Gerchberg-Saxton Based FIR Filter for Electronic Dispersion Compensation in IM/DD Transmission Part I: Theory and Simulation

Intensity-modulation and direct-detection (IM/DD) transmission over short-reach optical fiber links, require electronic dispersion compensation (EDC) at the transmitter and/or electronic equalization at the receiver. Recently, the iterative Gerchberg-Saxton (GS) algorithm was demonstrated for EDC in IM/DD systems, through treating the amplitude at the transmitter and the phase prior-to the direct detection receiver as a degree of freedom. In Part I of this work, three GS approaches using finite impulse response (FIR) filters for EDC in IM/DD systems are demonstrated. The first two are closely related and rely on a cascaded FIR structure, while the third offers a novel non-iterative EDC solution using a single GS optimized static FIR filter. This is achieved through decoupling pattern dependent aspects of transmission from the GS iterations by targeting a single impulse at the DD receiver. With every successive iteration an impulse response for the GS filter emerges and sets the FIR tap weights. It is also demonstrated that closed-form analytical expressions for the GS filter impulse response can be obtained through small-signal frequency-domain analysis. The FIR filter is simulated using 8-bit finite-precision arithmetic. An adaptive $T$-spaced post feed-forward equalizer (FFE) is utilized for mitigating residual chromatic dispersion. It is shown, that a $T/2$-spaced pre-EDC FIR filter with 417 taps can support 56 Gb/s non-return-to-zero (NRZ) on-off keying (OOK) transmission over 80 km of single mode fiber (SMF) with a chirp-free Mach-Zehnder modulator (MZM). Part II, presents experimental demonstration of the non-iterative GS FIR filter proposed and simulated in this article.

neck of the bandwidth crunch is thus moved to the metro, shortreach and access networks, which are expected to operate at 200 Gb/s to 400 Gb/s. The deployment of coherent transmission and detection in short-reach optical links is expensive. A more viable and cost efficient solution would be to utilize intensity modulation and direct detection (IM/DD) [3]. The combination of digital signal processing (DSP) [4] and high sampling rate digital-to-analog converters (DACs) [5], [6], [7], have enhanced both the spectral efficiency (SE) and reach in IM/DD systems. A great number of modulation formats have been explored and experimentally demonstrated for short reach IM/DD systems, such as, discrete multitone modulation (DMT) [8], Carrierless amplitude phase modulation (CAP) [9] and Nyquist subcarrier modulation (SCM) [10], [11]. However, the use of non-return-tozero on-off keying (NRZ-OOK), and four-level pulse-amplitude modulation (PAM-4) [12], remains the most favorable solution for implementation in extended reach 400 Gb/s Ethernet, due to their simplicity [8].
Irrespective of whether IM/DD systems employ externally modulated or directly modulated transmitters, invariably they suffer from the power fading penalty induced by the interaction of chromatic dispersion (CD) with direct detection. However, the utilization of coherent detection and/or dual-parallel Mach-Zehnder modulators (DP-MZM), enable complete compensation of liner effects, including chromatic dispersion, as both the real (in-phase) and imaginary (quadrature) components of the electric field are either fully controlled at the transmitter or fully recovered at the receiver. The same could not be said for direct detection, where a photodiode applies square-law operation |.| 2 and recovers the optical intensity only and does not recover the optical phase which is lost. Furthermore, the received symbol after direct detection suffers from inter-symbol-interference (ISI) due to both linear an nonlinear effects. The linear (first order) effect becomes fully manifested as strong spectral dips in the fiber frequency-domain transfer function [13], [14], while the nonlinear (second and higher order) effects include receiver power series expansion and signal-to-signal beating interference (SSBI) [14]. The focus of this current work is on compensating linear first order power fading effect using finite impulse response (FIR) filters.
is based on the Gerchberg-Saxton (GS) algorithm [21], where the fiber dispersion performs time-to-frequency mapping [22], while the unconstrained phase at the direct detection receiver acts as a degree of freedom. The iterative algorithm fundamentally linearizes the IM/DD channel effect including square-law detection. This algorithm was experimentally demonstrated to deliver 56 Gb/s over 50 km in [16], modified to reduce the implementation complexity and speed up convergence in [17], [18], [20], [23], [24], and implemented at the receiver as a data-aided decision directed equalizer, enabling 100 Gb/s over 100 km of single mode fiber (SMF) [19], [20], [24]. A 56 Gb/s PAM4 was transmitted over 80 km of SMF with only 5 iterations of the modified GS algorithm and a FIR noise shaping filter [23].
