rectangular and Nyquist sinc pulses obey this demand).

Since $f({\uptau ,0} )$ is given, then the shape after distortion,

(8)$$f({\uptau ,\upzeta } )= \int\limits_{ - \infty }^\infty {K({\uptau - \uptau^{\prime},\upzeta } )} f({\uptau^{\prime},0} )d\uptau ^{\prime}$$

is known as well, and therefore, after a distance $\upzeta$ the signal's field is readily written as

(9)$$E({\uptau ,\upzeta } )= {E_0}\sum\limits_{n ={-} \infty }^\infty {{a_n}f({\uptau - n,\upzeta } )} .$$

At the center of the mth symbols, i.e. for $\uptau = m$,

(10)$$E({m,\upzeta } )= {E_0}\sum\limits_{n ={-} \infty }^\infty {{a_n}f({m - n,\upzeta } )} .$$

Therefore, dispersion compensation is reduced to a deconvolution problem. We need, therefore, to find a filter $g({n,\upzeta } )$ that obeys

(11)$$\updelta (m )= \sum\limits_n^{} {g({m - n,\upzeta } )f({n,\upzeta } )} .$$

In general, there are many deconvolution methods, which can solve Eq. (11) (recursive methods, Fourier transform, etc. [24,25]). Specifically, the compensating filter has a very simple form when the relative distortion is very weak (which is one of the motivations of this study), in which case $f({n,\upzeta } )$ can be written

(12)$$f({n,\upzeta } )= \updelta (n )+ \Delta f({n,\upzeta } )$$

where $|{\Delta f({n,\upzeta } )} |\;< <\;1$, and then the compensating filter is straightforwardly derived to approximately

(13)$$g({n,\upzeta } )\cong \updelta (n )- \Delta f({n,\upzeta } )= 2\updelta (n )- f({n,\upzeta } ).$$

The derivation from Eq. (6) to Eq. (13) is generic and can apply to any pulse shape [provided Eq. (7) is valid], however, since in practice every optical channel is spectrally bounded, then in physical channels, it is advantageous to utilize the fact that any bounded signal can be written as a superposition of Nyquist-sinc shaped pulses (see, for example, Sec. 4.2.9 in Ref. [24]), i.e.

(14)$$E({\uptau ,\upzeta = 0} )= {E_0}\sum\limits_n^{} {{a_n}\textrm{sinc}({\uptau - n} )}$$

where $\textrm{sinc}(\uptau )\equiv \frac{{\sin ({\uppi \uptau } )}}{{\uppi \uptau }}$ is the well-known sinc function. That is, $f({\uptau ,0} )= \textrm{sinc}(\uptau )$.

It is important to emphasize that Eq. (14) is valid for any spectrally bounded signal regardless of whether the signal was intentionally designed to consist of a sequence of “sinc” pulses. As long as the initial signal was spectrally bounded to within the spectral band $- {({2T} )^{ - 1}}\;<\;f\;<\;{({2T} )^{ - 1}}$, then any signal (either the transmitted or detected) can be written in the same form [Eq. (14)].

It should be noted, that Nyquist-sinc pulses became very common in modern high-rate optical networks [26,27]. They have several advantages over ordinary pulses. On the one hand, they have high spectral efficiency and high resiliency against nonlinear effects, but on the other hand, they do not require complex detectors [28–31].

At the end of the fiber, i.e., after a distance z, the field is

(15)$$E({\uptau ,\upzeta } )= {E_0}\sum\limits_n^{} {{a_n}\textrm{dsinc}({\uptau - n,\upzeta } )}$$

where the “dsinc” (“dynamic-sinc”) function is(see Ref. [32] and Appendix A for derivation)

(16)$$\textrm{dsinc}({\uptau ,\upzeta } )\equiv \frac{1}{2}\sqrt {\frac{i}{{2\uppi \upzeta }}} \exp \left( { - i\frac{{{\uptau^2}}}{{2\upzeta }}} \right)\left[ {{\rm erf}\left( { - \frac{{\uptau - \uppi \upzeta }}{{\sqrt {i2\upzeta } }}} \right) - {\rm erf}\left( { - \frac{{\uptau + \uppi \upzeta }}{{\sqrt {i2\upzeta } }}} \right)} \right].$$

At the limit of short distances, the sinc function is regained, i.e., $\mathop {\lim }\limits_{\upzeta \to 0} [{\textrm{dsinc}({\uptau ,\upzeta } )} ]= \textrm{sinc}(\uptau )$. It should be noted that in case the initial pulses are rectangular ones, instead of Nyquist pulses, the “srect” [33] function should replace Eq. (16). The experimental validation of the “srect” function was confirmed at a sampling rate of 80GS/s [34].

A two-dimensional image of the dsinc function is presented in Fig. 1. In this figure, the absolute value, the real value and the imaginary value of the function dsinc [Eq. (16)] is presented (via false-colors) as a function of $\uptau$ and $\upzeta$. For $\upzeta = 0$ the function resembles a real sinc function, however, for $\upzeta\;>\;0$ the function disperses, gets widen and been distorted.

Fig. 1. A false-color presentation of the dsinc function. Lower panel: its absolute value $|{\textrm{dsinc}({\uptau ,\upzeta } )} |$. Upper panel: Its real (upper panel) and imaginary (lower panel) components, $\Re \textrm{dsinc}({\uptau ,\upzeta } )$ and $\Im \textrm{dsinc}({\uptau ,\upzeta } )$, respectively.