A multi-constraint version [20] and error-controlled version [24] of the GS algorithm has also been recently reported. In almost all demonstrations of the GS algorithm in IM/DD, dynamic real-time iterations are performed on the data or pilot symbols and additional equalization is needed at the receiver to combat residual ISI. In coherent systems, the bulk of CD compensation is performed using a static filter in conjunction with a low complexity adaptive filter to mitigate residual CD and other time-varying channel effects [25], [26], [27]. Adaptive equalization with a large number of taps, are slow to converge, requires high complexity and consumes high power, in comparison to static CD filters [25]. The static CD compensation is performed using either frequency-domain equalization (FDE) with over-lap and save [27] or time-domain equalization (TDE) with FIR filters [28]. The primary objective of this study is to investigate static CD equalization using symmetric (linear phase) FIR filters for mitigating the power fading effect. Although FDE is more efficient than TDE, the work presented here focuses on FIR filters because (i) they are unconditionally stable [29], and have no limitations on the maximum sampling frequency [30], (ii) fundamentally the application of the GS algorithm in this context is based on a time-domain effect where link dispersion performs real-time Fourier transform [31], as such insights obtained from this time-domain analysis can be easily translated into efficient frequency domain implementation, and (iii) the number of taps required for short reach links (order of 100's) allow for low complexity TDE, which would be comparable in complexity to FDE.
In Part I of this work, three distinct FIR filter implementations utilizing the GS algorithm are described and demonstrated. In particular, one of the implementations offers a non-iterative solution for EDC in IM/DD systems where the static digital FIR filter tap weights are optimized offline using the GS algorithm for transmitter based CD pre-equalization. The FIR filter is implemented using a non-recursive tapped delay line structure with an 8-bit finite-precision arithmetic. An adaptive T -spaced feed-forward equalizer (FFE) is subsequently utilized at the receiver for compensating residual ISI. In comparison to the existing solution for mitigating the power fading effect, namely, transmitter-side Tomlinson-Harashima pre-coding (THP), the GS based FIR is feed-forward and does not require any feedback path at the transmitter, or use nonlinear elements like modulo operation, requires no training symbols and does not need an additional multilevel demodulation step at the receiver.  [15], (b) implementation as a cascaded alternating CD compensation blocks with +D and −D dispersion parameters operating on target amplitude A T x and transforming it to the required amplitude for EDC A EDC .
In Part II of this work, an experimental demonstration of pre-EDC using the non-iterative GS based FIR filter is conducted, with system performance compared to that of the iterative GS. The influence of pulse shaping, binary modulation format, filter length and adaptive post-equalization is also experimentally investigated. This paper is organized as follows. The principle behind the proposed FIR filter designs is described in section II. In section III, the simulation set-up used for comparing the three approaches with highlights of the IM/DD transmission link and the various DSP steps employed at the transmitter and receiver, is described. In section IV, a proof-of-concept numerical simulation for a 56 Gb/s NRZ-OOK signal transmission over 80 km of SMF is performed and discussed, while a conclusion is provided in section V.

II. FIR FILTER OPTIMIZATION
The basic iterative GS algorithm for transmitter EDC is shown in Fig. 1(a). The amplitude at receiver |e(t)| is assigned the target amplitude |e t (t)| (here an ideal NRZ-OOK as in [15]), while the phase at the receiver is left unconstrained as a degree of freedom. After linear back-propagation with −D fiber dispersion to the transmitter end, the phase φ (t) is set to zero, while the back-propagated amplitude |e(t)| is assigned to the transmitter amplitude |e (t)| = |e(t)|. After N iterations, the required Efield at the transmitter is |e (t)| exp(i × 0). This EDC scenario can be implemented with a chirp-free single drive MZM. The iterative GS algorithm could be implemented using a cascade of frequency or time domain filters [15], [17], operating on the transmitted data pattern with 2 samples-per-symbol [15] or 1 sample-per-symbol [17]. The algorithm illustrated in Fig. 1(a), can be implemented as an alternating CD compensation blocks with dispersion parameters +D (zero phase input) and −D (target amplitude input), transforming the target amplitude A T x to the required amplitude for CD pre-distortion A EDC after K iterations as illustrated in Fig. 1(b) for K = 2 iterations and a single iteration i highlighted.
The complexity of implementing the basic GS algorithm arises from the need for iterative processing on the transmitted data for CD pre-compensation, which requires real-time processing of the pattern-dependent information bearing waveform. In this work, the focus is on time-domain digital filter implementations of the GS algorithm, in-part due to the fact that the basic GS algorithm utilizes time-to-frequency conversion of optical pulses, which enables optical fiber dispersion to act as a real-time Fourier transform [31]. Three distinct FIR designs for chromatic dispersion pre-compensation in IM/DD links, are outlined in Sections II-A, II-B, and II-C.

A. Cascaded Complex FIR
The first approach requires iterative processing of the transmitted symbols as would typically be performed with a cascade of complex FIR filters with alternating dispersion parameter signs and appropriate time-domain constrains at the transmitter (zero phase) and at the receiver (target amplitude A T x ), as illustrated in Fig. 2, for a single iteration i. This approach is refereed to herein as the cascaded complex-valued FIR (CC-FIR) filter. Although the output of the CC-FIR is a real-valued signal due to the existence of the modulus operation, the processing of the input data is still complex-valued as it uses the complex-valued fiber impulse response.
It is evident that the zero phase constraint imposed at the input of the +D CD compensating filter, although initially intended to match the chirp free MZM transmitter, has the added advantage of removing the need for filtering the imaginary components of signal. This, however, is not true for the −D CD compensating filter, which requires CD processing of both the real and imaginary components of the signal. The taps weights for the CC-FIR units are obtained through direct sampling and truncation of the fiber impulse response, which is in-turn acquired through the inverse Fourier transform of the fiber transfer function [25], [26]. The complex tap weight with index n for the FIR filter impulse response is [26]: The FIR filter is assumed to have a length of N , where N is an odd number. The parameter M is given by [26]: where D is the dispersion coefficient, L is the fiber propagation length, c is the speed of light and T is the sampling time, which is equal to T /2 or T depending on whether CD compensation is to be performed at 2 samples-per-symbol or 1 sample-per-symbol, with T denoting the symbol period. The FIR filter length N could not be arbitrarily long and must be less than N max = 2 2M + 1 to avoid aliasing [26], which is mitigated through truncating the impulse response and pulse shaping with raised cosine (RC) anti-aliasing filter [28]. In coherent systems, the CD coefficients could be calculated with the effect of the pulse shaping filter embedded. This would result in reduced complexity as only a single filtering operation is needed as oppose to two separate operations, one for CD compensation and the other for pulse shaping. The complexity saving are significant especially when the RC roll-off factor approaches zero producing a longer impulse response. However, this approach of combining CD taps and pulse-shaping taps could not be used with the CC-FIR as it would equate to filtering multiple times. Consequently, RC pulse shaping must be performed before the CC-FIR input.