Download Full Size | PDF

At the center of the mth symbol, the detected field reads

(17)$$E({m,\upzeta } )= {E_0}\sum\limits_n^{} {{a_n}\textrm{dsinc}({m - n,\upzeta } )} .$$

Equation (17) is valid for any spectrally bounded signal. In particular, it must be valid for the initial signal

(18)$$E({m,0} )\propto \textrm{dsinc}({m,\upzeta^{\prime}} ),$$

i.e., by choosing ${a_n} = \textrm{dsinc}({n,\upzeta^{\prime}} )$ and substituting in (17) we find the relation

(19)$$\textrm{dsinc}({m,\upzeta^{\prime} + \upzeta } )= \sum\limits_{n ={-} \infty }^\infty {\textrm{dsinc}({n,\upzeta^{\prime}} )\textrm{dsinc}({m - n,\upzeta } )} ,$$

since after an additional distance $\upzeta ^{\prime}$, the signal $\textrm{dsinc}({m,\upzeta } )$ evolves into $\textrm{dsinc}({m,\upzeta^{\prime} + \upzeta } )$. In particular, Eq. (19) is valid even for $\upzeta ^{\prime} ={-} \upzeta$, in which case

(20)$$\updelta (m )= \textrm{dsinc}({m,0} )= \sum\limits_{n ={-} \infty }^\infty {\textrm{dsinc}({n, - \upzeta } )\textrm{dsinc}({m - n,\upzeta } )} .$$

Therefore, following Eq. (11), the compensating filter is

(21)$$g({n,\upzeta } )= \textrm{dsinc}({n, - \upzeta } ).$$

This result is consistent with Ref. [22], for a specific case of a spectrally bounded channel (and infinitely long filter). However, this is a specific case, which can be generalized, as is explained above, for any sequence of pulses. Moreover, unlike Ref. [22] were the filter was optimized using the least-square method, it is proven here that this is indeed fundamentally the best filter. Therefore, it can be used to evaluate the fundamental performance limits of the filter (see below).

Unlike Eq. (5) there is no problem of aliasing, and therefore the filter should be as long as possible, i.e., in practical scenarios

(22)$$g({k,\upzeta } )= \left\{ {\begin{array}{cc} {\textrm{dsinc}({k, - \upzeta } )}&{ - Q \le k \le Q}\\ 0&{else} \end{array}} \right.$$

where Q is as large as possible (the effects of its length are discussed below).

In the case of weak dispersion, i.e., $\upzeta\;< <\;1$

(23)$$f({n,\upzeta } )\cong \updelta (n )+ i\upzeta w(n )\,\textrm{for}\, - \infty\;<\;k\;<\;\infty$$

and similarly

(24)$$g({n,\upzeta } )\cong \updelta (n )- i\upzeta w(n )\,\textrm{for}\, - Q \le k \le Q$$

where $w(m )$ is a universal dimensionless vector

(25)$$\begin{aligned} w(m )\equiv &\left[ {\begin{array}{ccccccccc} \cdots &{\frac{1}{{{3^2}}}}&{ - \frac{1}{{{2^2}}}}&1&{ - \frac{{{\uppi^2}}}{6}}&1&{ - \frac{1}{{{2^2}}}}&{\frac{1}{{{3^2}}}}& \cdots \end{array}} \right] = \\ & \left\{ {\begin{array}{cc} {{{({ - 1} )}^{m + 1}}/{m^2}}&{m \ne 0}\\ { - {\uppi^2}/6}&{m = 0} \end{array}} \right. \end{aligned}$$

We will note in passing that Euler’s identity emerges very elegantly from Eq. (25) (see Ref. [35]). However, unlike Euler’s solution, this derivation does not require infinite products, the legitimacy of which took Euler about a decade to prove [36,37].

4. Application to coherent M-ary QAM modulation

The coherent M-ary QAM optical communications system is presented in Fig. 2. The system consists of a laser source, a coherent modulator, a fiber, and a coherent detector. In such a communication system every carrier can carry 4 channels, i.e. the real and imaginary parts of the two polarization directions. However, since the dispersion equation in the first approximation [Eq. (1)] is independent of polarization, we ignore polarization and focus only on the real and imaginary parts of the field.

Fig. 2. System schematic.

Download Full Size | PDF

In M-ary QAM modulation, there are M possible symbols in every time allocation T. Since in optimal network the data can be regarded as a pseudo-random sequence, every symbol has the same probability to appear, i.e.,

(26)$$p({{a_n} = {a_{k,q}}} )= 1/M$$

where

(27)$${a_{k,p}} \equiv \frac{{2k - \sqrt M - 1}}{{\sqrt M - 1}} + i\frac{{2q - \sqrt M - 1}}{{\sqrt M - 1}}\,\textrm{for}\,k,q = 1,2, \ldots \sqrt M$$

are the M different symbols.