The following link parameters are assumed in this work: D = 16.75 ps/nm/km, fiber length L = 80 km with the reference wavelength set to λ = 1550 nm. Assuming a 56 Gb/s signal, the symbol period is T = 17.85 ps. The real and imaginary components of h(n) with 257-taps are shown in the inserts of Fig. 2. One of the main drawbacks of the CC-FIR filter design, beyond the growing complexity with increasing number of iterations, is the need for the 2(N − 1) sample delay buffer for the target amplitude samples A T x (k) per iteration. The next subsection describes a complexity reduced cascaded filter for CD compensation in IM/DD links.

B. Cascaded Reduced FIR
It is worth noting that the amplitude sample at the output of the iteration i, here, denoted A i+1 (k) is a result of a the absolute value or the modulus of the complex samples with real component real 2 and imaginary component imag 2 , as illustrated in Fig. 2. Correspondingly, A i+1 (k) = |real 2 | 2 + |imag 2 | 2 , and both components contribute to the value of A i+1 (k). In general, one would suspect that the informational content of a signal is equally distributed among its real and imaginary components. In fact, it is possible to reconstruct the complex signal from the Fourier transform (or, in our context time-domain fiber dispersion acting as a Fourier transform) of either the real component alone or the imaginary component alone. This can be achieved through noting that all signals can be expressed as the average of its even and odd decomposition, and knowing that the Fourier transform of an even signal is purely real and of an odd signal is purely imaginary, then the real or imaginary part of the Fourier transform output is simply the Fourier transform of the even and odd decomposition, respectively. This symmetry does not hold for the informational content of the phase of the complex signal, rendering phase retrieval a harder task. The basic GS algorithm imposes a zero-phase constraint at the transmitter, which implies a real valued signal with a zero imaginary component. This is indeed an additional implicit constraint at the transmitter-end forcing the imaginary component to zero, which would inevitably disrupt the symmetry between the real and imaginary and suggest the contribution of imag 2 to A i+1 (k) is less than the contribution of the real 2 . To verify this prediction, the root mean square (rms) of the imag 2 divided by the rms of real 2 as a function of the number of iterations is shown in Fig. 3(a). As the number of iterations increase it becomes possible to ignore the calculation of the imag 2 component and reduce hardware complexity, either throughout all of the iterations or only through the final few iterations. The former is adapted here with the removal of the imag 2 calculation path in all iterations resulting in a cascaded reduced FIR (CR-FIR) filter structure with a single iteration i shown in Fig. 3(b). The next subsection outlines a scheme for dispersion compensation using a single real-valued FIR filter.

C. Single Real-Valued FIR
The primary contribution of this article, is the decoupling of the pattern dependent and modulation format dependent aspects of the transmission from the GS iterative algorithm. The ultimate goal would be to design a CD compensation scheme that is based on an offline optimization method, which is agnostic to both modulation format and pattern-dependent effects, while primarily depending on the fiber parameters, digital extinction ratio (ER) and the sampling interval. The term digital ER is defined as the ratio, expressed as a fraction, in dB, between the highest and lowest optical power levels of the target digital waveform intended prior to the direct detection receiver [16]. The digital ER, in this context, is a DSP parameter and is markedly different than the conventional optical ER, which is a property of the pre-distorted waveform generated at the output of the optical transmitter. The third approach, attempts to achieved this goal through utilizing a single real-valued FIR (SR-FIR) filter as depicted in Fig. 4(a). The scheme presented here performs the basic iterative GS algorithm offline using a single impulse δ(t) function located at the center of the simulation window as input, as oppose to using the information bearing NRZ-OOK or PAM4 waveform as implemented in prior studies [15], [17], [23]. With every successive iteration of the GS algorithm the impulse response of required electronic dispersion pre-compensating filter at iteration i, denoted, h i GS (n), becomes manifest. The offline iterative optimization is shown in Fig. 4(b). The target amplitude at the receiver-end is set to with the direct current (dc) bias reflecting a predefined digital extinction ratio. The basic GS algorithm is then executed offline for a total of K = 15 iterations. The evolution of the impulse function h GS (n) over successive iterations of the GS algorithm is shown in Fig. 5(a).
The resulting taps were quantized to 8 bits, made symmetric to ensure a linear phase FIR filter and truncated to N samples, with N being an odd number. Unlike the CC-FIR or CR-FIR, anti-aliasing could be performed digitally through pulse shaping  with a raised cosine impulse response at a roll-off factor of 1.0. The resulting final taps are shown in Fig. 5(b). The magnitude response of the digital FIR filter (SR-FIR) with and without RC pulse shaping is shown in Fig. 5(c). The SR-FIR is an optimized FIR filter response to combat the power fading dips induced by chromatic dispersion. In fact a detailed analysis of the IM/DD transfer function has been examined in many works [13], [32]. It is shown in [13], [14], [32], that the ISI induced in IM/DD systems can be divided into two components one is linear and the other is nonlinear. The linear ISI is a first order interference, which is also known as the power fading ISI. The nonlinear ISI component consists of second and higher order terms resulting from the received power series expansion and signal-to-signal beating interference (SSBI). The frequency domain normalized power fading transfer function H(f ) causing the linear ISI can be expressed as [14], [32], [33]:  (1) can be inverted through changing the sign of the dispersion coefficient or equivalently, taking the complex conjugate of the taps. However, the same can not be said for the transfer function in (3), which operates on a real-valued input. Consequently, this hypothetical H −1 (f ) must approximate 1/ cos(2π 2 β 2 Lf 2 ).
The realization of such a pre-compensating filter requires multiple anti-notch frequency responses, which significantly degrades the signal-to-noise ratio (SNR), while demanding extremely high resolution in both the time and amplitude of the impulse response. However, it is possible to effectively mitigate ISI caused by power fading without actually eliminating the nulls in the frequency spectrum [13], through methods of pre-coding [14], [33] or spectral slicing around the power fading dips [34]. In particular, transmitter-side Tomlinson-Harashima pre-coding (THP) [14], [33], [35], utilizes a feedback path to subtract the predicted ISI induced by previous symbols [13], [14]. In effect, THP is a decision feedback equalizer (DFE) moved to the transmitter side with a modulo operation replacing the decision device and eliminating the problem of error-propagation present in receiver side DFE [14], [33]. The main drawback of the THP is the need for an additional decoding step at the receiver. Furthermore, the filter coefficients needed for THP must be known at the transmitter in advance, and consequently are calculated by means of a training based DFE at the receiver [35].
As of recently [34], [36], linear equalizers were thought to be inadequate for mitigating power fading ISI, although in principle power fading is caused by a linear system [14]. However, the results presented in this work challenges this notion through  the proposed SR-FIR which effectively mitigates the first order linear ISI. In comparison to transmitter-end THP or receiver-end DFE, the Gerchberg-Saxton based SR-FIR scheme, offers a lower complexity feed-forward solution without any need for a nonlinear feedback path or an additional multilevel demodulation step at the receiver.
In fact, Wu X. et al., has demonstrated in [16], the first theoretical application of the CD frequency domain small-signal transfer function to the analysis of the GS iterative algorithm convergence and amplitude distribution. The author analytically examined the first few iterations of the GS algorithm, while imposing appropriate constraints at the transmitter and receiver, producing a closed form recursive formula, which relates the small signal frequency response of the pre-distorted signal at iteration i, here, denoted ΔS i EDC to the small signal frequency response of the ideal target signal intended for transmission ΔS T x [16]. Assuming (θ = 2π 2 β 2 Lf 2 ), the first and second iterations of the GS algorithm produces: Consequently, the aforementioned recursive relationship is given by: In fact, at a low enough extinction ratio where small-signal analysis is applicable, closed form semi-analytical expressions for the SR-FIR taps can be obtained using the inverse Fourier transform of H i GS (f ) in (7), followed by direct sampling and truncation. The first two expressions for h i GS (t) at i = 1 and i = 2, scaled by L|β 2 |/4π, are given below: In essence, the SR-FIR taps can be obtained starting from the time domain impulse as in Fig. 4(b) or starting from the frequency response H GS (f ) in (7). However, the former approach allows the inclusion of the ER in the final tap weight calculation, while the latter assumes a low enough digital ER. The mean square difference (MSD), between the tap weights obtain via means of small signal analysis and those obtained via time domain impulse propagation as a function of the digital ER is evaluated with 8 bit fixed-point precision. The MSD averaged over number of taps and number of quantization levels is shown in Fig. 7(a) with 15 iterations and (D = 16.75 ps/nm/km, L = 80 km, λ = 1550 nm and T = 8.92 ps). The corresponding magnitude frequency response over the range of the first 5 spectral nulls is presented in Fig. 7(b) at digital ER = 10 dB.