To demonstrate the filter performance on a QAM protocol (see Ref. [1]), we simulated an optical coherent communication channel, where 2¹¹-1 symbols were modulated with 256-QAM before transmitted into a dispersive fiber. In this case, every symbol transmit 8bits. Therefore, a pseudorandom sequence of length 2¹¹-1 was generated with the values [Eq. (27)] and substituted in Eq. (17) (that is, a complex convolution was taken) to simulate dispersion and to simulate sampling at the symbols’ centers. It should be noted that comparison to other numerical methods has shown that this simple discrete convolution [Eq. (17)] predicts with high accuracy dispersion's effect. To calculate the dispersion effect the convolution was taken over the entire signal. However, to simulate the compensating filter the convolution was taken over $2Q + 1$ taps as Eq. (22) suggests.

The simulation results are presented in Figs. 3 and 4. As can be seen from the simulation, even after a distance $\upzeta = 0.03$ the dispersion’s distortions prevent data decoding since adjacent symbols’ clusters overlap. However, the filter [Eq. (22)] can eliminate the distortions, and restore the original constellation map. Moreover, even the implementation of the approximated (and simpler) filter [Eq. (24)] with only 3 taps, most of the distortions were compensated, which allow again for data decoding.

Fig. 3. Upper panel: the transmitted constellation of QAM 256. Lower panel: the received constellation after a medium with $\upzeta = 0.03$ (without compensating filter).

Download Full Size | PDF

Fig. 4. Upper panel: the reconstructed constellation after applying the filter (22). Lower panel: the reconstructed constellation after applying filter [Eq. (24)] with Q = 1, i.e., FIR with only three taps.

Download Full Size | PDF

5. Filter's length optimization

Before introducing the filter's effect, one needs to evaluate the maximum distortion created by dispersion. In case of a weak dispersion, the dispersion's effect can be evaluated by applying the filter [Eq. (23)]. Since $w(m )$ has alternating signs, then if $E({m,0} )$ is the center value of the mth cluster, the largest deviation from the cluster's center occurs for the initial signal

(28)$${E_W}({n,0} )= \left\{ {\begin{array}{cc} {E({m,0} )}&{n = m}\\ {{E_0}{{({ - 1} )}^{n - m}}}&{n \ne m} \end{array}} \right..$$

The subscript “W” refers to worst distortion.

Then, the deviation after a channel distance (without compensation) $\upzeta$ is bounded by a circle, whose radius is

(29)$$|{{E_W}({m,\upzeta } )- {E_W}({m,0} )} |= \upzeta \left|{\sum\limits_{|k |\ge 1}^{} {w({m - k} ){E_W}({k,0} )} } \right|\;<\;2\upzeta {E_0}\sum\limits_{k = 1}^\infty {\frac{1}{{{k^2}}}} = \upzeta {E_0}\frac{{{\uppi ^2}}}{3}$$

Moreover, since the initial distance between centers of adjacent clusters is $\Delta E = \frac{{2{E_0}}}{{\sqrt M - 1}}$, then error-free decoding is impossible without dispersion compensation (and without error correction) if the deviation [Eq. (29)] exceeds this distance, i.e., the medium's length cannot exceed (for error-free operation) the upper limit

(30)$$\upzeta _{\max }^{({NC} )}\;<\;\frac{3}{{{\uppi ^2}\left( {\sqrt M - 1} \right)}},$$

otherwise $|{{E_W}({m,\upzeta } )- {E_W}({m,0} )} |\;>\;\Delta E/2$. The superscript “NC” stands for “Non-Compensated” channel.

This result is consistent with Ref. [23] for the OOK case (equivalent to $\sqrt M = 2$), in which case ${\upzeta _{\max }}\;<\;{\uppi ^{ - 1}} \cong 3/{\uppi ^2}$ and with the quantum analogy (Ref. [32])

However, after applying the compensating filter [Eqs. (22) or (24)], the deviation from the original value decreases. The deviations are not eliminated completely due to the filter’s finite length.

The relation between the original transmitted field $E({m,0} )$ and the reconstructed (after compensation) one ${E_R}({m,\upzeta } )$ is

(31)$${E_R}({m,\upzeta } )= \sum\limits_{q ={-} Q}^Q {g({q,\upzeta } )\sum\limits_{k ={-} \infty }^\infty {f({k,\upzeta } )E({m - q + k,0} )} }$$

and therefore the deviation is now bounded by a smaller circle (see Appendix B)

(32)$$|{{E_R}({m,\upzeta } )- E({m,0} )} |\;<\;2\upzeta {E_0}\sum\limits_{k = Q + 1}^\infty {\frac{1}{{{k^2}}}} = 2\upzeta {E_0}{\uppsi ^{(1 )}}({Q + 1} )$$

where ${\uppsi ^{(1 )}}(z )$ is the first derivative of the Digamma (Psi) function [38], and the approximation was taken under the assumption that $\upzeta\;< <\;1$.

Since

\frac{1}{{Q + 1}}\;<\;\sum\limits_{k = Q + 1}^\infty {\frac{1}{{{k^2}}}} = {\uppsi ^{(1 )}}({Q + 1} )\;<\;\frac{1}{Q}

then

(33)$$|{{E_R}({m,\upzeta } )- E({m,\upzeta } )} |\;<\;2\upzeta {E_0}\frac{1}{Q}$$

In Fig. 5 the constellations of the compensated signals are presented for $Q = 1$ and for $Q = 3$, where the radius of the clusters’ boundary are $2\upzeta {\uppsi ^{(1 )}}(2 )\cong 1.29\upzeta$, and $2\upzeta {\uppsi ^{(1 )}}(4 )\cong 0.568\upzeta$ respectively. Note, that without the filters the radius is [following Eq. (29)] considerably larger $\upzeta {\uppi ^2}/3$. As can be seen, even a filter as short as $Q = 1$, i.e., filter with only three taps, can improve the signal's quality considerably.