The results in Fig. 7 indicate that within the range of extinction ratios where the GS algorithm offers BER improvement (digital ER between 1 dB to 5 dB) [16], both approaches produce nearly identical FIR taps. The taps can be expressed analytically in a closed-form expression for all iterations. In general, it is possible to express the GS filter impulse response h i GS (t), using the following compact formulation: where a k is a constant coefficient that is unique to each h i GS (t) function and changes from one iteration to another. It can be shown from (10), that for every additional iteration, two new terms are added to the analytical formula of h i GS (t) increasing the complexity of the overall expression. Although using symbolic math software can alleviate this issue, obtaining the analytical expression is not necessary, as the tap weights could be easily calculated numerically by one of two means: (i) performing the inverse fast Fourier transform (IFFT) numerically on the analytical expression of H i GS (f ) in (7) or (ii) using the impulse propagation approach of Fig. 4(b), which is inherently a numerical method. In either case, the offline calculation of the optimum taps do not require any significant computational resources and can be performed easily on a personal computer. The system level simulations, performed in the remainder of this article, utilize the basic iterative GS algorithm in [15], for offline computation of the optimum FIR tap weights.
Although the basic iterative GS algorithm is guaranteed to converge, there is no guarantee the obtained solution is at the global optimum, as oppose to trapping at a local optimum. In previous studies, efforts have been made to "escape" the local optimum trap through, for example, perturbing the solution via reverse correction factors [17], insertion of pilot symbols [19] or multi-constraint optimization [20]. These efforts have also the added advantage of speeding up convergence. In this work, the basic GS algorithm is implemented without these modifications due to the following reasons: (i) acceleration in the convergence is inconsequential to the non-iterative SR-FIR pre-EDC approach, as the FIR tap weights are optimized offline, and (ii) it can be argued that using the basic GS algorithm for only mitigating linear power fading, does not suffer from the local optimum problem. The evidence for the latter reason is the existence of close-formed analytical expressions for a family of impulse responses in (10), which approximates the globally optimum and unique inverse IM/DD channel response.

III. SYSTEM LEVEL MODEL
The system level simulation of the physical layer link is performed in Optisystem/MATLAB co-simulation for verifying and comparing the aforementioned filter designs for amplitudeonly EDC control. The simulation set-up is shown in Fig. 8(a). The transmitter employed an 8-bit DAC with a sampling rate of 112 GSa/s prior to the single drive MZM. The receiver employed an 8-bit ADC operating at 112 GSa/s after the PIN photodiode. All of the DSP filtering performed at the transmitter and/or receiver used a fixed-point arithmetic with a resolution of 8-bits.
A total of 2 17 bits were simulated with an analog resolution of 128 samples within the duration of the symbol period T . The first 1000 bits were allocated for training the adaptive FFE, while 36 symbols on either side of the simulation window were ignored, leaving 130,000 symbols payload for bit error ratio (BER) measurement and estimation. The binary information is up-sampled to 2 samples-per-symbol, quantized into 8 bits before undergoing RC pulse shaping and feed-forward filtering using either CC-FIR, CR-FIR or SR-FIR. In the case of the SR-FIR, RC pulse shaping was embedded in the filter weights. The generated electrical drive signal is used to modulate a chirpfree single-drive MZM, with 0.5 dB insertion loss, connected to a continuous wave (CW) laser operating at 1550 nm with 15 dBm launch power. An RC electrical lower pass filter (LPF) with a 70 GHz 3-dB bandwidth and a roll-off factor of 1.0 is placed at the DAC output to emulate the bandwidth limitations The receiver is composed of a variable optical attenuator (VOA), followed by a PIN photodiode connected to a 42 GHz fifth-order Bessel electrical LPF. The received electrical eye-diagram is then analyzed using an electrical oscilloscope. The received electrical current is sampled with a 112 GSa/s ADC with an 8-bit resolution, undergo synchronization and timing recovery before down-sampling into 1 sample-per-symbol soft-decision symbols A Rx . A T -spaced adaptive FFE with J = 21 taps, updated using the least mean square (LMS) algorithm with an optimized step size is used prior to the decision device and error counting and illustrated in Fig. 8(b). A hard-decision forward error correction (FEC) limit of BER = 3.8 × 10 −3 which corresponds to 8.5 dBQ [37] is assumed. The BER calculation was performed with direct error counting. Each reported BER is the average of 5 different runs, 4 of which uses a randomly generated 2 17 binary sequence with a uniform distribution, and 1 of which is a 2 17 deBruijn bit sequence (DBBS).

A. Transmission Performance
The primary objective of the simulations presented in this section, is to assess the fidelity of each of the FIR filter designs in generating the requires drive signal obtained from the basic GS algorithm. To aid with this task, an ideal GS simulation was performed without giving due consideration to the particle necessity of DSP and DAC implementations. The ideal GS simulation was performed in double-precision with 128 samplesper-symbol, and no limitation on the transmitter bandwidth or DAC resolution. However, the receiver modelling, noise sources and ADC resolution, were identical for all pre-EDC approaches including the ideal GS. Setting the ideal GS as a reference point, the optimum digital ER at the target waveform was found to be 2 dB, with an average BER = 2.1 × 10 −4 maintained with 15 iterations of the GS algorithm for a 56 Gb/s NRZ-OOK signal transmission over 80 km. The received electrical eye-diagrams, the RF spectra for the generated pre-EDC signal, and the RF spectra of the received electrical signal after direct detection, obtained for each of the ideal GS, CC-FIR, CR-FIR and SR-FIR are shown in rows (a), (b), (c) and (d) of Fig. 9, respectively.
The results indicate that the SR-FIR can compensate for the power fading ISI without the need for real-time iterations. Furthermore, the RF spectrum of the drive signal obtained using the SR-FIR Fig. 9(d) is closer to the ideal GS case in Fig. 9(a). The RF spectra associated with CC-FIR and CR-FIR suffer from CD equalization of out-of-band frequencies as observed in Fig. 9(b) and (c), respectively. This would be expected a pulse shaping can not be performed per iteration. The GS based pre-EDC filter uplifts the frequency content of the signal around the power fading notches, which remain bounded, while simultaneously, sharpening these notches with a higher damping factor, without eliminating the unbounded singularity at the null frequency [16]. Consequently, the RF spectra of the received signal in Fig. 9, indicate the persistence of these nulls, concomitant with successful pre-emphasis of the signal spectra around their vicinity. The BER plotted against the received optical power (ROP) for the three FIR implementations and the ideal case is shown in Fig. 10. Two additional scenarios are added to Fig. 10 as a reference for the worst case, in particular, (i) no pre-EDC and no post-FEE and (ii) no pre-EDC and 101-tap post-FFE. Using 101-tap FFE is unable to compensate for dispersion at the receiver. In Fig. 10 The number of taps were set to 257 for the both CC-FIR, CS-FIR and 513 for the SR-FIR. The T -spaced post-FFE adaptive equalizer was fixed to 21-taps, which only offered 1 dBQ margin improvement in the BER for all cases concerned. The influence of the number of taps N on the performance of each of the FIR implementations at the maximum ROP is shown in Fig. 11. It is observed that increasing the number of taps for the cascaded filters offer no improvement in the BER. This is expected as the filter response is fixed with the fiber impulse response, which decays away from the center-tap. This is not the case with the SR-FIR, as the filter approximates the inverse response of the IM/DD link for which increasing the N improves the accuracy of this approximation. It shown in Fig. 11, that 417 taps are sufficient to reach the FEC limit without any post-equalization.