Fig. 5. A) The original transmitted constellation (M = 16); B) The non-compensated signal after the dispersive medium with $\upzeta = 0.1$ (the dashed circles represents the cluster's boundaries). C) The compensated signal constellation with $Q = 1$; D) Same as C but with $Q = 3$.

Download Full Size | PDF

Again, following the reasoning that led to Eq. (30), since the initial distance between centers of adjacent clusters is $\Delta E = \frac{{2{E_0}}}{{\sqrt M - 1}}$, then error-free decoding is impossible provided the medium's length does not exceed the upper limit

(34)$$\upzeta _{\max }^{(C )} = \frac{1}{{2\left( {\sqrt M - 1} \right){\uppsi ^{(1 )}}({Q + 1} )}} \cong \frac{Q}{{2\left( {\sqrt M - 1} \right)}}$$

where the approximation on the right was taken for long filters, i.e., for $Q\;> >\;1$. The superscript “C” stands for Compensated signal.

Clearly, as the length of the filter (Q) increases, the filter’s performances improve, and the maximum distance increases.

Alternatively, for a given channel with a given distance $\upzeta$ and a given M, one can use Eq. (34) to calculate the minimum length of the filter, which is required to compensate dispersion effects in the given channel, namely

(35)$$Q\;>\;2\upzeta \left( {\sqrt M - 1} \right).$$

Thus far, it was assumed that no error correction was present. In case a Forward Error-Correction (FEC) is implemented, the system can operate with high fidelity even when the Bit-Error-Rate (BER) is as high as ${\sim} {10^{ - 3}}$. Therefore, by implementing FEC, these upper bounds can be extended.

When $\upzeta\;<\;\upzeta _{\max }^{(C )}$ there is no overlap between clusters and then clearly the BER=0, however, when $\upzeta\;>\;\upzeta _{\max }^{(C )}$ the clusters overlap and then the BER increases (BER>0). To evaluate the BER one can use the fact that unless the deviations from the cluster's center are smaller than the maximum deviation, i.e., when $\Delta E\;<\;2\upzeta {E_0}{\uppsi ^{(1 )}}({Q + 1} )$, then these deviations obey approximately a normal distribution (see Fig. 6) with the standard deviation (for a detailed derivation, see Appendix C)

(36)$$\upsigma ({\upzeta ,M,Q} )= \frac{1}{3}\upzeta E_0^{}\sqrt {2\frac{{\sqrt M + 1}}{{\sqrt M - 1}}{\uppsi ^{(3 )}}({Q + 1} )}$$

${\uppsi ^{(3 )}}(z )$ is the 3rd derivative of the Digamma (Psi) function (see [38]). When $M\;> >\;1$ and $Q\;> >\;1$, Eq. (36) can be approximated by

(37)$$\upsigma ({\upzeta ,M,Q} )\cong \frac{{2\upzeta }}{{3{Q^{3/2}}}}E_0^{}$$

which is practically independent of M.

Fig. 6. The comparison between the simulation's probability density of the (real part of the) deviation from the clusters centers $\updelta E$ (dashed curve) and normal distribution (solid curve). In this simulation, Q=3, M=4 and the number of symbols was 2¹³-1.

Download Full Size | PDF

Figures 7 and 8 present the value of the ratio between $\upsigma ({\upzeta ,M,Q} )$ and $E_0^{}$ for different values of fiber length $\upzeta$ and filter's length Q. In Fig. 7 there is a comparison between this ratio ($\upsigma /E_0^{}$), which was calculated using the simulation, and the result of Eq. (36). Despite the fact that Eq. (36) was derived for weak dispersion, i.e., based on Eq. (23), Fig. 7 shows that it is an excellent approximation even for relatively large values of $\upzeta$. Moreover, the effect of the filter's length Q on $\upsigma /E_0^{}$ is clearly shown. In Fig. 8 a contour plot of this ratio is presented as a function of $\upzeta$ and Q.

Fig. 7. Plot of the ratio between the standard deviation and the mean field as a function of the normalized distance $\upzeta$ and the filter's length Q. The solid curves correspond to the result of the simulation, while the dashed lines correspond to Eq. (36). Both were taken for M = 4.

Download Full Size | PDF

Fig. 8. Contour Plot of the ratio between the standard derivation and the mean field $\upsigma /E_0^{}$ as a function of the normalized distance $\upzeta$ and the filter's length Q. The labels indicate the value of $\upsigma /E_0^{}$.

Download Full Size | PDF

In QAM protocol, the BER can be evaluated as (see Ref. [39])

(38)$$BER\sim \frac{2}{{{{\log }_2}M}}\textrm{erfc}\left( {\frac{{{E_0}}}{\upsigma }\sqrt {\frac{{3{{\log }_2}M}}{{2({M - 1} )}}} } \right)$$

After substituting Eq. (36) in Eq. (38)

(39)$$BER\sim \frac{2}{{{{\log }_2}M}}\textrm{erfc}\left( {\frac{3}{{2\upzeta \left( {\sqrt M + 1} \right)}}\sqrt {\frac{{3{{\log }_2}M}}{{{\uppsi^{(3 )}}({Q + 1} )}}} } \right)$$

solving for $\upzeta$ we finally obtain the maximum distance for a given BER

(40)$$\upzeta _{\max }^{}({BER} )\sim \frac{3}{{2\left( {\sqrt M + 1} \right)\textrm{erfc}{^{ - 1}}\left( {\frac{{BER}}{2}{{\log }_2}M} \right)}}\sqrt {\frac{{3{{\log }_2}M}}{{{\uppsi ^{(3 )}}({Q + 1} )}}}$$

In case a FEC is present in the detection system, one should substitute $BER = {10^{ - 3}}$, i.e.,

(41)$$\upzeta _{\max }^{({FEC} )} = \upzeta _{\max }^{}({BER = {{10}^{ - 3}}} ).$$

In the presence of other (non-chromatic dispersion) distortions, these values, i.e. Eqs. (34) and (40), decrease accordingly.

It should be stressed that these maximum values of $\upzeta$ are universal in the sense, that they are independent of the dispersion coefficient ${\upbeta _2}$ and baud rate B. To convert to physical maximum fiber length, one should simply divide $\upzeta$ by ${\upbeta _2}{B^2}$, i.e., ${z_{\max }} = {\upzeta _{\max }}/{\upbeta _2}{B^2}$. In particular, for a $B = 50GB/s$ optical channel based on smf28 fiber (${\upbeta _2} \cong 20p{s^2}/km$) ${z_{\max }} = {\upzeta _{\max }}/{\upbeta _2}{B^2} = 20km \times {\upzeta _{\max }}$. In Table 1 a comparison between the non-compensated maximum distance $z_{\max }^{({NC} )}$ (right column), the compensated one $z_{\max }^{(C )}$ (in brackets) and the compensated one with an FEC $z_{\max }^{({FEC} )}$ is presented.

Table 1. The values of $z_{\max }^{({FEC} )}$, $z_{\max }^{(C )}$ (in brackets) measured in Km for different values of M and Q, as well as the value of $z_{\max }^{({NC} )}$ (NC column). The channels parameters are: $B = 50GB/s$ and ${\upbeta _2} \cong 20p{s^2}/km$.

View Table

The improvement in maximum channel distance as a consequence of using the filter is clearly shown. For example, in the Q=9 and M=64 the distance increases 160 fold (from 0.868Km to 140Km). It should be stressed that while $z_{\max }^{({NC} )}$ and $z_{\max }^{(C )}$ are fundamental accurate limits, since Eq. (38) is an approximation, $z_{\max }^{({FEC} )}$ is only an evaluation, which is less accurate for low M and Q.

6. Summary

In modern high capacity optical communications channels, adaptive filters are commonly used to mitigate multiple linear distortions. Since dispersion is a linear effect, it is expected that these filters can compensate for residual dispersions. In fact, they demonstrate superior performance in comparison to FD filters. However, the literature does not address some related questions: how these filters present such high performances without oversampling? what should be the minimal filter's length to compensate for a given dispersion? and how an FEC can improve the maximum distance the data can be transmitted to (for a given filter)?

In this paper, these questions are addressed. Weak dispersion compensation is crucial in M-ary QAM modulation protocol, where the maximum distance, beyond which no data decoding is possible $\upzeta _{\max }^{({NC} )}$, is substantially reduced with the modulation parameter M, namely, $\upzeta _{\max }^{({NC} )} = 3{\uppi ^{ - 2}}{\left( {\sqrt M - 1} \right)^{ - 1}}$. However, to compensate this dispersion, an infinitely long filter is required (in contrary to FD filters, i.e., Refs. [13,14] but in accordance with Ref. [22]).

It is shown that the minimum length of the filter (Q), which is required to decode the sequence data after a distance $\upzeta$ is $Q\;>\;2\upzeta \left( {\sqrt M - 1} \right)$. Furthermore, such a filter will extend the maximum distance to $\upzeta _{\max }^{(C )} \cong Q{\left( {2\sqrt M - 1} \right)^{ - 1}}$, which can be arbitrarily large. This distance can be extended even further with the aid of an FEC.

Appendix A: derivation of the dsinc function

The propagation of the sinc function can be formulated using Eq. (2), i.e.,

(A1)$$\textrm{dsinc}({\uptau ,\upzeta } )= \int\limits_{ - \infty }^\infty {K({\uptau - \uptau^{\prime},\upzeta } )} \textrm{sinc}({\uptau^{\prime}} )d\uptau ^{\prime}$$

Alternatively, it can be evaluated in the spectral domain, since the Fourier transform of $\textrm{sinc}(\uptau )$ is the rectangular function ${\rm rect}_{{2\uppi }}(\upomega )= \left\{ {\begin{array}{cc} 1&{|\upomega |\;<\;\uppi }\\ 0&{else} \end{array}} \right.$ then,