In SR-FIR approach, the number of taps N is primarily dependent on the transmission distance, as increasing the fiber length results in additional power fading nulls, which in turn, increases the number of taps required for synthesizing the precompensating filter. However, the iterative nature by which the optimum SR-FIR tap weights are determined, also influence  transmission distance and range of wavelength drift, is needed prior to deployment. The tolerance of the SR-FIR pre-EDC approach to dispersion mismatch, described in terms of fiber length mismatch [km], is shown in Fig. 12. This is achieved through changing the estimated transmission distance used in the offline optimization of the SR-FIR taps, while keeping the actual transmission distance fixed at 80 km. The SR-FIR pre-EDC used T /2-spaced 513-taps and no post-FFE. The dispersion tolerance of the ideal GS is simulated under identical conditions for comparison. It can be observed that the dispersion tolerance of the SR-FIR at the FEC threshold is around 1 km, which is  comparable to the dispersion tolerance of the ideal GS and is within range of the dispersion tolerances reported in recently published works [19], [24].
2) FIR Tap Quantization: In most studies on CD compensation in coherent systems, the filter taps are implemented using floating-point precision. However, in practice a finite-precision arithmetic is used for hardware implementation [25]. Consequently, the tap values are changed due to quantization errors, which result in a different frequency responses and, hence, different system performance. The tolerance of the SR-FIR pre-EDC approach to tap quantization is shown in Fig. 13(a). The results with double-precision arithmetic is also presented as a limit case. The change in the SR-FIR frequency response at different tap resolution is shown in Fig. 13(b), indicating the sensitivity to quantization error.

C. Implementation Complexity
A crude estimate of the hardware complexity involved in the different FIR implementations is summarized in Table I. The number of real multiplications per equalized symbol is used as the primary metric to assess complexity. To ensure a fair comparison, complexity is calculated with the minimum number of iterations K, and minimum number of tap length N , which are needed to reach the FEC limit without requiring any post-equalization. It is worth noting, however, that reducing K has a higher priority than reducing N in particular for CC-FIR and CR-FIR, as the offline optimization for the SR-FIR can be performed for a large K without any additional hardware complexity. As expected the CR-FIR reduces the complexity by 33% with the elimination of the final imaginary calculation path per iteration. This modification is algorithmic and thus can be also applied to the iterative GS algorithm performed with a FDE. The complexity of the SR-FIR is approximately 30 and 20 orders of magnitude lower than the complexity of the CC-FIR and CR-FIR, respectively. The complexity of pulse shaping is was not taken into consideration in Table I. V. CONCLUSION Three FIR filters based on the GS algorithm were described and demonstrated for CD compensation in IM/DD systems. The first two solutions presented were iterative, while the third offered a novel non-iterative solution for mitigating ISI induced by linear power fading. feed-forward static digital FIR filter was designed and optimized using the Gerchberg-Saxton algorithm for pre-EDC at the transmitter-end. This was achieved by decoupling the information bearing waveform from the GS iterations by time-domain impulse propagation or frequency domain small-signal analysis. The FIR filter was implemented using a non-recursive tapped delay line structure with an 8bit finite-precision arithmetic. An adaptive T -spaced FFE was utilized at the receiver for compensating residual CD. It was shown, as an example, that a 56 Gb/s NRZ-OOK signal could be transmitted over 80 km of SMF with a single drive MZM and a 417-tap FIR filter operating at 2 sample-per-symbol without post-equalization and a BER below the hard-decision FEC limit of 3.8 × 10 −3 . Enabling a T -spced 21-tap post-FFE, offers a 1 dBQ margin.