(A2)$$\begin{array}{l} \textrm{dsinc}({\uptau ,\upzeta } )= \frac{1}{{2\uppi }}\int\limits_{ - \infty }^\infty {{\textrm{rect}_{2\uppi }}(\upomega )\exp ({i\upzeta {\upomega^2}/2 + i\upomega \uptau } )d\upomega } = \frac{1}{{2\uppi }}\int\limits_{ - \uppi }^\uppi {\exp ({i\upzeta {\upomega^2}/2 + i\upomega \uptau } )} d\upomega = \\ \frac{1}{2}\sqrt {\frac{i}{{2\uppi \upzeta }}} \exp \left( { - i\frac{{{\uptau^2}}}{{2\upzeta }}} \right)\left[ {{\rm erf}\left( { - \frac{{\uptau - \uppi \upzeta }}{{\sqrt {i2\upzeta } }}} \right) - {\rm erf}\left( { - \frac{{\uptau + \uppi \upzeta }}{{\sqrt {i2\upzeta } }}} \right)} \right] \end{array}$$

and, in particular

(A3)$$\textrm{dsinc}({\uptau ,0} )= \frac{1}{{2\uppi }}\int\limits_{ - \uppi }^\uppi {\exp ({i\upomega \uptau } )} d\upomega = \frac{{\sin ({\uppi \uptau } )}}{{\uppi \uptau }} \equiv \textrm{sinc}(\uptau ).$$

Appendix B

In the first (linear) approximation, the distorted signal is [according to (23)]

(B1)$$\Re E({m,\upzeta } )\cong \Re \{{[{\updelta (m )+ i\upzeta w(m )} ]\ast E({m,0} )} \}$$

where the asterisk represents discrete convolution.

After applying the compensating filter on (B1), then, in the first approximation

(B2)$$\begin{aligned} &{E_R}({m,\upzeta } )\cong [{\updelta (m )- i\upzeta {w_Q}(m )} ]\ast [{\updelta (m )+ i\upzeta w(m )} ]\ast E({m,0} )\cong \\ &[{\updelta (m )+ i\upzeta \Delta {w_Q}(m )} ]\ast E({m,0} )\end{aligned}$$

where

(B3)$${w_Q}(m )\equiv \left\{ {\begin{array}{cc} {{{({ - 1} )}^{m + 1}}/{m^2}}&{0\;<\;|m |\;<\;Q}\\ { - {\uppi^2}/6}&{m = 0} \end{array}} \right.\,\textrm{and}$$

(B4)$$\Delta {w_Q} = w(m )- {w_Q}(m )= \left\{ {\begin{array}{cc} {{{({ - 1} )}^{m + 1}}/{m^2}}&{Q\;<\;|m |}\\ 0&{|m |\le Q} \end{array}} \right.$$

Then

(B5)$$|{{E_R}({m,\upzeta } )- E({m,0} )} |\cong \upzeta |{\Delta {w_Q}\ast E({m,0} )} |= \upzeta \left|{\sum\limits_{|k |\;>\;Q}^{} {w({m - k} )E({k,0} )} } \right|$$

This result is bounded by the worst scenario [Eq. (28)]

(B6)$$\begin{aligned} |{{E_R}({m,\upzeta } )- E({m,0} )} |\; & <\;\upzeta \left|{\sum\limits_{|k |\;>\;Q}^{} {w({m - k} ){E_W}({k,0} )} } \right|= 2\upzeta {E_0}\sum\limits_{k = Q + 1}^\infty {\frac{1}{{{k^2}}}} = \\ & 2\upzeta {E_0}{\uppsi ^{(1 )}}({Q + 1} )\end{aligned}$$

where ${\uppsi ^{(1 )}}(z )$ is the first derivative of the Digamma (Psi) function [38].

Appendix C

The standard deviation from the centers of the clusters is

(C1)$$\upsigma ({\upzeta ,M,Q} )= {\left\langle {{{|{E({m,\upzeta } )- E({m,0} )} |}^2}} \right\rangle ^{1/2}}$$

The triangular brackets represents averaging over all symbols in the sequence. Following B1-B5

(C2)$$\upsigma ({\upzeta ,M,Q} )= \upzeta {\left\langle {{{|{\Delta {w_Q}\ast \Im E({m,0} )} |}^2} + {{|{\Delta {w_Q}\ast \Re E({m,0} )} |}^2}} \right\rangle ^{1/2}} = \sqrt 2 \upzeta {\left\langle {{{|{\Delta {w_Q}\ast \Re E({m,0} )} |}^2}} \right\rangle ^{1/2}}$$

where $\Re E$ and $\Im E$ correspond to the real and imaginary parts of the field E. The equality on the right is a consequence of the pseudorandom nature of these components, i.e., they have the same mean square values. Therefore,

(C3)$$\begin{aligned}&\upsigma ({\upzeta ,M,Q} )= \sqrt 2 \upzeta {\left\langle {{{\left|{\sum\limits_{|k |\;>\;Q}^{} {w({m - k} )\Re E({k,0} )} } \right|}^2}} \right\rangle ^{1/2}} = \\ &\sqrt 2 \upzeta {\left[ {\sum\limits_{|k |\;>\;Q,|q |\;>\;Q}^{} {w({m - k} )w({m - q} )\left\langle {\Re E({k,0} )\Re E({q,0} )} \right\rangle } } \right]^{1/2}} \end{aligned}$$

and since the imaginary part receives the values [Eq. (27)] $\frac{{2q - \sqrt M - 1}}{{\sqrt M - 1}}$ for $q = 1,2, \ldots \sqrt M$

with equal probability, then

(C4)$$\begin{array}{l} \left\langle {\Re E({k,0} )\Re E({q,0} )} \right\rangle = E_0^2\updelta ({q - k} )\frac{1}{{\sqrt M }}\sum\limits_{p = 1}^{\sqrt M } {{{\left( {\frac{{2p - \sqrt M - 1}}{{\sqrt M - 1}}} \right)}^2}} = \\ E_0^2\updelta ({q - k} )\frac{{\left( {\sqrt M + 1} \right)}}{{3\left( {\sqrt M - 1} \right)}} \end{array}$$

After substituting (C4) in (C3)

(C5)$$\upsigma ({\upzeta ,M,Q} )= \sqrt 2 \upzeta E_0^{}{\left[ {\sum\limits_{|k |\;>\;Q}^{} {{w^2}({m - k} )\frac{{\left( {\sqrt M + 1} \right)}}{{3\left( {\sqrt M - 1} \right)}}} } \right]^{1/2}}$$

and since $\sum\limits_{|k |\;>\;Q}^{} {{w^2}({m - k} )} = 2\sum\limits_{k = Q + 1}^\infty {\frac{1}{{{k^4}}}} = \frac{{{\uppsi ^{(3 )}}({Q + 1} )}}{3}$, where ${\uppsi ^{(3 )}}(z )$ is the 3rd derivative of the Digamma (Psi) function (see [38]), then, we finally obtain

(C6)$$\upsigma ({\upzeta ,M,Q} )= \frac{1}{3}\upzeta E_0^{}\sqrt {2\frac{{\sqrt M + 1}}{{\sqrt M - 1}}{\uppsi ^{(3 )}}({Q + 1} )} .$$

Disclosures

The author declares no conflicts of interest.

References

1. G. P. Agrawal, Fiber-Optic Communication Systems (John Wiley & Sons, New-York) (2002)

2. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press) (2001)

3. X. Gu and L. C. Blank, “10Gbit/s unrepeated three-level optical transmission over 100 km of standard fibre,” Electron. Lett. 29, 2209 (1993). [CrossRef]

4. C. Lin, H. Kogelnik, and L. G. Cohen, “Optical-pulse equalization of low-dispersion transmission in single-mode fibers in the 1.3–1.7-µm spectral region,” Opt. Lett. 5(11), 476 (1980). [CrossRef]

5. D. S. Larner and V. A. Bhagavatula, “Dispersion reduction in single-mode fiber links,” Electron. Lett. 21(24), 1171 (1985). [CrossRef]

6. A. M. Vengsarkar and W. A. Reed, “Dispersion-compensating single-mode fibers: efficient designs for first- and second-order compensation,” Opt. Lett. 18(11), 924 (1993). [CrossRef]

7. C. D. Poole, J. M. Wiesenfeld, A. R. McCormick, and K. T. Nelson, “Broadband dispersion compensation by using the higher-order spatial mode in a two-mode fiber,” Opt. Lett. 17(14), 985 (1992). [CrossRef]

8. C. D. Poole, J. M. Wiesenfeld, and D. J. DiGiovanni, “Elliptical-Core Dual Mode Fiber Dispersion Compensator,” IEEE Photonics Technol. Lett. 5(2), 194–197 (1993). [CrossRef]

9. C. D. Poole, J. M. Wiesenfeld, D. J. DiGiovanni, and A. M. Vengsarkar, “Optical fiber-based dispersion compensation using higher order modes near cutoff,” J. Lightwave Technol. 12(10), 1746–1758 (1994). [CrossRef]

10. R. C. Youngquist, J. L. Brooks, and H. J. Shaw, “Two-mode fiber modal coupler,” Opt. Lett. 9(5), 177 (1984). [CrossRef]

11. J. N. Blake, B. Y. Kim, and H. J. Shaw, “Fiber-optic modal coupler using periodic microbending,” Opt. Lett. 11(3), 177 (1986). [CrossRef]

12. C. D. Poole, C. D. Townsend, and K. T. Nelson, “Helical-grating two-mode fiber spatial-mode coupler,” J. Lightwave Technol. 9(5), 598–604 (1991). [CrossRef]

13. S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express 16(2), 804 (2008). [CrossRef]

14. S. J. Savory, “Compensation of fibre impairments in digital coherent systems,” 2008 in Proceeding of IEEE European Conference on Optical Communication paper Mo.3.D.1. (Brussels, Belgium, 2008).

15. T. Xu, G. Jacobsen, S. Popov, J. Li, E. Vanin, K. Wang, A. T. Friberg, and Y. Zhang, “Chromatic dispersion compensation in coherent transmission system using digital filters,” Opt. Express 18(15), 16243 (2010). [CrossRef]

16. E. Granot, S. Bloch, and S. Sternklar, “Affordable Dispersion Mitigation with an Analog Electrical Filter,” Appl. Opt. 55(28), 7956–7963 (2016). [CrossRef]

17. S. Bloch, S. Sternklar, and E. Granot, “Affordable Dispersion Mitigation Method for the Next Generation RF-over-Fiber Optical Channels,” Appl. Opt. 56(24), 6777–6784 (2017). [CrossRef]

18. C. H. Cox, “Analog Optical Links: Theory and Practice,” Cambridge Univ. Press, New York2006.

19. K. Kikuchi, “Principle of adaptive-filter-based signal processing in digital coherent receivers,” IEEE 38th European Conference and Exhibition on Optical Communications, Amsterdam (2012)

20. E. Ip and J. M. Kahn, “Digital equalization of chromatic dispersion and polarization mode dispersion,” J. Lightwave Technol. 25(8), 2033–2043 (2007). [CrossRef]

21. T. Xu, G. Jacobsen, S. Popov, J. Li, K. Wang, and A. T. Friberg, “Normalized LMS digital filter for chromatic dispersion equalization in 112-Gbit/s PDM-QPSK coherent optical transmission system,” Opt. Commun. 283(6), 963–967 (2010). [CrossRef]

22. A. Eghbali, H. Johansson, O. Gustafsson, and S. J. Savory, “Optimal Least-Squares FIR Digital Filters for Compensation of Chromatic Dispersion in Digital Coherent Optical Receivers,” J. Lightwave Technol. 32(8), 1449–1456 (2014). [CrossRef]

23. E. Granot, “Fundamental dispersion limit for spectrally bounded On-Off-Keying communication channels and its implications to Quantum Mechanics and the Paraxial Approximation,” Europhys. Lett. 100(4), 44004 (2012). [CrossRef]

24. J. G. Proakis and D. G. Manolakis, Digital Signal Processing, 3rd Edition (Prentice Hall, New-Jersey1996)

25. K. R. Castleman, Digital Image Processing, (Prentice Hall, New-Jersey1996)

26. M. A. Soto, M. Alem, M. A. Shoaie, A. Vedadi, C-S Brès, L. Thévenaz, and T. Schneider, “Optical sinc-shaped Nyquist pulses of exceptional quality,” Nat. Commun. 4(1), 2898 (2013). [CrossRef]

27. R. Schmogrow, R. Bouziane, M. Meyer, P. A. Milder, P. C. Schindler, R. I. Killey, P. Bayvel, C. Koos, W. Freude, and J. Leuthold, “Real-time OFDM or Nyquist pulse generation – which performs better with limited resources?” Opt. Express 20(26), B543 (2012). [CrossRef]

28. G. Bosco, A. Carena, V. Curri, P. Poggiolini, and F. Forghieri, “Performance limits of Nyquist-WDM and CO-OFDM in high-speed PM-QPSK systems,” IEEE Photonics Technol. Lett. 22(15), 1129–1131 (2010). [CrossRef]

29. T. Hirooka, P. Ruan, P. Guan, and M. Nakazawa, “Highly dispersion-tolerant 160 Gbaud optical Nyquist pulse TDM transmission over 525 km,” Opt. Express 20(14), 15001–15007 (2012). [CrossRef]

30. T. Hirooka and M. Nakazawa, “Linear and nonlinear propagation of optical Nyquist pulses in fibers,” Opt. Express 20(18), 19836–19849 (2012). [CrossRef]

31. R. Schmogrow, D. Hillerkuss, S. Wolf, B. Bäuerle, M. Winter, P. Kleinow, B. Nebendahl, T. Dippon, P. C. Schindler, C. Koos, W. Freude, and J. Leuthold, “512QAM Nyquist sinc-pulse transmission at 54 Gbit/s in an optical bandwidth of 3 GHz,” Opt. Express 20(6), 6439–6447 (2012). [CrossRef]

32. E. Granot, “Information Loss in Quantum Dynamics,” chapter 5 in the book Advanced Technologies of Quantum Key Distribution, S. Gnatyuk, ed. (INTECH, Rijeka, 2017). Online: https://www.intechopen.com/books/advanced-technologies-of-quantum-key-distribution/information-loss-in-quantum-dynamics

33. E. Granot, E. Luz, and A. Marchewka, “Generic pattern formation of sharp-boundaries pulses propagation in dispersive media,” J. Opt. Soc. Am. B 29(4), 763 (2012). [CrossRef]

34. S. Marciano, S. Ben-Ezra, and E. Granot, “Eavesdropping and Network Analyzing Using Network Dispersion,” Appl. Phys. Res. 7(2), 27 (2015). [CrossRef]

35. E. Granot, “Derivation of Euler's Formula and ζ(2k) using the Nyquist-Shannon Sampling Theorem,” Am. J. Signal Process 9, 1–5 (2019). [CrossRef]

36. W. Dunham, Euler: The Master of Us All, The Dolciani Mathematical Expositions, no. 22, Mathematical Association of America, Washington, D.C., 1999.

37. V. S. Varadarajan, “Euler and his work on infinite series,” Bull. Amer. Math. Soc. 44(04), 515–540 (2007). [CrossRef]

38. M. Abramowitz and A. Stegun, Handbook of Mathematical Functions (Dover publication, New-York, 1965)

39. A. Goldsmith, Wireless Communications, Cambridge University Press, New-York (2005)

	Q = 1	Q = 3	Q = 9	NC
M = 4	15.0 (15.5)	49.7 (35.2)	219 (95.2)	6.08
M = 16	13.5 (5.17)	44.9 (11.8)	197 (31.7)	2.03
M = 64	9.59 (2.22)	31.8 (5.04)	140 (13.6)	0.868

Fundamental limitations of dispersion mitigation filters

Abstract

1. Introduction

2. General theory and the problem with FD filters

3. Filter designed for specific pulses

4. Application to coherent M-ary QAM modulation

5. Filter's length optimization

6. Summary

Appendix A: derivation of the dsinc function

Appendix B

Appendix C

Disclosures

References

Cited By

Figures (8)

Tables (1)

Equations (57)

Optics Express